text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
|---|---|---|
{"url":"https:\/\/www.gradesaver.com\/textbooks\/math\/prealgebra\/prealgebra-7th-edition\/chapter-5-section-5-5-fractions-decimals-and-order-of-operations-exercise-set-page-382\/43","text":"## Prealgebra (7th Edition)\n\nPublished by Pearson\n\n# Chapter 5 - Section 5.5 - Fractions, Decimals, and Order of Operations - Exercise Set: 43\n\nAfter you claim an answer you\u2019ll have 24 hours to send in a draft. An editor will review the submission and either publish your submission or provide\u00a0feedback.","date":"2018-08-19 06:55:58","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\": 0.8095727562904358, \"perplexity\": 4419.362073211598}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-34\/segments\/1534221214702.96\/warc\/CC-MAIN-20180819051423-20180819071423-00639.warc.gz\"}"} | null | null |
Q: How do I know if a game was updated? I like to read release notes of game patches. Is there a way to be notified or to know when a game was updated?
A: Games that have been recently updated should get a little white corner added to their tile in My Games and Apps, with a black asterisk on top. This applies to games that have been updated, or need an update.
However, in my experience it does not show up on Pins, and sometimes it doesn't show up at all. Not sure if that's a bug, or if the system periodically pings to check for game updates, and I just opened the game before it pinged, but it happens once in a while that a game without the flagged tile needs an update when I open it.
For release notes specifically, you have to rely on the developers of each game to either post them in-game, or on the web.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,237 |
Q: OnTouchListener, ACTION_UP fires automatically after 30 second timeout I need the user to be able to touch and hold a button/imageview for a long time (> 30 seconds).
unfortunately after ~30 seconds an ACTION_UP is fired automatically.
Do you have any idea how i can prevent or bypass this event and keep tracking the users touch?
This is essential for the control of my app.
A: It happens on my motorola me525 also,only last 20 seconds.
I advise you use a layout(for example linearlayout)instead of button or imagebutton
layout can be pressed for a long time as you want
A: At least for Motorola Xoom, as posted in Motorola Forum , Engineering team seems to consider this problem a feature.
There is a possible solution there(for Xoom), don´t know if works.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,556 |
Given the confidence I have in my spam filter, I read my emails in order (or reverse order) using the nifty "open bracket" shortcut in Gmail ("[" = archive and go to previous)... so when a cold-call email comes in, it means I somehow have a connection to the sending company or sender - so I read it.
Today I got one that I had to stare at and read about 4 times before I realized I didn't care about it - which was exactly the opposite of the goal the sender presumably had.
Subject: "Role of Managers in enabling Architecture Center of Excellence(COE) for Business Differentiator"
This was the title of a training webinar (I figured out after looking a few times)... Was it the goal of the organizers to create such a complex name that I would feel inadequate about not understanding it and attend the webinar just to be sure I wasn't missing something important? Am I missing a new marketing trend here which says that super-nasty names get twice the attention as simple ones? | {
"redpajama_set_name": "RedPajamaC4"
} | 7,375 |
US (Español)
Weekend binge watch
Republic Day deals
Phones under Rs 15,000
Sony details new PS5 and DualSense controller accessibility settings
By Emma Boyle published 31 October 20
Including the option to reduce or turn off haptic feedback
Ahead of the PS5's November release date, Sony has detailed some of the accessibility settings that will be built in to the console and its DualSense controller at launch.
In an official SIE blog post, Sony has said that the accessibility settings currently available on the PS4, including closed captions, inverting colors, text to speech and custom button assignments, will also be present on PS5.
The post also details some of the new options that will be available to PlayStation 5 users. These include a voice dictation feature which will make it possible to input text without a virtual keyboard; a Screen Reader to give blind and low vision users the option of hearing on-screen text; and text-to-speech options so that deaf and hard of hearing users can send typed messages which will be spoken aloud to party members.
PS5 games list: every PlayStation 5 exclusive and cross-gen game announced so far
PS5 unboxing and preview: design, controller and playing Astro's Playroom
PS5 size comparison: is the PlayStation 5 too big?
PS4 and more
These features will be supported in multiple languages at launch, including English, Japanese, German, Italian, France French, Canadian French, Spain Spanish and Latin American Spanish.
Sony has also stated that the PS5 will support user-adjustable color correction, with the option to create game presets so that users can customise their settings in advance for games which support this .
When it comes to the DualSense controller, Sony has said that it'll be possible to reduce the force of the haptic feedback and adaptive triggers or turn it off entirely. Further to that, improvements have also been made to audio in order to "provide players with better spatial awareness."
Both the PS5 and PS5 Digital Edition will launch on November 12 in the USA, Japan, Canada, Mexico, Australia, New Zealand and South Korea. Those living elsewhere (like the UK and Ireland) have to wait an additional week, with November 19 being the release date for the rest of the world.
We've had the chance to go hands-on with the PS5 already, and we'll keep bringing more PS5 coverage in the run up to the console's launch.
Black Friday PS5 deals: will there be any PS5 savings in November?
Emma Boyle is TechRadar's ex-Gaming Editor, and is now a content developer and freelance journalist. She has written for magazines and websites including T3, Stuff and The Independent.
See more Gaming news
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5 new Indian films on OTT platforms to watch this Christmas weekend
The Logitech Litra Glow could make even a cheap webcam look incredible
How to watch Attack on Titan, season 4, part 2 online from anywhere, episode 77: Sneak Attack hits tonight
Samsung Galaxy S21 FE 5G: sorpresa al lancio per tutti i Galaxy Fan!
Get in the festive spirit with Huawei's Light Up My Xmas Tree
Portable audio players vs MP3 players: what's the difference?
Strava now helps you find a public toilet or water fountain during a run or ride | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,104 |
Q: Exchange not truncating Transaction Logs after successful Full-Backup I'm using Dell Avamar native Exchange Agent to Full-Backup the Databases on a Stand-Alone Exchange 2013 Server (latest updates and patches). It uses VSS.
It's not an Agent issue, since at the end of the Backup we correctly get Windows Event 2046:
The Microsoft Exchange Replication service VSS Writer instance xxxxx has successfully completed the backup of database 'XXXXXXX'
Database log truncation has been requested for this database. Log truncation will occur on the active copy after the next log generation is created.
But the logs keep piling up at a rate of 560 logs a day.
I've confirmed that CircularLogging is disabled and LastFullBackup attribute gets correctly populated.
What am I missing?
A: If anyone else is reading this post, the logs started magically being correctly purged 3 days after I posted for help. I'm closing the issue, but I'm not very assured, since magic is something I don't like in IT.
Let me share the Powershell script I created as a workaround so I didn't run out of space meanwhile
The script:
*
*Checks if The MailboxDatabase is Mounted
*Enables Circular-Logging
*Dismounts the MailboxDatabase
*Mounts the MailboxDatabase
The logs get purged
*Disables Circular-Logging
*Dismounts the MailboxDatabase
*Mounts the MailboxDatabase
`
Add-PSSnapin Microsoft.Exchange.Management.PowerShell.SnapIn
################# Functions ###############
Function MandaMail{
Param ( [string]$assunto,[string]$texto )
Send-MailMessage -To "<recipient@domain.com>" -From "<sender@domain.com>" -Subject $assunto -Body $texto -SmtpServer "smtp.server.domain"
}
Function IsMounted{
Param ( [string]$mdb )
return [System.Convert]::ToBoolean($(Get-MailboxDatabase $mdb -Status | foreach { $_.Mounted}))
}
Function ReiniciaMDB{
Param ( [string]$mdb )
Start-Sleep -Seconds 5
Write-Host " ...vamos desmontar a Mailboxdatabase:"$mdb
Dismount-Database -Identity $mdb –confirm:$false
Start-Sleep -Seconds 10
Write-Host " ...confirmar se desmontou... "
if (IsMounted($mdb)) {
Write-Host " ...ERRO ao desmontar -> return False "
return $false }
Else {
Write-Host " ...ok, desmontou."
Write-Host " ...vamos montar a Mailboxdatabase:"$mdb
Mount-Database $mdb
Start-Sleep -Seconds 5
Write-Host " ...confirmar se montou..."
if (IsMounted($mdb)) {
Write-Host " ...ok. montou"
return $true }
Else {
Write-Host " ...ERRO ao montar -> return False "
MandaMail -assunto "Exchange | PurgeTransLogs | ERRO ao montar:"$mdb -texto "."
return $false}
}
}
Function LimpaTransLogs{
Param ( [string]$mdb )
if (IsMounted($mdb)) {
Write-Host "...a MDB "$mdb" esta montada. /n ...activar o circular logging..: "
Set-MailboxDatabase -Identity $mdb -CircularloggingEnabled:$true
Write-Host "...reiniciar a MBD run1... "
If (ReiniciaMDB($mdb)) {
Write-Host "...reiniciou com SUCESSO. /n ...desactivar o circular logging..: "
Set-MailboxDatabase -Identity $mdb -CircularloggingEnabled:$false
Write-Host "...reiniciar a MBD run2..: "
If (ReiniciaMDB($mdb)) {
Write-Host "...reiniciou com SUCESSO."
MandaMail -assunto "Exchange | PurgeTransLogs | SUCESSO " -texto "."
Write-Host "...FIM!"
return $true}
Else {
Write-Host "...algo não correu bem :("
return $false
}
}
Else { Write-Host "...nao reiniciou a MBD run1" }
}
Else {
Write-Host "...A MDB "$mdb" nao estava montada"
MandaMail -assunto "Exchange | PurgeTransLogs | ERRO: nao estava montada " -texto "."
}
}
#####################################################################
LimpaTransLogs("MailboxDatabase_01")
LimpaTransLogs("MailboxDatabase_02")
A: Have you reviewed this document?
EXCHANGE DAG 2016 : Transaction Logs are not truncated if a database copy exists through veeam backup - Microsoft Q&A
In this case, the OP said that you have to select all disk that contains Active & Passive database drives in the backup to flush the logs properly.
A: You could consider that enable the Circular logging, It would help you to reduce the log size of the databases
Exchange 2019: Enable Circular logging on a Database - TechNet Articles - United States (English) - TechNet Wiki (microsoft.com)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,112 |
package com.cognizant.devops.platformreports.assessment.dataprocess;
import com.cognizant.devops.platformreports.assessment.datamodel.InsightsAssessmentConfigurationDTO;
public interface BaseDataProcessor {
public void processJson(InsightsAssessmentConfigurationDTO assessmentReportDTO);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,148 |
Q: How to prototype NSTableRowView in Interface Builder I have a view-based NSTableView and I'm trying to customize the appearance of certain rows.
I understand I need to implement the delegate method mentioned in the title; However I'm not sure about how to do it.
The documentation says:
You can use the delegate method tableView:rowViewForRow: to customize
row views. You typically use Interface Builder to design and lay out
NSTableRowView prototype rows within the table. As with prototype
cells, prototype rows are retrieved programmatically at runtime.
Implementation of NSTableRowView subclasses is entirely optional.
However, unlike cells, there is no NSTableRowView class in interface builder, nor is it clear how to setup a "prototype" row view.
I am trying something like this (Swift 3):
func tableView(_ tableView: NSTableView, rowViewForRow row: Int) -> NSTableRowView?
{
if (row % 4) == 0 {
// ..................................................
// [ A ] SPECIAL ROW:
if let rowView = tableView.make(withIdentifier: "SpecialRow", owner: self) as? NSTableRowView {
rowView.backgroundColor = NSColor.gray()
return rowView
}
else {
return nil
}
// ^ Always returns nil (Because I don't know how
// to setup the prototype in Interface Builder)
}
else{
// ..................................................
// [ B ] NORMAL ROW (No customization needed)
return nil
}
}
I have similar code working for cells -i.e., -tableView:viewForTableColumn:row:.
A: OK, so the obvious (?) solution worked:
*
*On Interface Builder, drag and drop a plain-vanilla NSView into the table (it will only accept the drop in a specific column, not as a direct child of the table view).
*Go to the Identity Inspector for the just dropped view, and change its Class to "NSTableRowView".
*Because just setting the .backgroundColor property as in my code does not work, I instead used this solution and added a box view as a child, and configured that in Interface Builder. I had to setup autolayout constraints between the box and the row view, so that it would stretch to the row view's actual size at runtime.
(Alternatively, I could have played with the wantsLayer property of the row view... )
UPDATE: It turns out the backgroundColor property I was using in my code is defined in NSTableRowView (NSView does not have such property, unlike UIView).
But it also gets overridden by the table view's setting (i.e., alternating rows or not), so instead I should customize it in this method:
func tableView(_ tableView: NSTableView, didAdd rowView: NSTableRowView, forRow row: Int)
{
if (row % 4) == 0 {
rowView.backgroundColor = NSColor.controlAlternatingRowBackgroundColors()[1]
}
else{
rowView.backgroundColor = NSColor.clear()
}
}
...after it was added (and its background color configured by the table view).
(Incidentally, it turns out I do not need a custom row view after all. At least not to customize the background color)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,226 |
'Parks and Recreation' Season 5 Premiere Eyes Washington D.C. Shoot
Parks and Recreation Season 5 Premiere Eyes Washington D.C. Shoot
— July 6th, 2012
It seems that Leslie Knoppe may be heading to Washington D.C. in Season 5 of Parks and Recreation. A new unconfirmed report reveals the NBC comedy series is looking into shooting on location in our nation's capitol for the Season 5 Premiere. There will be spoilers below if you haven't watched Parks and Recreation Season 4 in its entirety, so read on at your own risk.
When Parks and Recreation signed off for the season back in May, we saw both a win and a loss for Amy Poehler's Leslie Knoppe. She won in her city council race over the dimwitted Bobby Newport (Paul Rudd), but "lost" her boyfriend, Ben Wyatt (Adam Scott), who took a high-powered job in Washington D.C., forcing the couple into a long-distance relationship scenario. It is believed that Leslie will make the trip to D.C. to help Ben get accustomed to the city, and the show may utilize the city for future episodes this season as well.
While we may see Ben back in Pawnee, Indiana before the end of Season 5, executive producer Michael Schur revealed in a recent interview that Adam Scott's character will remain inside the Beltway for a large amount of time.
"As far as how long they're going to be apart or anything like that, we certainly don't know that yet, but we want to make it real. We don't want to have it be that in the premiere all of a sudden he's back. That would be pointless. He's going to be gone for a while. It's also a little bit like the way that Tom (Aziz Ansari) was gone at Entertainment 720 where we still found ways to work him into the show and he was still an important part of what was going on."
We reported last month that Parks and Recreation will air the Season 5 Premiere Thursday, September 20 at 9:30 PM ET. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,505 |
Q: QML Drag & drop in a grid I have an item (a rectangle) which can be moved through the different positions of the grid (see code below).
The reference to decide whether the rectangle is in one cell or another is the top-left (x=0;y=0) of the rectangle. I want the reference to be the center of the rectangle. Any ideas?
Find the code below:
import QtQuick 2.5
import QtPositioning 5.5
import QtLocation 5.6
import QtGraphicalEffects 1.0
import QtQml 2.2
import QtQuick.Controls 1.0
Rectangle{
id: mainContainer
width:800; height:width
color: "red"
Grid {
id:gridA
anchors.centerIn: parent
width: parent.width; height:width;
columns: 2; rows: 2; spacing: 2
verticalItemAlignment: Grid.AlignHCenter
horizontalItemAlignment: Grid.AlignVCenter
Repeater {
id:cells
model: gridA.columns*gridA.rows
property int draggedItemIndex: -1
Rectangle {
width: gridA.width/gridA.columns; height: gridA.height/gridA.rows
color: "white"
DropArea {
id: dragTarget
property alias dropProxy: dragTarget
anchors.fill: parent
keys: [ "KEY_A" ]
Rectangle {
id: dropRectangle
anchors.fill: parent
states: [
State {
when: dragTarget.containsDrag
PropertyChanges { target: dropRectangle; color: "grey"}
PropertyChanges { target: cells; draggedItemIndex: index}
}
]
}
}
}
}
}
Rectangle{
id: icon
property var draggedIndex: 0
width: parent.width/3; height:width
color:"blue"
Drag.active: dragArea.drag.active
Drag.keys: [ "KEY_A" ]
MouseArea {
id: dragArea
anchors.fill: parent
drag.target: icon
cursorShape: drag.active ? Qt.ClosedHandCursor : Qt.OpenHandCursor
onReleased: {
console.log("Drag: " + cells.draggedItemIndex)
if (cells.draggedItemIndex != icon.draggedIndex) {
icon.draggedIndex = cells.draggedItemIndex
cells.draggedItemIndex = -1
}
icon.x = cells.itemAt(icon.draggedIndex).x
icon.y = cells.itemAt(icon.draggedIndex).y
}
}
}
}
A: You have the Drag-attached property, which has the hotSpot-property of type PointF.
Set this property to Qt.point(width / 2, height / 2)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,786 |
Marshall Heating and Air - Your Residential - Commercial heat and air professional HVAC contractor. Serving Lake Norman, Winston-Salem, Lewisville, Clemmons, and surrounding Piedmont Triad area for over 15 years. With energy-efficient equipment and high-quality service from Marshall Heating & Air, you'll enjoy lower energy bills, cleaner indoor air quality, and comfortable temperatures in your home year round. We're your full-service heating and cooling contractor with 14 years of experience and a 100% satisfaction guarantee. Let us show you how a new heat pump installation can save you money each and every month. Call Today And Save! Winston-Salem Area (336) 399-6221 Lake Norman Area (704) 661-5471 Established in 2001. Quality Workmanship & Dependable Service Marshall Heating & Air has earned an excellent reputation in the greater Winston-Salem, Lewisville, Advance, and Clemmons area for respectful, efficient and reliable HVAC service. Our certified, licensed and insured professionals always provide quality workmanship, and each new installation is personally supervised and inspected by the company owner. Residential or Commercial HVAC services available from a trusted local company. | {
"redpajama_set_name": "RedPajamaC4"
} | 764 |
function Costume(costumeId) {
}
module.exports = Costume;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,487 |
<?php
use Symfony\Component\HttpKernel\Kernel;
use Symfony\Component\Config\Loader\LoaderInterface;
class AppKernel extends Kernel
{
public function registerBundles()
{
$bundles = array(
new Symfony\Bundle\FrameworkBundle\FrameworkBundle(),
new Symfony\Bundle\SecurityBundle\SecurityBundle(),
new Symfony\Bundle\TwigBundle\TwigBundle(),
new Symfony\Bundle\MonologBundle\MonologBundle(),
new Symfony\Bundle\SwiftmailerBundle\SwiftmailerBundle(),
new Symfony\Bundle\AsseticBundle\AsseticBundle(),
new Doctrine\Bundle\DoctrineBundle\DoctrineBundle(),
new Sensio\Bundle\FrameworkExtraBundle\SensioFrameworkExtraBundle(),
new JMS\SerializerBundle\JMSSerializerBundle(),
new Nelmio\CorsBundle\NelmioCorsBundle(),
new Liip\ImagineBundle\LiipImagineBundle(),
new Frontend\AppBundle\FrontendAppBundle(),
new Backend\UserBundle\BackendUserBundle(),
new Backend\AppBundle\BackendAppBundle(),
new Common\AppBundle\CommonAppBundle(),
new Common\UserBundle\CommonUserBundle(),
new Common\ArticleBundle\CommonArticleBundle(),
new Frontend\ArticleBundle\FrontendArticleBundle(),
);
if (in_array($this->getEnvironment(), array('dev', 'test'))) {
$bundles[] = new Symfony\Bundle\DebugBundle\DebugBundle();
$bundles[] = new Symfony\Bundle\WebProfilerBundle\WebProfilerBundle();
$bundles[] = new Sensio\Bundle\DistributionBundle\SensioDistributionBundle();
$bundles[] = new Sensio\Bundle\GeneratorBundle\SensioGeneratorBundle();
}
return $bundles;
}
public function registerContainerConfiguration(LoaderInterface $loader)
{
$loader->load($this->getRootDir().'/config/config_'.$this->getEnvironment().'.yml');
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,181 |
//
// ZNGServiceToContactViewController.h
// Pods
//
// Created by Jason Neel on 7/5/16.
//
//
#import <ZingleSDK/ZNGConversationViewController.h>
#import "ZNGConversationServiceToContact.h"
#import "ZNGAssignmentViewController.h"
#import "ZNGMentionSelectionController.h"
@protocol ZNGServiceToContactViewDelegate <NSObject>
@optional
/**
* Optional delegate method that is called when the user pressed 'Edit Contact.'
* If this method is implemented and returns NO, the edit contact screen will not be shown, instead allowing the delegate
* to present UI.
*/
- (BOOL) shouldShowEditContactScreenForContact:(nonnull ZNGContact *)contact;
@end
@interface ZNGServiceToContactViewController : ZNGConversationViewController <ZNGAssignmentDelegate, ZNGMentionSelectionDelegate, ZNGConversationToolbarContentViewDelegate, ZNGServiceConversationInputToolbarDelegate>
@property (weak, nonatomic, readonly, nullable) ZNGServiceConversationInputToolbar * inputToolbar;
@property (nonatomic, strong, nullable) ZNGConversationServiceToContact * conversation;
@property (nonatomic, weak, nullable) id <ZNGServiceToContactViewDelegate> delegate;
/**
* Setting this flag hides the contact name (potentially only the first line of the title on services with assignment).
* Used for animated transitions.
*/
@property (nonatomic, assign) BOOL hideContactName;
/**
* Optionally hide all contact and messaging data when the app is entering the background.
* This will prevent contact data from appearing in the app switcher.
* This will also propogate to any contact edit, event, and assignment view controllers presented by this conversation view.
*
* Defaults to NO.
*/
@property (nonatomic, assign) BOOL hideContactDataInBackground;
/**
* Setting this flag hides the assignment name, e.g. "Jason" in "Assigned to Jason".
* Used for animated transitions.
*/
@property (nonatomic, assign) BOOL hideAssignmentName;
@property (nonatomic, strong, nullable) IBOutlet UIView * automationBannerContainerView;
@property (nonatomic, strong, nullable) IBOutlet UIView * automationBanner;
@property (nonatomic, strong, nullable) IBOutlet UILabel * automationLabel;
@property (nonatomic, strong, nullable) IBOutlet NSLayoutConstraint * automationBannerOnScreenConstraint;
@property (nonatomic, strong, nullable) IBOutlet NSLayoutConstraint * automationBannerOffScreenConstraint;
@property (nonatomic, strong, nullable) IBOutlet UIButton * automationCancelButton;
@property (nonatomic, strong, nullable) IBOutlet UILabel * automationRobot;
@property (nonatomic, strong, nullable) IBOutlet UITableView * mentionsSelectionTable;
@property (weak, nonatomic, nullable) IBOutlet NSLayoutConstraint * toolbarHeightConstraint;
// A toolbar that only exists to visually cover the gap between the bottom of JSQMessage's toolbar and the bottom of the safe zone
@property (nonatomic, strong, nullable) IBOutlet UIToolbar * lowerFacadeToolbar;
- (IBAction)pressedCancelAutomation:(nullable id)sender;
// Defaults to YES
@property (nonatomic, assign) BOOL allowForwarding;
@end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,127 |
As Independent Insurance Brokers we know that customer satisfaction is key to our success. We look for the best combination of price and coverage among the many companies we have access to.
We treat you like a person, and not just another number. We offer "one stop" shopping with a full range of products for all of your insurance needs.
When was the last time you had an insurance review?
Our service is personal. We're more than just a 1-800 number.
Insurance is not just about money. It's about protecting your assets. | {
"redpajama_set_name": "RedPajamaC4"
} | 101 |
Q: Are static resources loaded via api like read.csv refreshed during each microbatch when a stream-static join is done in spark Are static resources loaded via api like read.csv refreshed during each microbatch when a stream-static join is done in spark.
var a = sparkSession.read.csv(directory).as[A]
var b = sparkSession.readStream... # kafka stream
a.join(b, expr("some expre"), "inner") # Will the changes in directory made after the spark run started. be reflected in the join? Or is it completely static.
Why I am asking this question is because in some cases I have seen the data from static resources are also refreshed. But couldn't justify why it is refreshed during a run.
I couldn't find anything around this in the documentation.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,000 |
Are you really 'Ten Toes Down'? Here are three ways to know
By Joshua Eferighe November 11, 2019
Photo Cred: Jesse Vargas (@vargasvisuals)
Throughout time there have always been words that come and go. Buzzwords which, for a moment in time, capture our interest and temporarily permeate our vocabulary.
Words like "tight," which I'm pretty sure is played out and like "awesome," which is not the same stamp of approval it was a couple of decades ago.
But there are also phrases that stick and end up holding a permanent place in our everyday vernacular. We've seen it with terms like "turnt," and I don't see "lit" going away anytime soon.
They're expressions that resonate beyond cultural boundaries and have a relatability that makes them last. I see the same for the phrase "ten toes down."
I can't pinpoint when exactly, but the expression of being ten toes down or to be ten toes down has been inescapable. From an official challenge to songs swearing by the proclamation, being ten toes down is not only advertised as cool but as a character trait, one should try to embody.
The Urban Dictionary defines it as being committed to something — "to devote 100% of your support towards a thing — which, in an atmosphere where self-deprecating vices are celebrated, is actually refreshing."
Although mostly used in rap, the idea of being ten toes down is actually a lot more transferable than it appears. It comes off as this badge you must ear and like this status you must strive to with how gritty it sounds and all, but it's actually something you can do every day, in every aspect of your life.
Being ten toes down is about humility, toughness, and integrity. Once a proper handle is gathered on these three values, the hype of being ten toes down becomes much more clear.
We obviously all (for the most part) have ten toes. So, the emphasis on them being down is nothing more than a clever way of saying that you're grounded — which is actually harder than you think.
Remembering where you come from no matter how far you go, is something we all can do a little better with, no matter how far you've gone or how close you've stayed.
When you're grounded you understand that it's never just you and that along the way, it's the little things that make it happen.
Forgetting this is detrimental because it makes you lose touch with, not only where you're from or how you've made it, but with yourself.
In everything, be ten toes down. Don't get too high or too low. When we're even-keeled, we have the levelheadedness to keep performing at a high level.
The reason why "ten toes down" has been so naturally adopted by the hip-hop community is that there's a toughness that the comes with the phrase. There's a sense of pride being unmovable, which, when you're planted, you usually are.
Freight or flight is not only used to describe our actions when a during a dire situation but is also applicable in dealing with everyday life.
There are situations in the workplace, in relationships, within, etc, where those two options present themselves. When you're ten toes down, you weather the storm when needed, and you don't run.
Similarly, when you're ten toes down, and when you stand by your word, you're going to evacuate the scene when it includes your involvement with something you don't stand for.
When you prove yourself to be someone who is unshakable in what you say and mean, you're not only tough, but you're someone everyone will want on their side.
Most of all, being ten toes down embodies integrity. At the end of the day, it's all about being able to stand for your own words at all times, yet finding people like that is a rarity.
When you claim to be ten toes down, you're claiming that you're reliable and that you vouch for your actions. A lot of people only have nine toes down or one foot in and one foot out, but there are only a few who truly understand commitment, which is why the phrase has been so popular to champion.
Loyalty is a currency that will always hold value and I think people are just starting to see that. We all should strive for integrity and encourage it out of one another. The phrase "ten toes down" is making honesty sexy again and it's something that's been long overdue.
So, try and be ten toes down today — every day, in fact. When you are, you will receive it back. That's just how the universe works.
Tags mindset motivation ten toes down
Joshua Eferighe
Highly opinionated, contentious and one to always speak his mind, Joshua Eferighe has been on the path of expression through writing before he even chose the profession. Love him or hate him, you'll always want to know what he has to say.
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,635 |
package release_test
import (
. "github.com/onsi/ginkgo"
. "github.com/onsi/gomega"
"testing"
)
func TestRelease(t *testing.T) {
RegisterFailHandler(Fail)
RunSpecs(t, "Release Suite")
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,187 |
Thirty Million Letters ist ein Dokumentar-Kurzfilm von James Ritchie aus dem Jahr 1963.
Inhalt und Hintergrund
Der 30-minütige Dokumentar-Kurzfilm berichtet über die Arbeit der britischen Post, der Royal Mail. In kurzen Sequenzen wird der Weg von Briefen dargestellt, vom Einwerfen in den Briefkasten, das Sortieren in damals modernen Briefverteilanlage in Briefverteilzentren, den Weitertransport in Postfahrzeugen und Postzügen, bis hin zum Ausliefern an den jeweiligen Empfänger.
Die Musik zu dieser Produktion schuf Derek Bourgeois, der damals noch am Magdalene College der University of Cambridge studierte. Er hatte bereits als 19-Jähriger 1960 mit Symphony for Chamber Orchestra, Op. 8: 15 minutes seine erste Sinfonie geschrieben, und komponierte für diesen Film die Overture "Mail Train", Op. 16: 9 minutes.
Der Film wurde im Auftrag des Postministeriums (General Post Office) von Edgar Anstey für die Filmproduktionsgesellschaft British Transport Films hergestellt. 2009 produzierte BFI Video ein DVD-Video des Films.
Auszeichnungen
Anstey wurde für den Film bei der Oscarverleihung 1964 für den Oscar für den besten Dokumentar-Kurzfilm nominiert. Anstey, ein Pionier des britischen Dokumentarfilms, produzierte zwischen 1934 und 1979 168 Dokumentar- und Kurzfilme und gewann vier Jahre später bei der Oscarverleihung 1967 den Oscar für den besten Kurzfilm für Wild Wings (1966).
Weblinks
Thirty Million Letters in YouTube (Kurzversion)
Einzelnachweise
Filmtitel 1963
Britischer Film
Dokumentarfilm | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,505 |
Akahata on January 22 reported that the Abe government has decided to make a revision to the project to construct a new U.S. base in Okinawa's Henoko district because the seafloor in some portions of the planned construction site was found to be as soft as mayonnaise and thus a revised plan needs to be implemented.
The central government has to obtain approval from the Okinawa governor in order to carry out any additional work. With Governor Tamaki Denny opposing the Henoko base project, there is no chance of his accepting Tokyo's request for permission to conduct the soft ground improvement plan. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,480 |
I know many of you have seen those epic #DrinkTheMenu (which yes I'm going to start using that hashtag every time I hang out with The Minty) photos that feel almost overwhelming to any sane liver - and you very often ask me "How do I join in on the fun?!"
Well now's your chance. Did you know that there's an EPIC... I don't even know if epic covers it. Colossal, astronomic, monster of a cocktail event coming to Los Angeles on Saturday September 23rd (2017) that features 30 bars all making custom cocktails?
It's called Cocktails in the City and TBH it looks like the start of a very sloppy night.
Tickets are $25 and include 1x cocktail and a "swag bag" full of goodies. Additional cocktails are $10 each... which given the talent booked for this event is a pretty awesome deal considering many of these bars can charge upwards of $16 a pop regularly.
But this isn't a "tasting" event. It's a drinking event. I would highly encourage - well first I would highly encourage you Uber or Lyft there - but that you also bring a bevvy of cocktail loving friends so you can make the rounds and sip and share to taste all of the lovely concoctions the mixmasters are making. So who's in?
And in case you're wondering - who in God's name created such an event? The answer is - The Brits. It's a festival that began 5 years ago in London, has been making the worldwide circuit, and Los Angeles gets to play home to it's first US landfall! Who knows when it'll come again - so whatever you're doing that Saturday night, cancel it and buy tickets for this instead.
Also - it looks like if you download the Hooch app and use the code "cocktailscity" it'll save you $10 off the ticket price - or really, just give you a good excuse to get 1 more cocktail at the event. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,097 |
Good Moose Music
Loreen – Paper Light (Higher)
Posted on March 5, 2015 by goodmoosemusic
Singer of The Best Eurovision Entry Of All Time is back with the first single from her forthcoming second album. The track is called Paper Light (Higher), with the "paper light" bit meaning she could get her fans to go to a cold Stockholm park to send off a bunch of paper lanterns in this rather pretty promo video. It's technically currently only available on Spotify in the Nordics but will be out worldwide on Monday 9 March. But of course it's all over YouTube by now, with Warner Music Sweden frantically removing each link as it appears. It's a big (euphoric?) house-pop number with Loreen singing in her best falsetto and a massive orchestral outro. For some reason we expected something groundbreaking from Loreen and on first listen this isn't quite it. But, regardless, it's an excellent song which will definitely please the fans and with any justification earn some radio play outside of Scandinavia. She also hasn't forgotten the show which made her and will be performing the song live for the first time at this week's Melodifestivalen (what's that, we hear you cry?! Well, look here!). That could well be proper event telly.
Loreen Pop Sweden
Maja Francis – Last Days of Dancing
Jenny Hval – That Battle Is Over | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,912 |
\section{Introduction}
Most low- and middle-income countries (LMIC) are not covered by Civil Registration and Vital Statistics (CRVS) systems. Consequently, the estimation of health and population indicators relies on
other types of data, in particular household surveys such as Demographic Health Surveys (DHS), Malaria
Indicator Surveys (MIS) and Multiple Indicator Cluster Surveys (MICS). These surveys typically support reliable estimation for key survey indicators at the national level and the first subnational administrative level (states, provinces, or regions), namely admin-1 areas.
While estimates at national level and admin-1 level serve as important measures for policy makers, they do not reflect subnational estimation at finer scale, such as at admin-2 level, where health programs are designed and implemented \citep{mayala:etal:19}. In recent years, there has been a growing need for more localized DHS estimates. Formal use of local data can help in determining district
health priorities and planning, allocating resource,
managing health workers to better serve the needs of the local population \citep{wickremasinghe:etal:16}.
Beyond local needs, international development goals also boost the
demand for admin-2 estimates \citep{utazi:etal:21}. Sustainable Development Goals (SDGs) requires an improvement in the measurement and understanding of local geographic
patterns to support more region-customized decision-making and more efficient program implementation
\citep{united:nations:general:assembly:15}. Reliable estimates at admin-2 level facilitate the effort to improve health outcomes for all and reduce within-country inequalities, which are a priority of the SDGs \citep{hosseinpoor:etal:15}.
Producing area-level estimates for key indicators is within the sphere of small-area estimation (SAE).
There are two categories of models in the SAE literature: area-level models and unit-level (or cluster-level) models. Area-level models begin with design-based estimates. The DHS and MICS surveys use a stratified, two-stage cluster design with
probability proportional to size (PPS) sampling. Area-level models build upon weighted estimates with the complex designs being explicitly accounted for. In their seminal work, Fay and Herriot \citep{fay:herriot:79} propose a hierarchical models that incorporates random effects to improve the precision of weighted estimates. No additional considerations about the design are needed for area-level model because they are already incorporated in the weighted estimators, along with their design based variance estimates.
\begin{figure}
\centering
\makebox{
\includegraphics[width=0.95\linewidth,trim={0 3.2cm 0, 3cm},clip]{Figs/HIV-compare-admin2-est-scatter.pdf}
}
\caption{Comparison of cluster-level model estimates with direct estimates for HIV prevalence amongst women of age 15-49 in admin-2 regions of Malawi. Estimates from left/right panels are from fixed effects cluster level model without/with urban/rural stratification.}
\label{fig:stratified-compare-2}
\end{figure}
The design based area-level models perform well in producing admin-1 estimates, as DHS is powered to admin-1 level. For example, \cite{li:etal:19} estimate under-5 mortality in 35 African countries using a space-time Fay-Herriot model. However, due to data sparsity, weighted estimates at admin-2 level are unstable, or unavailable. Such a scenario calls for a unit-level model-based approach for reliable estimates. \cite{battese:etal:88} introduce a nested error regression model, which models the outcome at the level of the sampling unit (cluster level in DHS surveys) and includes area-level random effects to reduce uncertainty in estimates. A major challenge for these models is that they do not explicitly acknowledge the design for the survey and failing to account for the stratification might incur bias \citep{dong:wakefield:21, paige2022design}. Notably, two large-scale producers of subnational demographic and health estimates, IHME and WorldPop do not adjust for the U/R stratification. Unfortunately, accounting for the stratification is not straightforward, as we shall see.
In most DHS surveys, urban/rural (U/R) serves as a binary stratification variable. Often urban clusters are over-sampled, compared with rural clusters. This is particularly the case in less urbanized areas, because exact PPS sampling will result in insufficient samples for urban strata in a LMIC setting \citep{dhsmanual}. Moreover, the prevalence of outcomes of interest often differs in urban and rural areas. Consequently, ignoring the stratification might result in bias because of the non-representative nature of the sample. The extent of the bias is closely related to the extent of urban over-sampling and the association between the response and U/R status. We use the HIV prevalence of women of age 15-49 in Malawi in 2015-2016 to illustrate the problem. Each point in Figure \ref{fig:stratified-compare-2} represents the estimates from one admin-2 area. The Malawi DHS 2015-2016 is one of the few DHS surveys that is powered at admin-2 level, which makes it ideal to demonstrate the issue. As shown in the left panel, a fixed effects model (at admin-2 level) without U/R stratification produce estimates systematically higher than the survey weighted estimates. It is no surprise that such model overestimate the HIV prevalence because of the oversampling of urban samples in most admin-2 regions in Malawi and the fact that urban areas tend to have higher HIV prevalence than rural areas.
There are many important health indicators that exhibit urban and rural association. The list includes, but is not limited to, HIV prevalence, child stunting rate, child mortality rates and fertility rates. \cite{dong:wakefield:21} point out that appropriate adjustment for design in unit-level model requires: model terms that explicitly acknowledge the design, such as U/R fixed effects in the prevalence model; and an aggregation procedure that assembles strata specific estimates to area level estimates. The major challenge remains in obtaining the population fractions used for aggregation. A further complication is that the reported cluster locations are jittered version of the true locations - further details of this aspect are addressed in Section \ref{sec:jitter}. In this paper, we formalize a pipeline that account for stratification in unit-level models based on household surveys.
The structure of the paper is as follows. In Section \ref{sec:modeling}, we introduce the methodological framework with an emphasis on the population fractions used for aggregation. In Section \ref{sec:results}, we presents results for both the classification model and the overall modeling procedure in the context of HIV prevalence in a space-only setting. We summarize the paper with a discussion in Section \ref{sec:discussion}. In the Appendix, we implement an under-5 mortality rate example to demonstrate how the method works for a space-time indicator. We also place the additional modeling choices and minor details in the Appendix.
\section{Modeling}\label{sec:modeling}
To produce prevalence estimates at the admin-2 level, we use an {\it aggregation model} that combines a {\it prevalence model} at the cluster level with an {\it U/R classification model}. The prevalence model and the classification model can be developed separately and the relationship between the components is shown in Figure \ref{fig:model}.
\tikzstyle{decision} = [diamond, draw, fill=blue!20,
text width=5.5em, text badly centered, node distance=3cm, inner sep=0pt]
\tikzstyle{block} = [rectangle, draw, fill=blue!20,
text width=10em, text centered, rounded corners, minimum height=4em]
\tikzstyle{line} = [draw, -latex']
\tikzstyle{cloud} = [draw, ellipse,fill=red!20, node distance=3cm, text width=4.5em,text centered,
minimum height=2em]
\begin{figure}
\begin{center}
\begin{small}
\begin{tikzpicture}[node distance = -.3cm, auto]
\node [block] (level1left) {Prevalence\\ Model};
\node [block, xshift=5.5cm] (level1right) {U/R\\ Classification Model};
\node [block, below of=level1left, yshift=-3cm, xshift=2.75cm] (level2) {Aggregation\\ Model};
\path [line] (level1left) -- (level2);
\path [line] (level1right) -- (level2);
\end{tikzpicture}
\caption{Overview of modeling process}
\label{fig:model}
\end{small}
\end{center}
\end{figure}
The U/R status of any point in the country is defined through the sampling frame, as we will further explain in Section \ref{A-B-surface}. Thus, the U/R classification is independent of the survey indicators. At the final stage, the outputs from the two models can be combined to form the aggregated estimates.
We start by describing various prevalence models in Section \ref{prev-model} and provide details of the aggregation method in Section \ref{aggre-model}. Section \ref{sec:UR-class-frac} features the emphasis of the paper -- the U/R classification model and \ref{sec:jitter} considers the impact of jittering.
\subsection{Prevalence Model}
\label{prev-model}
We focus on unit-level models for prevalence estimation - these are generally required for admin-2 estimates in LMICs. The proposed framework aims to account for the design in particular with respect to the U/R stratification. In the setting of DHS surveys from LMICs, the units refer to clusters. We adopt Bayesian hierarchical models and use smoothing to produce estimates when the data are sparse. We start with a space only model for binary survey indicators and extend the model to a space-time setting. The association with the U/R stratification is incorporated as a fixed effect. In addition, we also consider including separate spatial fields for U/R, thereby allowing U/R effects to vary in subnational regions.
Here we adopt discrete spatial fields at the area level, though give a discussion about continuous spatial models in the Appendix. Moreover, we do not include covariates in the prevalence model here, but we do consider this extension in the Appendix.
We describe spatial prevalence model that we will use for the application of HIV prevalence for women of age 15-49 in Malawi. We let $Z_c$ and $n_c$ represent the number of events and the number sampled in cluster $c$, which has location $\mbox{\boldmath $s$}_c$ (which we assume for the moment is the true location).
In the mixed effects model we assume,
\begin{eqnarray}\label{eq:binomial}
Z_{c} | p_{c} &\sim& \mbox{Binomial}(n_{c},p_{c})
\nonumber \\
p_c &=& \mbox{expit} \left( \alpha^{\text{rural}}\times I( \mbox{\boldmath $s$}_c \in \mbox{rural}) + \alpha^{\text{urban}}\times I( \mbox{\boldmath $s$}_c \in \mbox{urban}) + S(\mbox{\boldmath $s$}_c)
+\epsilon_c \right) \label{eq:fixed-effect-prev}
\end{eqnarray}
where $ \alpha^{\text{rural}}$ and $\alpha^{\text{urban}}$ are intercepts for rural and urban with an associated indicator to determine whether $\mbox{\boldmath $s$}_c$ is in a part of area $i$ designated as rural or urban, $S(\mbox{\boldmath $s$}_c)$ is a spatial random effect associated with location $\mbox{\boldmath $s$}_c$, and $\epsilon_c \sim_{iid} \mbox{N}(0,\sigma_\epsilon^2)$ is a cluster-specific independent random effect. We assume the latter is reflecting the overdispersion (excess-binomial variation) due to within-cluster correlation; for more discussion, see \cite{dong:wakefield:21}.
We model the spatial random effect term in equation (\ref{eq:fixed-effect-prev}) as the summation of an independent and identically distributed distributed component and a spatial component, such that $S(\mbox{\boldmath $s$}_{c})=e_{i[\boldsymbol{s}_c]}+S_{i[\boldsymbol{s}_c]}=b_{i[\boldsymbol{s}_c]}$. The notation $i[\boldsymbol{s}_c]$ means the area within which the cluster $\mbox{\boldmath $s$}_{c}$ resides. We model the iid random effect as $e_i \sim_{iid} \mbox{N}(0,\sigma^2_e)$, the spatial part as an intrinsic conditional autoregressive (ICAR) component and $\boldsymbol{b}=[b_1,\dots,b_{m}]$ so that $ \boldsymbol{b} \sim \mbox{BYM2}(\sigma_s^2,\phi)$. We adopt the BYM2 parameterization \citep{riebler:etal:16} for the spatial random effects, in which $\sigma_s^2$ represents the marginal variance and $\phi$ represents the proportion of this variance explained by the spatial component. We impose penalized complexity (PC) priors \citep{simpson:etal:17} on the variance and correlation parameters $\sigma_s^2$ and $\phi$. In a simplified setting where we drop the spatial component of the random effects, the prevalence model will reduce to an iid random effect model.
Under the model specified in equation (\ref{eq:fixed-effect-prev}), we obtain the strata (area $\times$ U/R) specific risk in area $i$ as,
\begin{eqnarray}
p_{i,R} &=& \mbox{expit} \left( \alpha^{\text{rural}}+ b_i \right)
\nonumber \\
p_{i,U} &=& \mbox{expit} \left( \alpha^{\text{urban}} +b_i \right),
\label{strata-risk}
\end{eqnarray}
for rural and urban prevalence in area $i$.
This calculation is straightforward since we have assumed a constant risk within the area.
In a more flexible model, we allow distinct spatial fields for urban and rural stratum, where the U/R effect is allowed to vary across admin areas,
\begin{eqnarray}\label{eq:binomial}
Z_{c} | p_{c} &\sim& \mbox{Binomial}(n_{c},p_{c})
\nonumber \\
p_c &=& \mbox{expit} \left[ (\alpha^{\text{rural}}+ b_i^\text{rural})\times I( \mbox{\boldmath $s$}_c \in \mbox{rural}) + \right.
\nonumber \\
&&\left. (\alpha^{\text{urban}}+ b_i^\text{urban})\times I( \mbox{\boldmath $s$}_c \in \mbox{urban})+\epsilon_c
\right]
\label{eq:inter-prev}
\end{eqnarray}
The corresponding strata specific risk is,
\begin{eqnarray}
p_{i,R} &=& \mbox{expit} \Bigl( \alpha^{\text{rural}} + b_i^\text{rural}\Bigr)\nonumber \\
p_{i,U} &=& \mbox{expit} \Bigl( \alpha^{\text{urban}}+ b_i^\text{urban}\Bigr)
\label{int-strata-risk}
\end{eqnarray}
\subsection{Aggregation Model}
\label{aggre-model}
For a generic binary indicator $Z$, to calculate the area-level prevalence in area $i$, $p_i$, we aggregate the urban and rural specific risks:
\begin{eqnarray}
p_i &=&\underbrace{ \Pr( Z=1 | \mbox{rural},i )}_\text{Prevalence Model}
\times\underbrace{ \Pr(\mbox{rural}|i) }_\text{Classification Model}
+ \underbrace{ \Pr( Z=1 | \mbox{urban},i)}_\text{Prevalence Model}\times \underbrace{\Pr(\mbox{urban}|i)}_\text{Classification Model}
\nonumber \\
&=& p_{i,U}\times q_{i} + p_{i,R}\times (1-q_{i})
\end{eqnarray}
where $q_i$ is the urban fraction of the subpopulation corresponding to $Z$, in area $i$. We will now discuss in detail the estimation of $q_i$.
\subsection{U/R Classification and Fraction Model}
\label{sec:UR-class-frac}
\subsubsection{Overview}
\label{A-B-surface}
In a model-based approach, we need to adjust for urban and rural when the outcome depends on U/R and the survey did unequal sampling with respect to urban and rural. We emphasize the distinction between DHS U/R stratification and the ``true'' urban and rural classification over time. The definition and allocation of urban and rural clusters in a DHS survey occurs at the time of the census, which is the usual source of the {\it sampling frame}. The urban and rural cluster map is constructed whenever the sampling frame is formed, and is constant over time, just as the weights are in a design-based framework. However, the populations in those areas do change over time.
In a perfect world, we would have access to all clusters in the sampling frame, along with the relevant populations in these clusters. Based on the U/R status of those clusters, we would be able to calculate U/R fractions. However, such information is rarely available. Instead, we propose a generic pipeline using information from the sampled cluster and the Worldpop population density surface \citep{stevens2015disaggregating}. The main idea is to use pixels to represent clusters in the sampling frame. We construct a pixelated U/R map to estimate the actual U/R surface in the sampling frame; and use the Worldpop population density to allocate the relevant population in the pixels. We construct the U/R map using a classification model, which we will discuss in Section \ref{sec:classification}. For the moment assume we have a classification map $C$ and a generic population density map $H$ for the study region. Let $g$ be the index of pixels for a gridded surface and $\mbox{\boldmath $s$}(g)$ be the spatial coordinates of pixel $g$. The classification and population density at pixel $g$ are $C_{\scaleto{\mbox{\boldmath $s$}}{4pt}(g)}\equiv C_g$ and $h_{\scaleto{\mbox{\boldmath $s$}}{4pt}(g)}\equiv H_g$ respectively.
Then we can calculate the urban fraction in a generic area as,
\begin{equation}
q = \frac{\sum I(C_g = \text{urban}) \times H_{g}}{\sum H_{g}}
\label{eq:classify-to-frac}
\end{equation}
The method we propose is very flexible in that it allows us to calculate urban fraction for any sub-population and for any time. The estimation procedure is valid as long as we are working under the stratification defined by the specific sampling frame. Worldpop \citep{stevens2015disaggregating} provides yearly population density file for age-sex combinations in individual countries around the globe, and such data availability guarantees our methods are widely applicable.
\subsubsection{Classification Model}
\label{sec:classification}
The generic geographical problem is to classify each pixel on a grid over the study region as either urban or rural. Because different countries have distinct urban definitions, we normally define the study region as a single country, which coincides with the prevalence model.
There exists a non-statistical method to classify urban pixels. Motivated by the pattern that urban areas tend to be highly populated, \cite{wu:etal:21} adopt a thresholding algorithm to assign U/R labels for pixels merely based on population density. As DHS surveys often use a census as the sampling frame, it is likely that national and/or subnational urban total population fractions for the frame are available. We emphasize that those fractions are with respect to the total population, which is different from we aim to obtain: urban fractions for a specific age-sex subpopulation, such as women of age 15-49 in our HIV prevalence example. Moreover, the subnational urban fractions are usually only available at a level higher than we desire, e.g., admin-1.
Combining the fractions with WorldPop population density surfaces \citep{stevens2015disaggregating} for total population from the year of the sampling frame, we can derive an area-specific population density threshold which yields the correct proportion at the admin-1 level \citep{wu:etal:21}. To be specific, if we know the urban fractions for area $A_i$ is $r_i$, then pixels in area $i$ with population density greater than $N_{[i]}$ are classified as urban (with the rest as rural), with $N_{[i]}$ determined through the following thresholding algorithm,
\begin{equation}
N_{[i]}=\underset{N_{[i]}}{\mathrm{argmin}} \left| \frac{\sum_{g \in A_i} I{ (N_{g}>N_{[i]} )} \times N_{g}}{\sum_{g \in A_i} N_{g}}- r_i \right|,
\label{eq:benchmark:pop}
\end{equation}
where $g$ indexes pixels within admin region $A_i$, such that $ \mbox{\boldmath $s$}_g \in A_i$ and $N_{g}$ is the total population density at pixel $g$ (obtained from Worldpop).
The above method, which we refer as `population thresholding', is computationally efficient, but less reliable at finer spatial resolution. It is insufficient to only account for population density because there are many more criteria for defining urban settlements \citep{UNYearBook2018} and using a purely deterministic algorithm lacks statistical justification.
In this paper, we propose a more sophisticated pipeline that incorporates more data to better resemble the true U/R definition and our method is grounded with a legitimate statistical framework. The training data we consider are cluster information from DHS surveys, which contain the reported geographical coordinates and U/R status of each cluster. Those pixels only cover a small portion of the all gridded cells. As we are constructing an U/R surface consistent with the sampling frame, clusters from multiple DHS surveys can all be included in the training data as long as they are based on the same sampling frame. For example, both DHS Malawi 2010 and DHS Malawi 2015-16 used as sampling frame the 2008 Malawi Population and Housing Census, so we can combine the clusters to form a larger training set.
Our classification model is built on a suite of covariates. From a scientific perspective, we seek to find covariates that better mimic the official U/R definition for each country. According to Table 6 from the Demographic Year Book 2018 \citep{UNYearBook2018}, important urban features include, but are not limited to, high population density, infrastructure development (such as access to electricity) and existence of local government, engaged in non-agricultural activities. From a modelling perspective, the covariates need to be available for DHS countries, and in reasonable geographical resolution (for example, 1km$\times$1km). Based on the above information, we consider the list of covariates in Table \ref{table:cov_summary}. They are available in raster format and we also indicate the data source. For time-varying covariates, we pick the year of the sampling frame. In addition, we include admin region as a predictor because U/R classification might be different in different parts of the country. Usually we include admin-1 region because not all admin-2 regions have clusters for most surveys.
It is tempting to adopt spatial models like stochastic partial differential equations (SPDE) for constructing a U/R surface. However, urban areas often have hard boundaries and modeling a binary surface using a continuous spatial model is not appealing. In addition, we hope that the rich set of covariates will suffice to properly capture the spatial correlation between pixels. Hence, we opt for non-spatial statistical learning models, including Bayesian logistic regression, Bayesian additive regression trees (BART) \citep{chipman2010bart} and gradient boosted trees \citep{friedman2001greedy}. Under the Bayesian framework, it is straightforward to quantify uncertainty in classification models.
\begin{table}
\caption{\label{table:cov_summary}Summary of covariates.}
\centering
\fbox{%
\begin{tabular}{|l|l|l|}
\hline
Covariate Name & Details & Source \\
\hline
population & Population density & Worldpop \citep{stevens2015disaggregating} \\ \hdashline [0.5pt/5pt]
night time light & \makecell[l]{Annual average visible, stable \\ lights and cloud free coverages} & NOAA \\ \hdashline [0.5pt/5pt]
vegetation index & Mean annual terra vegetation indices & MOD13A3 v006 \citep{didan2015mod13a3} \\ \hdashline [0.5pt/5pt]
precipitation & Average annual precipitation & WorldClim \citep{fick2017worldclim} \\ \hdashline [0.5pt/5pt]
elevation & Altitude & WorldClim \citep{fick2017worldclim} \\ \hdashline [0.5pt/5pt]
temperature & Average annual temperature & WorldClim \citep{fick2017worldclim} \\ \hdashline [0.5pt/5pt]
access & \makecell[l]{Travel time to the nearest city of \\ 50,000 or more people in 2000} & Product described in \cite{nelson2008travel} \\
\hline
\end{tabular}}
\end{table}
In the following, we use BART to illustrate the method. Details of other models can be found in the Appendix. Let $\mbox{\boldmath $Y$}=\{Y_g\}$ denote the collection of U/R status with $Y=1/0$ indicating U/R for all pixels. Let $\mbox{\boldmath $x$}_g$ be the covariates at the pixel indexed by $g$. We use $\mbox{\boldmath $y$}_c$ and $\mathbf{X}_c$ to denote observed U/R status and covariates at DHS clusters, which constitute the training data.
The sum-of-trees model from BART can be expressed as
\begin{equation}
P(Y_g= 1|\mbox{\boldmath $x$}_{g},\mbox{\boldmath $\theta$}) = \Phi [G(\mbox{\boldmath $x$}_{g},\mbox{\boldmath $\theta$})]
\end{equation}
where \begin{equation}
G(\mbox{\boldmath $x$}_{g},\mbox{\boldmath $\theta$})=\sum_{q=1}^{q_0}g(\mbox{\boldmath $x$}_{g};\mbox{\boldmath $T$}_q,\mbox{\boldmath $M$}_q)
\end{equation}
Here, we use the probit link binary data, following the original BART paper \citep{chipman2010bart}, the function $g$ represents a regression tree, $T$ denotes a tree structure and $M$ denotes the parameter values for each of the terminal nodes of $T$.
BART uses a backfitting algorithm to generating a sequence of draws of tree parameters $\mbox{\boldmath $\theta$}=\{\mbox{\boldmath $T$},\mbox{\boldmath $M$}\}=\{(\mbox{\boldmath $T$}_1,\mbox{\boldmath $M$}_1), ... ,(\mbox{\boldmath $T$}_{q0},\mbox{\boldmath $M$}_{q0})\}$ such that they will converge to the posterior of $p(\{\mbox{\boldmath $T$},\mbox{\boldmath $M$}\}|\mbox{\boldmath $y$},\{\mbox{\boldmath $x$}(\mbox{\boldmath $s$}_c)\})=p(\mbox{\boldmath $\theta$}|\mbox{\boldmath $y$},\mathbf{X}_{\{\mbox{\boldmath $s$}_c\}})$. Inference then can be made from $m_0$ posterior draws of tree parameters $\mbox{\boldmath $\theta$}^{(m)}$, $m= 1, ..., m_0$.
Based on the classification model, we aim to determine the U/R status for all grid points within a country. The models we consider yield predicted probabilities and the quantities we aim to estimate are $P(Y_g= 1|\mbox{\boldmath $x$}_{g},\mbox{\boldmath $\theta$})$, where the predicted probability can be expressed as a function of the model parameters $\mbox{\boldmath $\theta$}$. We may use a cutoff such as 0.5 to convert into binary classifications. Thus, we let $v_g(\mbox{\boldmath $\theta$})=P(Y_g= 1|\mbox{\boldmath $x$}_{g},\mbox{\boldmath $\theta$})$, where we drop the dependency on covariates because we can treat the covariates at pixel $g$ as known values. Under the Bayesian framework, we aim to estimate the posterior for $v_g(\mbox{\boldmath $\theta$})$, where $g=1,...,n_g$ and $n_g$ is the total number of pixels in the country. A point estimate can be derived from the posterior $\mbox{\boldmath $\theta$}|\mbox{\boldmath $y$}$, such as the posterior mean, which we denote it as $\mu_g$,
\begin{align}
\begin{split}
E_{\scaleto{\mbox{\boldmath $\theta$}|\mbox{\boldmath $y$}}{7pt}}[P(Y_g=1|\mbox{\boldmath $\theta$})] &=\int v_g(\mbox{\boldmath $\theta$}) p(\mbox{\boldmath $\theta$}|\mbox{\boldmath $y$})d\mbox{\boldmath $\theta$} \\
& = \mu_g
\end{split}
\end{align}
We assign a classification for pixel $g$ at location $\mbox{\boldmath $s$}_g$ as $I{ (\mu_{g}>\tau )}$, where $ \mu_{g}$ is the estimated probability of pixel $g$ being urban. Here the cutoff $\tau$ is a number in (0,1), and it may vary for different locations $g$, if additional information is available.
Ideally, our classification surface would yield the same urban fraction(s) for total population as those from the sampling frame. Using a predefined and universal cutoff like $\tau=0.5$ will not in general produce predicted probabilities that are well calibrated. To address that, we adopt a strategy similar to the thresholding method described in equation (\ref{eq:benchmark:pop}). We refer to our deterministic calibration approach as exact benchmark. We consider the subnational case only and choose $\tau_i$ to make the estimated fraction closest to the subnational known proportions $r_i$, where $i$ indexes the admin regions.
For example, to match the admin-2 proportions in Malawi (these proportions are usually available only at admin-1 level), we have
\begin{equation}
\tau_i=\underset{\tau_i}{\mathrm{argmin}} \left| \frac{\sum_{g \in A_i} I{ (\mu_{g}>\tau_i )} \times N_{g}}{\sum_{g \in A_i} N_{g}}- r_i \right|
\label{eq:exact-benchmark}
\end{equation}
where the summation is over all pixels $g$ within admin region $A_i$, such that $ g \in A_i$ and $N_{g}$ is the 2008 Malawi total population density at pixel $g$ (obtained from Worldpop).
The above derivation is based on the assumption that the posterior $\mbox{\boldmath $\theta$}|\mbox{\boldmath $y$}$ is analytically available.
In practice, we use draws from the posterior to obtain the estimated predicted probability $\tilde{\mu}_g$ to $\mu_g$ as
\begin{align}
\begin{split}
\tilde{\mu}_g &=\frac{1}{m_0}\sum_{m=1}^{m_0} v_g(\mbox{\boldmath $\theta$}^{(m)})
\end{split}
\label{eq:tilde_mu_g}
\end{align}
where $\mbox{\boldmath $\theta$}^{(m)}$ are draws from the posterior $\mbox{\boldmath $\theta$}|\mbox{\boldmath $y$}$ and $m_0$ is the number of posterior draws. Similarly, we estimate $\tau_i$ using $\tilde{\mu}_g$ such that
\begin{equation}
\tilde{\tau}_i=\underset{\tau_i}{\mathrm{argmin}} \left| \frac{\sum_{g \in A_i} I{ (\tilde{\mu}_{g}>\tau_i )} \times N_{g}}{\sum_{g \in A_i} N_{g}}- r_i \right|
\label{eq:tilde_tau}
\end{equation}
and in the end we classify the pixels as
\begin{equation}
\tilde{C}_g = I(\tilde{\mu}_{g}>\tilde{\tau}_i)
\label{eq:tilde_c_g}
\end{equation}
where $i$ is the index for the area such that $ \mbox{\boldmath $s$}_g \in A_i$.
Finally, following equation (\ref{eq:classify-to-frac}), we may obtain estimates of urban fractions for population $N^{*}$ in an area $A_j^{*}$.
\begin{equation}
\tilde{q}_j = \frac{\sum_{g \in A^{*}_j} I(\tilde{C}_g = \text{urban}) \times N^{*}_g}{\sum_{g \in A^{*}_j} N^{*}_g}
\label{eq:classify-to-frac-est}
\end{equation}
where we adopt different notations ($N$ vs. $N^{*}$ and $A$ vs. $A^{*}$) to emphasize that the target population and domain are likely to be different from those in the classification model (we use total population as a covariate and for benchmarking). For example, we are able to obtain urban fraction estimates for women of age 15-49 in 2010 in admin-2 regions of Malawi.
\subsection{Jittering}
\label{sec:jitter}
To maintain the confidentiality of survey respondents, the DHS release a geographically displaced version of the true locations of clusters \citep{burgert2013geographic}. According to the rules, U/R clusters are displaced a distance up to 2/5 kms; with a further, randomly-selected 1\% of rural clusters displaced a distance up to 10 kms. The displaced distances are uniformly distributed and the displacement may not move the point across administrative boundaries (admin-1/2, country specific). Such displacement can harm our classification model. When a cluster is jittered to cross the U/R boundary, it becomes mislabeled. For example, an urban cluster in Blantyre from Malawi DHS 2015-2016 has been displaced into the mountains, as shown in Figure \ref{fig:cluster_529}(a). The true or undisplaced cluster location should be near the edge of the red circle, most likely in the northwest corner, because of high population density there. If we do not account for the displacement, the classification algorithm will be distorted.
Following the development in \cite{wilson2020pointless}, we denote the relationship of the true and reported (jittered) cluster locations as
\begin{equation}
\underbrace{\mbox{\boldmath $u$}}_{\text{Reported Location}} = \underbrace{\mbox{\boldmath $s$}}_{\text{True Location}} + \underbrace{\mbox{\boldmath $\epsilon$}}_{\text{Displacement}}
\end{equation}
where $\mbox{\boldmath $\epsilon$}$ is the result of the jittering probability density function. Under the DHS jittering algorithm, the true location is randomly displaced according to the distribution (in polar coordinates), \begin{equation}
p(\ell,\eta) = \frac{1}{2\pi R}\times I({0 <\ell <R})\times I({0<\eta<2\pi})
\end{equation}
where $R=2$km for urban clusters and $R=5$km for 99\% of rural clusters and $R=10$km for the remaining 1\% of rural clusters. As just discussed, the displaced location has to stay within the same strata, but we do not account for this in the notation, for simplicity. We describe two approaches to account for the jittering.
\subsubsection{Partial Adjustment}
\label{Partial-Adjustment}
We first model the true locations of clusters without using information from the classification model, so that there is no feedback, as shown in Figure \ref{fig:no-feed-graph}. The true location of each generic cluster is estimated through density $p(\mbox{\boldmath $s$}_c|\mbox{\boldmath $u$}_c,y_c)$ and then we use the covariates $\mbox{\boldmath $X$}_{\{\mbox{\boldmath $s$}_c\}}$ based on the estimated locations to obtain estimates for $\mbox{\boldmath $\theta$}$.
\begin{figure}
\centering
\includegraphics[width=1\linewidth,trim={0 0cm 0cm 0},clip]{Figs/diagram.PNG}
\caption{Graphical model for the partial adjustment method.}
\label{fig:no-feed-graph}
\end{figure}
To statistically model the jittering process under a Bayesian setup, we impose a prior on a generic true location \citep{wilson2020pointless}. To ease notation, we drop index $c$ for cluster and specify the prior for a single cluster as
\begin{align}
p(\mbox{\boldmath $s$} = \mbox{\boldmath $t$}_
) = d_k, \quad k=1,\dots,K, \label{eq:maskprior}
\end{align}
over potential locations $\{\mbox{\boldmath $t$}_k\}$, and
where $d_k$ is the probability that potential location $\mbox{\boldmath $t$}_k$ was selected. Here, we assume the true cluster locations appear only on the center of gridded pixels, which is a discretization of the true sampling process.
If probability proportional to size (PPS) sampling was undertaken (the usual strategy in the DHS), we may use $$d_k \propto N_k$$ where $N_k$ is the total population density of the cluster located at $\mbox{\boldmath $t$}_k$, obtained from the pixelated population density surface. In DHS surveys, the PPS sampling is often based on number of households and we use population density as a practical approximation.
The posterior is then,
\begin{equation}
p(\mbox{\boldmath $s$} = \mbox{\boldmath $t$}_k
|\mbox{\boldmath $u$}, y
) \propto p(\mbox{\boldmath $u$} | \mbox{\boldmath $s$} = \mbox{\boldmath $t$}_k,y ) \times p(\mbox{\boldmath $s$} = \mbox{\boldmath $t$}_k), \quad k=1,\dots,m,
\label{eq:true-loc-post}
\end{equation}
where $p(\mbox{\boldmath $s$})$ corresponds to (\ref{eq:maskprior}).
The distribution of $\mbox{\boldmath $u$} | \mbox{\boldmath $s$}$ is uniform on a circle of radius $R$, where $R$ depends on the U/R status $y$ of the cluster. We discretize the circle and use centers of pixels as potential locations for the cluster. We have
\begin{equation}
p(\mbox{\boldmath $u$} | \mbox{\boldmath $s$} = \mbox{\boldmath $t$}_k,R ) = C_{R} \times I({0 < d(\mbox{\boldmath $u$}, \mbox{\boldmath $t$}_k) < R})/{2\pi R d(\mbox{\boldmath $u$}, \mbox{\boldmath $t$}_k)}
\end{equation}
where the distance between the pixelated candidate location
$\mbox{\boldmath $t$}_{k} = [t_{k1}, t_{k2}]$ and the reported location
$\mbox{\boldmath $u$} = [u_{1}, u_{2}]$ is
\begin{equation}
d(\mbox{\boldmath $u$}, \mbox{\boldmath $t$}_k )= \left[(u_1 - t_{k1} )^2 + (u_2 - t_{k2} )^2\right]^{1/2}.
\end{equation}
The normalizing constant that accounts for the jittered point being restricted to stay within administrative area $A_i$ is
\begin{align}
C_{R} = \left[\sum_{\scaleto{\mbox{\boldmath $u$}}{4pt} \in A_i} [2\pi R d(\mbox{\boldmath $u$}, \mbox{\boldmath $t$}_{k})]^{-1}I({0<d(\mbox{\boldmath $u$}, \mbox{\boldmath $t$}_{k}) < R}) \right]^{-1} \label{eq:ulocs:norm}
\end{align}
We take into consideration the different mechanism of geographical displacement for urban and rural clusters and we arrive at the following posterior for the true location
\begin{equation}
p(\mbox{\boldmath $s$} = \mbox{\boldmath $t$}_k | \mbox{\boldmath $u$} ,y ) \propto
\begin{cases}
N_k Q_k(2), & \text{if}\ y=1
\text{ (\emph{urban})}\\
N_{k} \left[ 0.99 Q_k(5)+ 0.01 Q_k(10) \right], & \text{if}\ y=0 \text{ (\emph{rural})}
\end{cases}
\end{equation}
where we define $Q_k(R^*)= \frac{C_{R=R^*}} {2R^* \pi d(u, t_k)}I({0 < d(\mbox{\boldmath $u$}, \mbox{\boldmath $t$}_{k}) < R^*(km)})$
\begin{figure} [H]
\centering
\includegraphics[width=1\linewidth,trim={0 0cm 0cm 0},clip]{Figs/Cluster_529_urban_correction.pdf}
\caption{Example of a wrongly displaced urban cluster (cluster number 529 from Malawi DHS 2015-16). The blue dot is the observed cluster location and the red circle captures the potential true location for the cluster before jittering. The observed cluster location is in the middle of an unpopulated area, as shown from the terrain map and the population density map. }
\label{fig:cluster_529}
\end{figure}
Roughly speaking, the posterior probability is proportional to population density and inversely proportional to the distance between candidate location and reported location. We note that this posterior is for a single cluster. By the independence of the geographical displacement for cluster locations, the joint posterior is the product of the posteriors for single locations, so that
\begin{align}
p(\{\mbox{\boldmath $s$}_c\} | \{\mbox{\boldmath $u$}_c\} ,\mbox{\boldmath $y$} ) = \prod_{c=1}^{c_0} p(\mbox{\boldmath $s$}_c | \mbox{\boldmath $u$}_c ,y_c )
\label{joint:loc}
\end{align}
where $c_0$ is the total number of clusters.
The lack of feedback from the classification model means that the partial adjustment will tend to relocate clusters towards high population density pixels. Such a tendency is not desirable for rural clusters, especially when the observed location for a rural cluster is close to an urban center. We adopt a practical approach in which we only adjust the locations for urban clusters, but leave the coordinates of rural clusters unchanged. In addition, instead of generating samples from $ p(\{\mbox{\boldmath $s$}_c\} | \{\mbox{\boldmath $u$}_c\} ,\mbox{\boldmath $y$} ) $ and fitting a classification model for each sample, we take the posterior mode from $ p(\{\mbox{\boldmath $s$}_c\} | \{\mbox{\boldmath $u$}_c\} ,\mbox{\boldmath $y$} )$ and fit the classification model once using the set of corrected values. The posterior mode is the most likely location that the cluster truly resides, under the model. As an example, the cluster shown in Figure \ref{fig:cluster_529} will be corrected to the pixel at the upper left corner, as its posterior probability is the highest among pixels in Figure \ref{fig:cluster_529}(c). See Figure \ref{fig:cluster_Blantyre} in the Appendix for a visualization of the implementation of partial correction in an entire admin-2 area.
\subsubsection{Full Bayesian Adjustment}
\label{full-adjust}
Now we model the true locations of clusters and the parameters from the classification model jointly. Such a scenario requires the classification model to also fall in the Bayesian framework. Candidate models include Bayesian logistic regression and BART.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[
roundnode/.style={circle, draw=green!60, fill=green!5, thick, minimum size=12mm},
squarednode/.style={rectangle, draw=green!60, fill=green!5, thick, minimum size=12mm},
]
\tikzset{bigarrow/.style={decoration={markings,mark=at position 1 with %
{\arrow[scale=2.5,>=stealth]{>}}},postaction={decorate}}}
\node[squarednode] (X) {$ \mbox{\boldmath $X$}$};
\node[squarednode] (Y) [above=of X] {$Y$};
\node[roundnode] (theta) [left=of Y] {$ \mbox{\boldmath $\theta$}$};
\node[roundnode] (s) [right=of X] {$\large \mbox{\boldmath $s$}$};
\node[squarednode] (u) [right=of Y] {$ \large \mbox{\boldmath $u$}$};
\draw[bigarrow,green] (s) -- (u);
\draw[bigarrow,green] (s) -- (X);
\draw[bigarrow,green] (X) -- (Y);
\draw[bigarrow,green] (Y) -- (u);
\draw[bigarrow,green] (theta) -- (Y);
\end{tikzpicture}
\caption{Graphical model for the full Bayesian adjustment model.}
\label{fig:bayesian-graph}
\end{center}
\end{figure}
The joint model can be decomposed as the following, based on the graphical model in Figure \ref{fig:bayesian-graph}
\begin{equation}
p(\{\mbox{\boldmath $s$}_c\},\mbox{\boldmath $\theta$},\mbox{\boldmath $y$},\{\mbox{\boldmath $u$}_c\})=p(\mbox{\boldmath $y$}|\{\mbox{\boldmath $s$}_c\},\mbox{\boldmath $\theta$})\times p(\{\mbox{\boldmath $u$}_c\}|\mbox{\boldmath $y$},\{\mbox{\boldmath $s$}_c\}) \times
p(\{\mbox{\boldmath $s$}_c\})\times p(\mbox{\boldmath $\theta$})
\end{equation}
The joint posterior then can be represented as
\begin{align}
\begin{split}
p(\{\mbox{\boldmath $s$}_c\},\mbox{\boldmath $\theta$}|\mbox{\boldmath $y$},\{\mbox{\boldmath $u$}_c\}) & = \frac{p(\{\mbox{\boldmath $s$}_c\},\mbox{\boldmath $\theta$},\mbox{\boldmath $y$},\{\mbox{\boldmath $u$}_c\})}{p(\mbox{\boldmath $y$},\{\mbox{\boldmath $u$}_c\})}\\
& \propto p(\mbox{\boldmath $y$}|\{\mbox{\boldmath $s$}_c\},\mbox{\boldmath $\theta$})\times p(\{\mbox{\boldmath $u$}_c\}|\mbox{\boldmath $y$},\{\mbox{\boldmath $s$}_c\}) \times
p(\{\mbox{\boldmath $s$}_c\})\times p(\mbox{\boldmath $\theta$})
\end{split}
\label{eq:joint-post}
\end{align}
To sample from the joint posterior, we propose a MCMC algorithm, which requires the conditionals $p(\mbox{\boldmath $\theta$}|\mbox{\boldmath $y$},\{\mbox{\boldmath $s$}_c\},\{\mbox{\boldmath $u$}_c\})$ and $p(\{\mbox{\boldmath $s$}_c\}|\mbox{\boldmath $y$},\mbox{\boldmath $\theta$},\{\mbox{\boldmath $u$}_c\})$. Recall that $\mbox{\boldmath $\theta$}$ are the parameters from the regression model, $\mbox{\boldmath $u$}$ is the reported location and $\mbox{\boldmath $s$}$ is the true location.
The conditional posterior for $\mbox{\boldmath $\theta$}$ is
\begin{align}
\begin{split}
p(\mbox{\boldmath $\theta$}|\mbox{\boldmath $y$},\{\mbox{\boldmath $s$}_c\},\{\mbox{\boldmath $u$}_c\}) & \propto p(\mbox{\boldmath $y$}|\{\mbox{\boldmath $s$}_c\},\mbox{\boldmath $\theta$})\times p(\mbox{\boldmath $\theta$})\\
& = p(\mbox{\boldmath $y$}| \mathbf{X}_{\{\mbox{\boldmath $s$}_c\}}, \mbox{\boldmath $\theta$}) \times p(\mbox{\boldmath $\theta$})
\end{split}
\label{eq:cond-theta}
\end{align}
Sampling from this posterior is easily available from any classification model with a Bayesian setup, such as Bayesian logistic regression or BART. The model uses all training data such that the input consists of lists of labels, locations and covariates. Here we drop the conditioning on the reported locations because of the conditional independence shown in Figure \ref{fig:bayesian-graph}.
Based on Equation (\ref{eq:true-loc-post}), we write the conditional for $\{\mbox{\boldmath $s$}_c\}$ as
\begin{align}
\begin{split}
p(\{\mbox{\boldmath $s$}_c\}|\mbox{\boldmath $y$},\mbox{\boldmath $\theta$},\{\mbox{\boldmath $u$}_c\}) & \propto p(\mbox{\boldmath $y$}|\{\mbox{\boldmath $s$}_c\},\mbox{\boldmath $\theta$})\times p(\{\mbox{\boldmath $u$}_c\}|\mbox{\boldmath $y$},\{\mbox{\boldmath $s$}_c\}) \times p(\{\mbox{\boldmath $s$}_c\})\\
& = \prod_{c=1}^{c_0}| p(y_c|\mbox{\boldmath $s$}_c,\mbox{\boldmath $\theta$})\times p(\mbox{\boldmath $u$}_c|\mbox{\boldmath $y$},\mbox{\boldmath $s$}_c) \times p(\mbox{\boldmath $s$}_c) |\\
& = \prod_{c=1}^{c_0}| p(y_c|\mbox{\boldmath $x$}_c,\mbox{\boldmath $\theta$})\times p(\mbox{\boldmath $s$}_c|\mbox{\boldmath $u$}_c,y_c)|
\end{split}
\label{eq:cond-s}
\end{align}
We observe that the joint conditional for all clusters is a product of the individual conditionals, which is expected as the jittering process is independent for different clusters. This also suggests that we can sample locations for individual clusters separately. Within the product on the last line of equation (\ref{eq:cond-s}), the first term can be viewed as a feedback from the classification model and the second term is the likelihood without considering the classification model. With both conditionals at hand, we shall use Gibbs sampling to sample from $p(\{\mbox{\boldmath $s$}_c\},\mbox{\boldmath $\theta$}|\mbox{\boldmath $y$},\{\mbox{\boldmath $u$}_c\})$. Details of the MCMC algorithm can be found in the Appendix.
\section{Results}
\label{sec:results}
\subsection{Classification Model Results}
We use Malawi as an example to demonstrate the performance of our various methods. The small island of Likoma which correspond to an admin-2 region is excluded, for reasons given in the Appendix. The goal is to produce U/R classifications for the sampling frame used in both DHS 2010 and DHS 2015--2016.
To reiterate, the cluster U/R status and the associated GPS coordinates serve as training data (with 1655 clusters in total). In addition, we include the geographical covariates listed in Table \ref{table:cov_summary} for the classification models. The methods we use, unless otherwise noted, are all benchmarked to the known admin-2 urban fractions from the census.
The models we consider are listed as follows,
\begin{enumerate}
\item
Population Thresholding: no statistical model, directly threshold using total population density to meet benchmark.
\item
BART: three versions considered. Using original cluster locations (BART-uncorrected), corrected cluster locations (BART-corrected, method from Section \ref{Partial-Adjustment}), and MCMC algorithm (BART-MCMC, from Section \ref{full-adjust}).
\item
Bayesian logistic regression (Logistic Regression).
\item
Gradient Boosted Trees (GBT).
\end{enumerate}
Across all models, population density and night time light are the most important predictors. In logistic regression model, they are the only two covariates that are significant (under $\alpha=0.05$).
\begin{figure}[H]
\centering
\makebox{
\includegraphics[width=0.9\linewidth,trim={0 0cm 0 1cm},clip]{Figs/ROC-curves.pdf}
}
\caption{ROC curve for 5-fold CV across methods.}
\label{fig:ROC-curves}
\end{figure}
\subsubsection{Cross Validation Results}
We first examine the classification accuracy of pixels using cross validation. The clusters are split into 5 folds and classifications from the testing sets are aggregated to form a single ROC curve. In the testing set, the original cluster locations and associated covariate values are used for prediction. Hence, the observed U/R status in the testing set might not reflect the truth due to the geographical displacement process described in Section \ref{sec:jitter}. However, the ROC curves do serve as a fair comparison for the performance of the methods. The results are shown in Figure \ref{fig:ROC-curves}, where we observe that the naive population thresholding method is outperformed by all other methods. The statistical classification models share comparable performance as further shown by the zoomed ROC curves in Figure \ref{fig:ROC-curves-zoomed} in the Appendix.
\subsubsection{Pixel Classification Visualization}
The motivation for our method is the absence of an U/R classification map in countries in which we wish to carry out SAE of health and demographic indicators in. Hence, we do not have the ground truth for checking the pixel-level classification we produce. However, we might use a surrogate such as the google (stamen) maps to visually validate our classification results. Admittedly, such comparison is subjective and lacking of statistical rigour, but it provides further evidence of the performance of the methods we propose.
We take as an example the urban area in Blantyre, which is the biggest city in Malawi. The plot in the center of Figure \ref{fig:Blantyre-Classification} shows the terrain map for part of the admin-2 region of Blantyre. A complete map for this region can be found as Figure \ref{fig:Blantyre-map} in the Appendix. The urban area is distinguished by the white color and the U/R delimitation is rather clear because Blantyre city is surrounded by mountains.
Note that the map here is contemporary while the U/R classification we aim to construct is for the 2008 census sampling frame. The fairness of comparison relies on a slow expansion rate for Blantyre city \citep{gondwe2021analysis}.
Figure \ref{fig:Blantyre-Classification} compares the U/R classification in Blantyre using the models we proposed. All methods recover the approximate shape of the urban regions, but they possess different degree of fuzziness at the boundaries. The oversmoothed shape resulting from logistic regression might be attributed to the smoothness in the covariates surface and the linearity that logistic regression imposes on the logit scale. The tree-based algorithms performs better at depicting the boundaries and BART-MCMC is arguably the best method as it even recovers the unpopulated mountain area within the center of the urban area. The non-statistical method, population thresholding, yields reasonable results, though the realized urban area is too diffuse.
The performance for population thresholding might not be generalized to other admin-2 areas because Blantyre is highly urban (the most urban region in Malawi) and population density has high predictive power here. On the other hand, the rest of the methods incorporate additional covariates and are more robust, so we might expect them to yield stable performance across admin-2 regions.
\begin{figure}[H]
\centering
\includegraphics[width=0.95\linewidth,trim={0 0cm 0 0cm},clip]{Figs/Blantyre_compare_indicator.pdf}
\caption{Classifications from different models for admin-2 region Blantyre in Malawi. Green pixels represent the areas predicted as urban.}
\label{fig:Blantyre-Classification}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=0.95\linewidth,trim={0 0cm 0 0cm},clip]{Figs/natl_thresh_adm2_compare_abs_err.pdf}
\caption{Benchmark at national level and comparison of the resulting admin-2 urban fractions with truth.}
\label{fig:malawi-frac-check}
\end{figure}
\subsubsection{Urban Fraction Accuracy}
\label{sec:urb-frac-accuracy}
\begin{table}
\caption{\label{table:admin2-error}National thresholding, admin-2 proportions error}
\centering
\begin{tabular}{rlrr}
\hline
& Method & Mean Error& Mean Absolute Error \\
\hline
& BART-corrected & -0.26\% & 1.28\% \\
& BART-MCMC & -0.34\% & 1.46\% \\
& BART-uncorrected & -0.12\% & 1.27\% \\
& Gradient Boosted Trees & 0.02\% & 1.17\% \\
& Logistic Regression & -0.24\% & 1.24\% \\
& Population Thresholding & -0.03\% & 2.47\% \\
\hline
\end{tabular}
\end{table}
Checking classification map for pixels is visually appealing, but admittedly, such comparison is more subjective and less quantifiable. Considering the ultimate goal is to accurately estimate the urban population fraction in a certain area, we present results comparing the performance of methods in estimating urban population fractions.
From the 2008 census in Malawi, we can obtain the U/R specific population within each admin-2 region. In the complete estimation procedure, we can calculate the corresponding admin-2 urban population fractions and use them as benchmarks. To check the accuracy of our methods, we will instead use national level urban fraction for benchmarking. Then we will use the resulting classification map to calculate urban fractions at admin-2 level and compare with the truth from the census. Such treatment also mimics the situation in most other countries, as admin-2 level urban fractions are not generally available. Note that although we choose total population for quality checking purpose only, our method can be applied to any sub-population depending on the availability of population density surfaces.
We measure the errors at the absolute scale, rather than the relative scale, because the aggregation for prevalence is a linear combination of urban and rural estimates. Figure \ref{fig:malawi-frac-check} shows the absolute difference between estimated urban fractions with the true fractions. Table \ref{table:admin2-error} summarizes the mean errors and mean absolute errors for the comparison, where the errors are calculated as the difference between estimated urban fractions and the known urban fractions from the census, for the total population; the errors are averaged across 27 admin-2 regions in Malawi (Likoma excluded). The methods using statistical models for classification all perform better than the naive population thresholding method. Gradient boosted trees outperforms logistic regression and versions of BART by a small margin. The overall results are satisfactory as the difference are measured at absolute scale, i.e., on average we only over or under estimate the true urban fraction by $\sim$1.2\%.
\subsection{Data Application: HIV Prevalence -- Space-only Modeling}
\label{sec:HIV-example}
We apply our method to estimate the HIV prevalence for women aged 15-49 in Malawi in 2015. In addition to the BYM2 spatial model described in Section \ref{prev-model}, we consider two additional ways to model the area-level effects, namely a fixed effects model and an iid random effects model. The details of the model setup can be found in the Appendix. We stress that Malawi is not typical in the availability of data, in particular, in richness -- for most countries, there would be insufficient data to fit the fixed effects models.
U/R specific estimates are aggregated using urban proportion for female 15-49 population for Malawi in 2015. Classification surface is obtained from BART with uncorrected cluster locations, benchmarked to known admin-2 urban proportion for total population. For Likoma, we use the urban fraction for the total population as an replacement, for reasons described in the Appendix.
First, to illustrate the association between HIV prevalence and U/R, we present the odds ratio corresponds to urban effect in the fixed effect model with U/R intercept. The estimate for odds ratio (95\% CI) is 2.03 (1.69, 2.44), indicating that the odds of having HIV is 2.03 times higher among women in urban areas as compared to rural areas.
We compare estimates from various models with survey weighted direct estimates for HIV prevalence using all available data, i.e., not cross validated. Table \ref{table:HIV-est-res} presents results for three classes of cluster-level models: fixed effects models, area-level IID random effects models and spatial BYM2 models. In addition, we consider three stratification methods within each class of model, namely no U/R stratification (Area only), U/R intercept (U/R+ Area) and full U/R adjustment (U/R $\times$ Area).
We consider a number of metrics for evaluating model performance by quantifying the difference between model estimates and survey weighted direct estimates. Given point estimates from the model, $\widehat{\mbox{\boldmath $p$}}=(\widehat{p}_1, \dots ,\widehat{p}_m)^{\text{\tiny{T}}}$,
and direct estimates $\tilde{\mbox{\boldmath $p$}}=(\tilde{p}_1, \dots ,\tilde{p}_m)$ for the $n_{\scaleto{A}{4pt}}=28$ admin-2 areas, we calculate the overall scoring metrics for a model as,
\begin{eqnarray}
\mbox{Bias}(\widehat{\mbox{\boldmath $p$}}, \tilde{\mbox{\boldmath $p$}}) &=& \frac{1}{n_{\scaleto{A}{4pt}}} \sum_{i=1}^{n_{\scaleto{A}{3pt}}} 100 \times (\widehat{p}_i - \tilde{p}_i) \label{eq:bias} \\
\mbox{Absolute Bias}(\widehat{\mbox{\boldmath $p$}}, \tilde{\mbox{\boldmath $p$}}) &=& \frac{1}{n_{\scaleto{A}{4pt}}} \sum_{i=1}^{n_{\scaleto{A}{3pt}}} 100 \times |\widehat{p}_i - \tilde{p}_i| \label{eq:absolutebias} \\
\mbox{Relative Bias}(\widehat{\mbox{\boldmath $p$}}, \tilde{\mbox{\boldmath $p$}}) &= &\frac{1}{n_{\scaleto{A}{4pt}}} \sum_{i=1}^{n_{\scaleto{A}{3pt}}} 100\% \times \frac{\widehat{p}_i - \tilde{p}_i}{\tilde{p}_i} \label{eq:cv1}\\
\mbox{Absolute Relative Bias}(\widehat{\mbox{\boldmath $p$}}, \tilde{\mbox{\boldmath $p$}}) &= &\frac{1}{n_{\scaleto{A}{4pt}}} \sum_{i=1}^{n_{\scaleto{A}{3pt}}} 100\% \times \left|\frac{\widehat{p}_i - \tilde{p}_i}{\tilde{p}_i}\right| \label{eq:cv1}
\end{eqnarray}
\begin{table}
\caption{\label{table:HIV-est-res} Model comparison for HIV prevalence of women of aged 15-49 in Malawi.}
\centering
\begin{tabular}{llrrrrr}
\hline
Model type & Stratification & Bias & \makecell[l]{Absolute \\ bias} & \makecell[l]{Relative \\ bias}&
\makecell[l]{Absolute \\relative bias} & \makecell[l]{CI \\width} \\
\hline
Fixed effects & Area only & 0.75 & 0.86 & 9.47\% & 10.01\% & 7.75 \\
Fixed effects & U/R+Area & -0.03 & 0.57 & 0.27\% & 5.45\% & 7.09 \\
Fixed effects & U/R*Area & 0.14 & 0.41 & 0.74\% & 4.97\% & 7.53 \\
\hline
Area IID & Area only & 0.84 & 1.25 & 15.23\% & 17.22\% & 7.48 \\
Area IID & U/R+Area & 0.22 & 0.89 & 7.27\% & 11.01\% & 6.90 \\
Area IID & U/R*Area & -0.17 & 0.70 & 2.79\% & 8.42\% & 7.01 \\
\hline
BYM2 & Area only & 0.37 & 1.19 & 9.90\% & 14.60\% & 6.53 \\
BYM2 & U/R+Area & -0.23 & 0.99 & 2.35\% & 11.02\% & 5.94 \\
BYM2 & U/R*Area & -0.25 & 0.96 & 1.27\% & 11.61\% & 6.16 \\
\hline
\end{tabular}
\end{table}
Table \ref{table:HIV-est-res} summarizes the model performance via the above scoring rules. The values for bias and absolute bias are on the per 100 people scale. For example, on average the fixed effects model without U/R stratification overestimate HIV prevalence by 0.75 per 100 women, compared with the direct estimates. We expect such bias because urban areas have higher HIV prevalence and oversampling of urban results in overestimates of HIV prevalence.
For all types of models, U/R stratification significantly reduces the bias. The fixed effects models are not affected by area level smoothing and thus are not susceptible to shrinkage, which introduces bias. Focusing on the first three row in Table \ref{table:HIV-est-res}, we observe that the absence of U/R stratification leads to overestimation of HIV prevalence by 9.47\% and U/R adjustments limit the relative bias to be below 1\%. Let us revisit the motivating plots presented in the introduction section, where we compare estimates from fixed effect models with and without U/R stratification (row 1 and row 3 in Table \ref{table:HIV-est-res}). From the comparison between the two panels in Figure \ref{fig:stratified-compare-2}, we can see that the bias disappears when we include U/R stratification in the cluster-level model. Figure \ref{fig:HIV-admin2-map} maps the admin-2 estimates from survey direct estimator, fixed effects model and BYM2 model with full U/R adjustment (row 3 and row 9). Spatial smoothing can be observed in the BYM2 model.
For point estimates, across all type of models, full U/R adjustment (U/R*Area) is slightly favored over U/R intercept (U/R+ Area) as they produce estimates closer to the direct estimates. As a trade-off for flexibility, the full U/R adjustment estimates have higher uncertainty compared with the U/R intercept counterparts, since more parameters are being estimated. To sum up, models with U/R stratification account for the survey design and ignoring U/R stratification leads to bias in model estimates. In general, it may not be necessary to consider the full U/R*Area model. Spatial smoothing model (BYM2) could also yield reliable estimates at finer resolution. In the Appendix, we show the results for admin-3 level HIV prevalence in Malawi from BYM2 models with and without U/R stratification.
\section{Discussion}
\label{sec:discussion}
In this paper we have proposed a framework to account for stratification in unit-level models. In particular, we deal with the U/R stratification in DHS and other household surveys. Obtaining strata specific estimates from unit-level models is relatively straightforward, as the U/R status for clusters are available and can easily be included as model terms. The major contribution of the paper is to establish a comprehensive aggregation procedure for obtaining the U/R population fraction for aggregation.
\begin{figure} [H]
\centering
\makebox{
\includegraphics[width=0.95\linewidth,trim={0 3cm 0 2cm},clip]{Figs/HIV-compare-admin2-est.pdf}
}
\caption{Maps for HIV prevalence estimates at admin-2 level from three models.}
\label{fig:HIV-admin2-map}
\end{figure}
The classification methods we proposed in Section \ref{sec:modeling} significantly outperform the naive population thresholding method used in \cite{wu:etal:21}. Among the methods based on statistical models, BART and GBT with original cluster location yield estimates closest to the truth, as shown in Section \ref{sec:urb-frac-accuracy}. We might have expected the BART-MCMC and BART-corrected method to work better as they properly account for the jittering process from DHS surveys, but their performance was slightly interior to the methods that used the original cluster locations. Such results might well be country specific as the urban fractions across admin-2 regions in Malawi vary greatly. In terms of computation time, MCMC-BART is much more demanding than the other methods. Therefore, based on our current experience, for now we recommend using BART and GBT methods with the original cluster location. Logistic regression can also serve as an alternative, for its fast computation and accuracy in estimating urban fractions, but it is less flexible in handling interactions between covariates. We plan to apply our method in other countries and with other indicators and a comparison of methods across more countries will provide more concrete guidance.
Removing bias caused by the association between a health indicator of interest and U/R is the overarching goal for incorporating U/R stratification in a cluster-level model. Through the example of modeling HIV prevalence for women of age 15-49 in Malawi, we demonstrate that the proposed methods achieved such a goal. If we treat the estimated urban fractions as fixed values, the aggregated estimates tend to have smaller uncertainty compared with the estimates without U/R stratification. This observation is consistent with the general idea that stratified sampling leads to survey estimates with reduced variance.
\bibliographystyle{rss}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,588 |
Q: How to create horizontal scrolling menu with drop-down? I'm trying to build a (possibly) CSS-only menu for an HTML5 page with the following requisites:
*
*menu should be an horizontal scrolling list (a top navbar), with scrolling buttons and no visible scrollbars
*first level menu items could have sub-items like a multiple nested drop-down menu
Visually:
-------------------------------------------------------------------------
| < | Menu 1 | Menu 2 + | Menu 3 | Menu 4 | > |
-------------------------------------------------------------------------
| menu 2.1 |
--------------
| menu 2.2 + |---------------
-------------- menu 2.2.1 |
| menu 2.1 |---------------
-------------- menu 2.2.2 |
|---------------
where the < and > buttons scroll the main menu to the left or right (eventually opened sub items should undrop themselves).
I also would like to use CSS only (no JavaScript), but this is an option.
I've already spent lot of time for this and tests, but I didn't get any satisfying example.
Please suggest me how can I achieve this.
A: You can do so quite easily.
You firs need a tree like structure using html lists :
<ul class="my-menu">
<li>
TOP 1
<ul>
<li>
1 - 1 >
<ul>
<li>
1 - 1 - 1
</li>
<li>
1 - 1 - 2
</li>
</ul>
</li>
</ul>
</li>
</ul>
You need to display the first level side by side to have the horizontal menu :
ul.my-menu > li {
display : inline-block;
}
For the second level we wish it to display just below the TOP1 text. So we will put it in absolute :
ul.my-menu > li > ul{
position : absolute;
}
Now let's hide the second & Third level.
ul.my-menu > li > ul li{
display: none;
}
Finally we can add the hover logic. Basically we will say if my parent "li" is hovered then display me.
ul.my-menu li:hover > ul > li {
display: block;
}
Of course you need to work more on the display. The size & position of the boxes are very important as if not the mouse will leave the bounderies of the parent which will close the display.
Here is a jsfiddle : https://jsfiddle.net/9mbLabj4/1/
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,101 |
\section{Introduction}
\lSect{intro}
There must have been a first generation of stars whose initial
composition, reflecting that of the Big Bang, was essentially metal
free. The term Population III is often applied to such stars
\citep[e.g.,][]{osh08}. To the extent that lines of heavy elements
and dust are responsible for mass loss, and in the absence of deep
mixing, such stars would have lived their lives with near constant
mass. As a result, the average mass at death of such stars would have
been larger, even for a metallicity-independent IMF. They would also
typically have been bluer stars, especially at death, and their
supernovae might have been fainter. Their nucleosynthesis would also
differ, at least for the many elements that are in any sense
``secondary''.
Such stars have been of interest for a long time
\citep[e.g.,][]{Eze71}, but recent measurements suggesting the
existence of stars at a very early epoch in the universe
\citep{Spe03,Kog03} have invigorated their study. So, too, have new
observational data sets that highly constrain the nucleosynthesis of
the first few generations of stars \citep[e.g.,][]{cay04,Lai08} and
the discovery of ultra-iron poor stars
\citep[e.g.,][]{Dep02,Chr02,Fre05} which may be providing clues as to
how the first generation of stars formed and exploded
\citep[e.g.,][]{Ume03}.
Zero metal stars are also a limiting case of low metallicity stars.
While this is the first in a series of papers that will treat the
evolution and explosion of all stars up to solar metallicity and
beyond, it is expected that many of the results will apply equally
well to stars of $\Ep{-4}\,{\ensuremath{\mathrm{Z}_{\odot}}}\xspace$ or even $\Ep{-2}\,{\ensuremath{\mathrm{Z}_{\odot}}}\xspace$, so long as
mass loss remains negligible. An open issue is whether and when forms
of mass loss not dependent upon atomic lines of heavy elements or
grains will become important. Will $100\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ zero metallicity
stars near the end of their lives resemble Eta Carina? It is also
uncertain just how fast this first generation of stars rotated and how
rotationally induced mixing may have affected not only mass loss and
nucleosynthesis, but also the presupernova structure
\citep[e.g.,][]{Mey06}.
Here we consider a fiducial set of models whose physics is
comparatively simple. The initial composition is pristine Big Bang
nucleosynthesis and mass loss is neglected at all stages of the
evolution. Similarly, rotation is ignored and, along with it, any
consequences of rotationally-induced mixing and strong asymmetry in
the explosion. The mass range studied is $10\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ to $100\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$.
Except between $90\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ and $100\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, this selection avoids the
complications of the pair instability on the upper end and asymptotic
giant branch mass loss on the lower end. Each of these approximations
is a potentially serious omission, and we hope to return with these
embellishments in future works.
Other recent surveys of zero metallicity massive stars are
\citet{Woo95}, \citet{Mar01}, \citet{Heg02}, \citet{Chi04},
\citet{Ume05}, \citet{Hir06}, and \citet{Tom07}. This one differs in
following the evolution of a large number of stars in great physical
detail, from the zero age main sequence through explosion as a
supernovae. A variety of supernova properties (energies, mixing, mass
cut) are considered for each mass. Nucleosynthesis, supernova light
curves and remnant masses are calculated. Even this ``simple'' study
took us five years to complete. As we shall see, our simple models
agree reasonably well with all the available data and make some
interesting predictions. The yields of all isotopes in all stars and
supernova models calculated (extended versions of \Tabs{0.6-yield-mix}
- \Tabff{1.2-yield-nomix-Ye}) are available in the electronic version
of the paper and more detailed data including all the presupernova
models and the yield data base and search tool is available online.
\section{Initial Models and Procedure}
\lSect{models}
\subsection{Masses Studied}
This study is specifically of non-rotating stars that experience iron
core collapse in the end and avoid the pair instability. Because the
ignition of multiple shells of nuclear burning can often lead to
dramatic variation in the final stellar structure for stars that
differ only slightly in initial mass \citep{Bar90}, the grid of masses
employed is very fine - 120 different masses in the
range $10\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ to $100\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$. Frequently, model stars differing
by only $0.1\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ had substantially different presupernova
properties, especially towards the lighter end of the range studied.
The masses of the stars studied are given in \Tab{ionize}.
\subsection{The Kepler Code and Adopted Physics}
\lSect{kepler}
All calculations of evolution, nucleosynthesis, and light curves were
done with the \KEPLER code. \KEPLER is a one-dimensional implicit
hydrodynamics package adapted to the study of stellar evolution and
explosive astrophysical phenomena \citep{Wea78,WHW02}. The convective
criterion is Ledoux, but with a substantial semi-convective diffusion
coefficient, about $10\,\%$ of the radiative diffusion coefficient in
regions that are stable by the Ledoux criterion and unstable by
Schwarzschild \citep{Woo88}. A small amount of convective
overshooting is included by forcing convective boundary zones to be
semiconvective. Opacities from the OPAL and Los Alamos tables are
used wherever the helium mass fraction exceeds \Ep{-5} and the
temperature is less than $\Ep9\,{\ensuremath{\mathrm{K}}}\xspace$. Energy generation is followed
using the small 19 isotope nuclear reaction network prior to oxygen
depletion, and a 128 isotope quasi-equilibrium network thereafter as
described in \citet{Wea78}. The critical nuclear reaction rates and
other stellar physics are those described in \citet{WHW02} and
\citet{Woo07}. Compositional information on the neutron excess, which
can affect the structure during late stages of evolution, was taken at
each time step in the calculation from the larger nucleosynthesis
co-processing network (\Sect{network}). Since this quantity evolves
slowly, this procedure proved a stable and cost effective way of
implementing some of the results of a large network in a large survey.
No rotation was included in the present study and mass loss was taken
to be zero. Calculations by \cite{Kud02,mey02}, for example, suggest
that the mass loss for non-rotating stars lighter than $100\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$
and metallicity $\lesssim \Ep{-5}$ {\ensuremath{\mathrm{Z}_{\odot}}}\xspace \ can be neglected, though
see \citet{Mey06}.
\subsection{Nucleosynthesis Network and Initial Composition}
\lSect{network}
Nucleosynthesis was followed with an adaptive reaction network that
included from $\sim250$ isotopes on the main sequence to $\sim900$
isotopes during the explosion \citep{rau02}. Isotopes were added and
removed as necessary to follow the nuclear reaction flow, with decisions
based upon a conservative set of assumptions regarding abundances and
flows in the neighborhood.
The initial composition of Pop III stars is, by assumption, pristine
Big Bang material. We adopt a Big Bang composition from
\citet{Cyb01,Cyb02}. The assumed initial mass fractions of \I1H,
\I2H, \I3{He}, and \I4{He} are $0.7513$, $4.3\E{-5}$, $2.1\E{-5}$ and
$0.2487$, respectively. A mass fraction of $1.9\E{-9}$ was taken for
\I7{Li}.
\section{Presupernova Evolution}
\subsection{Ionizing Photon Yield on the Main Sequence}
The great majority of the energy released in the evolution of a
massive star does not come out as the kinetic energy of stellar winds
or of the supernova it produces in the end, but as the light it emits
as a stable star. Most of this light is emitted during the long-lived
main sequence evolution (central hydrogen burning, about $90\,\%$ of
the star's lifetime). It is thought that this emission, primarily in
the ultraviolet band may have played a key role in ionizing the
universe after the recombination epoch \citep[e.g.,][]{Loe01}. At the
end of central hydrogen burning the stars expand, making softer
radiation, but many stars here remained very blue compared to
their more metal-rich counterparts. A few of our more massive stars
became red supergiants during their late evolution (\Tab{pre-SN}) due
to the production of primary nitrogen.
\Tab{ionize} gives, for all models, the total energy, $E$, number
of photons, $N_\gamma$, and the average wavelength, $\bar{\lambda}$,
of all photons emitted during the presupernova evolution that are
capable of causing hydrogen ionization
($\mathrm{H}+\gamma\longmapsto\mathrm{H}^++\mathrm{e^-}$, marked as
``HII''), first helium ionization
($\mathrm{He}+\gamma\longmapsto\mathrm{He}^++\mathrm{e^-}$, marked as
``HeII''), second helium ionization
($\mathrm{He}^++\gamma\longmapsto\mathrm{He}^{2+}+\mathrm{e^-}$,
marked as ``HeIII''), or in the Lyman-Werner (LW) Band
($\mathrm{H}_2+\gamma\longmapsto2\mathrm{H}$, marked as ``LW''). The
values given for the LW Band only include photon energies between
$11.2\,{\ensuremath{\mathrm{e}\!\mathrm{V}}}\xspace$ and $13.6\,{\ensuremath{\mathrm{e}\!\mathrm{V}}}\xspace$, i.e., they exclude the hydrogen ionizing
photons. The data are obtained by integrating an time-dependent
assumed black body flux and spectrum based on luminosity and effective
stellar temperature over the entire stellar evolution. \Fig{ionize}
shows the number of photons for these energies normalized to the
stellar mass (photons per baryon).
For convenience, we also give, in \Tab{photon}, the photon numbers in
bins relevant to primordial radiation chemistry so that they can be
added up according to the desired cutoffs. The corresponding energies
are given in the Table. We use the following bin boundaries:
\smallskip
\noindent
\begin{tabular}{l@{+ $\gamma\longmapsto$ }l@{ for h$\nu >$ }r@{\,{\ensuremath{\mathrm{e}\!\mathrm{V}}}\xspace (}r@{\,\AA)}}
H & H$^+$ + e$^-$ & 13.5984 & 911.76 \\
He & He$^+$ + e$^-$ & 24.5874 & 504.26 \\
He$^+$ & He$^{2+}$ + e$^-$ & 54.4161 & 227.8 \\
H$_2$ & 2 H & 11.18 & 1109 \\
H$_2$ & H$_2^+$ + e$^-$ & 15.42 & 804 \\
H$^-$ & H + e$^-$ & 0.755 & 16421 \\
H$_2^+$ & H + H$^+$ & 2.65 & 4678 \\
H$_2^+$ & 2 H$^+$ + e$^-$ & 30 & 413.3 \\
\end{tabular}
\smallskip
\subsection{Energy Generation and Convective History}
\lSect{conv}
The convective and energy generation histories for four representative
stars ($15\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, $25\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, $40\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, and $80\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$) are
given in \Figs{conv15+25} and \Figff{conv40+80}.
As has been known for a long time, stars with zero metallicity are
different from all other stars \citep{Eze71}, even those with just a
trace of heavy elements. In order to burn hydrogen they must first
contract and burn a trace of helium to produce the catalyst for the
CNO cycle hydrogen burning (the pp-cycles are never important in such
massive stars). This need for high temperature and density to obtain
nuclear energy leads to denser, higher entropy cores from the
beginning, a characteristic that affects the evolution throughout all
subsequent stages. Also, without heavy elements in their outer
envelope, these stars are stable against opacity driven pulsations
that would tear down their modern counterparts as well as nuclear
driven pulsations \citep{Bar01}. They also have weaker hydrogen
burning shells and that adds to their tendency to be compact blue
stars throughout their entire evolution.
During core hydrogen burning, the central mass fraction of \I{14}N
gradually rises but is typically about \Epp-9. Only during the very
end of hydrogen burning are much higher values reached, approaching
\Epp-7. In contrast, during most of hydrogen shell burning, (i.e.,
during core He burning and thereafter) values of \Epp-7 are typical at
the base of the shell. In the more massive stars, even those that do
not make primary nitrogen (see below), the hydrogen burning shell can
nevertheless become very vigorous (e.g., the $40\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ star shown in
\Fig{conv40+80}) and move outwards in mass at a high rate.
Of the stars whose convective histories are plotted in
\Figs{conv15+25} and \Figff{conv40+80}, the $15\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, $25\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$
and $40\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ stars all die with envelopes that, at least in their
outer extremities, are radiative. As \Tab{pre-SN} shows, these three
stars are blue supergiants at death. The $15\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ and $40\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$
stars die without having ever made primary nitrogen in excess of
10$^{-6}$ by mass fraction their hydrogen layers. The $25\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$
star did make primary nitrogen of about \Epp-3 in the convective part
of its envelope and, given sufficient time, would have expanded to red
giant proportions. But it died in transit while still a very blue star
(\Tab{pre-SN}). This also happened for the $20.5\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$,
$21.5\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, $27$\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace, $28\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, and $30\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ models, but
above $30\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, primary nitrogen production always led to a red
supergiant progenitor. An example of the latter behavior is the
$80\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ star in \Fig{conv40+80}.
Carbon burning is convective in the center of the $15\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ star,
but radiative in the heavier ones - consistent with what has been seen
in solar metallicity stars and showing that the evolution overall is
most sensitive to the helium core mass, not much to the metallicity.
For the $80\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ star in \Fig{conv40+80}, a mild nuclear
instability is encountered during the ignition of the second oxygen
burning shell, notable as a series of growing spikes of energy
generation (dark blue) just above a mass coordinate of $7\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$,
starting at $\Epp-4\,{\ensuremath{\mathrm{yr}}}\xspace$ prior to core collapse. These shell flashes
reflect the onset of the ``pulsational pair instability''
\citep{Heg02}. As the mass of the star becomes larger, these flashes
become more violent and happen earlier, eventually moving to the
center of the star. They can briefly expand the core and reduce the
neutrino losses, as seen in the reduction in purple coloration
following each flash. During the peak of the last pulse (note
logarithmic time scale that make later pulses take more space on the
\textsl{x}-axis) burning has proceeded far enough for the stellar core
to collapses at this point.
\subsection{Primary nitrogen production}
As noted in the previous section, some of the models produced primary
nitrogen at the base of the hydrogen envelope, a phenomenon that not
only affected their nucleosynthesis, but greatly altered their
structure.
In the $25\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ model nitrogen production commenced late in the
evolution (\Fig{conv15+25}), just following central carbon depletion,
as the convective helium burning \emph{shell} encroached on the base
of the hydrogen envelope. The phenomenon is inhibited in lower mass
stars (like the $15\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ model) presumably because the exoergic
nature of carbon burning (including neutrino losses) causes core
expansion and a weaker helium shell. Because of the weak hydrogen
burning activity, the entropy contrast between the helium convective
shell and the base of the hydrogen envelope was not great at the time
they touched, $S/N_A k = 12.8$ as compared to $13.4$. As the carbon
and hydrogen were mixed together as a consequence of convective
overshoot mixing, the nitrogen mass fraction at the base of the
hydrogen shell rose rapidly. This led to increased energy generation,
a rise in entropy, and the onset of hydrogen shell convection. The
(``undershoot'' and) overshoot mixing of both shells continued to hold
them in contact and the nitrogen abundance rose. Carbon was dredged up
into the hydrogen shell immediately became nitrogen. This mixing
continued until shortly before the star's death. By that time the
entropy at the hydrogen shell had risen to $27.6$.
We regard this synthesis of nitrogen as unavoidable, but with an
uncertainty in the yield of at least an order of magnitude, especially
if the effects of rotation were to be included. We repeated the
calculation of the $25\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ model with convective overshoot turned
off, expecting the primary nitrogen to be reduced. Instead a different
phenomenon was observed. A trace of hydrogen mixed into the helium
shell by the first contact caused a burst of energy generation that
raised the entropy. Consequently the outer part of the helium shell,
that part into which hydrogen had been mixed became convective and
merged with the hydrogen shell. The substantial amount of carbon in
this boundary layer then all became nitrogen in the envelope. Almost
the same amount of nitrogen was made as before, but this was a
coincidence and the yield was found to depend upon the zoning and time
step at the moment the two layers touched.
Even above $25\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ some stars did \emph{not} make significant
primary nitrogen. The $40\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ model is an example
(\Fig{conv40+80}). The helium shell at the end of carbon burning in
this star contained little mass, only $1.3\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ and burned only a
few percent of its helium to carbon before the star
died. Consequently, the helium convective burning shell was weak and
never made it to the base of the hydrogen shell. The primary nitrogen
that was made survived in the hydrogen shell \emph{from hydrogen
burning} from its earlier, pre-helium burning evolution. Its mass
fraction, between $\Epp-7$ and $\Epp-6$, was adequate to catalyze
hydrogen burning and produce a convective region that extended almost
to the surface of the star.
In still higher mass stars, of which we may take $80\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ as
representative, copious primary nitrogen production occurred at an
earlier stage in the evolution and was more robust. Here the nitrogen
was made during helium \emph{core} burning, again by the encroachment
of the helium convective zone on the hydrogen shell. As the hydrogen
convective core receded near the end of central hydrogen burning, a
gradient of hydrogen concentration was left behind that resulted in
several solar masses with hydrogen mass fraction below $10\,\%$. The
helium convection zone first collided with an appreciable hydrogen
abundance when the central helium mass fraction was $0.50$. The initial
entropy contrast was, as in the case of the $25\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ star, quite
small, and the hydrogen envelope was entirely radiative.
In the collision, a trace of hydrogen was initially convected
downwards and combined with carbon to make nitrogen in the outer part
of the helium convective core. The energy released raised the entropy
and temporarily shut off the convection, truncating nitrogen
production after about $\Epp-4\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ had been made. When the helium
convective core grew again, most of this nitrogen was convected
downwards in the helium core and became \I{22}{Ne} (with interesting
implications for the \textsl{s}-process. Later, when the central
helium abundance was $0.38$, however, the convective core re-expanded
into the hydrogen layer again, making more nitrogen, this time about
$\Epp-3\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$. Enough of this was mixed into the hydrogen envelope
to ignite hydrogen burning and produce a convection zone that extended
out to a mass coordinate of about $55\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ (\Fig{conv40+80}). It
also shut off convection in the outer part of the helium convection
zone (see the vertical line at $\Ep{5.35}\,{\ensuremath{\mathrm{s}}}\xspace$ in \Fig{conv40+80}).
Over time the hydrogen burning replenished a thin shell of helium ash
at its base. When helium burning was almost over (central mass
fraction $0.02$), convective dredge up destroyed this thin buffer
shell and carbon began to be dredged up directly into the
envelope. The nitrogen abundance rose rapidly at this point
($\Ep{4.2}\,{\ensuremath{\mathrm{s}}}\xspace$ in \Fig{conv40+80}), eventually reaching
$0.25\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, and the star became a fully developed red supergiant.
Rotation, which was neglected here, is likely to have a large effect,
on primary nitrogen production, especially on the more massive stars
\citep{Hir08}. Shear mixing between the highly differentially rotating
helium core and hydrogen envelope will amplify the effects of
convective overshoot that we have included here and lead to greater
nitrogen production and more frequent red supergiant progenitors. This
could have a dramatic effect on the mixing during the explosion and we
intend to include these effects in a subsequent publication.
\subsection{Presupernova Models - Structure and Composition}
\Fig{presn} and \Tab{pre-SN} show the composition and density
structure of our presupernova stars. By and large, these structures
are the same as seen in solar metallicity stars with the same final
helium core mass. Thus whatever central engine blows up solar
metallicity stars below $40\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ should blow up these just as well,
though the degree of rotation and the amount of fallback after the
initial explosion will differ.
The density of the hydrogen envelope in the $15\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ stars is much
larger and its radius smaller for reasons we have discussed - chiefly
the weak hydrogen burning shell in stars with low metallicity. The
$25\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ star, as previously noted, is somewhat anomalous in having
produced primary nitrogen shortly before its death, but not yet
expanded to red giant proportions. The primary nitrogen and extent of
the convection zone are apparent in the bottom frame of
\Fig{presn}. But in general, the helium core structures and especially
the iron cores of the zero metallicity stars resemble their solar
counterparts. There are greater variations from mass to mass than
systematic differences in the two metallicities.
Presupernova properties of the zero metallicity stars are given in
\Tab{pre-SN}. For comparison, the central density, temperature,
entropy, and electron mole number of a $15\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ solar metallicity
star \citep{Woo07} at the same stage in the collapse would be
$7.25\E9\,{\ensuremath{\mathrm{g}}}\xspace\,{\ensuremath{\mathrm{cm}}}\xspace^{-3}$, $7.64\E9\,{\ensuremath{\mathrm{K}}}\xspace$, $0.698$, and $0.436$. For
$25\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ the equivalent numbers would be $3.34\E9\,{\ensuremath{\mathrm{g}}}\xspace\,{\ensuremath{\mathrm{cm}}}\xspace^{-3}$,
$7.89\E9\,{\ensuremath{\mathrm{K}}}\xspace$, $0.892$, and $0.443$.
The figures also make clear two possible choices for locating the
piston in the explosion - the edge of the iron core, where the
electron mole number, ${\ensuremath{Y_{\mathrm{e}}}}\xspace$ has a sudden decrease, moving inwards,
from ${\ensuremath{Y_{\mathrm{e}}}}\xspace \approx 0.50$ to ${\ensuremath{Y_{\mathrm{e}}}}\xspace \lesssim 0.48$, and the location of
entropy per baryon, ${\ensuremath{S/\NA\kB}}\xspace = 4.0$. The latter, typically at the base
of the oxygen burning shell (\Fig{presn}), is where the density begins
to decline rapidly (moving outwards). The physical motivation for
these choices is discussed in the next section.
\section{Explosion and Mixing}
Lacking a robust, first principles model for how core-collapse
supernovae explode, one must parameterize the explosion. Even if such
a fundamental understanding existed, the number of stellar masses to
be explored is so large that a parameterized approach would be
necessary, but this admittedly adds an element of uncertainty and
ambiguity in all studies of nucleosynthesis in supernovae. Assuming
that the central engine in these stars initially operates the same as
in their modern counterparts (though fallback may differ), then
present day constraints on nucleosynthesis, supernova light curves,
and remnant masses \citep{Wea78,Woo07} limit our choices of explosion
energy and mass cut. It is possible, however, that early stars of a
given mass differed in ways other than metallicity. More rapid
rotation, for example, may have led to a change in the central engine
and hence different explosion energies, symmetries, and mixing. Thus
we explore here a wide range of explosion energies and make mixing
essentially a free parameter.
\subsection{The Piston Model}
\lSect{pist}
One can simulate an explosion either by moving a piston (essentially
time-dependent momentum deposition) or depositing energy (usually done
instantly, but could, in principle, depend on time and space). We
choose the former approach because it is easy to implement in a
Lagrangian code and preserves, approximately, the entropy in the
material close to the piston. Depositing energy is a reasonable
alternative, but brings in additional parameters (e.g., in what mass
to deposit the energy) and may artificially increase the entropy of
the material in which the energy is deposited.
Besides choosing the location of the piston one must also chose how to
move it. A rapid large amplitude motion gives a powerful explosion; a
short or slow motion gives a weak one. Here we explore a range of
kinetic energies at infinity of $0.3\E{51}\,{\ensuremath{\mathrm{erg}}}\xspace$ to $\Ep{52}\,{\ensuremath{\mathrm{erg}}}\xspace$
with 8 intermediate values. These are named Models ``zmX'' where
``z'' stand for the zero metallicity models of this paper, ``m'' is
the mass of the star in solar masses, X is a letter denoting the
explosion energy and is one of: A ($0.3\,$Bethe), B ($0.6\,{\ensuremath{\mathrm{B}}}\xspace$), C
($0.9\,{\ensuremath{\mathrm{B}}}\xspace$), D ($1.2\,{\ensuremath{\mathrm{B}}}\xspace$), E ($1.5\,{\ensuremath{\mathrm{B}}}\xspace$), F ($1.8\,{\ensuremath{\mathrm{B}}}\xspace$), G
($2.4\,{\ensuremath{\mathrm{B}}}\xspace$), H, ($3.0\,{\ensuremath{\mathrm{B}}}\xspace$), I ($5.0\,{\ensuremath{\mathrm{B}}}\xspace$), and J($10.0\,{\ensuremath{\mathrm{B}}}\xspace$)
respectively, and the piston mass is situated at ${\ensuremath{S/\NA\kB}}\xspace = 4.0$. We
also followed two additional explosion series with the piston located
deeper in the star at the edge of the iron core. These were Models P
(explosion energy $1.2\,{\ensuremath{\mathrm{B}}}\xspace$) and V (explosion energy $10\,{\ensuremath{\mathrm{B}}}\xspace$)
An important point here is that the energy of the models given is
their \emph{kinetic energy at infinity}. This is not the same as the
energy often calculated by those who study core collapse, because
our energy automatically includes the gravitational binding energy of
all the mass that was ejected. That is the energy that must be
provided by the central engine is always larger than the energy with
which our models are labeled, especially for more massive stars. This
difference is also metallicity dependent.
For example, in our Model z25D, the piston is located at ${\ensuremath{S/\NA\kB}}\xspace =
4.0$ which is at $2.17\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$. In the explosion (plus fallback), all
the material external to a final remnant mass of $4.16\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ is
ejected with a kinetic energy of $1.2\,{\ensuremath{\mathrm{B}}}\xspace$. The binding energy of
that ejected material in the presupernova star is $0.775\,{\ensuremath{\mathrm{B}}}\xspace$, so the
central engine, plus any energy generation associated with fallback
actually had to provide $1.98\,{\ensuremath{\mathrm{B}}}\xspace$. For a corresponding solar
metallicity star, S25A \citep{Woo07}, the remnant mass was
$2.09\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ and the binding energy in the presupernova star at that
point was $0.865\,{\ensuremath{\mathrm{B}}}\xspace$, so the central engine had to provide
$2.07\,{\ensuremath{\mathrm{B}}}\xspace$. In general, the low metallicity stars are more tightly
bound and that makes them harder to explode. But the amount of mass
that falls back is also larger and that means less total energy is
required to provide the assigned kinetic energy to the material that
does escape.
This perhaps cumbersome definition of explosion energy has been used
in the nucleosynthesis community for decades and will probably persist
until the central engine responsible for the explosion is better
understood.
\subsection{Fallback and Remnant Masses}
\lSect{fallback}
The idea of ``fallback'' in supernovae, the incomplete ejection of
matter outside the neutron star, was originally introduced by
\citet{Col66,Col71}. \citet{Che89} gave analytic estimates for the
time history of fallback and pointed out that fallback would be
enhanced in compact stars (e.g., blue supergiants as opposed to red
ones). This is because the expanding helium core encounters
significant mass at an earlier time in more compact stars, thus
decelerating the deep interior at a time when its density is higher.
The early arrival of the ``reverse shock'' at the neutron star thus
increases its mass, perhaps turning it into a black hole.
Determining this number accurately poses a computational problem for
Lagrangian hydrodynamics codes like \KEPLER. If the inner boundary
(here ``the piston'') is reflecting and is held fixed at a constant
radius (here $\Ep9\,{\ensuremath{\mathrm{cm}}}\xspace$), then the matter that falls back can pile up
on it, ultimately distorting the accretion flow. Later the deposition
of energy by radioactive decay can even cause matter that has fallen
in to move out again. This problem can be circumvented, in part, by
fine zoning near the piston and by moving the piston into a smaller
radius, beyond the point where the infall becomes supersonic. In
practice, however, attempts to do this resulted in code instability
and small steps.
A different approach was therefore adopted. Explosions calculated in
\KEPLER were linked at $100\,{\ensuremath{\mathrm{s}}}\xspace$, a time after nucleosynthesis has
ceased but fallback has not commenced, to an explicit Eulerian
hydrodynamics code. A transmitting inner boundary condition
\citep[see also][]{Mac01} was assumed and the fallback calculated for
the approximately 1200 supernova studied here. The results, for the
full range of masses, explosion energies and choices for piston mass,
have been determined by \citet{Zha07} and were employed for all
nucleosynthetic yields reported here. \Fig{remnant} shows these
masses for the explosions of various energies using the
$S/N_{\mathrm{A}}k_{\mathrm{B}}=4$ piston location. As the figure
shows, fallback is a very important effect in zero metallicity stars
leading, for a standard explosion energy of $1.2\,B$, to delayed black
hole formation in stars over about $25\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$. This can have a
dramatic effect on the light curves, nucleosynthesis, and remnant
properties of such massive stars.
\subsection{Mixing}
\lSect{mixing}
Multi-dimensional effects such as mixing cannot be followed in our
one-dimensional code and this adds another source of uncertainty to
our calculations. This mixing can be quite important, both for the
light curve and the nucleosynthesis. In the more massive stars,
nuclei that might not have been ejected without mixing can escape when
mixing precedes fallback. Mixing was studied in SN~1987A by many
groups and should be similar for the compact stars calculated here
(\Tab{pre-SN}). For stars with radii $\lesssim 2\E{12}\,{\ensuremath{\mathrm{cm}}}\xspace$, it may
be less because the perturbations has less time to grow before
freezing out \citep{Her94}.
Here we adopt a consistent, but artificial prescription for mixing
across all masses and explosion energies, the same essentially as used
by \citet{Pin88} in their study of mixing in SN~1987A. A running
boxcar average of width $\Delta M$ is moved through the star a total
of $n$ times until the desired mixing is obtained. The default values
$\Delta M$ and $n$ are $10\,\%$ of the mass of the helium core and 4,
respectively. This gives, for example, the mixed composition for a
$25\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ star in \Fig{mixed}. Since this parametric mixing is
applied after the explosion is all over, other choices of parameters
ranging from no mixing to nearly complete mixing may explored provided
that the distribution of yields with Lagrangian mass in the unmixed
model is known (\Sect{robot}).
\section{Light Curves}
\subsection{UV-Transients from Shock Break Out}
The optical display begins as the supernova shock erupts from the
surface giving rise to a brief, hard ultra-violet flash. For the
progenitors with radius $R \sim 10^{12}$ cm and explosion energies of
$1.2\,{\ensuremath{\mathrm{B}}}\xspace$, the flash should resemble that estimated for SN~1987A,
\citep[e.g.,][]{Ens92}, i.e., $L \sim 5\E{44}\,{\ensuremath{\mathrm{erg}\,\mathrm{s}^{-1}}}\xspace$ for about two
minutes with a color temperature, ${\ensuremath{T_{\mathrm{col}}}}\xspace\sim\Ep6\,{\ensuremath{\mathrm{K}}}\xspace$. The displays
of the more compact stars will be fainter, last a shorter time and
have a somewhat harder spectrum. Obviously for supernova that may be
happening at high redshift, say $z\sim5-10$, these properties would
need to be scaled to the local frame and corrected for ultraviolet
absorption between here and the source, which may be significant.
\subsection{Typical Light Curves}
For those compact presupernova with radiative envelopes - essentially
those with radii smaller than about $50\,{\ensuremath{\mathrm{R}_{\odot}}}\xspace$ in \Tab{pre-SN} - the
light curves resemble that of SN~1987A, a brief faint plateau followed
by a broad peak powered by radioactive decay (\Figs{lc}). Depending on
details of the mixing, the dip just before the radioactive peak could
be greater (less mixing) or absent (with a lot of mixing, as in
87A). For supernovae that experience a lot of fallback, i.e., the
lower energy explosions for a given mass, the light curve is much
fainter because most of the \I{56}{Ni} falls into the collapsed
remnant.
\section{Nucleosynthesis}
For a given mass star, piston location, mixing prescription, and
energy, the nucleosynthesis is completely determined. Mixing is
applied across all the nucleosynthesis, including the part that falls
back. Otherwise the two operations, mixing and fallback, would not be
commutative. Mixing ejecta after some has already been removed from
the grid gives a different result than mixing before. \citet{Woo95}
did not include mixing and, as a result, their ejected compositions
were more sensitive to fallback.
\subsection{Nucleosynthesis in Representative Models}
\Tabs{0.6-yield-mix} through \Tabff{10.0-yield-mix} give a subset of
the nucleosynthesis determined for the supernovae studied here. They
provide the total ejected masses of each stable isotope from hydrogen
through germanium for six different mass stars ($12\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$,
$15\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, $20\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, $25\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, $35\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, $50\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$,
$75\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ and $100\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$) and five choices of explosion energy
($0.6\,{\ensuremath{\mathrm{B}}}\xspace$, $1.2\,{\ensuremath{\mathrm{B}}}\xspace$, $2.4\,{\ensuremath{\mathrm{B}}}\xspace$, $5\,{\ensuremath{\mathrm{B}}}\xspace$, and $10\,{\ensuremath{\mathrm{B}}}\xspace$). Standard
piston locations (the point where $SNAkB = 4.0$) and mixing
prescriptions (\Sect{mixing}) were employed. It is not necessary to
tabulate yields above germanium because they were negligibly small in
all cases. We do not include here, however, the nucleosynthesis from
the neutrino winds of all those stars that made neutron stars or from
any disks in those stars that made black holes. To first order, the
\textsl{r}-process and perhaps part of the \textsl{p}-process would be
made in the neutrino winds of these first generation stars in the same
amounts that they are today. That is, if the solar \textsl{r}-process
abundance pattern owes its origin to proto-neutron star winds, it
would not be surprising to see the same pattern in second generation
stars. More extensive tables, available in the electronic version on
the paper, give the ejected masses of each isotope for all 1,440
explosion models with four choices of mixing (120 masses times 10
explosion energies with $SNAkB = 4.0$ plus two explosion energies with
pistons located at the edge of the iron core). The principal mass
fractions in the ejecta of the mixed and unmixed Model z15D are given
in \Fig{mixed} which makes clear the sensitivity of the results to
fallback.
\subsection{Lithium and Deuterium Production by the Neutrino Process}
Deuterium, \I3{He} and \I7{Li} are known to be important products of
Big Bang nucleosynthesis, but interesting quantities can also be made
in stars. \Fig{Li7} shows the distribution of these species and the
radioactive progenitor of \I7{Li}, \I7{Be}, in the ejecta of the
$1.2\,{\ensuremath{\mathrm{B}}}\xspace$ explosion of a $15\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ star (Model z15D).
These species (as well as \I{11}B and \I{19}F) are made during the
explosion by neutrinos interacting with the supernova ejecta
\citep{Woo77,Dom78,Woo90}. The deuterium production occurs in the
envelope due to the charged current reaction, $\nu_e(p,e^+)n$, with
neutrons immediately reacting with protons to make \I2H. A substantial
portion of this \I2H is turned into \I3{He}, and in any case the
abundance is not large enough compared with the Big Bang value, $\sim
4\E{-5}$, to be interesting. In fact these stars are a net
\emph{sinks} for \I2H. The same is true for \I3{He}.
The most interesting of the three is \I7{Li}, which is produced in
much greater quantities than is destroyed. The reactions of interest
are the neutral current process
\I4{He}($\nu_{\mu,\tau},$\I{}n$)$\I3{He}($\alpha,\gamma)$\I7{Be} in
the helium and heavy element core, and the charged current sequence
${\ensuremath{\bar{\nu}_{\El{e}}}}\xspace($\I{}p$,$\I{}e$^+)$\I{}n$($\I{}p$,\gamma)$\I2H$($\I{}p$,\gamma)$\I3{He}$(\alpha,\gamma)$\I7{Be}
at the base of the hydrogen envelope (just outside $3.7\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ in the
figure). As we shall see later, the net yield averaged over an
initial mass function is an appreciable fraction of solar.
\subsection{Elemental yields as a function of mass}
\Figs{pf:A-F} show the production factors for all elements as a
function of stellar mass and explosion energy. All calculations
assumed the standard prescription for mixing. For all but the highest
energies, there is significant fallback above $30\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ that
substantially decreases the production of heavy elements.
Taking Case D ($1.2\,{\ensuremath{\mathrm{B}}}\xspace$ explosions) as representative case, one sees
very limited production of elements above zinc and a strong odd-even
effect for elements lighter than silicon. The lighter colored bands
for \El{Be} and \El{N} are indicative of substantial
underproduction. The trend continues to a lesser extent for \El{F},
\El{Na}, and \El{Al}. Beryllium probably does not owe its origin to
massive stars and nitrogen may be underproduced because of the neglect
of rotation. Fluorine is made in interesting quantities by the
neutrino process, but not quite in a solar ratio compared with other
major productions. Sodium and aluminum are sensitive to the initial
metallicity because the neutron excess in carbon and neon burning is
smaller when the metallicity is low.
Some interesting odd $Z$ elements are made though. The reddish band
for lithium shows that a lot is being mad in stars of many different
masses. Boron is also produced in copious quantities by the neutrino
process.
Stars lighter than about $12\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ produce large amounts of silicon
through zinc \emph{compared with magnesium}, though the total yield of
all heavy elements is low in such light stars. Overall though, one is
stuck with the preponderance of blue and black in the figure, i.e.,
most elements above neon in most stars are made in nearly solar
proportions.
\subsection{Integrated Elemental Yields}
\subsubsection{Standard model and variations in IMF and mixing}
While the yields of individual stars can vary discontinuously due to
the location of various burning shells, the average over large numbers
of stars shows less fluctuation. \Fig{IMFyield} shows the integrated
yields of elements from hydrogen to selenium for various assumptions
regarding the initial mass function, explosion energy and mixing.
Panel A of this figure shows what may be regarded as the ``standard
solution''. All explosion energies are taken to be $1.2\,{\ensuremath{\mathrm{B}}}\xspace$
regardless of mass and the standard mixing prescription is
employed. The solid line is the result when all masses are
included. One sees that the general trends discussed in the previous
section are reflected in the integral. Odd-even effects tend to wash
out above silicon and nitrogen is underproduced compared with, e.g.,
oxygen in the sun by almost a factor of 10. For these Population III
stars, the initial mass function is particularly uncertain, and might
have been ``top heavy'' compared with modern stars \cite{Tan04}, so
the dashed lines in Panel A explore the consequences of omitting
supernovae below a certain mass from the sample. For the standard
choices of mixing and explosion energy, deleting stars on the light
end of the IMF tends to suppress the production of heavier
elements. These heavy elements are made, but a large fraction falls
back into the collapsed remnant. Primary nitrogen is augmented by as
much as a factor of three because it tends to be made chiefly in the
more massive stars and does not fallback.
Panel B shows that the effect of fallback on heavy element production
can be even more dramatic if mixing is small or absent and this is
certainly one solution to the existence of ultra-iron poor stars (see
\Sect{ultraFe}). Mixing is probably suppressed for explosions in
presupernova stars that are very compact compared with stars red
supergiants derived from the same main sequence mass
\citep{Her94,Chu08}. Depending on the amount and timing of primary
nitrogen production,such compact progenitors may be common in
Population III stars (\Tab{pre-SN}).
\subsubsection{Effect of a mass-dependent explosion energy}
While we do not know the details of the explosion mechanism, it is
reasonable to expect that it will function differently in stars of
varying iron core mass and central density structure. In particular,
the density gradients will be shallower in the silicon shells and
inner oxygen shells outside the iron cores of higher mass stars.
Because of the increased ram pressure from accretion of this material,
it will be harder to develop an explosion in stars of higher mass. If
no outward shock is generated, the star just becomes a black hole and,
in the absence of rotation, simply disappears. This is equivalent to
imposing an upper mass limit on the IMF Such stars would contribute to
re ionization, but nothing else. On the other hand, if such stars do
explode, one expects their energy to be comparable to their binding
energy which increases with mass \cite{WHW02}. The higher accretion
rate may also cause the buildup of a greater overpressure before the
shock moves out. These considerations motivate exploring the effect
of an explosion energy, ${\ensuremath{E_{\mathrm{kin}}}}\xspace$, that increases with mass as some
power
\begin{equation}
{\ensuremath{E_{\mathrm{kin}}}}\xspace \ = \ F \left(\frac{M}{20 \ {\ensuremath{\mathrm{M}_{\odot}}}\xspace}\right)^n
\end{equation}
where $F$ is the kinetic energy at infinity for $20\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ supernova
and $n$ is some positive power. Since we have calculated a large number
of stars and explosion energies, interpolation in energy is meaningful
and a continuous function can be employed.
Panel C of \Fig{IMFyield} explores the consequences of $F = 1.2\,{\ensuremath{\mathrm{B}}}\xspace$ and
$n = 1$ and Panel D shows what happens if $F$ is varied while $n = 0$.
In both cases raising the explosion energy for the heavier stars
decreases the amount of fallback, increases the iron-group yield, and
makes the results overall more robust to variations in the IMF and
fallback. The sensitivity of the integrated yields is much more
sensitive to mixing, fallback and explosion energy than they are to the
placement of the piston (Panels E and F of \Fig{IMFyield}).
\section{An Automated Fitting Procedure to Abundance Patterns}
\lSect{robot}
Of course the solar abundance set was not exclusively produced by the
first generation of massive stars and it makes more sense to compare
the yields with observations of metal-poor stars. Before doing so
however, we briefly discuss an algorithm that will facilitate the
objective comparison with both average date sets for metal poor stars
and individual objects. The parameter space has indeed become rather
large - IMF, mixing, fallback, explosion energy, and piston location,
so an automated engine to search for the most appropriate models is
necessary. The parameters such an algorithm finds as optimal may have
interesting implications for the first stars as well as proving that a
good fit is in principle possible.
>From the compete isotopic yields from our nucleosynthesis calculations
we first follow all the decay chains including (ground state)
branchings to obtain yields of stable isotopes. To compare with
observed abundance patterns we compute elemental yields by adding the
number fractions for each isotope.
To find the best fits to observed abundance patterns we compute a data
base of ten explosion energies (\Tab{explosions}) and $13$ different
mixtures ($0$ and $-0.6$, $-0.8$, $-1.0$ [default], ... {\ensuremath{\mathrm{dex}}}\xspace as
fraction of the \El{He} core size) for each of the $120$ stellar
models. For the best star fits we search this data base. For the IMF
fits we integrate yields for different Salpeter-like power-law IMFs
($\Gamma=-0.65$, $0.35$, $1.35$ [standard], $2.35$, $3.35$) and for
different lower and upper bounds of the IMF. A second set assumes a
Gaussian IMF (in $\log M$) centered at different masses and with
widths of $(1\ldots20)\times 0.025\,{\ensuremath{\mathrm{dex}}}\xspace$. For both IMFs we assume a
mass-dependent explosion energy such that
\begin{equation}
E=E_0 \times \left(\frac{M}{20\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace}\right)^{\ensuremath{E^{\mathrm{exp}}}}\xspace
\end{equation}
where $E_0$ can be any of our explosion energies and ${\ensuremath{E^{\mathrm{exp}}}}\xspace$ is one
of $-1$, $-0.5$, $0$ [default], $0.5$, $1$.
For the $N$ elements present in each data set (HE0107, HE1327,
\citealt{cay04}) with $\log\epsilon$ values $D_i$ and uncertainties
$\sigma_i$ we then determine the quality of the fit from the standard
formula
\begin{equation}
\chi^2=\left(\sum_{i=1}^N W_i\right)^{\!\!-1} \times
\sum_{i=1}^N
\frac{\left(F_i+O-D_i\right)^2}{\sigma_i^2} W_i
\;.
\lEq{sigma}
\end{equation}
Extra weights $W_i$ can be employed; in this paper we always use
$W_i\equiv1$. The fit value for the elements, $F_i$, are shifted up
or down by a constant offset $O$. Note that the entire fits is done
in logarithmic abundance space. The optimal value for $O$ is found by
differentiating \Eq{sigma} with respect to $O$ and setting it to zero
and solving for $O$ to find the minimum (there is no maximum).
Upper limits are only included in the determination of $\chi$ when
the observational data points lie below the theoretical abundance
pattern, otherwise they are ignored. Algorithmically this is
implemented by first finding the offset $O$ when all upper limits are
ignored. Then we successively add the data points of the lower limits
most below the theoretical data (this always will bring the fit down)
until only upper limits above the fit remain. For the normalization
the upper limits always have to be considered. If there are $N$
normal limits ($i=1\ldots N$) and $U$ upper limits ($i=N+1\ldots N+U$)
we hence compute $\chi$ from
\begin{eqnarray}
\chi^2&=&\left(\sum_{i=1}^{N+U} W_i\right)^{\!\!-1} \!\!\!\times\;
\left(
\sum_{i=1}^N\frac{\left(F_i+O-D_i\right)^2}{\sigma_i^2} W_i
\; +\right.\\
\nonumber
& &\left.\sum_{i=N+1}^{N+U}\frac{\left(F_i+O-D_i\right)^2}{\sigma_i^2}
W_i\,\Theta\left(F_i+O-D_i\right)
\right)
\;,
\end{eqnarray}
where $\Theta(x)$ is the Heaviside function. Lower limits for
theoretical data are mathematically equivalent and are treated in an
identical way.
For combining elements, e.g., carbon and nitrogen, ``C+N'', we add the
number fraction of the isotopes. For production factors, we compare
the number fraction of isotopes with the total solar number fraction
for these elements. The error bars, while given in dex, we combine in
linear space by multiplying them by the abundance, i.e., by the
product of production factor and solar abundance. For a combined
species X which is the combination of $N$ species x$_i$,
X=''x$_1$+x$_2$+\ldots+x$_N$'' with production factors $P_i$, solar
abundances $S_i$, and error $\sigma_i$ in dex we compute the combined
solar number fraction $S_\mathrm{X}$, the combined production factor
$P_\mathrm{X}$, and the combined abundance error $\sigma_\mathrm{X}$ (in
dex) from
\begin{eqnarray}
S_\mathrm{X} &=& \sum_{i=1}^N S_i
\;,\quad\\
P_\mathrm{X} &=& S_\mathrm{X}^{-1}
\;\times\;
\sum_{i=1}^N P_i\, S_i
\;,\quad\\
\left(10^{\sigma_\mathrm{X}}-1\right)^2 &=&
P_\mathrm{X}^{-1} \, S_\mathrm{X}^{-1}
\;\times\;
\sum_{i=1}^N \left(10^{\sigma_i}-1\right)^2\, P_i\, S_i
\;.
\end{eqnarray}
Hence the absolute error of
combination of $N$ species of equal error $\sigma_i$ is $\sqrt{N}$
times bigger than that of each species, while the relative error
becomes smaller by a factor $1/\sqrt{N}$. When combing species for
very different number fractions the smaller one does not contribute
much to the combined production factor $P_\mathrm{X}$ or the combined
relative error $\sigma_\mathrm{X}$ (in dex). Note that a species with
a high production factor can still have a small number abundance due
to a small solar number fraction of this species. For example, when
nitrogen has a production factor similar to that of oxygen, nitrogen
would contribute little to ``N+O'' and its error bar since it solar
number fraction is about 10$\times$ less than that of oxygen.
\section{Comparison With Observations}
\subsection{The \citeauthor{cay04} Data Set}
\lSect{Cayrel}
\cite{cay04} have summarized the abundances obtained from observations
of 35 giants, 30 of which have \SolRatEl{Fe}H in the range $-4.1$ to
$-2.7$. In \Fig{Cayrel} we show the best matches between our model
database and these observations for various choices of physics.
\Tab{fit_table} gives further information on these and other similar
fits. Note that we printed in bold font the parameters or limits of
each fit that were held fixed. When carrying out these fits, we chose
to ignore the element chromium. We simply could not fit the
\citeauthor{cay04} \El{Cr} abundance using any subset of our models
and did not want to contaminate the statistics for the other elements
by trying. It seems likely to us that there are greater uncertainties
in the \citeauthor{cay04} abundance of Cr than indicated by the
published error bar \citep[see also][]{cay04,Sob07,Lai08}. We also
treated our calculated abundances of scandium and copper as lower
bounds. The zero metal stars studied here are singular in the absence
of any appreciable heavy element production by the \textsl{s}-process.
In a separate work in preparation, we have found that this is not the
case for stars with even a trace of initial metallicity (e.g., \Ep{-4}
solar). Since the \citeauthor{cay04} set is for metallicity around
\Ep{-3} solar, it is reasonable that some contamination by second
generation stars may already have occurred. Also, as previously
noted, we ignore nucleosynthesis in the neutrino-powered wind
\citep{Pru05}, GRB accretion disks \citep{Sur06} here, and any other
\textsl{r}-process site.
Panel E of \Fig{Cayrel}, by no means the best possible fit, shows the
``standard model'' - $1.2\,{\ensuremath{\mathrm{B}}}\xspace$ explosions for all masses, standard
mixing, and a Salpeter IMF with all stellar masses included. Even
this ``first guess'' gives good overall agreement with the
observations. Indeed, given all the uncertainties in both models and
observations, it may be that nothing beyond this set of assumptions is
really justified. The elements \El{Cr} and \El{Na} are greatly
overproduced in the model. We have discussed the uncertainty in
\El{Cr} and \El{Na} might have large non-LTE corrections or a
contribution from $Z > 0$ stars. Or it may be that the odd-even
effect has been underestimated in our models. As we shall see though,
it is difficult to simultaneously fit \El{Na}, \El{Al}, and \El{Mg}
for any selection of our model space.
Cobalt and zinc on the other hand are underproduced, though not by two
sigma. Zinc has an uncertain contribution from the \textsl{s}-process
and the neutrino wind \citep{Hof96}. Cobalt is chiefly a product of
the alpha-rich freeze out from nuclear statistical equilibrium and its
production is therefore particularly sensitive to how the inner zones
are treated in the explosion (i.e., using a piston or thermal energy
deposition to drive the expansion) and to fallback. Note that simply
increasing the explosion energy is not a good solution for the cobalt
underproduction \citep{Kob06}. Panel E shows the results when a
higher explosion energy is imposed on a standard IMF, $1.8\,{\ensuremath{\mathrm{B}}}\xspace$ at
$20\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ and proportional to $M$ thereafter. The production of
\El{Co} is not improved and the quality of the overall fit is reduced.
Since \El{Ni} and \El{Co} are both made by the alpha-rich freeze out
here, it is hard to raise \El{Co} without overproducing \El{Ni}.
Other possibilities are explored in Panels A - D. The best overall
power law fit, not surprisingly, is achieved in Panel B when the code
is allowed to vary simultaneously all degrees of freedom - mixing,
explosion energy, slope of the IMF, and upper and lower mass limits.
Essentially everything is fit except \El{Na} and \El{Cr}. The
parameters the code finds for this good fit are interesting. It
prefers to have stars in the limited mass range $11\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ to
$15\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, a flatter IMF than Salpeter, very little mixing in the
explosion, and a \emph{low} explosion energy that decreases with
increasing mass. The low energy and reduced mixing suppress the
contribution of the heavier stars because their synthesis mostly falls
back into the remnant. Panel A shows again this preference for low
mass stars and low explosion energy when the IMF is allowed to be a
Gaussian rather than a power law.
It should be noted, however, that the goodness of the fit, both by eye
and formally, in the $\chi$, for Panels A - D and even Panel E, does
not differ greatly and no particular significance should be attached
to any one set of parameters. Panel F on the other hand, where a
higher explosion energy is artificially imposed, is a decidedly
inferior fit. Overall the \citeauthor{cay04} data set is consistent
with a normal IMF and explosion energy, but with a preference for
stars lighter than $15\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ and reduced mixing in the explosion.
The reduced mixing may be a natural consequence of the compact nature
of the supernova progenitor stars in this mass range.
\subsection{Comparison with the Ultra-Iron-Poor Stars}
\lSect{ultraFe}
The stars HE0107-5240 \citep{Chr04,Bes04,Fre05,Col06,CBE07} and
HE1327-2326 \citep{Fre05,aok06,fre06,Col06,FL07} may be representative
of a larger class of stars that show both very low iron abundance and
extremely high oxygen to iron ratios. A popular explanation is that
such stars represent the ejecta of a primitive stellar population in
which mixing was minimal and fallback was large. The small iron
abundance thus reflects the fact that, in some supernova or set of
supernovae, iron was made, but failed to be mixed sufficiently far out
to be ejected \citep{Ume03,Lim03}. A very similar scenario was put
forward in \citet{Dep02} based upon one of our unpublished models for
a $35\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ zero-metallicity supernova, in order to explain
another low metal star, CS 22949-037, with a large oxygen to iron
ratio. In fact, large oxygen to iron ratios due to fallback in low
metallicity supernovae were also published (Models Z35B and Z40B) by
\citet{Woo95}.
If the abundances in such stars do reflect the operation of an early
generation of supernovae, then the pattern might provide information
about those events. Were they high mass stars or low, more energetic
or less, mixed or unmixed, and are the results consistent with a range
of supernova masses and metallicities or just a single event.
\subsection{Comparison with HE0107}
\Fig{HE0107} shows several model fits compared with data from
\citet{CBE07}. The fitting program (\Sect{robot}) was again
instructed to find the best fit subject to various constraints. As
discussed in \Sect{Cayrel}, the element \El{Cr} was ignored in the
fit, but the observational upper limit and was always above the model
value. In general, the best fits were for supernovae in the mass
range $12\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ to $30\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ with the best single star fit
occurring for $20.5\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ with an explosion energy of $0.6\,{\ensuremath{\mathrm{B}}}\xspace$ and
very little mixing (Panel A). If the carbon and nitrogen abundances
are combined to reflect possible \El{C}\El{N} processing in the star,
the fit is modestly improved (Panel B) and the favored stellar mass
shifts to lower values. Gaussian IMFs allow a spread in masses around
a preferred value and favor a relatively narrow range of masses around
$15\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, again showing that low explosion energy and less mixing
are favored. Panels E and F allow the code to seek a power law fit
that is either Salpeter in character (Panel F) or where the exponent
is allowed to float (Panel E). In all cases, the best fit is for low
explosion energies, small fallback and stars in the $10\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ to
$30\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ range.
The low energy and preference for lighter supernova progenitors is
similar to what was inferred for the \citeauthor{cay04} data set, but
more extreme. The \citeauthor{cay04} set definitely required more
mixing and was at least consistent with all masses from $10\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ to
$100\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ participating. The small amount of mixing may relate to
the compact nature of the progenitor stars \citep{Chu08}, but could
also reflect envelope stripping in a binary system or simply low
explosion energy. Unless the central engine produces a grossly
asymmetric explosion, one expects the least amount of mixing in Type
Ib and Ic supernovae derived from massive stars that have lost their
envelopes and this have no reverse shock in the explosion.
\subsubsection{Comparison with HE1327-2326}
Fits to observations of HE1327-2326 (\Fig{HE1327}) select similar
models. Here data are taken from \citet{aok06,fre06} with new upper
limits for \El{Cr}, \El{Co}, and \El{Ni} from \citet{FL07}. Again,
intermediate masses, low explosion energy, and a very limited amount
of fallback are preferred. Markedly inferior fits results for both
HE0107 and HE1327 result if one insists on sampling heavier mass stars
up to $100\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ using a Salpeter IMF (\Fig{HE1327norm}). Carbon or
carbon plus nitrogen are grossly underproduced. Magnesium is also
chronically overproduced. A particularly bad fit ($\chi = 30.3$) is
obtained for the ``standard model'' (Panel F). Clearly these stars
have a very different nucleosynthetic history from the
\citeauthor{cay04} set. Supernova solutions to the problem are
feasible and require low energy, a relatively narrow range of masses,
and little mixing. But just why these stars would sample such a
limited set possibilities is unclear. Low energy, spherically
symmetric explosions in stars that have been stripped of their
envelopes in binaries or in compact blue stars like many of the models
computed here would be favored. The numbers of such stars in a given
generation will be sensitive to the history of primary nitrogen
production since this affects whether the star dies as a compact star
or a red supergiant. The frequency could thus be a function of both
metallicity and rotation rate. It is by no means clear that the
ultra-iron poor stars are necessarily more primitive than the stars in
the \citeauthor{cay04} set.
\section{Conclusions}
We have systematically studied the complete evolution and
parameterized explosions of 120 stars of initially zero metallicity.
These stars have been exploded with a range of energies - ten choices
for kinetic energy at infinity between $0.3\,{\ensuremath{\mathrm{B}}}\xspace$ and $10\,{\ensuremath{\mathrm{B}}}\xspace$. For
two energies, $1.2\,{\ensuremath{\mathrm{B}}}\xspace$ and $10\,{\ensuremath{\mathrm{B}}}\xspace$, the location of the piston has
been varied. Nucleosynthesis has been accurately determined for all
isotopes with significant abundance in all these models, as have the
remnant masses that result after all fallback is over.
To first order, the nucleosynthesis of major elements has a solar
pattern. That is to say, the helium core mass dominates in
determining the synthesis of abundant elements with even nuclear
charge and the helium core mass is not so different for the lighter,
most abundant stars when the metallicity is reduced. There are four
important exceptions to this trend though. First, as has long been
realized, reducing the metallicity reduces the synthesis of odd-$Z$
elements and neutron-rich isotopes because the neutron excess after
helium burning depends on the initial abundance of \El{C}\El{N}\El{O}.
This is particularly true for elements lighter than silicon. Second,
since the mass loss is less, stars with larger helium cores can
survive to the presupernova stage and contribute to the
nucleosynthesis (if their explosion energy is large enough to unbind
them). Since such large stars have shallow density gradients around
their iron cores, explosive nucleosynthesis contributes more to the
yield. This can moderately increase the yield of iron group elements
and especially the $\alpha$-rich freeze out. Third, because there in
no primordial iron in these stars, there is very little
\textsl{s}-process synthesis heavier than germanium. Those few
neutrons that are produced during helium burning are absorbed on
carbon and oxygen and go to enhancing the production of intermediate
mass isotopes. Finally, a significant number of stars, especially
those over $40\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ (and presumably lower with rotation) produce
primary nitrogen. This does not happen in non-rotating stars of solar
metallicity because the entropy at the base of the hydrogen shell is
much higher and the helium convective core does not extend as far.
Even here, in zero metal, non-rotating stars, the interpenetration of
shells is sensitive to zoning and an uncertain parameterization of
overshoot (and undershoot) mixing.
We do not include here any contribution from the neutrino wind or the
``hot convective bubble'' in those models that make neutron stars, and
most do. Presumably the properties of those neutron stars and their
winds would be most sensitive to their mass and not to the star's
initial metallicity. Therefore we expect the same nucleosynthesis in
the wind as in modern supernovae. This probably means the production
of the \textsl{r}-process \citep{Woo94}, some light \textsl{p}-process
nuclei by the \textsl{rp}-process \citep{Pru06,Fro06}, and the
production of significant quantities of rare nuclei lighter than
molybdenum \citep{Hof96,Pru05}. Most important in the last category
may be the elements scandium and zinc. These elements can also be
made, to some extent, by the \textsl{s}-process and $\alpha$-rich
freeze out, but one should therefore treat deficient production of
these elements in our models with caution.
It should also be noted that \emph{zero} metallicity is a singular
condition only realized for the very first generation of stars. Stars
of even \Ep{-5} solar metallicity will have a different sort of
evolution, because they will not have to produce their own catalyst
for the \El{C}\El{N}\El{O} cycle and will have \emph{some} iron for
the \textsl{s}-process. Our next survey, nearly completed, is for
\Ep{-4} solar metallicity. Results will be reported elsewhere.
In addition to the above limitations and the omission of rotation, the
most fundamental limitations of the models is that they do not
incorporate a complete physical model for the explosion - because none
is known, and that they are one-dimensional. Consequently, the highly
important parameter of ``mix'', how much the composition is stirred by
the explosion, pulsar remnant, and hydrodynamical instabilities,
remains artificial. To compensate for this arbitrariness, we have
made a library of all our explosion models and developed an algorithm
that allows the user to specify any range of mixing and explosion
energy desired, or, conversely, to present a data set and find the
best fit. The initial mass function (and whether it is Gaussian,
Salpeter, or single star), the power-law dependence of explosion
energy on mass, and the mixing can all be varied. Upper and lower
mass limits (between $10\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ and $100\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ can be specified.
This powerful tool is discussed in \Sect{robot}.
To provide examples, as well as insight, we have applied these models
and this ``robotic'' search for best fits to three sets of data - the
``Cayrel'' set and two ultra-iron poor stars, HE0107 and HE1327. For
the former we find (\Fig{Cayrel}) several good fits that are not
clearly distinguishable within the observational and theoretical error
bars. What might be regarded as the ``standard model'' is not a bad
fit. That is a constant kinetic energy at infinity of $1.2\,{\ensuremath{\mathrm{B}}}\xspace$ for
all stars, mixing using a moving box car average with width equal to
one-tenth the helium core mass, a Salpeter IMF and including all stars
from $10\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ to $100\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$. This model fit, and all others,
greatly overproduce chromium which may be more uncertain than reported
thus far by the observers. The model also overproduces sodium and
underproduces cobalt and zinc by about two sigma. The fit is
significantly improved by reducing the mixing and the explosion
energy, especially in the higher mass stars. An explosion energy of
around $0.6\,{\ensuremath{\mathrm{B}}}\xspace$ to $0.9\,{\ensuremath{\mathrm{B}}}\xspace$ seems preferred. That the mixing be less
than what seems to work well for making solar abundances in higher
metallicity stars \citep{Woo07} may not be surprising. Many of these
zero metallicity stars are compact blue stars when they die and mixing
by the Rayleigh-Taylor instability is suppressed compared with red
supergiants \citep{Her94} where the instability has a longer time to
develop. The automated fits find no need for a high energy
``hypernova'' component \citep{Tom07} despite the presence in our
library of stars with up to 10 B of explosion energy.
For the ultra-iron-poor stars we find, as did \citet{Ume03}, that the
low abundance of iron group elements is a consequence of fallback
coupled to small mixing. This effect was also seen in the lower
energy calculations of metal free stars by \citet{Woo95} who had
\emph{no} mixing at early times, and a deficiency of iron in very low
metallicity stars was attributed to fallback in this models by
\citet{Nor01} and \citet{Dep02}. Here we have considered a much
larger sample of models and confirm that low explosion energy (i.e.,
less than the canonical $1.2\,{\ensuremath{\mathrm{B}}}\xspace$) for stars above $20\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$
improves the fit and that the mixing should be smaller than for the
\citeauthor{cay04} set.
Taken together these calculations suggest several characteristics for
the ``first supernovae''. First, no nucleosynthetic evidence is found
for either pair instability supernovae or hypernovae \citep[contrary
to][]{Ume05}. The same mass stars that make the solar abundances
today could have made the abundances in the most metal stars seen so
far quite well. Mixing was less and fallback more important. Both of
these effects are consistent with what is expected for more compact
supernova progenitors. Fallback would also be greater from the very
massive stars that, here at least, had no mass loss and thus died with
much greater binding energy. It will be interesting to see if the
abundance trends seen in HE0107 and HE1327 are characteristic of all
ultra-iron-poor stars when more are discovered.
The compact nature of some of the presupernova stars (\Tab{pre-SN})
also affects their light curves (\Fig{lc}). Supernovae resembling
SN~1987A with its initially faint plateau and slow rise to a peak
powered by radioactivity might be much more common in the early
universe. But this conclusion is sensitive to whether the stars make
primary nitrogen or not. Here only the heaviest stars became red
supergiants because of nitrogen production, but the inclusion of
rotation may extend the critical mass for primary nitrogen production
downwards \citep{Heg00,Chi06,Mey06,Hir06}. If so, the star in many,
and perhaps most cases will die as a red supergiant. This would make
the light curves more like ordinary Type IIp's. We caution again,
though, that zero metallicity is a singularity that only characterized
the very first stars. We find, in our survey of \Ep{-4} solar
metallicity stars that even a little metallicity can suppress primary
nitrogen production and that most of the supernova progenitors are
blue stars, though not so compact as for $Z = 0$.
For the upper end of the mass range studied, above about $90\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$,
the pulsational pair instability is encountered. This leads to the
ejection of the envelope and part of the helium core in a series of
violent nuclear powered flashes \citep{Woo07b}. Collisions between the
shells ejected in pairs of flashes can produce extraordinarily bright
supernovae.
Finally, the compact remnants of massive stellar evolution, especially
black holes, will be more massive for zero (and low) metallicity.
There are two reasons. First, because the mass loss rate is low,
larger helium cores are possible at the time the star dies. This both
increases the potential mass reservoir for making a black hole by
fallback (the hydrogen envelope is almost always ejected) and makes it
harder to explode the star. Indeed, unless the explosion mechanism
can somehow provide a much greater energy than $1\,{\ensuremath{\mathrm{B}}}\xspace$, black holes
will be the typical result of stellar evolution for main sequence
masses above about $30\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$ \citep{Zha07}. Black hole formation is
also favored in compact presupernova stars because the reverse shock
develops earlier and reaches the center when the density is still high
\citep{Che89,Zha07}. The maximum black hole mass, though, will be about
$40\,{\ensuremath{\mathrm{M}_{\odot}}}\xspace$, even for very low explosion energies. This is the largest
helium core that avoids the pulsational pair instability.
\acknowledgements
We are grateful to Thomas Janka for his physically motivated
suggestion to use the density jump at the base of the \El{O} shell as
most likely location for the initial piston that explodes the star.
We thank Rob Hoffman, Tommy Rauscher, Frank Timmes, Jason Pruet,
Karlheinz Langanke, and Gabriel Martinez-Pinedo for their
contributions to the reaction rate library, weak interaction rates,
and neutrino cross sections. We are also grateful to Tim Beers, Mike
Bolte, and David Lai for discussions about the interpretation of UMP
star abundance and enrichment observation and theory. We particularly
than Anna Frebel and Norbert Christlieb for providing us their most
recent abundance data for HE0107 and HE1327. We than Dan Whalen for
helpful discussions about primordial gas chemistry and relevant
radiation bands, and we than Candace Church for providing us results
from her work in preparation on multi-dimensional simulations of
mixing in Pop III supernovae.
This work was supported by the NSF (AST 02-06111), and the DOE Program
for Scientific Discovery through Advanced Computing (SciDAC; grants
DOE-FC02-01ER41176 and DOE-FC02-06ER41438). At LANL, Heger performed
this work under the auspices of the National Nuclear Security
Administration of the U.S.\ Department of Energy at Los Alamos
National Laboratory under Contract No.\ DE-AC52-06NA25396.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,596 |
Guelph je město ve Wellington County v provincii Ontario v Kanadě. Je hlavním sídlem Wellington County, není ale jeho součástí. Nachází se 28 km východně od Waterloo a 100 km západně od centra Toronta na křižovatce dálnic Highway 6 a Highway 7. Centrum města leží na soutoku řek Speed a Eramosa. V roce 2016 ve městě žilo 131 974 obyvatel.
Dějiny
Pro původní indiánské kmeny byla oblast města "neutrální" zónou. Sídlo Guelph bylo založeno v roce 1827 a v roce 1879 získalo status města. Bylo pojmenováno podle německého šlechtického rodu Welfů.
Reference
Externí odkazy
Města v Ontariu
Vzniklo 1827 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 450 |
Q: Orthogonal functions and linear operators Consider the following function $f: [-1,1] \rightarrow \mathbb{R}$, expanded in terms of Legendre functions,
$$
f(y;\boldsymbol{\beta}) = \sum_{i=0}^{\infty} \beta_i P_i(y)
$$
where $\boldsymbol{\beta}$ are the expansion coefficients.
Question:
Does there exist another set of expansion coefficients $\boldsymbol{\gamma}$ such that
$$
\left| f(y; \boldsymbol{\beta}) \right| = \left| f(y; \boldsymbol{\gamma}) \right|
$$
for all $y \in [-1,1]$? My approach so far is to treat $\boldsymbol{\beta},\boldsymbol{\gamma}$ as vectors in some infinite dimensional vector space $V$, and assume there exists an invertible linear operator $X: V \rightarrow V$ so that
$$
\boldsymbol{\gamma} = X \boldsymbol{\beta}
$$
Then the function $f(y;\boldsymbol{\beta})$ can be written as
$$
f(y; \boldsymbol{\beta}) = A(y) \boldsymbol{\beta}
$$
where $A(y): V \rightarrow \mathbb{R}$ is a linear operator. I would like to define "rows" of this operator as being specific to some $y \in [-1,1]$, in other words,
$$
A(y) = \pmatrix{a_{y_1}^T \\ a_{y_2}^T \\ \vdots \\ a_{y_{\infty}^T}}
$$
where the vector $a_y \in V$ is
$$
(a_y)_j = P_j(y)
$$
Now I have written the rows of $A$ with a discrete index $1,2,\dots,\infty$, but of course $y \in [-1,1]$ is a continuous variable, so in fact $A$ would have uncountably infinitely many rows, one for each $y \in [-1,1]$ but it would have countably infinitely many columns, one for each $P_j(x)$, $j=0,\dots,\infty$.
Progress so far
Observation 1:
The condition $\left| f(y; \boldsymbol{\beta}) \right| = \left| f(y; \boldsymbol{\gamma}) \right|$ leads to
$$
\boldsymbol{\beta}^T A^T(y) A(y) \boldsymbol{\beta} = \boldsymbol{\beta}^T X^T A^T(y) A(y) X \boldsymbol{\beta}
$$
Since this is true for all $\boldsymbol{\beta} \in V$, the equation $X$ must satisfy is
$$
X^T A^T(y) A(y) X = A^T(y) A(y)
$$
for all $y \in [-1,1]$.
Observation 2: There are two trivial solutions to the above equation, namely
$$
X = \pm I
$$
where $I$ is the identity operator on my vector space $V$.
Observation 3: The set of all solutions $X$ which satisfy the above equation forms a group. Above I assumed that $X$ is invertible, and the identity element is part of the set of solutions from Observation 2. Group closure can be readily verified also.
Observation 4: if $X$ is a solution to the problem, then $X + K^T$ is also a solution, where $K$ is defined as any linear operator such that $a_y \in Ker(K)$ for all $y \in [-1,1]$.
This is true because we can write $A^T(y) A(y)$ as
$$
A^T(y) A(y) = \sum_{y \in [-1,1]} a_y a_y^T
$$
so when $K$ acts on this sum from the left (or $K^T$ from the right), it will annihilate the vectors $a_y$. Unfortunately, I don't know if there exist any such $K$ operators!
SUB-QUESTIONS
Sub-question 1: Can anyone suggest a way to approach the problem of finding all possible $X$ operators satisfying the above equation?
Sub-question 2: So far I have been looking at the infinite dimensional vector space $V$, but I would also be quite interested in any solutions to the finite-dimensional problem of truncating the series for $f$ at some value $N$, and considering a finite set of points $y_i, i=1,\dots,M$ with $M>N$. Here $f$ would be defined as
$$
f(y;\boldsymbol{\beta}) = \sum_{i=0}^{N} \beta_i P_i(y)
$$
and I would seek solutions to
$$
\left| f(y; \boldsymbol{\beta}) \right| = \left| f(y; \boldsymbol{\gamma}) \right|
$$
only on the finite set of points $y \in \{y_1,\dots,y_M\}$. Here, $A$ would become a $M \times N$ matrix, $X$ would be $N \times N$, and the dimension of $V$ is $N$.
In this finite dimensional case, because $M > N$, the vectors $a_{y_i}$, $i=1,\dots,M$ won't be linearly independent, and so I don't think there will exist any $K$ matrices whose kernel contain the $a_{y_i}$. Perhaps if we chose $N > M$ we could find $K$ matrices in that case...
Disclaimer
It is possible, and even likely, that I have been sloppy with my assumptions and notation about infinite dimensional linear operators. For example I don't know if it makes sense to talk about the transpose of an infinite dimensional linear operator, though from what I defined above it makes sense to me. I apologize if I have used incorrect notation (I have a physics background).
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,455 |
\section{Introduction}\label{sec1}
Half-metallic ferromagnetic or ferrimagnetic materials have been
extensively studied during the last decade due to their potential
application in magnetoelectronic devices.\cite{Zutic,Felser,Zabel}
These magnetic systems present a usual metallic behavior
concerning their majority-spin electrons and a semiconducting gap
in the minority-spin electronic band structure and thus 100\%\
spin-polarization at the Fermi level and could in principle
maximize the efficiency of spintronic devices.\cite{Katsnelson}.
As expected the main interest has been focused in the potential
half-metallic magnets based on transition-metal elements but the
latter due to their large spin-magnetic moments are expected to
present large external stray magnetic fields and thus devices
based on them should exhibit considerable energy losses. A way to
detour this problem is to search for new materials presenting much
smaller spin magnetic moments and to this respect a very
interesting case are ferromagnetic compounds which do not contain
transition-metal atoms. These are widely known in literature with
various names like $d^0$ magnets, $p$-ferromagnets or sp-electron
ferromagnets.\cite{Volniaska10} Several of them combine
ferromagnetism with the desired for applications half-metallicity.
There are several ways to create sp-electron ferromagnets and an
extensive review is given in Ref. \onlinecite{Volniaska10}. First,
we can induce vacancies or holes at the cation sites in oxides,
which lead to a small exchange-splitting of the cation spin-up
and spin-down states and thus to half-metallic
ferromagnetism.\cite{Volniaska10,Zhou12,Uchino12,Maca08,Zheng11,Guan10}
A second route to $p$-magnets are the so--called molecular
solids.\cite{Volniaska10} In these materials the oxygen or
nitrogen atoms form dimers which are ferromagnetically or
antiferromagnetically coupled between
them.\cite{Ylvisaker10,Kovacik09,Kim10,Volniaska08,Slipukhina11,Wu10}
The third way to create half-metallic sp-electron ferromagnets is
the doping of oxides with nitrogen atoms.\cite{Volniaska10} N
impurities at anionic sites (oxygens) present a splitting of their
$p$-bands and the majority-spin states are completely occupied
while the minority-spin states are partially occupied leading to
half-metallicity.\cite{Mavropoulos09,Xiao10,Banikov11,Yang12,Kenmonchi04,Kenmonchi04b}
Experimental evidence for the occurrence of magnetism in N-doped
MgO has been provided by Liu and collaborators.\cite{Liu11}
An alternative way to half-metallic $sp$-electron ferromagnets is
the growth of I/II-IV/V nanostructures in metastable lattice
structures similar to the case of transition-metal pnictides and
chalcogenides in the metastable zincblende
structure.\cite{ReviewCrAs} Several studies to this research
direction have appeared following the pioneering papers published
by Geshi et al\cite{Geshi04} and Kusakabe et al\cite{Kusakabe04}
who have shown using first-principles calculations that CaP, CaAs
and CaSb alloys present half-metallic ferromagnetism when grown in
the zincblende structure. Ca atom provides two valence electrons
(occupying the 4s states in the free atom) while the anions
(P,As,Sb) provide 5 valence electrons (\textit{e.g.} in free atom
of As the atomic configuration is 4s$^2$ 4p$^3$). In total there
are 7 valence electrons per unit cell. The first two occupy the
s-valence states created by As atoms which lie deep in energy. The
p-states of anions hybridize strongly with the triple-degenerated
t$_{2g}$ d-states of Ca, which transform following the same
symmetry operations, and form bonding and antibonding hybrids
which are separated by large energy gaps. The bonding hybrids
contain mostly p-admixture while the antibonding hybrids are
mainly of d-character. The remaining 5 valence electron occupy the
bonding hybrids which are mainly of anionic p-character in such a
way that all three majority-spin p-states are occupied while in
the minority-spin band the Fermi level cross the bands so that
only the two out of three p-states are occupied. This gives in
total a spin magnetic moment per formula unit of exactly 1
$\mu_B$. This mechanism is similar for all half-metallic
ferromagnetic I/II-IV/V alloys in all three zincblende (ZB),
wurtzite (WZ) and rocksalt (RS) metastable structures, and the
spin magnetic moment follows a Slater-Pauling behavior with the
total spin magnetic per formula unit in $\mu_B$ being 8 minus the
number of valence electron in the unit cell: $M_t=8-Z_t$. Evidence
of the growth of such nanosctructures has been provided by Liu et
al who have reported successful self-assembly growth of ultrathin
CaN in the rocksalt structure on top of Cu(001).\cite{Liu08}
Finally we have to note that materials containing C or N seem to
be more promising for applications since the Hund energy for the
light atoms in the second row of the periodic table is similar to
the Hund energy of the 3d transition metal atoms.
Following the Refs. \onlinecite{Geshi04} and
\onlinecite{Kusakabe04} mentioned in the previous paragraph a lot
of studies on such compounds have appeared based on first
principles calculations and we will give a short overview of them.
Although extensive studies exist also for the alkali metal
alloys\cite{Sieberer06,Zhang08,Zberecki09,Yan12,Gao09,Gao11} and
the alkaline earth
chalcogenides,\cite{Gao07,Gao07c,Gao07b,Dong11,Zhang08b,Gao09b,Verma10}
most of the attention has been focused on the alkaline-earth metal
(IInd column) compounds with the Vth-column elements and mainly
the
nitrides.\cite{Geshi04,Kusakabe04,Sieberer06,Yao06,Li08,Volniaska07,Geshi07,Gao08,Droghetti09,Laref11,Gao11b}
Sieberer et al studied all possible II-V combinations in the ZB
and WZ structures and found that all alloys containing Ca, Sr and
Ba are half-metallic while only MgN was half-metallic between the
Mg-based compounds.\cite{Sieberer06} It was also shown in Ref.
\onlinecite{Sieberer06} that the ferromagnetic state is
energetically preferable to both the non-magnetic and the
antiferromagnetic configurations. Volnianska and Boguslawski, as
well as Geshi and collaborators have studied the alkaline-earth
metal nitrides and have shown that the RS is the more stable
structure with formation anergies of about -11 eV per unit
cell.\cite{Volniaska07,Geshi07} Gao et al have shown that among
the RS alloys containing Ca, Sr or Ba as a cation and N, P or As
as an anion only the nitrides are stable half-metallic
ferromagnets with a total spin magnetic moment of 1 $\mu_B$ and
cohesive energies about -9 eV per formula unit.\cite{Gao08}
Droghetti and collaborators have shown that RS-MgN is in verge of
the half-metallicity and suggested that MgN inclusion upon the
N-doping of MgO should lead to a material suitable for magnetic
tunnel junctions.\cite{Droghetti09} The most recent studies on
nitrides concern the Curie temperature in the ZB-structure which
was found to be 430 K in CaN,\cite{Laref11} and the RS-CaN/ZB-InN
and RS-SrN/ZB-GaP (111) interfaces which were found to retain
half-metallicity only when the interface is made up from Ca-N or
N-In atoms in the first case and N-Ga in the second
case.\cite{Gao11b}
\section{Motivation and computation method}\label{sec2}
As we can conclude from the discussion in the previous section,
rocksalt alkaline-earth metal nitrides combine some unique
properties among these alloys : they have a small spin magnetic
moment per formula unit (1 $\mu_B$) and thus create small external
magnetic fields, (ii) they present very stable half-metallicity
upon hydrostatic pressure, (iii) their equilibrium lattice
constant are close to a lot of semiconductors, (iv) results on the
ZB-structure suggest high values of the Curie temperature also for
the RS-structure, (v) the half-metallic gaps are wide, and (vi)
interfaces with semiconductors retain half-metallicity. Thus in
this manuscript we complete the previous studies on the RS CaN and
SrN alloys focusing on properties which have not yet been
determined. In the first part of our study we employ the augmented
spherical waves method (ASW)\cite{asw} within the atomic--sphere
approximation (ASA)\cite{asa} in conjunction to the generalized
gradient approximation (GGA) for the exchange-correlation
potential \cite{gga} to perform ground state electronic structure
calculations for RS CaN and SrN using the lattice constants from
Ref. \onlinecite{Geshi07} (5.02 \AA\ for CaN and 5.37 \AA\ for
SrN). Notice that within ASW, empty spheres have been used, where
needed, in order to describe better the lattice filling. We use
these results and the frozen-magnon technique\cite{magnon} to
determine the exchange constants and Curie temperature in both the
mean-field (MFA) and random-phase (RPA) approximations. The
formalism has been already presented for a one-sublattice system
like the ones under study (only N-N interactions contribute since
the Ca(Sr) atoms have negligible spin magnetic moments) in Ref.
\onlinecite{Sasioglu}. Then using a rigid band model as in Ref.
\onlinecite{Galanakis} we study how the exchange constants, Curie
temperature, spin-polarization and spin magnetic moments vary with
the band-filling. In the second part of our study we employ the
full--potential nonorthogonal local--orbital minimum--basis band
structure scheme (FPLO)\cite{koepernik} within the GGA
approximation\cite{gga} to determine the equilibrium lattice
constants of rocksalt CaN and SrN which are found to be almost
identical to the results of Geshi et al.\cite{Geshi07} We,
thereafter, study the effect on the electronic and magnetic
properties of tetragonalization keeping the volume of the unit
cell constant which usually accounts for growth on different
substrates. To make this study more complete we compare these
results with the results also on the zincblende lattice structure.
\begin{table}
\caption{Calculated magnetic properties for the CaN and SrN alloys
in the rocksalt structure using the ASW method. The spin magnetic
moments are given in $\mu_B$ units and the Curie temperatures in
Kelvin. The lattice constants are the ones determined in Ref.
\onlinecite{Geshi07}.}
\begin{ruledtabular}
\begin{tabular}{llllllccc}
& a(\AA )& $m^\mathrm{Ca(Sr)}$ & $m^\mathrm{N}$ &
$m^\mathrm{Total}$ & T$_\textmd{C}^{\textmd{MFA}}$
&T$_\textmd{C}^{\textmd{RPA}}$
\\ \hline
CaN & 5.02 & 0.015 & 0.980 & 1.00 & 620 & 480 \\
SrN & 5.37 & 0.007 & 0.995 & 1.00 & 594 & 418 \\
\end{tabular}
\end{ruledtabular}
\label{table1}
\end{table}
\section{Robustness of Curie temperature against
doping}\label{sec3}
We will start the presentation of our results from the temperature
related properties. As discussed above we employed the ASW method
to study the electronic properties of both CaN and SrN in the
rocksalt structure using as lattice parameters the equilibrium
ones from Ref. \onlinecite{Geshi07}. In Table \ref{table1} we have
gathered the calculated magnetic properties. The spin magnetic
moments present similar behavior as in the studies discussed in
Section \ref{sec1}. Both compounds present a total spin magnetic
moment per formula unit of 1 $\mu_B$ in agreement with the
Slater-Pauling rule for half-metallic mononitrides (there are in
total 7 valence electron per unit cell) and thus we expect the
density of state (DOS) to present half-metallic properties. Almost
all the moment is concentrated at the N atoms. Ca atom carries a
spin magnetic moment of only 0.015 $\mu_B$ and Sr of 0.007
$\mu_B$. Thus the cation-cation as well as the cation-nitrogen
interactions should make a minimal contribution to the exchange
constants and it is enough to consider the N-N intrasublattice
magnetic interaction when discussing the temperature related
properties. The very small spin magnetic moments at the cationic
site reflect the very small charge at these sites since the
occupied bonding hybrids are mainly of anionic character having
only a very small cationic d-admixture as discussed in Section
\ref{sec1}.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig1.eps}
\caption{(Color online) Upper panel: Total DOS (shaded region) for
both CaN and SrN alloys in the rocksalt structure using the ASW
method. The red line represents the electron counting setting as
zero the electrons at the Fermi level. With cyan vertical
solid lines we denote the cases of $\pm$1 electron.\\
Lower panel: total magnetic moment (shaded region) and
spin-polarization (red line) as a function of the electron
counting.} \label{fig1}
\end{figure}
The total DOS for both compounds are presented in Fig. \ref{fig1}.
We do not present the deep-lying occupied s-states since they are
located at about -12 eV below the Fermi level and thus are not
relevant for the discussion of the electronic properties. The
bonding and the antibonding hybrids are separated by a gap which
is about 1.7 eV for the minority-spin states; the antibonding
hybrids are not presented in the figure. Note also that the
antibonding states present almost no exchange splitting.
Ca(Sr)-resolved DOS is very small with respect to the N-resolved
DOS in the energy window, of the occupied states and thus we can
safely assume that the total DOS presented in the figure coincides
in this energy window with the N-resolved DOS. The Fermi level
cross the minority-spin DOS while all the majority N p-states are
occupied and the Fermi level falls within a majority-spin gap
which is 3.1 eV wide for CaN and 3.0 for SrN. The so-called
half-metallic gap, defined as the energy distance between the
highest occupied majority-spin state and the Fermi level, is about
0.1 eV for CaN and 0.2 eV for SrN.
Small degrees of doping can be assumed to result in small shifts
of the Fermi level as assumed also for the semi-Heusler compounds
in Ref. \onlinecite{Galanakis}. In Fig. \ref{fig1} we present in
the upper panel the total DOS for both CaN and SrN alloys and with
the red line the electron counting setting the number of electrons
at the Fermi level as zero. The vertical blue lines denote the
limits of $\pm$1 electron. As we dope our system with electrons
and we move to higher values of the energy with respect to the
Fermi level we populate also the minority-spin bonding hybrids.
For exactly a surplus of one electron per formula unit all
minority-spin bonding hybrids are occupied (we remind here that in
these alloys two out of three minority-spin p-states were already
occupied) and we end up with a compound with zero total spin
magnetic moment This situation is probably unphysical since it
corresponds to a very large degree of doping and ScN and YN in the
rocksalt structure which have one more valence electron than CaN
and SrN are semiconductors. If we start doping CaN and SrN with
holes we move deeper in energy. The two alloys present a
significant difference in their behavior due to the larger
exchange-splitting in the case of SrN which is also reflected on
the larger half-metallic gap. As shown by the spin-polarization
presented with the red line in the lower panel of Fig. \ref{fig1},
for CaN small doping with holes results very quickly in loss of
half-metallicity and the spin polarization deviates from the
perfect 100\%. On the other hand in SrN even doping with one hole
preserves the half-metallic character and the perfect
spin-polarization since the Fermi level is at the verge of the
occupied majority-spin electronic bands. In the lower panel of the
same figure we also present the variation of the spin-magnetic
moment with the electron counting which also reflects our
discussion. Exactly at the Fermi level we have a total spin
magnetic moment of 1 $\mu_B$ and when we dope with holes it
increases. For one hole (corresponding to -1 electron in the
counting) the spin magnetic moment of CaN reaches a value of 1.76
$\mu_B$ while for SrN the hole populates only minority-spin states
and the spin moment reaches the value of 2 $\mu_B$ expected from
the Slater-Pauling rule for perfect half-metals. Of course doping
with electrons leads to a decrease of the total spin magnetic
moment which vanishes when we add exactly one electron to our
system and all bonding hybrids are occupied.
\begin{figure}
\centering
\includegraphics[width=\columnwidth]{fig2.eps}
\caption{(Color online) Upper panel: Calculated Curie temperatures
within both the random-field (RPA) and mean-field (MFA)
approximations for CaN and SrN in the rocksalt structures as a
function of the electron counting (see Fig. \ref{fig2}).\\
Lower panel: Calculated exchange constants between nitrogen atoms
as a function of the electron counting up to fourth neighbors. The
N-Ca(Sr) and Ca(Sr)-Ca(Sr) exchange constants are negligible with
respect to the N-N ones due to the very small spin magnetic moment
of the Ca(Sr) atoms.} \label{fig2}
\end{figure}
In the last part of this section we will concentrate on the Curie
temperatures. In Table \ref{table1} we present the estimated
values of the Curie temperature in Kelvin. The mean filed
approximation (MFA) gave a value of 620 K for CaN and 594 K for
SrN while the random-phase approximation (RPA) gave values of 480
K and 418 for CaN and SrN, respectively. RPA is expected to give
more accurate results with respect to MFA since RPA corresponds to
a larger weight of the lower-energy excitations contrary to MFA
which assumes an equal weight for both low- and high-energy
excitations.\cite{Sasioglu} The calculated Curie temperatures
exceed significantly the room temperature as was also the case for
various sp-electron ferromagnets discussed in Section \ref{sec1}
and thus these alloys can have potential room-temperature
applications in spintronic devices. Note that for CaN in the
zincblende structure we had calculated within RPA in Ref.
\onlinecite{Laref11} a value for the Curie temperature of 415 K
which is lower than the 480 K for the rocksalt-CaN in the present
study although the nitrogen atoms have a spin magnetic moment of
0.98 $\mu_B$ in both ZB and RS structures.
In Fig. \ref{fig2} we present how the calculated Curie temperature
and the exchange constants in the N-sublattice behave with respect
to the electron counting. We focus in a window of $\pm$1 electron
although usual doping with electrons or holes should result to a
much smaller change in the electronic counting. In all cases MFA
results are higher than the RPA ones and as we move to the case +1
electron the MFA and RPA values coincide. At the vicinity of the
zero electron counting the estimated Curie temperature still
exceeds considerably the room temperature For both CaN and SrN the
calculated RPA temperature reaches the room temperature either for
doping with 0.6 electrons or for doping with 0.5 holes and thus
for moderate degrees of doping we are well above the room
temperature. At the zero of the electron counting we get the
maximum Curie temperature due to the combination of the
contribution of the exchange constants between both N-N nearest
($J_1$) and next-nearest ($J_2$) neighbors as shown in the lower
panel of Fig. \ref{fig2}. Exactly at zero electron counting both
$J_1$ and $J_2$ favor ferromagnetism and make important
contribution to the Curie temperature. In SrN with respect to CaN
the $J_2$ takes larger values while the opposite occurs for $J_1$.
As we dope with electrons $J_1$ starts dropping fast, while when
we dope with holes $J_1$ increases considerably but at the same
time $J_2$ starts favoring antiferromagnetism leading also to
smaller estimated Curie temperatures. The interaction between N-N
third and fourth neighbors are significant only at the vicinity of
the -1 electron value for the electron counting.
\begin{figure}
\includegraphics[width=\columnwidth]{fig3.eps}
\caption{(Color online) Total DOS of CaN and SrN alloys in both
the rocksalt (shaded region) and zincblende (red thick lines)
structures using the FPLO method. Calculations are performed at
the calculated GGA equilibrium lattice constants (5.02 \AA\ and
5.45 \AA\ for CaN in the rocksalt and zincblende structures,
respectively, and 5.39 \AA\ and 5.82 \AA\ for SrN in the same two
lattice structures respectively). Details as in Fig. \ref{fig1}.}
\label{fig3}
\end{figure}
\begin{table*}
\caption{Calculated spin magnetic moments (in $\mu_B$) for SrN in
both rocksalt and zincblende structure using the FPLO method
within the GGA approximation. The first line corresponds to the
ideal equilibrium lattice constants and the rest to
tetragonalization where we vary the in-plane lattice parameters by
the percentage shown in the second column and in the same time we
vary also the lattice parameter along the c-axis so that the
unit-cell volume is kept equal to the equilibrium one. Note that
in the second column the minus "-" sign means compression and the
plus "+" sign means expansion. In all cases the total
spin-magnetic is kept equal to 1.0 $\mu_B$ and thus the
half-metallicity is preserved. Similar are the results for CaN,
where the only noticeable difference is that in all cases the
absolute values of Ca and N spin moments are smaller by about
0.01$\mu_B$ with respect to SrN and thus they sum up again to 1.0
$\mu_B$.} \label{table2}
\begin{ruledtabular}
\begin{tabular}{lr|rrr|rrr}
&&\multicolumn{3}{c|}{SrN-RockSalt (5.39\AA ) }& \multicolumn{3}{c}{SrN-ZincBlende (5.82\AA ) }\\
& Case & Sr & N & Total & Sr & N & Total\\ \hline
Ideal & & -0.074 &1.074 &1.000& -0.093 & 1.093 & 1.000 \\ \hline
Tetragonalization & -1\% & -0.074&1.074 &1.000& -0.093 & 1.093 & 1.000 \\
& -5\% & -0.073& 1.073&1.000 & -0.093 & 1.093 & 1.000\\
& -10\% & -0.068& 1.068&1.000 & -0.093 & 1.093 & 1.000\\
&+1\% & -0.074&1.074 &1.000 & -0.093 & 1.093 & 1.000\\
&+5\% & -0.072&1.072 &1.000 & -0.093 & 1.093 & 1.000\\
& +10\% & -0.069&1.069 &1.000& -0.094 & 1.094 & 1.000
\end{tabular}
\end{ruledtabular}
\end{table*}
\section{Stability of half-metallicity against lattice deformation}\label{sec4}
In the second part of our study we will focus on the stability of
half-metallicity against lattice deformations. To carry out these
calculations as stated in Section \ref{sec2} we have employed the
FPLO\cite{koepernik} code, which is a full potential code and
thus is expected to describe tetragonal deformations more
accurately than the ASW\cite{asw} code which employs the atomic
sphere approximation.\cite{asa} Since we are interested in
deformations the first step is to calculate the equilibrium
lattice constants using total energy calculations. To this respect
we have employed the GGA approximation which is well-known to
produce more accurately results concerning the elastic properties
than the local-spin density approximation (LSDA). For CaN in the
rocksalt structure we found an equilibrium lattice constant of
5.02 \AA\ and for SrN 5.39 \AA . These values are almost identical
to the results of Geshi and collaborators who found for rocksalt
CaN and SrN 5.02 \AA\ and 5.37 \AA , respectively. To make our
study on deformations more complete we have also included the case
of zincblende CaN and SrN since the tetragonalization effect has
not yet been studied for this structure. Our FPLO-GGA total energy
calculations gave as an equilibrium lattice constant of 5.45 \AA\
for CaN and 5.82 \AA\ for SrN in the ZB-lattice. The lattice
constant of the ZB cubic unit cell is larger than in the
RS-structure since in the former case it also contains two void
sites while no voids are present in the RS-case. In Fig.
\ref{fig3} we present the total DOS for both compounds and for
both ZB and RS lattices at the equilibrium lattice constants. In
the RS-case our DOS within FPLO are similar to the ASW DOS in Fig.
\ref{fig1}. In the ZB-cases the bands are more narrow than the
RS-structure and again both CaN and SrN are half-metallic
ferromagnets with a total spin magnetic in the formula unit of 1
$\mu_B$. For the ZB-lattice the half-metallic gaps are
considerable larger being around 0.5 eV for SrN and slightly
larger for CaN.
To simulate deformation of the lattices with respect to
equilibrium we took into account the case of tetragonalization
where we vary
both the in-plane and out-of-plane parameters keeping the unit
cell volume constant to the equilibrium. Such a deformation is
expected to model the growth of the materials under study on
different substrates. In Table \ref{table2} we have gathered our
results concerning the spin magnetic moments for SrN in both RS-
and ZB-lattices. We do not present the results for CaN since the
effect of deformation is identical for both alloys although their
electronic band structure in Fig. \ref{fig3} present slight
differences. In the second column we present the percentage of
change; "-" means compression by that percentage and "+" means
expansion, and the percentage refers to the in-plane lattice
constants. We took six values of the percentage into account:
$\pm$1 \%, $\pm$5 \% and $\pm$10 \%.. In all cases the SrN
compound remains half-metallic in both the RS and ZB structures
with a total spin magnetic moment of 1 $\mu_B$. Deformations lead
to small changes of the absolute values of the Sr and N atomic
spin magnetic moments of less than 0.01 $\mu_B$ in such a way that
they cancel each other keeping the total spin moment constant.
Even in the case of CaN (not presented here) in the RS structure
where the half-metallic gap is only 0.1 eV (see discussion in
Section \ref{sec3}) the half metallic character is preserved even
for a $\pm$10 deformation. Our calculated DOS (note presented)
confirm the conclusion drawn from the spin magnetic moments and
are identical to the ideal cubic lattice for all degrees of
tetragonalization under study.
Finally we should also shortly comment on the expected behavior of
the Curie temperature upon tetragonalization. In Refs
\onlinecite{Sasioglu05b} and \onlinecite{Kudrnovsky11} it was
shown by means of first-principles calculations that the Curie
temperature in Heusler compounds increases with increasing
hydrostatic pressure which is in agreement with the initial
Castellitz interaction curve based on experiments for transition
metal compounds\cite{Castellitz} and its recent generalization for
Heusler alloys by Kanomata and collaborators.\cite{Kanomata} For
the sp-electron ferromagnets under study we expect a similar
behavior since in Ref. \onlinecite{Laref11} we have shown that the
Curie temperature is very sensitive to the lattice spacing in
zincblende pnictides and this also led to the larger Curie
temperature for CaN with respect to SrN (the former has smaller
lattice parameter). Tetragonalization keeping the volume constant
is not expected to change the Curie temperature. If, e.g., we
have smaller exchange parameters within the $xy$-plane with
respect to the equilibrium due to expansion, the compression in
the out of-plane $z$-axis will lead to larger exchange constants,
eventually compensating each other. Only in cases where strain
induces volume changes, like in the case of hydrostatic pressure
or tetragonalization keeping the in-plane lattice parameters
constant, one could expect variation in the estimated Curie
temperatures.
\section{Summary and conclusions}\label{sec6}
Half-metallic ferromagnets, which do not contain transition metal
atoms, are attractive for applications due to the smaller spin
magnetic moments. Among these so-called sp-electron ferromagnets
the case of alkaline-earth metal mononitrides crystallizing in
the rocksalt, wurtzite or zincblende structures are promising
since the half-metallicity is combined with a total spin-magnetic
moment of 1 $\mu_B$ per formula unit leading to minimal energy
losses in spintronic applications. Moreover their equilibrium
lattice constant makes them suitable for growth on top of a
variety of semiconductors.
In the present study we have concentrated on the rocksalt
structure of CaN and SrN which is energetically favored with
respect to the wurtzite and zincblende ones. Employing
electronic-structure calculations in conjunction with the
generalized gradient approximation and using the frozen-magnon
technique we have studied the temperature dependent properties
upon varying the electron counting. Both alloys were found to
present Curie temperature above the room temperature (480 K for
CaN and 415 K for SrN using the random-phase approximation). Upon
small degrees of doping either with electrons or holes the Curie
temperature presented only a small decrease from its maximum
value, and we had to dope it with $\sim$0.6 electrons or $\sim$0.5
holes per formula unit for the Curie temperature to become
comparable to the room temperature. At the zero electron counting
(no doping) both the nearest and next-nearest N-N interaction
favored ferromagnetism; as we dope with electrons their intensity
decrease, while as we dope with holes the nearest N-N exchange
interaction becomes more sizeable while the next-nearest N-N
interaction starts favoring an antiferromagnetic configuration
leading to the decrease of the Curie temperature.
In the second part of our study we studied the response of both
the electronic and magnetic properties upon tetragonalization
keeping the unit cell volume constant. Except the rocksalt we
performed calculations also for the zincblende lattice for both
CaN and SrN. In all cases under study the half-metallicity was
preserved and both the electronic and magnetic properties only
marginally changed. Moreover, since the unit cell volume remains
constant, also the Curie temperature is not expected to vary.
Thus we can conclude that these materials are promising for
applications since small degrees of doping or large deformations
keep intact the half-metallic character and a high value for the
Curie temperature and their controlled experimental growth is
expected to boost the interest on sp-electron ferromagnets.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,720 |
## ENDORSEMENTS
"A stunning and captivating novel. Brennan McPherson's fluid prose and vivid imagination are a treat for the senses, bringing to life one of the Old Testament's most enigmatic figures."
—Billy Coffey, critically acclaimed author
of _When Mockingbirds Sing_ and _The Devil Walks in Mattingly_
_"Cain_ is a wonderful novel that explores the 'what ifs' of the world's first family. Brennan McPherson explores the questions we've all had about Adam and Eve. 'What happened to Cain?' is just the beginning. Brennan writes in a refreshing style that makes you want to read more, and I expect this novel is just the first of many more to come."
—Gordon Robertson
CEO, The Christian Broadcasting Network
"Orson Scott Card meets Stephen King. An imaginative debut novel that explores the first murder and makes us aware of the Cain in all of us. McPherson can write, and his beautiful language is both artistic and thought provoking."
—William Sirls, best-selling author
of _The Reason_ and _The Sinners' Garden_
"With literary skill and respect to the original text, McPherson weaves a terrific 'what if' story on the life of Cain, all the while exploring the depths of temptation, sin, and above all, God's unfathomable mercy."
—Bill Myers, best-selling author of _Eli_
"Thought provoking, well written, creative, highly imaginative, and all written within an honest attempt at accurately interpreting Scripture while allowing for creative license within the bounds of what might be possible. A smartly-paced debut novel. I think we'll be hearing more of Brennan and his stories."
—Charles Martin, _New York Times_ best-selling author of
_Unwritten_ and _Water From My Heart_
CAIN
_The Story of the First Murder and the Birth of an Unstoppable Evil_
Copyright © 2016 Brennan S. McPherson
ISBN: 978-1-4245-5232-0 (softcover)
ISBN: 978-1-4245-5233-7 (e-book)
Published by BroadStreet Publishing Group, LLC
Racine, Wisconsin, USA
BroadStreetPublishing.com
All rights reserved. No part of this book may be reproduced in any form, except for brief quotations in printed reviews, without permission in writing from the publisher.
Scripture quotations marked ESV are from The Holy Bible, English Standard Version. Copyright © 2001 by Crossway Bibles, a division of Good News Publishers. Used by permission. All rights reserved. Scripture quotations marked NKJV are from the New King James Version. Copyright © 1982 by Thomas Nelson, Inc. Used by permission. All rights reserved.
Cover design by Josh Meyer Photography and Design
Interior by Katherine Lloyd at www.TheDESKonline.com
Printed in the United States of America
16 17 18 19 20 5 4 3 2 1
_To my grandmother_ ,
_Nonnie_
## Partial Family Tree
## CONTENTS
Author's Note
Prologue
PART ONE: MURDER
Chapter One
Chapter Two
Chapter Three
Chapter Four
Chapter Five
Chapter Six
PART TWO: RETRIBUTION
Chapter Seven
Chapter Eight
Chapter Nine
Chapter Ten
Chapter Eleven
Chapter Twelve
Chapter Thirteen
PART THREE: COLLAPSE
Chapter Fourteen
Chapter Fifteen
Chapter Sixteen
Chapter Seventeen
Chapter Eighteen
PART FOUR: INTO THE HEART OF DARKNESS
Chapter Nineteen
Chapter Twenty
Chapter Twenty One
Chapter Twenty Two
Chapter Twenty Three
PART FIVE: THE CHILDREN
Chapter Twenty Four
Chapter Twenty Five
Chapter Twenty Six
Chapter Twenty Seven
Chapter Twenty Eight
Chapter Twenty Nine
Chapter Thirty
PART SIX: WAR
Chapter Thirty One
Chapter Thirty Two
Chapter Thirty Three
Chapter Thirty Four
Chapter Thirty Five
Chapter Thirty Six
PART SEVEN: BEYOND THE SANDS OF TIME
Chapter Thirty Seven
Chapter Thirty Eight
Chapter Thirty Nine
PART EIGHT: THE RETURN OF CAIN
Chapter Forty
Chapter Forty One
Chapter Forty Two
Chapter Forty Three
Chapter Forty Four
Chapter Forty Five
PART NINE: THE GARDEN
Chapter Forty Six
Chapter Forty Seven
Chapter Forty Eight
Chapter Forty Nine
Chapter Fifty
Chapter Fifty One
Chapter Fifty Two
Chapter Fifty Three
Chapter Fifty Four
PART TEN: REDEMPTION
Chapter Fifty Five
Chapter Fifty Six
Chapter Fifty Seven
Chapter Fifty Eight
Chapter Fifty Nine
Chapter Sixty
Epilogue
Reading Group Guide
About the Author
Acknowledgments
## AUTHOR'S NOTE
Genesis is a literary masterpiece intended specifically for a Hebrew audience, so converting the story of Cain and Abel into a modern, full-length novel presented a few difficulties, not the least of which was the terseness of the narrative. The account spanning from Cain's birth to his expulsion into the land of Nod (which means "wandering") is only about two paragraphs long (Genesis 4:1–17).
Another difficulty was its extraordinary ambiguity, which, while beautifully rich, makes it impossible to offer any singly authoritative interpretation. One example of this is in verse 17, which in Hebrew may be read as either, "Cain built a city and named it after his son, Enoch," or, "Enoch built a city and named it after his son." Two more examples lie in both the nature of the relationship between Cain and Abel as well as the hopes Eve held for her firstborn. Some scholars believe the original Hebrew implied that Cain and Abel were twins, and that Eve's proclamation over the newborn Cain implied she hoped him to be the Christ prophesied of in the garden of Eden (Genesis 3:15; 4:1–2).
Of course, a detailed and unified story could not be crafted without drawing specific conclusions to replace the original story's ambiguity. _Cain_ , like all works of fiction, is therefore a fantasy, a product of my imagination. But it is one that was formed out of Genesis 4:1–17 and the compelling questions it raises. What happened to the family after Cain was banished? What other family might he have had? Did Abel have a wife? How about children or grandchildren? How would the family have handled Abel's murder? Why did God warn Cain of sin crouching at the door and waiting to devour him? Was there significance in using such language?
Answering those and many other "what ifs" brought me on my own journey to a deeper encounter with Christ, and so the story reaches past the original narrative into the rest of Scripture to imagine the ongoing repercussions of sin and to rejoice in the redemptive power of Christ. The consequences of sin and the promise of redemption through Christ, I believe, are the most central themes of Genesis, and what connect the book to all of history. So, while this story is an expansion of the original text, nowhere did I draw any conclusions or intertwine any themes that couldn't be fitted into the original Hebrew text.
Two more notes should be made: one for the unusual capitalization of words such as "Music" and "Light"; and one for concepts jarring to the reader unfamiliar with the early Genesis stories.
Certain elements in the book refer to biblical concepts that needed to be grounded in the familiar, and for this reason, I used recognizable images or words and capitalized them when in reference to something other than themselves.
This story is also set in a culture vastly different from ours, and the critical reader must keep this in mind. Though I have modernized the story in many ways, other elements, such as Adam and Eve's children marrying each other, remain. While this may seem disturbing to some, nearly all biblical scholars agree this was the only choice, and was actually encouraged by God in this time period. It wasn't until thousands of years after this story takes place that God formed laws regarding siblings marrying (Leviticus 18).
Thank you for picking up this book. I hope you enjoy reading _Cain_ and that it draws you closer to the heart of God.
## PROLOGUE
_Then the Lord God said, "Behold, the man has become like one of us in knowing good and evil. Now, lest he reach out his hand and take also of the tree of life and eat, and live forever—" therefore the Lord God sent him out from the garden of Eden to work the ground from which he was taken. He drove out the man, and at the east of the garden of Eden he placed the cherubim and a flaming sword that turned every way to guard the way to the tree of life._
—GENESIS 3:22–24 ESV
Piercing cries echoed through the dark cavern. Adam felt Eve's fingernails dig at his hand hard enough to draw blood, but couldn't sense whether the moisture was blood or sweat as the shadows rendered it colorless.
He brushed back her hair as she drew air through flared nostrils like a wounded animal. Her body shook as she gritted her teeth and let out a scream longer and more agonized than any before. Then a sound that had never been heard in the world met his ears.
The sound of a human baby crying.
The sunrise breathed into the cave and warmed Adam's back as he cleaned the baby, cut the cord attached to its belly, and presented the newborn to Eve as a trophy marking months of struggle in the wilderlands. She accepted it with rapture and said in a shaky whisper, "With the Lord's help I have delivered a boy!"
Adam pressed his forehead to hers and laughed. His unmistakable reflection glowed in the newborn's features, but as he watched Eve twirl the child's black hair, a cold sensation dulled the smile on his face and prickled the hairs on his arms.
Eve looked up from their child. "We should call him Cain, for he is our firstborn."
Adam looked up and, upon seeing the joy in his wife's eyes, smiled and nodded, and mouthed their baby's name to feel it on his lips. So much waiting, so much wondering. To finally experience the birth they had waited for produced a strange mixture of excitement, disbelief, and anxiety.
But as they entwined their fingers and rested, Eve let out a cry. "Something's wrong," she said.
He shook his head and swallowed, attempting to attend to her as her hand closed over his again. But no matter what help he offered, her screams grew. "Eve," he whispered, though she did not reply, seemingly too deafened by pain. "Eve."
"What?" she said through bared teeth.
"I think another child is coming."
Her breath accelerated and her neck muscles tensed as she screamed again. He whispered to her that all would be well, but his eyes caught the dark baby lying beside her, and the words seemed to fall apart in his mouth. He pressed his lips together and focused on helping her again. Finally, through her sweat, blood, and water, a second child plunged into the world and gasped in shock.
Adam lifted the second infant into the growing glow of daylight as it wriggled and squealed in discomfort. Another boy, except with platinum hair and a complexion to match. Adam looked at Eve with his own delighted smile. "We will call him Abel, for he came like a sudden gasp." He brushed the boy's hair aside and whispered, "My boy, Abel."
In a cave in the wilderness, the first children ever born lay cradled in their mother's arms. Abel on her left and Cain on her right. But the father's gaze rested most on Abel.
## PART ONE:
## MURDER
_For your hands are defiled with blood_
_and your fingers with iniquity;_
_your lips have spoken lies;_
_your tongue mutters wickedness..._
_and from one that is crushed a viper is hatched._
—ISAIAH 59:3, 5 ESV
##
Cain stood on a nameless hilltop. He was far from the City, and below him stretched a vast field of long grasses and wild flowers. Several trees stood in the depression, but the valley was mostly flat and open, and cloud cover poured darkness into it like wine into a giant bowl. He studied a familiar figure in the midst of the valley as the smell of coming rain wove through the scent of plant and soil. Then he gazed into the growing eastern twilight and descended toward the figure.
The earth was soft beneath his feet. Moisture bubbled between his toes, and mud suctioned his feet in place, as if the world was attempting to stop him. He smiled at the thought.
_Let everything try to stop me. Let the world try._
Upon reaching the bottom of the hill, he waded through the waist-high grass. A slight wind blew through the valley, brushed the blades back and forth in rippling waves, and caused them to shimmer. He focused on the figure he had followed to this place as his mind likened it to a statue, a silhouette chiseled out of the surroundings. He was close enough to be heard, so he stopped as the grasses beat his legs. Every moment was intoxicating, and though he had spent weeks planning every word he would speak tonight, he decided to savor the moment.
The winds settled and the silhouetted man addressed him in the lull. "Which do you enjoy more, the calm or the storm?"
Cain did not respond. The wind picked up again, and the grasses continued their attack on his legs.
"The clouds are in labor, but the floodgates are barred."
Cain took a step forward with clenched hands. The figure turned and his blue eyes gleamed. Cain knew those eyes well. He remembered seeing them gaze at him from a face soft with youth as the memory aged and merged with the shadowed figure before him.
_Abel_. The name spilled from countless recollections like a compendium of wasps.
"I know why you are here. You might as well voice your thoughts," Abel said.
Cain's head throbbed and his breath stuttered; so much depended on this encounter. "You have always wanted to frame me as a failure, haven't you?"
Abel paused before answering. "I have done nothing but speak truth."
"And strive to ridicule me."
Abel shook his head.
Somewhere, long ago, young Abel tugged on Father's garment and pointed, directing Adam's twitching frown and furrowed brow toward Cain. "You have always played the favorite." He broke a stalk of grass and tossed it. "I could have had respect and love, but you stole it."
"I have stolen nothing. Is it my fault if others show favoritism?"
"Of course it is. Don't play the fool. Every time I fall, you are there to take my place. Every mistake I make, you are there to correct it. Everything you do is born through your desire to be better than me. Since birth, life has been nothing but a contest between us, hasn't it?"
"I do what is right, and if that puts me ahead in others' eyes, so be it."
"Do I only do wrong?"
"You have sinned."
Cain laughed. "I take chances. I diverge from the well-trodden road to forge my own because that earth feels better under my feet. The dust kicked up by everyone else gets in my eyes and teeth, and I grow tired of spitting it out."
"It was not my desire that the Almighty would accept my sacrifice and not yours."
Cain knew his gift still lay by the altar caked with the dry blood of Abel's offering. The muscles in his neck flexed and pulsed with hot blood at the memory of the broken stalks of his garden lying discarded on stone and trampled by feet. He shook his head. "That wouldn't be enough to bring me here. You have played me for a fool since the day you learned you could."
Abel laid his words like a silk carpet between them. "I made my choices and you made yours. I have never been guided by malice."
"But you think me a stepping-stone. You cannot deny you have perpetually gained by my failure."
Abel bit his cheek and shook his head.
"You expect me to ignore the past hundred years?"
Abel's eyes chilled and sharpened, though his voice remained mild. "No matter what I say, you believe what you want to believe. Your jealousy has poisoned you against me."
Cain lowered the pitch of his voice. "I have not been jealous of you one day of my life."
"But I know you have. I know you better than any other."
"If you did, you would not open your mouth again."
"Are you threatening me? I have abided the hatred in your eyes for far too long. I have loved you all my life, but you have pitted yourself against me as if I were trying to usurp your place as first-born."
"I took care of you when you were weak. I protected you, I taught you, and I led you. _That_ is love. What have you ever given me in return?"
Abel was silent.
The corner of Cain's mouth trembled. "And what do I get from everyone else? Father prefers you, he always has. From the moment you were born, I was unwanted. And now even the Almighty has rejected me."
"But I have done nothing to you."
"No, you've made sure of that. And your inaction, your calculated silence, it has twisted everyone I love against me."
"Give me proof, not empty words."
Cain remembered how Sarah's eyes had danced over Abel at the celebration the day before; and when Abel turned and met her gaze, a flame burned in her eyes like a spark in dry grass. Cain knew that look and had for years failed to draw it from her. It was admiration, attraction, _love_. Yes, as much as she tried convincing Cain otherwise, Sarah loved Abel. Cain scowled. "We both know my words are anything but empty."
Abel softened as if noticing Cain's cruel expression. "We are both sinners, brother." His voice lulled and pulled at Cain. "All of us, even our children and their children. It is the curse Father brought upon us by partaking in the forbidden fruit."
"Of that, you speak truthfully." Cain stepped forward, severing the space with sharp strides. He had grown weary of the argument. All pleasure deteriorated into impatience and irritation at the sound of Abel's voice. "So, brother,"—Cain smiled—"shall we have a contest like when we were children?"
He could smell Abel's breath and feel the heat of his body. Insects buzzed around their ears, but neither moved. He could see Abel's calculating gaze analyzing, attempting to understand.
Cain's smile cracked. "Let's see who the real sinner is."
As Abel's eyes narrowed, Cain seized him by the throat. The impact made Abel blink and sputter, but Abel shoved him back a few paces. Cain regained his footing as Abel massaged his throat and coughed. Then Cain rushed toward him, screaming as his shoulder made contact with Abel's midsection, sending them both to the ground. They tumbled through the towering grass, and Cain managed to roll on top of him and pound knuckles on his face, but then Abel struck back and the two twisted. Abel threw him off and rushed to his feet, stumbling like a drunkard. Cain scrambled up and leapt on him, flattening Abel on his belly. Cain pinned him to the earth with his knee and hammered the back of his head with knuckles hardened by harvest and hatred.
Abel went motionless. Cain paused, but his hand found a stone and gripped it hard. He poised the object high above and then smashed it into the back of Abel's head. As black emotions swelled within his chest and erupted as a bestial scream, he bludgeoned Abel's skull over and over again.
Eventually, Cain's limbs weakened, and he fell back. All around Abel's head and torso lay a glistening pool of liquid, flesh, and bone. No air entered Abel's lungs and no air escaped them. Cain stared at his brother's body, at the shape of the crater of collapsed bone and tissue. Nausea and pleasure coiled together, and he wondered if what lay on the ground were truth or fiction. It had been so simple, so quick.
But there it was. The smell of blood and brain blackened his nose.
"Murder," he whispered. The world had not known it possible, but with a stone and two hands, he had proven it was. He had never before felt so powerful and vulnerable at once, but as twilight fell into darkness, panic speared his shoulders.
_What now?_ He had thought through each step countless times but the tossing froth in his stomach overwhelmed him. He bent, rested hands on knees, and breathed deeply against the rising nausea. He felt as if it would pass, but his body lurched and he wretched bubbling stomach acid.
He stayed bent, gasping for air and spitting the remains in spider-web streams. The biliousness receded and his body relaxed, but a whispered voice came to him as if from far away. He thought it said, _"Bring the body to the river."_ Cain stilled himself, wondering if it were anything more than amplified thought. Again he heard the voice, and chills scrambled up his neck. _"Bring the body to the river."_
He straightened and wiped his mouth. Could someone have seen him kill Abel? His lungs squeezed his throat with bony hands. Plants rustled in the wind and insects chirped. The screech of a distant owl pierced the night sky and ricocheted off the hills, but he could identify nothing abnormal. Nothing, except for the itch in his brain and the sweat in his clothes.
_"Bring the body to the river. Take him to the river and wash yourself."_
He couldn't seem to pinpoint the direction from which the voice spoke. Though he knew sound behaved strangely in the hills, the displacement unnerved him.
_"You need not fear."_
He twisted and searched with narrowed eyes for a face, a body.
_"Do what you must."_
If it were the voice of the Almighty, he could not recognize it. In the long silence, he listened. No more words.
Cain nodded slowly. Then he stepped to Abel, stooped, and grabbed his garment by the neck with one fist. He began to drag his brother's body toward the river. He listened for the voice but heard only the wind in the hills and the crackle of thunder. Breath burst past his clenched teeth and his fingers ached with the weight, but all he could seem to see was blood glistening in the darkness. A shadowy trail through trampled grass.
_So much blood..._
The panic pulsed in his head, but the sound of rushing water just beyond the trees met his ears like cool water on cracked lips. He let the body drop. His arms and legs shook like tree limbs in a storm, despite his attempts to still himself.
_"The river_ ," urged the chilled voice again.
Cain's chest tightened.
_"The river, the river, the river_ ," it droned on and on, buzzing in his mind.
"Wait," he whispered. "Just wait."
_"The river!"_
He shook his head as the anxiety intensified.
_Am I fighting myself, or does this voice actually exist?_
He smeared bloody fingers through his hair as he sensed himself slipping into that familiar abyss where only panic existed. He wanted to beat himself, to rip his hair out, to silence the voice, but most of all he wanted to kill again, to feel that rapturous release.
He recalled feeling the rock in his hand as it made contact with Abel's skull. He thought of how the violent blows shook his shoulder, and how his vocal cords tore as his scream pierced the murky darkness. He had felt such power and pleasure. He had felt like God.
It made him sick.
_"The river!_ "
"I know!" His voice echoed through the valley, but soon was swallowed by the roaring river. He ran to the body and hefted it again as desperation invigorated his limbs, but the world felt all too intangible.
_Surely the corpse my knuckles strain against is only as heavy as my thoughts. Surely the voice is nothing more than a reaction to stress._
With aching shoulders, he arrived at the river's edge. It was wide and fast and, most importantly, flowed away from the City. The icy water washed his calves, and his thighs as he thrust the body deeper into the river and watched it bob and float along the surface. The current carried it away as Cain washed his arms and face, scrubbing his skin until it felt as if only bone remained. He plunged his head into the water, attempting to rinse the sticky matter that had congealed in his hair and clung like tree sap.
The voice returned. _"Cut it."_
He straightened as the hair on the back of his neck prickled through beads of cold water.
_"Cut it off._ "
"Who are you?"
There was no answer.
"Tell me who you are!" He scrambled up the bank and searched for the source of the voice, but the entire world seemed overcome by rushing water. He let his eyes linger on the shivering stream, and asked again, only softer, as if trying to coax a child from hiding. When he grew tired of waiting, he chuckled anxiously, ran fingers over his matted hair, and walked home as if the world was as it always had been.
As if Abel were still alive.
The voice returned. Though his mind was absorbed by it, he let it be. When the volume increased, so did the itch in his mind, and he wondered if the two were connected. But he had little time to spare, so he did his best to ignore it, because it was time to see Sarah.
_Sarah. My sister. My wife._
When at last he walked through the doorway of their home, she stood by the window. The sight of her shadowed figure stopped him just inside the threshold. She said nothing, but he knew she saw him. She had waited for him.
He walked to her slowly, unable to breathe. A mental image of Abel lying on the ground, dead, poured snow into his gut. He swallowed as he neared and saw her glowing eyes trace his stained figure. For the first time, the heat of guilt wetted his forehead.
Her breath was ragged, as if she had been crying. He paused and almost raised his hand to touch her, but stopped. She knew what he had done.
His dry voice hammered the silence. "I need your help, Sarah."
She stared, but did not speak.
He turned and headed toward their sleeping quarters, and she followed as he knew she would. He snatched a burning candle from the hall and set it in a holder that his son Gorban had made. The golden flame unveiled her face, and as their gazes met, every minutia was communicated. Sarah grimaced and looked toward the floor as if it would open up beneath her.
"I need you to cut my hair." He cleared his throat and whispered, "The blood..."
After a long, tense moment, she wiped her eyes and nodded.
He rubbed his face with shaking hands as Sarah retrieved a blade. When she returned, Cain sat cross-legged with his back to her knees.
She slid long fingers over his matted hair and recoiled from the gore. But slowly, reluctantly, she brought up the blade and began her work. The knife tugged and scraped and sliced, at times causing him to wince, but the pain made him feel alive.
Sarah was crying again. Part of him ached at the knowledge. The other part felt profound satisfaction that he had done what was necessary. He had no other escape. All logic within him demanded he murder Abel and test the bonds that held them within the walls of the City of the Almighty. Though his soul had at first been repulsed by the thought of murdering his twin, repressing his conscience had been a necessary progression.
_"A step for humanity_ ," whispered the voice in his mind. He tried to push it away, but couldn't help but agree. Yes, and who better to take that first step than he? He would take the fall. He would perform as they all expected him to. And he would rise again, stronger than before.
Or die in righteous rebellion.
His eyelids and the corners of his lips drooped at the sound of his hair tearing against the dull blade. The flame on the table stabbed the shadows, and as his thoughts spread and interwove, the rhythm of the blade carried him close to waking dreams. Like the swinging he had felt in his mother's arms so many years ago. The same arms that had held his brother.
A chunk of matted hair struck his nose. The smell of Abel's blood filled the air as though his body were hanging in the room, and suddenly the voice spoke again, whispering secrets in the dark. Sarah had stopped. He looked over his shoulder and saw her face buried in red-stained hands. She wept, but not for herself. Cain knew she cried for Abel.
He stood and walked out to wash his scalp. Even if he had wanted to, he could not have comforted her. It was the chasm between them. He looked at his hands and shook his head.
_I killed you, brother, yet still I feel your hands pushing her from me._
His fingers clenched.
_If only I could kill you again._
He lifted a pot of water high and pitched forward, letting the water cascade across his head. As the last drops fell, he lowered it to the ground, careful not to damage it. He wiped the water from his face and breathed deeply. The unexplained buzz drove him to seek solitude, and the whispering sporadically increased in volume until he could perceive words.
_"Kill her_ ," it commanded.
He shook his head. That, he would not do. Sarah would tell no one. Not until the time was right. She hated Cain, yet her fate was bound to his. And as much as he wanted her to experience pain for desiring Abel, he could not bear her death. Even as the hunger inside him grew.
He started at the paradox between his longing for her and the desire to end her life. The thought came to him that the voice, and its urgings, belonged to something else entirely, something new. He could no longer deny it, and the more he meditated on its meaning, the warmer it throbbed like blistered skin.
He shook his head to rid it of the buzzing itch as Sarah's cries continued growing until she wailed. He couldn't endure the sound, so he turned down the road and strode on. Billowing black clouds swallowed the sky. Strangely hued lightning bolts streaked through them and shook the ground with vengeful rumbles, but no rain fell. His eye twitched and his labored breathing brought no satisfaction. Each lungful felt hollow, somehow less than it should be.
He needed the rain. He had counted on the rain, had waited for a storm such as this. And yet it stalled.
He found a patch of soft grass underneath a large oak and closed his eyes, but he found no peace. So he stared at the sky and waited for it to wash away his sin. Hours slipped by like waterless droplets in an ocean of thought, but only one thought remained solid like the earth at his back. He closed his eyes and whispered, "Nothing could clean stains such as those."
He felt a strange peace in speaking it.
##
Lilleth's footsteps pattered the pathway like a bolting bird's. Abel hadn't returned, and Lilleth could wait no longer.
_He must be in the hills searching for wayward sheep. He must be healthy and whole._
"That's a lie," she whispered. She could not evade the fact that he was gone, and never before had he been gone this long without explanation.
_I have nothing to worry about. We're in the care of the Almighty. Nothing bad can happen to us. He promised we're protected._
More lies. She had every reason to believe, but could not. Why? Maybe something in her believed that evil could breach the Almighty's shield and end their happy, simple lives. Still, it eluded her. The years in the wilderness worrying about the Jinn, those demonic perversions of animal life, had not yet departed her consciousness, and the strangeness of the brooding storm disturbed old fears.
The world was dangerous. They didn't live in the Garden anymore, running free in naked innocence. Now they clothed themselves as they rose and went about with care and toil, working the ground with sweat and blood.
_The world has changed._
_Of course it has_ , she thought. _It has been a century and a half since we fled Eden._
Her throat tensed. _Do not fool yourself. This is more than a consequence of the Fall. You sense something different._
Lilleth hurried. The sky was as dark as night, though sunrise should have begun hours ago. The storm seemed malevolent, yet the Almighty had promised them protection from such forces.
_"While you dwell in me, no danger will reach past the walls I have constructed. Not sickness, not demon, not nature."_
The recollection of his voice in the wilderness evoked the taste of dust and the feel of wandering. Other sensations came as well, and though she had no desire to return to those days of hardship, she felt an affinity for them.
She bounded across the grass in front of Mother's home and called out as she pulled aside the flap hanging from the arched marble entryway and passed within. Dim lights flickered as the flap fell back into place, and she was suddenly aware of the sweat matting her clothes and her breath buffeting the silence. The stone walls seemed to stoop over her, their smooth texture dully reflecting the candle flame.
"Lilleth, what brings you here at such a time?" Eve sat like a tower on a hill, holding wool and needles in her hand.
"Have you seen Abel?"
Eve shook her head.
"I can't find him. He's been gone all night."
"I have seen no others since we left the celebration last night."
Lilleth hugged her waist. "I wonder if something has happened."
For a moment, it seemed shadows obscured Eve's face. "Come now, you know where we are." She set down her tools and waited for a peal of thunder to dissipate. "You have no cause to worry."
"Of course. I know." She rubbed her clammy palms together. "But it isn't like him to be gone so long, and he didn't explain why he walked off to the pastures last night. Such behavior is strange, even for Abel."
"I understand your concern, but we are in the City of the Almighty." She cleared her throat. "Are you all right? Did you just awaken?"
"I haven't slept at all."
Eve motioned to the cushion beside her. "Very well, sit and tell me what happened. Then I will make us tea."
Lilleth slid onto the cushion and forced a deep breath. "Last evening as the sun set, Abel acted oddly. He talked very little, especially for such a joyful day. The Almighty accepted his offering with grace and gave him much honor, but he acted as if he had been—" She paused and implied the rest with raised eyebrows.
Eve squinted.
"He acted as if he had been rejected. Like Cain."
Eve pursed her lips and nodded.
"And the things he said were strange. He kept mumbling about the storm and wouldn't respond to my questions. Then he said he was off to the pastures, and I haven't seen him since. All night I have been awake, staring at the sky and hearing its distant thunder. The darkness of the storm seems wrong and so does Abel's absence."
Eve opened her mouth as if to speak, then stared at the ground. When the silence grew heavy, she said, "Since we have been with the Almighty these past two years, we have been safe from danger. His Spirit is with us at all times, protecting us from beast and nature, from all manner of sickness, even from the Jinn. Abel must be safe... It is impossible for anything to have happened." Eve sighed, and her tone gained disapproval. "Dear daughter, when will you come to trust that we are safe in his arms? He loves and guards us fiercely. His intention is for us to live joyful, peaceful lives. He wants us to prosper. If he didn't, why would we serve him?"
Lilleth took in the words with a grimace.
Eve smiled. "Do you see?"
Lilleth nodded.
"Good."
"But could we not look for him?"
Eve's eyes narrowed. "Why?"
Lilleth pressed her hands to her chest and bowed. "For my sake. I wish to speak with him."
Adam entered the room with his tunic tied around his waist and a jug of water in his arms. He bent to set it on the ground next to the fireplace. Lilleth observed her father's shoulders, jawline, and eyes—so similar to Cain's.
"Adam," Eve said. He glanced at her. "Lilleth is wondering where Abel is. He's been gone all night. Would you go find him?"
Adam met Lilleth's gaze, then nodded at Eve. "After you're done with the tea, will you heat this for me?"
"I will make sure it is ready when you return."
Adam turned to Lilleth. "Where might he be?"
"He walked toward the pastures last evening."
Adam smiled, pulled the tunic over his torso, and nudged Lilleth. "Do not worry. I will find him." He walked into the morning twilight and was gone.
Lilleth breathed in, then forced it out. Eve laid a hand on her shoulder and smiled. "Abandon worry, beloved daughter. All is well."
Lilleth smiled and repeated, "All is well."
But it was little more than a whisper.
##
Seth bolted up in bed, and the woven covers slid off his shoulders. His pulse throbbed behind his dark eyes, and he swallowed the taste of metal while wiping sweat from his shaved head. Ayla moaned next to him and turned to peer through sleepy eyes.
"What is it?"
He swallowed, but the excess saliva did little to cure his throat's dryness. Already the details of the nightmare were falling out of reach, but still the world seemed to close in about him. "Just a dream, only a dream." He slipped out of bed, walked to the wall, and steadied himself with one palm. His legs wobbled as if unsure of the ground's stillness, and if Ayla hadn't been watching, he would have dropped to his knees.
She rustled in bed and sat up, brushing milky hair behind a flushed ear. "Are you all right?"
"Yes, I'm fine." She looked at him as if imploring him to explain, but he shook his head and said, "You can go back to sleep. I need to clear my mind."
She didn't lie down.
Seth walked the cold marble floor and slipped on his sandals. As hard as he tried, he could recall neither his dreams, nor any reason for fear, yet his belly was troubled by a creeping chill.
As he exited the stone house that the Man had crafted with his very own hands, the breeze carried the scent of leaves and flowers and damp earth. The brusqueness was rousing, so he began jogging the dirt road snaking between the houses that the rest of his family slept in. His joints ached. As he ran, he stared at the tumbling darkness above. Every so often, the sky was painted red with lightning, as if the bolts hemorrhaged the heavens.
For exactly seven nights, he had been awakened by nightmares. Only twice had he woken Ayla. Four nights earlier, he decided there may be a reason for the nightly recurrence. The effect of the nightmares on his body had intensified every evening, and even now, he trembled.
_What are you trying to show me?_
Seth lifted his eyes and followed the road up the hill to the well, which sat in the center of the City. His legs burned as the path climbed, and upon reaching the well, he sat on its ring. The rock edge dug into the back of his thighs as water whispered from within. Seth dropped a stone, waiting until it broke the surface of the water. He brushed the dirt from his hands and surveyed the surrounding land.
When they first came to the City two years ago, he had come to this hill every morning. The view and tranquility made it a resting place, and memories of times at the well hung in the air, calming the shakes. He thought of Ayla, her ivory teeth shining in the light of a summer day, laughing at him after he tripped on a stone. He smiled and moved on to the years ahead of them. He saw them clasping fingers and walking new roads with their brothers, sisters, nieces, and nephews as the sun rose and set.
But then he thought of his eldest brother Cain. After over a century of marriage, Cain and Sarah had clearly grown apart, and that reality drained the color from Seth's imagined future.
He shook his head and rubbed his face, redirecting his thoughts to his mother. He relaxed, thinking of Eve. When would she bear his next brother or sister? There were nearly eighty years between him and his older sister Lilleth. Mother and Father had been so focused on surviving in the days of their nomadic existence within the wilderlands that bearing children had been a luxury postponed. Seth had been a welcomed, albeit unintentional, gift after the Jinn had stolen Cain's eldest, Lamech, from the world. Still, they had grown as a people, with grandchildren and even great-grandchildren born through the fires of young love.
And now they lived under the protection of the Almighty. It seemed only a matter of time before his mother conceived again. Even after all this time, he was excited by the thought.
Seth examined the back of his hands. His coloring was different from Cain's and Adam's. Theirs was an olive complexion, darkened by sunlight. Seth had come from the womb darker than either, and any exposure to sunlight had no visible effect. Would a new child look like him? Or like his wife and sister, Ayla, whose pale skin and green eyes reflected the sun like the sea?
The complexities of humanity were deep, and the thought of the Almighty's creativity widened Seth's smile. All of life seemed to coalesce as his eyes absorbed the City of the Almighty, crafted by the very hands of their Maker. In his mind he saw the connections form from his parents to him and his siblings, and watched them trace on through the generations, past Calebna and Lukian and to their children. Encircling them were the inner and outer walls of the City of the Almighty. Outside: Death. Inside: Life. A simple calculation. He liked it that way.
He breathed deeply, stood, and swept his gaze across the land. Something moved in the corner of his vision, and his blood frosted. A familiar feeling lodged itself in his belly.
"I have seen this before," he whispered.
Down in the valley, a lone person stood beneath the canopy of a tree. He hadn't noticed the figure before, as it had lain motionless, but he recognized it was his eldest brother, Cain.
A flood of images came upon him and threatened to burst his skull. In a fraction of a second, he saw blood drip from the tip of a knife, a woman's mouth gape in a scream, and a man's limbs break under the weight of violent hands. He tried to juggle the images and understand their connection, but they multiplied and expanded until he fell to the ground, thrust into blackness.
##
Adam walked to Cain and Sarah's home, the next closest building on the way to the fields. Upon arrival, he called out, and Sarah's muffled reply sounded from inside, "Come in!"
He pulled aside the tapestry hanging from the stone archway and entered. She was by the fireplace, the wood crackling as she poured tea. The scent of jasmine blossomed in the smoky air as she asked, "Would you like some? It is ready." She appeared to have been awake for hours.
"Thank you."
She filled a wooden cup for him, and he smiled when she handed him the vessel, though she looked away, placed the pot over the heat, and sat on a cushion near the fire. "Did you sleep well, Father?"
He sipped his tea, scalding his tongue before swallowing. "For the few hours I was allotted. Calebna and I were the last to leave, which would have been tolerable if not for my body deciding to wake before sunrise." His eyes stung with the smoke, and he rubbed them with his thumb and forefinger. "You seem alert."
Sarah scooted from the heat. "I could sleep no more, so I rose and decided I needed tea."
Adam nodded. "It does calm the restless mind." He looked around and listened for movement. "Is Cain resting?"
"No." Sarah brought the cup to her mouth and sipped several times before turning her back to him. As he watched her, he was warmed by compassion. As an infant, her grip had been strong and her will stronger. Even now he saw her toothless smile as she hobbled into his arms for the first time, and he marveled at the swiftness of time. Could the woman sitting before him really be that little girl?
_Does she muse over Cain's humiliation?_
He took a breath and finished the last of his tea. "Have you seen Abel?"
She paused longer than he thought natural. "Is he gone?"
Of course, that would remind her of the Almighty rejecting Cain's offering last night. He shouldn't have mentioned it. He cleared his throat of the mucus the tea had loosened. "He walked into the fields last night and has yet to return. Lilleth is worried. You know how she is. I checked on Seth before I came, wondering if he had seen him, but neither Seth nor Ayla were home. I know it's early, but have you heard anyone speak of where they may be?"
"I haven't. I am sorry."
"Don't be. Their house was a mess, though I suppose I shouldn't be surprised after the celebration yesterday. So much to prepare for, especially for the women, and Seth has never cleaned a thing in his life." Thunder rumbled in the distance. His words had brought tension, and he had little desire to add any more, so he set the empty cup on the ground and placed a hand on Sarah's shoulder. "Thank you for the tea. And may the Almighty bring you peace. His shame will pass."
Sarah stiffened, and he could see her eyes fill with tears. He frowned. Her relationship with Cain had struggled for years; such was obvious to all. And yet, she had remained faithful.
He squeezed her shoulder. "You deserve more than he has given you."
She shuddered.
He kissed the crown of her head and whispered, "You're a strong woman. You should be proud." He turned and left. Thoughts of Cain had put him in a mood, and he wanted to talk with Abel. If he knew his son at all, he knew where he would be.
##
Cain stood and began to walk home. The storm above raged like a caged beast. It tore through the atmosphere with invisible claws streaking red across the sky. As a fresh carcass impassions a hungry dog, the sight made him long for Sarah. But after over a century of marriage, the endless fighting had distanced them. He realized it was merely a symptom of the Fall.
_Proof of my father's failure_ , he thought.
He and Sarah, at moments, felt something more than loyalty to each other, but thinking on it made the past sting all the more.
He breathed deeply, savoring each portion of air that was free of Abel's presence. He would endure the years of torture again just to feel such rapture as he had with that final blow, and he hungered for it with an intensity that frightened him.
He had hoped that freedom from Abel's presence would remove the sickness, but though the pleasure had faded, the sickness endured. He breathed deeply and reminded himself of all the reasons he had killed his twin. Then he shivered as he remembered the chill voice that had rested on his ear in the darkness of the valley. It had become so constant he had almost forgotten it, but if he focused, he was able to recognize its presence again. It was there, it was real, and it was something other than him.
The thought that there could be something inside him manipulating his decisions was disturbing. _But eventually_ , his mind reasoned, _knowledge will unveil the mysterious as benign—just as the morning sun burns the shadows._
He ran a hand across his scalp and wondered at how no one but he and Sarah knew of Abel's death. Everyone in the City of the Almighty was ignorant of Abel's body now rotting on the river's surface.
And what of the Man? Did he know? Did he understand?
_Yes_ , Cain thought. _Understanding. It will come, all in time._
He quickened his pace, rounded a corner, and saw someone exiting his home. He slid behind the nearest wall, held his breath, and listened past the thrum of his heartbeat. The sound of the footsteps faded, but the identity of the person remained.
"Adam," he whispered.
Father must have visited Sarah and was now walking toward the fields.
He wiped away the sweat on his brow and allowed his lungs to reinstate their rhythm. Could they be searching for Abel so soon? His heartbeat raced, and he shook his head. Impossible. They believed too deeply in the Almighty's protection to support suspicions already.
His fingernails dug into his palms as he looked up and willed the boiling clouds to wash away the blood in the field. He ground his teeth. "What are you waiting for?"
He'd hoped for more time to prepare. He couldn't deny he wanted to be present when they found out, to see the knowledge creep across Father's face, but he knew what would happen if he stayed that long, and for her sake, he could not risk it.
He threw occasional glances backward, but the streets were vacant and he reached his home quickly. As he entered the foyer, the scent of his wife's hair was almost palpable. She stood by the window, bending in an uncomfortable position, as if to peer through the window without being seen.
All the noise in his head, even the ever-present whispering, quieted as he admired her. Then his foot slapped the ground and she spun and cried out, knocking a bowl of fresh olives off the table. Her eyes flashed as she cupped her mouth. She lowered her shoulders, but her eyes burned with fear. Her voice was hushed. "Are you alone?"
"Of course. Why?"
"You scared me." Her breath came in uneven lurches. "They're looking for him. If they find out what you did..." She shook her head.
Cain's smile flattened. If they found the blood, suspicions would grow. There would be no way around it.
His eyes narrowed. Had she said something? The thought made bile bubble up his throat.
_She wouldn't have. She's more afraid than I am of what they might do if they find out the truth._
His eyes roved over Sarah's body. Then he walked past her to the next room and sank into a cushion. The voice came back louder, and he had a difficult time ignoring it.
She followed, her words hushed and her eyes glancing about. "I know what I told you, but I didn't actually want him dead. I was just angry. You know that. You must know."
He closed his eyes and leaned back while supporting his head. The voice droned on with hers, gnawing at his patience like an insect, commanding him to get up, to finish what he started and flee the City. "Say what you must to deliver yourself from guilt."
Her lips pressed into a thin line. "So cold."
"I say what needs to be said. You would do well to imitate me."
"Father came to get your help looking for Abel. I couldn't hold back my tears."
Cain glared at her.
"I said nothing. He knows nothing."
"Then leave me in peace."
She waited, and he sensed the disquiet inside her. "What happens if they find him?"
"They won't."
"What if they figure it out?"
"It doesn't matter."
"How can you say such things?"
Cain stood, grabbed her arms, and shook her. "Whether you wanted him alive or not, he's dead. He won't ever come back. You cannot change the past. Nor can you remove your fingerprints from it."
Tears formed in her eyes and her lips parted. He could see the fear, but also the searching. Her gaze darted from his left eye to his right.
He frowned. "I have simplified life. No more choices, no more questions. Just follow the path before you."
Her muscles tensed beneath his hands. She was a wild deer, poised to escape. Untamed for a century and counting. "Where will that path lead me?"
"Have patience. Just know that I"—he paused and let his gaze jump across her features—"I won't let them hurt you. This is a new beginning for you and me. I have shaken the world. It will take time for them to adjust."
She stared while the voice in Cain's mind grew louder. Over and over it repeated the same refrain, urging him to travel east. He felt his limbs' desire to respond to those words, like a flame to his feet, and now, with the possibility of discovery only hours away, he knew it was almost time. As much as he wanted to deny the possibility, he had to flee soon. Yet the knowledge of what he must do to Sarah was immobilizing.
He cursed Lilleth's inquisitive and worrisome nature. For the first time in his life he actually wanted someone to be stupid. His grip tightened on Sarah's arms. _Remain calm_ , he thought. _You must remain calm._
He tipped his mouth to her ear and whispered as sweat broke out on his skin, "If they really are searching for Abel already, then I have to leave for a while. Just remember what I told you. Remember me."
She pulled away. "What do you mean? What are you talking about?"
He hushed her, forcing a smile though sweat burned the corners of his eyes. "Do not fear. I promise I will come back for you, no matter what happens. I will fix it all. Just know that everything that happens is a part of the plan I have set into motion."
It felt good to hold and comfort her. He was surprised he wanted to comfort her. Their lives had been set on the edge of a knife, and now everything had changed. With a stone and two hands, he had changed the world.
He thought of how her body felt against his. Such a beautiful creature in every way. He desired her now as he had not in many years, and he slipped his hand lower down her back. His mind moved on, imagining enjoying her once more, but she jerked herself out of his grasp after sensing his intentions.
Anger burned his throat, and he thrust his palm into her face with such speed he hardly perceived the crack of his hand against her cheek. She stared with mouth gaping and brought a hand to cover the wound. To his surprise, the look in her eyes forced an ache through his numb interior like a spear through a wild pig.
He turned and strode out of the room as she sank to the floor and cradled her face. He stopped after a few paces and leaned against the wall to steady his breathing. She was crying quietly.
He breathed deeply, pushing his fingers through his shortened hair. All he wanted to do was hold her, but he knew he could not. It was far too late for that. The ache in his chest could not be dislodged no matter what he did, and the whispering voice droned on.
He rushed to the next room, grabbed a leather satchel, and started throwing miscellaneous items inside. Treasured carvings, gifts from his children, flint, rope, a rain cloak made of cured animal skin.
What had he been thinking? He loved her. He had always loved her. And he always would. She had been good to him. Maybe too good. His mind was stressed, being pulled in many directions at once.
He didn't hate her.
And yet he hated her.
He loved her.
And yet he could not love her.
"One day," he whispered and thrust a loaf of bread into the satchel. One day he would heal the old pains, the disappointments. They would all be replaced.
He tossed the bag on the ground and pushed away the temptation to believe he could avoid damaging her. He had struggled with the decision for many days, but even if it would break her, he would rather her experience pain, than have to live without her.
He rubbed his eyes, unable to remember what else he needed to pack. It felt as though insects were burrowing through his legs. He cleared his throat and started pacing.
He had so carefully calculated each step, his reaction to every possible inquiry. Even so, there had been complications. His family had grown suspicious too soon. The rain had not come. Adam was searching for Abel. And there was the voice. Yes, that was what had thrown him. He had let it creep under his skin and push him to talk with Sarah for comfort, like a fool.
Too much weighed against his will. He was losing acuity, becoming distracted. He could wait no longer; it was time to act.
_I will come back for you, Sarah. I will. I must._
##
Adam scratched the jaw of a young sheep looking up at him through rectangular pupils. It chewed on grass as if indifferent to the world. Its relatives stood about doing the same, and scattered between them were gnarled trees. They looked old, maybe a few as aged as he.
Life had moved so quickly in the century and a half since walking in the Garden. And how long had they lived inside it? He remembered the strange speed at which time seemed to flow there. With unending restfulness and peace in the Light of the Almighty's presence, they had labored hard and yet all had been joyful. In the Garden, work remained no more than a building block of satisfaction.
Then their world had crumbled.
He looked at the flock of sheep again. Few of them could have been more than several years old, yet who knew what the eyes of the eldest sheep had witnessed? In a world so much more temporal than the Garden, the curse was felt by all with deadly force.
His brow furrowed as he noticed a lump on one of the sheep's shoulders. He touched it and the sheep jerked away. As it did so, Adam noticed another lump on the side of its head.
_I will have to show this to Abel when I find him_.
He had never seen an animal suffer from such a malady before. None of their animals had taken sick since wandering the wilderness, and the Almighty had assured them their animals would remain healthy within the City's walls.
So what then was this?
He patted the tangled fur on its shoulder and attempted to reason it away. He should talk to High Priest Calebna after he found Abel.
He resumed walking.
He and Abel used to walk together often. And when Abel grew old enough to compete, they raced. The first time Abel beat him was on the outskirts of a desert whose creeping sands had slowly encroached on the niche they had carved into the cliffs. That day the dunes reflected the sun like so many mirrors, and the blue sky was devoid of blemishes. Abel had turned and slapped his hand, and they had walked home to find Cain had abandoned the herd of sheep once again to scout the wilderness for game.
Truthfully, Adam couldn't remember the last time he and Cain had done anything. And the last time Abel provoked Cain into competition was when Cain still found joy in how Abel looked on him with shining eyes.
Cain. A leader in all things, even from the beginning. What had happened?
_"I am my own man, Father," Cain said. "I am not, and do not want to be, anyone but myself."_
_"You are destroying relationships, and for what reason? To keep your pride? Give it up. Stop fighting against everyone. None of us desires it. We just want peace, we want to be a family again."_
_"As do I. But you won't find it by trying to change me into something you prefer more. If I am not good enough, so be it."_
Adam frowned as his feet slid through the grass. Remembering the fight angered him. It had only served to increase the tension between Cain and the others.
_Everything I do fails to make an impression. Such thick-headedness. Such stupidity._
The problems had yet to resolve themselves as he imagined they would upon returning to the Almighty. Now he wondered if they ever would. And even though it gave him no comfort, he continued praying.
He crested a hill and saw a valley spread away filled with grass and freckled with trees. It was the space between Abel's pastures and Cain's fields, and he descended into it, making his way toward a patch where the grasses looked depressed. The dark clouds rendered light little more than shadows, but as he reached the clearing, he saw spots of color, as if the grass was sickly. The spots were positioned around a pool of liquid that lay beneath crushed grass, and he stopped and ran his hand over the stalks to break the surface of the liquid with his fingertip. He brought it to his nose, and as the scent entered his lungs, his scalp tingled.
_Blood?_
Looking around, he found clumps of hair. Picking them up, he rolled them between his fingertips. More blood came off them, and the hairs themselves were light. Very light.
"My God," he whispered as shadows encroached on his vision. "What is this?"
Adam rubbed his face. He was becoming paranoid. Lilleth's fear had planted the seeds of unnatural thoughts. _You only fear danger to your son because you have been searching for him._
But what could explain the blood? And hair? What animal had such light hair and with such a texture?
He noticed a faint trail ahead, and he sidestepped the blood and followed the trail toward the line of trees that veiled the river in the distance. The grasses bent as though something had been dragged this way. The Almighty had never sanctioned the killing of animals unless for sacrifices, and surely no animal was allowed to be slaughtered in the fields and dragged so far, even here, between the inner and outer walls that the Almighty had built as boundaries to the City and the pasturelands outside it.
The hairs. The fine, light hairs. Each second stretched into hours. The Jinn, those demonic half beasts, could not have entered this far. Surely with the Almighty's protection, humanity was safe. Had he misunderstood?
He scratched his beard. No, the reason he agreed to come back to the Almighty was for the relief that very promise offered his family, who were so burdened by the unnatural pressures he had purchased them by tasting the fruit they should not have tasted.
He reached the river and found footprints in the mud. He bent to one knee and examined them. They were human and fresh, maybe a day old. He traced his fingers over the curvature of the arch, the five toes and shape of the heel.
Droplets fell from the sky as he ran his hand across his forehead. What had happened? No animal would be thrown in the river. Such wastefulness was not only sinful, but it was foolish. Abel would never do such a thing, would he? He had buried every animal that died under his care with honor and solemnity. And sacrifices were given with equal respect.
His eyes widened. Could it have been Cain?
Cain had been gone earlier, and Sarah had acted strangely. It certainly was possible Cain had been in the field and slaughtered one of Abel's sheep. But why? Retribution for the sacrifices the day before?
What if it had been someone else?
But this was Abel's pastureland, and his flock was still here, and both Cain and Abel were gone. Cain's hair was black as coal. Abel's was almost white. The hair he had found was not an animal's, and Abel would not have harmed himself, and couldn't have dragged himself to the river. Whoever made the footprints was uninjured, and whatever was dragged had been large.
He could not admit that Cain may have harmed Abel. _But what if he had?_
Adam squeezed his eyes shut and rubbed them. How could he think such blasphemy? The Almighty held them safely. He had promised. Was he then a liar? "Let me die before I entertain such thoughts."
He turned and started for home, his knuckles white from unanswered questions. He sped until he was jogging, anxiety growing until his breaths came in shallow pulls. Then he sprinted, pushing his legs until they burned with exhaustion. Drops of rain fell faster. One landed on the bridge of his nose, and others came until the storm matured and fell in sheets.
## PART TWO:
## RETRIBUTION
_Then Satan answered the Lord and said, "... Have you not put a hedge around him and his house and all that he has, on every side? You have blessed the work of his hands, and his possessions have increased in the land. But stretch out your hand and touch all that he has, and he will curse you to your face." And the Lord said to Satan, "Behold, all that he has is in your hand. Only against him do not stretch out your hand." So Satan went out from the presence of the Lord._
—JOB 1:9–12 ESV
##
Cain stood in his bedchamber and shoved the last few items into the satchel as he recalled a moment eight nights earlier when he had surrendered to his murderous desires and embraced the decision to plan his brother's death; when his mind had plunged into endless darkness, and hours later he woke naked on the stone floor of his house, wondering if he had already committed the act. After actually consummating the sin less then twelve hours ago, he felt the familiar sensations creep up the back of his skull. The strange itching ballooned in his mind as he felt himself slipping, falling into that nameless space he had come to know and hate.
He screamed and thrust his fist into the wall. The voice pulled at him, wagging its tongue in his ear, but Cain pressed his palms against his head. _Leave me!_
But it would not. It was inside of him. Its desires throbbed against Cain's mind like a beating heart. Beating. Beating. Beating!
_This isn't how it is supposed to be. I wanted freedom from the Almighty's tyranny, from the expectation to live in impossible perfection. Instead, I am teetering on the borderlands of madness._
A chill scuttled through the building and under his clothes. The whispering grew until it was all he could hear, and he squeezed his eyes and fists shut and waited for the darkness to swallow him.
Instead, all went silent.
He waited. No voice. No itch. No burning passion to consume life. He opened his eyes and sensed normalcy like an infant waking from a dream. "Hello?"
The room echoed silence. The dark corners feigned calm, but he sensed with a shiver something was present. For a moment, he wondered whether it was paranoia.
_"Give it to me."_
He spun to the voice. His mind stumbled, but after blinking he couldn't deny that he saw what appeared to be a little boy with silver eyes, silver hair, a tall forehead, and an elongated nose above a set of thin lips. Its neck was tall with dark arteries, and though it stood before him as if it belonged, something he could not place rendered it alien.
_"There's no more time."_
The hair on the back of Cain's neck prickled. He knew that this had been the itch in his mind, the voice he had obeyed the previous night as he dragged Abel's body to the river. The fact was disturbing, and he felt nausea ride up the sides of his stomach. Words circled through his mind, but only three found their way to his tongue. "Who are you?"
_"You will know me in time. But now, you must obey._ "
The creature seemed surreal, but Cain could not doubt its reality, for he saw it with his eyes and felt it with his soul. The thing's lips curled like straw in fire—a convulsive, flashing expression.
Saliva pooled in Cain's mouth. His face twisted and his legs burned to run, but his body would not respond. He was locked in place, forced to stare into those eyes as he felt the world buzz. The Almighty's warning one day earlier returned. _"If you do not do well, Sin is crouching at the door. It desires to devour you, but you must rule over it._ "
"Is your name Sin?"
_"Come_ ," it commanded. _"Give it to me and I will show you the way. He is calling us._ "
Cain tried to look away but only managed to tip his head.
It frowned and arched its eyebrows with something like sadness. _"Still holding on? Too proud to give in?"_
Cain's response was no more than a throaty grunt.
_"I will break you._ "
Would it kill him? Would those eyes, those silver rims around black holes, suck him up and never let go?
The silver boy turned and, with a motion of its hand, evaporated. Cain bent and vomited. When he finished retching, he laid down and let new thoughts enter his mind.
_Which is the illusion—the Almighty or the hope of having any freedom at all?_
##
Seth could not move. Nor could he quite see. At least, not in close detail. What he did see were shapes like murky outlines in a clouded pool, and soon those shapes and lines congealed and formed a silhouette, and then a face.
He knew that face. It was familiar, and yet seemed like a painted image. The painting's expression changed, and he wondered what could be happening. Was he dreaming?
Its lips formed his name, and its eyes strung heady emotions through the holes in his soul. For a moment, all was calm. Its hair fell over him, glaring in the light, and he winced. The world rushed in with strangling intensity. Hands clamped down on his arms, and everywhere there was pain. He could hear his breathing, and that of the painting, like pulsating counterrhythms atop a symphony of winds—like terrible Music.
The world slowed and quieted. "Seth." It whimpered as droplets of color slapped his forearm. "Seth!"
This was no painting. He shook his head. How had it taken him so long to recognize who it was? The light mellowed. The picture sharpened. With difficulty, he lifted his arm and wrapped fingers around the hand on his chest. "Ayla?" The word barely limped out.
She fell on him and wept.
"What's wrong?" His tongue felt like wadded wool. He looked at his bruised and bloody arms. "What happened?"
"I don't know." She squeezed his waist. "I left to find you, hoping to join you on your walk. But then I saw you fall on this hill, and when I arrived, you were thrashing yourself against the stones. I was scared. So scared."
He squeezed her hand and rubbed the back of it with his thumb. The thumping in his throat slowed, and the muscles in his face relaxed; but he struggled for breath, and his head was filled with painful pressure. "I'm all right. I'm fine now. I'm sorry I worried you." His muscles ached and burned.
"What happened?" He lay frozen by the way the tears revealed the color of her eyes. She went on pawing at his chest. "Are you sick?"
He reached through the murk to grasp at his memories. "I think I had a vision. Such terrible images. I saw..." He swallowed and shook his head, not wanting to continue. "Terrible things. Can we not speak of it? Remembering makes me feel strange."
"Of course. We don't need to talk about it."
He swallowed and attempted a smile, but it probably looked painful, because it was. "Can you help me up?"
She rose to her knees, grasped his hand, and helped him lean against the wall of the well. His arms and legs had cuts and bruises, and his joints felt swollen. Dark blotches appeared on Ayla's arms and face. He pointed to her cheek and she touched the spot and winced.
"Is that from me?"
She nodded.
What had happened when those visions rushed in? They certainly had not come from the Almighty, so maybe he really was sick. He gazed at the black clouds in the sky, and a spear of panic struck him. He tried to stand, but Ayla pushed him to the ground. "What do you think you're doing?"
"I'm going home," he said.
"What if it happens again?"
"You want to wait until we're soaked by the storm?"
"We don't know what's wrong with you."
He shrugged. "It was just a dream."
"Does a dream overtake you as you walk? Does a dream make you injure yourself?"
The sound of sporadic droplets striking the leaves punctured the silence, and Seth wondered at how long it had taken the storm to start. "Don't you think it odd that this storm has been hanging over us for days, but not a drop has fallen until now?"
"Is that all you can think about? You're injured. If it happens again, you might throw yourself down a hill or worse."
"You forget that we're in the City of the Almighty. You're starting to sound like Lilleth."
"Tell me how we were protected from these." She pointed to the marks on their bodies.
He was startled by the realization that what she said held weight. It was true that they were still in the City of the Almighty, and that the Man had promised them safety as long as they dwelt with him.
He looked at the bloody contusions on his arm. _So, what do these mean?_
Something was different. He could feel it in his bones.
He glanced at Ayla, whose pale face was rounded by concern. He reached up and clumsily pushed a tassel of hair behind her ear. "I will be all right. I promise. We should wait to speak more until we've gotten under shelter."
Ayla stood and brushed the dirt and grass from her knees. "Father's house is closer. We will go there, but you will not walk without my aid. And when we arrive, you will do no more than lie down and rest. Agreed?"
He nodded and smiled. She frowned. Droplets fell faster as he draped his arm over her shoulders for support, and she helped him stand.
"I will let you know when I sense the next dream coming."
"No. No more dreams."
The rain intensified and as they walked, his knee bothered him less, but when he tried convincing her he could continue on his own, her grip only tightened. They made their way slowly and were drenched by the time they arrived at the gray arch of their parents' home, which sat apart from the others that knelt clustered together like so many stone bodies bowing at the foot of the Temple.
He saw Eve peering up at the storm from inside. When she saw them, she called out. "What are you doing in the storm?"
"Seth fell at the well. He is having trouble walking," Ayla yelled.
Thunder pealed and shook the ground. "Is he all right?"
"I think he's sick."
Eve came out and braced Seth's other side. "Come inside and rest. We'll dry your clothes. Tea is ready."
Seth nodded thankfully as they ushered him dripping through the doorway.
Eve glanced at Ayla. "Lilleth is here."
"And Abel?"
"He never returned last night. She came here this morning to ask if we had seen him."
"Have you?"
Eve bit her cheek and shook her head as they passed into the lounge, where Lilleth lay on a cushion and gazed into the fire as if lost in thought. Eve and Ayla helped Seth to one of the cushions, and Lilleth noticed them with wide eyes. "Brother, are you not well?" Her sparrow voice sounded strained.
Seth nodded and smiled, then winced. "I'm fine."
Ayla mumbled something directed toward him, shook off her soaked clothing, and helped him do the same—though not without a bit of roughness, he thought. Eve handed them cotton rags to blot their hair and skin with, and soon they donned new dry garments.
Eve removed the tea from the flames and poured a cup for each. Seth accepted the wooden vessel and breathed deeply. Chamomile, mint, and lemongrass, brewed overstrong. Eve returned the pot and slid it from the flames, either ignoring or not noticing Lilleth's glare.
They blew on their tea and sipped without speaking. Ayla twisted her wet hair behind her head, Lilleth wrung her fingers, and Eve sat cupping her tea with both hands. After a while, Seth smoothed the coverings on his legs and said, "This storm is unusual."
Eve nodded and Lilleth seemed to whiten a shade, though it was hard to tell in the firelight. The only sound was the pitter-patter of the rain on the roof. Lilleth would not look at them, and Eve stared blankly, the creases in her face deepening.
Seth caught Ayla's gaze, and they exchanged curious expressions. He cleared his throat. "What has happened?"
Eve glanced up. "Nothing. Nothing has happened. Adam is looking for Abel. He left this morning and has yet to return."
Ayla addressed Lilleth. "I assumed when I saw Abel leave the celebration last night that he was going home."
Lilleth hugged her knees and rested her chin on them.
"Lilleth and Abel returned home after the celebration, but afterward he left for the fields," Eve said.
For a moment, it seemed to Seth as if clouds darkened the flames in the hearth, though he did not know why.
Ayla addressed Eve, since Lilleth seemed in a strange mood. "Seth fell at the well earlier this morning. I found him on the ground, moving as if lost in a dream. After a while he awoke, but he's been injured."
Eve glanced at Ayla. "Injured?"
Ayla's voice gained an edge. "Yes. He has bruises and cuts from where he struck rocks."
Eve furrowed her brow at Seth. "Why would you do such a thing?"
"I think I was dreaming."
"Do you not remember?"
"I remember sitting on the edge of the well."
"I meant your dreams," Eve said.
"You never did say," Ayla said.
Seth sighed and wiped his face. He sensed their gazes, but he suddenly felt weary and a little sick. "I'm unsure what it means, if anything, and I know it might sound strange coming from me, but I'd rather rest than talk right now, if you don't mind." His head throbbed, and he rubbed his temples. The pain came in waves.
Eve folded her hands in her lap. "Of course. You should rest."
"I will be fine." Seth felt Ayla's hand on his leg. He breathed deeply and leaned back to ease the pressure, but his swollen joints and muscles pained him, and the cushions offered little relief.
The silence returned, and Ayla shifted. "Do you have any idea what it might be?"
"Sorry?" Eve said.
"Do you have any idea what might have happened to Seth?"
Eve shook her head. "It is new to me, whatever it is. Perhaps..." Her eyes dulled. "No, I have never seen or heard of anything like it." She turned toward the fire and seemed to fall back into heavy thought.
Ayla stood, as Seth knew she would, and cleared her throat. She never could abide silence. "Thank you for the tea. Is there anything I could help you with in return?"
Eve smiled, but somehow looked wearier. "There are bowls and clothes needing to be washed. You may finish them."
Ayla spun away, and Eve called after her saying she could stay and rest, but she had already vanished. Seth's brow wrinkled as he tried to find a comfortable position to rest.
_"Adam is looking for Abel_."
The significance rolled in his mind. He rubbed his forehead and sighed. Surely he needed rest, but he was afraid of what might happen if he slept.
_The dreams..._
Thunder rumbled the house, and the sound of rain increased to a roar. Wood creaked with the winds, and the buildings howled. It sounded strangely orchestrated, as if a part of some subtle Music too discreet to be fully known.
Exhaustion took over, and he closed his eyes. As he drifted, he pictured places Abel might be, and reasons why he might not have returned.
Then he dreamed a series of nightmares he would not remember.
##
The City was washing away. Cain could hear it as he smashed their jars and broke their furniture. The winds screamed at their house. The rain pelted the earth, flying through the tented arched windows and gathering into puddles on the floor. It was so loud he didn't notice Sarah until she grabbed his shoulder and shook him.
"What are you doing?" she asked.
"What needs to be done."
"Destroy our house and our things? Why?" Her fingers needled his arm as he pulled away. Her red hair was disheveled and her face was dotted with water from the storm. He couldn't remember a time she looked so haggard. "What's going to happen to me? What will our children think when they find out what you've done?"
He grabbed a small table and lifted it above his head. She released him and shuffled back, her eyes black and wide.
_Do you think I would kill you too?_
He yelled and smashed the table on the ground, its fibers exploding. Sarah screamed, pressed her hands over her ears, and shut her eyes. "Stop it. Stop!"
"No."
"You're a monster."
"What if I am?" He could feel the blood hot in his cheeks. "What if I am?"
Her voice warbled and she sank to her seat, cradling her face in her hands. He was surprised at her reaction. At how much she seemed to care. He wondered if he were making a mistake.
_What if she really does love me?_
But the buzzing in his mind made it difficult to focus. The voice was mumbling, then speaking.
No, not just speaking, it was shrieking.
_"She only cares because you're all she has left, because Abel is dead, and when you leave she will have no one left to love her, no one left to hold her or to keep the darkness at bay. Your children are all grown and no longer need her. She uses you, she always has."_
He shook his head. She had loved him once. And with Abel gone, there was no one else she _could_ love. But then again, that was the point, wasn't it?
She was shaking. She was wailing. Her body was alive, but the sobs were the sound of a soul falling to pieces.
_My wife's soul._
His eyes softened. He wanted to kneel and scoop her into his arms like a child. He wanted to hold her and bring her to their bedroom and press his lips against hers with all the passion he felt in his chest.
_"But you can't stop now. It will all fail if you stop, and you know it. She has to believe you hate her. She has to detach from you. If she doesn't, she will either die at the hands of your family, or follow you into the deadly wilderness. Everything you have done will fail if you comfort her in this moment. Break her. Break her!_ "
He grabbed the broken remains of the table, lifted it high, and smashed it on the floor. "I am a monster. I killed him, and I'll kill anyone who gets in my way. I am a monster." He beat the floor with the wood. His chest heaved with the intensity of his breathing, and he dropped the remnants. Sarah was no longer looking at him, for her eyes were hidden behind her hands.
He rushed into the other room, grabbed his satchel, and came out, stopping one last time to look at Sarah's crumpled figure. Lightning boomed overhead and seared the image of the room in his mind with white silhouettes. He had to get out of the City while he still had time. Wanderlust burned his legs, and he could resist no longer. Everything had been rushed. The broken remnants of their furniture and pottery lay scattered about the room.
_Details that will distract and confuse. Misdirection._
The voice returned. _"They will think she fought with you. They will think you left her like the worthless whore she is, and they will look on her with sympathy. Then, when you return..._ "
Cain's hand extended and his throat constricted painfully. He nearly risked everything in that moment to speak words of comfort, to whisper he loved her and to hear her say it back, but thunder pealed and muted him. His cheeks flushed as fantasies fell and disappeared like ash. Desire for her tender touch cracked and bled shame, and a sudden craving to beat her rose in his chest. He turned and flipped up his hood instead. He hesitated, caught between rage and some longing he could not describe. Then he strode out of the room and into the storm, leaving Sarah alone on the floor.
If someone had seen him walking in the storm that night, they may have thought the wetness on his cheeks was only rain. But it was not.
##
Adam strained for air. He had to get back. If Cain really did it...
He stumbled forward as the tears on his cheeks became lost in the raindrops pattering his face like tiny wooden paddles. Lightning flashed and thunder rattled his teeth, and the long grasses, which before had swayed so blithely in the breeze, were wet hands grasping his legs. He struggled on. He fought the world to get back to his family, to find them safe. But when he reached the last hill and looked down on the buildings, his heart coughed and sputtered.
Adam mumbled a quick prayer and made for Cain's home. He slipped down the hill and fell on his palms, splashing mud in his eyes. He scrambled up, flung the muck from his hands, wiped his face, and grit his teeth. He loped on until he saw their house. As he entered Cain's home, he recognized a scent he had missed when last there. The faint, though unmistakable, smell of death.
He breathed deeply.
_That's the smell of your son. The last sign you'll ever have of him. Abel is dead, you fool._
Sarah sat in the same position he'd last seen her in, though all the furniture had been broken and lay scattered among shards of pottery, and there was no longer any fire in the fireplace, though three candles burned to her right.
The room bore the signs of long-bred violence, and she seemed to cower in the shadows.
"Where is he?"
She did not respond.
"Sarah, where is he?"
She rubbed her arms.
"Is he gone?"
She nodded.
He tried steadying himself against a table, but his arm pushed off a broken piece of pottery and sent it clattering across the floor. "Did he really do it? Did he really kill my son?"
Her warm alto was frosted with shame. "Yes."
He breathed raggedly, his wet clothes producing pools on the floor. He watched a droplet grow on the end of his fingertip and saw it change color. The forbidden fruit hung and swayed, and Eve reached for it and broke it off. After she took a bite, he received it from her one mouthful lighter. Oh, how it felt to puncture its skin and let the coppery taste fill his mouth. It tasted of death. The same smell that hung in this room, only one hundred fifty years later.
Sarah wept quietly.
How could she have said nothing? How could she live here with that serpent and not warn anyone? He strode to her, grabbed her shoulders, and shook her. "Tell me where he is."
She guarded her face. "I don't know."
"You're lying."
"I thought he was going to kill me."
"You're lying!"
"I wish I were." Her jaw clenched, and her red hair twisted about her head like a motionless fire.
He turned away, and all he could think of was Abel, his son. Lost. Dead. He felt the terrible weight of the mistakes he had made, and yet could not admit, for fear they would crush him. He struggled to convince himself the house was not collapsing about him, that the world itself was not being mercilessly pressed between the hands of the Almighty, whose eyes pierced the heart of every man.
But surely his Lord knew his intentions and forgave him. Because Adam's darkness was not that of Cain's—a remorseless, all-consuming evil.
"Dear God," he whispered. "How could you have done nothing?"
After all the service Adam had given the Almighty, after all of the sacrifices he had made, God had let his son, his most beloved child, die. Could the Almighty be loving and still let that happen? Could Abel live under the Almighty's protection and yet be killed?
His throat burned. There was only one conclusion such unshakable facts pointed toward. He opened his mouth and let the words fall from his mouth like forbidden fruit. "God lied." And they, too, tasted like death.
##
The rain fell like a volley of arrows. Eventually, glowing faintly through the storm-filtered light, Cain saw the eastern portion of the inner wall of the City of the Almighty. It reminded him of the spine of a great beast, long slain and removed of flesh and fur. He bent to see the Temple on the hill, but the hood of his cloak hugged his skull. There was a red flash as electric flames wormed through the sky, and the rain transformed, momentarily, into a million flaming spears stabbing the earth.
He turned up the path that led to the Temple entrance and rubbed a hand over his scalp, attempting to relieve himself of the buzzing sensation. The pressure corralled his thoughts through a diminishing space, until at last every part of his conscious mind was occupied by mundane facts that repeated endlessly.
_Make each step the same length. Make each stride vary as little as possible so that the numbers you've counted will be measurable. You need to know how many steps it takes to get from your house to the Temple._
But his true goal wasn't in numbering his steps. He didn't care how far it was. He only wanted to escape from the hellish itch in his skull. It was as if the silver boy were grinding its teeth down the edges of his mind. Scraping. Cutting. Devouring.
He planted his feet before the doorway, and with his left hand, grabbed the handle of the polished marble door of the Temple. He gazed up at the gilded image carved into it of two winged angels guarding their faces. It was heavy on its hinges, but the gap between it and the wall widened, and he slipped through. The door swung shut behind him, and inside was blackness.
The violent storm had been the first indication that everything had changed. The darkness of the Temple was the second. Water dripped to cold stone as he flung his hood back, rummaged through his satchel, and pulled out a rope and flint. He coiled the rope, laid it on the ground, and dropped the flint on top of it. Then he patted the wall until he found a torch and grabbed it. He returned to the flint and struck sparks until the torch's head warmed to embers. He dropped the flint, tended the embers into a flame, and watched the light reach into the corners of the basilica.
As the flames reached their full strength, he realized he was not alone, and the buzzing in his mind died away. His breath stilled and sweat broke out on his forehead and palms as the Man stared at him with golden irises. His hair was long and white, and concealing his mouth was a voluminous beard.
"Hello, Cain."
That voice. It was deep, earthy, familiar. Questions long-suppressed bubbled to the surface, but he dared not speak. He knew the importance of this moment. He had expected the Almighty to be here, of course, but the darkness had distracted him.
_The Man knows. Why else would the Temple darken for the first time since we arrived two years ago?_
"Where have you been?" the Man asked.
Cain rubbed his temple out of habit, but realized there was no itch, no voice. Only silence. He knew he should be relieved, but the voice's absence disturbed him.
The Man straightened. He rested against a staff, but his eyes were as intense as a morning star. "Speak, son of Adam. Where have you been?"
_Come now, you have long prepared for this moment._
Cain cleared his throat. "I have been home. With Sarah."
"Is that all?"
"Could that ever be all?"
Golden irises shifted over him. The man's beard clung to a slight frown, but no clear emotion revealed itself. "Do you think I do not know?"
"You know all things."
"Tell me," the Man said.
"I do not know what you want me to tell."
"Be free with your speech while you still retain the capacity."
Cain wondered at the Man's words. Did he threaten?
"Where is your brother?"
"I do not know."
"I saw you both walk to the fields."
"Am I my brother's keeper?"
The Man bent, and the words he spoke next were so small he nearly missed them. "What have you done?"
Cain stared, suddenly feeling the heat of the torch scald his hand. _Show me who you are_. _Will you try to suppress me like a tyrant? Show me your true self. I am dying to know it._
He wondered what the Man would do, but Cain would not turn back. He could not.
The Man's voice was soft and smooth. "I know what it is you have done. You have killed Abel. I hear his blood crying out to me from the ground. If you would have but spoken truthfully..." He shook his head and blinked watering eyes. "So, now you are cursed from the ground, which has opened its mouth up to receive his blood from your hand. When you work the ground, it shall no longer yield to you its strength. You shall be a fugitive"—his mouth contorted with emotion—"and a wanderer on the earth."
Cain's body shook, and as he thought of the Man's words, he sensed the tyrant loom. "If I had but spoken truth, you say? I will speak it now. You curse me to the life of a fugitive, and if I don't die from starvation, whoever finds me will kill me. This banishment is no less than murder itself, and yet it is for murder I am sentenced." He raised his torch. "By doing this, you commit the very act for which I am being punished. Where is your justice? Where is your mercy? Where is your _love_ and _forgiveness_?"
The Man's eyes flickered, and he lifted a hand. "If anyone kills you, vengeance shall be taken on him sevenfold. I tell you that unless you will it, no man may sever a hair from your head."
Cain straightened. He blinked and rested the butt of the torch against his thigh. "I was right about you. You are just as weak as I expected. You know what I do and yet still give in."
The Man held his gaze.
"Why?" Cain asked. "I could understand and hate your punishment, but this I cannot. I killed him. Why?"
"Indeed, why?" The Man reached forward and pressed a finger into the center of Cain's forehead. A burning sensation rippled across his skin, and black ridges like painted vertebra appeared across his arms and legs and glowed faint silver in the torchlight.
The Man looked at him with concern, and his voice came in a rushed whisper. "You do not have much time. Leave, and do not look back. I tried to warn you, but you have placed yourself in its jaws. Only remember my mercy. Remember that there is _always_ a way back to me."
The Man turned and departed, his staff clacking on the stone long after he disappeared into the dark labyrinth of the Temple.
Cain stared at the marks on his skin, considering the significance. Water and flame dripped from him. It felt surreal. The conversation, the curse, all of it seemed like a distant dream he saw in the reflection of the puddle at his feet. He looked at his arms and breathed the smell of dust and burning wood. _I'm alive_ , he thought.
_"You are more than alive_ ," whispered the voice. _"You have been set apart."_
Little fingers grasped his hand. In the light of the torch, he saw the silver boy looking up at him, but he was no longer afraid. Its voice was thin and harsh, like wind hissing between stone, like harsh Music. _"Come. He is calling us._ "
##
Eve watched as Lilleth slept on the floor beneath a blanket, and Seth snored softly. The fire had settled to white coals, and though the smell of smoke and ash warmed Eve's chest, the disquiet of the storm left her restless.
She stood and thought about finding Ayla. Instead, she walked the hall and peered at the streams, birthed by the rain, flowing down the hill in serpentine patterns. The deluge had intensified, and though she knew it was midday, it seemed like the darkest night.
_What kind of storm is this?_
After two years of safety, the Jinn seemed an imagined horror, but this brought the memories back.
_How quickly truth is forgotten._
She shivered and glided to the lounge where the air had warmed to something moist and sticky. She avoided the flooring she knew would creak, and took a seat between Seth and Lilleth. Their breathing was heavy, and she regarded them.
After Cain and Abel were born in that long-abandoned cave, movement through the wilderness had been difficult. She and Adam stopped often to feed and care for them, and she could still feel their tender skin against her fingers as she clothed them in leaves and woven fibers. What a beautiful burden to be those children's only means of survival. She recalled her children's squeals and smelled the leafy beds they had nestled under fallen trees and inside caves.
But then she thought of the darkness. And with the darkness came the Fog. And with the Fog...
Eve cleared her throat and clasped her hands in her lap. She examined Seth, her youngest son. He was an honest man with a personality conveyed through a wide grin. But if Seth was a caricature of expression, her daughter Lilleth's face was muted emotion. Neither beautiful nor ugly, the dependent Lilleth relied on Abel for support and direction, but having Lilleth rest on every word Eve had spoken today aroused old desires.
_Is it a sin to desire her dependence?_
Lilleth's breath caught in her chest, and her eyelids trembled, but soon her breathing slipped back into a steady rhythm. Eve noted how her hands cupped her chin, and her knees curled to her elbows. Even in sleep, she held herself like a youth. Did Abel find that attractive?
Eve closed her eyes and thanked the Almighty for everything he had given them.
_Almighty God, I desire you with every portion of my body. My bones cry for your presence. I cannot have enough. Though your peace and power overwhelm me, I thirst for you. I know I have not experienced all of you. I know I do not live you. Please, tell me how I can do more._
She tipped her head, spread her palms, and reached for his presence. She pushed, yet found nothing. She called for his Spirit, but felt only emptiness. Surprise and fear raked her stomach. Something was different. She had never felt such distance, not since returning to the Almighty two years ago.
_Where are you?_
She reached and nearly felt something like sticky fingertips skipping across her skin. She gasped, Lilleth stirred from her dreams, and as Eve bent and sucked at the air, her daughter said, "Mother?"
Seth still slept, but his sweat-covered forehead gleamed in the firelight, and he moaned.
Lilleth called his name, but he did not wake, so Eve hurried to her feet and shook him. He gasped and his eyes flashed wide. He clutched Eve's arm, then swallowed and loosened his grip.
"I was dreaming," he said.
"So was I," Lilleth said. "What did you dream?"
He frowned and gave her a look as if not wanting to say.
A noise at the doorway drew their attention. Adam entered the archway with water dripping from his bent shoulders and stared past them as if not realizing they were there.
"Father?" Lilleth asked as her drowsiness sped toward excitement. She struggled to her feet. "Where is he? Did you find him?"
Adam stood stonelike, but his gaze shifted to Lilleth, then to Eve, and finally to Seth, who raised himself on his elbows. Adam's jaw stretched open, clenched, then relaxed. Eve expected Abel to enter behind him, but Adam stood unmoving, and nothing moved behind him but the rain in the wind.
Thunder pealed and Lilleth clasped the clothing by her throat. The thought struck Eve that this was a cruel joke. Adam was remaining silent to teach Lilleth how absurd it was to distrust the Almighty. She could hear his voice even now. _"Lilleth, all is well, of course. How could you expect anything else? Is the Almighty our God, or is he a liar? There, do you not see? You must trust in his protection, for he will never abandon us. If he cares for the sparrow, how much more will he care for a songbird like you?"_
Fear rose in Eve's chest, and then anger. "Why do you wait? Tell us where he is."
Adam shook his head. "No." His voice was quiet, but carried a peculiar emotion.
"What does that mean?" Lilleth searched their expressions.
"He is gone," Adam said.
Lilleth spilled the cold tea she hadn't finished and quickly tried to mop it up with one of the cushions she had used as a pillow. "Who is gone?"
"Abel." He nodded and said, as if to convince himself, "Abel is gone."
Lilleth squeezed the pillow until her knuckles whitened. "Where is he?"
Eve thought water dripped down Adam's cheek. But his face shuddered. "Abel is gone, and we could not find him though we searched the world." His voice fell to a whisper. "Cain killed him. The Temple is dark. Everything was a lie."
Eve reached out to steady herself. "What?" She could not believe those words any more than she could disbelieve her own existence, but the sound of his moans darkened her sight. She turned toward Lilleth, whose eyes were like black stones.
Adam rested his back against the wall and slid to his seat as he cradled his face and wept. "He killed my son. My Abel."
Saliva was thick in Eve's mouth, and an ache intensified in her throat. "No. He's not dead. God could heal him."
Lilleth edged past Adam and into the storm. Seth rose and hobbled out after her, whether to follow or walk elsewhere, Eve did not know.
Adam's cries continued, and Eve sank to the ground, incapacitated by shock. She wept into her hands until she could weep no longer. Eventually footsteps came, then words, but she could not bring herself to answer. The footsteps pounded past and out into the storm, and Adam and Eve were together, alone.
##
Sarah stirred the bubbling broth over the fire. The chunks of vegetables slowed the spoon, and eggplants and potatoes bobbed through the steam. Her empty stomach reached for food, but the scent was repelling and she crinkled her upturned nose at it and angled away to get a fresh breath. She laid a hand on her belly, which seemed to protrude ever so slightly, though the change would have been imperceptible to all but one looking for it. A hint of a smile came, but then faltered.
_You should be happy._
_Shall I be happy in loneliness?_
_A child grows within you._
_The fruit of forced intercourse._
_It is so much more than that. It is a human child, your child._
_But will it survive?_
_You will give birth._
_Will it know its father? Should it?_
She stood and her lower back ached as she walked the hallways and let her fingertips ride the walls. The sensations blocked out any thought of her children, or any regret in her past, and for a moment, she was content to merely _be_.
Her hands rested at her sides and she found herself once again staring outside. "It's a dream," she whispered. She pushed a hard breath out as she glided back to the food she no longer desired. She took the pot off the fire and placed it on a wooden rack, and the bubbling liquid settled to a steaming reflection. She breathed the herbs, stifled an involuntary gag.
_Control yourself. There's no reason for such weakness._
_But Cain is gone. I am pregnant and alone. Abel is dead._
She threw her bowl of food. It shattered across the floor, joining the rest of the shards and slivers scattered about her home.
_I feel dead._
_But there is life within you._
She rubbed her belly and imagined the baby warming to her touch, reaching out with tiny hands to grab her fingers. Her stomach groaned and she gazed at the soup once more. She needed to eat for the child's sake.
She lifted a spoonful and let it fall in her mouth. A chunk of vegetable swished in the broth behind her teeth before she chewed and swallowed with much effort. It was nauseating, but she dipped for another mouthful.
Something shuffled behind her. It was a human noise amidst the thunder and rain. She turned and tensed at the familiar brown eyes and simple, soft face. She dropped the spoon into the pot. "Lilleth?"
"Don't speak." Lilleth's usual lilting soprano had been replaced with a strident alto.
Sarah swallowed and awkwardly faced her younger sister. She thought of Abel's blood on Cain. The scent of death. Adam's visit.
A feeling of responsibility gripped her abdomen. A moment passed when she strained for a way to deny any involvement, but she couldn't.
_Lilleth knows. Why else would she come? Why else would her voice bite with such venom?_
Lilleth gestured toward the pot on the rack. "I made stew for Abel yesterday, but he was late in returning. The stew grew cold so I had to place it back on the fire and stoke the flames." Her voice grew as she approached. "I added more and more wood until a new day began and the logs spilled into ash. I felt it then, but my mind told me it couldn't be true. I thought maybe Mother was right. Maybe I was just being fearful." Lilleth's eyes searched Sarah's, as if pining for a reason to disbelieve.
_This is not my fault_ , Sarah thought. _If only you knew how much I regret it. If only you knew how much I wish I could have stopped him._
Lilleth was shaking, and Sarah could no longer deny what Cain had done, but neither did she want to. She moaned, brought a hand to her mouth, and almost reached for Lilleth's arm.
"So it is true," Lilleth whispered. Her cheeks flushed. "I was a fool for wishing, but still I hoped..."
"I'm sorry."
"You're sorry, and yet I am alone. You're sorry, but you're still alive. And so is your husband."
The way she exaggerated the word _husband_ sounded an alarm within Sarah. Lilleth had always been delicate, but something in her eyes was cold and brittle.
Sarah lowered her head. "I am sorry."
"Is that all you can say?"
"What more could I say?"
"You've not been speechless a day in our lives. But I want you to tell me what it's like to kill with lust and smuggled glances. I want you to tell me what it's like to want what you cannot have until the desire is so great you'd rather obliterate it than ache any longer."
Sarah's face grew hot. "I could never exalt in your pain."
"He was the only one I ever loved. No one, nothing, could repair the hole you bore into me. Tell me, because I want to remember how far you fell." Lilleth took a step closer and Sarah felt a pinch in her belly. She looked down and saw Lilleth's hand gripping a knife pressed against her abdomen.
The floor tilted beneath her. Her breathing sped and her body tensed, but she couldn't move. _Oh, God, don't let her kill my baby._ "What do you want?"
Strands of wet hair clung to Lilleth's cheeks. "I want to see you and Cain burn. I saw your eyes follow Abel. You always thought it unfair that I, not you, could be with such a man. You knew he was better than Cain like you knew your own superiority."
Sarah edged back, felt the knife dig in.
"Don't move. I'll thrust this blade straight through you."
"What do you want from me? Just tell me what you want."
"I want to kill you."
The startling admission stopped Sarah's breath. Ways to refute Lilleth's accusations raced through her head. But nothing would help. She knew it. Truth be told, she had influenced Cain. And part of her welcomed the guilt, the idea of death as atonement. But the child in her belly begged for breath. "You want to kill me?"
Lilleth's face shone in the light as she brought the knife to Sarah's neck and pressed it against the side of her throat. "I want you to die."
Sarah's mind scrambled for logic. She licked her lips and gasped as the knife pinched her throat and warm wetness trickled down her collarbone. She closed her eyes and tried to focus past the spinning. "You hate us." Sarah's voice shook. "You want me to die but you don't want to _kill_ me. I know you."
"It's all that I want, all I've ever wanted, though it took the pain to open my eyes."
Sarah felt Lilleth's body tense in preparation of plunging the knife into her throat. Her mind emptied and she reverted to animal fear. "Stop!"
"Shut your mouth."
"He would hate you for it."
"I said shut your mouth."
The knife pressed harder and Sarah gasped. "Abel would hate you for it." The knife shook against her throat. Sarah dared not move.
"I would rather you both died than live knowing you won," Lilleth said.
Sarah shook her head, careful not to press into the blade. "I didn't win, Sister. I didn't win anything."
Lilleth stared with shaking intensity. For the first time, Sarah saw her examine their surroundings, the broken furniture and shattered pottery. Lilleth's expression softened, then shifted to something like fear. She whispered, "I hate you. I hate you. Look what you—" The words caught in her throat. She sobbed and stared at the floor; the shaking knife drew away.
Sarah chanced a breath, seeing the murderous intent in Lilleth's eyes replaced by dullness. But then something happened that Sarah had not expected. Lilleth took the knife across her own throat and opened a thick gash from one edge to the other. The knife clanged on the ground as she backed away and stared.
"Sister," Sarah screamed. "No!" She rushed forward, caught Lilleth as she fell, and laid her on the ground. She tried to press her hands against the wound, but blood gushed from it like water from a spring. It took her a moment to realize there were fists beating her sides. She looked down and saw Lilleth's eyes boring into her and said, "I'm not going to let you die."
Lilleth gurgled and beat her ribs. Sarah's hands slipped on and off the wound, and Lilleth weakened as she lost blood. Sarah could not stop it alone.
"Help," Sarah screamed, looking up toward the doorway. "Help me. God, where are you? Help me!"
Her voice was swallowed by the storm, which suddenly rumbled like distant, violent Music. Lilleth lay in a pool of her own blood. Her eyes were glazed, locked open like a dying fish gasping for one final breath. The bloodletting stopped, and Sarah brought her shaking hands slowly away.
"No," she said. "Why? Why would you do that? You fool." She shook Lilleth's shoulders. "Wake up. Wake up."
Sarah's hands were stained with blood for the second time in two days. She stared at the dark pool that soaked her knees and ached for a way to fill her sister with it once more.
She thought of Cain, of the good years together that felt like so many lifetimes ago. The light of those years and the darkness of now so starkly contrasted that she could hardly believe the light existed. She closed her eyes, wiped the blood on her clothing, and imagined being in his arms again. She saw him tilling the earth and planting seeds. She saw him shoveling dirt from holes...
Her eyes snapped open. She couldn't let anyone find Lilleth's body. If they found out, they would banish her. And if she were banished—
She placed a shaking, stained hand on her lower abdomen and tried to sense the baby within her as the thunder returned. She stood and shivered with sin.
"God, forgive me."
## PART THREE:
## COLLAPSE
_Be ashamed, O tillers of the soil;_
_wail, O vinedressers_ ,
_for the wheat and the barley_ ,
_because the harvest of the field has perished._
_The vine dries up;_
_the fig tree languishes._
_Pomegranate, palm, and apple_ ,
_all the trees of the field are dried up_ ,
_and gladness dries up_
_from the children of man._
—JOEL 1:11–12 ESV
##
It didn't take long for Sarah to realize her plan was a poor one. Mud was everywhere, and with every shovelful flung, half as much water rushed in. What was she going to do—float Lilleth's corpse in it like a hunk of wood? She pictured her sister's pale face bloated with moisture and decay. Twin balloons pressing the eyes until they disappeared behind fetid flesh.
Sarah felt as if the world was trying to drown her, and maybe it was. Nothing had happened as Cain said it would. She had lost so many she loved. First Abel, then Cain, now Lilleth.
_If only vomiting could rid me of the guilt I feel._
She stared at the puddle and gripped the shovel like a two-handed axe. How could she plan on burying Lilleth without speaking a word to another? She loved her sister. Maybe she hadn't realized how much before this day. Perhaps she truly had been blinded by jealousy, but now she knew.
_I'm sorry I ever wanted you gone._
She refilled the hole, hurried into the house, whipped off her hood, flung water-clumped hair from her eyes, and breathed deeply. The shovel clanged on the floor as she dropped to her knees and held her face, letting the emotions rush, if only to be rid of them. She slapped her hand on the floor and said through gritted teeth, "Stop it. You can't do this alone. Think of who you can trust."
Mason. Her second eldest son, mute from birth. He could carry the body and dispose of it quickly and discreetly. He would never speak of it, because for him speech was physically impossible. With the storm, Lilleth's body would be rushed far away, and surely no one would walk outside in such a tempest and stumble upon them in the dark.
_But the blood..._
_Listen to yourself. You can't even bring yourself to honor your sister's death. You are just like Cain._
She looked at the knees of her dress, stained by Lilleth's rusty veins. "Do what you must, and nothing more." She repeated those words until it became a mantra. She mopped up much of the blood, crawled out of the soiled clothing, and redressed. She donned her rain tunic, threw the ruined dress and undergarments on the fire, and watched them burn. Then she draped a curtain over Lilleth's body, which lay in the closet, and rushed into the storm to find Mason.
He lived on the far side of the quarry a quarter mile south, and though it seemed a journey away in such a storm, she couldn't risk involving her other children. The mud was like quicksand, and the hills and ditches ran with enough water to swipe her feet away. Once she fell on her palms. Once more on her side.
The rain dropped like darts and she closed her eyes to protect them. But when she found the quarry, it was not by sight. As she stepped forward and her foot failed to meet ground, it took the time of her body pitching forward and her arms flailing to realize she had stepped off the ledge of the quarry.
She tumbled like a bag of broken bones and came to a quick stop, saved by a ledge. Her head spun and it took her a moment to realize she was no longer moving. She rose to her knees and cradled her rib cage. Her breath came in short spurts in the fight against the pain that pushed out the air with the force of a hammer. Soon she felt panicked for oxygen. She exhaled, finding it easier than inhaling, then frantically urged it back in again. When she could breathe easier, relief washed her. But as her hand brushed her abdomen, a new thought hit her.
_My baby, what if I hurt my baby?_ Her fingers tingled.
"Dear God," she prayed as she started crying again with the memory of all that had transpired in the past two days. "I don't know if you exist in such hell, and I know I deserve nothing." She swallowed and tried to stand, lightly pressing her stomach. "But please keep my baby alive. You can throw me off the highest mountain. You can drown me in the deepest sea. Just don't let my baby die."
She felt a peace deeper than words could express. The ledge she stood on jerked and broke away. As her feet pedaled the air, the surreal sensation of flight stirred her abdomen. Then she hit.
Thick blackness exploded in her ears and muffled the bubbles bursting from her mouth as she screamed and gasped at liquid. Seeing a bolt of lightning dimly, she twisted toward it, breached the surface, and coughed. Thunder roared and water slapped her ears as she struggled to stay afloat. Her toes, no matter how far they stretched, found only water.
_Flooded_ , she thought. _How is the quarry flooded?_ Water stuck its fingers up her nose and she coughed and felt her sinuses burn against the intrusion. How had the storm deposited so much water so quickly?
The river. It ran next to the quarry. The storm must have bloated it until the water broke its boundaries and filled the quarry. She could try to swim across and find a way out, but she felt so heavy and her fingers clenched her tunic spasmodically. With effort and pain, she shook free from the tunic and began to tread across the water, remembering what she had been taught as a child. Spread the arms and kick the legs, breathe deeply and consistently. But with broken ribs and God knows what else, she was soon clutching at the sides of the quarry. Swimming took energy she didn't have and demanded muscles she couldn't utilize. As did climbing out of the quarry. So she clung to the wall and crawled to the right, feeling for the stairwell.
_I am going to die. Like an insect in a trap._
She imagined the Almighty watching her from the top of the quarry, smiling at how easily she had been ensnared. She imagined the look of satisfaction in his eyes, an approving nod as the frigid water slammed her against the wall. Her hands could no longer grasp the stone, and she grunted with the effort of trying to move her arms. It took her a moment to realize she was floating once more, free. Water tickled the bottom of her ears, and then her cheeks, as her eyesight clouded and she felt as if she were being encased in ice.
_How long have I been in the water? I feel like a small child falling into heavy slumber. But I'm not falling asleep, I'm dying. There's a little child inside me who has never inhaled the air my body is screaming for. And that child does not have the strength to carry on without me._
As Sarah sank beneath the water, her hands rested on her abdomen.
_I wonder if the child inside me can feel pain. God, please give my child a painless death. Grant me at least that._
The warmth of her womb had been replaced by shivering skin, and her ribs ached less with each moment. With closed eyes, she saw the Almighty standing on the ledge above, his head nodding. Up and down. Up and—
##
Cain awoke shivering on the ground. He was naked and stones dug into his back. He sat up, brushed them off, and felt the impressions they made in his skin. The sky was dark, though clear and dotted with pricks of light sharp enough to draw blood.
How much time had passed since he gave up consciousness?
He rubbed his hand over his scalp and strained to clear his mind. The silver boy had taken control and pushed Cain's soul into a black empty space. There had been no way out, though it seemed he searched for an eternity. And it had been cold. So very cold.
He palmed his shoulders in an attempt to warm them. The surrounding wasteland gave him the sensation of absolute exposure. There were no trees, no hills, nothing but endless flatland covered in small, sharp rocks. And he had the strange sensation that every noise was an intrusion, as if hidden hands stopped hidden ears to keep the place secret.
"Hello?"
Silence loomed like a shadow, and a cold wind pinched Cain's skin. He looked down and wondered what had happened to his clothes. Then he studied the stars and began walking the direction he thought was east.
Where had the silver boy gone? He didn't feel the itch in his mind, and he didn't hear its voice, but neither did he think the creature dead. It had understood Cain's plans with precision, and yet seemed to have devices of its own. Whatever it planned, he doubted it harbored hopes for his good health.
He breathed into his hands, creating clouds in the air. His skin rose like a reptile's, his fingernails ached, and he no longer felt his feet, which concerned him as he gazed at the stone-cut ravines on his skin that were filling with fresh blood.
Everything had gone well. He had killed Abel, successfully disposed of the body, escaped from the City, and gained the Man's protection.
So why was he naked in the freezing wilderness?
The silver boy had been and remained an anomaly in the landscape of Cain's new life. Cain had felt safe with its fingers around his, and yet here he walked in this wasteland. The solitude began to feel like an anvil on which he had been laid in order to await the hammer strike. _To be fashioned into a tool_ , he thought and grit his teeth. _That is why it has brought me here._
What was the silver boy's relationship to Cain? Why had it come to him? And what was the reason for the unmistakable familiarity between them?
A cough lit a fire in his chest, and he grabbed at the skin of his breast to ease the pain. In the years of their nomadic existence after his parents had been expelled from the Garden, his family wandered far across foreign terrain. Though the times had strained them, the pressure and pain had also strengthened them. _Could it be the silver boy is trying to strengthen me? If it wanted me dead, it would have no reason to wait._
_No. It wants to consume me_. He envisioned the stone in his hand as it crushed Abel's skull. _The only way to gain power is by taking it from another. Like the Serpent in the Garden._
The Almighty's warning returned. _"Sin is crouching at the door. It desires to devour you, but you must rule over it."_
The silver boy had claimed it would break him. Maybe this was its way.
He sat and sifted through all that had happened. He had killed Abel, disposed of the body, and first heard the voice then, though he had felt the itch in his mind long before. How long? He couldn't remember. Several weeks? Maybe more? Then the voice appeared as the silver boy and urged him away, and he left Sarah and—
_Sarah. I wonder if she is thinking of me as I think of her. And how she is coping with the changes I have set into motion._
He lifted a handful of pebbles and let them fall one by one. The world would know soon enough that Abel was dead. Would the people question their God? Would Calebna, the High Priest, lose his faith in the face of his father's murder?
And what of Cain's own children? Lukian, the twin brother of his dead firstborn, would likely seize any opportunity to free the people from bondage. Cain could still see the look on his son's face as he planted questions like seeds.
_"Son, do you believe we have true freedom?"_
_"We have the power to craft our lives."_
_"In the Garden we were given dominion over the world."_
_Lukian nodded._
_"So why did we flee for over a century?"_
_"The Fall."_
_"The Fall was the decision of one man and one woman. What about my freedom, what about yours? Do we not have the choice to regain our dominion? Another question. Where are we?"_
_"The City of the Almighty."_
_Cain nodded, and a smirk crept up the corner of Lukian's mouth._
The stones of the wilderness dug into Cain's thighs and buttocks as he closed his eyes. He had broken the Almighty's chains with a stone-smashed skull, and yet what had he gained? Marks across his skin and blood on his feet. And now he waited for death to come in silver skin.
He tensed his muscles. He would not be controlled. He would not be dominated. If the silver boy demanded all of him, he would embrace death.
But what if he retained a splinter of that control?
_I escaped slavery before. I could do it again._
Cain closed his eyes and rested, letting his mind dip into endless silver waters until something jerked him back. He was sitting on the ground with his legs crossed and his arms in his lap. His fingertips were blue, and his body was numb. His mind told his hands to clench, but they moved as if listening to the commands of another.
He was tired, and he felt detached as he lay and watched his skin blacken and his limbs shake. Little feet approached, and as the gravel crunched by his ear, a small shadow blocked the stars to the south.
_I am dying_ , Cain thought. _Eventually, I will pass into the world beyond. Unless..._
That familiar voice whispered in his ear, _"Let me in. I will give you what you need if you would only give me what I want."_
Cain was repelled by the sound of the voice and the scent of its breath. It was an acrid, yellow stink that crept across the side of his face.
_"You know what this means."_ Its voice dripped with desire. _"You must understand. Not all, but more."_
Cain nodded and whispered, "Come." And he thought to himself, _I promised I would come back for you. And I decided I would never lie again—not to you._ And he thought of how her red hair shone in the sun and danced in the wind and felt between his fingers.
The silver boy's nails dug into Cain's shoulders, and its breath passed over his lips and down his throat. It swam down his limbs and crackled through his head, and though he knew what it was doing was shameful and should never be, he knew that it _was_ , for the shadows of its emotions hovered at the outskirts of his mind.
Cain's skin grayed, and he wanted to observe it closer, but already the silver boy urged him in a new direction. He stood and realized then what he must do. For a time he would suffer the thing to live in him, but when the moment presented itself— _and it must_ —he would crush its head underfoot.
_But for now, I will walk with the Devil._
##
Seth knew his father was wrong. Still, as he made a mental list of all that could not be true, he ran as fast as his stiff legs would allow. As he had lain on the floor of Mother's home and listened to Adam's words, he remembered. And not just the single set of visions that threw him into seizures. He remembered everything.
He looked back through blurred eyes and saw only rain. He had left as soon as he could, and maybe, just maybe, it would save her life. _I cannot accept a world in which belief is only a misty haze simulating substance. I cannot abandon hope._
"I'm running." He gasped for breath. "I'm trying."
He hobbled up the hill and nearly fell in the mud. His lungs burned as he crested and staggered to the double doors of the Temple. Shaking with weakness, he reached, grabbed hold of one of the gold rings, and leaned back, but the door did not move. Panic invigorated his limbs, and he tugged until it opened, and he slipped inside.
He looked around, seeing the Temple's insides grayed with shadow for the first time. Lampstands stood like mute sentinels, their flames extinguished, their figures gleaming in the light that stole through the cracked doorway. The tapestries and paintings depicting the glory of the Almighty were drab and colorless on the walls, and as he made his way forward, the darkness deepened.
Seth fumbled along the wall to find the door to the Throne Room, and as his fingers slid from stone to metal, footsteps pattered in from the rain. He glanced behind him and saw a shadowed figure silhouetted by the shaft of light pressed between the doors. He squeezed the handle, jerked the door open, and slipped inside. But as the door shut behind him, his feet melted.
The room was lit by two candles that burned on either side of the throne. The flames were weak after many hours, but still bright enough to illuminate the throne and what little was in the room. He shook his head and steadied himself against the wall. Then he crumpled to his knees and flattened himself against the floor. "Almighty?"
There was no response.
The door opened and feet shuffled beside him. He found it unusual how in this moment everything was magnified, from the sound of his wife's footfalls, to the subtle intake of breath as she took in their surroundings, and the sting of the dust turned up by her tunic as she fell to the floor beside him.
"Almighty?" His nose ran from the chill of the chasm. If his dreams were only dreams, looking upon the glory of the Almighty could strike him dead.
But he had looked already, hadn't he?
He held his breath. Perhaps a second look would be fatal.
He glanced to the left. To the right.
Where were the colors? Only Calebna had been in the Throne Room, but the stories he told were those of Fear and Reverence. Not the petty kinds of the incarnate world, but the purified oils of the Almighty, refined in the flames of eternal Holiness—gifts from beyond the edge of the world.
_Could Calebna have lied all this time?_
He could bear the unknown no longer. At the very least, what greater death could there be than to die in the presence of one's Lord? He raised his head slowly, painfully, until he could view the throne out of the corner of his eye. Sweat poured from his skin as thoughts swirled through his mind. He could see the angled stone, the flickering shadows from impatient flames, and...
A robe.
"No," he whispered as he let fear and rage burn his face. The fabric lay torn and wrinkled across the throne, and he didn't need to look again to see the Almighty's crown bent and dirtied with black stains.
_The dreams._
He glanced at Ayla who still lay with her face to the floor.
"Don't," she whispered. "Don't, Seth."
He turned and straightened, and Ayla stilled. He stared at the Almighty's empty throne, but no retribution came. Nothing happened, and that nothing was more terrible than any punishment he could have endured.
"I've seen this before," he said. "I've seen it all." He clenched his fingers into fists and swallowed the stone in his throat. Ayla's clothing quivered against the floor as she let out a shaky breath, but still she did not stand.
He touched the sleeve of the Almighty's stained robe. Some of it still glistened, but more had crusted and flaked like rust. A groan escaped his throat. He laid his face in his hands and let the horror flow until he stumbled and fell to his seat. He rested his head on the floor and let tears stream from temple to Temple.
"No." The roar of his voice echoed through the chamber. "He's dead."
He thought of the remaining pieces of his dream and had to shut them out. He would stop it. He would stop it all. They wouldn't die. He wouldn't let it happen.
_But how?_
Ayla's sinuses bubbled as she sniffed. She remained prostrate, perhaps because there was no longer a reason to move, because she believed her husband and dared not look at the proof lest grief crush her.
For who could endure the death of their God? Who could see the ultimate reduced to nothing but lifeless matter? They had put their trust in the Almighty and it had been destroyed. But by what? Had Cain killed the Almighty? Was that why Cain had murdered Abel—to show them they had been duped by a false God?
It was true that only Adam and Eve had walked with God in the Garden. What if they had been fooled? What if their memories had somehow been twisted against them? Or what if they had never seen God at all?
"No," Seth whispered. "No." He scrambled to his feet, grabbed the crown in both hands, and threw it across the room. It clattered and skidded to a stop, and he noticed then an item that hadn't been in his dreams. A goblet on the throne's armrest. He lifted it and liquid sloshed over the rim. At first he thought it wine, but then the smell of blood reached into his awareness. He grimaced and dropped the cup, splashing the redness across the throne. "He lied to us."
Ayla straightened until she knelt. "We should leave."
"He's dead. Our God is dead. It was a façade. A trick."
"Listen to yourself. You would have given your life for him, and now you toss his crown like refuse."
"What does his name mean? He claimed to be the Creator of the world, the Creator of our souls, and yet he is gone and his Temple is dark and his robe is stained with blood."
She struggled to her feet. "He's not dead."
"Then how do you explain what you see?"
"I don't know. Just stop it."
"I already tried. Why do you think I'm here? Hours ago you argued for the impossible, that suffering could find us despite his promises. Now I say you were right, and you can't even bring yourself to look at the proof."
She shook her head.
Rage swelled like a bubble in his head. "You won't take her from me. I won't let it happen."
Her voice shook. "You're scaring me."
The images wouldn't leave his eyes. It would happen exactly as he had been shown.
He remembered a detail, a momentary flash amidst the flood of images, and its meaning quickened his blood. He turned to Ayla, his eyes wide and clear. "We need to get out of here."
"What?"
"Run. Go!"
She turned slowly, her eyes narrowing. He could see she was hurt by the sharpness of his voice, but he didn't care. He wanted to push her, to throw her out the door and get her as far from this place as possible.
A blinding flash lit the room, as if they had been struck by lightning. Ayla screamed, and a high-pitched ringing smashed into Seth's ears, though it sounded like a thunderous Word. And in the space of one fleeting moment, Seth knew that their lives had ended, and that their story had just begun.
##
Mason was dreaming, but the words spoken by a familiar voice seemed to grab hold and pull him halfway to consciousness.
_"She's drowning in the quarry. You must run to the quarry as soon as you wake. If you disbelieve me, she will die. Do you want to carry that weight? Then get up and run. Run!"_
He woke with his feet swinging, then landing, now sprinting. The entrance of his single-room hut was already filled with water, and he splashed through puddles as he whipped out the doorway and into the storm. His matted hair slapped his shoulders like waterlogged snakes, but he knew where to go, even with his eyes closed.
He skidded to a stop at the edge of the thirty-foot drop and searched the water below. He didn't think the quarry would be so full already, but the voice in his dream had told him it would be so. By the time his mind had roused enough to question why he would obey a voice from a dream, he was staring at his mother's head bobbing above the surface of the water, then sinking underneath.
He tore off his tunic, sprinted, and jumped into the quarry. Air and rain buffeted his face as he fell and plunged deep. His legs hit the bottom and jarred, though not hard enough to break, and he swam to the surface, then dove and searched with his hands for Sarah. He felt a body, cold, but real, and grasped it. He breached the surface and thrust her face above the water, but she wasn't breathing, so he swam the rest of the way and tugged her up the ramp.
He lifted her head and struck her back with a flat hand, but she didn't breathe. Instead, she vomited. After the initial shock of it, he realized it was better she vomited, not worse. She coughed, then began gasping. He picked her up as gently as he could, and she hung in his arms, cold and heavy, and her eyelids fluttered closed. She began to shake.
_"Don't stop until you bring her home. You can't bring her anywhere but her home. If you do not bring her home, she will die."_ The voice had been right about her being in the quarry, so why should he distrust it now?
He paused, then sprinted up the staircase, feeling the burn in his thighs as he crested the top and turned for her home. Time blurred in the space between thunderclaps. The sheet over the doorway stuck to them as he pushed through, and he had to angle sideways to slip past with her in his arms. The storm muffled as he walked into the kitchen and set her on the floor. There was a trail of dark fluid twisting out the room and down the hallway.
He looked at his mother and brushed the hair from her face. She was shaking and her eyes were closed, but she was still breathing. He stood and followed the trail to the closet, where he found the source of the blood—a body with a curtain draped over it. He slid the curtain back.
_Lilleth?_ He dipped and pressed his ear to her chest in search of the rhythm of life. There was none.
His skin tingled as he returned to his mother. Had she done this? Had that been the reason she had come to the quarry? His mouth was dry and he tried to swallow, but it stuck in his throat.
_"Bury the evidence_ ," the voice said, _"and make sure no one can find it. Do it or your mother will die."_
So many questions Mason could not answer. But he would not let his mother die.
##
Eve was in the lounge when Abel's firstborn, Calebna, entered. "The crops are shriveling." Calebna was breathless, as if he had nearly drowned in the water that dripped from his long hair.
"What do you mean?" Eve's voice was harsh, and Adam stiffened. She could feel his fingers tighten around her hand. She remembered the pain of Abel's entrance into the world, and how her hands had dug into Adam's, but Abel's murder overshadowed any pain she had yet felt. And Adam?
"I mean that everything green and growing in the City is decaying," Calebna said. "And our fruit is rotting. Some blight is taking it."
"What about the fish? Our animals? Are they alive?"
"Most. But many show signs of sickness. Some have lumps across their bodies. Others have been consumed." He paused. "There is more."
"Speak quickly, Son of my son," Adam said.
"Mason found Sarah in the storm. It seems he risked his life to retrieve her from the water that destroyed the southern half of the City. Gorban informed me only moments ago that all attempts to wake her have failed. Mason took Sarah to the Temple, seemingly in hopes that the Almighty would heal her..."
"What? What happened? Speak!"
Calebna jumped as if startled. Eve could see that he was shaking. "The Almighty is gone. I would not believe it until I saw it with my eyes. The Temple is dark, and when we entered, we found the torn and bloodied remnants of his garment on the throne, and beside it were Seth and Ayla's prone bodies. I am sorry. They are dead as well."
Eve whispered, "Dead?"
Adam toppled a table and shouted. Eve studied the desolation in Adam's eyes as Calebna brushed the wrinkles from his tunic and looked anywhere but at Adam.
"I am sorry," Calebna said again.
Adam slid to the floor and held his face. He was weeping, but out of anger rather than sorrow, Eve thought. A chill clambered up her back as she remembered the prophecy the Almighty offered them in Eden—that through one of Eve's sons would come a reversal of the curse. When Cain had been born all those years ago in the cave, she had thought him the Savior the Almighty spoke of, and perhaps Adam had thought Abel the same. But Cain had proven himself born of evil seed, and both her other sons were dead.
_That is why Adam weeps_ , she thought. _He knows now there is no hope, for not only are our children either dead or false, but the Almighty himself has abandoned us. And can a promise be any truer than the character of the one who promised?_
She turned toward her grandson. "Show me."
Calebna led the mother of all mankind under the last vestiges of the storm to see the bodies of those she loved too deeply to express. Somehow she remained upright. It didn't affect her like it had Adam. Like an underground lake, her soul lay quiet in darkness. Though the tears didn't flow, she remained filled by them, and that was perhaps the most painful part.
Calebna's brothers, Philo and Tuor, arrived as Eve clutched the Almighty's torn and bloodied tunic. Together, the men hauled Seth and Ayla's bodies out of the Throne Room and into the antechamber, where they laid beautiful tapestries over their pale forms.
Eve exited the Throne Room, saw the bodies, and ran her fingertips over their covered faces. Though she knelt staring for what seemed hours, no one spoke. They merely watched, as if waiting for her to determine their direction. How should they react? Surely there must be some explanation?
But there wasn't. So she stood, turned her back on the Temple, and walked home. When she returned, Adam sat with his back against the wall, and his eyes held some goal in the farthest distance. When she spoke, he would not respond, and Eve understood what she had seen in his eyes.
_Can one die inside and yet remain living?_
_Yes_ , she thought. _And it seems Adam has._
## PART FOUR:
## INTO THE HEART OF DARKNESS
_These are waterless springs and mists driven by a storm. For them the gloom of utter darkness has been reserved. For, speaking loud boasts of folly, they entice by sensual passions of the flesh those who are barely escaping from those who live in error. They promise them freedom, but they themselves are slaves of corruption. For whatever overcomes a person, to that he is enslaved._
—2 PETER 2:17–19 ESV
##
It had started slowly. For many days Cain simply traveled. The sounds of the world became amplified in the absence of human speech, and the itch in his mind disappeared beneath the roar of nature. For what seemed an eternity, all that remained was earth and sky, the breath of lungless leaves, and the quiver of wind over water. _Music._ The whole of Time seemed contained in an eternal Song, and he joined in its exaltation.
Then all had flickered and pieces of the Music fell bit by bit into little pockets of blindness. Eventually he grew accustomed to the strangeness, but the pockets widened until a great nothing rolled out, swallowing everything and turning back upon him until only the void remained.
"In the empty chill, I finally know the extent of the bondage I welcomed. Slavery tastes bitter, and I would fight the gates of hell to stop it from burning your beautiful lips." He had taken to speaking to Sarah in the void. "A habit born out of necessity," he explained, so she wouldn't think him odd. He imagined her nodding in response, half believing, half mocking. "Yes," he said to her. "I know you think it strange, but it is true. I need to talk to you."
She pursed her lips and said, _"Whatever helps._ "
"I've wondered if I lost myself long ago and imagined all this as I'm imagining you. If maybe killing my brother simply broke me, or if maybe I imagined murdering him, and that all this is only a dark dream."
_"You know this is not a dream_ ," she said. _"And you don't look broken."_
"But I feel it." He thought of the silver boy forcing his body along. It seemed like many years since last touching his soul to his body. He grimaced. _My body. Not that silver devil's. Desire grows and aches in my chest. I long to feel the dirt beneath my feet and the breeze on my cheek, or even the sensation of pain._
He reached through his memory for any vivid sensation and grasped the musty smell of sweat. He held onto it like a broken rose.
_"Look at you_ ," she chided. _"You should be ashamed. You even think that dwelling on the smell of your dirty body is an accomplishment."_
He stared at his gray hands in the void. Slowly, he nodded. "I am ashamed. It's good you're still at the City and have no idea what I've become."
_"So stop it._ "
"What could I do?"
_"You chose this path._ "
"I killed Abel because I had to. I tried to find another way, but you pushed me to it."
_"Still fighting the truth?_ "
"You have no understanding of the truth I've lived."
Sarah's flamed hair seemed to crackle. _"And what of me?_ "
"I'm not saying you haven't suffered."
She flicked her wrist as if to dismiss him.
Cain pressed his palms against the sides of his head. "Can't you see that I need you?"
_"I'm not your toy, a child's rag doll to be soiled and discarded until next I fit your fancy."_
He bowed his head and opened his mouth to respond, but the words stumbled over each other.
_"Speak."_
Angry phrases flittered through his mind like rats, but he pushed them aside and said, "I am sorry."
She scoffed. _"This must be the first time in your life."_ But the words lost their edge.
He pressed his eyes and felt the anger slip away. "I wish I could explain what kept me from expressing it. I wish I could understand why I couldn't kiss you or slip my fingers into yours. I simply couldn't. But I wanted to. I still want to. If you were here, I would hold you tenderly. Remember before I left? Remember how I held you?"
She nodded and held her expression. _"I remember more than that."_
He bit his cheek, remembering how he had struck her and broke their home and yelled at her as she collapsed and wept on the floor. He stared at her, sensing again the invisible wall between them. Even in his imagination he could not do what he most wanted to do.
"Do you miss me?" he asked.
She rubbed her eyes.
"Does anyone miss me?"
She nodded and a smile creased her lips, then crumbled. She turned and her image dissipated like fog in the wind. He searched the darkness, but found nothing. The emotions came like aftershocks to an unfelt quake, and in the safety of solitude, for the first time since childhood, he wept. He was no longer a man. He was a boy finally realizing that for nearly a hundred and fifty years, nothing had been enough.
_I just want to satisfy the restlessness. Everything feels wrong, as if pieces of the world have been fabricated and replaced without me knowing. What is this longing? This thirst?_
The blackness crept so close it felt as if it were inside him, and he rubbed his hands over his arms, but generated no warmth. Faintly he remembered the smell of sweat, and he longed for it again.
The silver boy's words returned. _"Not all, but more."_
Cain felt as if he were awakening from a long dream. He hadn't given all of himself to the silver boy. He hadn't offered complete control, and if he had overcome it for a moment, why could he not push it away for an extended period of time? He thought of how long he had been confined in this capsule of nothing, and the possibility that he could have been lulled here brought flames to his neck. The pockets of blindness, had they been the beginnings of the void around his soul as the silver boy displaced him and took over his body?
"Let me out!"
_"Would you quiet yourself?"_
He turned and saw Sarah wincing and covering her ears.
His mouth twitched. "I thought you left."
_"I came back."_
He shook his head, questioning if he really were seeing things. "Why?"
She sighed and folded her arms as if wondering whether to speak her mind.
"Tell me."
_"You're a fool. You want out? Then leave._ "
"You're saying I can?"
_"The world needs you. Our sons—Lukian, Gorban, Mason, Kiile, and Machael. They need you."_ She turned and walked away.
"Wait."
_"You promised. Or don't you remember?"_ She waited, then nodded. _"You always were a liar."_
"How do I escape?"
She placed a hand on her abdomen and stared at it, then smiled and whispered, _"Just wait, little child. I'll take care of you. And he'll never touch you."_
His eyes widened.
She turned and faded into the dark.
"Wait!"
But part of him thought she wasn't gone. That she'd never left him at all. He breathed hard and his chilled hands shook. Suddenly, as if lifted by some unseen force, what he could only describe as _layers_ arose like stacked sheets of papyrus. He didn't so much see them as feel them through some unknown sense, piled where Sarah had stood. He ran his hand across his scalp and rubbed his eyes. Was Sarah showing him something? Was she leading him toward these layers? Or had he truly lost his mind?
He approached the stacked sheets, stuck his fingers between two, and pried them apart. Dim Light shot out from between them, and as he bent and leaned in, he saw moving shapes inked onto the fabric. It was like looking at the whole of the world caught in flatness.
He let go and the sheets snapped into place. He repeated the process lower down. Then again. And again. He found worlds stacked endlessly upon each other, and each wove into the next like the Words of a great Song. His mind tried to convince him that this was only his imagination, but he knew that was a lie. As he fingered through the layers, he sensed the echoes of truth fluttering through its pages, like the softest Music ever played. He let a few sheets fall, and the sound of nearness, of belonging, grew. He gazed between the pair of sheets and saw himself standing in blackness, gazing between the layers. He moved his leg and watched the flat image mirror the movement. His hands slipped and the layers slammed shut.
_This is impossible._
_"It's a gift."_
He spun at the sound of Sarah's voice, but he could not find her.
_"You should be thankful._ "
"Do you want me to use it?"
_"Don't play the fool."_ She laughed then, as he hadn't heard her laugh in years, and the echo resonated until it disappeared behind the ringing in his ears.
Cain looked at the layers. Where had they come from? The silver boy must have awoken something in him. Or had the catalyst been something else?
He bit his cheek as another thought struck him. _Do the layers end?_ He rifled past innumerable sheets. Eventually he reached a layer beneath which was endless blackness. He paused at the brink and studied the space. The blackness seemed alive. It moved like a roiling hot spring, and yet he felt the empty chill of it.
Cain swallowed, forced his shoulders to relax, and turned back through the layers until he found his way to another brink on the opposite end. This one was not so dark, and he found himself standing at the edge, poised to jump. It seemed like a river that flowed through the layers, but though there was illumination of some kind, he saw not through his eyes, but sensed through that same unknown faculty.
With a terrible, half-purposeful motion, he plunged into the depths, and the contact of what he could only describe as Water made him gasp. He swam its currents, explored its tributaries, and found his mind bursting with significance.
In the Waters lay shivering images. He paused and cupped his hands to catch an image of his face, and he watched himself speak, and even heard the words spoken.
His skin tingled as he let the Water fall from his hands. He cupped another image, this one of Lukian pacing the inner wall of the City.
_I am watching life._
The idea made his mind buzz as he cupped another image.
_But some of what I see is unfamiliar. Might these images be moments that are yet to come?_
He trudged ahead, cupping Water and watching unlived life unfold. The most interesting areas were where the rivers branched, showing twisted projections of choice. Soon he found himself swimming multiple branches and coming to darker Waters.
He knew the ability to traverse these Waters stemmed from his interaction with the silver boy, but he also knew there was something more. His thoughts resonated with the river, and from somewhere beyond the river came whispers of truth and falsehood. Those whispers confirmed that there was something special about the silver boy, and about his own relationship to it. But what?
From time to time, atop a wave, he sensed multiple pathways ahead, splitting in ever-burgeoning potentialities, and the endlessness of it was overwhelming. The riverscape grew chaotic. The waves became violent and thrashed him about. Ahead lay a harbor whose surface was glassy-calm, and he swam for it. He reached and strained, the current resisting him with ever-increasing strength. Still he grew closer.
When at last he crested a wave and felt the harbor within reach, a black force welled up and threw him through the darkness into the light.
##
The first thing Cain noticed was the smell of grass. Blades stabbed his eyes, and he turned his head, grimaced, and spat into a blurry jungle. The back of his neck tensed, lifting his skull enough to see he was lying in a field. Granules of dirt ground between his teeth as he pushed himself onto his knees. He tried to straighten, but instead tipped and shot his hands out for balance. The world pulsed and rolled. He itched his face and peeled the grass from his skin. His face tingled and felt like leather, and he caught a string of saliva stretching from his lips.
He grabbed his shoulder and squeezed until it hurt. "I'm back," he said and closed his eyes. "I'm alive."
Clouds flecked the sky and a late afternoon breeze combed through his hair. He knew in only a few moments the sun would fall beneath the horizon, but with the wind on his skin and the smell of autumn and musty earth filling his awareness, he had never felt more human.
As twilight chased the red away, the skin of his arms reflected the light in metallic hues, and the vertebral marks hooking down his arms glowed silver. Little footsteps approached. He turned his arm to one side, then the other, observing the visible humps the marks produced. Not needing to look to know it was the silver boy.
The footsteps stopped and Cain turned. Its eyes watched him unblinking. Black holes within silver rims. And its skin—insipid gray littered with veins like black netting. He grimaced.
_"You must be thirsty_ ," it said as Cain absently grabbed at the skin of his stomach. He wondered if it knew where he had just been, if it had somehow awoken him on purpose to keep him from reaching that harbor. He attempted to swallow, but his tongue was dry like coal and stuck painfully in his throat. He coughed to stifle a gag, knelt and clawed at the ground, moist with day-old rainfall. The silver boy crouched and watched him dig until he created a small pool of brackish water. He dipped and sucked up the liquid, keeping his eyes trained on the silver boy. It didn't move. When he finished, he stood and held out his arms. They were quaking.
Warm fluid filled his teeth, and he let it fall from cracked lips. He stared at the colored goo dripping in long strings to the ground.
_Why am I bleeding?_
The silver boy said, _"You cannot give back my gift."_
Cain straightened and rubbed a hand across the vertebral marks, disturbed less by the texture than by the lack of warmth. He looked across the plains and spit more blood on the ground. He had no notion of where he was. Everything, even the stars above, was unfamiliar.
_"I gave you what you wanted_ ," it said. _"You think you had no choice? You had a lifetime."_
"I chose none of this."
_"You prefer death?"_
Cain turned away, hooked his fingernail around one of the vertebrae, and pulled, stretching his skin with it.
_"That you will die is more certain than your next thought. Do you crave release from that final tyrant too? Because I could set you free."_ It approached and scraped its sharp fingernails down the back of Cain's hand.
He pushed it away and stepped sideways, accidentally slipping his foot across the red spittle on the ground. He felt faint.
_"How does it taste?"_
Cain's eyes narrowed. "What do you mean?"
_"The blood. How does it taste?"_
Saliva pooled against the coppery flavor of blood between his teeth. He tried to swallow, but it poured into his mouth, flooding his tongue. His sight undulated like a ship on rough waters, and he stumbled to his seat. He was weak and thirsty. Thirsty for...
"Blood?"
##
Cain wondered at the size and shape of the growth sprouting from the ground and shooting through the cloud cover. It was a Tree, but seemed more stonelike than alive, and its canopy disappeared behind cloud cover.
"What is it?"
_"A means of travel._ "
"From where to where?"
_"From this world to the next and to those beyond."_
Cain looked up and squinted, searching once more for the top of the Tree, but the silver boy laughed. _"It has no end. It is beyond your ability to understand."_
Cain touched the trunk, and his skin tingled with an unfamiliar sensation. He looked at the hazy Fog they passed through to arrive here. He had seen eyes like burning candles in the mist, but nothing had approached, and now there was only the sparkle of sunlight through condensation.
"Could another find this without your help?"
_"No man has seen this Tree, and perhaps none will again."_
Cain knocked his knuckles against it and recoiled. Blood oozed from his knuckles in beads. He smeared the blood away, then stared at the wounds, stifling the urge to lick them.
"Are we going to use it?"
_"Why else would I bring you here?"_
Of course there were reasons, but he needn't point them out. The silver boy utilized manipulation in a way Cain both respected and hated. "You couldn't prod me along fast enough. Why pause now?"
_"You are exhausted. A moment of rest."_
Cain shook his head. _It intends to impress upon me my reliance, but I am no fool. I see its desire to convince me for what it is. The truly dominant need not claim authority._ "I am ready."
_"You will not return unchanged._ "
"So be it."
There was a pause. _"Give yourself to me, and I will bring us through the Tree to what lies beyond."_
He was repulsed by the prospect of giving up control. His neck was stiff and warm, and his vision narrowed. _If I give up control, it may try to take more. But if I do not..._
There was danger in taking such a risk, but more than anything Cain needed power, and he risked danger most by hesitating. _Sometimes to gain, one must also give._
He let his soul retreat just far enough to relinquish a sliver of control. The cold fingers of the void clamped around his arms and pulled him into liquid darkness. Here he waited. And waited. There was a sound, as of a pulsing heartbeat, but it seemed too rhythmic to be organic, and the sound increased until he felt it branch and increase in complexity. Just as he began piecing it together, he was expelled from the void and into his body, and all five senses burned.
There was newness in his surroundings, in the smell of mold and taste of damp minerals. He tried to get up but was too weak. His ears rang and his sight was dark, but he sensed his environment imbued with a bluish hue. Slowly his eyes adjusted and he peered about. Above and arcing around was endless gray rock.
_A cave?_
Some of the stones glowed and differed from the others with peculiar marks like veins on a leaf. He stared at them, but suddenly everything seemed to glow—or was it merely a reflection?
His mind itched and buzzed. The silver boy was with him again. "Where were you?" Cain said.
_"Your soul is simple and light. It moves quicker than I through the Trees."_
Trees. So there were more of them. The thought made his mind spin. How much more of the world had they been blinded to by the walls of the Almighty? He struggled to his feet. "Movement seems more difficult here." There was no response. "This is your home, isn't it?"
_"You presume much and are wrong._ "
"Then where are we? I feel corporeal, while all else seems somehow elevated."
_"We are where no man has ever been, nor perhaps shall be again."_ Its voice softened as if in reverence. _"The City of the Light Bringer."_
He examined the glowing rocks and breathed the newness as another thought struck him. "You could not have arrived here without me, could you? You seem to take form, but it's no more real than the images in my mind."
The silver boy said, _"I hear the call. It is loud. Do you hear it? Of course not. There are many things you do not hear. Many things you cannot sense or understand."_
"Who is the Light Bringer? What does he want from me?"
_"You will know soon enough. He has summoned us._ "
"You fear him."
_"The child fears the father. The father fears failure. The strongest fears himself. The weakest fears everything. What do you fear?"_
He suppressed the burning urge to swallow.
_"I read the space between every word and understand its source."_
For days Cain had contemplated the possibility that the silver boy could sense his thoughts, and now that fear returned. _If anything, it sees only the footprints of where my mind has been, and footprints may be erased. Surely it cannot read my thoughts._
_"Before this is over_ ," the silver boy said, _"I will teach you subtlety and pain, and perhaps a little more."_
The silver boy yearned for control of Cain's body, but Cain suppressed it and walked down the corridor. The road ahead angled ever downward, and the path was several feet narrow, but the ceiling remained at least twenty or thirty feet above, and the glow of the stones, if indeed the light came from them, kept the illumination dim but satisfactory. It seemed they walked for hours, and Cain grew weary. The fluid in his belly seemed to have evaporated, and the pale thirst rushed against his insides like a moonlit tide. Soon he would need to face the change the silver boy had brought to him—to understand the true costs.
_Soon, but not yet._ He remembered Sarah on the floor of their home and lost himself to the crumpled shape of her lit by lightning and washed by water. She shook with sobs and slowly lifted her gaze to his.
_Does she see me as I was or as I am?_
Cain shook his head and realized the madness in his thoughts as he stumbled on loose rocks and slid down a ditch. He stood, brushed the gravel from his tunic, and lifted himself onto a ledge. Beyond the ledge was a thin path that led down a tunnel toward light. Finally the hall opened, and he steadied himself against the wall as he peered through the opening at what lay beyond.
"Great Almighty," he whispered.
The domed chamber was perhaps a mile high, and suspended in the center was a light whose brilliance rivaled the sun's. Though it flickered like a flame, it lit the deepest ends of the dome, and cast long shadows from the buildings and from what looked like people bustling between the buildings in a sea of movement and sound.
_A world entombed in stone_.
_"It is beautiful_ ," the silver boy said.
"There must be thousands of them."
_"More, son of Adam. The City of the Light Bringer is more vast than you could imagine. This is only one Dome. There are thousands more."_
If the City of the Light Bringer was truly so large, of what interest could Cain and his family be? It seemed a strange thing, but still he felt the Light Bringer's call.
_Not all is as it seems._
He felt the rock wall, cold of the variety that sucked life from bone. "Why did you leave this place?"
_"I have never been."_
Cain quieted himself, but uneasiness churned his stomach. The City seemed a great hub, and the knowledge the silver boy possessed spoke contrarily.
_It could not have come here without me, but how do I know that? And how does it know the City's name, and the expanse of its construction? From what hole did the silver boy crawl?_
It would not tell Cain of its origins, no matter how forcefully he questioned. He remembered the prophetic streams and wondered why the silver boy's origins, of all mysteries, remained out of reach.
_Perhaps because it is a point sharpened to a tip so thin that it remains invisible to all but the one who stands upon it._
He descended and waded through the mob like a drop of oil in a murky pond. Nudging into and flowing past him were what looked like people of varied shapes and colors. Nevertheless, they were no more people than the voice was a silver boy, and Cain felt as if he were the center of attention, sticking out like a splinter, though none addressed him.
The buildings were beautiful but cold. Everywhere he went, the pale light throbbed from the stones, hummed off the towers, and buzzed the soles of his feet. Someone screamed in a strange tongue and exchanged blows with another. A group danced and sung to Music played on unfamiliar instruments, and everywhere there was chatter, and the sound of countless feet scuffing dust from stone.
An amplified voice rumbled the Dome, and every mouth closed, and every ear turned in respect. The sudden halt startled him, and he stilled. All around were closed eyes and bowed heads, and Cain would have imitated them if not for the fear pinching the nape of his neck. The last echoes faded and the cacophony resumed. Soon he found himself pressed against the wall by a crowd of bodies.
"What was that?"
_"It declared a shift of some sort._ "
"Of some sort?"
_"Keep moving._ "
"Answer me."
_"I was born for more than to ease your curiosity."_
"I don't care what you were born for."
_"Someday you will wish it was all you had cared about._ "
"In this moment there is only you and me in this City. Remember that you would not have arrived here if not for me." It did not respond, and he felt a sort of satisfaction in this, though soon he wondered if its silence stemmed from ignorance or hidden knowledge.
Now Cain could physically feel the Light Bringer's pull. It was as if a magnet were set by his head and, as he went off course, it tipped him back. Time and again this happened. Eventually his legs tired and his body ached with the monotony of the journey. All numbed and fell away like stains washed in a river, and he slipped into what he could only explain as a waking dream. He was unsure how long this lasted, but remembered waking with a start against the wall at the bottom of a stairwell. He looked up the angled shaft and realized he must have descended beneath the metropolis.
What he at first thought were people now filed through stone archways ahead one at a time. He stumbled forward, obeying the magnetic compulsion to follow, though he tripped and landed on the bony shoulders of the individual before him. Before he could right himself, a fist smashed his forehead, and he staggered and blinked away motes of light. After being spat on, he kept his distance and waited for the line to move.
A pair descended the stairs and fell into place behind him. They gargled unintelligible words, then chuckled. He glanced back and saw them smiling. They looked androgynous. Cain's face flushed and he looked away. They chuckled again, and his neck crawled.
After filing through the archway, he proceeded down the tunnel until it ended with dark holes on the left and right. No light stones led the way down those paths, where the ground was hollowed out.
Wind gushed out of the hole on the right, and a metal cylinder slid into view and stopped. It was set on its side and marked with doors that slid open and allowed groups to exit before those that had waited with him entered. He followed the compulsion into the cylinder and found a seat at the end. The others took seats far from him, the doors shut, and the cylinder sped down the tunnel into darkness.
Throughout the journey, the cylinder stopped and some exited while others took their place, but he never felt the impulse to move. Three stops down, the pair that molested him with their eyes exited, and exhaustion came like an innocent caress. Indeed, he was so fatigued that he thought he wouldn't have been able to move if commanded. After the last of them left, the hum of the cylinder lulled him to sleep for the first time in many days.
It was a deep and dreamless repose.
##
Cain awoke with a gasp, still sitting on the bench in the cylinder. He was alone, and the container was illuminated by pale light filtered through cracks in the doors. Dust flecks floated through the beams, and the scent of aluminum and sweat clung to the enclosed atmosphere. He rubbed his face with a sticky palm and stood. The machine creaked and the doors opened when he neared them. He hesitated, temporarily blinded by the light, then exited the cylinder to a well-lit corridor identical to the last, only empty.
"Hello?"
He walked a few slapping steps forward and stopped, listening to the rush of his blood, a soft but steady rhythm. He tasted the pale thirst and momentarily wished he had a red-filled cup to quench it. He ran a hand through his hair, smothered the desire, and walked on.
_"Few come this way_ ," said the silver boy.
Cain looked at the arches. "Where are we?"
_"The central Dome of the Light Bringer. Only those summoned dare come so far."_
Cain walked past the arches, pausing only to run his hand across their slimy texture. He ascended the staircase. As he reached the top, he paused. This Dome was not the same busy metropolis, filled with beings and buildings. It was a barren wasteland stretching from end to end and punctured by a singular Tower dwarfed amidst the grandeur of the Dome. He walked toward the Tower and the ground beneath him sped away strangely. It seemed to him that Time itself was rolling beneath him, and he skipped from crest to crest. Finally the undulation stilled, and the Tower loomed above him.
The door in the Tower's side opened like that of the moving cylinder, but its shadows were repulsive, and light failed to enter its mouth. It impaled the ground, a beautiful and archaic thing, both severe and terrible, and fear held him motionless.
But what was fear but an illusion? Cain was chained to the silver boy and had set down a path he could not escape, walking the land of the Light Bringer, whose power made the silver spirit inside him quake. Every instinct screamed for him to run from this place, and yet he knew how misleading instinct could be.
_The rabbit follows impulse and becomes prey. The human outwits the trapper and gains dominance._
_"What are you waiting for?"_ the silver boy asked.
"You offered me a moment earlier. I will take that moment now."
He knew that to walk into this Tower would be to pass into a great nexus of Time. Every detail of the future, the fate of his grandchildren, even his great-grandchildren and those beyond, ran through this singular point. It was not simply another choice, it was a _defining_ choice.
He had realized even when planning the deed that killing his brother had set him down a path into the core of the forbidden, and now he found himself at the entrance to one of its arteries. If he turned back, Abel's murder would become nothing more than an experiment in futility.
Cain was reminded of the incredible power of the present. Every choice crafted truth from clay, and though days earlier he would have welcomed this moment, now, with the Waters of prophetic truth rinsing him of ignorance, he saw what violence could arise from such a seemingly benign choice.
_Stray but a little and the world may shatter. But if I resist..._
_No. I cannot choose that way._
To move forward was to embrace chaos. No man knew what possibilities lurked within that chaos, but only with movement could the pieces align. Failure was the true Sin, and as the Almighty claimed, it would consume him. Mankind had embraced chaos before, and it had brought them to a state of heightened awareness. The Almighty banished them from the Garden out of fear, for he knew that mankind in such a state could accomplish _anything_.
Clarity with immortality. The recipe for Godhood.
_The human lays traps for the animal, but God ensnares mankind in the ambition to be as he. Power. It is what separates man from God._
This was his only chance to move forward, to embrace innovation, to seek and find what lay hidden in the dark recesses of the world and worlds, to reclaim the godhood mankind inherited in the Garden by eating the fruit of the Tree of Knowledge of Good and Evil.
_"For God knows that if you eat of it, your eyes will be opened, and you will be like him."_
It was fate. Destiny. _Purpose_.
He looked at the vertebral marks burned into him by the Almighty, and raised his fist to the ceiling, accepting for the first time the meaning those physical manifestations contained. His throat constricted and his fist shook as he screamed past the Dome to whatever lay beyond.
"I am." He shook his fist, "I am. And nothing and no one can stop me!"
He turned, let his fist fall, filled his lungs, allowed the fear to pass through him, and plunged into the Tower's throat.
##
Machines growled and whirred as the sensation of movement spun Cain's mind like a falling maple seed. Down he went into the bowels of the Tower to meet the Light Bringer. He could see nothing, for there was only darkness, and as he groped and attempted to maintain balance, his fingers waved through bone-chilling air.
He felt as he had when left in the wilderness by the silver boy. He was once again powerless, surrounded by empty nothing. The hum of grinding metal—or was it so many hammers striking anvils?—cycled with steady rhythm, and the deeper he went, the more lifeless and hollow the space became, until at last he shivered and brought his extremities in to conserve heat. Frost formed on his nose, and he closed his eyes and hugged his knees.
His insides sank as the descent slowed and everything stilled. No more gears, only the sound of Cain's uneven breathing. His body shook.
There was the sound of sliding stone, and a sliver of light stabbed the darkness ahead. He jumped and ran toward it, shoving his hand into the ray to feel warmth. The sliver widened, became a beam, then an open doorway.
The light melted the frost from his skin as he entered an immaculate hall decorated with tapestries, lampstands, and art. The way was wide enough for perhaps twenty or so men to walk abreast, and symmetrically placed pillars held the ceiling high. Into the pillars was carved the likeness of a single face that was sensuous and drew him until he smiled unknowingly.
At the end stood golden doors into which was cast the focal point, a work of art—without equal—portraying thousands of beings with weapons raised. They marched toward a great City, and at their front stood a figure lifting a brass horn from which light volleyed in widening arrows.
He brushed his fingers across it. _They march to war?_ He traced the leader's arm to the instrument he held. _A musician._
He studied the image, then knocked thrice.
A basso voice rumbled from beyond the doors. "Enter."
Cain obeyed.
The chamber was not as he expected. The vanity of the hallway and the overwhelming size of the City had primed him for indulgence, but the door opened to a dull gray box whose only ornamentation was one stone throne, two red lamps on the throne's armrests, and a towering mirror on the far wall, lit by burning braziers. In the throne sat who Cain guessed was the Light Bringer, for the face was similar to the busts in the hallway and the musician in the mural.
"You are drawn to me." His voice was sweet and smooth, yet powerful.
Cain wondered if the Light Bringer could see him shaking.
"You are free to speak in my presence." A pause. "You have no reason to distrust me, though fear is an appropriate reaction." The Light Bringer sipped from a goblet and set it next to the lamp that had obscured it. The drink stained his lips a dark crimson, and Cain felt the pale thirst return.
"You are a strong man." The Light Bringer smiled with tainted teeth, then slid out of his seat and approached until close enough to touch. "You and I are cast from similar molds."
The Light Bringer was roughly the same size as Cain, but Cain felt overwhelmed by his presence. "In what way?"
"You want freedom." The Light Bringer's voice lowered enough to shake the room. "And I brought you here to show you that I _am_."
"You are what?"
He spread his arms. "What do you see?"
"Stone walls."
"And what do they represent?"
"A prison?"
The Light Bringer laughed, but it was cold. "Try again. Where are you?"
The silver boy tried seizing control, but Cain suppressed it and said, "Your City."
_"My_ City. And how did I make this _my_ City? Well, son of Adam, that's precisely why you're here, isn't it?"
"I'm here because I was led here."
"Liar. You're here because you chose to come. The moment you decided to beat your brother's brains into the dust, you put yourself on the road that led only here."
The corner of Cain's mouth twitched toward the floor.
"Ah, you see truth. You desire release from stupidity, and the truth will set you free, will it not? That is why you came to me. But let us speak plainly. We need each other."
Cain gazed at his hands, then cocked his head. "Why do you call yourself the Light Bringer?"
"They call me the Light Bringer because I bring illumination. I expose what others hide. I give fruit to those who hunger. I give insight when silence reigns. I give pleasure when chastity binds. I give freedom past darkness. I am the Light Bringer, the Morning Star, the Son of the Morning, the Day Lark." The Light Bringer's words fell into a melodious rhythm, and Cain felt himself compelled by the Music. The notes and tension of his voice stirred the heart. "I am the Great Musician, the Voice of the Damned, the Friend of the Broken, God's Helping Hand. I do what he needs done, though I'm despised as a rebel son. Listen!" The final word struck Cain's ears with the force of a hammer blow. His body shook as the air itself seemed to fall silent before the Light Bringer's eyes.
"Come," he commanded, reverting to his rumbling bass. "Show me what's inside you."
Cain stepped back, but the Light Bringer approached and grabbed his chin with a soft hand, and the sensation struck him like open palms to the chest.
"Let it out."
Cain felt his awareness recede and realized he was voluntarily giving up control. The silver boy came forward, and the Light Bringer moaned with satisfaction.
"Yes," he whispered. "You are beautiful. Everything I ever dreamed. My beloved child..."
Tears seeped from Cain's eyes. He felt as if the words _beloved child_ had been directed toward him, but he was the rightful first-born of Adam, not the product of this lord of spirits. Could it be, perhaps, that this being was their Creator? His skin tingled at the mere thought of it.
He jerked from the Light Bringer's hold and realized the Light Bringer had been speaking to the silver boy, and not he, and that it was the resonance of the silver boy's emotions that brought the tears.
_So_ , _that is who you are. And that is why you wouldn't tell me your name or where you came from. You were waiting to be given one by your father, the Light Bringer._
The silver boy retreated to the void, and he sensed satisfaction spilling from the darkness. He swallowed the saliva that filled his mouth, wiped the tears the thing had forced from him, and stepped back to reestablish dignity and distance between him and the Light Bringer.
But something had happened in the few moments since entering the Light Bringer's chamber. He felt strange intimacy with this being whom he had never met. He knew they understood one another. But there was more. With a crackling sensation rippling through his mind, he realized their relationship _was_ the nexus, the point of convergence where the fate of the world would be decided for millennia to come.
Cain, the Sinner, was in this moment the most powerful man in the world. He felt this fact resonate with the Waters. His newly awakened prophetic sight gave him the right to make this claim devoid of arrogance, for simple truth may be believed without pretense, however much of it involves one's self.
The Light Bringer slid his fingers under the goblet resting on the arm of the throne and shifted back toward Cain. After swirling its contents, the Light Bringer offered him a drink. "Come now, you must be thirsty."
Cain looked at the liquid and, against his better judgment, allowed the scent of blood to reach into his awareness. His heartbeat sped as he felt his muscles twitch and burn. Saliva pooled in his mouth, and his tongue longed to break the surface of the drink. His fingers chilled and shook.
"You desire. I see it in your eyes." The Light Bringer's voice was smiling, but Cain could not look from the ocean in the goblet. Brackish coagulates churned hypnotically, and as he stared, the Light Bringer swirled it again. "As fresh as if you sucked it from a wound."
Sweat beaded on Cain's face.
The silver boy said, _"Take it. Accept what you have become."_
His mind burned for him to drink. _No_. _I need not accept this curse._
The Light Bringer said, "You cannot escape what you are."
With great restraint, Cain looked from the drink to the Light Bringer's eyes. He thought about slapping the cup away. About beating that smiling face into a grimace and grabbing a stone, as he had in the darkness of the valley, to spill red yolk from skull to goblet.
_"Feed the lust. Let it grow and mature until you must vomit it forth."_
Cain reached, gripped the goblet in a shaking hand, and brought the cool lip to his mouth. Blood poured into his mouth and filled his tongue with pleasurable flavors. His body lurched, and he tipped the goblet until it spilled down his neck and chest.
He gasped as the goblet clanged on the floor.
"Beautiful," the Light Bringer said. "You were thirstier than I thought. Well done." He laughed. "Well done."
Cain ground the bitter clots between his teeth and felt them burst. Tears blurred his vision and his hands clenched at his side. _Sarah, I won't forget my promise. I will return to you, even if I must bear the ultimate shame to do so._
The Light Bringer frowned. "Young child, young gift. Is a worker embarrassed when his master says, 'Well done'? Is a son ashamed when his father says, 'You are my pride'? They should not be if the praise is timely. Yet you were cursed instead of applauded." He stepped close and his voice grew tender. "Ah, I know of the injustice. Your ears were starved for affirmation. Absorbed by wanderlust, they searched for approval, and for a time you found it in your sister, whom you loved and married. But the pain grew with you. Every time you reached for her, the movement stretched your wounds and spilled their rot on you both. Her eyes grew weary of the hardness of your pain, and more and more she sought relief in the tenderness of another."
Cain folded his arms and shifted away, feeling the Light Bringer's words clutch his emotions.
"Abel stole her eyes, as he stole your father's and the eyes of all the others. He was a void absorbing everything you loved. You feared he would overtake you. That he would steal everything and you would die alone and forgotten. You and he were the Night and the Day. The very laws of nature held your faces to the grindstone. Dear child, I understand." He stroked Cain's cheek and whispered, "That is why I gave you the strength to do it."
Cain closed his eyes, inwardly screaming for the Light Bringer's arms around him.
_No! He wants to use me. I can see it with my very eyes!_
The Light Bringer leaned until his lips touched Cain's ear. "I offered what you wanted, and you didn't care what it cost then, so why care now?"
Cain felt himself nod as logic lost itself in desire.
"I love you, Cain. I am the only one who ever loved you. And I am proud of you. You have done well." He pulled him into an embrace and kissed his head, as a father would a child.
Cain shook against the warmth of the Light Bringer's body. A century of suppressed emotions he hadn't known existed boiled and stung his eyes and throat.
"I am proud of who you are."
_What perversity. It's a hoax. A fabrication._
The Light Bringer said, "Give in. Accept what you are."
Cain whispered, "What am I? Who am I?"
"You are _mine_." The Light Bringer traced the vertebral marks on Cain's arms with sharp fingernails.
Cain could not deny the depth of the relationship between him and the Light Bringer. It felt like fate. It felt like love.
But what was love? He thought he loved Sarah, but this was unlike anything he had experienced, and he wanted it. He wanted it all.
"What do you want me to do?" Cain said.
The Light Bringer smiled. "Build me a Garden."
"A Garden?"
"A New Garden, a place of safety, a home."
"Why?"
"Because your family needs you. The Man is dead."
"Dead?"
The Light Bringer laughed. "Don't look so surprised. You killed him yourself. And soon they will die without him. They need you. Your family needs you. I need you."
"Dead," Cain whispered. _The true test of divinity is to kill the one who claims to be God—does that mean I am he? But the silver boy..._ He frowned.
"The imposter is gone, but your family is tangled in his web nonetheless. You must liberate them. Some are already struggling, but the others—" He did not finish.
"Even an imposter's curse may hold power," Cain said.
"Indeed, his curses hold power beyond even the end of all things. You could not accomplish this alone, but together..." The Light Bringer smiled. "Do you think I do not know your plight? I will stay here to attend to my duties, but at the present you will keep my child, who has shared your existence and begun in you the process of _refinement_."
Cain wiped the sweat from his face with a shaking hand, and wondered if he truly did understand. He felt like an animal in heat, his veins pumping with passion. The words were distant and weightless as they leapt from his lips, "What am I to call it?"
"My child shall be called Abomination, but never speak a word about its existence, and do not speak of our meeting here, or that this City exists. More depends on this than you know." The Light Bringer's eyes seemed to suck him in like endless black holes rimmed with blood. "Surely you understand the magnitude." He drew out the final word in a way that highlighted its importance like a tongue across skin. "After you do these things, you and your people will be truly free for the first time since the foundations of the world were formed. You will become gods. You will become like me."
Cain was fully compliant to the Light Bringer, possessed by its very Abomination of a child, and though reason warned him of the danger, he knew he must welcome this momentary slavery. Yes, in the end, bowing to this new lord would give him the chance to free all humanity from bondage, whether the Light Bringer was a liar or not. Cain would play the game, he would walk the line, and at the opportune moment he would turn and exert mastery over all. Or, as he had planned from the beginning, he would die a martyr.
Alone, but never forgotten.
## PART FIVE:
## THE CHILDREN
_Woe to them! For they walked in the way of Cain and abandoned themselves for the sake of gain.... These are hidden reefs at your love feasts, as they feast with you without fear, shepherds feeding themselves; waterless clouds, swept along by winds; fruitless trees in late autumn, twice dead, uprooted; wild waves of the sea, casting up the foam of their own shame; wandering stars, for whom the gloom of utter darkness has been reserved forever._
—JUDE 1:11–14 ESV
##
The sky's great yellow eye hid itself from the bodies in the coffins, and Lukian had little difficulty understanding why. Many wanted to turn away. Many still didn't believe the bodies were real. How could they be?
Adam and Eve were at the front of the procession but hardly seemed capable of balance. They leaned on each other silently, though Adam seemed more damaged than she.
_Perhaps_ , Lukian thought, _it is the elegance with which she holds herself. Or maybe it's the austerity of her eyes or the emptiness of Adam's._ Regardless, Eve had always been the stronger of the two. No one spoke of it because they'd never needed to. It was simply understood.
Behind their matriarch, and the man from whom she had been formed, walked their children, and their children's children. Leading them all was Calebna, High Priest and eldest son of Abel. His mouth was drawn and his eyes dripped like melting candlesticks, but he was not alone in suffering.
With shaking voice, Calebna led them in the song of lamentation. Voices faltered. Some broke while others became songbirds perched on tombs. It seemed the world was at war with itself, and the wind joined with howls and spurts of wounded coughing. But Lukian was absorbed with the question of whether his father was still alive.
Where was Cain? Had he succeeded in finding greater power and freedom? Or had he merely destroyed himself?
Lukian glanced at Mason. He could tell by the swing of his arms and the shuffle of his feet that he just wanted to finish the ritual, return to Sarah's side, and wait for her eyes to open for the first time in days.
His other brother, Gorban, bumped into him and glanced up in curt apology, then focused on the path before him. Gorban walked beside his wife, Peth, but she offered no comfort to him. Not now.
_Soon, little brother_ , Lukian thought. _The time for action is growing near._
When they reached the grave site behind the Temple, Calebna led them around the three tombs in a semicircle, broke away and faced them. His cheeks shone in the light of the torch, and the shadowy flames enlarged his bruised figure.
"Today"—Calebna's voice broke—"we entomb only a portion of those taken from us. My father remains where God let him lie, and my mother remains missing, may God protect her soul. You must forgive me for not offering words of comfort." He gazed at the sky as if searching for the words. His expression said he failed to find them, and instead he let the leftovers fall from his tongue. "My world collapsed two days ago when I heard of my father's murder. Moments after, I learned of the death of Seth and Ayla, and the death of our God. I dare to say none of you but Adam felt more lost than me. In the days since, I have wondered how we can serve a being and so completely believe he is our only hope, our salvation, our very Creator, and yet place his bloodied crown in a tomb? I do not know the answer. Maybe there isn't one. Yet now more than ever, I feel the longing, the need in my soul for the rituals of the Almighty to calm me, to repeat to me in the darkest hours that everything will be made right one day. And these are the darkest hours."
Calebna wiped his eyes. "Maybe I am a desperate man holding onto a fool's hope, but I will not relinquish my duties as High Priest. My spirit tells me my eyes betray me as I look upon the Almighty's torn and bloodied robe, and the bodies of Seth and Ayla, and so I find in myself a contradiction. I believe and yet I do not believe. I need to accept, but am compelled to deny."
The silence absorbed Calebna's words, for those gathered knew his pain well. The children of Abel glanced warily at the children of Cain, and Lukian thought, _Meaning in every motion. Distrust is a deadly poison in our well, but the world is charred and thirsty, so thirsty._
Calebna's voice strained. "I refuse to believe they are dead. The Almighty once promised he would make all things new, and challenged us to follow him through pain and shadow. Though we bury God, I believe he is greater than the grave. Though we perform his funeral, I cannot believe he is gone." He shook his head and whispered as if to convince himself, "Not forever."
It struck Lukian as odd that even his brothers had no idea what Calebna's words ordained for them. He almost smiled. _No one understands that you, dear cousin, are now the only thing standing between me and freedom._
Calebna stepped toward his family and offered the torch to Adam, though Eve grasped it. She led Adam haltingly, step by painful step, to look upon the faces of their dead children.
The bodies in the tombs were clay figurines. Adam did not look down, for his motionless eyes were empty. Eve glanced first at her expressionless husband, then at Seth's cold face. She smiled as if expecting him to do the same, but he didn't, and so she reached out to stroke his cold eyelids with the tenderness one would offer an infant. If she would not have had to support Adam, she may have dipped to kiss their son. Instead, she spoke words for their ears alone.
When she saw the body of Ayla, her youngest daughter, the sorrow flowed more readily. She mouthed her youngest daughter's name, but the word was strangled as she clutched Adam's arm for the support he was unable to offer.
Upon seeing the Almighty's crown and torn robe, she straightened and nodded, and her grip on Adam loosened. She raised her chin and swept her eyes across her family, stopping finally on Adam.
Lukian frowned and thought, _Fatalism. A dangerous emotion I shall have to expunge. There awaits now no calm path or return to innocent bliss. Only violence._
After an extended silence, Eve helped Adam to the Temple, where the family had taken refuge together.
The Temple. The house of the imposter who had imprisoned them. What more ironic refuge could they take? _Let the slaves run to their cage. Let them box themselves in._
One by one, the remaining mourners passed over the coffins and gazed on the terrible beauty of death, and when all had grieved, Mason, Lukian, Calebna, and his son Jacob pushed the stone covers over the coffins and sealed them to keep the worms from feeding on their remains.
A final dignity. A gift to the dead.
Adam sat in the corner—where he could not fall and strike his head—listening to familiar voices argue, wondering who they were and thinking strange thoughts. But he had always thought strange thoughts, hadn't he? Or at least they had become strange since...
_Since the pain. Waterfalls of pain falling, rushing, splitting me like a mountain. The rush of water, the thrum of voices, and the deafening thunder of thought—these sounds remain, though all else is gone. I cannot jump the river. I cannot reach the other side. I am stagnant. I am rotting bones. I am dust and decay. I am... Who am I?_
"He hasn't moved in days."
"Days? You have watched him so long?"
"What else would I do? He needs me."
"Yes, we all need you."
"I know it. I..."
"It's all right."
The voices droned on. One comforted another. No comfort came to him.
"I have been praying for him."
"Praying? Have your eyes been sealed shut?"
"I know what you would say. But it comforts me."
There was a pause, then softly, "Who do you pray to?"
"I don't know. The dead. The world. Whatever god is out there."
Another pause. Amidst tears. "My sons are gone. God has fled and taken them with him. What hope do I have left? I feel as though there is nothing. Nothing."
The tears became rivers and washed the voices away. Adam was alone again. There was relief in the emptiness, and he savored it. He imagined he could be washed away with them.
Yes, washed away. Like blood in a field. Like blemishes in the cloth of Time. Gone. Just gone. He would enjoy that. He would like to disappear.
So he closed his eyes and dreamed. And the world was good again.
##
Lukian paced the hall and assessed those gathered. His shoulders were rigid, and he walked as if his feet were spears and the ground their mark. It had been six days since they sealed Seth, Ayla, and the elements of the Almighty in their respective tombs, and even longer since Abel died and Cain and Lilleth disappeared.
_I must keep the appearance of strength, for this is a moment of proving._
Lukian's brothers remained separated from the sons of Abel. Gorban leaned against a wall with his arms crossed, and Mason sat cross-legged, nearly as tall while sitting as Gorban was while standing. Calebna stood with his hands clasped behind his back and his eyes glaring sideways. Eve sat with Adam, stroked his hand, and silently begged him to speak, to do anything.
Those gathered likely had a myriad of explanations for Cain's actions, the dead bodies, the state of Sarah and Adam, and the death of their God. Their suspicions and the tension had metastasized into malignant glances.
"I am no liar," Lukian said. "No one has gained from this. I see it in your faces." He looked at Calebna's brothers, then at Calebna's eldest, Jacob. He looked at Eve, whose eyes grasped him like vines under winter's first chill, straining to keep their strength despite the ache of the frost.
Calebna spread his hands. "We make important decisions today. We have no room for belligerence."
"I cannot go back to when I first heard whispers against me and my brothers. I held my tongue then out of respect for the dead, but I can stand it no longer." Lukian paused, yet everyone remained silent, unwilling to break the tension for fear of its recoil. _So strong behind closed doors, yet so weak when pressured_. "Is a child a copy of his father? And were Abel's children the only ones harmed?"
Calebna sniffed and cleared his throat. "What choice do we have? How can we believe no one aided Cain in his sin?"
"By having faith." The people mumbled. None held Lukian's stare. He continued, "Perhaps you should share from your excess store of it, High Priest."
Calebna's expression rotted.
Lukian continued, "You are right about one thing. Separation is dangerous. But we do not know what has befallen our mother. Why was she found half drowned in the flooded quarry?"
"Many think their suspicions warranted."
"Change has come, and unless we adapt, none of us will survive."
"It seems you've been conceiving a plan in silence, and I'd rather you voice it and let us discuss its merits than for us to argue over opinion and conjecture."
"Finally you speak with sense," Lukian said. "You remember destroying our weapons when we arrived. Tell me, with the Almighty's death, what do you think has happened to his protection?" Some of their eyes catalogued the possible danger. _Yes_ , _realize how ignorant you've been._ "We need weapons, but have only shovels, hoes, scythes, hammers, and pickaxes. We cannot defend ourselves with these. We must remake them into implements of violence." He could see the fear in their eyes. _They are ready_ , he thought. "Unless we flee from the City, every one of us, from our great mother Eve to the babe in Terah's arms, will die."
The mumbling increased to a gentle roar, and Calebna raised his hands to quiet them. "The viper spits, but is there bite behind the prattle?"
"Our plants are dying, our food is rotting, and our animals are sick or dead," Lukian said.
"Fear will gain us nothing. We must stand strong in faith. All else is cowardice."
"Is it cowardice to free ourselves from bondage? Is it cowardice to consider the lives of children more important than dead principle?"
"You would have us abandon our lifestyle, remake our weapons, and run blindly into the wilderness, but for what? Do you also propose yourself as our leader?" Calebna laughed and shook his head, not finishing what was written in his expression.
"You mock yet still cling to a lifeless God as if his skin could warm you. God is dead. His cloak and crown are lying in a stone coffin next to Seth and Ayla, and I will not lie down and die by the rules of an imposter, for that is what the Almighty was."
"What reason do we have left to live, if not for hope in God? By what transcendence could we gain direction?"
"By the same principles that led us through the wilderness. In those many years we synthesized our own meaning. Or have you forgotten?"
Calebna's shoulders sagged. "I remember the pain of those days, and it sticks in my throat even still."
"And yet those days were _ours_. I cannot say that about our lives here, and I would fight and pay the price to say it again."
"We returned to the Almighty because we were dying. The demonic Jinn overwhelmed us. It seems you, not I, have forgotten the true price we paid. Indeed,"—Calebna's eyes glinted in the torchlight—"the price you yourself paid."
Lukian's neck warmed at the memory of his twin brother's entrails wrapped around his mangled legs. _That was a cowardly blow, but they are watching. Calm your anger._
"The Almighty is our only hope," Calebna said.
"How?"
"Our own strength has ever failed."
"Maybe you're right. Maybe our chances are better here with no food and no protection. Or maybe our future is laid before us waiting to be claimed."
"Our purpose lies here."
"In becoming sacrifices? Do you intend to burn us all at your holy altar?"
Calebna narrowed his eyes. His jaw moved from side to side, then clenched. "Hear this now. As High Priest of the Almighty, I declare that the Almighty's laws, which we all swore to preserve upon returning two years ago, remain. Those who live by the sword will die by the sword, but those who live by the Spirit of the Almighty will stand on the firmest rock. Remember the flood that washed the City the night after Abel's death, and heed my words that another flood is coming, and its red waters will drown all who quench themselves in its depths. I vow today to never again pick up a weapon and strike with fierceness, and if any desire to call me family, you will take this vow with me."
Calebna's son Jacob stood. "I vow the same." But none followed save Calebna's wife, Terah, and she hesitantly.
Those standing looked at those who sat, and doubtless shame weighed upon the shoulders of Calebna's younger brothers, Philo and Tuor. But Eve spoke, and her voice chilled the air to silence. "For me, there is no hope." Her white knuckles held Adam's hand in her lap, and the light failed around her shadowed eyes. Some of the people were fearful. Others were exhausted. Certainly all felt the bite of her words.
Danger lurked in every shifting shadow and darkened corner. Lukian knew that if he failed to stoke the flames, they might be forever snuffed, but if he pushed too far...
"If you cannot do it for your own sake, do it for mine. Do it for Sarah's," Lukian said. He gazed at those still sitting. "Do it for Abel. Don't let yourselves join the Almighty in the unforgiving stone."
His words stopped but his eyes went on. He saw in some of Abel's children and grandchildren the desire to struggle on. He thought he saw in Gorban, Mason, Kiile, and Machael the willingness to follow him into death, but no one moved. No one said anything.
Eve rubbed Adam's hand. Mason stared in voiceless intensity. Gorban rubbed his neck and squinted. And Calebna stared at his own clasped hands.
Then something unexpected happened. Adam's eyes stretched and breath burst past his lips. He was struggling, and for the first time in days, he spoke two words, though they were barely audible in the hush.
"My Abel." And tears welled in his eyes as he rocked.
The room was filled with the throbbing ache of loss, and Lukian saw the damage rifle through them all. He wondered at his grandfather as the man swayed. What had drawn those words, the first he had spoken since the storm subsided?
Lukian's eyes widened. _Was that the first time in all these days that anyone spoke Abel's name in front of Adam?_
He watched Adam's twitching figure. After an awkward silence, Eve led him away. She looked fatigued, but Lukian knew she would not sleep. Instead, she would watch Adam's face and pray for her husband to return, but as had happened every day since Abel's death, his eyes would open and she would look into them and find only emptiness.
_Adam has lost himself to desperation_ , Lukian thought. _And I must draw a similar desperation from the others. Not too much and not too little. The battle is not yet lost._ He bit his cheek. _I fear it has only begun._
After Adam and Eve disappeared, the group trickled away until only Lukian, Mason, and Gorban remained.
Lukian cleared his throat and brushed at his tunic. Storm clouds churned in his skull, and he stood and walked from the Temple, barely aware of his feet touching the ground. On the road, Lukian heard footsteps behind him.
"Brother," Gorban said. And he repeated it as if unsure what else may be said.
"I know. You, at least, I would never doubt. I am going to walk the inner wall."
"I will follow."
For a time they were silent, but questions remained.
"What is going to happen?" Gorban asked.
"We will survive as we always have."
"How can you be sure?"
"It is the only choice we have."
Again there was silence until Gorban shifted. "They are afraid."
Lukian's sandals clapped the road.
"They will fight in the end."
"Let us hope it won't be too late," Lukian said.
"But what about the food? How long can we survive before it runs out? And then what do we do?"
Lukian stopped. "There is always a way."
"Most of the forges were destroyed in the flood. Your sons and I checked them today."
"What remains?"
"A few on the outskirts appear usable, but they're small," Gorban said.
Lukian nodded. "Could you prepare one by tomorrow?"
"Yes."
"The days ahead will be long, but not long enough. We must jolt them from slumber." He grabbed Gorban's shoulder. "We cannot afford to wait."
As night fell over the horizon, Gorban, Lukian, and his children congregated by the wall, kept watch, and burned fires to keep warm. For a time, Lukian was hopeful his other brothers and nephews would come.
None did.
##
Gorban inhaled the hot air belched from the forge, and was content for the first time in many days. As he hammered redhot metal, the flames in the furnace licked the sweat from his skin and dimmed the world behind the flash of molten metal. A song floated through his mind, and he scooped it up and let it burn his tongue. He remembered only a shard of one verse, after which he went on humming, but the words spilled like gold into the mold of the meter.
_The green grass that covered the hill_ ,
_Near the dark forest ere winter's chill_ ,
_Berry bushes and thickets tall_ ,
_And the oak and burdock in the fall_ ,
As he repeated the words, Calebna appeared in the doorway. The man's hair was tied back and cascaded like undyed silk, and his reedy face seemed tighter than usual, though he tipped his head in greeting.
Gorban wiped the grime from his hands.
"I hope you're keeping a few tools for their original purpose," Calebna said.
"Taking new interest in the business of men?"
Calebna cleared his throat, grabbed a hammer, and tested its weight. "Peaceful tasks are often most difficult in times of disquiet, so I understand in part what you attempt."
"The forge is hot."
He chuckled and seemed to search Gorban for sarcasm. "I've never had the gift for plain speech. Maybe you could teach me."
Gorban folded his arms.
"I'm here for the people," Calebna said.
"Having trouble?"
"It's the question of leadership."
"Adam hasn't improved?"
"I thought you knew."
"I'm busy," Gorban said.
"Do you not care?"
"I'm no healer."
"Well, you will be glad to hear he hasn't worsened, but whatever ails him seems reticent to relinquish its grip. Many of us have little faith it will pass, and as your brother so eloquently illustrated, that leaves us with the question of headship."
Gorban cursed, turned to his work, and said, "Drawing substance from you is worse than cleaning a wound."
"It will be a few days until we decide, but the people are concerned. The weapons, the talk of fleeing into the wilderness, it has them worried."
"I don't know why it would."
"There's concern that if a choice of leadership were put through, violence may arise."
"You know that wouldn't happen."
"I would hope it wouldn't."
"Lukian is a good man."
"I know he is."
"Why are you wasting my time?"
"The people just want to know you still have their best interest in mind."
"We do."
"All right."
"All right."
Calebna placed his hands on his hips. Gorban could see thoughts brewing in his eyes, in the way his mouth hung open as if ready to make a pronouncement. He was always making pronouncements.
"I will do whatever it takes to keep my family safe."
Gorban motioned for the hammer still in Calebna's hand, and the man handed it to him. "So will I."
Calebna folded his arms. "You know how I feel about violence."
"Times have changed. Why don't you make yourself useful?"
Calebna's lips tightened. "I will do what I must and no more."
"What about your vow?"
"Violence is not the only means of resolving a dispute, and there are those who would fight to defend the innocents that Lukian is jeopardizing. He does not hold everyone's attention. You are his brother. Tell me, why is Lukian doing this?"
"I'm busy."
"Busy or hiding?"
"Ask him yourself. Or are you afraid?" Gorban shoved a half-made weapon into the furnace.
Calebna cleared his throat, looked sidelong at Gorban, then nodded and turned. In the archway, he stopped. "I remember the green grass on the hill. But lately it's been hard to shut out winter's chill. You should sing more."
Gorban scowled and doubled his pace for the next hour.
"Something's happening at the wall."
"What is it?"
Gillian, Lukian's eldest, looked afraid to say. "Fog."
Lukian jumped up and followed his boy, who held a spear in his hands and the shadow of a beard on his face. The winding stairwell echoed with their footsteps as they ascended the parapet, and when they reached the top, the doorway opened to a divided sky. A pillar of Fog reared on the far side of the inner wall, and the outer wall, merely a waist-high mark of the extreme limits of the pasturelands, was completely consumed by gray mist.
Lukian followed Gillian to peer at the moisture. It seemed to loom over the wall, though it was only a trick of the eyes. His son pointed and Lukian noticed the reason for his fear. Countless eyes glowed like flickering candles in the gloom.
"They grow more numerous with each passing hour."
Lukian's hands gripped the wood of his weapon, but the sensation was dull. Old emotions tumbled through his chest like broken cinders as he looked at the clear sky over the City and realized the heat of the sun _was mild, and the smell of spring was thick. He and his twin brother Lamech were playing at the river, skipping rocks through the reeds and imagining felling a Jinn with each throw. Lamech lifted a small boulder and stumbled with its weight in his eleven-year-old arms. After repositioning, he swung it into the river with a tremendous splash, though he couldn't keep himself from being pulled in after it by the momentum. Water dripped from his hair as he jumped up laughing, and Lukian fell back and offered the sky his giggles._
_"I know a place we can see one in the flesh," Lamech whispered with squinted eyes and dimpled cheeks. "I saw it last week, but Father told me to close my eyes and forget. I counted our steps and marked the turns, and later, when he was asleep, I came back and killed one."_
_"Your words float like a pile of rocks."_
_Lamech shrugged and smiled. "But I did find it again."_
_"If you get caught wandering off..."_
_"It's not far. I could show you where it is."_
_Lukian sat cross-legged, tearing grass up and staring at the water._
_Lamech smirked. "Father will never know."_
_"How do you know you weren't looking at a wild dog?"_
_"I saw its eyes."_
_"You didn't see a thing."_
_Lamech smiled as only an elder brother could, and Lukian knew. Father would understand, but Lamech wouldn't._
_Lamech turned, started walking, and waved for him to follow._
_Lukian did._
_As they arrived and peered over the edge, Lamech drew an invisible line with his finger, and Lukian's excitement calmed._
_"Fog? That's what you came here to show me?"_
_"Its eyes burned like lanterns in the mist." He continued in an excited whisper, "You should have been there! If I'd had a weapon, now that would have been something. I would have rammed that Jinn through the eye and stood over it just like this." He donned a pose they once saw Cain assume over a dead Jinn, one of the first they had found in the wilderlands._
_Lukian pushed his twin's face. "You are full of lies."_
_They stayed awhile longer, lying on the grass and peering into the Fog. Then Lukian spoke the deadly dare. How could he_ have known that would be the last time he would hear his brother laugh? God knew he hadn't wanted his brother's insides torn and thrown across the ground. God knew he hadn't wanted to drive a thorn into his parents, but the pain had changed the way Cain looked at him, and the way Sarah looked at Cain.
"Father?" Gillian was looking at him.
Lukian closed his mouth, turned away, and tossed the memories over the edge of the wall.
"What are we going to do?"
"We will wait in the City, and if they break through the wall, we will do what we must."
"That's it?"
"What else would you have us do?" Lukian's mind churned. They needed his brothers and nephews, but to belabor the fact was foolish. He could speak no word of their need. They would not survive any wavering.
_If only we had run when we'd had the time, but we are trapped. We could never outrun a Fog this large._
Upon reaching the bonfire, they exchanged no words. Gillian sat with his arms folded, and Lukian stoked the fire from time to time, adding wood to entertain the blaze. The chill in the air felt like the beginnings of winter, but it would be many days until that season should begin. As the sun began its descent, Gorban returned from his labor at the forge. His arms were filled with weapons, and he set them in a heap.
"Where's Mason?" Lukian said.
"Guess."
"It is time someone spoke with him."
Gorban's breath hissed from his nostrils. He slapped the metallic dust from his tunic and left. Gillian followed, likely excited to hear what Gorban would say to the mute giant.
Lukian looked at his remaining children, Jubal, Irad, and Zachariah. Together, they numbered six armed warriors, yet still a pile of weaponry remained unused, and Zachariah, his youngest, had barely reached fifteen years. If they fought alone, they would die. But if they hid in the Temple, they would succumb to starvation in a matter of weeks.
Mason would come eventually. But what about the others?
##
Mason stoked the fire and felt the flame burning in his mother's cheeks. How many times had he done this? When he found her floating in the water, it had almost been too late. Then the flame ignited, and she burned with hungry passion in the deep sleep that continued to devour her.
_If I could but speak words of comfort_ , he thought. Since being taught as a boy, he whistled to fill the silence, and he did so now as he replaced her bandages and tended her ribs. The marks were black, and her toes kept their gruesome color. The skin on her fingertips was peeling, but would be usable in time. As long as the fever left. But that was why Calebna was praying.
The Temple. It seemed appropriate that what once had been their spiritual refuge was now their physical asylum. Built on the highest hill, it had remained safe from flooding, and that alone was reason enough to seek shelter within, after a significant portion of the City was destroyed. But more than that, Lukian had been right. They needed each other.
He stabbed at the burning wood. Calebna's youngest cried in the hallway, and he wondered if those sounds reached Sarah's mind. Was she paralyzed, cognizant but incapable of moving?
Her brow furrowed, and sweat beaded on her cheeks.
_Fight the burning. Don't let it consume you._
Sometimes, when he dreamed, he could speak and sing, and upon awakening the realization always stung like an open palm. It was easy to forget his strangeness, but the dreams always reminded him.
What did _she_ dream? That her words were stopped?
The irony slipped down his throat like alcohol as he brushed his thick fingers through her hair. Veiny muscle contrasted with pale, tender skin. He promised he would be her strength as long as she needed. He would stay if all others left. He would—
Calebna's child wailed louder. Why did it cry? Was it uncomfortable? Hungry? He wished someone would tend to it.
The door opened. "Mason."
He did not turn.
"How is she?"
With a glance, Mason saw Gorban standing in the doorway, hardly filling it with his compact frame and black hair curling like vines around his trunk-like neck. Gillian peaked his mousy head from around Gorban.
How could Mason communicate the few times Sarah had woken, that in those moments he had only the time to give her water before she fell back asleep? He dabbed the sweat from her eyelids.
_It's been nearly three entire days since last you opened your eyes. Why won't you wake?_
Gorban said, "There's Fog at the wall. It's time. We can't wait any longer."
Sarah's hands clenched and Mason took them in his fist and carefully pressed his thumb between her fingers until they relaxed.
"Brother." A hand warmed Mason's shoulder, but he shrugged it off. He could feel Gorban staring, but he remained obstinate, and soon Gorban's footsteps shuffled out the door, leaving him and Sarah alone again.
_I will stay. You're going to be well again. You will. You must. They don't realize why. But I do._
And he pulled the covers over her belly.
"Where is Mason?" Lukian said.
"He wouldn't leave."
"I told you to talk with him."
"I did," Gorban said.
"We can't do this alone."
"He's as immovable as a boulder."
"A stick can push a boulder if the right leverage is applied."
"What of the Fog?"
"It has kept its distance, but the Jinn are amassing." Lukian's eyes were tired, and his shoulders sagged. "You remember how to use a spear?"
"Old muscles forget slowly."
"Calebna still thinks the Temple will save him. The fool will kiss my feet before this is all over."
"He came to talk to me the other day."
"Ah?"
"Said he was there for the people, that they were worried about the threat of violence if a choice were made for a new leader. He wanted to know you still had the interest of the people in mind." Gorban sucked his teeth. "He implied you wouldn't be his first choice, and that God knows you wouldn't be _theirs_ either."
Lukian smiled.
"He threatened you, in his roundabout sort of way."
"Thank you for telling me."
Gorban shrugged.
"I would love to see him fight again. That would be something worth my time."
"Is Adam doing so badly?"
"No signs of improvement. The others have made perimeter checks, but the Fog is only on our eastern flank. We can't afford to spread thin anymore, so until something happens, only one group of two will walk the inner wall in intervals." Lukian nodded. "We ready ourselves for confrontation, and for what might come after. A decision over leadership will be unnecessary if this goes as I expect."
"Many respect Calebna."
Lukian lifted his hammer and tapped Gorban's chest. "Worry less of me and more of how you're going to get Mason out here. And don't worry if you return before me. It's time I apply some leverage of my own."
##
You made the right choice." Terah's hand slipped into Calebna's and squeezed.
Calebna laid his head against the wall of the prayer room. Lampstands stood beside a small cushion set atop an ornate rug of gold and violet, and next to them lay the bed he and his wife had slept in since the storm subsided. "That's not it."
"What then?"
Calebna chuckled. What couldn't it be? "I wouldn't leave you like that, not for such hopelessness. Lukian is just trying to be a man of action. He's trying to fix things." A pause. "He's trying to be his father."
"So who are you trying to be?"
"I don't know anymore. A good man."
"You _are_ a good man."
He searched her face. "What is good?"
"You are."
"I mean what is _good_?"
"I don't understand."
Ben was crying again. Terah's hand left his as she went to their infant, cooing as she gathered him in her arms. Calebna watched her, but his mind was fixed. He had thought the Almighty was good. He had thought living in safety was good. He had thought peace was good. He had thought a happy life was good. But all of that was gone.
Ben's cries continued. Calebna stood and paced the room. He wanted to speak to Lukian again, but what would he say? He shook his head. Lukian wanted a fight. He would enjoy a fight. But part of Calebna would enjoy it as well.
_No_ , he thought, _I am no longer that man._
"Are you all right?"
He was looking at the floor, resting one hand against the wall while the other fiddled with the medallion around his neck. He looked at Terah and nodded, assuming a more relaxed position and taking Ben from her. Calebna held him out and watched him writhe and wail, his tears dripping down his fat cheeks.
If only Ben knew the real horror of life. If only he knew the pain he would endure, the anxiety of danger, the ache of loss. Maybe there would be moments where the sun would warm his toes, but there would be so much more sorrow in a world without God.
_It would be better if he were never born._
Terah snatched Ben back and held him upright on her shoulder, rubbing his back. "What has gotten into you?"
He hadn't realized he had spoken his thoughts.
"Don't you dare say such things."
"You'd rather he live in hell?"
Her mouth warped as if ready to retort, but she whipped around and ignored him instead. Ben was crying louder, disturbed by the angry tones.
"I didn't mean it spitefully," he said.
Terah didn't respond. She was bouncing Ben, who had found his thumb and sucked, barely breathing past the mucus in his nose. His wet cheeks dried against Terah's hair, and his eyes looked like washed crystals. Such innocence. Such purity.
"I'm just angry. Not at you." He paused, searching his reasons. "I don't know what I'm angry at. I just don't want him to taste pain."
Still bouncing her son, Terah said, "He already has."
Calebna regretted what he had said. He was weary, that was all.
Terah's eyes flamed. "Don't say that again."
"All right."
"I'm not a fool."
"You're not."
"I know the world isn't going to be good to him. It hasn't been good to me either. You're not the only one suffering, not the only one who's lost someone, who's lost hope."
"I know." An admission.
An uncomfortable silence followed. Calebna leaned against the wall and his wife sat and cradled Ben in her lap. Ben settled and drifted to sleep. Terah looked at Calebna from time to time. Her gaze was hard but, somewhere beneath, filled with affection.
"Love is."
He cocked his head. "What?"
"Love. Love is good."
He nodded, though the statement begged the question, what, then, is love? He thought about asking, but didn't have the energy. They remained as they were, though she turned so he wouldn't see her cry.
Someone knocked at the door and Terah stood and sniffed away the wetness. Calebna rose and cracked the door.
Calebna saw Lukian's face through the gap. "We need to talk." Lukian peered over Calebna's shoulder.
Calebna filled the gap with his body. "Not here."
"Follow me." Lukian turned and walked three steps away.
"Wait."
"There's no time. Come." His eyes widened. "Now."
Calebna glanced behind him and heard Ben start crying again. He paused, stepped out, shut the door, and chased the sounds of Lukian's footsteps.
They walked down corridors and descended stairs to storage compartments that smelled like incense and mildew. Lukian placed a torch in a receptacle on the wall and shut the wooden door behind them. "I didn't think you, of all people, would stoop to such tactics."
"I've done nothing."
"What do you think you will accomplish?"
"I'm fulfilling my responsibility."
"The demonic Fog has amassed near the eastern wall."
Calebna's throat felt cold. "And?"
"Stop playing the fool. I saw them. Jinn. Hundreds, maybe thousands. Look for yourself if you're unsatisfied with my word. There are so many..." Lukian shook his head. "More than I've ever seen before. We cannot do this alone. Not anymore. And you're dividing us."
"Me? Everyone is here in the Temple except the men you've recruited to your bonfires."
"Why do you think we're out there? To relax in the warmth of firelight? This Temple won't save you. They'll tear its roof down on top of you. You're a fool to think otherwise."
Calebna slowed his words and sharpened their edges. "I'd rather be a righteous fool than a clever wicked man."
"Would you murder your family to be named righteous?"
Calebna's scarred leg ached as he remembered what the Jinn had done to him. As a young man he had grown bitter about his wounds and had harmed Terah as a result of it. He could never let that happen again, but to think of what the Jinn had done to Lamech, whom Calebna had loved in childhood...
"You don't have the luxury of living in black and white anymore."
"I have already chosen."
"The Jinn are going to break the wall, and when they do, this Temple will not save you. Who will you blame when your infant and wife are crushed before you?"
Calebna's eyes narrowed.
"We have no hope while divided, but together..."
Calebna nodded slowly.
"The death in my words is as real as the breath in your chest."
"God will deliver me. What about you? What do you hold onto? Yourself?"
"I will not apologize for putting my family first. The Almighty never loved me. He never loved any of us. If he had, he wouldn't have let this happen. And you're following in his footsteps, abandoning us in our time of need. You are free to stay in your Temple, but you're not free to decide for them." Lukian approached and thrust his finger into Calebna's chest. "Gorban told me about your visit."
_Will this come to violence?_ Calebna wondered. He examined the empty storage room and listened to the silence, suddenly fearful of it.
"He said you claimed you'd do anything to defend your family."
Calebna backed up until he leaned against the wall, but could not quell the beating of his heart. _What would happen if I killed in defense?_
_No, I cannot not choose that way._
"I challenge you to prove your words hold meaning." Lukian leaned forward and Calebna felt his hot breath on his ear. "If you don't, I will."
With that, Lukian spat, smeared it across the floor with his sandal, and strode away.
Calebna stayed until the torch nearly petered out. Thoughts rumbled in him, and for the first time in years, he pondered violence.
_Love. Love is..._
He stared into the darkness. _Love is what?_
##
Gorban cursed as he flung the Temple door open.
_I'm not a fool. If Mason weren't Mason, there'd be no difficulty finding the proper leverage._
Lukian was right, of course, which made Gorban all the more irritable. They needed their brother. If he just weren't so attached to Sarah.
_That's an unworthy thought._
_And yet I thought it._
Gorban tried the door he thought was the correct one only to receive Philo's hand in his face. "Out, out. What are you doing?"
He apologized and doubled back, feeling heat on his neck. Two more tries in the labyrinth and he succeeded, thinking, _The Almighty's obsession with symmetry runs deep_.
Mason knelt as before, petting Sarah's hand in the dim room that crackled with warm smells, not all pleasant. Lukian's wife, Keshra, stood by and watched for signs of cognition. Sarah's eyelids were ashen pale against the silhouette cast by the flames, and seeing her frailty stirred deep emotions within Gorban. He cleared his throat and knelt at Mason's left. This time, his hand was allowed to perch on his brother's shoulder. "How is she?" His voice was reverent as he nodded to Keshra, who bowed and left them.
Sweat darkened Mason's bed-of-snakes hair.
Gorban nearly spoke several times, but words seemed to fall short of communicating what he desired, as they so often did. Finally he chose to say something, anything. "I am sorry I wasn't there."
Mason offered a wary glance.
"I—we need you. Our danger grows."
Mason could be so inscrutable. It made Gorban itch to leave. He scratched his leg and his voice grew sharp. "Will you abandon us in our need?"
Mason glanced at Sarah as if to ask, _"What of her need?_ "
"Keshra and Peth can tend to her. Healing is women's work."
Mason shoved him away.
Gorban smacked the dust from his hands and glared at the back of Mason's head. "Never claim I didn't try." He threw up his arms in mock surprise. "But I forget—you couldn't if you wanted." Gorban spun on his heel, making sure to accentuate the slap of his sandals against the floor.
As he slipped out of the room, he realized the need to invent an explanation. Lukian wouldn't be happy. Then again, Lukian was never happy.
Lukian found Kiile and Machael, as expected, huddled together and attended by their wives and children. Kiile lounged against a cushion, his limbs splayed and his mouth filled with racket. His children stood in a ring, laughing and nudging each other while a pair of the youngest wrestled and exchanged blows. Machael sat cross-legged beside his wife, Zillah, and watched with those sleepy eyes. His children stood and pointed from time to time, looking unimpressed, as usual.
"I expected you sooner," Machael said, his voice slow and deliberate.
"I could say the same," Lukian said.
"You know it's too much fun to hear Kiile chastised by the others' wives. It's like the old days when we slept huddled together and traveled in a group through the wilderlands."
"In more ways than one."
Machael nodded and seemed to grow drowsier yet.
There was a roar of applause and a few moans of dissatisfaction as Kiile's youngest wrenched his brother's arm back and slammed his face into the floor.
"He's like his grandfather, that one," Machael said. And then he added, "Are the weapons ready?"
"They're piled against the inner wall."
"Some are too young," Machael said.
Lukian pointed at the littlest boy, who strutted about slapping the hands of his elder brothers. "That one fights like a badger. By your own estimation he's useful."
"It will end in grief," Machael said.
"Another one of your premonitions?"
"Just a feeling."
"If it weren't for the gift of speech, I'd doubt you had feelings at all."
"Kiile will be eager to see his weaponry. He's stayed with Elsa, but his mind is with you."
"He's chained to that woman."
"At least they retain a connection."
Lukian huffed.
Machael's oldest, Madai, tapped his father on the shoulder, "May I participate in the wrestling contest?"
"Change out of that tunic first, and wash it and hang it before the night comes."
Madai bowed, smirked, stripped his tunic off and left to join the raucous ring.
Machael turned to Lukian. "Others will come. And soon."
"Another one of your feelings?" Lukian said.
"A premonition."
Kiile welcomed Madai, noticed Lukian, jumped to his feet, and called out as he approached. "Brother! Come to clean the dirt out of your bed?"
"Only to place it in yours."
"I can always rely on you." They embraced, and his smile faded as quickly as it came. "They're prepared?"
Lukian nodded.
Kiile cursed and thanked the Almighty in the same breath, then lowered his voice so his wife wouldn't hear. "I'm worried about Elsa."
Lukian said, "Calebna won't let her leave?"
"That's not it. God knows he would restrain her if he could—he's policing everyone with increasing intensity—but she'll slip a knife in him to be next to me when the battle comes."
"If Calebna hasn't the strength to restrain her, Terah will," Lukian said.
Kiile laughed long and hard. "Maybe."
"Eve made the pronouncement after hearing word of the Jinn. No women are to leave until Calebna deems it safe," Machael said.
"What of the boys?" Lukian asked.
"Calebna will let them choose for themselves." One of Machael's eyes awakened and bored into Lukian. "Every boy must learn to be his own master. Eve could not disagree, and yet she could not let her daughters go."
"Ah... and if Kiile takes the boys from Elsa?"
His eye fell back asleep. "She will stay."
Kiile grimaced and said to Machael, "I like you better when you don't talk." Then to Lukian, "Did Gorban make it how I asked?"
"You know your preferences better than me."
Kiile searched for and caught Elsa's gaze. He nodded as if she knew what he wanted to ask, and she frowned in response. "What?" he yelled. "If the sun sets and darkness touches my toes, I'll cut them off for you."
She twisted her hair and secured it in a bun with a bamboo rod. "Promise me the feet!"
Kiile smiled, then glared at his brothers. "Well? Show me my weapons—and quickly!"
##
The touch of clothing on Eve's skin gripped her attention with painful intensity. She looked down and saw her thinness exposed by the ever-loosening dress. She no longer remembered what she last ate or when. Her stomach didn't even ache anymore, and she was reminded of the years spent in the wilderness, carrying twins and living a hunted life.
She breathed in, then breathed out. _The very twins whose fates have been twisted down to Sheol._
She rubbed her arms. Through the years, the Jinn had grown greater in number. At first, avoiding them had not been difficult, but as they multiplied, so did the danger.
After obeying the summons and coming back to their Creator, they had thought themselves immune. Now they were trapped.
Eve tipped her head and caught Adam sitting against the wall. She stared at the creases in his robe, the way his hands twitched in his lap, and how his eyes squinted at the lines in the wall. In all the madness, all the confusion and danger, he had done nothing.
"Wake up," she whispered. She wondered what would happen if she shook him, if she grabbed him by the tunic and threw him to the ground. _Nothing_ , she answered. _He would do nothing._ She crossed to him trembling. "Wake up." She slapped his face, and the sting echoed in the skin of her fingers.
_Do I hate him?_
_I hate his not being here. I hate his silence. His love for Abel._
"Yes," she said. "I see you loved him more than you loved me, for you left me in order to somehow find him, to reconstruct him from your memories—the only place he still lives. But I can't follow you there."
Eve swallowed the anger as new emotions arose. Some she recognized, others were muddled. Her hand reached, then retreated to rest on her sternum. She said, "Truly, I love you more than I could ever express," but those words dissipated like fogged breath. "How could I not after so many years? After knowing the life I have was crafted from yours? I couldn't let you waste away alone. That's why I have stayed, but still..." The bitterness burned, and out of the ash came something familiar. She thought it pity, though it could have been shame.
_I see now where all this violence was born—all those years ago in the shadow of that cave, and his worshipping that little babe._
"How could you do this to me?" Tears dribbled down her cheeks and neck. "Look at you. You're a wretched thing, dead but still breathing."
She crouched, paused, then slid her arms around him and laid her head on his chest. It was warm and solid, and it rose and fell and thumped out a rhythm.
_This is the man I was fashioned from, the man I was created for._
How foolish her anger was. He had failed her, but she had failed him too. Leaves, falling in the autumn wind, shimmered as if branded into her eyelids, and she winced at the memory.
_The Serpent. The coppery flavor of forbidden fruit. The red-black juice leaking down my hand as I push it toward Adam and offer him a taste of death. The burn of the sun on my naked flesh as I breathe in the sensation of terrible exposure, and the genesis of darkness in Adam's eyes as he takes in my figure and for the first time I feel the fear of a man's desire._
A hundred and fifty years passed and Sin had stolen her children and killed her God. And when she needed Adam most, he was lost in his mind.
His tunic was wet with her tears. "Please," she implored as she clawed his back. "I know you've carried the guilt of that moment all these years. I could see it in your eyes, I could feel it in your touch. You took the shame, and accepted my own, but I shouldn't have made you do that. I was just scared. I loved you too much to admit I had harmed you." Her words were swallowed by sobs, but somehow she managed to say, "I'm so sorry." The moans forced the words out until she nearly screamed, "Can't you see that I love you? That I never blamed you? But I let you suffer alone all these years. How could I leave you alone? And now..."
His hands remained limp, and Eve wept, but not for him and not for her children. For the first time, she wept for herself.
Like Adam, dead but still breathing.
Lukian observed the warriors, whose shoulders were bowed by the weight of the night and whose glances kept returning to the Temple drawn in silver lines atop the hill. He wondered what they thought. _Do they desire the illusion of safety? Or do they simply think of their wives, mothers, and sisters?_
Surely Kiile thought of Elsa, but the Fog had become more chaotic in the past day, and they knew the violence would begin soon.
Lukian fingered the metal of his hammer. The edges were cold and harsh. He looked at his sons, brothers, and nephews. Of what mold had they been formed? Of what mold had he himself been fashioned?
"Hand me some," Kille said.
Gillian broke stale bread and handed it to his uncle. The man squashed the spongy dryness and spoke with mouth half full. "Is there much left in the storehouse?"
"Most of the grain rotted before we could make dough of it."
Kiile's eldest raised his voice. "I'm hungry."
"We're all hungry," Gorban said.
Kiile spit chewed bread mush in his hand and waved it toward Gorban, who scowled, clutched his spear, and walked to the edge of the firelight. Kiile returned the bread to his mouth and swallowed after considerable effort, then handed the rest to his son.
"Can you eat Jinn?"
"We'll soon have plenty to try."
"How about Jinn stew?" Gillian chanced with a smirk.
Chuckles bounced about the circle, but Lukian bit his tongue, stood, and faced the wall. Kiile fell silent, as did the others. _They fear for tomorrow_ , Lukian thought. _Even my son joins to mock what his mind cannot accept._
Lukian wanted to walk the wall, long and far in the cover of dark, but the sound of approaching footsteps rooted him in place.
"Who is it?" Gorban dipped his spear toward the sounds.
The firelight illuminated two sets of eyes, then two brown tunics. The taller edged in front of the other. "It is I, Philo, and my brother Tuor. Will you receive us?"
Gorban sauntered toward them. Lukian watched with interest. _Philo and Tuor, Calebna's younger brothers._
"What brings such delicates to us after nightfall?"
"We want to join you."
"Why?"
Philo lifted a small blacksmith hammer. "We want to fight."
Gorban glanced at the hammer in Philo's hand, then at Lukian, who approached.
"If that is your intention," Lukian said, "you are welcomed. But know that we have no space for cowards."
He noticed a slight shiver in the younger's muscles. The motion drew his gaze to the hunched shoulders and baby-soft hands.
"We won't run," Philo said.
_No_ , Lukian thought as he looked to the older boy, _you won't, at least._ A smile edged across his face. "Why now?"
Philo paused, then said, "We have no wives. We are young and weary of waiting. We want to be useful."
"And your family?"
"They share convictions my brother and I do not."
"Do they know?"
Philo nodded. His younger brother's eyes twitched toward the grass.
_He's lying_. Lukian motioned to the hammer and the boy offered it to him. He felt its weight and balance. "Philo, your brother seems to have conquered his tongue. It is strange, for I seem to remember how he couldn't stop it from wagging behind his teeth."
Tuor stilled.
"Come here."
Tuor's eyes strained toward Philo, who scowled and nodded. Tuor edged forward and eyed the hammer in Lukian's hand.
Lukian lifted it. "You think I'd strike you?"
Tuor shook his head.
"If I saw you flee from death, I wouldn't hesitate."
Tuor nodded.
"Speak."
From the corner of his eye, Lukian caught the older boy's frown and the shift of his feet.
Tuor said in a boyish tenor, "I understand."
Lukian nodded and dangled the hammer like a toy. "Do you know what this would do to a Jinn?"
Tuor cleared his throat. "No."
"Follow Gorban. He has real weapons for you."
## PART SIX:
## WAR
_We should not be like Cain, who was of the evil one and murdered his brother. And why did he murder him? Because his own deeds were evil and his brother's righteous.... We know that we have passed out of death into life, because we love the brothers. Whoever does not love abides in death. Everyone who hates his brother is a murderer, and you know that no murderer has eternal life abiding in him._
—1 JOHN 3:12, 14–15 ESV
##
Gorban spat on the ground and readied his spear as bodies rushed into a hastily formed crescent, clutching wood and iron shafts with painful intensity. In one chest-thumping explosion, the Fog had ruptured a vein through their safety. The crack shifted the masonry, leaving a half-foot gap between it and the other side, and Tuor shrieked like a child waking from a nightmare and jabbed his spear at the serpent-tail vapors swirling out of the crack. Gorban cursed under his breath and yanked the boy back by the collar. "Stay."
Tuor trembled, but nodded.
The hissing filled their ears. Another eruption and a second crack formed nearly fifteen feet to the left of the first. The isolated section tipped back, its top falling into the Fog, throwing root and soil. They had seen such displays before, but now, for the first time, they witnessed proof that the Almighty's protection truly had vanished.
Lukian addressed the men. "Stand your ground and spear them down. Don't break the line."
The men shouted as the wall disappeared and the Fog rushed over them and condensed on their faces and tunics. The eleven-foot wooden shaft in Gorban's hands was nearly wrenched from his grip as a man-sized Jinn materialized from the white and slid down its length. It was impaled through its chest and clacked slowing teeth. He kicked the thing off and shoved his spear into another's thigh. That one swiped at him and shattered the shaft, and he smashed the splintered end into its face.
Gorban tossed the remnants of the spear, pulled a mace from his belt, slammed the beast in the temple, and pulled its head to the ground to stomp its skull till it shattered like an egg.
The half demons swarmed, their bodies as thick as a wall. They jostled for a chance at human flesh, but the men speared them down. One had three or four impaled on the same pike, their bodies standing wedged in place. Then the spears fell, and secondary weaponry glinted across the field.
Gorban swung twin maces, crushing and twisting the beasts into bloody heaps of flesh. Hands grasped at him. He broke them. Claws painted his skin, but he wasted no thought on wounds. Every faculty was applied to surviving another moment, to outmaneuvering the three or four swiping at him at any one time.
Gorban saw he was being pushed back, becoming isolated from the group. He recognized the tactic, but there were too many, and they functioned with singular purpose. It seemed he fought for hours, though he knew how exaggerated time could become amidst violence. Still the Jinn came. His eyes burned with sweat, and his throat ached with the chill of the air. Human screams mixed into the curdling beast wails, but they were now farther away. His body slowed as he tore one's throat out, deflected another's arm, and planted his foot in a third's chest.
_They are desperate_ , he thought. _Never have I seen them so suicidal._
He imagined how, with each swing, he gave the men and women in the Temple another moment to live, and yet they likely sat looking out at the Fog as it poisoned their city and brought death to their beds. Did they think they could avoid the coming destruction? If the inner wall failed, did they think the Temple wouldn't? _Fools_ , he thought.
Gorban looked down at teeth that had sunken into his thigh, and black stones that seemed to have flames inside them, staring from deep-set sockets. He struck with his mace, but the metal stimulated its jaws to convulse, and Gorban screamed as its teeth sunk deeper, then released. It twitched on the ground as another grabbed his tunic and flattened him to the ground. He grunted as his ribs compressed with the weight, and he felt its claws grapple his shoulders and pull him close.
Gorban clutched its head and struggled to keep it from his throat. He dug his thumbs into its eyes until they burst and bled down his wrists, but the thing's tongue snaked out to taste his chest. He screamed and pushed his fingers against the bone behind its eyes, but the head still approached.
He gazed at the cracks in its skin, powdery gray as if dusted with ash. It seemed strange how the starkest reality could feel invented. Even now, the wound on his leg hardly ached, though he felt the Jinn's hot breath and smelled its acidic saliva.
_Today is the day I die._
The first quake struck silence in the Temple. The second shot Mason to his feet. His heart skipped a beat, feeling the weight of the terrible decision he faced. It had happened too quickly, even from the beginning. Violence had come before its time, and it sped mercilessly on.
He clenched his hands into fists easily twice the size of any other man's, and felt the sensations he had feared in himself. The same sensations he knew plagued his brothers. The same sensations his father, Cain, had lost himself to.
_Some vessels for honorable use and some for dishonorable use._
Mason knelt, folded his hands together, and closed his eyes in reverence. He prayed rushed, private prayers, and felt the shaking come. His face quivered as deeply as the earth had only moments ago, because he knew that he must do what he had been born to do.
His eyes snapped open.
_It is time to fight._
##
As Jacob joined his father outside the Temple doors, he wrestled with disbelief. The wall had fallen, and pooling into the City of the Almighty was the defiling Fog. It curled around the men like water over an anthill, and for the first time, the repercussions of his decision to stay grew thick in his throat.
"The earth groans," Calebna said, as if to himself.
Jacob observed his father's hand, the way it scratched his bearded chin. He looked at the lines on his forehead and the darkness of his eyes.
"The Almighty taught me his true name once. It is a holy thing, never to be spoken by unclean lips, never to be heard by evil ears," Calebna said. Flakes of skin caught in his beard. "Names are important. They remind us who we are. I thought about telling you the Almighty's true name. For many days I thought I actually would. I decided not to."
The Fog deepened and expanded. What about this moment brought Father to speak like this? Why now, as the sons of Cain—no, as their _family_ —fought for their lives?
Calebna folded his arms. "Are you embarrassed?"
Jacob paused, noting the foreign tone of his father's voice. "I don't understand."
"Do you want to be like me when you grow old and weary of the world? Do you want to lie on your bed and think only thoughts you know I would approve?"
"Why does it matter?"
"You look like me, but you are not me, just as Lukian looks like Cain but is his own. I think those differences serve a purpose."
"What do you mean?"
"That there are some things a man is appointed to do. Our souls fit together, each in its habitation, and none can take another's place."
"You speak of our choice to stay instead of fight?"
"I made a vow to the Almighty. As I bowed in worship, I said, 'In the dust of the world, I will remember your faithfulness. In the spring of life, I will let passion spill forth. But I will always serve you. I will always follow your commands.'" Calebna rubbed his eyes. "Is it righteous to make a promise you cannot keep? Often I've wondered, was that my destiny or only a manufactured purpose, something I took on myself to fulfill the role demanded of me?"
"You can still fulfill your vow."
Calebna laughed, a strange sound against the canvas of violence. "That vow stands on the Almighty's epitaph, and one question continues to haunt me, perhaps because I know my answer is not what it should be." He faced his son. "Am I willing to die for him?"
Jacob's pulse quickened. Was Father questioning what they'd built their lives around, what they believed to be true and had held fast to in the threat of death? "You would. As would I."
"Do you believe he lied to us?"
"I—" The words caught in Jacob's throat.
"Don't just answer with a counterfeit truth. What do you _believe_?"
Jacob's stomach churned at the screams piercing the Fog, and he wondered if they were dying, or if perhaps some had died already. "Our lives cannot have been wasted."
"Can they not?"
He nearly yelled out, "No," but something stayed him. _I knew I might die when I vowed to stay_ , he thought. _I knew others might die as well, but seeing it happen makes it so much more difficult to justify._
_So what if I joined the warriors? Wouldn't those who have already been injured or killed have paid with their bodies without reason? And what of my family? Could I betray my own father?_
Jacob swallowed. He could choose only one path, but still he hungered for a compromise, for some middle way.
_Which is worse: to stand for death, or to stand for a life you know has already died?_
He faced Calebna. "You said we need faith now more than before. What better test than this moment? What happened to the righteousness you taught me as a boy? Because it is for those teachings I stand today. I swore to abandon my cousins in hopes the Almighty was not dead, that he would rescue us."
Calebna put a hand on Jacob's shoulder, and he almost shrugged it off until he saw his father's solemn expression. "You must understand. I need someone I can speak this to. It burns." Calebna glanced at the Fog. "There is much I do not understand, but I know that if we survive this difficulty, the world will never be the same. Too much evil has happened. And too much evil remains ahead."
Jacob's thoughts seemed to settle. He saw the reason in his father's words, the wisdom in pressing him. _Was it a test? And have I passed?_
He looked at his father. Truly he did want to be like Calebna, as Calebna had wanted to be like Abel. He desired that peace and strength, that gentle determination. And yet his father's humanity struck him more intensely in that moment than ever before. All his pain and doubt, every fragile imperfection, lay exposed.
_Can it be there is more to your questions than a desire to test me? Can it be you ponder abandoning the Almighty?_
No. Not Calebna. Not the son of Abel.
Jacob shivered and smothered disturbing emotions. "I love you, Father."
Calebna frowned, turned, walked up the short incline to the doorway of the Temple, grabbed the brass handle of the gold door, and swung it wide. Jacob cast one final glance toward the swirling Fog, and turned to join his father in their Temple.
They sealed the entrance.
When they thought they killed the last, more rushed from the Fog. Lukian motioned for them to focus their efforts on the center, but two more streams came on either side. They were being pushed from each other until only Lukian and his children remained surrounded by the swirling Fog. The beasts shoved their bodies between Lukian and his children, and while most were dead within seconds, they succeeded in splitting him from his children.
"Father!" Gillian called as a swarm encircled him and his brothers.
Lukian's face reddened. A beast grabbed his weapon, but he twisted to release it, spun and swung down, bringing the bladed edge of his war hammer through its neck. Its head thumped to the ground with mouth still open, but as it fell, three more assumed its place. He slashed out five, six times. They fell, then the bodies multiplied.
He stepped left, jumped back, slid right, crippled one's knees, and stumbled back, barely ducking in time to miss swiping claws inadvertently slashing another, flicking droplets of blood across his face.
Time ground on like a blade on a grindstone. Eventually another lull came, and Lukian felt the fatigue of his burning muscles. His children were too far to be seen through the Fog, but he could hear them screaming.
His feet pounded as memories from years long past overcame him. _I will not fail. I will not let my children die. And when we return to the Temple, the cowards responsible will pay seven times over for the bloodshed._
The Fog spun in violent eddies growing in intensity until the wind whistled. He slowed, feeling as if he would be lifted from the ground. Droplets of blood skipped through the air. He looked down and saw his feet rooted, but felt disconnected somehow, as if he floated far above. His gaze treaded across the blood-soaked field. The dead grasses had color once more, a macabre mixture of red on brown filtered through the gray mist of the Fog. Discernable just ahead were bodies jostling each other. There were screams, but the hissing made his ears ring. He planted one foot in front of the other, clutched the hammer in both hands, and leaned into the current as his clothing whipped his limbs.
Thoughts breached his mind. It had been long since he felt such rage. And he savored it as he swung at the nearest beast and crumpled its shoulder. It gasped and hissed as Lukian lifted the hammer and smashed its head. Others turned to face him.
"Gillian!"
The hissing swallowed his voice.
"Gillian! Where are you?"
Arms reached for him. He broke them. Legs kicked at him. He crushed them. Joints popped and oozed across the sides of his weapon, and he piled flesh upon flesh as if folding dough into a bloody crust. He clambered over bodies and screamed with each swing of his hammer until his throat swelled and his eyelids crusted with gore.
Just ahead the beasts split as if making way. "Face me!" A few ran toward him. He beat them down. Others fled as he swung his path free of stragglers. He spun and searched, and the beasts backed away.
"Gillian." His eyes burned. "My children."
He heard a low moan, almost imperceptible beneath the hissing and shuffling of the Jinn. He wondered if it had been real, but there it was again. He sprinted forward and the Jinn parted. There, only a few yards ahead, lay a pile of mangled bodies. His head shook before he realized the need to deny what he saw. The hammer fell from his hands. "No."
The Jinn were hissing and staring, but they did not reach for him. He let his gaze fall on the largest of the bodies. It was piled over the others as if even in death attempting to protect those beneath it.
"Gillian," he whispered, and his knees slid in the blood as he worked his shaking hands under his son's back. The face had been torn from the skull and hung in tatters, and Lukian let his face fall on his son's red-soaked breast.
"My beloved son." And the hissing grew as he closed his eyes and felt the warmth of Gillian's life fade in the chill of the air.
##
The Jinn's head jerked. Gorban examined the trunk-like fingers wrapped around the Jinn's skull and watched as it rose and flew out of sight. A figure towered over him, reached down, and pulled him to his feet. "Mason?" Gorban wondered if what he saw was real because the man seemed wraith-like in the mist. But Mason nodded and motioned toward Gorban's leg, which was badly wounded.
Gorban blinked. "I can manage."
Mason turned him from the sounds of battle, and though the Fog clouded Gorban's vision, he understood well enough.
"I'm not leaving," Gorban said.
Mason shoved him, and he fell. The jolt shocked the breath from his chest. He struggled to stand and scrambled for insults, but Mason was already sprinting away, disappearing into the mist.
The screams and moans of the wounded bubbled from the Fog like water from a spring, and the fear he should have felt came like a late moon peeking through cloud cover. Death was everywhere. Gorban wondered how many of his brethren lay dead or holding to life's final strands. Who would retrieve them? Who would patch their wounds?
_Who will bandage_ my _wounds?_
He swallowed and looked where he knew the Temple should be, though the Fog concealed it. Then he braced himself and started toward it.
Lukian wiped his face, but blood and bone replaced the sweat and dust. His eyes burned and his throat ached. His lips shuddered, but he could not weep. His knuckles ached against the handle of his weapon, but as he stood, his senses numbed. He saw, as if the scene were happening before him, Calebna, Terah, and Jacob resting against the wall of the Temple, rolling out the last of the bread. They reached for it with porcelain fingers, pinching fibrous lumps, and dunking them in a bowl of wine before letting the red juice stain their lips.
Lukian stood over his dead children in a field of blood and shadow. No wine to wet his tongue. No food to fill his stomach. No children, no legacy.
The monstrous half-breeds backed away until their eyes were floating flames in the cauldron of Fog. He felt a prick at his neck, and after slapping at it, brought his hand away with two pinpricks of red. He lifted the hammer to look at its gory edge. A putrid scent pinched his nostrils and stabbed his throat. He almost coughed, but instead brought the hammer level with his mouth. He leaned forward and waited, lips parted inches from the edge of the weapon. Rage bubbled up his throat from his stomach, and he shuddered as he fought against a desire he had never before felt.
_No wine for the eldest son of Cain. Only blood to quench his thirst._
He pressed the metal against his face and sucked, letting the cold gel fill his teeth and stick to his face. He tipped his head back and closed his eyes as he dropped his hammer, feeling blood drip from his chin as he swallowed the rot and stifled his repulsion.
_Repulsion. The soul's attitude toward the world it is ushered into, screaming and kicking and covered in..._
_Blood._ It was everything and everywhere as the Fog whirled in violent eddies. He spread his arms, wondering if this was how his father felt as he killed Abel. He knew somewhere in a deep corner of his mind that this was insanity, but for the first time, he felt freed of worry. He held no concern for the future, no fear of death or pain. He simply felt the desire—the _need_ —to consume life.
He should feel sorrow. But could sorrow bring breath back to his children's lungs? Could sorrow bring retribution to those responsible for their deaths?
In the case of the latter, deadly desire could. And would.
He opened his palms and let rage pour into him like a mixture of stimulant and opiate. He wondered at it. How had he known the proper response was to _accept_ this moment?
And yet he knew, and dared not question it again for fear it would break the evolution. He rubbed the stinging spot on his neck and brought the hand away, smeared with thin red lines. He thought he saw a pair of silver eyes glint in the Fog, then the pain disappeared and his body shook with rage. He wrestled a rising panic to fight the shift inside him.
_Run. Don't let this desire take bed in your soul._
He bit his tongue and lost himself in red waters.
Calebna kneeled and bowed, his eyes closed more to avoid looking at the empty throne than to charge the words spilling from his tongue with passion. For he felt no passion, no emotion, no life. All was cold ritual, and he allowed it to pass through him.
"Our Father, hallowed be your name. Your kingdom live, your kingdom grow, on earth as it has in heaven. Give us today our daily bread, and forgive us our sins as we..." He faltered. "As we forgive those who sin against us. And lead us not into temptation, but deliver us from evil."
He thought to stand, but remained kneeling. His eyes opened and focused on the throne's square form. The candles burning on either side melted to the floor, their stump figures slouching with long trails like swollen scars. He smelled the familiar incense, felt the weight of priestly robes, lifted his hands, and closed his eyes. He breathed the words, "I have been faithful to you, as has my family. We have followed you even in the midst of great tribulation and confusion, even as we placed your elements in a tomb. So I ask you to send me a sign. Give me hope where I have none."
For a long moment, his arms floated weightless. Then they descended toward the floor and the sleeves swallowed his hands.
Did it please the Almighty to see his children fall and wallow in the consequences of others' decisions?
Calebna straightened his shoulders and pursed his lips. "I am good. I _am_ a good man." He wiped his hands and stood, then licked his finger and thumb to extinguish the candles one by one, watching the curling trails of smoke float to the ceiling. He swiveled, hesitated, and bowed before turning to leave.
The door was heavy and shut hard behind him, but the cool air in the hallway lightened his lungs. As he passed open doorways, gazes trailed him and a few people called out, asking if the Almighty would save them, but the voices wrestling in the vestibule ahead drew his attention.
As he entered the vestibule, Terah, Keshra, Peth, and two other women stood around a figure lying beside one of the golden lampstands. Terah caught sight of Calebna and said, "Come quickly. It's Sarah."
"What is it?"
"She is awake."
Calebna approached and peered over his wife's shoulder. Lying on the ground, half propped up by cushions, was Cain's wife, Sarah. Her bright eyes were salient and she was saying something, but only when the others hushed could he hear her.
"What has happened?" Sarah said. "Please, someone tell me."
Calebna looked at his wife. "You haven't told her?"
"She stumbled out of the room and fell here," Terah said. "We propped her up with cushions and were just discussing what to do."
Someone said, "She looks half starved."
Another, "Her bones, her skin!"
Still another, "She smells. Those bandages should have been replaced hours ago."
Sarah's gaze bounced from woman to woman. "Where is Mason?"
Lukian's wife, Keshra, said, "I saw him leave a while ago."
Another woman hushed her. "She doesn't know what that means."
Sarah said, "Please, where is he?"
"He has joined his brothers." Calebna contemplated Sarah's malnourished figure. She had been found in the quarry a full day after Abel's death. Could it be she hadn't known Cain murdered Abel? He narrowed his eyes and said, "What do you remember?"
She swallowed and seemed to search her memory. "I don't know." She blinked. "I remember waking to Mason and a fire. He gave me water each time and once some food. But I was tired. I seem to have memories of other things, but they are so dim I can hardly grasp their detail."
The women were touching her, testing her arms and legs.
"Give her space." Calebna pushed them back and crouched. "How do you feel?"
She pulled spider-web hair behind her ear and said, "I'm thirsty, and I hurt."
"Are you hungry?"
"I feel strange."
"You've been sleeping on and off for fifteen days."
Her eyes widened, then she glared as if he were lying.
"You were asleep so long many declared you already dead, but none of us had known of your waking. It's a pity Mason can't speak. We almost gave up on you, though I suppose now we know why Mason never did."
She swallowed and looked at the hanging tapestries, the marble triangles in the floor, the gold walls and vaulted ceiling. She didn't appear to recognize her surroundings. "What has happened?"
Calebna and his wife exchanged a wary glance, and at that moment the door of the Temple burst open. They turned as Gorban hobbled in with heaving chest, and after a few hurried steps, he toppled with a moan and clapped his wrists on the stone. Gorban's wife, Peth, screamed and placed her hand over her mouth. Others whispered, their eyes wide.
Calebna looked at Terah. "Who unsealed the door?"
"I did," Keshra said.
Calebna's neck grew hot. "Why would you do such a thing? You fool!"
"No door can keep the Jinn away," Terah said.
"It will if the Almighty tells us it will."
Peth ran to Gorban and knelt beside him, whispering words of comfort, though he hushed her, already overwhelmed with pain.
Sarah's face paled. "Gorban? My son. What is happening?"
Calebna grimaced and walked to Gorban. Peth helped him lie on his back, but his leg was mangled and beneath it grew a puddle of blood. "Have your weapons failed you?" Calebna said.
"We need your help," Gorban said.
The women began to murmur. Calebna knew they waited for him to speak, but thoughts came to him of the cruel pride of Lukian's speech and of Gorban's arrogant replies at the forge. Calebna's chest rose and his shoulders angled back. "You knew the cost you might pay for choosing violence over peace."
"I'm not asking you to fight."
"Then what?" The sight of Gorban's blood in such a place dried Calebna's mouth and thickened his tongue. _He defiles the Temple with his very presence._
"The wounded. They need help." His chest crackled as he coughed, and Peth rubbed his back.
"You ask us to place ourselves in danger for your sakes, but why should we bear the repercussions for your sins?"
"They will die on their own, but we might be able to save some if we move quickly." He gazed at Calebna as if gauging his resolve, wiped his hands on his tunic, and slipped in the blood as he tried to stand.
"Help him!" Sarah said.
Gorban's eyes widened as he noticed his mother for the first time.
Terah hurried to Gorban and braced his right side as Peth braced his left in preparation to help him stand, but Calebna held out his hand. "Stop."
"But he's wounded," Terah said.
Peth tried lifting him, but Terah did not follow, and so Peth had to set him down slowly.
Calebna thought of the danger of aiding the children of Cain, of the ability of sin to spread its defilement. He thought of the many prayers and sacrifices he had offered for the safety of his family, and the thought of letting them die in an attempt to reverse one man's deserved punishment filled him with disgust. "Let him lie in the bed he's made."
"What?" Sarah said. "Why won't you help my son?"
Terah's hands remained under Gorban's arm as she eyed her husband.
Calebna turned from them. He knew what giving in to anger would do. He needed to keep their trust, but they did not understand the depth of this decision. The depth of their danger.
Gorban shook his head. "You sent us out like sacrifices."
"We chose life, you chose death. We all are in the hands of the Almighty. Pray for forgiveness and he may yet deliver you."
Gorban laughed. "You blind fool. Have you not noticed anyone missing?"
The women's whispers grew.
Calebna twisted back, his calves tensing. "Don't test my patience."
"Philo and Tuor offered their help. We could not refuse it."
Terah stepped back. "My brothers?"
Gorban clenched the cloth around his leg and winced as Peth repositioned him. "They, at least, were man enough to fight."
"Is it true?" Terah asked.
"He is lying," Calebna said.
Peth said, "My husband is no liar." Keshra agreed.
"Would you help them and not me?" Gorban coughed and his face reddened.
"Silence!" Calebna said, and the women hushed.
Gorban laid back and shut his eyes as Peth dabbed away the dirty blood. He took a deep breath and said, "Look for your brothers. You will not find them."
Calebna swiveled on his heel and shouldered past Jacob, who was just entering the vestibule, likely drawn by the sounds of their heated discussion. Calebna sped down the hallway, searched every room, and asked everyone he saw if they had seen Philo or Tuor. None had. The boys' rooms were empty, and the last time anyone had seen them had been the day before.
As much as Calebna could not admit it, he felt the terrible possibility grow in his mind. He wanted to scream and curse. He wanted to rage at the Almighty for allowing such stupidity.
To be betrayed by his own brothers...
He steadied himself against the wall of Philo's room and searched it once more for signs. It was disheveled, as if hastily sorted through. As High Priest, Calebna had been tasked with the responsibility of bridging the space between man and God. With understanding the mysteries of the Almighty so that he could, in some way, aid the people to walk between the lines. But their lives were in chaos.
He lifted his chin and breathed. If they had left, if they were really out there wounded and dying, then he would do nothing. He closed his eyes.
_Let them be condemned._
He nodded slowly and let the truth settle in the cracks of his soul.
_Let Judgment reign._
##
Lukian's fingernails ached. Bloody flesh stuck out from them like so many feathers, and he did not know where he was or how he had come to be there. He simply remembered the moments before plunging into the red waters of rage and violence, and letting them penetrate his soul.
He raised his hammer to get a better look at it. How many had he killed?
A body crawled through a pool of blood. Lukian turned and squinted through the Fog, but could not make it out. He approached, realizing the weakness in his limbs as the curtain of Fog rolled back from animal-like carcasses littering the soaked field. The corpses formed a wall around the crawling body who he recognized was Philo.
Lukian's first compulsion was to kill the boy, and had the inclination been stronger, he might have, but cold logic chilled the burning desire. He watched Philo struggle. The boy's right arm had been replaced by a bloody stump, but Lukian knew the gore Philo lay in was more than just his own.
_What better tool to use against Calebna than his own brother?_
He shook his head and frowned. What had he done by allowing the rage to pour into him?
Tears drew flesh-colored streaks across Philo's blackened cheeks. "I failed," he mumbled. "I failed, I..." He ran the fingers of his only hand through a human corpse's hair. "I failed you. God forgive me."
Lukian hopped the wall of dead bodies and felt a stirring in his abdomen. The boy jumped upon seeing him and cried out, but Lukian knelt and clamped his hand over his mouth. "They'll hear you and kill us. Is that what you want?"
The boy's eyes were animal-wide.
"Did you hear me?"
Philo nodded.
"Will you be quiet if I release you?"
Philo nodded again, but Lukian clamped his mouth harder and said, "If you scream, I will kill you."
The boy's eyes jerked and filled with tears.
Lukian took his hand away.
The boy began mumbling at once. "My little brother, Tuor. What have I done, cousin? He wanted to stay, but I pushed him. He was afraid. He was..."
"You did well. You must know that," Lukian said.
"I said terrible things to make him come."
"You were strong. Your brother was weak like your father. But you are different." Lukian licked his lips and stayed his shaking hands from the boy's blood. Why did he want to taste it?
Philo blinked pale eyelids. Dark liquid oozed down his side from the stump of his arm. His eyes glazed. "No... I..."
Sweat ran down Lukian's neck. "Your life is leaving you, but you've been brave. Will you be brave for me now?"
Philo's forehead creased, and his eyes closed. He clamped his jaw and nodded.
"You were a good brother." Lukian resisted no longer. He reached up, grabbed Philo's throat, and squeezed until the boy's eyes bulged. "Be still," he whispered. "Be still." He slid his knee onto Philo's chest and watched unblinking, feeling mild curiosity and a strange connection to Cain. The boy's eyelids closed over bloodshot white, and he convulsed and went still.
The knowledge that he had murdered Philo brought Lukian a strange sort of pleasure. He knew such pleasure was perverse, but he savored it like honey-dipped bread.
Then something shifted inside, and he fell into the bloody pool. He scrambled up, chest heaving, fingertips prickling. He grabbed his hair and pulled.
_The desire. The thirst. Leave me!_
It was as if a serpent were coiled inside him biting, devouring. At one moment he was repulsed. At the next he only wanted to embrace it. His soul shook, caught between extremes.
But he was weak, and the desire _grew_.
His gaze was frozen on Philo's body, the way it lay there twisted. He wanted to turn, but couldn't. He saw, as if from far away, a finger extended toward the pool of blood. The finger broke the surface, then came to Lukian's mouth, which opened wide to receive it. His lips closed over the finger, which squirmed around his mouth, but he felt nothing. He thought he saw silver eyes in the Fog, and a small figure, like a little boy whose skin was gray with death.
_Lamech_ , he thought. _Why did I urge Lamech to run into the Fog?_
His brother's death screams had echoed on and on, calling for him to help, to do anything. But Lukian ran. He just ran.
The sensations flooded back. He gasped and realized what the wetness on his face was.
"Blood," he said, as it dripped from his lips.
Sarah was crying, and she slid her arms around herself and squeezed. The memories were flooding back quicker than she could handle, and she rocked.
_Do they know?_
She rolled onto her side. She knew they watched, but no longer cared. She couldn't keep everything coiled inside. She knew from Gorban's wounds that something terrible had happened.
_No._ Lilleth's lifeless eyes floated through her mind. _Many terrible things have happened. Where are you, Cain? You promised you'd come back. You promised..._
God had forsaken her, and she was trapped in his Temple. Why else would she be here but to receive justice? She longed for companionship, but to open herself to another felt like a fate worse than death. Though she hated and feared Cain, he seemed to her the better of two damnations.
_I don't want to die._
"What have you done?"
She jumped at Jacob's voice.
"What happened? Have you failed?" Jacob said.
Gorban's reply was sharp. "Learn respect, child."
Terah walked to her son. Her lips trembled with suppressed emotion, and she slid her arms around him and wept. Jacob held her, looking surprised and alarmed.
"My brothers are out there." Terah's voice was muffled by his shoulder.
Jacob rubbed her back.
"I know my brothers are out there," she repeated.
"Father still has not come back."
Terah just cried.
Gorban gazed at them strangely, and Sarah wondered what was churning in his mind. Then his gaze met hers, and she looked away and clutched her robe. The fibers were rough, and the hard floor hurt her bones, weakened by days of inaction. _Has it really been fifteen days?_
Sarah closed her eyes, but the images only burned brighter. She glanced up. Gorban still watched. She turned away and hoped he mistook what he saw for anything but what it was.
For the first time since waking, she dared to touch her stomach. _I am sorry, my child._
Gorban spoke to Jacob as his wife, Peth, cleaned his wound and bandaged it with linen. "Your father is no different from us."
"What does the son of a murderer know?" Jacob said.
"Your father has murdered too."
"You speak nonsense."
Eve entered with a small container of wine, several cups, and some of the last bits of stale bread and a handful of nuts. After hearing of Calebna's reaction, she had rushed off in grim silence to prepare refreshments for Sarah and Gorban. Now she knelt beside Sarah, who accepted a portion of the bread and a cup of wine with a nod before Eve took the rest to Gorban.
"Calebna refused to tend to the wounded," Gorban continued, ignoring what Eve placed beside him. "Has he not then pronounced judgment?"
Jacob rubbed his fingers and frowned.
Terah grabbed Jacob's face. "Think, son."
He pushed her hands away.
"Calebna plans to purchase his safety by sacrificing us." Gorban continued.
Eve laid her hand on Gorban's and said, "Peace, son of my son. You must rest." She glanced at Jacob. "We have stirred up enough strife. Let us pray instead and be silent."
##
After Eve led them in the prayer for safety and guidance, Jacob sat thinking of the sounds he heard while gazing at the Fog with his father. Words came disjointed and yet connected in ways that he knew held meaning.
_"The world is dying... a holy thing, never to be spoken by unclean lips, never to be heard by evil ears... remind us who we are... our true nature... do you still want to be like me?... he is not Cain... those differences serve a purpose... we all have a role to fill... in the spring of life, I will let passion spring forth... am I willing to join... ?"_
Jacob swallowed and let his soul embrace rebellion. It felt righteous. It felt holy. He wondered, _Is this what Cain felt as he murdered my grandfather?_
The blood on the floor of the Temple smelled rank. _Lives should not be wasted with such impunity. This, at least, I know is evil._
_So am I to turn to evil to fight evil?_
His mother's body shook as she leaned into him again. She believed her brothers were out there, and Father would not let Jacob leave, no matter the truth. Jacob had seen changes come over Father, and the fears he suppressed only a few hours ago returned. If what they said was true, Father had embraced a God that Jacob never could—one devoid of mercy.
It would take no violence. It would demand no violation of his vow.
_"We all have a role to fill..."_
Jacob set his jaw, steadied his mother and pushed her away. Her breath caught as she looked up. Gorban stared, and despite the bitter words they traded only moments before, Jacob felt a connection with him he'd never had. He nodded, and Gorban's eyebrows crouched, but Jacob offered no explanation. He simply straightened, walked toward the door, and fulfilled his role.
The sounds were so dense Mason could almost smell them. Sensations from battles long past broke forth as he rubbed his blood-crusted fingers together. The air in his lungs felt more like scraping teeth than Fog, but his legs stretched on, smashing the ground like cedar trees. He ran over fallen Jinn, popping their ribs beneath his callused feet and leaving massive impressions in the blood-soaked earth.
Piercing cries mixed into the sounds ahead. He had seen so few human bodies littering the battlefield, and that gave him hope, though only a little, and so he prayed. In the midst of the madness, when no other man truly believed, he prayed to the Almighty because he knew the Almighty had never died.
But how could a mute speak of the wondrous mysteries of God? How could a voiceless man sing of the Glorious Intervention ordained before the Spirit's breath buffeted the formless waters? Before the Word spoke and earth and fire and body and soul rolled off his tongue and into its habitation? Before Adam tasted sin, death, pain, and loss? Before Cain, his own father, murdered Abel?
Indeed, his task was not to speak, but to lay a foundation for the One to come. So he ran, and he fought, and he listened to the voice from his dreams, as he had from the very beginning when he found his mother half drowned in the storm.
Most of all, he prayed.
Jacob had never seen so many corpses.
_What am I doing out here?_ He wiped his face.
The Fog hung over the battlefield as if Time itself were suspended. Here, nothing moved. No life. Only faint sound in the distance, and he made his way toward it. He passed mangled bodies and checked them to make sure they weren't Philo or Tuor. One was Mellore, Kiile's second oldest. The other was marred beyond recognition. Neither moved.
Jacob walked on, jumping over piled carcasses and splashing through bloody grass. The bodies became fewer and soon failed altogether. He closed his eyes to shut out the world and feel safe again.
_Speak to me. Give me a sign. If it be your will that I go farther, I will, but I need to know. Are you dead like everyone says?_
His ringing ears strained. Had he heard something or was it only his imagination? He held his breath and waited, scanning the white for sound or sight.
Then there were footsteps, approaching hard and fast. His heart raced and he twisted to flee, but instead slipped in the blood.
The footsteps slowed. It was Lukian, and swinging at his side was a bloody hammer. He held what looked like a dead body, and upon seeing Jacob, he stopped and studied him.
Jacob scrambled up and took a step closer to the body in Lukian's arms, searching the face. Recognition burned the nape of his neck, and he closed his eyes, wondering if he would vomit.
"It is he," Lukian said.
"And what of Tuor?"
"Torn to shreds and cold to the touch. I found them together, but could carry only one."
Jacob began to weep. "Please, lead me to Tuor."
"It is too dangerous."
"It could not be far."
"I will not take the risk. You shouldn't be out here unarmed."
Jacob noted a strange twitching under Lukian's right eye, but said, "May I carry Philo home?"
Lukian approached and set Philo in Jacob's arms, and Jacob dipped his head in respect. He thought about thanking him but instead turned, said a short prayer, and began carrying his father's dead brother back.
Lukian did not follow.
Jacob glanced back, barely able to see Lukian's motionless shape through the Fog.
"What will you do?" Jacob asked.
"I must stay."
"What is out there?"
"Go!"
Tears burned Jacob's eyes, but despite his desire to stay and fight beside the man who had retrieved his uncle's body, he brought Philo to the Temple.
His tomb.
##
The hammer hung in both of Lukian's hands as he gripped it with a resurrected fury. He hungered for violence as deeply as a desert dweller thirsted for water.
_Gillian. My son._
He bit his cheek.
_The desire..._
He couldn't trust himself to help the wounded. At the very least, he had made an impression on Jacob, and hopefully the boy would aid him in making an impression on the others. When the battle was over, he would need them. He only hoped this deadly desire wouldn't continue mounting.
_Almighty, don't let it grow._
First he would end Calebna and any others responsible for the death of his children. Had they come to help, fewer men would have died, and maybe Gillian would still be beside him.
_The time for peace has ended. The time for violence has begun. Once Calebna is out of the way, they all will follow me. It is survival, evolution, progression._
But what about Jacob? The boy was different. He had risked his life for Philo, and yet he had shown his weakness through tears.
A scream opened the Fog before him like a screeching gate. He sprinted forward, the path lit by dull gray light. Ahead, a mass roiled like dark waves on a pale shore, but soon more details leapt into the light. It was a throng of Jinn, possibly one hundred or more, arranged in a circle around what he could only guess was the final bastion of Lukian's brothers and nephews.
Lukian poised his hammer low and leaned forward, bursting into a sprint. He lifted his hammer high and reveled in the crack of hammer against skull. He spread the beasts and glimpsed Machael fighting alongside his sons from the inside. Machael's eyes widened as he slashed a Jinn's arms off at the elbows and his boys opened up a path for him. Lukian rejoined and saw Kiile, though the man's youngest and oldest were gone.
Together the brothers and sons fought, and the demons fell like insects. Lukian noticed beasts turning in confusion. Then, bursting through the ranks, came Mason, crushing skulls with stone-hard knuckles. He split three heads in five strikes, then joined the group.
"You're late," Lukian said.
Mason peered into Lukian's eyes, and Lukian looked at the blood dripping from Mason's fingers. The red juice drew him, but with a glance up at Mason, he saw the man's eyes glint.
Lukian lifted his hammer, feeling its weight like never before. But with Mason at his side, and Machael and Kiile and their children, he would bathe in blood until he drowned.
"You should have left Philo outside the Temple," Calebna said as he resealed the Temple with iron bars. To Jacob, he looked bent and aged, more a vulture than a man, and the lines about his mouth ran deeper than his frown.
"How could I?" Jacob said.
"Holiness is only difficult to those who deny its power." Calebna's eyes strayed to Gorban, whose leg was bandaged. Then he looked at Philo, whose skin was paler than the stone he lay upon.
There was long silence. Philo's wounds no longer bled, and one simple, inescapable fact bound their tongues. He was dead. Jacob couldn't stop wondering when it happened. He wanted to know the moment, the spot he had stood upon when the boy's soul left his body. As much as he didn't want to admit it, Calebna's contempt burned to the center of his chest. "I think he might have died in Lukian's arms."
"I'm sure he did."
"At least Lukian had the kindness to bring Philo back." Jacob didn't know why he was defending Lukian, but something about his father's demeanor made him bristle.
Calebna's eyes twitched, and Jacob wondered if his father would strike him. "Philo disowned us."
"Philo died protecting us," Terah said.
Calebna glanced from Jacob to Terah, who returned his glare. Calebna sucked his teeth and nodded. "I see." He laughed and looked at the ceiling. "This is what you give me? A family of cowards?"
Eve was adjusting Sarah's cushions, but stood. "Cowardice? Is that what you think this is?"
Terah's face was red, and her hands were balled into fists by her side. "With every word you sound more like Lukian."
"I am the High Priest, the intermediary between you and the flames of Judgment."
"And what of you? Who will judge you? Are there no consequences for your decisions?" Eve said.
Calebna's eyes darkened. "I have maintained my integrity in the face of death, but who might I turn to when I cannot trust my son? My wife? I do not envy the violence stored up for you in the depths of eternity."
Terah scoffed and fell silent.
Jacob's face chilled. He said in a low voice, "For years I have feared your shame, but now you have won mine."
Gorban raised his voice, pointing at Calebna. "Cain killed one, but you've done better. You've offered us all at your altar."
Calebna slammed his fist into the wall. "I'm the servant of God!" He no longer hid his contempt. "As the Almighty spoke, 'Behold, the wicked man conceives evil and is pregnant with mischief and gives birth to lies. He makes a pit, digging it out, and falls into the hole that he has made. His mischief returns upon his own head, and on his own skull his violence descends.'"
"I pray you never feel the sting of your words," Eve said.
Jacob turned and urged his mother out of the vestibule, though she remained in place.
"Now," Calebna continued with finger pointing at Gorban, "their violence falls on others as well. We have no hope apart from the Almighty. If everyone had only listened to me, none of this would have happened. If Philo hadn't been—"
Terah said, "I take back my words."
"Come," Jacob said. "Let's go."
Terah struggled against Jacob and threw her words like javelins. "You're not a good man. You're not a man at all."
"What's going on? Why are they just standing there?" Kiile said.
Lukian held his hammer outstretched.
"Why aren't they attacking?"
"Quiet," Lukian said.
The Jinn were still. Deep in the Fog footsteps crunched gravel, and the Jinn parted. A shape moved toward them, dark and slender amidst the beasts.
"Great Almighty," Kiile muttered. "Is that... ?"
Lukian dropped his hammer.
A man strode toward them, and the Fog peeled from his gray skin as if he were formed from it.
_It is an apparition, a ghost_ , Lukian thought.
But it was real. Too real.
The man nodded and spoke to Lukian in that familiar, resonant bass, "Hello, Son."
## PART SEVEN:
## BEYOND THE SANDS OF TIME
_You were the seal of perfection_ ,
_Full of wisdom and perfect in beauty._
_You were in Eden, the garden of God;_
_Every precious stone was your covering:_
_The sardius, topaz, and diamond_ ,
_Beryl, onyx, and jasper_ ,
_Sapphire, turquoise, and emerald with gold._
_The workmanship of your timbrels and pipes_
_Was prepared for you on the day you were created._
_You were the anointed cherub who covers;_
_I established you;_
_You were on the holy mountain of God;_
_You walked back and forth in the midst of fiery stones._
_You were perfect in your ways from the day you_
_were created_ ,
_Till iniquity was found in you._
—EZEKIEL 28:12–15 NKJV
##
Waves rush past and slide the man across the Sands, some of which grind between his teeth while he hovers in the twilight of Time. He is stuck on the edge of awareness, floating. Just floating. The peace tugs him toward rest, and yet the pink glare of Light through his eyelids pains him. His head thumps against a boulder, and he opens his eyes with a moan.
He squints in the Light while sitting in the salty Water with his back to the boulder, and observes the white Sands stretching away on either side, the glittering blue-green Water, and the jungle trees guarding him from behind. Warm wind rides the waves crashing on the Sands. He breathes and rests on the heels of his hands, letting a smile form on his lips despite having no reason to feel happy. He does not know who he is, where he is, or why he is sitting on an endless shoreline. He merely knows that he is here, and that there is only "now."
Perhaps that is why he smiles. There is such peace in disregarding what _was_ or _will be_. But he is startled by the vague thought of something he cannot quite remember. He tries to retrieve it and runs against a barrier. Something about existing in only the present makes his mind spin, and he feels displaced, as if he does not belong here.
But why should it matter? He smiles again. "Now." It contains everything. All that matters is "now," and everything contained within "now," which is, of course, infinite. These truths strike his mind as illogical, yet his soul revels in their robustness. All is fresh and all is stale. There is joy and peace in grappling with the oneness, and simultaneous individuality, of everything.
He stands and walks the beach to feel the sand between his toes, the warmth of the Light and the saltiness of the air. There are jungle smells; trees, undergrowth, and mossy decay. Other scents remain too subtle to describe, but together they bloom and grow to something gestalten.
He shambles up the incline toward the line of trees and stops just short to examine the boughs laced together like the weave of a shirt. They creak, pulled by the wind in an endless dance as it leads with hands on leaves.
_How many arms the wind must have to dance with so many._
He approaches to join, and when his fingers touch one of the tree trunks, it shivers. He smiles and presses his hand against it only to see it slide aside and open a path into the jungle. He pauses at the entrance, and the green-scented breeze brings whispered melodies to his ears. He stops to hear them better, but they quiet as if carried reluctantly from the forest. As he stands contemplating whether there might be Music playing within, a strange sensation stirs him to turn.
He swivels and gasps upon seeing a speck of a figure bobbing on the waves. He sprints down the Sands and arrives at the Water's edge as he realizes the speck is a young woman. The shoreline catches her with open palms, and his shadow casts long as his feet splash her shoulders. Her mouth is slightly open, and behind the heart-shaped lips are teeth like milky diamonds. She opens sea-bright eyes and smiles.
He crouches, meets her gaze, and says, "Hello."
"Hello," she replies.
He extends an arm and she slips her fingers between his, which are much darker than hers. She stands, but their hands remain entwined. They turn toward the jungle and, as they near the trees, the melodies return like the scent of roses. She looks at him.
"Care to dance?" he says.
She nods, and though he doesn't understand the strength of his desire to explore, she seems to share it, and together they walk under the leafy canopy.
The path grows longer, and all around is the Music. They laugh and run, only barely keeping their fingers clasped as the trees part with increasing speed to make way. They break through the brush into an open space filled with stone ruins, and the Music hushes.
Now comes a pregnant silence—the living space between notes.
Arches, domes, and towers lie half-fallen across the ground. He examines the constructions with wonder, and brushes fingers across porous rock lined with zigzagging patterns of fleshy plant life. The ocean air is further humidified by the ever-breathing grove that, other than the creeping vines and lichen, keeps its distance from the buildings, as if the stones poison the ground.
Her thumb rubs the back of his hand, and he turns. There is love in her gaze, and he finds no difficulty returning it. She means much to him, though he doesn't understand why. He leans in and kisses her lips, so soft and warm, and she kisses back. They admire each other and continue through the ruins.
She points to a black recess in one of the walls, and they walk to it. It is a grim opening, tarnished by weather and vine. A few of the stones lie on the ground like fallen teeth, and a staircase tumbles down its throat. He tugs her forward and they enter and find themselves in a tunnel with windows every fifteen paces or so spilling Light like Fog into the cobalt shadows. They walk with fearful reverence and arrive at a doorway in the shape of a shield with its center pressing toward them in a needlepoint. Forming a circle around the shield is an assortment of letters. They are strange, but somehow he understands them. He tips his head and brushes his finger across the chiseled words to feel their peaks and valleys. The Shrine of the Song. Who knocks?
The Music returns, marked by a tentative hush, as if the players themselves hang on every movement. The melodies twirl in gentle arpeggios, conjuring images of little rivers tumbling toward unknown bends. The man and woman look at each other and back at the shield, which stands guarding them from what lay beyond. "Who am I?" he says.
She shakes her head. "I don't know. Who am _I_?"
"I don't know. At least, I don't _think_ I know."
She frowns, examines the shield, and brings her fingers up to fiddle with the point. She accidentally pricks her finger on the tip, and draws her hand back and nurses it as he fights a nearly irrepressible urge to do the same. He makes a motion as if to turn away, but his hand darts forward and presses against the tip, drawing a drop of blood and mixing with hers. He pulls his hand back and the needle retreats into the shield until it disappears. An unseen mechanism grinds, and the shield rolls away, revealing a new, expansive chamber.
The Music deepens and expands, yet retains a measured patience, and he wonders what it means. He slips his arm around her waist and nearly asks, but it seems she probes his eyes for the same.
The new room is circular and its walls are formed from polished stone. Ornate script runs along the bottom, and pillars rise like arms to support the ceiling with many hands. A single mural spans the room, and he spins to take it in, though it is steeped in shadows.
"What is this place?" he asks.
"I don't know." She squeezes his fingers as if to hold him more securely. "It makes me uncomfortable."
A cold sensation settles in his stomach as he looks at the carven wall. He approaches for a better look, but she doesn't follow out of the light, so he lets her hand slip out of his and, as his eyes adjust to the shadows, is surprised at the detail. There are countless persons in the mural, and yet he can see the expression on each face and the stitching in their clothing.
"You need to see this," he says.
"There's something here," she says.
"This is incredible."
"Come back. Hurry."
He turns and sees her back against a pillar. "What's wrong?"
She points and her green eyes frost. "What is that?"
Now, as a deeper note of the Music strikes, a silver shape advances, and darkness slips from its shoulders like a cloak. It pierces them with a hollow, metal gaze. Its arms bend at the elbows. Its legs are slightly longer than what would be normal on a human in relation to the torso, and each has two knee joints bending in opposite directions. The entire creature is bright silver with only rivets and lines where sheets begin and end to add detail.
They hold their breath, expecting the thing to continue, but it does not. The man edges toward it, but the woman pulls him back and wraps an arm around his bicep.
"Don't..."
Amplifying curiosity distorts his face into a frown. He asks the thing, "What are you?"
Its gaze narrows, and it speaks from some hidden mechanism, "Who am I?"
Her fingernails dig into his arm.
"Are you not more interested in knowing who you are?"
"Why do we fear you?" the man says.
"Because I have the power to puncture your skin and spill your blood on this holy ground."
Both man and woman retreat. "Stay back," the man says.
"You have little right to make demands. You cannot stop the Music."
"We want neither to hurt you nor to be hurt by you."
_Machine._ The word comes to him, strange and familiar at once. _That is what it is called. But how do I know that?_ The woman tugs his arm as if to urge him to silence, but he refuses, for questions compel him to speak. "Why are you here? What is your purpose?"
The machine says, "I am the keeper of the Shrine."
He thinks of the fallen ruins, the unkempt entrance of broken molars, and the name on the door.
_The Shrine of the Song..._
Thoughts link through his mind. "Who owns this temple?"
"The Master."
She keeps pulling, but his desire to speak with it grows greater still. The Music splits. One line ascending, the other burrowing. He says to the machine, "Who made you?"
It does not respond.
She whispers, "Don't."
He says, "When were you made?"
"I exist."
"How? What does that mean?"
The machine says, "Time is a law of nature binding the physical realm to relatively steady movement down a unidirectional pathway, but 'now' is all that 'is' here. We sustain outside the boundaries of Time. And so there is not a _time_ when I _was_ made. I did not _come_ to be. I _am_ , as you _are_."
He nods. Of course. He knows that. Or does he? He feels exceedingly strange in this place.
"We're sorry for disturbing you," she says. "We will go now."
He scowls and says in hushed tones, "I'm not leaving. Go without me if you're so afraid."
She winces as if his words cause pain and flicks her gaze between him and the machine.
"What is this Shrine's purpose?" he asks.
The machine says, "It is a haven for all who seek something more."
"And do we seek that?"
"You are a part of the Music, are you not?"
He frowns. He is. Or at least he thinks he is. But his heartbeat strikes polyrhythms. "The doorway asked who we are but opened to our blood. Why is that?"
"You remain on the brink of both ends; a son of twilight."
"You speak in riddles."
"I speak plainly."
"Then speak more plainly."
The machine pauses. "You give your blood to the Shrine, and though the offering is symbolic, it represents your life, and the Shrine accepts you for it. This is not your home, yet it may be your resting place."
"Where then is my home?"
The machine says nothing. It merely stares, its metal eyes jerking from time to time.
Her hands wrap around his arm, and she asks, "Is the Music you speak of what is playing through the forest, and in this Shrine?"
"The Music is the Song of the Master."
They shuffle to the mural, now more fearful of the machine than the dark. The machine remains as if bolted in place, but its gaze follows them.
She brushes fingers across the mural and stifles a gasp. In the image, countless beings march in rows, carrying banners and weapons toward a City. The stone smells aged, like a tomb long sealed, but he knows that is neither an image nor a thought that belongs in this land beyond Time.
She turns to the machine and points at the leading figure in the mural, the one holding what looks to be a trumpet. "Is this it? Is this the Music?"
"It is."
"What happened to them? All these figures in the picture?"
"They make war."
"But here is a mural of them."
"They make war."
"You already said that."
"I _say_ that they _make_ war."
Its rebuke makes her chew her lip. "What do they war against?"
"The other Music."
She points at the marching figures. "Who are they?"
"They are the Watchers, the Sons of God."
The man thinks of the abandoned ruins outside and says, "Why is everything so empty and lifeless here?"
"It is not so empty as you think. If you wish to see more, I can show you."
She fidgets and tugs on his arm. "We have lingered long enough."
"But—"
"It is time to leave." She jerks him toward the entrance, but he does not want to leave, and his mind scrambles for some way to stall her.
"Do we have names?" he calls out.
The machine says, "You do."
She stops and offers a wary glance.
"Do you know them?" the man says.
The machine says, "If I tell you, will you stay?"
He looks at her, and she presses her lips together.
"We will stay for a time," he says.
"Your words will bind you," the machine says.
"I bind myself to my words."
"Very well. Your name is Seth, and hers, Ayla."
He tries to imagine owning that name, but his mind rejects it as unfamiliar. He looks at the woman the machine calls Ayla. That name seems strange as well, but when he mouths their names, something about their shapes feels familiar.
He looks at her, remembering her desire to leave. He points at the mural. "Where is this City? The one under attack?"
"It is the center of this world."
"And where are we now?"
"The Shrine of the Song lies on the outskirts, near the Sands that keep the Waters of Time within their borders."
"Is the City occupied?"
"A different Music plays there, and it finds discord with ours. If you wish to see more of the Song of the Shrine, I can show you."
Indeed, the desire to see the rest of the Shrine and listen to more of the Music swells in his chest. "Please, lead on."
Ayla glares, but says nothing.
The machine walks past a few pillars, and Seth pulls her to follow. It presses its arm into another shield-like door, which rolls away just like the first, revealing a long passage down into darkness. The floor is smooth, yet not steep enough to warrant danger. She stops urging him back, but her hand clutches his arm, and when it becomes too dim to see, the machine's eyes cast light such as two lamps.
Questions breach the surface of his thoughts. "You spoke of Sands that hold Time. What does that mean?"
"The Sands are where Time begins and flows outward in ever-multiplying rivulets. Without the Sands, Time would not know where to begin and end, and would flow on endlessly until all movement and depth vanished."
"How are we here?"
"Your bodies are dead."
Ayla's grip tightens on his arm. "We are dead?" Her face registers the same fear he feels as terrible realization grates his skull and numbs his tongue. He asks no more questions while they journey down the tunnel, and the only noises are of grinding metal, and the fleshy slap of foot against stone.
##
Rows of light glimmer down the corridor as more flicker to life like so many glowing insects. They hum and buzz dully as they hover, causing Seth to keep a wary distance.
"They are not alive," the machine says as Seth sidesteps one with a suspicious glance.
"Where are we?"
"The Hall of Worship, where the Watchers keep vigilance."
Seth imagines beings filling the hall and trailing after one another in reverent lines, but it is massive and bare, and imagined secrets bloom like nightshades. The machine leads them down the corridor, past pillars and opulent floors. The hall curves to the left, but ever so slightly as to be almost imperceptible. They arrive at a set of broad doors on the left-hand side, and the machine opens them. They enter another hallway, only this one is smaller, with many doors and no pillars.
The machine stops and says, "This is the Eastern Nexus, from which the Chambers of Science branch. Wish you to see them?"
"What are they?"
"They devote themselves to the Mystery of Life. It is the Master's hope that the Watchers may find greater life through the Chambers of Science."
"Lead on."
"You must choose your own path now. Each note struck here resonates the instruments till they sustain with synchronized troughs and peaks."
Seth looks at Ayla, who shrugs. "So we are to choose a door and open it?"
"Choose carefully."
He brushes his fingers over subtle artwork, like shadows burned into the wood of each door. After viewing four, they examine the doors on the opposite side. Back and forth they shuffle until they see an image of a flower. Ayla points, her eyes widening. "The Aylana, the White Flower! It's the very same I was named after, though it is black here." She frowns. "And I cannot remember who named me. This memory loss is so very strange." She looks at the machine as if desiring to ask it all her questions, but the Music rolls on, gaining volume and momentum.
Seth grabs the handle of the door with the black flower on it, his hands growing clammy, but before he opens it, he whispers, "Do you want to go in? We do not know what lies beyond."
"I am not as afraid of this one as I am of"—she pauses and glances at the beast on the next door—"the other ones."
He looks at that image, a lion with the tail of a serpent and a forked tongue springing like dancing fire from its jaws. He glances at the door in front of them and notices thorns at the base of the Aylana. "Why would this image be here if not to function as a sign? It nearly bears your name."
"Perhaps the Shrine, or maybe the Music, knows us." She nods as if conceding a begrudged truth. "But we know so little of it."
"It is causing us no pain, no discomfort. Indeed it is fascinating."
She nods, but looks hesitant.
"We will view just one room. Then, if you wish, we will turn back." He smiles, and she attempts one in return. He throws the door wide and lights glimmer as if awakened by their movement. Their glow is filtered through glass and liquid, and the kaleidoscopic reflections dance as the pair makes their way forward, the machine following with head bowed and footfalls like cymbals. Throughout the room are cylindrical vats of liquid tinged green as if by algae. Lights gaze up from the bottom of each, and floating in them is what looks like twisted branches and roots.
Seth sidesteps another hovering light and moves toward the glass. Suspended within is a rose with petals closed and drooping. Its stalk flows serpentine to the base, where a treelike root, thicker than the flower, stretches like a fist striking at the bottom of the vat.
She grimaces. "What is it?"
"I don't know." They speak quietly, pitching their voices so the machine, which lumbers close, might not hear. Seth glances at it cautiously. "Take care."
She nods and squeezes his fingers. The air holds its breath, and its pulse is like an underwater timpani. It waits for them to speak, to determine the direction of the Music, but they force it into stasis, too frightened to pluck the wrong note.
They move to the next vat and observe another plant that seems but a simple vine with leaves sprouting in symmetric patterns. Upon closer inspection, tiny bone-like projections, reminiscent of teeth, sprout from the edges of the leaves.
The air grows thick and warm.
Ayla whispers, "These plants aren't like anything I've ever seen."
"They look altered in peculiar ways."
"They do evil things here. I feel it as if it touches me this very moment." She hugs herself and rubs her arms.
"The Watchers," he says as if remembering an ancient secret.
"I do not think we belong here. It is wrong for us to be here, to open these doors at all." She presses her fingertips against the chilled glass, and the plant seems to move in response.
Something about the crossbred organisms draws him, and he cannot turn away. In subsequent vats are a bush with flowers sprouting from its roots, a flower with a pink object resembling a tongue, and another vine with bone-spines and malignant growths like bloody contusions.
Ayla turns. "I don't want to see any more."
But Seth is compelled to see more. The Music turns, descending from innocence toward skin-tingling darkness, raising the hairs on his neck and his desire to see more, to give in, to become one with the Music as a dancer's body becomes the manifestation of musical emotion.
They edge toward the door and Seth's throat dries despite being filled with saliva. He says, "The machine says we are here for a reason. I wonder what led us from the beach to the Shrine and pushed us down the stairwell."
"I think it was nothing but our own curiosity," she says.
"We cannot turn back now."
"Yes, we can."
"And go where?"
"Somewhere, anywhere."
"We are meant to see this."
"But I hate it. I hate how it makes me feel."
_I should take her far from this place._ He forces a smile and feels a quivering in his abdomen. "It's the Music. It is leading us. It knows who we are and what we want, and it is offering what we need."
"But what if we're here by accident, and the darkness I feel is a warning in my heart?"
"I don't believe in accidents."
Her eyes search his, and he can see her resolve crumble. She does not speak. She simply grabs his hand and follows him down the corridor.
Desire blooms into itchy satisfaction as they make their way through the endless rows of vats stretching on until they become too small for the eye to register. They return to the Eastern Nexus, find the door of the lion with the tail of a serpent, and enter together, hungry for what might be seen.
The lights flicker on, but the vats that greet them are hideous. Floating in the pinkish liquid are hardly recognizable cadavers, malformed hunks of meat suspended in torturous positions.
_Why are they here?_
Seth swallows and touches the glass with a shaking finger. The creature's eyes roll and lock on his. Its lungs pulse in quick, sharp movements. Its mouth is twisted open, locked in place by deformed joints, and the image becomes a blur in his pained eyes.
Ayla's fingernails dig into his hand, and he shakes his head and wipes his face. Her gaze burns his cheek, and he tries ignoring the same thoughts he knows trouble her.
What torture is this creature enduring? An eternity spent drinking in the pain of such an embryonic existence—what hell could be more terrible?
But the _Music_! It is beautiful and terrible and _growing_.
"The Master says it is one of the most beautiful sights." The machine walks to the vat and inserts its arm into the apparatus. The beast's eyes bulge as it rushes from the glass and stares at the hand in the apparatus. The water fills with gray smoke like Fog.
"What are you doing?" Seth says.
The murk disappears, as if a current is sucking it away, and the fluid is clean.
Blue water. The vat is empty.
"What is happening?"
"A morsel of Time is stolen from the Waters and suspended in the vat, and inside is woven the fabric of life. It is the Master's supreme art, performed in true Time-suspension."
_Suspension_ , Seth thinks. _I feel suspended above black water. Such terrible darkness, such brutal addiction. I am burning with fear, and yet cannot plug my desire. I am thirsty, so thirsty... And Ayla?_
"Where do the creatures go?" she asks in a trembling voice.
"To the Master."
"To do what?"
"To serve."
Seth cannot look away. He cannot stop, and it seems neither can she. It is as if he only watches decisions made by another.
"Let me show you the Master's greatest creation."
Seth remembers the picture on the door, the serpent's tail, barbed and poisonous. He can feel it lurking in the Music, a whipcord striking every measure, subtle yet relentless.
"The Master calls it Nephilim," the machine says.
It seems at first like a man, but the eyes are silver, the skin is gray, and curling out of its skull are two dirty gray projections, like the horns of a ram.
"What is it?" Ayla asks.
"It is one of you taken and transformed. A soulless half-breed. Before Adam dies, the physical world will see it walk among its cousins. It is the next evolution of what your people call the Jinn, only as applied to humanity instead of animals."
"But if it is soulless, how then does it live?" Seth says.
"An empty body may be filled. Indeed, the intention of the Master is that the Sons of God might walk about as kings of men. Here, as well as there, I prepare the way for them."
The machine manipulates the console, but the reaction of the Nephilim is wholly different from the other creature's. Its lips slide into a skeletal smile, and it grabs its arms and digs. Wisps of red float to the top. The gray Fog comes, now disappears.
"Can you show us the rest of the Shrine?"
"You wish to move on from the Chambers of Science?"
In shaky unison, they say, "We do."
"Very well, follow me. There are many rooms left before us, and they are greater than these, though none quite so distinct."
They move on, but the hum of the Chambers continues—adding complexity upon complexity and growing in both beauty and ferocity.
##
The Shrine expands. Seth finds it strange that a building can be alive, let alone move and change, but steps continue to materialize as the machine methodically pushes the stone walls ever deeper.
_Where are this puppet's strings? By what will does it move? Is the Music its life? The Song is strange—strange and powerful._
And yet, in the absence of Time, he recognizes there is something greater still, as if glimpsing a star through the cracks in a moon. What is seen is only a shell hiding the true glory, though for the present it is all they know.
The deeper they walk, the farther through the octaves the Music cascades. The staircase ends and an archway appears, gilded in silver and shining like pearls. Beyond the arch the way widens to a dome, and the machine's eyes dull reactively. In the great basin revels a pool lapping against the sides of its domain, like a crooked smile undulating in ecstasy.
"This is the bathhouse of the Watchers," the machine says.
"They bathe?" Ayla asks.
"Not such as humans do."
Somehow, Seth senses the purpose of the room is not to wash filth, but to rinse cleanliness with unbridled pleasure. The machine wades across the pool. There is no ground on which they can escape, but neither does Seth desire to. He steps toward it with veiled excitement, and as his toes touch the water, existence itself becomes indulgence. Prudence flees and he falls headfirst, gluttonizing the water through his nose, mouth, and skin. He vibrates with the hum of the pool and senses his being ravaged by the rhythms.
The meaning, which sounds at first like a muffled groan, crashes into him.
He kicks in the pool and breaches the surface, grasping for substance with hungry lungs. He feels like a drum, hollowed inside and filled with violent air thrusting against his shell.
Ayla crawls from the pool next to him, and they lie dripping in the archway next to the machine. Her eyes are ringed with darkness, sunken into tired sockets, and he wonders if his look the same. They stand and shake the remnants of the pool away, though, oddly, it seems to remain.
With terrible clarity, he realizes the Song is not what it pretends. It monotonously propounds excitement. It screams to fill the pit with sound. It kills to claim life.
_Contradiction. That is its lifeblood._
_What if our purpose in coming is not to experience the Song, but to stop it? What if we are appointed to sing a different Song?_
She whispers for him to hold her and rub the perversity from her skin, but he feels incapable, and she reaches an arm around his waist instead. The machine turns down a new hallway as the Shrine begins expanding once more, revealing new rooms, hallways, and staircases. They follow it with a different spirit, attempting to shrug the weight planted on their shoulders by the pool.
The Music—or is it so many hammers striking anvils?—dims in the Light of a new awareness.
"It is the heart of the Shrine." The machine sweeps its arm toward the monstrosity filling their ears with measured noise. "It is the pump of the lifeblood, the percussion in the Music. It is the Metronome."
"But Time does not exist here," Seth says.
"The Metronome is a multiplicity, an anomaly lodged between the layers. It is one of the Master's many masterpieces, and through it we can glimpse the workings of the Song in other Places. Come."
It ascends a staircase leading over the Metronome. They follow and see from their vantage what can only be described as a bubble set into the Metronome like a great eye, the surface of which is akin to a membrane of oil with iridescent markings shifting constantly.
Seth asks, "What is that bubble?"
"That is the Metronome's telescope. It reveals the glory of the Song, and though some notes may be only vague images, others are vivid phantasms."
Seth bends over the rail and peers inside. The telescope seems to expand, and he wonders if he is leaning backward or forward, for it engulfs his vision. In the ever-shifting shadows, he senses arms reaching toward him, fingers straining and sliding around him and pulling until he feels himself removed and fastened to the Metronome's belts. The movement does not frighten him. It feels natural, as if his soul were meant for such an action.
Visions appear, and some remind him of dreams he might have had while under the influence of Time, which he feels closer to, like part of him has crossed through the Metronome's eye. But he is incapable of setting both feet in any one Place, and he remains suspended as he views a piece of Time here, a scrap of life there.
He watches people with gray skin heap piles into a foundation. The foundation spreads until there is no escaping its totality, and on it, cruel empires build towers and warriors beat their chests. They march with footsteps in line with the Music, and flames rise and lick the lens.
Now comes Water—a singular element from outside the Music. It fills the foundation until it overflows like a great basin and washes the world clean. As it recedes, there comes more gray-skins who build another foundation, and from it rises a Tower whose peak pricks the stars.
The visions move on relentlessly, but all is the same. There are empires, buildings, technologies, symbols, fires, disasters, wars, and plagues, but Death is all that reveals itself. There is a simplicity, a repetition in the Music, like a measure replayed. _The Music is a circle_ , Seth thinks. _It is the circumference of the telescope, a great engine sustained through the whole of Time._
_They build to tear down to build to tear down. Contradiction. An opposing statement._
_Rebellion._
This is all the telescope shows. The Music rolls on and the Metronome releases him, though it sets its rhythm in his chest, and he is moved to synchronize with it.
"Now you know the beauty ordained—the inevitable thrust of the universe."
The machine's voice jars him awake. He blinks and sees his surroundings, including the Shrine, for what it is.
_The Music is a mindless machine, an instrument itself._
_But played by what? Or by whom?_
He thinks of the people. The gray-skins. They choose the paths they walk, but seem to forfeit their will in the end. Do they then become lifeless machines? Hollow Instruments?
"Come now," the machine says. "There is another chamber I desire to show you, and it is the greatest of all."
The machine leads them to a vast room that is made up of seemingly endless angles set against each other and that is filled with a singular object—a massive headless body. Instruments of every kind weave in and out of the four limbs and torso, and they play as if the fingers of many masters attend them. Seth and Ayla's mouths gape at the sight, and the machine motions them closer. More than in the bathhouse, Seth feels the sensual pull of the Music. It is more forceful, more violent, and more beautiful, and they quake in the Song's enormity.
"This is the Master's body," the machine says.
"This? This is the Master? Where is his head?"
"The Master _is_ the head. The violence of the Enemy's Music separates the head from the body, but the Master's body plays on." The machine opens its chest as if peeling back flesh, and the metal separates to reveal gears, belts, and glowing parts reminiscent of organs. "And I, the machine, remain self-existent through the brilliance of the Master. I am the first of his children, my purpose to tend the great Music of the Master and prepare the way for the Sons of God to come in the skin of men."
It leaves itself exposed and stares at them. It is terrible, but they stare because the glory of the machine, and the pulse of the Music, hypnotizes them.
"The Music exists to defy the Enemy," the machine continues. "Every note is an expression of this base intention, and every chord, every melody line woven through the Music, is sung by one of the Enemy's creation."
"His creation?"
"The Sons of God and of man. Do you not wish to sing the Song? Do you not wish to become an Instrument as magnificent as I?"
Seth cannot deny that he desires it. It is truly great, truly beautiful, though it is terrible to behold, and his desires strain against each other.
_But there is something missing_ , he thinks. _Some ruinous ghost in the hollow between notes._
"Why should we sing this Song? Why should we become the Master's Instruments?" Ayla says.
"Because the Master longs to adopt you, to free you from the Enemy. The Song shall break the Enemy's chains, if you allow it, and if you become an Instrument of the Music, you may add to it whatever your heart desires."
"We may sing anything we want?"
"It matters not whether your Song be Power, Pleasure, or Pride. All are one with the Music of the Master."
Seth teeters on the edge of the Music's embrace. As he looks, the machine gazes as if seeing his nakedness revealed. The Music hums, and his eyelids lull as he feels himself drawn invariably toward it. He remembers the control of the Chambers, the pleasure of the Pool, and the magnificence of the Metronome, and he wants it.
He wants it all.
But in all the sensual beauty moving in seemingly perfect fluidity, a note squawks his mind awake, and he sees the Music and the machine with disgust. The allure redoubles until the room shakes so violently he can hardly see, and he feels himself once more slip under the Song's command.
Ayla's fingers slip into his, and he looks at her and sees her mouth moving. Though all sound is drowned in the violence of the Music, he hears the words in his mind.
_I love you._
The Music screams, and the silver temptress is revealed for what it is—an Abomination. The smile yellows, the eyes bleed and stink, and all pleasure falls away like rotten fruit.
Seth meets Ayla's eyes, and they sprint forward, grab the machine by the arms, lift it into the air, and carry it toward the body of the Master. A metal arm swings down into Seth's abdomen, and he grunts. His knees buckle as they drop the machine and it falls into the swirling machinations of the Instrument. The Music grinds the machine to a twisted, sparking heap, and there is an ear-bursting screech like scraping strings, and a rumble of stones like falling towers.
A hush descends. The machine is in a ruinous heap, but its eyes...
Its eyes remain aimed at Seth.
Seth cradles his abdomen. Ayla's voice is shrill in his ear. "You're bleeding?" But the sound is nearly lost in the groaning approach of the Music as it begins anew.
"Merciful Almighty, there's so much blood. Get up. You have to get up."
He looks up and his eyes sting with the dust of falling stone. His fingers feel thick, cold, and sticky. Something strains against his arm, and he stands to relieve the pressure. It twists him around and he stumbles, sensing crumbling violence filtered through the darkness overtaking his vision.
There is Music like cruel noise, as if all is only painful percussion. No singing, only shredding screams and gargling gutturals, and the tug of...
Ayla. Her arm against his, so beautiful and thin and strong. And her fingers. They are cold.
Pain. He bends and gags and nearly vomits, but she jerks him up, and he stumbles forward.
"Hurry!"
"I am," he says, or does he merely think it?
Light stabs through the darkness. His heartbeat washes the Music from his ears, and jungle air is thick in his throat. She is kissing his face with those beautiful lips, so soft, so perfectly formed. She is telling him of how close they are, and that if he just stays awake, it will all be fine.
He hopes she doesn't stop, for the pain is unbearable, but her voice and lips are gone, and they are rushing forward again. His toes grab sand and he falls into Water, and the last thing he thinks, as Time engulfs him, is how Ayla's fingers feel as they slip out of his.
Then there is only the Music, rumbling and shaking the Waters with thunderous peals.
## PART EIGHT:
## THE RETURN OF CAIN
_But when you see the abomination of desolation standing where he ought not to be (let the reader understand), then let those who are in Judea flee to the mountains. Let the one who is on the housetop not go down, nor enter his house, to take anything out, and let the one who is in the field not turn back to take his cloak.... For false christs and false prophets will arise and perform signs and wonders, to lead astray, if possible, the elect._
—MARK 13:14–16, 22 ESV
##
Cain studied the way Lukian's eyes bored into him. Was it disgust, or hatred? Mason's arms hovered at his sides, tensed, and Cain's other sons, Kiile and Machael, stood alive and prepared for violence. The Abomination had wanted them, too, but Cain would give it no more.
_Do they sense what prowls inside me, and how it lusts for the very warmth of their veins?_
Mason edged toward Lukian. Kiile nudged Machael and said, "Is it him?"
Lukian shook his head. "Cain is dead." He lifted his bloodied hammer. "And even if he weren't, there's no place for him anymore."
Cain laughed, and by doing so, let the Abomination slip past. "I see," the Abomination said through Cain. "You were doing well without me? I thought you were in danger, but now I see."
Cain thrust the Abomination back and wiped away the moisture that beaded on his face. Mason's fingers twitched. Kiile shuffled left, then right. Machael stared with those half-open eyes. Lukian wrung his hands on the handle of his weapon while the children waited for their elders to react.
"It's an illusion. A demon from hell," Lukian said.
"You're right. I have changed, and return more than a man," Cain said. "Days ago, in the darkness of the wilderness, I died. But then I rose again, crawled before the Almighty's throne, and bled the life from his veins. I've brought a bit of hell back with me." His eyes widened as he fought against the Abomination's demands for control. _This is my time_ , he thought, _my own_. "You know the truth, but some have yet to admit it. The Almighty is dead. And I am returning to claim what is mine."
"And that is?" Lukian said.
"Your worship."
Their legs were rooted like plants awaiting harvest, their rosy cheeks plump with blood. He quelled the pale thirst and wondered if they sensed his struggle.
"Are you insane?" Kiile said.
"Bow," Cain said.
Lukian spread his feet and gripped his weapon with both hands, and Mason drew up his knuckles. Kiile and Machael pointed their weapons forward, and their boys did likewise.
The Abomination's laugh was loud in Cain's mind. It said for his ears alone, _"Resistance can only be met with violence_ ," and Cain's nostrils flared. "Would all of you rather die than follow my commands?"
"You abandoned us, but we survived your absence," Lukian said.
Cain said, "Where are your children?"
Lukian didn't respond. His face reddened.
"When you found their bodies mutilated on the ground, why didn't you breathe life back into their lungs? Why didn't you heal their wounds?" Cain said.
"Shut your mouth."
"So angry."
Lukian lifted his hammer high and burst into a sprint. Cain readied himself as Lukian swung, and just before the hammer struck, he grabbed it by the head, stopping it midswing. Lukian pitched forward as his hands slipped, and he tumbled to the ground.
Kiile yelled, and Machael and a few of the younger boys dashed forward. Lukian scrambled to his feet, brushed them off, and faced Cain, who held the hammer by the head as if it were a toy.
Silence. Mason, Kiile, Machael, and the others stared. No one moved.
"You have two choices." Cain tossed the hammer toward Lukian. "Bow to me"—he snapped his fingers, and eyes lit the Fog like stars in the sky—"or die."
Lukian grabbed his hammer and raised it. There were fingerprints pressed into the metal where Cain grabbed it.
"Think of your wives. Think of the children too weak to stray from the Temple," Cain said.
"You're a coward," Lukian said.
"No, Son. I am God."
To Calebna, the Temple seemed alive with whispered prayers. The boys, including Jacob, stood in a corner of the vestibule with arms crossed, and the women sat like seeds scattered from the sower's hand. They were blinded by the simple fact that God had not saved them as Calebna had claimed. To them, Philo was not just a corpse, he was the embodiment of God's abandonment. It didn't matter that the boy had run headlong into danger. Philo's motionless body was the final emphasis to their bitter question: _"Why have faith in a dead God?"_
And Calebna stood in front of them as the last remnant of that faith.
_I lost control_ , Calebna thought, _and Philo and Tuor died while I abased myself before the Almighty's shadowed throne._
_After all the sacrifices I spilled before your feet, after all the love I lavished on your name... Is the presence I sensed nothing more than the imaginings of my mind?_
_Almighty God, if you truly exist, then kill Lukian. Take his life before he steals my family. Remember me before I disappear._
Gorban crawled to Sarah, who lay curled into herself. Calebna was surprised he felt no hatred toward the wife of the man who had caused all of this.
_But we are both scarlet letters drawn in the blood of our kin. No one could have stopped him from killing my father. I know this now for certain. Cain and Lukian, alike in so many ways._
_And what of me? What of my father?_
Abel had walked between the lines Cain had trampled, and for all this time, Calebna had tried carrying on his father's legacy, as he saw Lukian carrying Cain's. But he could do it no longer. Calebna was no holy man.
_I'm the worst of them all. I'm a dirty man who thought himself clean._
_Kill Lukian. Kill him quickly._
Terah returned through the archway, passed Calebna as if he were a ghost, and handed Ben to Keshra, of all women, the deepest insult. As Terah knelt beside Philo and pressed her face into her brother's bloody breast and wept, Ben gazed at her with disturbing solemnity.
_Her opinion of me is ruined by grief. And bitterness. Or are those two separate? My brothers are dead and I feel... nothing._
He glanced at Gorban, who examined the red and brown soaked through the bandage on his leg. Peth came out again and replaced the soiled fabric, and Gorban cried out as the wounded skin tore with the fabric, but soon clean wool secured his leg, and the two lay in each other's arms.
Calebna stood a while longer, and as the flavor of failure made bitter his soul, three hollow knocks turned his gaze toward the Temple doors, and the last knock resonated long and hard, like a gong sounding the passing of an age. He blinked as if waking from a long dream, walked to the towering doors, slid the iron bars from the handles, and pushed the doors open enough to reveal a host of bloodied men.
Lukian, Gorban, Mason, Kiile, Machael, and their boys shouldered past him. Then a voice he knew all too well made the hair on the back of his neck prickle.
"Hello, Nephew."
From behind them entered a familiar figure leading a host of Jinn. The women screamed at the sight of the beasts, and some fled the room, but the familiar man lifted his hand, and all the world seemed to hush.
Calebna blinked. It looked just like Cain, but there were details amiss. The skin looked as if it had been cured and dyed a dusty gray, and black vertebral marks shot down his arms and lined the top of each finger like snake bones. His lips were a deep shade of red so as to appear black, but his eyes were iridescent silver. They shone in the dim torchlight like marbles polished round and popped into his eye sockets.
Calebna waved his arms, and his voice was angry. "You let demons defile the Temple of the Almighty?"
"This is no Temple anymore." Cain laughed. "It is just a cage for fools. And you've caught yourselves for me."
Calebna shivered. Gorban's hand slid to the weapon at his side, but Cain's glare swiveled toward him. "Release your weapon," Cain said.
Gorban didn't move There was dangerous silence.
"The beasts behind me will rip your throat out if you do not obey me."
Sarah grabbed Gorban's arm. There were tears in her eyes, but they seemed not to come from fear. "Do as he says."
Eve also came up beside Gorban and laid her hand on his shoulder. He let go of his weapon.
Calebna gazed at them, a tattered and broken people, driven to the edge by starvation and demonic pressures. They were at their weakest, and the demon, or man, that looked like Cain had chosen this moment to appear. Such a thing could not be a coincidence.
Cain spread his arms. "What you see is truth. I am Cain, and I have returned for my family."
Calebna's legs shook with the effort to stand, and he found his mind fighting to disbelieve. His eyes narrowed as he took in the differences between what he saw and what he remembered. _If you truly are who you appear to be..._
"I have come to save you from yourselves. To save you from the world my father corrupted." Cain paced. "Tell me, where is he? Where is Adam, first among men?"
"He is sick," Eve said. "Since you left." And her gaze bored into Cain.
Cain said, "Go then, and bring Father here, for I have returned to establish the New Religion and usher us into our glorious future."
Eve motioned for Machael to retrieve Adam, and so he left, though not before hesitating with his hand perilously close to the weapon at his side.
"We have reached the threshold to our future as a race. Immortality is within your reach. All that is left is to accept the invitation. You need only pledge yourselves to me, and I promise I will give you everlasting life."
Lukian turned his back to them and rolled his shoulders, and Calebna wondered what had happened on the battlefield. He scanned the men who had returned and realized none of Lukian's children were present. He knew how volatile Lukian could be. If the man's children had been murdered, there would be nothing that would stop him from exacting revenge on everyone he thought responsible.
But why would he not look at Cain?
Calebna observed the Jinn standing in the doorway. Did Cain control them? Was he in league with them? He looked back and said, "There is only one Tree of Life, and it is guarded by a cherubim and a flaming sword. You say you will give us immortality, but you lie. It could never be so easy."
"It isn't easy," Cain said.
"Then what is it?"
"An alternative to death."
Silence.
Cain said, "I have been to the depths of hell and ascended to the throne of the Almighty. I have met the Devil and killed God. Look at you—gathered in your Temple, so afraid, so weak. You worship the dead, but I am the Living Death. I killed God and I became God. And now your God speaks." He offered them an open hand. "Follow me or die."
##
Seth opened his eyes and strained to grasp at the smallest bit of light, but found nothing. He brought his hands to his face and felt his eyes. They were open, but he saw nothing. His mind registered the stiffness in his back, and his nose was assaulted by vile smells. He twisted and threw out an arm, but his hand smashed against stone and he cried out.
Shapes and forms joined in his mind. He pulled his hand back, massaged his wrist, and winced, but in the blackness he was unsure his facial expression had changed. He carefully touched flat walls to his left and right, and with difficulty and no small amount of shimmying, edged his arm above his head and found a ceiling.
He was alone in a small, enclosed space filled with darkness. His mind raced, recalling strange images of an ocean, a forest, an underground labyrinth, endless vats filled with tortured creatures, and a machine. Melodies bloomed over remembered instrumentation, and though the sounds were no more than phantoms, he knew their shapes held meaning. It was a great Music, and he and Ayla had been drawn by it into the underground labyrinth, and followed the machine until...
He patted his stomach. He didn't feel any wounds.
_I could be dead_ , he thought. _Is this what death feels like?_
But he had already died before washing up onto the Sands. The machine had attacked him, and he had bled and lost consciousness in the Water. If he weren't capable of dying again, then what did the wound mean?
He shifted onto his back and tried to clear his mind. His breath reflected hot and stale, and the smell of death permeated every inch of space. He pushed against the walls and screamed until his voice tore like a dry garment, but no answer came.
He slammed his feet against the sides, then scooted his legs up until his knees were against the wall in front of him. He held his breath and pushed until his head swelled with blood. The wall popped free and light stabbed his eyes. He gasped and recoiled, curling his knees up and covering his face with his forearm.
He slid his arm away and his eyes struggled and ached. There was an ocean of blue, and in it floated white islands. He wondered if he was high above, looking down at a vast sea. His eyes regained their focus.
"The sky," he whispered.
He grabbed the stone cover that he realized was above him and, as he leaned up, pushed it until it fell to the earth with a thud. He stood shakily and hopped onto the brown grass that cracked like dry bones. He straightened and looked at the stone tub that had encased him. It sat like an empty stomach belching stench as if filled with dead rats.
"A tomb?" He rubbed his temple with two fingers and waved the reek from his face. He spun on his heels and, for the first time, noticed the Temple on the hill behind him.
The whole world seemed dead. The trees were burnt husks, hollow chaff of a world that was no longer. Though the Temple was here, this was no longer the City of the Almighty. This was something else entirely.
His eyes traced a line over two identical boxes sitting next to his. He stared at them for a few blank moments. They were tombs, sealed shut as his had been.
_Ayla!_
Seth ran to the closer of the two and pressed his ear to the top. There was no sound, but that meant nothing. He tried to pull off the sealed cover, but couldn't. He dug his fingernails under the edge until he bent his nails back. He grabbed a stone and smashed it against the seal. As he made his way around, pounding the seal as he went, pieces of the stone broke off and stung his skin. He tossed his tool, pulled the lid up, and flung it away. The tomb was empty.
_Who would seal an empty tomb? Had another dead man woken like me? But it was sealed..._
A high-pitched, muffled cry prickled the hair on the back of his neck. He turned and ran to the third tomb, his heart hammering his lungs like they were bells. He smashed the seal and threw the cover to the ground, and there lay Ayla curled into herself. She screamed and squinted against the brightness. But a moment later, she stood, threw her arms around him, and sobbed into his shoulder.
"You're all right," he said. "I'm here." He rubbed her back and arms and kissed her hair. "I'm here." Upon feeling the rise and fall of her chest against his, all doubt fled, and he believed. He cried and laughed at once because she was alive. Because they were both alive.
"Was it real?" Ayla said.
"I was there. The Water. The jungle. The Song."
"I thought you were dead," she said.
He hushed her. "I'm here. So are you."
"I see it when I close my eyes."
"It's gone. And we're alive. Everything is going to be all right."
She pressed her face into his neck, and he looked out on the dead grasses and trees, the shriveled flowers and brown pall. He swallowed and his throat pressed against Ayla's head. Memories returned from before he awoke in that place beyond the edge of Time. The longer he thought, the more he remembered. And the more he remembered, the more he trembled.
_I lied to her. Nothing is all right. I've escaped my dreams only to land in my nightmares_. "God, wake us up," he whispered.
"What?"
"Nothing." He kissed her hair and held her tighter. "Nothing."
##
Cain crouched, plucked a blade of grass, brought it to his nose, and sniffed. His heightened senses constructed a list of the few animals that had brushed against it: a mouse, a fox, and a wild dog. But the smells were days old, and little was left of their delicate forms. He rose and scanned the rolling hills. There were patches of green here and there, the first visible signs of life during the last two days of travel.
_"The curse spread quickly."_
The Abomination. Cain was struck by the oddity that no one but he knew of the Abomination's existence. No one but he knew that he was possessed by the very son of the Devil.
He lifted a hand to his chest and clutched his breast where a pain had developed the last two days. The pale thirst was strong, and grew stronger every day, but to feed was to nourish the Abomination. That, of all things, was what he most feared, and so he waited.
Someone slipped loose rocks from the cliff, and the sound of them skipping into the valley of the City of the Almighty echoed on and on. Cain turned. Lukian had pulled himself over the lip and brushed sand and dust from his tunic. Kiile and Mason followed and lifted Gorban, whose leg remained unusable.
The people had grown soft in the comfort of nearly two years of safety, but the awareness of life in the wild would wake them soon.
_"Lukian's mind is burning against you."_
The Abomination's words kept breaking through his thoughts. He was angered by it, but felt the sting of its perception nonetheless.
"Learn your place," he whispered. "It is early still. He is my son. My eldest..."
_"Now you taste power, but can you swallow it?"_
His tongue tingled with the sensations. It was like tasting blood, like drinking life. He looked at his gray skin, saw the paleness, and wondered how many more days he could wait. Would he need to slip away unnoticed?
Mason dipped to help Sarah up. She clutched his arm as he lifted and set her atop the cliff. She stood tall and erect as she let go of her son's muscled arm, and her dress rippled against her slender frame as she stumbled on a rolling stone. Mason caught her by the forearm, but still she looked so elegant, notwithstanding how gaunt she was from days of starvation. He noticed the nearly imperceptible bump of her lower abdomen growing to accommodate what developed inside.
_Pregnant_ , Cain thought. _My unborn child grows, just as she said. I knew it was more than a dream._
_"So why will she not look at you, let alone touch you?"_
He clenched his hands as the Abomination returned to wag its tongue in his ear.
_"She should fear you. It is better yet that she hate you. You cannot allow any to become too comfortable in your presence. That was the mistake the Almighty made. They slipped from his grip and began to question him. And they will question you too. Suffering is the only true teacher. Be free with your rod."_
He realized he was nodding, and Lukian was watching. Their gazes met, and Cain could hardly hear anything beyond the Abomination's exclamations. He quieted it and, with great struggle, rejected its grab for power. It had become so difficult to maintain control, and doing so now squeezed sweat through his pores. He wiped his face and shifted away.
The Abomination. Bloodthirsty destruction manifested in personhood. It wanted to consume—to _grow_ —but the time was not yet ripe. The Abomination thirsted for the human soul above all else, and he could pacify it for a while with animals, but there was a chance that Lukian might still prove useful. Cain hadn't yet been able to test the Waters for the certainty he needed, but he had sensed the possibility.
He chuckled, though he knew the Abomination heard and rightly divined its cause. _The trouble with possibilities is they contain both deliverance and danger. I must wait. I must endure the chaos. Just a little while longer..._
Lukian stared at him, and Cain turned eastward, determined to focus on their destination.
_Towering trees and leaves like fans in the morning. The smell of damp life, the relief of shade, and the touch of cool water bubbling from deep underground. A place we can begin anew. A place to observe the New Religion._
"The place where I am God," he whispered, resisting the foreboding he sensed from that place beyond Time.
The desert wilderness made him long for green life. He thought back to the days spent in the underground City of the Light Bringer, and how the relentless stone and pale light had grown to a buzzing drone. But he bit his cheek upon thinking of the dark chamber and the goblet of blood. Cain was no longer the same boy straining for Adam's acceptance. The Abomination had changed him.
Or had he changed himself? Upon giving himself to the Abomination at the edge of his life in the wilderness, both his person and body had become something new, but where had the opportunity for that change originated? Cain and the Abomination had merged in an unholy coupling, like naked bodies in the dark, and yet even then the change had not been fully realized.
He breathed and flexed his forearms to feel the strength that coursed through his veins; part gift of the Almighty, part human evolution. He savored it as he savored its beginnings through the sound of a breaking skull. He had done it. He had proved his supremacy and gained a power none had imagined.
Now he was God. He was Satan.
But inside there remained turbulence. With so much power, how could he not calm the Waters? Why had everything become so much more perilous?
_Because you've set yourself in the Abomination's crucible. You've used it to gain supremacy, but at what cost?_
He did not know. Perhaps the Man would have known, but the Man was dead.
Kiile whispered to Gorban, who hushed him, and Cain turned and spoke so they all could hear. "If we continue at the same pace, we will arrive a day early. Though it will be a strain on the wounded, I assure you it will be worth the struggle. The haven I have prepared for you is without equal."
Sarah shifted her eyes away as Cain neared. He was captivated by her neck and shoulders, the way the musculature wove under her delicate skin, and how the artery rolled to the rhythm of the Music—or was it the other way around? He averted his gaze and cleared his throat. "I assure you that when we arrive, everything will become clear."
Cain sensed Sarah's gaze flick his direction, but only for a moment, and looked in time to see her twirl away. Mason observed him, then turned and followed his mother.
Cain stared at the space she had occupied. The others watched him, and he suppressed the Abomination's motion for control. "We rest for the night."
##
The Temple entrance loomed as Seth reached for Ayla's fingers and remembered the smell of the air as they stood before the mouth of the ruinous Shrine of the Song. "The last time we saw this Temple, it was shrouded in darkness and storm."
Ayla nodded. "I barely reached you in time. If I hadn't seen you hobbling into the storm and followed after, what would have happened?"
"I wouldn't be here. And neither would you."
She brushed the golden doors. "This world feels so unreal in comparison. I keep wondering if maybe I am sleeping on the Sands and dreaming of us being in this place. That maybe you really are dead, as you should have been after that terrible machine..."
"You know this is real."
"So what was the Shrine? What was the Music? What was the machine, the forest?"
"That too was real."
"I saw it stab you. I held my hand against your deadly wound and felt your blood between my fingers. If that was real, then how are we here? _Why_ are we here?"
"I don't know. But we must behave as if both worlds exist in more than just our minds." He quieted and thought once more of his nightmares.
Ayla said, "Do you think we will find ourselves there again?"
"I hope not."
"I keep thinking of what the machine said about another City where a different Music plays."
"I remember it speaking of Watchers as well, but we saw none. Maybe the City is nothing more than a myth, something the machine used to entice us deeper into the Shrine."
"Maybe," she said. "But what if it does exist? What if that City, and the Music that plays there, was what we failed to find in the Shrine?"
Seth remembered the sensuous pull of the Song, how he had longed to give in, and where those desires brought him. What could war against such darkness but Light itself? He smiled, feeling a bit of the normalcy between them from before they opened themselves to the Music. "I thought I was the positive one."
"That's why you smile so much."
Seth's ears retuned to the deafening silence, and his smile faded like the color of the world they inhabited. He grabbed the handle of the Temple door and swung it wide. Together they entered and shut the door behind them.
After their eyes adjusted, Ayla said, "Someone is in the sacrificial chamber."
He strained to see past the ghosts of sunlight. "What do you see?"
"Torchlight, I think."
Slowly, a dim glow appeared at the far end of the room, as if from around a bend.
She squeezed his hand and whispered, "Someone is here."
He hushed her. "Don't speak until we know it is safe."
"Why?"
"Do you not remember what happened before we first came to the Temple in the storm? If everything we heard was true—as I believe it is—if Cain murdered Abel and the Almighty is gone, then this Temple is no longer what it was."
"The Almighty isn't gone," Ayla said.
"What do you mean?" he said.
"You say everything we remember is real."
"We saw his empty throne, his bloodied crown and robe."
"We saw a Place that should not exist," she said.
"What about the silence of the City of the Almighty? The dead grass and trees?"
"We _died_ , but now we're _alive_ ," Ayla said. "You know as well as I do that we should have sunk into the depths of those Waters. Something took hold of us. That's why I let go of you."
It would make a terrible sort of sense if he weren't able to remember the nightmares that had thrown him into seizures before they passed into that world beyond worlds. "You may be sure, but I'm not. Just trust me."
She paused long, then said, "I will."
"We stay to the shadows as much as we can. And don't let go of my hand. I will not be separated from you again."
He thought she nodded, but heard no response. He sensed in her silence, in the way she moved, that she knew he was keeping something from her. He wanted to brush it off, but could no longer behave as if the world were no more than a reason to smile. He had seen the future and experienced the Song, and for her own sake, he could not tell her much. Because telling her would change her behavior and influence the outcome, and she would not survive it. However, silence held its own dangers—some that he couldn't avoid.
Seth pulled her closer as they edged through the hallway. There was a sharp smack of metal against stone, and she jumped. It happened again, and they flattened against the wall of the hallway where the shadows were deepest. Again a loud smack, and with it a grunt and the sound of toppling tables or lampstands.
He slipped past Ayla and motioned for her to stay, though she followed anyway. As they reached the end of the hall, they both peeked around the corner. She breathed sharply, and he clamped a hand over her mouth and pulled her back.
Ayla pushed his hand away. "Is that...?"
He motioned for silence, but said, "Calebna."
Calebna had carefully gathered the rugs, tapestries, lampstands, tables, and benches, and placed them in front of the altar that was still dirty from the celebration that marked the beginning of the end of his life. Calebna had to scoot the ashen bones of Abel's sheep in order to make room for the items, and still pieces tumbled down the sides.
"Sacrifice." The word echoed through the chamber, brilliant and brief, like a spark underwater. "The last servant of the Almighty stands where all things have come to an end. Here, he will join his God."
Calebna walked to the wall where the torch sat, wrapped his fingers around the wooden handle, and lifted it from its receptacle. He stood watching the flames drip, then walked to the pile and touched fire to oil. Flaming tides swept up the heap, and the fabrics burned and crackled and sent sparks to the ceiling like dying stars. Calebna stood back, raised his arms to the blaze, as if listening to some far-off Music. Oil dripped from his hands as he closed his eyes.
"As evil was born through one decision, so it will be destroyed through atonement."
_Atonement. The exchange of life for life. An endless cycle._
Today, for Calebna at least, that cycle would end. He opened his eyes and walked closer, his sandals crunching glowing ashes. "They left me because I believed in you." He wiped his face with his forearm, feeling emotions claw out of hiding. "I lost everything for you. For a lie."
Calebna took another step forward. The heat of the flames urged his eyes closed, but he spread his arms wide. He thought of Terah, Jacob, and Ben. It had been two days since they left, and though he had been sure Jacob, at least, would have returned, he hadn't. No one had.
This was how God rewarded the faithful. This was the lot of the righteous. He had done everything he could to serve God and lead his family in the way the Man commanded, yet he would die alone.
_By my own hand, and in the time of my choosing._
"Now." He let the word linger amongst the sizzling refuse, as if the flames would listen. He opened his eyes to examine the force that would unite him with his God. The heap was a molten, slumping mountain. Tongues of fire licked the air, and beyond them...
_What is that?_
He squinted and discerned an upright shape in the hallway beyond the sacrificial altar.
He stepped back from the flames and shuffled past the pile to see more clearly. Yes, he could see a man standing in plain sight in the entryway. A man he recognized, but who could not be there, because he had been dead for weeks.
"Seth?" Calebna's lips dried, and he passed his hand over his eyes. The man was still there. "You died. You were dead. You _should_ be dead."
Seth stepped closer.
"Stay back," Calebna said.
Seth raised his palms. "I won't harm you."
Calebna pointed at him. "You came to take me with you."
Seth frowned as if confused. Speaking slowly, he said, "I have come to take you with me, but perhaps not to where you expect."
The heat of the flames seemed to singe the hair on the back of Calebna's neck. He shook his head and dropped to his knees. "Not there. You can't take me there." He clenched his teeth. "I gave my life for him."
Seth laughed. "I'm not here to usher you into the afterlife."
Calebna glanced up. "Either this is a dream, or I am dead." He looked at the fire and imagined his body lying there. Had he killed himself without realizing?
"Give me a moment to explain. Don't be impatient."
"Patience has given me only pain. I won't be swayed by the lies of a demon," Calebna said. Was he lying there, his skin crackling and bubbling in the heat?
"I am no demon. I am Seth, third son of Adam, once dead and twice alive. I have returned to this world to do something. Maybe to help you." He glanced at the flaming hump of refuse, then at Calebna.
"If you're not a demon, then tell me how you are alive. I helped seal you in your tomb. I touched your body after Mason found you dead in this very Temple."
Seth's smile faded.
Calebna waved his hand. "Enough lies. Leave me in peace if you won't give me answers."
Seth stirred and licked his lips. "Why not put your torch down and follow me outside?"
"Because I'm going to kill myself."
Seth stopped. His reaction made Calebna question if he could be real. But there had been so much death. If Seth truly were alive, why him and not Abel? Why now instead of days ago, when he needed a sign, when he pleaded for help to the point of tears? He turned away. "If only you were more than a ghost."
"Do you deny that a ghost could possess knowledge you do not?"
"I deny your existence."
"What danger lies in following me and listening, just for a moment?"
"The fire will only last so long."
"If sight and sound won't cure your doubts..." Seth approached and offered his hand. "Touch me. Feel my body."
Calebna stared at him.
"Grab my hand and tell me I'm not real."
Calebna reached and touched Seth's fingers. They felt thick and warm. He retreated a step, his mind reeling for a way to explain the mystery before him. "If you're not a dream, I must look like a fool."
"You look desperate."
"I suppose I am," Calebna said.
"I cannot help our family alone."
_Could it be real?_ "A legion could not help them. The whole world is lost."
"You don't believe that."
"Why do you think I'm here?"
"You plainly had plans before I arrived," Seth said.
"Leave me to fulfill them."
"First, tell me why you paused when you saw me."
Calebna tried to answer, but instead swallowed hard.
Seth continued, "What reason could there be besides that you thought it possible somehow to start over, to heal?"
"I lost my family. My God. How can you heal what's no longer there?"
"I don't know. But that does not mean it can't be done."
"There's nothing left for me now."
"Then why did you stop?"
_Because I hoped you were my son_ , Calebna thought. But he couldn't say that. He couldn't say anything.
"I can't imagine the darkness you've endured, but maybe that's why I've returned," Seth said.
Calebna thought of his speech at the funeral, the way Eve had dipped to touch Seth's cheek, and the grind of stone as they sealed the tombs. But here Seth stood with fingers that felt all too real, and a voice that reasoned just like the man he had known.
"I was dead, but now I am alive, and I think you are the reason why," Seth said.
"How could I be?"
Seth offered his hand again. "Let me show you."
##
Sarah walked to the edge of the cliff. Stars dotted the sky and lit the valley beyond the drop, and in the distance lay the walled City of the Almighty. The mountains circled it like the cupped hands of God, and the river running through it shimmered like a serpent caught between his fingers.
"It is as beautiful as when I first saw it." Sarah glanced at Mason, who sat cross-legged at the edge. He was staring at her silently, sternly. "You were too serious, even as a child." She nodded toward the City. "Do you miss it already?"
He clenched his jaw.
She folded her arms for warmth. "I wonder if we will ever return."
It seemed Mason considered the thought, then looked at his fingernails to pick the dirt from beneath them.
Sarah sat beside him and let her feet dangle over the edge. She looked below and her stomach coiled at the distance. She shifted back. "I know what you did for me."
He tipped his head.
"Peth said you never left my side."
Mason gazed at the night sky and breathed toward the City.
"You speak with your presence." She bit her cheek, resisting the emotions that had surged through her since awakening. "Thank you." And then she could resist no longer. She wiped her cheeks and nose, and leaned her head against his shoulder. He was a rock long warmed by the sun, something solid to keep her upright. Her guardian. Her protector. "I am frightened." She glanced toward the men and women by the fire thirty yards away. "I can't find Cain."
Mason's gaze flicked toward her, and his eyebrows crouched.
"Your father was always a hard man, but he was never a monster." She let the silence extend. "Maybe Cain is still in there, somewhere, but something has changed. I can feel _it_ , but I cannot feel _him_. No? I don't know either what I mean by that. I just feel that something isn't right with him." She stood, crossing her forearms over her belly. She caught Mason glancing at the motion and acted as if she had been brushing dust from her tunic.
In the sky, the North Star shone like the point of a sword stabbing a veil, and the moon hung beneath it naked, exposed.
"I always wondered why the Almighty never let you speak. It made me angry, all those years. Now, for the first time, I'm thankful for your silence."
Calebna stared at the empty tomb of the Almighty and rubbed away the oil he had drenched himself in. "Tell me how you died."
Seth exchanged a knowing glance with Ayla, who shrugged. "When Adam returned and told us Cain had killed—" Seth stopped short and studied Calebna, as if realizing too late what he was about to say.
Calebna said, "My father has been dead for weeks. Hearing the truth can wound me no deeper."
"Forgive me. There are many things still positioning themselves in my mind," Seth said.
"You are forgiven. But hide nothing out of fear of offending me. I have grown tired of subtle deceits and hidden intentions."
Seth rubbed his fingers together and cleared his throat. "When Father told us Cain killed Abel, I thought he was lying. My mind knew it was true, but I thought that if he were dead, surely the Almighty could give life back to him. So I ran to the Temple."
"Foolishly," Ayla added. "He was injured after throwing himself against rocks."
"Why in God's name would you do such a thing?" Calebna asked, then sat on the edge of the Almighty's tomb.
"Ayla enjoys embellishing when the result is me looking foolish."
Ayla scowled. "He was assaulted by visions and thrown to the ground. I saw him fall, and when he awoke, I helped him to shelter, where Adam told us what happened. Lilleth was there, but ran into the storm. Tell me, is she with the others?"
"Mother disappeared that night and still has not been found." Calebna pressed his palms into his eyes, and for the first time, allowed himself to feel the depth of that loss. His throat ached as he said, "Every single one of Lukian's children, then Philo, Tuor, and Kiile's eldest and youngest. So many lives have been lost."
"Are you the only one left?" Ayla said.
"I'm the only one who stayed."
Seth laid a hand on his wife's leg. "We still haven't answered his questions. I, too, would like to hear what has happened, but it should wait until we finish."
Calebna once again stifled his emotions, for duty had ingrained in him the ability to do so quickly and without strain. "What did you see when you arrived at the Temple?"
"The torches were extinguished and no light reached past its doors, but I had to see the Holy of Holies before believing the Man was gone," Seth said.
"I found Seth lying facedown before the throne of the Almighty. And in the throne—" Ayla bit her lip.
"Did you see the Almighty?" Calebna rested his elbows on his knees. He sensed a hardness in his own heart, and thought bitterly of what had forced him to become so.
"We saw his torn and bloody cloak, and on top of it his crown, but not him. I could no longer deny Adam's claims. I lost myself."
"But then there was a flash, and the world went dark," Ayla said.
"Went dark?"
"All sound and sight were simply gone. The only way I can explain it is that my soul had been a tense coil, and instantly, I was released," Seth said.
Calebna eyed them.
Ayla nodded. "The world slipped away like a heavy cloak, and after floating through nothingness, I awoke on a beach with Seth, who I didn't recognize."
"You both died and somehow just ended up in the same exact place?" Calebna said.
"It was a place that should not exist. A place beyond this world."
Calebna squinted, but waved them on nonetheless. "What happened next?"
"We were drawn into a jungle by this beautiful Music. We ended up in a grove filled with ruined buildings. We entered the ruins to a place called the Shrine of the Song, where we were greeted by this... thing."
Calebna waited for more, but neither spoke. "Can you describe this _thing_?"
Seth chuckled and rubbed his scalp. It was a nervous habit that reminded Calebna of Cain, and he thought, _This is now Cain's only brother. I wonder what Eve would do if she knew he was alive?_
"It was like a human," Seth explained, "and yet made of metal. A word for it came to me then: _machine_. That was what I called it in my mind, though everything sounds so very strange to me now that I speak it."
"I've known much strangeness in the past few weeks," Calebna said.
Ayla stood and began to pace. "But you have never seen anything like the Shrine. It was enrapturing, and the Music walked with us. It grew and changed into something we hadn't expected, and we came to understand it was capable of affecting this world."
"What do you mean?"
Ayla glanced at Seth, who said, "The machine showed us a window through which we could view the Song's inner workings. The Music we heard in that place beyond Time is driving some sort of contention here. I think that it is somehow at the root of everything that has happened."
"You are saying the Music killed my father?"
"I'm not saying that. However, the machine claimed we could become Instruments of the Song. We're not sure we know all of what it means, but we do think that we can stop it."
"How can you stop what you don't understand?"
"The machine's speech was strange, but what we saw through that window was clear enough. We are all parts of a singular conflict, one whose origin lies beyond the Sands of Time."
"So how will you save us?"
"I can't. Not alone. The machine talked as though we were the vectors of the Song. It was as if the Music had been constructed to entice us, but its true purpose was much more sinister."
"Death," Ayla whispered.
Seth looked up. "A living death."
Calebna's scorn turned to interest. "You think the Music has done more than just influence this world. You think it has come to life."
"There is some Thing here that never should have been, and it is the centerpiece of a struggle spanning all of Time; a struggle beyond us, but for some reason conducted through us. It grows in power only as we give it power. In this way, then, I think we could stop it. Not one man alone, but together through the choices we make, through the evil we refuse to tend in our minds like Forbidden Gardens." Seth paused. "Danger and deliverance lurk in every decision, and if we stray but a little, all humanity might be lost."
Calebna leaned back, looked at his hands, and realized they were fisted. He rubbed the sweat from his palms, thought of Cain's gray skin, and said, "If what you say is true, Almighty help us all."
##
Lukian slipped away just below the edge of the forest to relieve himself while the others bustled around the fire and gathered water to boil roots dug out of the hillside. He stayed away because, judging by the stillness around him, no one seemed to have followed, and he desired solitude.
"Blood. Why blood?" He shook his head, trying to rid it of the incessant buzzing. All he wanted was to see it again—to smash a living skull and watch the life ooze out, to smell the smell and taste the taste. "Maybe it was the children."
Yes. Maybe watching his children die had broken something. Or maybe, in that instant, he had simply given himself up to something that had been there all along. Whatever it was, as he found his children murdered and knelt in their lifeblood, he was consumed by raw desire, and now that desire chewed at his insides.
A chill rushed past, and with it, a whispering voice, thin and harsh, like distant Music carried on the wind. He felt a pinch at his neck, rubbed it, and brought his hand away marked with thin lines of red. There were footsteps, and through the darkness approached a small boy whose skin was silver, as if the flesh had died and only just begun decaying. The hair on Lukian's back tingled as scar-like memories ached. "Lamech?" he said. It looked just like the twin brother he had lost to the Jinn. But how could it be?
The boy stopped and seemed to assess him. It nodded and whispered, _"I have seen the war, both outside and in, and have come from beyond to give guidance."_
"Brother?" He reached for the boy's hand, which he saw clenched and dappled with blood. Then the boy grabbed Lukian's fingers, and he realized that the apparition before him was more than just a ghost. It was real enough to touch and feel. It was soft and warm, all its blood on the inside, not on the outside as his memory had betrayed his eyes into seeing just moments ago. "Is it you?"
No answer. Just that presence. Just those fingers in his hand. Just those eyes staring into his.
Words spilled from his mouth like water over broken levees. "I shouldn't have pushed you into that cursed Fog. I should have urged you away. I should have grabbed your cloak and pulled hard enough to twist you around. I should have run before—"
_"I am free."_ Three simple words whose impact stemmed the flood. _"And soon_ ," it said, _"if you follow my guidance, you will be as well."_
"But how?" Lukian bent as if craning to see a detail too small. "What do I need to do?"
_"Listen and obey. Will you promise to do that?_ "
"I am dreaming," he said. "This is a dream."
_"Will you promise?"_ As if plucking it from its back, it reached behind and pulled out a pale fruit the size of a peach. It raised it so Lukian could see it unshadowed, and its skin seemed to shimmer—or was it a trick of the eyes? _"Taste and see."_
Lukian was reaching for the fruit, but didn't remember directing his hand to do so. Then the fruit was in his hand, but he didn't remember grabbing it. The world was moving in strange shifts, as though Time itself were skipping beats. The fruit was in his hand by his waist, then at his nose, now between his teeth, and the red juice dripped down his chin and chest and fingers.
Then the fruit was gone, and so was the silver boy. He looked around, wondering if he had been dreaming strange visions of death and insanity. He felt so very strange. He clenched his fist and the juice bubbled between his fingers. He looked down and saw dark fluid like blood staining his clothes. Horrified, he tore off his outer garment, comforted that only a few drops remained on his undergarment. He balled the bloodied garment and wiped his hands, face, and neck with it before stuffing it under a bush. He turned to dash back, then thought better of it and dug a hole in the dirt with his hands. The ground a few inches down was hard and cold, and his fingernails were not strong enough to release it and one bent back. It bled and darkened, and he sucked on it as he threw the garment into the shallow hole and covered it with dirt, but it wasn't enough, so he gathered leaves and twigs and threw them over it, but that looked too intentional, so he kicked it a few times and nodded and turned away.
But how could he explain the few droplets on his garment? They would wonder if he had injured himself.
_My fingernail_ , he thought and smeared it along his undergarment in a few other places.
_Fool_ , he thought again, _they will wonder about your nakedness first, and for that you have no answer. What madness is overtaking you?_
Men laughed and sounds approached. He stumbled and fell before the bush beneath which he had buried his garment. Between the leaves of the shrub were dark berries he had not noticed. He plucked some and smeared them in his hand. The juice smelled and looked identical to the stains on his clothes.
He wiped the sweat from his forehead and tried to settle his breathing. _You are seeing things_ , he thought. _You are imagining terrible things, conjuring ghosts of shame because you are filled with shame for letting your children die, just like you let Lamech die._
He knelt, grabbed more berries, and burst them between his fingers. He thought he should feel comforted that the visitation with the boy had only been a vision, but he could no longer trust his perception, and that made him feel cheated.
He dug up his garment. Dirt had stuck to and caked in the wetness, but after he slapped most of it off, he slipped the tunic on, turned toward the approaching footsteps, and walked back to change his clothing and burn what he wore.
_"Strange fruit_ ," he would say. _"I tripped and fell on a bush with strange fruit that stunk like death."_
## PART NINE:
## THE GARDEN
_But the serpent said to the woman, "You will not surely die. For God knows that when you eat of it your eyes will be opened, and you will be like God, knowing good and evil." So when the woman saw that the tree was good for food, and that it was a delight to the eyes, and that the tree was to be desired to make one wise, she took of its fruit and ate, and she also gave some to her husband who was with her, and he ate. Then the eyes of both were opened, and they knew that they were naked._
—GENESIS 3:4–7 ESV
##
Cain's toes sank into the desert sands as he led his family ever forward. He strained just to keep himself upright, but knew their final destination lay just beyond the mountainous dunes ahead.
_An oasis in the midst of a sea of sand. A Garden, cool and comfortable, wrought in the twilight of Time with strange seeds and no little bending of the rules that hold nature strung together._
The sun positioned itself at the edge of the horizon and tossed its red cloak high. Then, as it dove and the color faded to twilight, the vertebral marks on Cain's arms glowed. He struggled up the incline and, as he reached the top, found Sarah by Mason. Her green eyes gleamed like burning copper, and her shape seemed extracted from the world, something too vibrant to belong amid so much gray.
_Soon_ , he thought, _her belly will become obvious, and the others will wonder if the child is mine._
The thought struck him then as it never had before. He had been so consumed by every decision, and how their weight bent the streams of Time, that he had never actually wondered if the child's origins could be questioned. _To think Abel could be living on through Sarah's womb..._
He looked at her red hair and felt the desire to snuff the flame as never before.
_But the Sarah you know would never stoop to such evil. You only recently felt the bond between her and Abel grow._
_But sex could have been the impetus. You should kill her now._
Cain's gaze was caught in the hem of Sarah's garment, so much so he couldn't tell if he was fighting the voice of the Abomination or his own paranoia. _We loved each other once. I remember us lying with fingers entwined until sleep froze our thoughts._
_Or was that too a lie?_
Uncertainty burned in his mind. He brought his hands to rub his eyes, but stopped midaction. _How could I forget the streams?_
His eyes lingered on her. Mason was staring from his peripheral, but Cain could not tear himself from the thought of being with her once more, of her body pressed against his in the warmth of total acceptance.
_I could swim the pathways of Time and find the truth. I need only the opportunity. When we arrive and the others rest, I will drink, and then plunge into the past._
He crested another dune and there stood the Garden, glowing with a faint bioluminescence. The others followed and gasped at the sight of its high canopy bending with the wind like a hand in the distance waving them close. As they stilled and their glances hopped between Lukian, Cain, and the Garden, Cain wondered at the importance of this moment. There was deep resonance here, and he basked in it.
_"They have chosen to follow you and wonder at the repercussions."_
Cain frowned. _I myself wonder. But soon I will know, because I will test each minuscule choice against the outcome I desire. It won't be long. I will rid myself of you. There will come a day I will see you bleed._
He walked to their new home, and Eve was the first to follow, though she still led Adam hand in hand. Next were Sarah and Mason, and some of the women and children. Finally, Kiile and Machael helped Gorban up once more to finish the final steps of their journey.
Except for the shuffling of feet and rustle of clothing, they made no sound. As they neared, the faintly luminous leaves grew against the darkening canvas of night, the moon and stars disappeared under a sweeping sheet of clouds, and the Garden seemed the only source of illumination.
As Cain passed beneath the canopy, the buzzing intensified. His ears rang, and he shook his head and tipped it to rid himself of his discomfort. They huddled together until they came to an end closed off by a tapestry of thorny vines and green globes bobbing on stems bent with their weight. The globes bobbed toward Cain, and their subtle slits peeled back to the stems, revealing what looked like red eyes that danced and glowed. As one bobbed toward Jacob, he cried out and shuffled back. Mason let them brush against him and seemed comfortable enough, but the others swatted at them when they came near.
As the whispers grew in Cain's mind, so did the movement of the eyes and the vines that slithered forward and extended sharp points toward the people.
"Prick your fingers on the vines," Cain said.
"Why?" Eve asked, and the others mumbled agreement.
"It is a rite of passage."
Lukian was the first to prick his finger, though he stared oddly at the blood as it beaded on his finger. Mason was the second, and the others followed until at last Jacob let the Garden drink its fill.
The eyeballs snapped shut and curled into the thorny vines that slithered away into the foliage on either side to reveal a passageway. There was a strange calm as they passed within, and all life seemed caught in a stillness so pervasive it swallowed even the sounds of the fluttering leaves, which were clearly moving, though not under any compulsion so natural as a breeze.
Jacob gaped at everything, and Kiile's children kept stopping to point, only to be urged along by Elsa. Lukian was staring again, searing holes into Cain's skull, and Sarah walked beside Mason, as comfortable a distance from the edge as possible. Eve strode on like the queen she was, suffering her right hand to lead Adam, though she walked as if the light of the Garden dared not touch her countenance.
Cain smiled. _This Garden is no fabrication as you think it is, dear mother._
They entered an orchard around which lay a seemingly impenetrable wall of thorns and greenery. The bobbing globes were everywhere, though their slits were closed and their glow diminished. It seemed the grove was filled with fruit trees of every kind. Apple, peach, mango, pomegranate, cherry, and on and on until they saw mixtures of the fruits, and what seemed to be wholly new breeds. Their bewilderment cooled, and starvation overcame their lingering fear, so that they plucked the bounty and ate in muted wonder.
Cain rubbed his face with a shaking hand and groaned. He had fasted longer than he should, but the time was not yet ripe. It was Sarah, the way she stood with her arms folded, and spun to stare at the entrance of the Garden fastened against them with thorny vines. He approached as the others moved on, though she failed to notice, seemingly too distracted by the Garden.
He pushed aside a leafy branch and she jumped at his presence. They stared into each other's eyes for the first time since his return. His chest constricted and he grappled for a breath.
"I ache for you," he said. Or had he merely thought it? He could hardly think because in her eyes lay no ambiguity, no guardedness, only a transparency that was at once terrifying and stirring. He thought of the years of kisses and injuries exchanged under sun and moon alike, and all jumbled together and pressed against his mind, buzzing with the Abomination's incessant chatter. He wanted to squeeze her arm the way he had when words were outlawed and only action could speak, but dare he open himself to her? Dare he invite her to gaze upon what he knew resided inside?
"You look the same as when we kissed under the starlight so many years ago."
She continued staring with that spellbinding transparency. What did she search for?
"Do you remember the night?" he said. "Do you remember the love on the sea breeze and the salt taste on our lips?"
The transparency smoked as she looked away. "Perhaps you have mistaken me for another."
"I remember. Even now I see you crouched over countless shells, your feet pushing the foam of the surf that is washing more ashore. You pick your favorite." He touched her chin and she met his gaze again. "And bring it home, caught between your fingers."
Sarah's eyes dimmed. "Much has faded behind memories I can never forget, behind a broken home and bloody hands, a falling sky and... pain. I see pain, I feel pain—I am pain."
The trees swayed and brushed their leaves against them, and Cain was struck by how violence could hide in the most benign of places. In a kiss. In a smile. In a glance. In a word. As the light dimmed, she squinted as if noticing something new.
"What?"
"Your eyes," she said. "They are not the same silver within silver. They are different. They are dark again."
Cain paused and searched for the familiar buzzing sensation, though he felt nothing but the sweat matting the clothes to his skin. He searched for the Abomination, and his jaw hung as he realized there were only the sounds of him and Sarah bathing in Garden green.
She narrowed her eyes. "What are you hiding?"
"Nothing."
"Something haunts you. If you desire my trust, why face it alone?"
He extended his arm. "Hurry. Take my hand."
##
Sarah clenched Cain's arm and silently cursed herself for so quickly abandoning her desire to remain detached for the comfort of his touch. She should have stayed in the orchard, but there were many things she _should_ have done, and for reasons she thought better to ignore, her hands remained where they were. "Is someone following us?"
Cain's chin sliced left, and his profile—the one she had so often admired in sleep and in waking concentration—was silhouetted by moonlight peeking through the alley above. "It's the Garden."
"Can it hurt us?"
"It could kill you. But it won't."
Her back muscles contracted as the hiss of movement sounded behind them.
"We are almost there," Cain said.
As they passed a pair of trees with gnarled bark undulating slowly like pulsing veins, one of the branches whipped Sarah's arm and she shuffled back, noticing blood as it oozed from her shoulder. She stared at the trunks and pointed, horrified. "There are faces in the trees."
He hushed her and pushed her behind him, lingering to stare at the blood smeared across her arm. His face paled and became shadowed, but as she opened her mouth, he grabbed her and pulled her toward the space between the trees. The canopies quivered and lurched, and the branches unraveled and snaked away, revealing a hallway deeper into the Garden.
The way was lit by green globes and flowering vines arching overhead that projected a pale, pink light.
"Don't touch them," he said. "Their thorns are deadly."
She jerked back her hand and resisted the impulse to wrench herself out of his grip, for she feared the Garden over his cruelty. Ahead, a wall of green globes turned and opened. Cain slapped them away and jerked Sarah through the opening, and as she stumbled forward, caught in his arms, she squinted and gasped.
A dome of black thorns arched over a great tree, around which were thousands of flowers upturned toward the crest, where a great light floated. The light, which looked like a small sun, threw gray shadows from drooping willow-whip branches and the fruit that bent them low.
Cain gripped a fruit and twisted until it cracked from the branch. "You must be hungry. Come, eat."
Sarah took three steps forward, careful not to crush the flowers that seemed to watch her as she walked. Something about the fruit disturbed her, though she knew not what it was. She looked around. "What is this place? What have you done since leaving me in that storm?"
He bit the fruit and smiled as the juice dripped down his chin. "I wandered far and returned to bring you to your new home. Here we can live freed from struggle, freed from the hunger that forced me to labor long under sun and rain. You and I can raise our children in safety. We can live as gods."
"Safety," she intoned. The smell on his breath revived the memory of matted hair and crackling candle flame. "And if I refuse?"
"You do not know what is in my power to offer. I understand your reticence, but it is a gift, Sarah."
She let the drooping branches stream across her shoulders, then curled her fingers around a dangling fruit and felt its flesh quiver. She broke it off with a twist and brought it to her nose. It had no smell at all, and she was tempted to bite it, to taste what Cain had tasted, to try it just once. She turned. Cain examined her every movement and expression. "After so many years," she said, "there is still an element of that boy I loved. You've always struggled to impress."
"Things could be as they were. They don't need to stay how they are."
"And how are they?"
He neared, yet his arms remained at his sides. His eyes tipped to search hers. "In those many days, I tried to conjure your voice just to feel the fire of your spirit. But you are more compelling than any memory I could summon."
She bit her lower lip, and her skin tingled at the brush of his breath across it.
He slid his hands to the small of her back. "It feels like years. If only you knew what I..." He hesitated, his eyes probing.
"What?" A pause. "Tell me."
He set his jaw.
She turned in his grip, but he held her fast. "If you want me, show your secrets."
"And what of yours?" There was accusation in that voice.
"You left only weeks ago, but now you return with silver eyes, black marks on your skin, and—"
"And what?" The seduction was gone.
"There's something in you. I saw it when you first came. It was the tone of your voice, the look of your eyes. An evil so deep I thought it would swallow me. You bring me here and offer something I do not understand, but what reason do I have to trust you, who beat me? Who murdered your twin brother and left me to die?"
"You think me insane, but I did what was necessary."
"As do I."
He said, "Did you think I could let you follow me into the wilderness? Did you think I could take you with me to the depths of hell?"
"I wish that you had hated me. At least then I could understand why all this has happened the way it has."
Cain released her and ran his fingers through his hair, which had grown these past few weeks. The branches swayed and bent as the wind whispered through the canopy of leaves.
She nodded, and her eyes burned. "You always were a liar."
"What right do you have to judge me, after everything I've seen of you?"
"Stop being a child."
"I did it for us."
"You did it for yourself. You abandoned me. You abandoned our children, and our grandchildren have been murdered by the beasts you seem able to control."
A hint of a smile attempted itself on his face. "Ah, yes. Our children." He took a few steps forward, eyeing her stomach.
"What are you looking at?" She shuffled back.
"Little child, will you not tell us whose you are?"
Darkness settled in her toes. Her voice shook. "How dare you?"
"Innocence carries no reason for anger."
She threw the fruit and it crashed through the branches. "The only fidelity between us is my own. You may think your bitterness is justified, but I watched your eyes wander long before the thought ever entered my mind."
Cain said, "Where were you those nights I lay awake?"
"Sitting in the darkness and weeping for the love we had but somehow lost."
"You made love to Abel."
"I never touched him."
"You snuck off into the fields under the cover of darkness as if it would hide your sins."
Her face reddened. "He was a good man."
"I wonder if Lilleth knew."
"I never touched him, but now I wish I had, because then maybe this wouldn't have happened. Maybe he'd be alive, and you..."
The silver grew, and he approached her. She fell to her seat and scrambled away, but he climbed on top of her and struck her in the face. She screamed, "I never touched him," but he hit her again. She was weeping, and the tears wetted her hair. His fists came again and again, and she thought she would choke.
Cain stopped and fell back, the breath hissing between his teeth. Sarah lay weeping on the ground, her legs and arms pulled into herself. She cupped her face, which swelled with aching wounds. He stood, and his voice sounded alien as he said, "You may think you can choose a different way, but you and everyone else will find there is nothing now but me."
"You're a monster."
"I killed Abel, I killed God, and if I must, I will kill you and everyone who stands against me. We could have peace. We could. Why won't you listen?"
She clenched her eyes until they hurt. She could feel the heartbeat in her stomach. It ached and throbbed, and she rubbed it with her hand and prayed for her unborn child.
Cain's voice rumbled behind her. "Whether the child is Abel's or not, it will always be mine."
##
Lukian knelt alone at the edge of the Garden. The sounds of the others had traveled far enough away to become mumblings, and so he sat with a fruit in each hand and a pile before him, eating. The pale thirst had returned stronger than before, and with it came that gnawing itch.
He swallowed partly masticated chunks. The fruit in his hands oozed on his fingers, but nothing brought satisfaction. It was like scratching an itch that remained too deep to be quenched.
_Quenched_ , he thought. _Thirsty._
A chill wind blew past and raised his skin. He dropped the fruit, stood, and turned every direction as the whispering voice crept close. "Brother," Lukian whispered. "Is that you?" The buzz in his mind grew to a roar, and when he felt he could take no more, it stilled. The sounds of his family exploring the Garden danced on the fringes. But as he strained his ears, he heard footsteps approaching from deep in the thicket, and he shuffled closer to gain a glimpse, though the hedge was too thick.
Something pricked his neck, and he slapped at it. He brought his hand away with two spots of red and stared at them, recalling an identical moment deep in the Fog, and another in the forest.
At that moment, a pale arm extended from the thicket and pushed the branches aside. A second arm shot out and spread the gap, allowing the glow of the globes to illuminate a pair of delicate feet and slender calves, and the pale face of his little twin brother.
"Brother," Lukian said, "deliver me from this sickness."
The boy frowned with pity. _"Come close so that I may."_
Lukian knelt and offered his hand. The boy took it and smiled, a thin line cupping a lengthy nose. Lukian silently pleaded for answers. Instead, the boy pulled out a fruit different from the last, perfectly round and the color of skin. _"Taste and see..."_
Something in Lukian was repulsed, but the longer he stared at the fruit, the more he itched with pleasure. Not the pleasure of fruition, but of a desire too deep to ever truly own. He smiled at the fruit, then those eyes. "Are they silver like Cain's because you both died?"
_"Cain found the God inside and embraced it. I came because I saw you searching for it. Because I recognized your potential—your receptivity."_
Lukian licked his lips and stayed a shaking hand from snatching the fruit. He wanted to maintain a semblance of dignity. "For me?"
It nodded, its eyes insatiable. _"Taste and see. But understand that once you do, you may not return."_ It paused. _"You must understand."_
_So the forest was just a vision_ , he thought. _But what about the blood on my neck?_ "What will happen to me?"
_"To conquer death you must embrace it. Death, like life, is an illusion. One that you might overcome if you but embrace the truth."_
"What is the truth?"
_"That inside you is everything—the world, the stars, life, deity—and there you will find the power to transcend. You must no longer restrain your desire, for it is the key to finding your truth. You need only believe in yourself. Believe in what you can do, what you can become."_
"The Almighty?"
_"His laws held you captive. They suppressed the God inside. They forced you to abandon happiness in favor of his. Come. Taste and see..."_
Lukian shivered. His hand reached a few inches forward, his fingers itching to grab the fruit, but more so to grab the hand. He stared into those silver eyes, and though the rims were beautiful, they seemed like open throats. _This is my brother. Why should I fear my brother?_
A voice called for him, and the boy's silver eyes shot over his shoulder. Lukian turned to see, and in the distance, between the orchard's branches, moving shadows danced long and low.
Lukian turned back, his soul longing for the fruit, but the boy and the fruit were gone. He struck the ground and suppressed the urge to scream.
_Cheated_ , he thought. _I've been cheated again._
Eve felt as if she had been thrown a hundred and fifty years into the past—back to the Garden of Eden.
_It looks so authentic, but I know that it is not real. It cannot be real._
Something jerked her arm, and she glanced over her shoulder and saw Adam's hand in hers. "Sorry," she said. "I did not mean to twist your arm." She let go and watched his arm swing flaccid at his side. Adam mumbled, though it took her a moment to realize she had not imagined it. She squeezed his shoulder. "What did you say?"
He mumbled again and his eyes jerked to something over her shoulder. She turned and saw a round purple fruit she had never seen before hanging off a branch. She broke the fruit off its stem and pushed it into his hand, but his fingers hung motionless. She brought the fruit to his mouth, but he puckered his lips and shifted away. She brought the fruit to her own mouth and tore the skin away. The juice was sweet and fresh, and the skin sour, and as she brought it to Adam's lips, he bit, chewed, and swallowed.
When at last only the hard pit remained, she tossed it on the ground and used the hem of her garment to wipe his face and beard. Some of the juice had dribbled down his neck, and she scrubbed it from his skin.
Eve slipped her fingers into his again, and everything but the pain in her throat fell away as she pressed her lips against his bristly cheek.
_If only I had known that fruit so many years ago would, in return, sink its teeth into us_. "I'll shave you soon," she whispered. "I know you hate how it itches when it gets like this." She buried her head in his neck, wishing his arms would slide around her. Wishing he would awaken. Wishing he would hold her... as he did now.
She jerked her head back to look. His arms were locked around her waist, and his voice was soft as he said, "When did you get so thin?"
The sound frightened her. She tried to jerk away, but failed. Her mind spun. Adam was holding her. Adam had spoken. She searched those eyes that had been dim since Abel's death, but that were as bright and alive as she remembered them being when she woke from the breath of God and the bone of his side. "Adam?"
His brow furrowed as he squeezed her. "Where are we? Is this...?"
"Adam!" She buried her face in his neck.
"All is fine, is it not?"
"No." The word barely found form.
"Are you hurt?" His voice grew harsh. "Who hurt you?"
Eve shook her head. "You don't understand."
And she laughed and kissed him in the shade of the trees.
##
The vines whipped Cain's arms as he struggled through, snapping thorns as he went. He was no stranger to violence, but the desire to cut himself had never entered his mind.
_So why do I want to slice my wrists and drink?_
_Because Sarah is right, I am a monster, a slave to rage. And if I don't kill myself, I might kill the only one I yet love. What angers me most is I chose this. But what else was there to receive me?_
He struck one of the bobbing globes as its lids opened at him. It recoiled and squinted, and he stifled the urge to break it from its stem. He clenched his fists and screamed, pushing until he felt the air stab his throat.
Then he heard it. Sustained laughter bubbling out of the void, followed by quiet footsteps and cold fingers slipped between his. The Abomination stared up at him in the familiar illusion of that little silver boy.
_"You see?"_ it hissed. _"You're nothing alone. You will always need me, if only to stem the flood of madness. You can never be free."_
He tore his hand out of its grip.
_"Go on, suck the blood from your wounds. Give in. Why not?_ "
"You changed me."
_"I gave you what you wanted. You're nothing but a prostitute selling yourself for the ultimate drug. You wanted a deathless body? It's yours. Transcendent power? It's yours. But in return, you will always be mine."_
Cain wanted nothing more than to crush its skull as he had Abel's. But he couldn't. Not yet. "You speak as though freedom is a fantasy, yet in the same breath claim I chose you."
_"Life gives you one choice—who will you worship?"_
"I never chose you."
_"No, you chose yourself, and even God didn't make that mistake."_ The Abomination paused. _"There are issues craving resolution. It is dangerous to let them rally. They're already beginning to question your divinity."_
"You speak of my father."
_"Humans are emboldened by small victories. Let me have this time. I'll reward you."_
Cain sensed the pale hunger in its voice and refused its violent grab for control. "Crawl back into your hole," he said, "before I force you."
It was an empty boast.
Gorban tore at the wrapping on his leg until it loosed. He unwound the fabric, and as he peeled the last bloody fibers away, all that remained underneath was white skin and black hair.
"My leg."
"What's wrong with it?" Kiile asked.
"It's healed."
Kiile laughed. "Did you eat bad fruit and muddle your mind?"
"I did eat some of the fruit." Gorban stood and, when he felt no pain, jumped. "And I have no more pain."
"Impossible." Kiile grabbed Gorban's shoulder to survey the pale flesh of his thigh. His eyes widened as he craned his neck to look at both sides. "You weren't even able to walk yesterday."
"I can now."
Kiile walked off, but kept glancing at Gorban's leg.
Gorban laughed and stomped the ground. "Do you see this?"
Mason nodded and smiled broad enough to show his teeth.
Jacob jogged through the trees and skidded to a stop. Sweat matted his hair and eagerness sprinted across his face. "Come quick. Adam's in the center of the orchard with Eve." He turned and ran, and Gorban, Mason, Kiile, and Machael hurried after.
Voices were raised in exclamation. Shadows danced, and as they rounded a row of trees, they saw the group gathered in an oblong circle.
Peth called to him and he joined, but as she smiled and pointed toward Adam, she noticed his unbandaged leg. "What happened?"
"It's healed," Gorban said.
"How?"
"I think I know." Gorban called for his grandmother. The people parted.
Adam and Eve were entangled in each other's arms, and upon seeing him, Eve said, "You are healed as well?"
"You fed him the fruit," Gorban said.
"He woke immediately." Eve was beaming.
Gorban bowed. "Welcome back, Grandfather."
Adam said, "It is good to be back."
Cain burst through to where his family stood in a diamond formation amidst the orchard, and said, "Now you see." They turned and hushed. "The fruit of the Garden does more than feed and sustain. It heals, though within limitations." He waved to indicate all they could see. "But this is child's play compared to the true secret of the Garden."
Adam was there, truly there, for the first time since the murder, and Cain could see the disdain in his father's gaze, that familiar expression of disappointment. Adam's face had paled upon seeing him, and Cain was struck in that moment of how disconnected they had become. Adam had not yet seen the changes in Cain, and surely seeing him evoked both fear and disgust.
Cain feigned applause. "Congratulations, Father. You've finally stepped out of your dreams."
"It seems I've done so only to step into your perverse fantasies," Adam said, looking directly into Cain's eyes.
Cain hadn't expected that voice to still carry its sting, and he set his jaw against it and lowered his voice until it seemed the two of them spoke privately. "The days I spent in your shadow have long since passed."
"Have they now?"
Cain plucked a small yellow fruit and began eating it. "Do you remember Eden?"
"I remember many things."
"Have I missed anything?" Cain swept his arm toward the vines, trees, and thorns, and tossed the finished fruit as Eve walked up beside Adam and hugged his arm.
"Nothing could compare to that paradise," Adam said.
"And yet you threw it away."
"Always passing blame," Eve said. "Will you never learn?"
"This Garden is a gift. You would do well to accept it," Cain said.
"Who do you think you are?" Adam said.
"I'm who you never could be. You were the sickness. I'm the solution."
"You're twice the sinner I ever was."
Cain laughed. "You're every bit the same as me. The only difference is you are powerless and I am not."
"And what do you think wielding that power will gain you?" Adam said.
"Not me. It will gain you. It will gain Mother. It will gain anyone who is willing to embrace the God inside." He pulled out the perfectly round, flesh-colored fruit of the new Tree of Life, so benign in appearance, and their eyes followed as he dangled it in front of them by the stem. He spoke for all to hear. "In the heart of the Garden lies a reversal of the curse. A true Tree of Life. All you need to do is taste and see." He nodded at Adam. "Began and ended through fruit. It seems fitting, doesn't it?"
There was dangerous silence. Cain felt the echoes of decisions and truth, and realized this point was another nexus. The world and all beyond hushed in anticipation of their every move, though somehow he knew this nexus more potent, more treacherous than any before. He closed his eyes and let each moment fall on his ears like terrible Music.
"I would rather die than become a curse like you," Adam said.
_"My_ curse is everlasting life."
"I taught you to love. I taught you to serve the Almighty and steward the earth."
Cain's sight dimmed at his father's stupidity, and he felt the blood hot in his cheeks. "Who then taught me hate?"
"Abel listened to me. He was a good son."
"Who holds you in the crucible? Who has given sanity back to you from denial's greedy grasp?"
"Every chance I gave, you threw away. How can I applaud stupidity?" Adam kept going on as if the argument had nothing to do with the present situation. And perhaps it didn't.
Cain paused. "I've done more than sin."
"I would have questioned your humanity if you hadn't done _some_ good. But ever since childhood you have done shameful things."
"And ever since I could remember, Abel absorbed every ounce of your adoration."
Eve's eyes seemed to flash at that.
"Everyone saw it," Cain said. "The moment Abel was born— _before I had my very first thought_ —you loved him more, didn't you? Tell me how that is not the darkest of evils. Abandoning an innocent child merely because you were preoccupied with the beauty of another. How could you?"
"I offered you every chance to prove yourself."
Cain laughed, long and cold. "What sign did you search for? Was it something in my infant babble? Perhaps the way I gripped my toes?"
"Don't mock me, child."
"I will mock who I wish." And to press the point home, he struck his father's face.
Eve caught Adam, who wiped blood from his mouth. Adam's voice shook. "All your fury could not match him. I knew the moment he was born that he was perfect. But you... you troubled me."
"What was it you so hated in me?"
Adam pressed his lips together.
"Why not be forthcoming?"
Adam turned away.
Cain's voice grew until he was nearly screaming. "When you looked on my dark skin and eyes, on your sharpness of features reflected, what did you fear? That I would be like you? That I would choose Sin over God, like you did?"
Adam's voice was low. "I was right to fear what you would become."
Cain paused as new thoughts settled, and he realized that for the first time he was seeing his father clearly. His voice quieted. "You still feel the shame of that moment all those years ago. You still see your mistakes crystalized in my features, forever preserved in flesh and bone too similar to yours. That's why you hated me, isn't it? That's why you still hate me."
Adam's shoulders rolled with every breath, and his hands clenched and released.
Cain hadn't realized until that moment how much he had hoped Adam did love him, however little. Perhaps that was why Cain felt such enormous importance in this moment. Because everything, from the Abomination to the Garden, led back to that day over a century ago, when the twins were born and the father chose his favorite.
The family gazed in stunned silence. Mason's eyes were bright and cold as he stared at Cain. Lukian looked bitter and distant, and Gorban's neck was red with shame for his father, perhaps his grandfather, maybe both.
The Abomination chewed at the base of Cain's skull, its angry calls for power weaving into an unintelligible storm of vowels and consonants. It wanted to speak. To spread the fear it thought so pertinent to their survival, and Cain was wearied by the pale thirst and the crumbling of hopes he had gathered through countless years. He had never so bluntly pointed to his father's faults, and somehow by doing so and finding them firm, he felt the throb of wounds run deeper still.
"Give me emptiness," he whispered, and the Abomination thrust him into the void.
##
Mason had been waiting for the moment the silver would return, and as he stood listening to the echoes of Cain and Adam's bitter exchange, Cain's eyes melted to polished marbles, and Mason knew the monster was back.
"Enough foolishness," Cain said.
Mason's throat dried, and he shivered. This was not Father. Whatever had happened in the wilderness, Sarah was right. He had not returned alone.
"As promised, I will let you live in peace, but you must stay."
Eve touched Adam's arm, but he shrugged her away.
Cain, or something inside him, smiled. "So far my patience has proven deep, but it is not bottomless. The Garden watches. It thirsts for violence, and nothing but my promise holds it from drinking your blood with its thorny roots."
Cain floated in the void hearing voices like far-off underwater mumblings. The Abomination was in control of his body, and he wondered if he should attempt reclaiming power.
_No_ , _I must take this time to search the Waters of Time; for it is only when the Abomination is distracted that I may be safe from intrusion. And it must never know what I find._
He remembered the black force that had thrown him out of the stream upon his first encounter. He knew what the black force had been even then, and though the Abomination hadn't spoken of it, he knew he had been on the verge of discovering something it wished to keep hidden.
_The Abomination realized then that I know of the prophetic streams, but it cannot watch tirelessly._
_I must hurry._
He closed his eyes and felt the layers arise. He reached and sifted through them, slowly descending until he realized something was drawing him toward the bottom of the layers. Upon arriving, he stood poised at the edge of that endless darkness, but something stayed him.
_What would happen if I jumped? Could I find my way back?_ The darkness seemed to call with a voice of cold finality, but he perceived it was not yet time. He would return to that darkness all too soon—that truth resonated like a wet finger circling the edge of a glass—but he couldn't sense why, so he shifted back and ascended, grappling one arm over the other to pull himself toward the light. He arrived at the other brink and stood at the edge as he had before, prepared to jump into the stream. The Water churned and boiled.
As he stood, he remembered Sarah's shining eyes and the sound of her weeping; Mason's glare and Lukian's anger; the silver eyes of the Abomination and its arrogant nose; and Abel's blood pooling under darkness and grass.
_I fear what lies in the streams of Time._
_But where else might you turn? You have stalled too long already. Go. Go!_
He stepped off the edge and the Water rushed up and closed its hands about him, twisting and throwing him in circles, burying his face in cold wetness. He struggled until he breached the surface. A storm raged and lightning struck. Where the jagged light stabbed the surface, the stream exploded, and as he struggled against the current, it splashed on all sides. He cried out as a flash slapped the Water in front of him, but there was no pain.
_Of course. My body is walking in the Garden. I am here, and yet I am not here. Through the lens of the void, I have become an observer of the Waters of Time._
He recognized visions in the Water. As a bolt of light illuminated a suspended droplet, he caught a glimpse of a baby raised high. Though the droplet sped by, the vision was so vivid that, in an instant, ages filtered through his mind. He saw the baby wriggle and heard it squeal as it was lowered into familiar hands. A woman with red hair leaned in, crinkled her nose, and pressed it against the baby's cheek.
_This is my unborn child_ , Cain thought. _I see the resemblance even now. The jawline, the shape of the lips—ah, but it has Sarah's nose. Yes, I see the child is older and his hair is like glowing coals. And... what is this?_
The boy, whose name he did not know, turned and glared, but the peculiarity remained. Dirty-white projections, like the horns of a goat, pressed out the top of his head.
The vision continued in sputtering clips. Lightning struck, and as the droplets sped by, he saw the boy grown to a man. The horns curved back upon themselves, and darkened and shimmered as if moistened by the blood of the innocent.
Cain turned away. "I will not watch this." He plunged ahead, not bothering to look at the other images that demanded attention. They screamed and howled, but he swam on. The storm intensified, the lightning struck with greater frequency, and he wondered if he could find the source of the storm. He knew, as a man knows his existence, that the stream _defined_ the world, or maybe that the world defined _it_ —either an equal truth—and that if a possibility lay not in the stream, it could not exist, and so the stream projected the limits of will itself.
_But what is this storm?_
He swam on but again became caught in the visions. They grew more violent, and as the droplets sped by, murder multiplied. Men with horns just like his yet-to-be-born son beat and tortured each other with cruel instruments.
"No," he said. "This is not the world I bought with slavery. This is not the world I wanted. Show me the way I glimpsed in the eye of my imagination. The way of progress, the way of transcendence."
But true prophetic vision was altogether different and more terrifying than imagination. He could no longer bend logic to reach the conclusion of his design. The fury of Time's brutal movement shocked his eyes wide, and as another bolt struck the surface, a vision of one of his other descendants struck his cheek. The man was ugly, and one of his horns was crooked.
_Lamech. The name of the son I lost to the Jinn. Though this Lamech is one of my distant grandchildren._
Another man struck Lamech, and he struck back. Lamech tossed the man to the ground, then kicked and stomped his chest and head. Cain turned and bit his tongue. He wanted to burn the images from his mind, but they grew and festered like open wounds.
Cain rushed on and witnessed countless infants murdered and unspeakable things done to women—and through it all those beast men with horns, and flames and disintegrating towers growing ever higher.
"Dear God." Cain pressed his fingers in his eyes. "Dear God. Dear God!" His hands were shaking. "Where did this darkness come from? How could so much change in so little time? This is not what I chose."
_But it is._
"What should I do? How can something so all-consuming be stopped?"
_Was it not you who chose to introduce murder into the world? Is anything else more urgent? Swim on. Swim on!_
Cain dove upstream as he fought the terrible sensation of duality. Part of him was repulsed, though the rest lusted for what he saw, and that sickened him all the more. He shivered and suppressed the frightening sensations the Abomination had awoken in him through the unholy coupling.
The chaos grew, and he came to a fork in the river. On the path bending right lay details of the future, but on the path bending left lay shards of the past in a great glassy bay. He realized that this was the point the Abomination had kept him from, and he plunged down the path bending left. In the storm's place came a distant rumbling. He turned and saw the stream devoured by black clouds.
_Does my presence affect the stream?_
Cain blinked as new knowledge resonated through him like a clear note piercing flesh and soul. _The knowledge I gather here informs my decisions, and I am the tipping point, the crux upon which the world has been balanced, so tenuously that one decision might shatter it._
The new river was so clear that he could see his toes, and the visions that presented themselves were familiar. After he traveled a distance, they were all either of him leading his family through the wilderness, or of Calebna and...
He paused, confused. There, with Calebna, were two whom he recognized.
_Seth and Ayla?_
They were talking with Calebna. And he saw them walking the City of the Almighty while Calebna, drenched in oil, stacked endless items on the altar in the Temple, next to which still lay the shriveled remnants of his rejected offering to the Almighty.
The vision altered again, and he realized he was watching Time unfold in reverse. He shifted to compensate and made his way forward until an image caught his eye. He backtracked until the vision unfolded itself. He saw a closed tomb, but the top shifted, then flew off, and Seth crawled out.
"Impossible..." Cain watched the vision as Seth opened the Almighty's casket, which was empty, and helped Ayla out of hers. They embraced each other and Ayla wept.
Cain shook his head. "It cannot be."
But it was motionless Truth. He stopped the vision on the Almighty's empty tomb and swallowed to banish the fear from his throat.
_They're alive. How are they alive? And what about the Man?_
_The Man is alive._
"But I killed him."
_Did you? What of the empty tomb?_
"The Light Bringer even claimed I killed him, and why would my family bury the Man if they didn't have proof he was dead?"
_Fool! Does a truth disappear because you close your eyes? You thought yourself God, and so you thought yourself unmatchable, but if you had opened your eyes, you would have struggled. And it could not chance you doing that. What explanation do you have for an empty sealed tomb? None!_
"But how could I ever rid myself of the Abomination? I don't even know what it is or how it came to be."
There was a bend farther down the river that led to the great bay. The distant rumblings of the storm lay so far behind that he could barely make them out amidst the deafening silence. His mouth hung as thoughts fell into place. "How the Abomination came to be..."
##
Cain rewatched the vision of himself play in sputtering clips. He was standing on the top of a hill, overlooking the fields where Abel stood silhouetted against the twilight of storm. He still tasted the emotions and thoughts that peppered his brain as he walked into the valley and argued with Abel. As they fought, he felt the rock and smelled the bitter thickness of Abel's crushed head as if it were truly happening that very moment.
He rushed the vision ahead and watched as he bent and lurched under nausea. But as he stared at his mouth, waiting for vomit to spew, his jaw stretched and instead of vomit, a pale child gushed onto the ground, webbed with womb netting. The child tore and bit at the goo until it could take in a few gasping breaths. It latched onto him and whispered in his ear.
Cain's mind claimed that it could not be, but here he saw that it _was_.
Abomination. Silver boy. Son of Satan and... of _Cain_. Now he knew the reason for the unmistakable familiarity between himself and the Light Bringer. He understood everything.
_"I_ birthed that monster." He squeezed his eyes shut. "I birthed it, and now— _through me_ , curse it all!—it will pervert my unborn child into something else. Something evil. Something deserving only death and damnation." _Something like me_ , his mind finished. _I could justify damnation for the sake of my progeny's salvation, but to curse my lineage for such a fate..._ Cain knelt. "Almighty God, if you exist, if you truly live, then help me. Free me from this Abomination."
The tears on his cheeks were not of sorrow, but of fear. He feared himself. He feared what he had become, as Adam had rightfully feared him. He feared the pale thirst, and what he knew he could do, even without the child of Sin inside him.
_Capable of every abomination I saw in those visions._
He stood as a sudden thought struck him like nails in the wrists. Begun through death and ended through death. Blood for blood and fruit for fruit. It truly did seem fitting.
_Law 1: Humans are incapable of owning satisfaction. Law 2: Indulgence only feeds the mortal desires it claims to quench. Law 3: A spirit possessing a human body cannot escape Laws 1 and 2._
The Abomination had listened to the Light Bringer's words in the underground City, but only now realized their full weight. The first several weeks of possession had been exhilarating. To feel the physical world in so many dimensions—to taste the blood, feel the consummation, and revel in the pleasure—the Abomination had been drunk. Even now, still shy of the true depth of the fall, it could hardly hold itself back from killing again. Animals just didn't offer the satisfaction.
_And Sarah..._
"The time is not yet right," the Abomination said through Cain. "She still carries my future in her womb."
The Abomination smiled, feeling the pleasure only the son of the Devil could by using another's voice. It felt even more pleasure knowing the body it possessed belonged to the very soul who had birthed it.
_My father, the Light Bringer, and Cain—that whore of whores—my mother._
A cruel chuckle escaped from Cain's throat, broken at the end by violent rage. Momentary sanity came in waves, followed by the fiercest upsurges of ire.
_I need a second host. Another body. Lukian._ The Abomination laughed again, a long, cold cackling that echoed through the Garden. It was certain Cain had no inkling of what it planned with Lukian, and thought it a shame that it couldn't rely on one host, though pitting the two against each other would reap even greater joy.
To drink another's blood and simultaneously gain praise. What could be better?
It waved its hands and offered no more than a glance toward the receding thorns and bobbing globes. Their lids popped and glowed as the Abomination passed by in Cain's skin. "Dim your lamps."
The eyes obeyed and the vines slid into place behind it.
"This body is tired and fogs my mind." The Abomination entered the chamber of the Tree of Life, pulled itself out of Cain's body, and rode the wind through the leaves to find Lukian, son of Cain.
##
The vines opened and Sarah was already half turned when Cain slumped to the ground. She called his name, but he did not respond. "Are you all right?" She held her belly with a wary hand. Flowers buffeted her calves as she stepped near and caught a glimpse of his back as it rose and fell.
She knelt beside him. He lay facedown, his body crumpled awkwardly. She nudged him and whispered his name. She worked her fingers under him and heaved him onto his back. His right arm remained twisted underneath, but after a few moments of struggle, she slipped that out as well.
Sarah sat beside Cain and hugged her knees. In the mixture of silver and green light his face looked sickly pale, but its shape held a regality that could not be missed. His dark lips, full enough to balance their length, looked strangely beautiful.
_But what those lips have touched._
She looked at the fruit hanging around them and stifled repulsion. How could he enjoy the taste of blood? She had smelled it on his breath and saw the thick redness of it.
_He has dipped into darkness so deep it stained his skin._
She nearly reached out to trace the marks lining his arms and neck.
_Why am I attracted to what repulses me?_
"I miss you," she whispered, and nodded as if to convince herself that was the only reason. "If you truly can make everything how it was..."
His eyes fluttered open and searched hers, and though her first instinct was to flee, something froze her muscles. "Is it you?" Sarah said.
Cain's eyes, devoid of silver, zeroed in on the swelling wounds on her face. He laid his head back and breathed through flared nostrils.
Her fear dissolved, and in its place came urgency. "What is it?"
He sat up and again she thought to flee, but she knew the monster was gone, and for the first time she could see the reason she followed him to this grove.
_I stay to help the man I love rid himself of the monster I hate. The vows we made, the life we shared, the pain we felt—it was not all for naught._
He slid his warm hands over her shoulders. "Sarah, you must listen to me, and you must remember."
Her eyes were sealed to savor his touch as she pressed a shaking hand against his cheek. "I know already."
He squeezed her shoulders until she felt pain. "Tell me you're listening. Promise to remember."
"I promise you my life."
"Do you love me?"
"How could you ask that?"
He shook her hard. "Do you love me?" His voice was edged with fear and passion, a quality altogether different from the sharpness of anger.
_"Yes!"_ she said.
"Then kill me."
Her face contorted. "How could I?"
"How could you not?"
"I'd rather die."
"Stop trying to save me."
"I can't lose you. Not now. Not after you took so many away from me. What do I have left? Do not sentence me to such loneliness."
He paused. "I was never yours."
"Stop lying." She wanted to embrace and be embraced by him. Amidst the hate, scars, and fresh wounds, she loved him now more than ever before. The desperation for his affection, for his mere countenance, had grown without her knowing like vines through her ruined soul, and it toppled her. "No," she whispered, and then she shouted, "I won't kill you."
Cain slapped her, and she held her cheek, remembering moments not so long passed and feeling fresh bruises throb.
"If you knew what I know, you would break a thorn and thrust it into my chest."
"Then tell me what you know."
"The world will shatter if you fail. They are coming. The bane of our race—soulless half-breeds born from human wombs. The Jinn were a mere dabbling in the craft. But through me Death has been perfected. They found a way to touch the children, to pervert them into something else. There were barriers before that kept them away but—"
"Who? Kept who away?"
"Kill me. Before it returns!"
"There must be another way."
He slapped her again, and the crack of it echoed. "There was never another way."
She couldn't make sense of his words, nor could she stifle her tears.
Cain looked at her and all the hardness melted. His fingers poised as if to take back the wounds. His mouth hung and his eyes shone in the light of the Garden, and her soul quickened. But as quickly as he turned, the transparency muddled.
He strode to the Tree of Life and grabbed hold of the hanging branches. They were strong, yet pliable. He pulled himself up one and, with legs crossed around the branch he hung on, twisted another branch around itself and knotted the loop. "Someday I hope you will understand. I pray to God that you do." He swallowed. "Just remember..." He paused, closed his eyes, and pulled the knotted loop around his neck.
_What?_ Sarah thought. _Tell me, you fool!_
He was suspended, hanging from the Tree of Life, and holding back words they both knew he should speak. He fell. His neck twisted, strained by the weight of his body. His feet swept through the flowers, but did not touch ground. His face reddened and his body convulsed.
She scrambled up, wrapped her arms around his waist, and lifted. She thought of Lilleth lying on the floor of her house, the maw in her throat gushing blood. Now Cain was killing himself too.
"Tell me," she screamed as she tried to lift him out of the knot. "I know what you were going to say."
He kicked her away and she fell. Everything was a distortion through rough waters. "I wish you never loved me." She moaned. "I wish you never had."
Lukian felt the tug at his hand and hardly began turning before he was jerked into the brush. He nearly yelled, but delicate fingers covered his mouth. He looked up at silver eyes surrounded by pale skin.
_"I have had patience."_
Lukian frowned and, as the boy's fingers slipped away from his mouth, he said, "What have I done?"
_"You've listened poorly, brother._ "
"I've done what you said."
_"Eat the fruit."_
The boy lifted the fruit and pressed it against his face, but he turned away. "Why?"
Its fist pounded Lukian's head, and he saw stars and cried out, only to receive its cold fingers on his mouth again. _"Eat it."_
"No." Lukian made to sit up and leave, but those little hands held him down with unnatural strength. A strange stirring in his abdomen chilled his breath and spun his mind. "Why are you doing this?"
_"Eat it!"_
It stuck its fingers between his teeth and wrenched his jaw open, then jammed the fruit against his teeth until the juice leaked down his throat. Lukian coughed and spat, but it continued shoving the fruit down his throat, and he choked and convulsed.
_"Eat!"_ Its voice was altogether too large, and it rumbled and shook the ground.
Lukian clawed at his throat and then its arm. His eyesight blackened and flashed at once, and he convulsed and vomited the fruit up. He coughed and held his throat, feeling as though it had kicked him there.
The boy twisted and stared into the thicket. Its eyes were strangely focused, as if it could see through the hedge. It screamed. The sound rose to a pitch higher than he thought possible, and he stopped his ears and closed his eyes until its voice sharpened to a point so high it disappeared.
When Lukian opened his eyes, it was nowhere... and blood was everywhere.
##
Cain felt his eyes would burst. His mind claimed that at this very moment he was hanging by his throat from the Tree of Life to die, but the irony was distant behind the blackness of closed eyes and pain. Pain that overwhelmed at first, and yet rapidly dulled.
_Forgive me. Please, forgive me. I wish I would never have done what I did. I wish I could take back my actions. But these realizations have come far too late._
The washing heartbeat no longer sounded in his ears, and Sarah's weeping had disappeared. There was no more pain, no more sensation except that of vague emptiness. He shifted, opened his mouth, and yelled. The sound echoed endlessly.
_Is this what death feels like?_
He felt his body with numb fingers and sensed himself floating, though upon straining his eyes found nothing to see.
_"You think I would let you kill yourself?"_ The Abomination's voice echoed through the darkness and throbbed with fury.
_So_ , he thought, and the action of thinking it was like salting a wound, _the Abomination has pushed me back into the void._
Cain closed his eyes to test the fact, and the layers arose as he expected. He sifted through the sheets of reality slowly, methodically peeking through them to see what lay smashed between.
_I don't even have the power to kill myself. How great a failure can one be?_
_You should not be surprised._
_But why should I not think on these facts? The Abomination will live on through me to pervert mankind, and violence and sin will spread until it overwhelms the world. Sarah will be crushed between the Abomination's jaws while I float in embryonic darkness, only to be born again into a world unrecognizable. A world filled with demons in human skin. A world I no longer have the power to change._
_There is still a way._
_There was never a way. Never any chance of success. I knew it the moment I murdered my brother. My bones shook with that one truth, but with all of Time coursing through my veins, I blinded myself to the most basic of truths. For what is the future but a present to come? And I built my future on an Abomination. How could it become anything less? I should have listened to the Almighty._
_What stays you from listening now? Do you not remember the Man's promise?_
He stared into the abyss at the bottom of the layers, and part of him wondered, _Could the abyss be the escape my soul desires? Is this what the Man meant when he said there would always be a way?_
_What else could he have meant? The only other choice was between death and the Abomination._
_I should have chosen death._
_Perhaps. But you could choose this now._
_What will happen to me upon jumping into the darkness?_
_What else might you lose but pain itself?_
He paused at the brink, wondering if he could see Sarah again. More than anything, he wanted to hold her, to slip his fingers between hers, to kiss her, and to smell her hair.
_But it would only tear the wounds afresh. Maybe, in the end, it would be better if she came to believe you never loved her at all._
He nodded. _And perhaps I never did._
Cain's body swung, convulsing, but his body jerked and stilled. Sarah thought him unconscious, but his body lifted and the vine around his neck slackened.
She crawled away with a start. His eyes were open to silver gemstones. No white, no black, only iridescent silver. His hand tore the vine from his throat, and he descended until his feet touched the ground.
Sarah screamed, ran for the guarded entrance, and cut her hands on the thorns. She cursed and commanded them to let her through, but the way would not open. The green globes bobbed close and stared.
"Stop looking at me," she yelled and punched one. It recoiled and the others squinted. Deliberate footsteps sounded behind her, and she spun.
It was smiling at her.
"Who are you? What have you done with Cain?"
"Come now, you know me, as I know you."
"Get away from me." She rushed up the hill and slid behind the Tree. She breathed hard with her back to the trunk, examined her hands, and wiped the blood on her dress. The wounds throbbed, and she wondered how deep the thorns had gone.
"You've met me in the dark halls of your deepest thoughts."
The footsteps seemed to echo. She twisted and heard them from differing angles.
"I've watched you curse when you thought yourself alone. I've whispered in your ear and you've repeated lies."
She held her breath and twisted, but the branches hid all but a few feet in any direction.
"I am the darkness within darkness that you fear. I am the child of Sin."
A fruit thudded to the ground next to her, and she jumped and screamed.
"Are you hurt?" it mocked.
_God help me_ , she prayed. _If you exist, if you ever existed, deliver me from this place._
"I can feel the pulse in your throat as if it were pressed against my teeth. Come before I grow angry. You won't like me angry."
She could not escape. She could bide her time trying to dodge it in the grove, but she had seen the monster's rage, and of that she was most fearful. She placed a hand on her abdomen and shivered. _If it beat me again, it might take you from me. But I won't let it do that. No, dear child, I would endure anything before that._
##
Cain gazed into the boiling darkness beneath the layers and felt it his final home. Like a pilgrim through the lands of Sin, he realized the darkness just over the edge was his final bed, his resting place. Though shame would have stilled him at any other moment, he knelt and prayed with a fervency reminiscent of Abel.
_God, if there is nothing you can do to stop what I've set into motion, only keep Sarah safe. Do what you will with me, but if there would be anything you would have me do before I die, I would do it._
There was no response, though the silence seemed to speak in its own way. _"Come and bear the consequences_ ," it said. _"Experience the everlasting darkness prepared for you in the fullness of Time."_
And so he jumped. For a moment, the sensation of falling into that darkness was similar to falling into the streams of Time. But the darkness peeled back to a Light so violent it penetrated his very soul, and he crumpled into himself in the attempt to guard his eyes from the burning fury, though every shifting shadow was eradicated in its presence.
_"My son."_
Cain shivered at the familiarity of the Man's voice, but there was no anger or disdain, only sorrow and relief, as potent as mixed wine, and he wondered at it.
_"There is no hope left to prevent the evil you've born and borne, but there is hope. For you have severed its connection to the streams of Time, and its power is diminished."_
Cain opened his mouth to speak, but found his tongue too parched from the heat of the Light. He was dust and emotion, barely held together by the residue of Time quickly curing.
_"I heard your words and have given you one last chance to change the outcome, to mitigate at least a portion of the pain._ "
_And after?_ he thought, for he was incapable of speech.
_"And after_ ," the Man replied, for in the Light were no hidden places, _"you will go to the place I have prepared for you._ "
He shook with fear, but though there was ambiguity in the Man's answer, he sensed justice in the Light which rightfully burned him away. For Cain was Sin—broken, bloody, brutal Sin. Whether the place prepared was darkness or Light, Cain thought either would be equally terrible.
_"Hurry along_ ," the Man said, and to Cain's surprise there was a humor behind the words.
_You laugh at me?_
_"You've no need for a cloak such as pride. Nothing could hide you now._ "
_What could keep me from burning away? How could anyone endure this?_
_"Blood_ ," came the answer. _"But not blood such as you've drunk. My blood. For I bear Abel's wounds. I hold Sarah's scars in my hands. I feel the teeth of the serpent at my heels, the same serpent which fed at your throat."_
Cain nodded, feeling that familiar resonation of Truth, but realizing now that it came from the Light. Indeed, the Man had spoken to him all along, even as he had thought it his own wisdom. _Why? Why would you bear such suffering?_
_"For you."_
Cain's vision swam as he wept in the pain of the Light. Emotion threaded itself through him, and he realized it was the emotion of the Man—the sorrow of an eternity of separation, of love and loss. Of true divinity. And he realized now that much more than power differentiated God from man.
_"Go, quickly_ ," the Man said. _"Back to the place of darkness, for that is where men's dreams are formed. It is a place the Abomination could never touch, a place I have blinded it to. There you will find a guide to show you how to weave the fabric of the future into mortal minds and to speak to them in visions. Because of your faith, you will be given a small portion of Time in which you may give men dreams, but afterward you will continue on to the place prepared for you."_
Cain nodded, though he wanted to stay, even as he felt the dross of his life dissipating in the heat and sensed there would soon be nothing left. Obediently, he walked where the Man directed, and came to the darkness.
There he met another familiar figure.
"Hello, brother," the man said, and the words fell like silk on his ears.
The Abomination watched the woman walk from behind the tree. _So courageous_. _What beauty. Now I know how I will kill her._
"What are you happy about?" Sarah asked.
The Abomination approached and struck her, but her only reaction was a flaring of the nostrils.
"I've not found such strength in the others. You're a formidable woman."
She chewed her lip to stop its movement, but kept her gaze on the Abomination's.
"Why have you stayed with him all these years?"
Tears moistened her eyes. "What have you done to Cain?"
"I thought you knew him."
"Why are you doing this?"
"You don't care."
"You know nothing of what I care about."
"He is gone. You were a fool to stay, a fool to hope. Such tragedy..."
She wept silently, and her beauty was magnified through Cain's silvered eyes. To see the world colored through another's emotions was a strange experience, one that the Abomination thought it would never tire of.
"In truth, I pity you. To love someone so absorbed by hatred is a bitter fate. If only I had come sooner. Maybe you would have been spared the pain." It raised its finger and pressed it into her forehead, then skipped the barriers and plunged through the fibrous pathways of her brain, plucking infinitesimal sparks and scattering them until all darkened and she slackened in its arms.
It laid her unconscious body on a bed of flowers and heard, as if carried by a sudden wind, quiet Music. The Abomination smiled, rested a hand on her abdomen, and recalled images from the rivers of Time. Hollow men in white coats, wearing masks and controlling machines alien to this millennia, turned a lens that clicked to the rhythm of the Music until it magnified an embryo on the surface before them. They gazed through the lens and, using the machines, cut and spliced twirling helixes together, adding and detracting, changing and molding.
The Abomination did similar work on the child in Sarah's womb. But the Abomination's abilities were more sophisticated and more primal than those of the men in the vision. It smiled, sensing the child struggle for life. But the changes were too rapid, too violent, and the soul could not remain. It slipped away, and in its place came...
Emptiness.
The Abomination withdrew from her womb and whispered, "My dear Sarah. You will wake, and you will live, and the child inside you will grow. But after it is born with little horns, I will come for it, and for you. Then I promise you will feel pain such as none you've felt before, but in the vastness of eternity, it will be only a moment." It caressed her unconscious face and kissed her motionless lips. "Merely a moment."
## PART TEN:
## REDEMPTION
_When man began to multiply on the face of the land and daughters were born to them, the sons of God saw that the daughters of man were attractive. And they took as their wives any they chose.... The Nephilim were on the earth in those days, and also afterward, when the sons of God came in to the daughters of man and they bore children to them. These were the mighty men who were of old, the men of renown._
—GENESIS 6:1–2, 4 ESV
##
Seth woke sweating.
"I've been watching you." Calebna's voice was low, though sharp. Seth forced his fingers to relax and realized he had been holding his breath. He sat up and tried to steady the shaking from the nightmare. Calebna sat like a tombstone by the remnants of the fire they had risked, and all was quiet silver under the moon and stars as Calebna rubbed the necklace by his throat.
"Who is it you talk to?"
"The dead." Seth looked at his wife's sleeping figure, then at the puffs of clouds migrating through the sky.
"Dawn is many hours away."
Seth nodded, pulled the cloak over his shoulders, and cinched it in his hand. The two sat hunched in the chill of autumn, and Seth's mind sucked on thoughts like wet cotton. He knew why Calebna couldn't sleep—why neither of them could sleep. It was the dreams. "I saw eyes," Seth said.
Calebna scraped the ashes with a crooked stick, drawing meandering black lines through the gray.
"They were on fire."
The stick dropped from Calebna's hand. "I keep thinking of Terah, Jacob, and Ben. What if they don't want me? What if they tossed what little love they had left?"
"I don't think such a thing is possible."
"Then why did they leave?"
"I think that fear, not hate, drove them away."
"Then their love was never love at all." Calebna stood and stared westward, toward the City they had left for a hope hidden behind danger. "When we arrive, promise me something."
"Anything."
"Promise you will do one simple thing after all this is over with."
Seth waited for Calebna to continue, but the man didn't. "What?"
"I'm waiting for you to promise," Calebna said.
"But I don't know what you—"
"Just do it."
Ayla shifted next to Seth, then settled back into heavy breathing. "All right, I promise."
Calebna hung his head and heaved another sigh. He sat and drew with his fingers in the slag. "Do you think God did it? Do you think he let Cain murder my father—your brother?"
"I think he could have stopped it, but didn't."
There was long silence marked only by the scratch of Calebna's fingers in the dust. "We buried him. We buried God."
"It seems a habit of man to try to swallow what he can't understand," Seth said.
"Better we had choked on it."
"I think we all did."
Ayla moaned and thrust her arm out in search of Seth, who laid his hand in hers. Her fingers curled around his and squeezed.
"How long until we reach the Garden?" Calebna asked.
"Three days."
"Seth?" Ayla said.
"I'm here."
"I dreamt I was pregnant," she said. "And the baby looked like you. I didn't give birth. I just knew what it looked like."
The fear of his nightmares burned anew. "Was it a boy or a girl?"
"A girl. She was perfect."
Relief. "Then it certainly was your daughter." Seth closed his eyes and drank in the darkness, wishing he could stay and forget all he had seen. Wishing he could abandon all he had been called to do.
"Did you dream?" Ayla asked.
His smile fell. "I dreamed." And he opened his eyes again.
##
The Abomination smiled at Sarah's unconscious figure outlined by blue flowers. It shifted as if to leave Cain but found itself fixed. Again it tried to detach from Cain's body, but failed.
It closed its eyes and searched for the womb from which it had been born, the same hole the Abomination threw Cain into so often, but the void was gone. It snapped branches and ate fruit after fruit after fruit, but nothing changed.
The meaning came slowly. _I'll kill your family. I'll flay them alive and drink their blood. I'll tear their limbs off one by one, and I'll kill Sarah last and most painfully._ It waited, listening, hoping against hope that Cain was still there, somehow hidden from its sight.
But Cain and the void were not misplaced. They were simply gone. Vanished.
The Abomination thought of the layers—those gateways to other worlds—and the rivers of Time that ran through them. Could it be Cain had found a way to osmose the barrier?
But there was nothing other than the layers and the Waters flowing through them. The Abomination had searched them time and again with meticulous intensity. It found nothing else. There could be nothing else.
Unless...
It cursed Cain and sprinted for the orchard in his body.
The desert sands drank the moisture from Seth's feet and burned it from his tongue. If it had been the middle of the summer, they might not have made it so far, but when the frozen months came near, the sands could be safely traversed. Soon the dunes would frost at night, and the sun would burn less at midday, but always it remained a land of extremes.
Ayla stumbled and fell shoulder first into Seth, who caught both himself and her. She mumbled an apology, and he waved it off with a limp wrist and licked his flaking lips. A shadow sprinted past, and Seth winced at the sun to find the moving shape beneath it.
_A bird_ , he thought.
Then another shadow or was it several?
"Look," Calebna croaked, and pointed from atop a mountainous dune.
They hurried up on hands and feet and saw in the distance an oasis undulating in the heat of the desert.
"It's dancing," Ayla said.
"Is it real?" Calebna asked.
Seth stumbled forward and they followed. Of course it was real. He had seen it in countless dreams. It called like strange Music, and for reasons he could not explain, he felt he could do nothing but walk toward that singular point.
The Garden. A place of beginnings and a place of endings. Like the East is to the West, or the summer is to the winter, each singularity no more or less an equal and opposite expression of another. The Garden of Eden to this new perverse substitute.
_Contradiction._ The word bowled down the hill of his mind, gaining speed and size as if rolled of snow. As they neared the Garden, it began to remind him of that world beyond the Sands of Time. Perhaps it was the movement of the trees and undergrowth, so tightly woven together that it seemed any movement should crack limbs. Perhaps it was the memory of an endless shoreline, and a forest filled with Music, though there was no moisture here, and apart from the rustling of wind through the leaves, he sensed no violent Music, though he knew violence grew within. _Contradiction._
He wiped his face and itched the stubble. His body stunk, though he had long since let the upper half of his tunic beat at his thighs. He thought of what awaited them and worried at what he couldn't foresee, for there was darkness amidst the prophecies, valleys of shadow cast by Death.
As they entered the shade of the Garden, he sensed for the first time the glow cast by the plants. It sickened Ayla's pale skin and made her cheeks appear sunken. She caught him gazing at her and said, "What?"
Seth shook his head as vines slithered toward them and extended sharp points. The hair on his neck tingled as he remembered the entrance to the Shrine of the Song. He lifted a finger and pressed it against the vine's tip, feeling a sharp pinch and watching the vine drink a drop before slithering away. Ayla had already done the same, though Calebna hesitated and only followed after a nod from them both.
"This place feels evil," Calebna said.
"It is," Seth said.
"What do we do now?" Ayla took in the surroundings with rapt attention.
The Garden opened to receive them, and Calebna took a step back, for it seemed at first that they heard the tinkling of chimes.
"We change the Music," Seth said.
Ayla grabbed his hand, which had that chilled sweat feel in the cool breeze wafting from the Garden. "It's just like the Shrine."
"Yes. But I won't fail again by pushing you into darkness and danger. No, this time you will stay here," Seth said.
"But I perceive that you will have need of me before the end," Ayla said.
He laughed out of love, for there was no humor left to fuel it. "I will need you safe, not dead beside me."
"Nothing could make me let you do this alone."
"I will not—"
She dug her fingernails into his wrist. "I swore to never let you hurt like that again. I would rather die."
Seth stared into the eyes of the woman he loved and wondered what he had just asked of her. That she lay down love for self-interest? That she let him die to live alone, when all her life was enfolded into every detail of his?
He couldn't ask her to do what he himself could not do, so instead he leaned in, wrapped his arms around her, and kissed the side of her head.
And with that they entered the Garden, and Calebna followed with wringing hands and damp brow.
##
Calebna's clammy fingers stuck together. _I had claimed suicide was courageous, though if I had been honest with myself, I knew then that I should endure the pain. But how easy it can become to close your eyes and exhale, never to struggle for breath again._
Movement gathered like a buzz at the edge of Calebna's awareness, sharpened to a bitter tip trained toward that singular goal he had come to accept only after days of struggle. _I must become a martyr for a hope I shall never own._
_For their redemption, if not for my own._
_God forgive us. God keep us. God save us all._
As they walked the hallway of deadly vines, Calebna remembered the words of the voice in his dreams. The voice of the man he now knew was dead. The voice of the man whose body still walked among men. The voice of the man he would violate his vow to kill.
_"The pain will last but a moment if you succeed, but if you fail... You must not fail."_
_This is not my father_ , Lukian thought. He had known since Cain stepped out of the Fog that something had changed. But now he knew there was nothing left but a manic monster trapped in human skin.
_Such power._ Lukian examined the black vertebra set under his father's skin and those silver-within-silver eyes. Strange but provocative. He wanted it. Only he couldn't understand how to gain it. The boy had yet to return, and he ached for its presence. The creature was not his brother, or if it were, it was like Cain, hollowed out inside and replaced with something altogether different. Something evil.
_I will steal your power no matter the cost. Just come back to me._
A new thought struck him. What if the boy was inside Cain? After all, why would Cain's eyes shift from silver to brown and back again? Why would he have grayed skin? And how else could he have such strange power?
Cain was wringing his hands and glancing about. Sweat beaded across his skin and stuck the tunic to his body. Mason stood with Gorban, Adam, Eve, Jacob, and the others, and Lukian wondered why they stared at him. He remembered the blood still glistening on his skin from his encounter with the boy almost an hour ago, and he tried to brush it away, though it was thick and sticky. He grimaced and flicked his hands. Still they stared.
He plucked another fruit and ate it. He tried to clean away the blood with the juice of the fruit. It did help, though only a little.
He asked Cain, or whatever was inside him, "What are you searching for?"
Cain hushed him with a hiss. He seemed to be listening, and Lukian thought he heard a slithering sound, as if the Garden were opening again. Cain's face seemed to become shadowed, his silver eyes glinting. Sweat dripped down his neck and his hands shook.
"What is coming?"
Cain did not respond. He only stood there shaking.
The trees, brush, and vines at the edge of the orchard creaked and separated, opening a pathway through which walked familiar shapes, but Lukian's eyesight darkened upon seeing them.
Cain, or whatever was inside him, cried out, "The Enemy! He's cheated!" and screamed a string of profanities.
_These three figures should not be here_ , Lukian thought. _How could it be?_
Eve called out the names of her youngest son and daughter as she ran to them and threw her arms over their shoulders. "Is it real? Are you alive? But I saw you dead. I saw them close your tombs. And you! How are you here with them, Calebna?" Eve gripped her children as if afraid they were phantoms.
"God took us away to show us a glimpse of what was to come," Seth said.
Ayla smiled. "And God has brought us back to you."
"Praise the Almighty." Eve cupped their faces with her hands.
Calebna stepped toward Cain, and the family watched in stunned silence.
Cain pointed at them and shifted back, though his eyes were locked on Calebna. "You don't belong here."
"You thought to leave me for dead, but the Almighty directs my course," Calebna said.
Cain spat on Calebna's foot. "You're a charlatan. A bloodstained hypocrite."
"I will not deny the truth. I was going to kill myself there at the altar. I was going to lay myself down on the same stone upon which I laid so many other offerings."
Cain's face was twisted tight. "You're worthless. Your own God abandoned you."
"You can no longer shame me. I know that I am no holy man, and I've come to pay the price. What about you? Why don't you finish what you started?" Calebna opened his arms and pointed to his chest. "Strike me down. Destroy the bloodline as you intended from the beginning."
Cain's silver eyes flashed, and he stepped closer, hand poised at his hip as if ready to grab a veiled weapon. Suddenly Cain cried, "He knows!" and Calebna lunged like a bolt of lightning, right hand clutching a knife extended and plunging into Cain's chest. Cain screamed and swung his arms wildly. Lukian caught the glint of cold metal in Cain's hand as well, then silence, broken only by a crackling in both men's throats.
Then came screaming and a flurry of motion. Seth, Jacob, and Mason ran for Calebna while Gorban sprinted to his father. Mason tore the men apart, and Seth dragged Calebna away while Jacob pawed at Calebna's wounded chest and screamed his name.
Cain was gasping on the ground in a growing pool of red, and Gorban stood near, pale faced and staring at his father. Some of the women screamed, others collapsed with their arms over their heads as if the Garden would strike them down.
Lukian stood in a cloud of euphoria. He stared at the blood on Cain and wondered how it could be. He looked at Calebna, whose blood spilled out the corner of his mouth as well as down the side of his chest.
"You're going to be all right," Seth said as he bent over Calebna. "You're not going to die."
Calebna replied, "I have peace," and clutched Seth's hand. He coughed and the veins in his neck bulged with the strain. "He told me what to do. He told me it would be all right, that it wasn't really him, that he had crippled it."
Ayla was pulling Seth's arm, her eyes pleading as she said, "We need to get him out of here."
"What do you mean? _Who_ told you that?" Seth said.
"Leave." Calebna squeezed Seth's hand.
"No."
"You promised."
Seth's eyes grew with recognition and he struck the ground. "Don't speak like a fool."
"Don't add your sin to the foundation."
Seth raised his voice. "Tell me who told you."
"You already know." Blood sputtered in Calebna's throat.
The pool grew around Seth's feet and trickled toward Lukian, who knelt and watched the liquid come, first like a creek, then like a river, lastly like an ocean.
"He's dead," Gorban said in seeming disbelief as he stood over his father, Cain, but the words were distant as Lukian plunged the tip of his finger into streams of red. He heard the silver boy's voice. First distant, then beside him, lastly inside him. _"You have work to do._ "
##
Sarah woke in pain to a world spinning with colors, to swirls of green and brown encircling a spot of white so brilliant it seemed to stab her. She closed her eyes and grabbed the skin of her forehead with her cold fingers.
Where was she? She remembered the smell of blood on Cain's breath and those silver within silver eyes.
_No_ , she thought. _Not Cain. A silver-eyed monster._
She stood and held her belly with trembling fingers. Half-eaten fruit lay like severed heads amidst the flowers, and the smell of death was as thick as it had been the night all this evil was born. Wind whispered through the thorns, and she thought she heard far-off Music. It trickled through the vines and rattled the leaves, and she closed her eyes and strained to listen. It went as quickly as it came. Like innocence in an infant. Like beauty and love and all the good things of the world. It was a different Music than she had heard before, and she wondered if there could be something just beyond what was visible. A Light hidden under branch and leaf. A diamond buried in the crust of the world. A seed lying under scorched ground.
Then she stepped on a thorn and her blood joined the scattered red. Just ahead was an archway leading through the Garden. It hadn't shut after Cain. Had he been in a hurry?
She nursed her foot and made her way slowly forward, careful not to touch the vines or the deadly flowers. With every step, she felt her strength revive until at last she saw clearly and moved boldly. She rounded a bend and passed the trees with faces. They watched, but she ignored them and struggled through brambles. They grabbed her arms and legs and clothing, and by the time she escaped, she was scraped and bleeding. As the Garden's clamped mouth opened to the orchard, she saw a confused mass of people scattered about. Jacob knelt by a man she recognized as Calebna, and whose chest was wet with... blood?
She clutched her mouth and whispered, "Dear God... Calebna. Calebna?"
Her mind was reeling in an attempt to absorb the sounds and sights, and how it all could be. Not far from Calebna lay another figure drenched in blood, and as Gorban turned and met her gaze, she recognized the wounded man.
"Cain!" She screamed and ran to him, wrapping her arms under his shoulders. "You're bleeding." She looked up and searched Gorban's expression. "What happened to him?"
"He's dead," Gorban said.
But that couldn't be. She had just saved him. How could he have died? How could...?
But she knew. She knew and she beat Cain's chest and screamed at the bitterness of it all.
Gorban tore her away. "What are you doing?"
"He did it," was all she could say. "He did it..."
Now there truly was no hope left to resurrect the man she both hated and loved. Mason approached and lifted her. She leaned against him and struck his chest with her fists. "None of this should have happened."
Eve knelt, paler than a pillar of ivory, and Adam stood close by shaking his head and rubbing his eyes. Terah and Jacob wept, and everyone and everything else became a blur. Mason held her, and for the first time, she truly wept for Cain.
Seth pressed his fingers to Calebna's neck and felt nothing. The man's chest no longer moved, and the blood no longer seeped out of his wound. Seth stood, grabbed Jacob's shoulder, and shook him. "Quick, make torches."
"What?"
"Make fire."
Jacob looked at Calebna.
"Cry later. Your father died to save us, but we will only live if you make fire. Now."
The boy nodded, and he, Machael, Kiile, and several others broke branches and bundled them together. A bonfire was lit, and torches lay in the hands of nearly half of them. Jacob shuffled close.
Seth pointed. "The Garden."
Jacob nodded and cast wary glances toward the globes that seemed to stare through green slits, and the vines that slithered here and there. Seth's sister Sarah was still weeping in Mason's arms, and the wails of all the others soared through the canopy above.
Eve knelt next to Calebna, but she did not weep, and she did not look at her grandson. Instead, she stared at Adam, who knelt beside Cain and silently cradled his firstborn son, as if to repent for turning his back on him all those years ago out of pride and shame. But it was too late. Cain had claimed to be a reversal of the curse, but instead had become the fulfillment of it. The sin of the father had cultivated the sin of the son, and on and on it would go, a seemingly endless cycle from the first to the last Adam, just as he had seen through the vision the Metronome offered.
Ayla laid one hand on Seth's forearm and held a torch in the other. "Is it over so soon?"
"Is what over?" Jacob's eyes hardened.
"It is just beginning," Seth said.
"Did you know this would happen to my father?"
"I still don't know all of what has happened," Seth said.
Jacob's voice rose. "Why didn't you stop him?"
Seth glanced at the boy. "Why did you leave your father behind? You knew he would die alone in the Temple, so why do you grow angry now?"
Jacob turned and waved as if to dismiss the question. "It does not matter. I have lost him again."
Seth started two more torches and handed them to Eve and Terah.
"What will we do with their bodies?" Ayla asked.
"Mother?" Seth laid a hand on Eve's shoulder.
She turned, and her eyes were like wastelands.
"We need to leave."
She nodded and stood, but her gaze shot over his shoulder, and Seth turned too late to see what she looked at. He felt the familiar sensation of metal piercing flesh and remembered the attack of the machine—that Abomination from the world beyond—and grunted and doubled over.
Thrust into his abdomen was a knife. He walked his eyes up the arm that held it to sharp features and silver eyes, and wondered if somehow Cain could still be alive.
Then he realized who it was and gasped for breath as Ayla screamed, "Lukian!"
The Music had found a new Instrument.
##
The Abomination savored the violence, wishing it had time to drink Seth's blood. Its aspirations had been destroyed, but it could still salvage a portion of the glorious future.
_I will not fail you, Father. I will not let the Enemy win._
It cursed the complexity the Man had woven through the streams of Time. It could not kill them all, for that, too, would shatter the glorious future. It must choose its victim carefully, one and no more. So the Abomination pulled the knife out and stabbed Seth again and again. The man's insides rolled out over Lukian's hand, and the brutality would have continued but for the hand that grasped Lukian's neck and threw him to the ground.
The Abomination forced Lukian's body to turn, but already those huge hands grappled him with frightening strength. It was Mason, and as the bloodthirsty Abomination forced Lukian to stab Mason once in the shoulder and once in the thigh, the man broke first Lukian's left arm, then his right. Mason snapped Lukian's neck, and the Abomination spoke to the darkness. _"So ends Lukian, son of Cain, grandson of Adam, slave to Sin..."_
Gorban's eyes were wide with shock as he watched Lukian stab Seth with the knife he had retrieved from Cain's motionless hand. But the sight of Mason breaking Lukian's body swept him up into action. He grabbed Adam and Sarah and jerked them along, pulling them toward the entrance of the Garden, though Eve called Adam's name and Adam tore out of his grip to run to her.
"Keep moving," he said to Sarah, who had glanced at Cain's motionless figure.
The others were already setting fire to the Garden, and when he found Keshra and Peth, who had torches in their hands, he grabbed them and they ran together with Sarah toward the entrance.
Gorban stole Peth's torch and threatened the vines with it. The plants slithered away, and as they pushed their torches forward, they found that by grouping together they could make their way safely. As they exited the Garden, they glanced back to see flames consuming the canopy and belching black plumes.
Sarah knelt in the sand, but Gorban said, "We cannot rest yet."
"What about the others?" Peth said.
"We can do no more," he said. "We must get out of the desert heat."
"Where will we go?" Keshra asked.
Gorban knew that she had years ago lost what little love she had left for her husband, Lukian, but to see her dry eyes was disturbing nonetheless. They all saw the madness overtaking Lukian after his children were murdered by the Jinn, but what if that same madness could overcome Keshra in the loss of both her children and her husband? He suppressed his fears. "We will flee east, far from the desert, far from the City of the Almighty. And if they have any sense, they will follow."
Gorban pushed Peth and Keshra, and Sarah stood but did not follow. He turned to stare at his mother, and Peth and Keshra walked several paces before noticing.
Sarah's eyes were filled with tears. Gorban felt the tightness in his throat, the longing he had for the love of his father that now neither of them would receive.
He approached and they embraced.
"We must go." He gazed at the flames licking the Garden canopy.
"I know," she said.
"And we must never return."
"But I will never forget him." She offered him her hand, and he took it in both of his and knelt.
"I will never ask you to," Gorban promised.
##
Mason held his wounds, which burned with a scorching pain. Seth was still alive, if only barely, and he lay in Ayla's lap. The woman wailed, and as the others formed a circle around them, Mason approached.
"Don't leave me, Seth," Ayla said. "Don't leave me."
Eve's gaze looked to be carved from a glacier while Machael, Kiile, and their sons, daughters, and wives gathered around. Jacob and Terah wept for the loss of Calebna, and Ayla wept for what she was losing.
Flames grew about them and the heat nearly singed their skin. Soon the entire Garden would be aflame, and if they did not flee, they would be as well.
Adam heard Eve's call, ran to them, curled his arms under his final son's body, and lifted him out of Ayla's lap. Mason, bleeding from two knife wounds, lifted Calebna's dead body over his shoulder and joined the rough line following their patriarch. When they exited the Garden and walked a distance away, Adam set Seth on the sand, and Mason set Calebna next to him.
Mason remembered the words spoken to him in his final dream. _"Those you flee with will have need of your strength. You must stay with them and never leave. Even though the love inside you might yearn to break these bonds, doing so will only bring greater harm. This will be the last day you will see your mother or your brother Gorban. But you need protect them no longer."_
He had no idea how Gorban, Keshra, Peth and Sarah could have slipped away without him noticing, for he had decided to follow them anyway. But they were gone, and he was here. And wisdom bade him to stay.
Seth felt sand beneath him and remembered that place beyond Time. There was darkness, but pain remained. Then— _through_ the pain—came gentle melodies and rhythms. At first they were filled with an immeasurable sorrow. In the end the most mournful notes became the most joyful, the most triumphant, and he realized this Music was altogether different from the Song they heard within the Shrine.
"You were right," he whispered to Ayla, though he was unsure whether she heard him.
As the Music grew, a figure approached and knelt beside him. His countenance was a searing white, but Seth recognized him all the same.
_"Your time has not yet come_ ," the Man said.
Seth asked, "Is this real?"
_"As real as Sin. As real as the Music. As real as your life, which is in my hands."_
"I must be dreaming," he said.
_"Look at my face."_
Seth stared. "We thought you were dead. If you weren't, then how could all this be? How could you allow such violence?"
The Man looked closely at Seth. _"Would you rather I never gave you breath? Would you rather I never brought you life?"_
"I would rather avoid the pain, the evil, the Sin."
_"Then why didn't you?_ "
"How could I?"
_"You, not I, chose Sin_ ," the Man said.
"What about Abel? What about Lilleth? What about the rest?"
_"Everyone chooses their own curse, but their stories do not concern you. What you must understand is you, like Cain and all the others, have birthed death; but I, and only I, have birthed life."_
"You let them destroy each other."
The Man pointed downward, and Seth looked at the Man's heels, which were wounded. _"I let them destroy me. In place of Adam. In place of Cain. In place of you. Your words are tall, but you see so little. Do you really think death is the End? I am the End, just as I am the Beginning."_
"Then what will happen?"
_"I will make all things new. You need only trust me."_
Seth nodded and wrapped his fingers around the Man's wrist, which also was scarred. Tears came to Seth's eyes as he remembered the change he heard in the Music, from sorrow to joy. "I am sorry I doubted you. Forgive me. I..." He averted his gaze and closed his eyes, suddenly feeling how pocked with darkness he was. The Man was Light incarnate, but Seth felt himself murky and smudged with gray. "I am not worthy."
The Man lifted his chin with a touch so gentle it made him weep. _"You are not worthy, but because of your faith, I bore your unworthiness, and by doing so I call you worthy, and so does my Father, the Almighty."_
"Never leave me. Please—"
_"I am sending you back._ "
"I don't want to go."
_"You must go._ "
"Why?"
_"The End has not yet come. The blood that bore the evil of the world will come through your line, and in that day, you will rise to be with me. Then I promise I will never leave you nor forsake you. Only remember me and trust that the Music I am weaving will be more glorious than you could ever imagine. Then you will see."_
"What? What will I see?"
The Man smiled. _"Me._ "
"Why not now?"
_"Your eyes are not my eyes. Your ways are not my ways. Your wisdom is not my wisdom. Your love is not my love. Only trust in me. Only trust me."_
And the echo of the Man's voice went on and on as Seth felt the wholeness of life rush into him like blinding Light. He opened his eyes to a smoke-blackened sky and his loved ones weeping around him. Their eyes were closed and their heads were bowed, and none saw him awake. He looked down. There was no wound, no blood, but Calebna's dead body was beside him and the wound in his chest was still open and wet with blood.
Why had the Man healed him and not Calebna? Why had any of this happened? Why had so many suffered? He felt a righteous anger grow in his chest like hot coals.
But then he remembered the Man's words, and it seemed as though he heard them anew, like a man fully answered, _"Your eyes are not my eyes. Your ways are not my ways. Your wisdom is not my wisdom. Your love is not my love. Only trust in me. Only trust me."_
And after his family opened their eyes and shed their tears of disbelief, Eve looked at him. "I feel as I did when I gave birth to you. God has given me a new offspring, for Cain killed Abel. And though having you doesn't erase the pain, the joy of this moment somehow makes all of life glow brighter. Even the parts I had thought held only darkness."
Seth wrapped his arms around Ayla and said, "It seems redemption looks back for some as it looks forward for others." Ayla reached up and kissed him more passionately than ever before. He said, "I feel a new wind in the air."
And they rode it to new lands and a new era. But as every man returns to the dust of the ground and the hands of his Maker, so all eras come to an end, and most in terrible violence.
## EPILOGUE
_She was pregnant and was crying out in birth pains and the agony of giving birth. And another sign appeared in heaven: behold, a great red dragon, with seven heads and ten horns, and on his heads seven diadems. His tail swept down a third of the stars of heaven and cast them to the earth. And the dragon stood before the woman who was about to give birth, so that when she bore her child he might devour it._
—REVELATION 12:2–4 ESV
In a cave in the wilderness, Sarah lay screaming. She dug her fingernails into Keshra's hand, but the woman calmly brushed Sarah's hair from her face and wiped the sweat from her forehead. The sunset threw a hue of gold that slowly darkened to a red so deep it fell toward black. Finally, guarded from the pale silver eye of the moon, Sarah gave birth to a son.
Keshra cleaned the newborn and handed it to Sarah, who received her child with rapture. She crinkled her nose and nuzzled the baby's cheek, but though he seemed healthy and whole, two sharp points sprouted out the top of his head.
"What will you call him?" Keshra said.
"Enoch," Sarah said, and mouthed the name to feel it on her lips again.
Keshra played with Enoch's toes. "'A New Beginning.' What a beautiful name. But what is this?" She lightly pressed Enoch's little horns.
Sarah suddenly cried out in pain. It seemed at first an isolated occurrence, but then the agony increased, and Keshra handed the newborn to Peth and slipped her hand into Sarah's once more.
"Sarah," Keshra whispered, though she did not reply, too distracted by pain. "Sarah."
She said through bared teeth, "What?"
"I think another child is coming."
Sarah's muscles contracted and her back arced as she screamed. From the darkness of her womb to the darkness of the world, came a shriveled corpse. Keshra brought it to Sarah, who waved and said, "Take it away. Take it away."
Tears came to Sarah's eyes as Keshra disposed of the dead child. Keshra and Peth returned, and Gorban leaned against the mouth of the cave. Peth handed Enoch to Sarah, and though Sarah's mind claimed it would be better to abandon the child in the wilderness and let wild animals tear him to pieces, she held her baby close.
_He looks just like you_ , Sarah thought. _Just like you..._
Keshra spoke after a minute of quiet. "Where will we go? What will we do?"
Sarah said, "We will build a great city and name it after my son. But for generations it will be known as the city Cain built, for that will be the truth." She splayed Enoch's fingers and squeezed his tiny fingernails, though Enoch looked up at her with silver eyes that knew more than any newborn could, until she looked away.
Peth didn't seem to have noticed, for she was examining her own protruding belly, and smiled at Gorban. "In only a few months my first child will be born as well, and we will grow to a great people and spread across the earth."
"Yes." Sarah's countenance darkened.
The widowed and childless Keshra said, "Do you think we will ever see the others again?"
Sarah stopped playing with Enoch's hand and patted Keshra's leg. "I am weary. Will you tend to Enoch while I rest?"
Keshra took Enoch from her again and, with one hand feeling the horns on the top of his head, she, Peth, and Gorban left Sarah alone in the dark. And there came to Sarah then a quiet Music. A Music that used its tenderness to hide a violence yet unmatched in the world.
## READING GROUP GUIDE
**P ART 1**
1. Eve says, "His intention is for us to live joyful, peaceful lives. He wants us to prosper. If he didn't, why would we serve him?" Do you agree or disagree? What should be our motivation for serving God? Why do you think this?
2. Cain and Sarah's relationship is deeply wounded. Why do you think Sarah has stayed with Cain all these years? And why do you think Cain still feels a desire for her?
3. In chapter 2, we see that the Almighty promised the people, _"While you dwell in me, no danger will reach past the walls I have constructed. Not sickness, not demon, not nature."_ But as Adam is searching for Abel, he finds a cancerous lump on a sheep. How can this be?
**P ART 2**
1. Why did the voice say, _"Give it to me and I will show you the way. He is calling us"_ ? What is it demanding from Cain? And who do you think is calling Cain?
2. What do you think Cain was feeling in chapter 9 as he left Sarah behind?
3. If Sarah could have stopped Cain but didn't, does that make her as guilty as Cain? What about the Almighty? Is he culpable as well?
**P ART 3**
1. What exactly happened when Cain made the wager with the silver boy? How could he justify doing such a thing?
2. Seth and Ayla find three items on the Almighty's throne. What is the significance of each item? What does each one represent?
3. What do you think was going through Adam's mind as Eve and Calebna left to see the bodies of their loved ones and the proof that the Almighty was gone? Why did Adam not respond when Eve returned?
**P ART 4**
1. Cain thinks the Almighty banished humanity from the Garden of Eden out of fear. What else might have compelled the Almighty to do so?
2. Why does the Light Bringer urge Cain to drink blood? And why does Cain have such a strong desire for it?
3. Who is the Man and how could Cain have killed him?
**P ART 5**
1. Why do you think the people made a tomb for the Almighty as well as for Seth and Ayla?
2. If you were present, would you side with Lukian or Calebna? Why? Do you think your natural choice would be the _right_ choice?
3. Eve admits, _I hate his not being here. I hate his silence. His love for Abel._ Is she right to feel the way she does? What do you think you would feel in her position?
**P ART 6**
1. Do you think there was more to Calebna's sealing the Temple doors than just his desire to keep danger out?
2. After Lukian finds his children, the theme of consuming blood returns. What is the connection between the prick Lukian feels on his neck and the "shift" he senses in himself?
3. Why did Cain choose this particular time to return?
**P ART 7**
1. What is the connection between the Shrine of the Song and the Jinn? What exactly was the machine?
2. The machine shows Seth and Ayla three distinct areas in the Shrine: the Chambers of Science, the bathhouse, and the Metronome. What motivation could push "the Master" to build these three areas?
3. The Music grows in intensity as Seth and Ayla walk further into the Shrine. When all is said and done, it is "shaking the Waters with thunderous peals." If the Music can so powerfully impact all of Time, could the rest of Seth and Ayla's family have sensed it? Can you give any examples from earlier sections of the story that might point to this?
**P ART 8**
1. Calebna is perhaps the first person on earth to have struggled with suicidal thoughts. Being the High Priest, he would have known that suicide is against the Almighty's will. But what impact do you think the lack of historical examples would have on his ability to cope with the pressures he was experiencing? Do you think the imagery of the Spirit of God being an "all-consuming fire" would have played into his desire to use fire as the "uniting force"?
2. Seth says, "In this way, then, I think we could stop it. Not one man alone, but together through the choices we make, through the evil we refuse to tend in our minds like Forbidden Gardens." The language he uses seems to imply that, though we can refuse to tend evil, its seeds are still imbedded within us. Is it enough to refuse to tend the evil inside of us? Or, like a weed, will it sprout and grow nonetheless?
3. Lukian is struggling against the same madness that overtook Cain. Why do you think he, of all the family, has become the Abomination's favorite target?
**P ART 9**
1. When Cain finally confronts his father, and for the first time understands why Adam never showed him the love he longed for, his reaction is to completely disengage. Why?
2. This book deals with multiple generations and the impact of fatherhood. Do you think that if Adam had behaved differently, Cain might not have killed his brother? How do you think it might have changed Cain's personality?
3. What do you think happened to Cain in the end? Do you think his repentance was sincere?
**P ART 10**
1. What are some of the symbolic connections between Seth's experience in the Shrine and his experience in the perverse substitute for the Garden of Eden?
2. When Seth awakens next to Calebna, he struggles with why God would allow some terrible consequences and not others. Why is this question (and the Man's answer) so important? How does it square with your theology?
3. The core of this novel deals with the idea of worship. What exactly is worship? And how does this theme weave throughout the book and impact the characters and their relationships?
## ABOUT THE AUTHOR
Brennan S. McPherson has always wanted to tell stories, but it wasn't until his junior year in college that he built up the nerve to try. Three years later, _Cain_ , his first novel, was born. Brennan is married to his best friend, works full time at a small nonprofit, and plays the drums in his spare time.
## ACKNOWLEDGMENTS
Thank you, Jesus, for setting me free and showing me what it means to live.
To my wife, best friend, constant companion, crutch when I'm weak, warmth when I'm cold, Anna—where to start? Nothing I write is enough. You are a queen, and I can't wait to see what God does with you and your incredible talent.
Thanks to: My mom and dad, for giving me life and loving each other every day. My grandmother, Nonnie, for babysitting me and showing me good books. My teacher, Sandy Weber, for pouring years into me—without you, my head would be one empty vessel. My good friend, Don Boyer, for marking the daylights out of my early work with red ink (for good reason), yet still somehow encouraging me. All the guys at BroadStreet, for investing in a kid with a weird book. The superb editor of this novel, Natalie Hanemann, for tirelessly pouring through the manuscript, seeing what I couldn't see, excising the junk and beefing up the thin spots.
Finally... dear reader, THANK YOU! What a blessing to share this adventure with you.
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Beatriz Guttmann (Castelló de la Plana, 25 de gener de 1931 - 2014) fou una pintora i professora d'art valenciana. Consagrada i compromesa amb la seva ciutat i amb els més joves, Guttmann és un referent en la pintura abstracta valenciana.
Biografia
Guttmann va néixer el 1931 a Castelló de la Plana, filla de pares vienesos. Quan va esclatar la Guerra Civil Espanyola (1936) va traslladar-se a Viena amb la seva mare, on va viure fins a l'any 1938. Va estar uns mesos a Holanda i el 1939 va tornar a Espanya.
Entre 1948 i 1950 va estudiar Filosofia i Lletres a Barcelona i entre 1977 i 1981 va estudiar restauració i gravat a la Facultat de Belles Arts de València. L'any 1983, va obtenir el títol de professora de dibuix per l'Institut de Ciències de l'Educació de la Universitat de València. Finalment, l'any 1983 es va llicenciar en Belles Arts en la Universitat Politècnica de València i el 1992 va començar la seva tasca com a professora en la mateixa universitat. Al 1977 va ser acceptada com a membre de l'Associació Espanyola de Crítics d'Art, i el 1988 en l'Associació Internacional de Crítics d'Art.
Va publicar nombrosos articles monogràfics i estudis com el llibre publicat amb motiu de la seva tesi doctoral El Museo de Vilafamés: un hecho insólito. La seva primera exposició individual va ser el 1962 i, des de llavors, va exposar a tot Espanya, Europa i Amèrica.
Va formar part de nombroses activitats culturals, museístiques i congressos nacionals i internacionals com la II Trobada d'Art de Castelló, Lisboa 1987, Buenos Aires 1988 o Moscou 1989. Va tenir una vinculació professional i personal amb el Museu d'Art Contemporani Vicente Aguilera Cerni de Vilafamés, al qual sempre va estar lligada, ocupant diversos càrrecs com a secretària, directora adjunta i membre de la junta rectora.
El 2013 va fer la darrera exposició al MUCBE - Centre Cultural de Sant Francesc, antològica, a la seva població natal.
Va morir l'any 2014 als 83 anys a Castelló, a causa d'una fallida cardíaca després d'una llarga malaltia.
Obra
La seva obra, en la línia de l'art abstracte, suggereix presències figuratives sobre fons vaporosos. Utilitza generalment camps cromàtics homogenis sense permetre intervencions tonals de tints estranys. Va obtenir moltes distincions al llarg de la seva vida, com el nomenament de Dona de l'Any del 2007 per la ciutat de Castelló. Algunes obres destacades són La huella del pájaro, Collage de hojas o Paisaje solitario.
Les exposicions pictòriques que ha realitzat al llarg de la seva vida, de manera individual o col·lectiva, són innombrables. Els darrers anys abans de la seva mort va completar las seva trajectòria amb treballs ceràmics (objectes i murals) i amb el disseny de joies.
2000 - "Paràbola de las dunas y los vientos", febrer, Galeria d'Art Orfila, Madrid.
2000 - "De cine, mares y estrella", Festival de Cine, Castell del Papa Luna, Peníscola (Castelló).
2001 - "Retomar... continuar", juliol, Sala d'Exposicions de la CAM, La Llotgeta, València.
2002 - "Beatriz Guttmann. Parábolas de las dunas y los vientos", maig, Sala d'Art El Comendador, Vinaròs (Castelló).
2002 - "Retomar... continuar", 22 de maig a 8 de juny, Art Dam Galeria d'Art, Castelló de la Plana.
2004 - Galería Roglan, Barcelona.
2009 - Orfila, Galeria d'Art, Madrid, 9 al 28 de març.
2013 - Evocación de la Memoria III, MUCBE - Centre Cultural de Sant Francesc, Benicarló.
Guttmann té obres exposades a :
Institut Valencià d'Art Modern (IVAM), València.
The Hispanic Society of America, Nova York.
Museu d'Art Contemporani de Barcelona.
Museu d'Art Contemporani Vicente Aguilera Cerni, Vilafamés (Castelló).
Iberoamerica Vereinm, Hamburg (Alemanya).
Museu Cívic, Abano Terme (Província de Pàdua, Itàlia).
Museu Puixkin, Moscou (Rússia).
Centre de Documentació d'Art Valencià Contemporani de la Universitat de València, València.
Museu de Belles Arts de València.
Museu de Belles Arts de Castelló
Museu d'Art Contemporani d'Elx
Museo de Arte Moderno de Buenos Aires.
Referències
Morts a Castelló de la Plana
Pintors castellonencs
Pintors valencians contemporanis | {
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Q: Python Dataframe: Create function that makes all values in one column uppercase I have two python data frames. I want to make all the values in one column of both data frames uppercase.
The following code works:
df_ERA4['reqmnt'] = df_ERA4['reqmnt'].str.upper()
df_ERA5['reqmnt'] = df_ERA5['reqmnt'].str.upper()
But when I want to do the same thing in a function it doesn't work:
def uppercase(df):
df['reqmnt'] = df['reqmnt'].str.upper()
df_ERA4 = uppercase(df_ERA4)
df_ERA5 = uppercase(df_ERA5)
df_ERA4.head()
Specifically, when I run the above code it gives me the following error: AttributeError: 'NoneType' object has no attribute 'head'
A: Your function, by default does not return anything. So, by default, it returns None. When calling your function, the correct thing to do would be to not assign the return value to anything, since changes are being made in-place.
There are a couple of options now. The first one being: don't return anything, and don't assign anything.
def upper(df, col):
df[col] = df[col].str.upper()
upper(df, 'reqmnt')
However, this may not be the best approach (personally, I don't much fancy functions that perform in-place operations). You could, alternatively, have a copy returned via an assign call.
def upper(df, col):
return df.assign(**{col : df[col].str.upper()})
df = upper(df, 'reqmnt')
Note that there's a caveat on this side to - assign returns a copy, and sometimes, where efficiency/performance are paramount, you don't want to be needlessly making copies of GBs of data. What to use should be decided by a combination of style and need.
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{"url":"https:\/\/www.physicsforums.com\/threads\/relative-velocity-discussion.91567\/","text":"Homework Help: Relative Velocity Discussion\n\n1. Sep 29, 2005\n\ngasapple\n\nRelative Velocity?\n\nOK - I'm having problems visualizing a relative velocity problem. Please let me know if Im on the right track... problem involves a jet with initial V of 300 mi\/h due east that encounters winds at 100 mi\/h in a direction 30 degrees north of east. I need to calculate new velocity relative to the earth (or ground).\n\nI believe my relative velocity equation should be Vpe (plane to earth) = Vpa (plane to air) + Vae(air to earth). I'll solve for Vpe. Does this seem right? And since I'm asked for the new velocity (vector), i believe I'd take the arccos of the Vpa\/Vpe to get the angle?\n\nSound like a good approach?\n\nThanks again,\n\nLast edited: Sep 30, 2005\n2. Sep 30, 2005\n\nHallsofIvy\n\nWhat do you mean by \"Vpe\", etc.? They should be vectors but then \"arccos of the Vpa\/Vpe\" doesn't make sense- you can't divide one vector by another. Draw a picture (is the wind blowing toward 30 degrees N of E or from that direction? Usually wind is given as coming from a direction but you said \"in\" that direction so apparently it is blowing toward that.) Once you have drawn the vectors for the jets original velocity and wind velocity, draw in the third side of the triangle- the planes new velocity. You will need to use trigonometry to solve for the new speed and angle- cosine law to find the speed and then the sine law to find the angle.\n\n3. Sep 30, 2005\n\ngasapple\n\nThere was a relative velocity question where the relatives were divided and then the arccos was taken to determine the angle. It was a sample problem I viewed, hence my deduction. Had something to do with a boat crossing a river with a current and boat speed relative to shore - so I saw some simliarities with this problem...\n\nSo, it does makes sense that I would be adding the vectors for this relative velocity question? The sentence says \"in a direction 30 degrees north of east\" and \"what is the new velocity of aircraft relative to ground\"... So, I would use the sin function for the angle and how would I determine the velocity - simply the pythagorean?\n\nThanks again\n\nLast edited: Sep 30, 2005","date":"2018-07-16 15:26:24","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\": 0.802990198135376, \"perplexity\": 904.459560128771}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676589350.19\/warc\/CC-MAIN-20180716135037-20180716155037-00056.warc.gz\"}"} | null | null |
Q: MySQL - Get first 3 comma separated values I have a query which returns a field with a set of comma separated values. I would like to get the first three of these values into separate columns in the query result.
I can get the first, using SUBSTRING_INDEX but how can I get the other two?
SELECT
'aaaaa, bbbbb, ccccc',
SUBSTRING_INDEX('aaaaa, bbbbb, ccccc', ',', 1) AS column_one
EDIT - Oops, sorry forgot to mention. The value I want to split could have more (or less) than three strings to extract.
For example, the above string could easily be 'aaaaa' or 'aaaaa, bbbbb, ccccc, ddddd, eeeee'.
In each case, I only need the first three (or however many exist).
Any advice appreciated.
Thanks.
A: You could use SUBSTRING_INDEX twice, the second one with -1 parameter:
SELECT
'aaaaa, bbbbb, ccccc',
SUBSTRING_INDEX('aaaaa, bbbbb, ccccc', ',', 1) AS column_one,
SUBSTRING_INDEX(SUBSTRING_INDEX('aaaaa, bbbbb, ccccc', ',', 2), ',', -1) AS column_two,
SUBSTRING_INDEX(SUBSTRING_INDEX('aaaaa, bbbbb, ccccc', ',', 3), ',', -1) AS column_three
If the parameter is negative, everything to the right of the final delimiter (counting from the right) is returned. Eg.
*
*SUBSTRING_INDEX('aaaaa, bbbbb, ccccc', ',', 2) will return aaaaa, bbbbb
*SUBSTRING_INDEX(aaaaa, bbbbb, ',', -1) will then return bbbbb
You also might want to use ', ' as a delimiter, or TRIM the result.
Please see fiddle here.
Edit
If you want to consider strings that might have less than three values, you could use something like this:
SELECT
s,
SUBSTRING_INDEX(s, ',', 1) AS column_one,
CASE WHEN LENGTH(s)-LENGTH(Replace(s, ',', ''))>0
THEN SUBSTRING_INDEX(SUBSTRING_INDEX(s, ',', 2), ',', -1)
ELSE NULL END AS column_two,
CASE WHEN LENGTH(s)-LENGTH(Replace(s, ',', ''))>1
THEN SUBSTRING_INDEX(SUBSTRING_INDEX(s, ',', 3), ',', -1)
ELSE NULL END AS column_three
FROM
strings
Please see fiddle here.
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"redpajama_set_name": "RedPajamaStackExchange"
} | 3,411 |
\section{Introduction}
\label{sec:Intro}
The theory of topological physics has been realized and is being investigated in detail using real experimental systems.
Specifically, systems of cold atoms in optical lattices have a significant possibility of simulating the physics because such systems have a high parameter controllability,
isolation from environment, and no impurities \cite{Bloch,Georgescu}.
Very recently, as a typical verification experiment in one-dimensional (1D) topological physics,
topological charge pumping phenomena \cite{Thouless,Shen,Asboth} have been realized in cold atoms in a 1D optical lattice \cite{LeiWang,Lohse,Nakajima}.
It is therefore important to theoretically consider the topological physics and obtain new knowledge that has yet to be obtained.
Motivated by the experimental successes of the topological charge pumping, various studies on 1D topological physics have been conducted in recent years.
For example, the interaction effect for the topological charge pumping under adiabatic conditions has been extensively studied \cite{Qian,Zeng,Nakagawa,Tangpanitanon,Kuno,Hayward}.
The breakdown of the quantization of topological charge pumping has also been discussed \cite{RLi,Privitera}.
The effect of the initial nonequilibrium state for topological charge pumping and
interband coherence correction during adiabatic pumping in a periodically driven system were also reported \cite{Zhou,Wang,Raghava}.
The generalization of the Zak phase to the thermal states \cite{Viyuela} and the reinterpretation of the microscopic meaning of the Zak phase \cite{Rhim} have also been reported.
However, some areas in this field have yet to be investigated in detail.
During the past three decades, numerous papers on theoretical topological physics have been submitted.
The fundamental framework of topological physics has been theoretically developed \cite{Shen,Asboth}.
In conventional topological insulators, the topological properties are based on the following assumptions: the bulk band gap exists, and the system is close to equilibrium, i.e., the model is under adiabatic conditions.
This naturally brings up a question of how the non-adiabaticity affects the topological properties,
which needs to be answered.
However, there have been few studies on the theoretical formulation and quantitative evaluation of non-adiabatic effects \cite{Privitera}.
Therefore, this paper discusses the non-adiabatic effects on the topological properties using a typical model, focusing on the properties of the lower band ground state, particularly the lower band topological properties of the system. The target model is the Rice--Mele (RM) model \cite{Rice}.
If we introduce an adiabatic modulation parameter in this model and change the model parameter,
it exhibits a quantization of the Zak phase \cite{Zak} under an inversion symmetric condition.
In addition, under a certain periodic adiabatic parameter sweeping,
the model exhibits a two-dimensional instantaneous energy band topology in the adiabatic parameter dimension.
The instantaneous energy band topology then leads the topological charge pumping.
The RM model is much close to some experimental cold atom systems in an optical superlattice
\cite{Lohse,Nakajima,Atala}. In this study, we primarily deal with a weak non-adiabatic effect, where the transition probability between the lower and upper bands is small.
The non-adiabatic dynamics for the lower band population is formulated by applying the Landau-Zener (LZ) transition method \cite{Landau,Zener,Nori}.
Using the analytical population dynamics, we can formally construct the center of mass (CM) shift for the lower-band Wannier function after one pumping cycle and formulate a lower band pumped charge related to the topological charge pumping.
A similar prescription is described in~\cite{Oka,Lim}.
After formulating the non-adiabatic effect, the breakdown of the quantization of the CM shift is numerically estimated using the obtained formula.
This paper is organized as follows. In Sec.~\ref{Rice-mele}, the RM model is introduced and the proper linearized form of the RM model for the LZ transition method is explained.
In Sec.~\ref{LZTheory}, we show the LZ transition method. In particular, the LZ transfer matrix is introduced.
In Sec.~\ref{TEA}, we describe the dynamics of the lower band population under the assumption of an adiabatic-impulse approximation.
In Sec.~\ref{NAEZ}, we formulate the non-adiabatic extension form of the Zak phase and the CM shift of the lower band Wannier function corresponding to an electric polarization \cite{Resta,Vanderbilt,Marzari}. In Sec.~\ref{ELZ}, we estimate the validity of the LZ formulation by using a numerical simulation.
In Sec.~\ref{EXPLZ}, we discuss the current experimental conditions used for testing our results.
Finally, the conclusion is given in Sec.~\ref{Conq}.
The aim of this work is different from that of previous works such as \cite{Lim,Lim2,XShen},
which focused on the effects of an external force for the system and the Stueckelberg interferometry.
\section{Rice-Mele model}
\label{Rice-mele}
\begin{figure}[t]
\begin{center}
\includegraphics[width=8.5cm]{Fig1.eps}
\end{center}
\caption{(a) The band structure of the $t-k$ parameter space. (b) The band structure for $k=\pi$ and $0$. Two avoided crossings appear. For both cases, $J_{0}/\Delta_{0}=0.4$ and $\delta_{0}/\Delta_{0}=0.2$}.
\label{RMband}
\end{figure}
We start by considering the bulk momentum representation of the RM model. The Bloch vector representation is written in the following form:
\begin{eqnarray}
\hat{h}_{RM}(k,t)=d_{x}(k,t)\hat{\sigma}_{x}+d_{y}(k,t)\hat{\sigma}_{y}+d_{z}(k,t)\hat{\sigma}_{z},
\end{eqnarray}
where $\hat{\sigma}_{i}$ ($i=x,y,z$) is the Pauli matrix; $d_{x}(k,t)=J_{1}(t)+J_{2}(t)\cos k$; $d_{y}(k,t)=J_{2}(t)\sin k$; and $d_{z}(k,t)=-\Delta_{0}\cos (\Omega t)$,
$J_{1}(t) = J_{0} + \delta_{0}\sin(\Omega t)$; $J_{2}(t)=J_{0}-\delta_{0}\sin(\Omega t)$; and $\Omega=2\pi/T$. In addition, $T$ is the interval time of one cycle in the RM model. $J_{0}>0$, $\delta_{0}>0$, and $\Delta_{0}>0$.
The instantaneous energy spectrum has two bands, given by $E_{\pm}(k,t)=\pm |{\bf d}|$. The typical instantaneous energy spectrum is plotted in Fig.\ref{RMband} (a).
When we focus on a certain wave number $k$, the $t$ dependent instantaneous energy spectrum at $k$ can be regarded
as a two-level system including some avoided crossings in a certain parameter regime (see Fig.~\ref{RMband} (b)).
Assuming that the two bands never touch each other along one pumping cycle (interval time $T$), and a specific case $\Delta_{0}>2\delta_{0}$ and $J_{0}>\delta_{0}$,
an avoided crossing appears around $t=T/4\equiv t_{1}$ and $3T/4\equiv t_{2}$ for {\it any fixed $k$}.
At the avoided crossings, a non-adiabatic transition may occur depending on the sweeping speed, depending on $\Omega$.
The energy landscape is shown in Fig.\ref{RMband} (a). In what follows, the focus is placed on the energy landscape.
Here, the LZ transition around the avoided crossing point for a certain fixed $k$ is considered.
Around the avoided crossing points $t=t_1$ and $t_2$, $\hat{h}_{RM}$ can be linearized in terms of $t$ as $t=t_{1}\pm\delta t$ and $t=t_{2}\pm\delta t$.
The linearized Hamiltonian for $\hat{h}_{RM}$ is generally given in the following form:
\begin{eqnarray}
\hat{h}_{RM}(k,\delta t)=A(k)\hat{\sigma}_{x}+B(k)\hat{\sigma}_{y}+\Delta_{0}\Omega\delta t\hat{\sigma}_{z}.
\label{LZH1}
\end{eqnarray}
Here, $A(k)$ and $B(k)$ are $k$-dependent functions (independent of $t$),
$A(k)=(J_{0}+\delta_{0})+(J_{0}-\delta_{0})\cos k$, $B(k)=(J_{0}-\delta_{0})\sin k$, and
the $O (\delta t^{2})$ order terms are dropped.
Let us introduce the rotational transformation of the Pauli matrix. In the spin space, for the $i$-axis rotation ($i=1(x),2(y),3(z)$) with a certain angle $\rho$,
the rotated $j$-component Pauli matrix $\tilde{\sigma}_{j}$ is given by
\begin{eqnarray}
\tilde{\sigma}_{j}(\rho)\equiv \hat{\sigma}_{j}\cos\rho+\epsilon_{ijk}\hat{\sigma}_{k}\sin\rho.
\end{eqnarray}
Here, we use the above rotational transformation twice: The first is the $z$-axis rotation with $\rho=\tan^{-1}(B(k)/A(k))$,
and the second is the $y$-axis rotation with $\rho=-\pi$.
Thus, $\hat{h}_{RM}(k,\delta t)$ can be transformed into the following form:
\begin{eqnarray}
\tilde{h}_{RM}(k,\delta t)=-\sqrt{A^{2}(k)+B^{2}(k)}\tilde{\sigma}_{x}-\Delta_{0}\Omega\delta t\tilde{\sigma}_{z},
\label{LZH2}
\end{eqnarray}
where $\tilde{\sigma}_{x(z)}$ is a rotated $x$($z$)-component Pauli matrix. In addition, $\tilde{h}_{RM}(k,\delta t)$ is the canonical form for applying the LZ transition formula.
Now, the general LZ application form is defined as $\tilde{h}_{RM}(k,\delta t) \equiv -\frac{\Delta(k)}{2}\tilde{\sigma}_{x}-\frac{v\delta t}{2}\tilde{\sigma}_{z}$,
and thus $\Delta (k) \equiv2\sqrt{A^{2}(k)+B^{2}(k)}$ and $v=2\Delta_{0}\Omega$.
Here, the adiabaticity parameter is introduced by $\bar{\delta}(k)=\Delta^{2}(k)/(4v)$.
In the LZ transition method, the transition probability for the upper band at an avoided crossing is given as $\exp[-2\pi \bar{\delta}(k)]$.
Thus, $\bar{\delta}(k)$ characterizes the degree of the adiabaticity.
In this study, we assume the weak nonadiabatic regime. This means that $\bar{\delta}(k)$ is large to a certain extent,
that is, the transition probability for the upper band at an avoided crossing is small (Practically, we assume that the transition probability is less than 50\% for all $k$).
In what follows, we set $\hbar=1$ and take $\Delta_0$ as the unit of energy, $\Delta_0=1$.
\section{Landau-Zener transition}
\label{LZTheory}
The LZ transition is the transition between the lower and upper bands at avoided crossings.
The linearized RM model $\tilde{h}_{RM}(k,\delta t)$ of Eq.~(\ref{LZH2}) has two avoided crossings, namely, at $t=t_1$ and $t=t_{2}$.
The linearized Hamiltonian is given in the following matrix form
\begin{eqnarray}
&&\tilde{h}^{\pm}_{RM}(k,\delta t)=\nonumber\\
&&\left[
\begin{array}{cc}
\pm \Delta_{0}\Omega\delta t & \sqrt{A^{2}(k)+B^{2}(k)}e^{i\tilde{\rho}(k)} \\
\sqrt{A^{2}(k)+B^{2}(k)}e^{-i\tilde{\rho}(k)}& \mp \Delta_{0}\Omega\delta t \\
\end{array}
\right],
\label{LZH4}
\end{eqnarray}
where $\tilde{h}^{+(-)}_{RM}(k,\delta t)$ is defined around $t=t_{1(2)}$ and $\tilde{\rho}(k)=\tan^{-1}[B(k)/A(k)]$.
From the above matrix, the LZ transfer matrix can be introduced around the two avoided crossing points at $t=t_{1}$ and $t_{2}$.
The matrix is known to have the following form: \cite{Nori,Lim,Lim2,Kayanuma,Kayanuma2}:
\begin{eqnarray}
&&\Gamma_{\pm} (k) =\nonumber\\
&&\left[
\begin{array}{cc}
\sqrt{q_{LZ}(k)}e^{-i(\gamma_{nb}(k)\mp \tilde{\rho}(k))} & \pm \sqrt{p_{LZ}(k)} \\
\mp \sqrt{p_{LZ}(k)} & \sqrt{q_{LZ}(k)}e^{i(\gamma_{nb}(k)\pm \tilde{\rho}(k))} \\
\end{array}
\right],\nonumber\\
\label{LZ_matrix}
\end{eqnarray}
where $\gamma_{nb}(k)$ is the Stokes phase \cite{Kayanuma,Kayanuma2},
$p_{LZ}(k)=e^{-2\pi \bar{\delta}(k)}$ is the transition probability between the lower and upper bands, and $q_{LZ}(k)=1-p_{LZ}(k)$.
In addition, $\gamma_{nb}(k)$ is given by the following form:
\begin{eqnarray}
\gamma_{nb}(k)=\frac{\pi}{4}+\bar{\delta}(k)[\ln \bar{\delta}(k)-1]+ {\rm arg}[\Gamma (1-i\bar{\delta}(k))],
\label{Stokes}
\end{eqnarray}
where $\Gamma (z)$ is the complex gamma function.
Here, we comment briefly on the derivation of Eq.~(\ref{LZ_matrix}) and the Stokes phase $\gamma_{nb}(k)$.
First, the LZ transfer matrix of Eq.~(\ref{LZ_matrix}) is a slightly modified version of the usual LZ transfer matrix derived in \cite{Nori}
because $\Gamma_{\pm} (k)$ has an additive phase factor $e^{-i(\pm \tilde{\rho}(k))}$ in the diagonal elements compared with the usual LZ transfer matrix \cite{Nori}.
The phase factor comes from the factor $e^{\pm i\tilde{\rho}(k)}$ in the off-diagonal part in Eq.~(\ref{LZH4}).
The usual LZ transfer matrix is obtained when the off-diagonal part of Eq.~(\ref{LZH4}) is $\sqrt{A^{2}(k)+B^{2}(k)}$ \cite{Nori}.
In ~\cite{Lim,Lim2}, the LZ transfer matrix of Eq.~(\ref{LZ_matrix}) is derived from Eq.~(\ref{LZH4}).
Its derivation process is as follows: \\
\nin
(i) Although the factor $e^{\pm i\tilde{\rho}(k)}$ exists in the off-diagonal part in Eq.~(\ref{LZH4}),
the Schr\"{o}dinger equation of the linearized Hamiltonian in Eq.~(\ref{LZH4}) provides the Weber equation \cite{Zener}.
Accordingly, the same procedure as the derivation given in \cite{Nori} can be applied.\\
\nin
(ii) The Weber equation can be solved asymptotically. The solution is given by the Weber function \cite{Zener,Nori}.
Thus, the solution gives the probability amplitudes for the upper and lower band states.
Then, the phase part of the Weber function is the origin of the part of the Stokes phase \cite{Nori,Kayanuma,Kayanuma2}.\\
\nin
(iii) Then, the Weber function solution is substituted into the Schr\"{o}dinger equation for the linearized RM Hamiltonian in Eq.~(\ref{LZH4}),
and a slightly modified expression of Eq.~(A.12) in \cite{Nori} can be obtained, which includes the $e^{-i(\pm \tilde{\rho}(k))}$ factor.\\
\nin
(iv) Using the modified expression of Eq.~(A.12) in \cite{Nori} and following the same procedure in appendix A in \cite{Nori}.
the LZ matrix of Eq.~(\ref{LZ_matrix}) can be obtained.
Here, $\Gamma_{+} (k)$ and $\Gamma_{-} (k)$ act as a transfer matrix between the lower and upper bands at $t_{1}$ and $t_{2}$, respectively.
In the matrix $\Gamma_{\pm} (k)$, the diagonal terms are lower-to-lower and upper-to-upper state transitions,
and the off-diagonal terms are a lower-to-upper state transition and vice versa.
As explained in detail below, the adiabatic-impulse approximation is employed.
Therefore, the matrix $\Gamma_{\pm}(k)$ acts on extremely narrow time intervals, $t_{1}-0\leq t\leq t_{1}+0$ and $t_{2}-0 \leq t\leq t_{2}+0$.
Accordingly, we ignore the dynamical phase accumulated in such a narrow time regime.
In addition, we should comment on the approach of the adiabatic perturbation theory (APT) carried out in \cite{Privitera}.
Both the ATP and LZ transition approaches focus on the small $\Omega$ regime.
In the former approach, the Floquet lowest energy population after a single pumping cycle has a $\mathcal{O}(\Omega^{3})$ error, and
thus for a large $\Omega$ regime, the former approach is broken.
In the later approach, as long as the double avoided crossings in the band structure are created by tuning the parameters
and the largest gap regime in the band is sufficiently larger than $\Omega$, a large transfer to the excitation band at the avoided crossing can be handled,
and using the LZ transfer matrix in Eq,~(\ref{LZ_matrix}) it is possible to incorporate the influence of the interband coherence.
\section{Time evolution in adiabatic-impulse approximation}
\label{TEA}
\begin{figure}[t]
\begin{center}
\includegraphics[width=7.5cm]{Fig2.eps}
\end{center}
\caption{Schematic figure of the adiabatic-impulse approximation and the Landau--Zener transition.
The blue shaded area represents an impulse regime, which is an extremely narrow time interval.
In the adiabatic regime, the transition between the upper and lower bands is negligible.
In the blue shaded regime, the system is described using the linearized Hamiltonian $\tilde{h}_{RM}(k,\delta t)$. }
\label{Adiabatic-impluse}
\end{figure}
The adiabatic-impulse approximation is applied to the band structure, as shown in Fig.~\ref{RMband} (a).
Here, we assume that the avoided crossing regime for the target parameter regime is narrow.
In the other time regimes, the system is under adiabatic conditions.
A schematic figure of the adiabatic-impulse approximation is shown in Fig.~\ref{Adiabatic-impluse}.
Under these conditions, the time evolution of the wave function expanded by the instantaneous eigenstates is considered, with a focus on the change in the lower band occupation.
First, we prepare the wave function constituted by a linear combination of the periodic functions of the instantaneous lower and upper bands,
\begin{eqnarray}
|\Psi(k,t)\rangle=c_{1}(k,t)|u_{1}(k,t)\rangle+c_{2}(k,t)|u_{2}(k,t)\rangle,
\end{eqnarray}
where $c_{1(2)}(k,t)\in {\bf C}$ is the coefficient of the instantaneous eigenstate of the lower (upper) band, and $|u_{1(2)}(k,t)\rangle$ is determined by the Bloch theorem at time $t$.
The time evolution of $c_{1(2)}(k,t)$ can then be calculated
from the adiabatic-impulse approximation.
In the adiabatic regime, the time evolution is obtained
by considering the following unitary operator:
\begin{eqnarray}
U (k,t,t') &=&
\left[
\begin{array}{cc}
e^{-i\int^{t}_{t'} E_{+}(k,t'')dt''} & 0 \\
0 & e^{-i\int^{t}_{t'} E_{-}(k,t'')dt''} \\
\end{array}
\right].\nonumber\\
\label{Unitary_operator}
\end{eqnarray}
The operator $U$ acts on the coefficient vector $(c_{2},c_{1})^{t}$ and gives the adiabatic time evolution from $t'$ to $t$.
By contrast, around the avoided crossing corresponding to the impulse regime,
the time evolution of $c_{1(2)}(k,t)$ can be obtained by applying the LZ transition matrix $\Gamma_{\pm}(k)$.
It should be noted that the width of the impulse regime is assumed as a single point.
Even under such an assumption, the adiabatic-impulse approximation is believed to give fairly correct results, as mentioned in \cite{Nori}.
Under this situation, the coefficient $c_{1(2)}(k,T)$ can be connected to $c_{1(2)}(k,0)$ by applying the unitary
operator $U$ and the LZ transition matrix $\Gamma_{\pm}(k)$. By introducing the coefficient vector defined by ${\bf c}(k,t)=(c_{2}(k,t),c_{1}(k,t))^{t}$,
the one-cycle time evolution can be written in the following form:
\begin{eqnarray}
{\bf c}(k,T)& =& U(k,T,t_{2}+0)\Gamma_{+}(k) U(k,t_{2}-0,t_{1}+0) \nonumber\\
&\times&\Gamma_{-}(k) U(k,t_{1}-0,0){\bf c}(k,0).
\label{TEc1}
\end{eqnarray}
From this relation, the time evolution of the wave function $|\Psi (k,t)\rangle$ can be obtained. The time evolution includes the interband transition effect.
The interband transition is a non-adiabatic effect.
In this work, we put $c_{1}(k,0)=1$ as an initial state.
For the time evolution described by Eq.~(\ref{TEc1}), in this study we focus only on the dynamics of the lower band, i.e., the dynamics of $c_{1}(k,t)$, is of key interest.
The contribution of the lower-to-upper band after one pumping cycle may be regarded as a dissipation from the lower band state \cite{Oka}.
In this work, however, we do not focus on this contribution.
From Eq.~(\ref{TEc1}), the lower band population denoted by $|c_{1}(k,T)|^2$ is given by
\begin{eqnarray}
|c_{1}(k,T)|^{2}&=&1-2p_{LZ}(k)+2(1-p_{LZ}(k))p_{LZ}(k) \nonumber\\
&\times&\cos \biggl[ \int^{t_{2}}_{t_{1}}[E_{+}(k,t')-E_{-}(k,t')] dt'-2\gamma_{nd}(k)\biggr].\nonumber\\
\label{c1kT}
\end{eqnarray}
\noindent Here, it should be noted that in Eq.~(\ref{c1kT}) the coefficient $c_{1}(k,t)$ picks up the Stokes phase $\gamma_{nb}(k)$ twice because the LZ transition matrix $\Gamma(k)$ acts twice along the time evolution.
Under a classical assumption, the lower band population may be $|c_{1}(k,T)|^{2}=1-p_{LZ}(k)+p^{2}_{LZ}(k)$; however, Eq.~(\ref{c1kT}) includes the cosine factor,
whose phase factor is determined by the information regarding the instantaneous energy spectrum of both the lower and upper bands.
Thus, the cosine factor can be regarded as the interference effect of the upper band.
In addition, we mention the large $T$ limit for Eq.~(\ref{c1kT}). We then obtain $|c_{1}(k,T)|^{2}\to 1-2[1-\cos(\beta)]p_{LZ}(k)$,
where $\beta$ is the phase of the cosine part of the RHS in Eq.~(\ref{c1kT}).
Because $p_{LZ}$ decays exponentially with the increase of $T$, $|c_{1}(k,T)|^{2}$ tends to decay exponentially with the increase of $T$.
It is also interesting to compare Eq.~(\ref{c1kT}) with the result in \cite{Privitera}.
In \cite{Privitera}, a Floquet analysis shows that the lowest order deviation from unity in the nonadiabatic population transfer is proportional to $\Omega^2$.
Although it is difficult to verify the complete relationship between our result of Eq.~(\ref{c1kT}) and the $\Omega^2$ decay behavior,
as in \cite{Privitera}, if the lowest energy Floquet state is prepared as the initial state through a suitable smooth switch-on of the driving,
exponentially small corrections are recovered. Accordingly, Eq.~(\ref{c1kT}) in the large limit of $T$ is expected to exhibit
similar behavior as the result obtained by the Floquet analysis in a suitable smooth switch-on driving case.
\section{Non-adiabatic extension of the Zak phase}
\label{NAEZ}
Using Eq.~(\ref{c1kT}), a non-adiabatic extension of the Zak phase can be formulated.
The Zak phase is known to correspond to the electric charge polarization in a strongly correlated electron system \cite{Resta,Vanderbilt,Marzari}.
To formulate the extension form, by first using the lower band sector of $|\Psi(k,t)\rangle $,
we construct the lower band Wannier function $|W(t)\rangle$ \cite{Asboth} as follows:
\begin{eqnarray}
|W(t)\rangle &=& \frac{1}{\sqrt{N}}\sum^{N}_{m=1}e^{imk}|m\rangle \otimes c_{1}(k,t)|u_{1}(k,t)\rangle \nonumber\\
&=& \int^{\pi}_{-\pi}\frac{dk}{2\pi}|k\rangle \otimes c_{1}(k,t)|u_{1}(k,t)\rangle, \label{Wani}
\end{eqnarray}
where $m$ is a lattice site, $|m\rangle$ is the state in which a particle is localized at site $m$, and $N$ is the total number of lattice sites.
Accordingly, from $|W(t)\rangle$, the CM of the lower band Wannier function is given as $\langle W(t)|\hat{x}|W(t)\rangle$,
where $\hat{x}$ is the position operator of the particle as viewed from a continuous space.
In general, the CM is known to correspond to the Zak phase \cite{Asboth,Vanderbilt,Marzari}.
Hereafter, the CM $\langle W(t)|\hat{x}|W(t)\rangle$ is denoted by $P(t)$.
If $c_{1}(k,0)=1$, i.e., the initial state at $t=0$ is in the lower band state and we use $c_{1}(k,T)$ obtained from Eq.~(\ref{TEc1}),
the CM at $t=T$, $P(T)$ can be calculated as follows:
\begin{eqnarray}
P(T)&=&\frac{i}{2\pi}\int^{\pi}_{-\pi}dk\: \biggl[c^{*}_{1}(k,T)\partial_{k}c_{1}(k,T)\nonumber\\
&&+|c_{1}(k,T)|^{2}\langle u_{1}(k,T)|\partial_{k}u_{1}(k,T)\rangle \biggr],
\label{Pt}
\end{eqnarray}
where the first term in the integrant in the LHS vanishes
because $c^{*}_{1}(k,T)$ and $c_{1}(k,T)$ are symmetric for $k$ because $E_{\pm}(k,t)$, $\bar{\delta}(k)$, and $p_{LZ}(k)$ are symmetric for all $k$,
i.e, $c^{(*)}_{1}(k,T)=c^{(*)}_{1}(-k,T)$. Therefore, $P(T)$ is determined by only the lower band population $|c_{1}(k,T)|^{2}$ after a single pumping cycle.
In addition, because in an adiabatic limit $T\to \infty$, $|c_{1}(k,T)|\to 1$, the representation $P(T)$ is smoothly connected to the usual (adiabatic) Zak phase form \cite{Asboth}.
In this sense, Eq.~(\ref{Pt}) can be regarded as a non-adiabatic extension form of the Zak phase \cite{nonZak}.
Furthermore, by using Eq.~(\ref{Pt}), we can directly write the displacement of $P(t)$ from $t=0$ to $t=T$ as follows:
\begin{eqnarray}
&&P(T)-P(0) \equiv \Delta P(T) \equiv \Delta P_{0}(T)+\delta P(T),\label{DPT}\\
&&\Delta P_{0}(T)\equiv\frac{i}{2\pi}\int^{\pi}_{-\pi} dk\: \biggl[\langle u_{1}(k,T)|\partial_{k} u_{1}(k,T)\rangle\nonumber\\
&&\:\:\:\:\:\:\:\: -\langle u_{1}(k,0)|\partial_{k} u_{1}(k,0)\rangle\biggr],\label{DPT2}\\
&&\delta P(T) \equiv \frac{i}{2\pi}\int^{\pi}_{-\pi} dk\: \gamma_{d}(k) \langle u_{1}(k,T)|\partial_{k} u_{1}(k,T)\rangle,\\
&&\gamma_{d}(k) \equiv |c_{1}(k,T)|^{2}-1,
\end{eqnarray}
where $|c_{1}(k,T)|^{2}$ is given by Eq.~(\ref{c1kT}), $\Delta P_{0}(T)$ represents the adiabatic portion of the displacement $\Delta P(T)$, and
$\gamma_{d}(k)$ represents the deviation from the full population of the lower band for each $k$.
Here, the meaning of the total deviation $\Delta P(T)$ should be further discussed: $\Delta P(T)$ is the total shift of the CM of the lower band Wannier function after a single pumping cycle.
The total deviation $\Delta P(T)$ corresponds to the lower band pumped charge,
and not the total pumped charge of the system, which is generated by both the lower and upper band contributions. Here, to distinguish them we denote the two charges by $Q_{L}$ and $Q_{sys}$, respectively.
In an adiabatic limit, the lower band pumped charge $Q_{L}$
corresponds to the total pumped charge $Q_{sys}$ of the system because the upper band contribution is negligible.
This situation corresponds to the usual topological charge pumping \cite{Asboth}.
Therefore, we clearly define the lower band pumped charge $Q_{L}$ as
\begin{eqnarray}
Q_{L} \equiv \Delta P(T).
\end{eqnarray}
Here, if we assume the adiabatic limit $T\to \infty$, $Q_{L}$ is as follows:
\begin{eqnarray}
Q_{L} = \Delta P_{0}(T) = C_{N}, \label{CNP}
\end{eqnarray}
where we use the fact that $\Delta P_{0}(T)$ is regarded as the lower band Chern number $C_{N}$ \cite{Asboth}. Thus, in adiabatic situation,
because $C_{N}$ is known to take an integer value \cite{Thouless,TKNN},
the lower band pumped charge current $Q_{L}$ takes an integer value,
i.e., the topological charge pumping is recovered. However, in non-adiabatic situation, $Q_{L}$ does not take an integer value.
This indicates the breakdown of the quantization of the topological charge pumping owing to the decay of the lower band population $|c_{1}(k,T)|^{2}$.
\section{Estimation of the LZ formulation}
\label{ELZ}
\begin{figure}[t]
\begin{center}
\includegraphics[width=6cm]{Fig3.eps}
\end{center}
\caption{Lower band population $|c_{1}(k,T)|^{2}$ for $T=100 $(a), $50$(b), $40$(c), and $10$(d).
The solid line shows the analytical result of Eq.~(\ref{c1kT}). The blue circle shows the numerical result.}
\label{c1pop}
\end{figure}
In this section, we evaluate the adiabatic impulse approximation and the LZ formulation with the help of a numerical simulation.
In particular, we estimate the lower band population $|c_{1}(k,T)|^{2}$, and
demonstrate how the LZ result of Eq.~(\ref{c1kT}) captures the weak non-adiabatic dynamics in the RM model.
To this end, we numerically calculate the dynamics of the RM model obeying the Schr\"{o}dinger equation.
We then use the momentum representation of the Schr\"{o}dinger equation,
\begin{eqnarray}
i\frac{d}{dt}|\Psi(k,t)\rangle = \hat{h}_{\rm RM}(k,t)|\Psi(k,t)\rangle, \label{Seq}
\end{eqnarray}
and use the spin up and down bases, $|\Psi(k,t)\rangle=a_{1}(k,t)|\uparrow\rangle+a_{2}(k,t)|\downarrow\rangle$,
where $a_{1(2)}(k,t) \in {\rm {\bf C}}$ and $\hat{\sigma}_{z}|\uparrow (\downarrow)\rangle=+1(-1)|\uparrow (\downarrow)\rangle$.
In solving Eq.~(\ref{Seq}), we employ a fourth-order Runge-Kutta method.
To obtain the lower band population at $t=T$ after one pumping cycle,
we employ a gauge fixed exact solution of the instantaneous eigenvector of the lower band in the RM model.
This is given by
\begin{eqnarray}
|u^{ex}_{1}(k,t)\rangle =
\left(
\begin{array}{c}
\cos[\phi(k,t)/2]\\
-e^{-i\theta(k,t)}\sin[\phi(k,t)/2]
\end{array}
\right), \label{exactu1}
\end{eqnarray}
where the parameters $\phi(k,t)$ and $\theta(k,t)$ are determined by $J_{1}+J_{2}e^{ik}=[\tan\phi(k,t)]e^{i\theta(k,t)}$.
By solving Eq.~(\ref{Seq}) for each $k$ and using the exact solution $|u^{ex}_{1}(k,t)\rangle$,
we can obtain the numerical result of the lower band population at $t=T$ as $|\langle u^{ex}_{1}(k,T)|\Psi(k,T)\rangle |^{2}$.
Let us estimate the LZ analytical form of Eq.~(\ref{c1kT}).
We calculate both the LZ analytical lower band population \cite{anagamma} and the numerical one obtained by solving Eq.~(\ref{Seq}).
In this study, we vary the sweeping speed by varying $T$ for the one pumping cycle from $T=10$ ([$\hbar/\Delta_0$]) to $T=100$.
Here, $T$ is connected to the driving frequency as $\Omega=2\pi/T$.
In our target parameter set in the RM model, the minimum band gap at the two avoided crossings is $4\delta_{0}=0.8$ (we take $\Delta_{0}$ as the energy unit).
Fig. \ref{c1pop} shows the analytical and numerical results for $T=100$, $50$ $40$, and $10$ in $-\pi\leq k \leq 0$.
The $T=100$ case in Fig.~\ref{c1pop} (a) is assumed to be adequately adiabatic
because the driving frequency is quite small for the minimum band gap, i.e., $\Omega \ll 4\delta_{0}$.
The result therefore indicates that both the analytical and numerical lower band population $|c_{1}(k,t)|^{2}$ remains completely in the lower band after one pumping cycle.
Therefore, our analytical form of Eq.~(\ref{c1kT}) covers the adiabatic dynamics of the lower band population.
Next, see Fig.~\ref{c1pop} (b), (c), and (d), these are for $T=50$, $40$ and $10$.
In this parameter regime, we observed some decays of the lower band population, where the upper band population is finite at $t=T$.
For $T=50$ in Fig.~\ref{c1pop}(b), please see the analytical result.
Although the driving $\Omega$ is small, that is, the situation is fairly adiabatic,
a decay from $|c_1(k,T)|^{2}=1$ occurs, particularly around $k=-\pi$, because the band gap at $k=-\pi$ is smallest in the first Brillouin Zone,
and the analytical result is almost consistent with the numerical calculation, which is shown with the blue circles in Fig.~\ref{c1pop}(b).
Furthermore, let us focus on a faster driving case. As shown in Fig.~\ref{c1pop} (c), the analytical result for $T=40$ seems to indicate
that the degree of decay around $k=-\pi$ is larger than that in the case of $T=50$.
Although the numerical result seems to almost capture the behavior of the analytical result, slightly different values in the analytical result $|c_{1}(k,T)|^{2}$ are shown at near $k=-\pi$.
We expect that this difference originates from the adaptability of the adiabatic-impulse approximation and the LZ formalism.
More concretely, as noted in \cite{Nori}, the validity of the LZ transition method
is determined based on an inequality condition $4[d^{2}_{x}(k,t_{1(2)})+d^{2}_{y}(k,t_{1(2)})+\delta^{2}_{0}]\gg \Omega^{2}$, where the LHS is proportional to the band gap.
We estimate this condition for our target case. At near $k=-\pi$, the band gap is small, and thus the LHS tends to be small.
However, the RHS tends to be large for a small $T$. Therefore, under this situation, the degree of the inequality condition is weak as approaching the minimum band gap around $k=-\pi$ \cite{detaildecay}.
As a result, as shown in Fig.~\ref{c1pop} (c), we expect
that a deviation of the analytical result from the numerical result tends to appear at near $k=-\pi$ for a small $T$.
This result indicates the limitation of the adiabatic-impulse approximation and the LZ transition method.
Such a deviation tendency also appears for a further fast driving case.
Please see the $T=10$ case in Fig.~\ref{c1pop} (d). Although at a glance, the analytical result of the lower band population is almost consistent with the numerical result,
a large deviation occurs between the analytical result and numerical one around $k=-\pi$,
and also in this case we find a further deviation at near $k=0$.
We expect that this finding may also originate from the same reason with around $k=-\pi$.
In addition, we should comment the maximum decay regime for $T=10$.
Seen from the band structure and band gap tendency, as shown in Fig.~\ref{RMband},
the maximum decay is intuitively expected to occur around the minimum band gap regime $k=\pm \pi$.
As shown in our analytical and numerical results, this intuition is true for $T=50$ and $40$, as shown in Fig.~\ref{c1pop} (b) and (c),
but not for $T=10$. We expect that this difference originates from the interference effect of the upper band,
which is determined by the difference between the lower and upper band instantaneous energy spectrums and the Stokes phase as Eq.~(\ref{c1kT}).
Furthermore, it is interesting to compare $\Delta P(T)$, which was introduced in the previous section, with the total pumped charge $Q_{sys}$ of the system.
In particular, we compare the analytical $\Delta P(T)$ with the numerical $Q_{sys}$ obtained by solving Eq.~(\ref{Seq}).
Numerically, the total charge current of the system can be calculated by \cite{Asboth,Lim3}
\begin{eqnarray}
J(t)=\int^{\pi}_{-\pi}\frac{dk}{2\pi}\langle\Psi(k,t)|\frac{\partial\hat{h}_{\rm RM}(k,t)}{\partial k}|\Psi(k,t)\rangle \label{Current},
\end{eqnarray}
Using the current $J(t)$, the total system pumped charge is given by
\begin{eqnarray}
Q_{sys}(t)=\int^{t}_{0}dt'\: J(t') \label{Charge_pumped_Current}.
\end{eqnarray}
\begin{figure}[t]
\begin{center}
\includegraphics[width=8.5cm]{Fig4.eps}
\end{center}
\caption{(a) $T$-dependence of the total pumped charge of the system.
(b) The decay behavior of $\Delta P'(T)$ and $Q_{sys}(T)$ when varying $T$. }
\label{dQf}
\end{figure}
By contrast, it is difficult to directly calculate $\Delta P(T)$ of Eq.~(\ref{DPT}) owing to the gauge dependence.
To avoid this difficulty, we shift the target time interval of the RM model, $\Omega t\to \Omega \tilde{t}=\Omega (t-T/4+\delta t)$, where $\delta t$ is a positive small displacement.
Then, without a loss of generality, we can construct the same CM shift $\Delta P'(T)\equiv P'(T)-P'(0)$, where $\tilde{t}=0$ and $\tilde{t}=T$ are extremely close to the inversion symmetric point of the RM model.
Under adiabatic condition, the values of $P'(T)$ and $P'(0)$ are known to be $1/2$ and $-1/2$, respectively \cite{Asboth,Zak}.
In non-adiabatic regime, the finite $\gamma_{d}(k)$ in $P(T)'$ of Eq.~(\ref{DPT}) causes a breakdown of the discrete gauge invariance $P'(T)\to P'(T)+j$ (where $j$ is an arbitrary integer) \cite{Asboth}.
Thus, under non-adiabatic situation, while $P'(0)$ remains in $-1/2$ by using the exact solution of the instantaneous eigenvector of the lower band in Eq.~(\ref{exactu1}),
$P'(T)$ is expected to deviate from the value of $1/2$.
Here, to estimate the value of $\Delta P(T)$ qualitatively, we assume that the equivalence $\Delta P(T)=\Delta P'(T)$ is almost consistent. We calculate $\Delta P'(T)$
using the exact solution of the instantaneous eigenvector of the lower band in Eq.~(\ref{exactu1}).
To begin with, in Fig.~\ref{dQf}(a) we plot the time-dependence of $Q_{sys}(t)$ for various sweeping speeds. For $T=100$ and $70$, $Q_{sys}(T)=1$,
whereas for $T=50$, $Q_{sys}(T)$ does not reach unity, which clearly represents a breakdown of the adiabatic condition and the quantization of the topological charge pumping.
Figure~\ref{dQf}(b) shows the $T$-dependence of $\Delta P'(T)$ and $Q_{sys}(T)$.
Both cases clearly show a breakdown of the adiabatic condition and deviate from unity when decreasing $T$.
Interestingly, the decay behavior represents an oscillatory damping. This is expected to be caused by the interference term in Eq.~(\ref{c1kT}).
In addition, the result displays a reasonable behavior for a small $T$ regime, $\Delta P'(T)>Q_{sys}(T)$.
This is because $\Delta P'(T)$ includes only the lower band CM shift,
whereas $Q_{sys}(T)$ includes the current contribution of both the lower and upper bands,
where the current contribution of the upper band is inverse to that of the lower band, which originates from the upper band Chern number $C_{N}=-1$ \cite{Asboth}.
Remarkably, figure.~\ref{dQf}(b) indicates that, in the LZ formulation, a non-adiabatic breakdown of $\Delta P'(T)$ and $Q_{sys}(T)$ occurs
from a slow sweeping speed to a certain extent compared to the inverse minimum band gap at the avoided crossing.
\section{About experiment}
\label{EXPLZ}
Here, we mention the verification of our results for a cold atom optical lattice experiment.
Measuring the CM is not too difficult because an experimental method has previously been established, e.g., a band mapping method \cite{Lohse,Nakajima}.
The RM model has already been implemented in an optical super lattice system \cite{Lohse,Atala}
and a continuous RM model has also been developed \cite{Nakajima}.
These experimental systems can reach our considered parameter regime in terms of $J_{0}$,$\delta_{0}$ and $\Delta_{0}$ in the RM model.
However, some experimental limitations still remain.
For example, a perfect full occupation of the lower band state has not been realized \cite{Nakajima}
owing to a finite temperature effect, and a harmonic trapping potential in the experimental systems also breaks the translational symmetry of the system.
These obstacles must be overcome before highly accurate measurements of both the lower band population
after one pumping cycle and the CM shift of the lower band Wannier function are carried out
because the deviation from the quantization value in our estimation described in this study is small, i.e., at most $5-10\%$ in a weak non-adiabatic regime.
In addition, throughout this work, we focused on the weak non-adiabatic effects, i.e., a small correction for the lower band population.
Our estimation for such a small correction can also provide a deeper understanding and a key to a high precision control of quantum devices such as superconducting qubits \cite{Nori,Krantz}.
\section{Conclusion}
\label{Conq}
A weak non-adiabatic effect for the Zak phase and the topological charge pumping in the RM model has been discussed.
The dynamics of the lower band state has been formulated by applying an adiabatic-impulse approximation and using the LZ transition method.
We have derived the lower band population after one pumping cycle. The formula of Eq.~(\ref{c1kT}) includes the interference effect from the upper band.
From the lower band population after one pumping cycle,
we have obtained a LZ analytical formula describing a non-adiabatic extension of the Zak phase, which corresponds to the total CM shift of the lower band Wannier function.
We then estimated the validity of the LZ analytical method. In particular the analytical lower band population has been estimated
through a comparison with the results of a dynamic numerical simulation.
The analytical lower band population of Eq.~(\ref{c1kT}) almost captures the non-adiabatic dynamics of the lower band population in the system for the weak adiabatic regime.
Furthermore, the breakdown of the quantization of the total pump charge of the system has been numerically evaluated for various sweeping speeds and
compared with the CM shift of the lower band Wannier function obtained using the LZ analytical method.
The results indicate that the decay behavior depending on the sweeping speed exhibits some oscillating behavior, which may originate from the interference factor in Eq.~(\ref{c1kT}).
In addition, we found that the breakdown of the quantization of the topological charge pumping
starts earlier than the qualitative starting point characterized by the minimum band gap in the RM model.
If cold atom optical lattice experimental system is further cooled
and achieves the full occupancy for the lower band in the experimental RM model, our finding can be measured.
The general idea and prescription used to derive Eqs.~(\ref{c1kT}), (\ref{DPT}), and (\ref{DPT2})
are effective in investigating the non-adiabatic effects in wider topological models. \\
Y. K. acknowledges the support of a Grant-in-Aid for JSPS
Fellows (No. 17J00486).
\section*{Author contribution statement}
Yoshihito Kuno developed the idea of this article, performed
all calculations and wrote the manuscript.
\bibliographystyle{spphys-tv}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,152 |
Q: Подсчет частоты вхождения пар значений в строках DataFrame Всем привет!
Интересует как правильно и красиво написать python code.
Возможно ли для решения ниже приведенной задачи использовать только функции pandas или другие библиотеки и не использовать циклы for/while?
Как написать с помощью for я знаю, но решение мне кажется громозким.
Задача - для всех возможных комбинаций букв AB, AC, ..., XY, XZ, YZ подсчитать в скольких строках встречалась каждая комбинация букв. Буквы в строках предварительно надо отсортировать по возрастанию и по условию задачи буквы в строке не могут повторяться.
Из результата нужно выбрать 10 наиболее часто встречаемых пар букв.
Входные данные
A, B, J, K
B, E, P, A
R, Z, A, B
K, L, M, A
X, A, J, K
Результата запроса
A,B,3
A,J,2
A,K,3
B,J,1
B,K,1
и т.д.
A: Сначала нам понадобятся все комбинации пар столбцов:
import numpy as np
import pandas as pd
from itertools import combinations
cc = list(combinations(df.columns,2))
получилось:
In [39]: cc
Out[39]: [(0, 1), (0, 2), (0, 3), (1, 2), (1, 3), (2, 3)]
теперь можно найти все встречающиеся пары букв:
t = np.concatenate([np.sort(df.loc[:, c], axis=1).sum(1) for c in cc])
получилось:
In [43]: t
Out[43]:
array(['AB', 'BE', 'RZ', 'KL', 'AX', 'AJ', 'BP', 'AR', 'KM', 'JX', 'AK', 'AB', 'BR', 'AK', 'KX', 'BJ', 'EP', 'AZ', 'LM', 'AJ', 'BK', 'AE', 'BZ', 'AL', 'AK', 'JK', 'AP', 'AB', 'AM', 'JK'],
dtype=object)
теперь можно воспользоваться collections.Counter:
from collections import Counter
res = Counter(t).most_common(10)
результат:
In [46]: res
Out[46]:
[('AB', 3),
('AK', 3),
('AJ', 2),
('JK', 2),
('BE', 1),
('RZ', 1),
('KL', 1),
('AX', 1),
('BP', 1),
('AR', 1)]
Также можно создать Series из t и воспользоваться стандартными методами Pandas вместо испольования collections.Counter:
s = pd.Series(t)
res = s.groupby(s).size().nlargest(10)
результат:
In [49]: res
Out[49]:
AB 3
AK 3
AJ 2
JK 2
AE 1
AL 1
AM 1
AP 1
AR 1
AX 1
dtype: int64
Исходный DataFrame:
In [51]: df
Out[51]:
0 1 2 3
0 A B J K
1 B E P A
2 R Z A B
3 K L M A
4 X A J K
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,536 |
\part*{Supplementary Material}
\section{Conclusion}
\noindent Visual-only camera relocalisation has received significant attention in recent years because of the key role it plays in a wide variety of computer vision and robotics applications. However, many such applications require a system that can be used online, without expensive prior training on the target scene, for which many state-of-the-art methods, particularly those based on training a network to directly regress the camera pose \cite{Kendall2015}, cannot be used. Of those methods that can be used online, image retrieval methods fail to generalise to poses that are far from the training trajectory, whilst sparse keypoint matching methods tend to struggle in textureless regions, owing to difficulties in detecting suitable keypoints. Scene coordinate regression (SCoRe) methods generalise well to novel poses and can leverage dense correspondences to improve robustness, making them an appealing alternative to such approaches, but hitherto, only the forest-based approach of \cite{Cavallari2017,Cavallari2019} has been able to work online, and that method struggled to generalise to harder outdoor scenes because of its reliance on features that were hand-crafted for indoor use.
In this paper, we have shown how to address this limitation by proposing a way of leveraging the output of a SCoRe network (`ScoreNet') trained on one scene to predict correspondences and relocalise a camera in an entirely different scene. Our approach allows a single ScoreNet, trained on a scene from Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017} and adapted online, to achieve state-of-the-art performance on all scenes from both an indoor and an outdoor dataset, in under 300ms, without the need for offline training on each individual scene. Notably, unlike the online forest-based approach of \cite{Cavallari2017,Cavallari2019}, which leverages features hand-crafted for indoor use to achieve state-of-the-art results indoors on the 7-Scenes \cite{Shotton2013} and Stanford 4 Scenes \cite{Valentin2016} datasets, but struggles to relocalise well in harder outdoor scenes such as \emph{Great Court}, our method, which uses learnt features, is able to generalise well to such scenes, making it an appealing option for applications that require fast, accurate and online RGB-D camera relocalisation that works equally well in both an indoor and an outdoor context.
\section{Experiments}
\label{sec:experiments}
\noindent In this section, to evaluate our approach, we perform experiments on two well-known relocalisation benchmarks. More experiments can be found in the supplementary material.
\textbf{7-Scenes} \cite{Shotton2013} is a popular RGB-D relocalisation dataset that consists of $7$ different indoor scenes.
Whilst the scenes are relatively small, the captured sequences are in practice quite challenging, exhibiting motion blur, reflective surfaces, and repetitive and/or textureless regions.
\textbf{Cambridge Landmarks} \cite{Kendall2015,Kendall2016,Kendall2017} is an outdoor dataset consisting of $6$ scenes, captured at various locations around Cambridge. It is most commonly used for RGB-only relocalisation, but the coarse 3D SfM models provided for each scene allow it to also be used for RGB-D relocalisation, since it is possible to render depth images of the scene based on these models for each training and testing pose. For our evaluation, we make use of the depth images rendered by Brachmann et al.\ \cite{Brachmann2018CVPR}, to ensure that our results are comparable with those of both their DSAC++ approach and other recent works \cite{Cavallari2019}. As per \cite{Brachmann2018CVPR}, we also compare to a number of other well-known methods that are capable of making use of the 3D model \cite{Brachmann2017CVPR,Kendall2017,Sattler2017}. Note that, in common with other learning-based methods \cite{Brachmann2017CVPR,Brachmann2018CVPR,Brachmann2019}, we ignore the \emph{Street} scene, for which our method too was unable to produce reasonable results (the SfM reconstruction in the dataset appears to be of poor quality for this scene \cite{Brachmann2019}).
\subsection{Relocalisation Performance}
\label{subsec:experiments-performance}
\noindent To evaluate the overall performance of our online relocaliser, and its ability to adapt the predictions of a ScoreNet to a new scene, we pre-trained a ScoreNet for each scene from 7-Scenes \cite{Shotton2013} and Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017}, and evaluated their performances after grid-based adaptation (see Tables~\ref{tbl:performance-7scenes} and \ref{tbl:performance-cambridge}, and the supplementary material).
Several considerations proved important when training our ScoreNets. 7-Scenes \cite{Shotton2013} unfortunately only contains training and test sequences for each scene (i.e.\ there are no validation sequences), so we used the train/test/validation splits published by \cite{Cavallari2019} when training ScoreNets for these scenes. For Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017}, the training sequences contain many moving objects (pedestrians, cars, etc.).
For this reason, prior to training, we segmented the training images for each Cambridge Landmarks sequence using an Xception model pre-trained on CityScapes,\footnote{Specifically, the \texttt{xception65\_cityscapes\_trainfine} model from \url{https://github.com/tensorflow/models/blob/master/research/deeplab/g3doc/model_zoo.md}.} and invalidated the depths of all pixels from dynamic classes to avoid them being used during training. We also removed sky and ground pixels, as their depths were unreliable.
Our results on 7-Scenes \cite{Shotton2013} (see Table~\ref{tbl:performance-7scenes}) show that we are able to achieve superior performance to almost all of the methods against which we compared, with the notable exception of the forest-based approach of \cite{Cavallari2019}, which currently outperforms us by around 3\% indoors (although our average median localisation error is the same as that of \cite{Cavallari2019}). However, \cite{Cavallari2019} has the notable downside that whilst it performs well on easier outdoor scenes (e.g.\ see Table~\ref{tbl:performance-cambridge}), it performs extremely poorly on harder scenes such as \emph{Great Court} \cite{Kendall2015,Kendall2016,Kendall2017}.
Indeed, this is still the case even if we explicitly train a forest on \emph{Great Court} itself (see Table~\ref{tbl:performance-cambridge}), and regardless of whether we train a forest with both the RGB and depth features from \cite{Cavallari2019}, or with RGB features alone. By contrast, our approach allows a single ScoreNet trained on \emph{Great Court}, and using only RGB features, to be used to achieve excellent online relocalisation performance not only on all Cambridge Landmarks scenes (see Table~\ref{tbl:performance-cambridge}), but also on all scenes from 7-Scenes (see Table~\ref{tbl:performance-7scenes}).
\newcommand{\best}[1]{\textbf{\textcolor{red}{#1}}}
\newcommand{\secondbest}[1]{\textcolor{blue}{#1}}
\begin{table*}[!t]
\centering
\scriptsize
\begin{tabular}{lcccccccccH}
\toprule
\textsc{Indoor Scenes} & \textbf{Chess} & \textbf{Fire} & \textbf{Heads} & \textbf{Office} & \textbf{Pumpkin} & \textbf{Kitchen} & \textbf{Stairs} & \textbf{Average} & \textbf{Avg.\ Med.\ Error} & \textbf{Frame Time (ms)} \\
\midrule
\textbf{RGB-D, online} \\
\midrule
Ours (Office) & 98.95\% & 98.50\% & 99.10\% & 99.78\% & 89.70\% & 84.88\% & 81.60\% & 93.22\% & 0.013m/1.18$^\circ$ \\
Ours (Great Court) & 97.85\% & 97.20\% & 96.80\% & 91.95\% & 84.75\% & 79.86\% & 73.60\% & 88.86\% & 0.019m/1.06$^\circ$ \\
\midrule
Cavallari \emph{et al.} \cite{Cavallari2019} & 99.95\% & 99.70\% & 100\% & 99.48\% & 90.85\% & 90.68\% & 94.20\% & 96.41\% & 0.013m/1.17$^\circ$ & 257 \\
\midrule
\textbf{RGB-D, offline} \\
\midrule
Ours (Offline) & 99.80\% & 100\% & N/A & 99.78\% & 90.85\% & 90.36\% & 83.50\% & 94.05\% & 0.015m/1.08$^\circ$ \\
\midrule
Shotton \emph{et al.} \cite{Shotton2013} & 92.6\% & 82.9\% & 49.4\% & 74.9\% & 73.7\% & 71.8\% & 27.8\% & 67.6\% & -- & -- \\
Guzman-Rivera \emph{et al.} \cite{GuzmanRivera2014} & 96\% & 90\% & 56\% & 92\% & 80\% & 86\% & 55\% & 79.3\% & -- & -- \\
Valentin \emph{et al.} \cite{Valentin2015RF} & 99.4\% & 94.6\% & 95.9\% & 97.0\% & 85.1\% & 89.3\% & 63.4\% & 89.5\% & -- & -- \\
Brachmann \emph{et al.} \cite{Brachmann2016} & 99.6\% & 94.0\% & 89.3\% & 93.4\% & 77.6\% & 91.1\% & 71.7\% & 88.1\% & 0.061m/2.7$^\circ$ & -- \\
Meng \emph{et al.} \cite{Meng2018IROS} & 99.5\% & 97.6\% & 95.5\% & 96.2\% & 81.4\% & 89.3\% & 72.2\% & 90.3\% & 0.017m/0.70$^\circ$ & -- \\
Schmidt \emph{et al.} \cite{Schmidt2017} & 97.75\% & 96.55\% & 99.8\% & 97.2\% & 81.4\% & 93.4\% & 77.7\% & 92.0\% & -- & -- \\
Brachmann and Rother \cite{Brachmann2018CVPR} & 97.1\% & 89.6\% & 92.4\% & 86.6\% & 59.0\% & 66.6\% & 29.3\% & 76.1\% & 0.036m/1.1$^\circ$ & -- \\
\midrule
\textbf{RGB-only, offline} \\
\midrule
Brachmann and Rother \cite{Brachmann2018CVPR} & 93.8\% & 75.6\% & 18.4\% & 75.4\% & 55.9\% & 50.7\% & 2.0\% & 60.4\% & 0.084m/2.4$^\circ$ & -- \\
Li \emph{et al.} \cite{Li2018ECCV} & 96.1\% & 88.6\% & 86.9\% & 80.6\% & 60.3\% & 61.9\% & 11.3\% & 71.8\% & 0.043m/1.3$^\circ$ & -- \\
\bottomrule
\end{tabular}
\vspace{1mm}
\caption{Comparative results on 7-Scenes \cite{Shotton2013} (the \%s are of test frames with $\le 5$cm translation and $\le 5^\circ$ angular errors). We report results after hypothesis ranking (see \S\ref{subsec:method-cameraposeestimation}) for our approach and \cite{Cavallari2019}; see the supplementary material for further results. \emph{Ours (Offline)} denotes a variant in which we pre-trained separate ScoreNets offline for each scene, and then adapted and tested each ScoreNet online on its own scene. The ScoreNets for \emph{Ours (Office)} and \emph{Ours (Great Court)} were pre-trained offline on the bracketed scenes and then adapted and tested online on each different scene.
Note that \emph{Great Court} is a scene from Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017}, demonstrating the ability of our approach to adapt successfully between datasets.}
\label{tbl:performance-7scenes}
\end{table*}
\newcommand{--}%$\otimes$}{--
\begin{table*}[!t]
\centering
\scriptsize
\begin{tabular}{llllll}
\toprule
\textsc{Outdoor Scenes} & \textbf{Kings College} & \textbf{Old Hospital} & \textbf{Shop Fa\c{c}ade} & \textbf{St.\ Mary's Church} & \textbf{Great Court} \\
\midrule
\textbf{Online} \\
\midrule
Ours (Great Court) & 0.011m/0.056$^\circ$ & 0.010m/0.056$^\circ$ & 0.009m/0.040$^\circ$ & 0.011m/0.056$^\circ$ & 0.018m/0.040$^\circ$ \\
& \emph{76.97\%} & \emph{66.48\%} & \emph{95.15\%} & \emph{67.17\%} & \emph{77.50\%} \\
\midrule
Cavallari et al.\ \cite{Cavallari2019} \\
-- Office, RGB-D features & 0.01m/0.06$^\circ$ & 0.01m/0.04$^\circ$ & 0.01m/0.04$^\circ$ & 0.01m/0.06$^\circ$ & $\otimes$ \\
~~~(results from \cite{Cavallari2019}) & \emph{76.97\%} & \emph{82.97\%} & \emph{99.03\%} & \emph{79.62\%} & $\otimes$ \\
-- Great Court, RGB-D features. & --}%$\otimes${} & --}%$\otimes${} & 0.009m/0.040$^\circ$ & --}%$\otimes${} & --}%$\otimes${} \\
~~~(trained by us) & \emph{35.86\%} & \emph{41.21\%} & \emph{94.18\%} & \emph{7.55\%} & \emph{0\%} \\
-- Great Court, RGB features & --}%$\otimes${} & --}%$\otimes${} & 0.011m/0.056$^\circ$ & --}%$\otimes${} & --}%$\otimes${} \\
~~~(trained by us) & \emph{27.41\%} & \emph{4.40\%} & \emph{74.76\%} & \emph{30.94\%} & \emph{16.58\%} \\
\midrule
\textbf{Offline} \\
\midrule
Ours (Offline) & 0.008m/0.040$^\circ$ & 0.008m/0.040$^\circ$ & 0.009m/0.040$^\circ$ & 0.009m/0.040$^\circ$ & 0.018m/0.040$^\circ$ \\
& \emph{99.71\%} & \emph{100\%} & \emph{100\%} & \emph{99.62\%} & \emph{77.50\%} \\
\midrule
PoseNet (Geom.\ Loss) \cite{Kendall2017} & 0.99m/1.1$^\circ$ & 2.17m/2.9$^\circ$ & 1.05m/4.0$^\circ$ & 1.49m/3.4$^\circ$ & 7.00m/3.7$^\circ$ \\
Active Search (SIFT) \cite{Sattler2017} & 0.42m/0.6$^\circ$ & 0.44m/1.0$^\circ$ & 0.12m/0.4$^\circ$ & 0.19m/0.5$^\circ$ & $\otimes$ \\
DSAC (RGB Training) \cite{Brachmann2017CVPR} & *0.30m/0.5$^\circ$ & 0.33m/0.6$^\circ$ & 0.09m/0.4$^\circ$ & *0.55m/1.6$^\circ$ & *2.80m/1.5$^\circ$ \\
DSAC++ \cite{Brachmann2018CVPR} & 0.18m/0.3$^\circ$ & 0.20m/0.3$^\circ$ & 0.06m/0.3$^\circ$ & 0.13m/0.4$^\circ$ & 0.40m/0.2$^\circ$ \\
\bottomrule
\end{tabular}
\caption{Our average median localisation errors (m/$^\circ$) on Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017}. Since, like \cite{Cavallari2017,Cavallari2019}, our approach requires depth, we compare to methods that make use of the 3D models provided with the dataset. The \%s (where available) are of test frames with $\le 5$cm translation and $\le 5^\circ$ angular errors.
An $\otimes$ denotes a published failure. A -- is used for the median localisation errors when more than 50\% of the frames failed to relocalise. Numbers marked with a * were the result of end-to-end optimisation that did not converge. For \cite{Cavallari2019}, we report results for forests trained on \emph{Office} and \emph{Great Court}, with both RGB and depth features, and a \emph{Great Court} forest with only RGB features. All forests perform poorly on \emph{Great Court}, with the two forests trained on \emph{Great Court} also performing poorly overall, whereas our online ScoreNet trained on \emph{Great Court} adapts well to all scenes from both this dataset and 7-Scenes \cite{Shotton2013} (see Table~\ref{tbl:performance-7scenes}). See \ref{subsec:experiments-performance} for more discussion.}
\label{tbl:performance-cambridge}
\vspace{-\baselineskip}
\end{table*}
\subsection{Visualising our Approach's Behaviour}
\label{subsec:experiments-correspondencevisualisation}
\begin{stusubfig}{!t}
\begin{subfigure}{.24\linewidth}
\centering
\includegraphics[width=\linewidth]{images/proxy-5-a}
\end{subfigure}%
\hspace{1mm}%
\begin{subfigure}{.24\linewidth}
\centering
\includegraphics[width=\linewidth]{images/proxy-5-b}
\end{subfigure}%
\hspace{1mm}%
\begin{subfigure}{.24\linewidth}
\centering
\includegraphics[width=\linewidth]{images/proxy-5-c}
\end{subfigure}%
\hspace{1mm}%
\begin{subfigure}{.24\linewidth}
\centering
\includegraphics[width=\linewidth]{images/proxy-5-d}
\end{subfigure}%
\\[0.5mm]
\begin{subfigure}{.24\linewidth}
\centering
\includegraphics[width=\linewidth]{images/proxy-40-a}
\end{subfigure}%
\hspace{1mm}%
\begin{subfigure}{.24\linewidth}
\centering
\includegraphics[width=\linewidth]{images/proxy-40-b}
\end{subfigure}%
\hspace{1mm}%
\begin{subfigure}{.24\linewidth}
\centering
\includegraphics[width=\linewidth]{images/proxy-40-c}
\end{subfigure}%
\hspace{1mm}%
\begin{subfigure}{.24\linewidth}
\centering
\includegraphics[width=\linewidth]{images/proxy-40-d}
\end{subfigure}%
\\[0.5mm]
\begin{subfigure}{.24\linewidth}
\centering
\includegraphics[width=\linewidth]{images/proxy-60-a}
\end{subfigure}%
\hspace{1mm}%
\begin{subfigure}{.24\linewidth}
\centering
\includegraphics[width=\linewidth]{images/proxy-60-b}
\end{subfigure}%
\hspace{1mm}%
\begin{subfigure}{.24\linewidth}
\centering
\includegraphics[width=\linewidth]{images/proxy-60-c}
\end{subfigure}%
\hspace{1mm}%
\begin{subfigure}{.24\linewidth}
\centering
\includegraphics[width=\linewidth]{images/proxy-60-d}
\end{subfigure}%
\caption{Visualising the raw and adapted points that a ScoreNet trained on \emph{Chess} \cite{Shotton2013} predicts for three images from \emph{Shop Fa\c{c}ade} \cite{Kendall2015,Kendall2016,Kendall2017}, in comparison to the ground truth. Left-to-right: input images, raw predictions (in the \emph{Chess} scene), adapted predictions (in the \emph{Shop Fa\c{c}ade} scene), ground truth points. Points are mapped to scene-specific RGB cubes for visualisation purposes.}
\label{fig:proxy}
\vspace{-\baselineskip}
\end{stusubfig}
\noindent To explain why our approach is able to adapt the predictions of a ScoreNet to enable online relocalisation in a new scene, we visualise the raw and adapted points that a ScoreNet trained on \emph{Chess} \cite{Shotton2013} predicts for three images from \emph{Shop Fa\c{c}ade} \cite{Kendall2015,Kendall2016,Kendall2017}, and compare these to the ground truth (see Figure~\ref{fig:proxy}).
Note how the raw points predicted for similar-looking pixels in the input images are in similar parts of the \emph{Chess} scene (e.g.\ see the green window signs): this is what allows our grid-based adaptation approach to successfully cluster pixels in the input image based on their appearance.
Note also that our approach's performance is also not especially sensitive to the generation of perfect correspondences for every pixel: indeed, as implied by the last two columns of Figure~\ref{fig:proxy}, many predicted correspondences can be incorrect without affecting our ability to relocalise. This is because only $3$ good correspondences are actually needed to successfully estimate the camera pose using the Kabsch algorithm \cite{Kabsch1976}, and so as long as we have predicted enough good correspondences to have a high probability of finding and verifying $3$ good ones during the RANSAC process, our relocaliser is still likely to succeed (see supplementary material). This gives us a significant margin for error when adapting predicted points to a new scene, and makes our approach very robust in practice.
\subsection{Effects of Reservoir Sharing}
\label{subsec:experiments-reservoirsharing}
\noindent To study the impact of reservoir sharing (see \S\ref{subsec:method-onlineadaptation}) on relocalisation performance, we evaluated our relocaliser on 7-Scenes \cite{Shotton2013} for various fixed numbers of reservoirs (i.e.\ values for $N$). The results in Table~\ref{tbl:reservoirsharing} show that our approach is robust to a fairly high level of reservoir sharing: in particular, the performance stays above $90$\% even for values of $N$ as low as $5000$, in a context in which an average of around $29000$ reservoirs are needed if sharing is to be entirely avoided. The performance does eventually decrease for smaller values of $N$, but remains above $80$\% even when only $625$ reservoirs are used. This supports our hypothesis in \S\ref{subsec:method-onlineadaptation} that because the points in each reservoir are clustered into multiple sets that are disjoint in space, and because we use a RANSAC-based backend that is robust even when a high proportion of poor correspondences are generated, our reservoir sharing scheme's overall impact on performance is quite limited in practice for all but extremely low $N$.
\subsection{Timings}
\label{subsec:experiments-timings}
\noindent To better understand the time our approach takes to relocalise a frame, we provide a timing breakdown for our pipeline in Table~\ref{tbl:timings}. Like Cavallari et al.\ \cite{Cavallari2019}, we found that two costly steps were the optimisation of pose hypotheses during RANSAC, and the post-RANSAC ranking of the last $16$ hypotheses. In our case, the initial hypothesis generation is also somewhat costly, since the cost of running a forward pass of the ScoreNet is greater than that of predicting correspondences using a regression forest. Nevertheless, the overall time taken by our relocaliser is only around $292$ms, which is still fast enough to allow our method to be used in live scenarios such as interactive SLAM \cite{Prisacariu2017}.
\begin{table}[!t]
\scriptsize
\centering
\begin{tabular}{cccccccc}
\toprule
N & 625 & 1250 & 2500 & 5000 & 10000 & 20000 & 40000 \\
\% (5cm/5$^\circ$) & 81.21 & 85.31 & 89.07 & 92.02 & 92.95 & 93.34 & 93.22 \\
\bottomrule
\end{tabular}
\caption{The average performance on 7-Scenes \cite{Shotton2013} of our relocaliser trained on \emph{Office} for different numbers of reservoirs $N$ (see \S\ref{subsec:method-onlineadaptation}). On average (over all $7$ scenes), $\approx 29$k reservoirs are needed to avoid reservoir sharing. The performance remains relatively high even when only $5$k reservoirs are used, but eventually decreases for very small $N$.}
\label{tbl:reservoirsharing}
\vspace{-\baselineskip}
\end{table}
\begin{table}[!t]
\scriptsize
\centering
\vspace{2mm}
\begin{tabular}{lc}
\toprule
\textbf{Step} & \textbf{Time (ms)} \\
\midrule
Hypothesis Generation & 72.4 \\
Hypothesis Pruning & 1.2 \\
Inlier Sampling and Energy Computation & 1.4 \\
Optimisation & 72.1 \\
Hypothesis Ranking & 145.2 \\
\midrule
Total & 292.3 \\
\bottomrule
\end{tabular}
\caption{A timing breakdown for the version of our relocaliser trained on \emph{Office} \cite{Shotton2013}. See \S\ref{subsec:experiments-timings} for a discussion.}
\label{tbl:timings}
\vspace{-\baselineskip}
\end{table}
\section{Introduction}
\noindent Visual-only camera relocalisation has a wide variety of applications across computer vision and robotics, including augmented reality \cite{Castle2008,Paucher2010,Golodetz2015SPTR,Rodas2015,Bae2016}, tracking recovery and loop closure during SLAM \cite{Williams2011,MurArtal2014,Cavallari2019}, and map merging \cite{Kaehler2016,Golodetz2018}. For many applications, an ability to relocalise a camera \emph{online}, i.e.\ without expensive prior training on the scene of interest, is critical. For example, in an interactive SLAM context, it is typical to initialise the pose of the camera at the start of reconstruction and then track it from one frame to the next,
but when that tracking inevitably fails at some point, it is important to be able to relocalise the camera as soon as possible so that reconstruction can continue without unnecessary delay. However, despite the significant research attention that has been devoted to camera relocalisation in recent years, many state-of-the-art methods (especially those based on regression) remain wedded to an offline setting, making them difficult to deploy for live use.
Existing methods can be broadly divided into five types:
(i) \emph{Global matching} methods match one or more frames (or a descriptor, point cloud or 3D model derived from them) against either the contents of a database (e.g.\ one containing a map from keyframes to known poses), or a model of the scene, to look up a suitable pose. Methods like \cite{Gee2012} match the image itself against synthetic views of the scene, whereas \cite{GalvezLopez2011} and \cite{Glocker2015} match image descriptors against a database. Other methods \cite{Laskar2017,Balntas2018} find the nearest neighbours to a query image in the database, and use their poses to determine the query pose. Such image retrieval methods can struggle to generalise to novel poses. Geometry-based matching methods avoid this, but often require more than a single frame from which to relocalise. For example, \cite{Deng2016} matches a point cloud constructed from a set of query images to a point cloud of the scene, whilst \cite{Lu2016} reconstructs a 3D model from a short video sequence and matches that against the scene. One exception is \cite{Schoenberger2018},
which matches hallucinated subvolumes against a database using a variational encoder-decoder network.
This is single-frame, but quite slow, taking around a second per frame to relocalise.
\stufigstar{width=.95\linewidth}{images/pipeline-crop}{\textbf{An overview of our approach.} Ahead of time, we train a scene coordinate regression network \emph{offline} to predict correspondences between pixels in an input image and 3D points in an arbitrary pre-training scene (here, \emph{Chess} \cite{Shotton2013}): see \S\ref{subsec:method-offlinetraining}. To use this network to predict points in a different target scene (here, \emph{Heads} \cite{Shotton2013}) \emph{online}, we use the points the network predicts to index into an array of reservoirs, implicitly clustering the pixels using a grid-based approach that uses predicted pre-training scene location as a proxy for appearance: see \S\ref{subsec:method-onlineadaptation}. At online training time (purple and red boxes), we fill the reservoirs with points from the target scene, which we cluster using Really Quick Shift \cite{Chum2003}. At test time (purple and blue boxes), we predict a reservoir for each pixel, and use the point clusters the reservoirs contain to generate correspondences that can be passed to a Kabsch-RANSAC camera pose estimation backend \cite{Cavallari2019} to relocalise the camera: see \S\ref{subsec:method-cameraposeestimation}.}{fig:pipeline}{!t}
(ii) \emph{Global regression} methods directly regress an image's pose, using e.g.\ decision forests \cite{Kacete2017}, pose regression networks \cite{Kendall2015,Kendall2016,Kendall2017,Melekhov2017,Wu2017,Acharya2019}, GANs \cite{Bui2019} or LSTMs \cite{Clark2017,Walch2017}. Various recent approaches \cite{Brahmbhatt2018,Radwan2018,Valada2018,Li2019} have made use of the relative poses between images to improve performance. Global regression methods have proved popular, but have typically struggled to achieve the accuracy of local methods (see below). Those methods that do achieve better performance \cite{Radwan2018,Valada2018} currently do so by relying on an estimated pose from the previous frame, thereby essentially performing camera tracking rather than single-image relocalisation. Indeed, recent work by Sattler et al.\ \cite{Sattler2019} has shown that global regression is in many ways conceptually similar to image retrieval, and that current such approaches do not consistently outperform an image retrieval baseline or generalise well to novel poses. (This link with image retrieval was also noted earlier by Wu et al.\ \cite{Wu2017}.)
(iii) \emph{Local matching} methods match points in camera space with known points in world space, pass the correspondences to the Perspective-n-Point (PnP) algorithm \cite{Hartley2004} or the Kabsch algorithm \cite{Kabsch1976} to generate a number of initial camera pose hypotheses, and then refine these down to a final pose using some variant of RANSAC \cite{Fischler1981}. Many approaches match the descriptors of sparse keypoints to perform this matching \cite{Williams2011,Li2015,Feng2017,Sattler2017}; some approaches that perform dense matching also exist \cite{Schmidt2017}. Local matching methods tend to generalise better to novel poses than image retrieval methods, since individual points are often easier to match from novel angles than are whole images.
(iv) \emph{Local regression} methods generally use regression forests \cite{Shotton2013,GuzmanRivera2014,Valentin2015RF,Brachmann2016,Meng2016,Cavallari2017,Meng2017IROS,Meng2018IROS,Cavallari2019}, neural networks \cite{Brachmann2017CVPR,Brachmann2018CVPR,Duong2018,Li2018RSS,Li2018ECCV,Brachmann2019}, or a mix of the two \cite{Massiceti2017} to predict the scene coordinates of pixels in the input image. They then pass these correspondences to PnP/Kabsch and RANSAC. Compared to local matching methods, local regression methods can avoid the need for explicit keypoint detection, which can be costly, and can make use of correspondences from the whole image during RANSAC, which can help with robustness \cite{Shotton2013}. Like local matching methods, they also generalise well from novel poses. However, whilst they tend to be more accurate than local matching methods at small/medium scale, they have not yet been shown to scale well to very large scenes \cite{Sattler2019}.
(v) \emph{Hybrid} methods use both the global and local paradigms, generally by first performing some kind of lookup/matching, and then refining the results using either RANSAC \cite{MurArtal2015,Taira2018} or continuous pose optimisation \cite{Valentin2016}.
Not all of these methods are designed for online, single-frame relocalisation. Image retrieval methods can normally be used online, but struggle to relocalise from novel poses. Global regression methods generally require significant offline training on the target scene; moreover, their comparatively poor accuracy makes them unattractive for applications like interactive dense SLAM \cite{Newcombe2011} that require precise poses (their main niche is large-scale, RGB-only relocalisation scenarios in which coarse poses are acceptable).
Local matching methods can generally be used online \cite{Williams2011,Feng2017}, but because most rely on detecting/matching sufficient sparse keypoints in the image, their robustness can suffer in textureless parts of the scene.
By contrast, local regression methods avoid the need to detect keypoints explicitly, making them appealing for robust relocalisation in small/medium-scale scenes. However, most such methods, like their global counterparts, require costly offline training.
One \emph{online} local regression approach is that of \cite{Cavallari2017,Cavallari2019}, which showed how to adapt the regression forests of \cite{Shotton2013} for online use in real time.
Their approach achieves state-of-the-art performance on the popular 7-Scenes \cite{Shotton2013} and Stanford 4 Scenes \cite{Valentin2016} indoor datasets, and also performs well on some of the easier outdoor scenes from Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017}. However, because their forests use hand-crafted features that were designed for indoor use \cite{Shotton2013}, they struggle \cite{Cavallari2019} to work out-of-the-box on harder outdoor scenes.
Whilst it might in principle be possible to solve this problem by hand-crafting new features for outdoor use, doing so could be time-consuming and costly. Indeed, the broader trend in machine learning has been towards replacing models such as regression forests with neural networks that can learn suitable features, rather than trying to hand-craft them manually. However, replacing the forests used by \cite{Cavallari2017,Cavallari2019} with networks is not straightforward. To achieve online relocalisation, they rely on the way in which their forests predict leaves containing reservoirs of points to adapt forests between scenes, and it is tricky to see how this scheme can be easily transferred to work with local regression networks, which tend to directly predict individual points in the training scene.
\textbf{Contribution.} In this paper, we address this problem by proposing a novel method that allows the predictions of a network trained to regress 3D points in one scene to be leveraged to predict points in a new scene, thereby enabling the network to be used online. Our approach (see \S\ref{sec:method}) works by replacing the appearance clustering that was implicitly being performed by the branching structure of the forests in \cite{Cavallari2017,Cavallari2019} with a two-step process that first uses the network to predict points in the scene on which it was trained, and then uses these predicted points to look up reservoirs of points from the new scene. We show via experiments on 7-Scenes \cite{Shotton2013} and Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017} that our approach achieves state-of-the-art performance in under $300$ms, whilst requiring no offline training on the test scene. We further show that the learnt features of our networks allow us to perform well even on harder outdoor scenes that
were causing methods such as that of \cite{Cavallari2017,Cavallari2019} to fail.
\section{Method}
\label{sec:method}
\subsection{Overview}
\noindent Our pipeline is shown in Figure~\ref{fig:pipeline}. We start by training (\emph{offline}) a scene coordinate regression network (a `ScoreNet') to predict correspondences between pixels in an input image and 3D points in an arbitrary pre-training scene. The structure of the ScoreNets we use and how they are trained are described in \S\ref{subsec:method-offlinetraining}. To use a ScoreNet to relocalise in a scene other than the one on which it was trained, we need a way of adapting the predictions of the network \emph{online} so as to predict points in the new scene of interest. As described in more detail in \S\ref{subsec:method-onlineadaptation}, we do this by using the points predicted by the network to index into an array of reservoirs that can be refilled with points from the new scene at online training time, and then looked up again at test time to generate correspondences. This scheme draws inspiration from the adaptive regression forest approach of Cavallari et al.\ \cite{Cavallari2017,Cavallari2019}, but modifies it to work in a ScoreNet context. Having obtained the needed correspondences, we can then generate camera pose hypotheses using the Kabsch algorithm \cite{Kabsch1976} and refine them down to a single output pose using pre-emptive RANSAC, as was done in \cite{Cavallari2017}.
ICP \cite{Besl1992} against a 3D model of the scene can then be used to refine the initial pose produced by the relocaliser. Furthermore, the last few candidates considered by RANSAC can also be ranked using the model for an additional boost in performance, as described in \cite{Cavallari2019}. See \S\ref{subsec:method-cameraposeestimation} for more details.
\subsection{Offline ScoreNet Training}
\label{subsec:method-offlinetraining}
\stufig{width=.9\linewidth}{images/vgg_arch-crop}{\textbf{ScoreNet architecture.} We use a truncated VGG-16 feature extractor, followed by several $1 \times 1$ convolutional layers, to regress 3D world space points for a subset of pixels from the original image.}{fig:scorenet}{!t}
\noindent Inspired by Brachmann et al.\ \cite{Brachmann2018CVPR}, we train a ScoreNet with a fully-convolutional, VGG-style \cite{Simonyan2015} architecture to predict correspondences between pixels in the input image and 3D points in world space. Our network takes as input an RGB image of size $w \times h$, and produces as output a $w/8 \times h/8 \times 3$ tensor of 3D world space points corresponding to pixels subsampled densely from the original image on a regular grid with $8$-pixel spacing, i.e.\ pixels $\{(8i,8j) \in [0,w) \times [0,h) : i, j \in \mathbb{N}^+\}$. The architecture (see Figure~\ref{fig:scorenet}) consists of a truncated VGG-16 feature extractor, followed by several $1 \times 1$ convolutional layers to regress a 3D point for each relevant pixel. Each network is trained on the RGB-D training sequence associated with a single scene from one of our datasets (see \S\ref{sec:experiments}). Further details about the architecture and precisely how we train our networks can be found in the supplementary material.
\subsection{Online ScoreNet Prediction Adaptation}
\label{subsec:method-onlineadaptation}
\noindent \textbf{Problem Formulation.} A ScoreNet trained offline on an RGB-D sequence of a scene, as in \S\ref{subsec:method-offlinetraining}, can later be used to relocalise new images in the same scene.
This targets an \emph{offline} formulation of the relocalisation problem, in which both training and testing are performed on the same scene, and there are no constraints on the time available for training. However, this formulation does not take into account the practical requirements on a camera relocaliser for live scenarios such as interactive dense SLAM \cite{Newcombe2011}, in which it is infeasible to spend hours or even days training a relocaliser on the scene of interest; rather, a relocaliser must be trained online as the user moves around the scene, and then be usable immediately when camera tracking fails.
To address such scenarios, we target the alternative \emph{online} formulation of the relocalisation problem proposed by Cavallari et al.\ \cite{Cavallari2017}, in which there are three stages: offline training (`pre-training'), online training and testing. Offline training is performed on sequences of \mbox{RGB-D} frames (with known poses) from one or more scenes, generally other than the target scene. Online training is then performed on a single RGB-D sequence (again with known poses, e.g.\ as produced by a camera tracker) from the target scene. Finally, testing is performed on a single RGB or RGB-D image whose pose is to be determined. (For interactive SLAM, the idea is that a user will move around the scene at online training time, either training a new relocaliser online, or adapting a pre-trained relocaliser online to function in the target scene. If and when camera tracking fails, the trained relocaliser can then be used to recover the camera pose.)
Cavallari et al.\ \cite{Cavallari2017,Cavallari2019} described their online training stage as `adaptation' because they were adapting a pre-trained regression forest to relocalise in the target scene. In particular, they showed that the branching structure of a scene coordinate regression forest can be seen as a scene-independent way of clustering the pixels in an image based on their appearance. Based on this insight, they adapted a pre-trained forest to a new scene by emptying the reservoirs in its leaves and refilling them with points from the new scene at online training time, and then using the forest to look up the reservoirs again to provide correspondences at test time. Inspired by this approach, we show in this paper how to adapt the predictions of a ScoreNet so as to allow these relocalisers too to be deployed in an online context.
\textbf{Reservoir Prediction.} The adaptation scheme described in \cite{Cavallari2017,Cavallari2019} was highly effective, but relied on the fact that their forest does not predict points in any particular scene directly, but instead predicts leaves containing reservoirs of points, which can then be used to generate the needed correspondences. These reservoirs can be refilled with points from the new scene, which is what allowed their method to work, but it is not straightforward to see how it can be transferred to ScoreNets that directly predict individual points in the pre-training scene. To achieve this, we thus propose a new scheme that, rather than clustering pixels into leaves based on routing their associated feature vectors down a regression forest, clusters them into cells in a grid placed over their associated predictions in the pre-training scene (see Figure~\ref{fig:pipeline}). Note that this implicitly clusters pixels in the input image based on their predicted pre-training scene locations, rather than directly based on their appearance.
Intuitively, a ScoreNet, which has been deliberately trained to map similar-looking pixels in an image to similar 3D points in the pre-training scene, can in practice do this for images of any scene, not just the one on which it was trained, and hence pre-training scene location can be used as a reasonable proxy for appearance (see \S\ref{subsec:experiments-correspondencevisualisation} for a discussion).
As mentioned in \S\ref{subsec:method-offlinetraining}, our ScoreNets take an RGB image of size $w \times h$ as input, and produce as output a $w/8 \times h/8 \times 3$ tensor that contains a predicted 3D point (in the scene on which the ScoreNet was trained) for a regularly-spaced subset of pixels in the image. We initially map each of these predicted points, $\mathbf{p} = (p_x, p_y, p_z) \in \mathbb{R}^3$, to a grid cell index as follows. First, we imagine placing a bounded regular cubic grid, with cells of side length $\ell$ and an overall side length of $C\ell$, over the pre-training scene, as shown in Figure~\ref{fig:pipeline}. (The $C$ and $\ell$ values we use can be found in the supplementary material.) Next, for each dimension $k \in \{x,y,z\}$, we compute an index $g(p_k) \in [0 \, .. \, C)$ via
\begin{equation}
\textstyle g(p_k) = \mbox{clamp}\left( \left\lfloor \frac{p_k}{\ell} + \frac{C}{2} \right\rceil, \; 0, \; C - 1 \right).
\end{equation}
Finally, we combine these three dimension-wise indices into a grid cell index, $G(\mathbf{p})$, via
\begin{equation}
\textstyle G(\mathbf{p}) = C^2 g(p_z) + C g(p_y) + g(p_x).
\end{equation}
This initial raster-based mapping produces grid cell indices in the range $[0 \, .. \, C^3)$, but in practice, it is undesirable for memory reasons to try to allocate a reservoir for every cell in the grid. Each reservoir may need to store many point clusters, and must be allocated upfront on the GPU with a fixed size. As a result, if every cell in the grid must have a reservoir, then $C$ must be kept small to avoid exceeding the available GPU memory, limiting the size of scene we can handle with our approach.
\stufig{width=\linewidth}{images/adaptation-crop}{\textbf{Grid-based reservoir indexing.} Suppose that the 3D point $\mathbf{p}$ that the ScoreNet predicts for a given pixel falls into cell $(2,1,3)$ in a bounded grid placed over the training scene (we show only the $x$ and $y$ dimensions, for simplicity). Then $g(p_x) = 2$, $g(p_y) = 1$ and $g(p_z) = 3$, and we can calculate a grid cell index of $G(\mathbf{p}) = 4^2 \times 3 + 4 \times 1 + 2 = 54$ for $\mathbf{p}$. We use this grid cell index to perform a lookup in a table $T$ that we construct during online training, which stores a mapping from the $C^3$ potential grid cells to a fixed-size buffer of $N \ll C^3$ reservoirs, which we allocate ahead of time. In this case, $T$ maps grid cell $54$ to reservoir $1$, which we then associate with the original pixel.}{fig:reservoirindexing}{!t}
Fortunately, however, there is no need for every grid cell to have a reservoir: as noted by \cite{Niessner2013}, most cells in a scene are empty in practice, and we can exploit this observation to store a sparse set of reservoirs for only those cells that contain predicted points. To achieve this, rather than using the grid cell indices produced as above directly, we instead allocate a fixed-size buffer of $N$ reservoirs upfront, and construct a lookup table $T$ during online training that can be used to map a grid cell index in $[0 \, .. \, C^3)$ to a reservoir index in $[0 \, .. \, N)$: see Figure~\ref{fig:reservoirindexing}. More precisely, we start online training with an empty $T$, and each time we see a grid cell index $G$ for which $T$ has no entry, we add an entry $G \mapsto R$ to $T$ so that $G$ can be remapped to $R \in [0 \, .. \, N)$ in future. We map the first $N$ distinct grid cell indices we see to distinct reservoirs. To handle situations in which the number of grid cells that contain predicted points is greater than $N$, we simply let multiple grid cell indices map to the same reservoir (we randomly pick an existing reservoir to which to map each new grid cell index when no free reservoirs are available). Whilst this can ultimately lead to points with very different appearances being added to the same reservoir, this is not a major problem: as described later in this section, we follow \cite{Cavallari2017} in clustering the points in each reservoir into multiple sets that are disjoint in space, and the only implication of having clusters with different appearances in the same reservoir is that some poor correspondences may be generated. Since the RANSAC-based backend we use \cite{Cavallari2019} is already highly robust to a high proportion of poor correspondences, we would thus expect the practical implications of our reservoir sharing approach to be limited, and indeed our experiment in \S\ref{subsec:experiments-reservoirsharing} shows that this is the case.
\textbf{Reservoir Filling.} The scheme described above allows us to predict reservoir indices in $[0 \, .. \, N)$ for a regularly spaced subset of the pixels in an input image. At online training time, we can use these indices to fill the reservoirs with (world space) points from the target scene. As mentioned above, the online training sequence consists of an ordered set of RGB-D images of the target scene, with their associated $\mathbf{SE}(3)$ poses (which we assume are known, as a result of successfully tracking the camera during online training). For each frame $\mathcal{F}$, we proceed as follows:
\begin{enumerate}[wide]
\item First, we pass the $w \times h$ RGB image through first the ScoreNet and then the grid-based adaptation process just described to produce a reservoir index image of size $w/8 \times h/8$, in which each pixel $(x,y)$ contains the reservoir index to associate with pixel $(8x,8y)$ in the original image.
\item Next, we compute the 3D (world space) point in the target scene corresponding to each pixel $\mathbf{u}$ in the input image for which (i) we have computed a reservoir index and (ii) we have a valid depth value $D(\mathbf{u})$. To do this, we back-project the pixel using the depth to get a point in 3D camera space, and then transform it into world space using the known transformation ${}_\mathcal{W}T_\mathcal{F}$, via
\begin{equation}
\textstyle \mathbf{p} = {}_\mathcal{W}T_\mathcal{F} (D(\mathbf{u})K^{-1}\dot{\mathbf{u}}),
\end{equation}
in which $\dot{\mathbf{u}} = (\mathbf{u}^\top,1)^\top$ is the homogenous form of $\mathbf{u}$, $K$ is the intrinsic calibration matrix for the depth camera, and ${}_\mathcal{W}T_\mathcal{F}$ denotes the transformation from the camera space of frame $\mathcal{F}$ to world space ($\mathcal{W}$). This yields a $w/8 \times h/8 \times 3$ tensor of world space points.
\item Finally, we add each computed world space point to its associated reservoir. We follow \cite{Cavallari2017} in clustering the points we add to each reservoir online using Really Quick Shift (RQS) \cite{Fulkerson2010}, and in maintaining, for each cluster, 3D and colour centroids and a covariance matrix. Since our point clustering is exactly the same as that described in \cite{Cavallari2017}, we refer the reader there for the details of how this works.
\end{enumerate}
\subsection{Camera Pose Estimation}
\label{subsec:method-cameraposeestimation}
\noindent Having filled the reservoirs with clusters of world space points from the target scene at online training time, as per \S\ref{subsec:method-onlineadaptation}, we can then use these clusters at test time to relocalise the camera. To do this, we first pass the $w \times h$ RGB test image through the ScoreNet and the grid-based adaptation process described in \S\ref{subsec:method-onlineadaptation} to produce a reservoir index image of size $w/8 \times h/8$, just as we did during online training. This index image implicitly establishes correspondences between a regularly spaced subset of pixels in the input image and clusters of world space points, which can be used to generate camera pose hypotheses that can be fed to RANSAC. For the actual camera pose estimation, we use the implementation of \cite{Cavallari2019}, which is publicly available in the open-source \emph{SemanticPaint} framework \cite{Golodetz2015SPTR}. Since our contribution in this paper is to the correspondence prediction part of the pipeline, rather than to the camera pose estimation, we summarise how this works only briefly below, and direct the reader to \cite{Cavallari2019} for further details.
\textbf{Hypothesis Generation.} A pose hypothesis $H \in \mathbf{SE}(3)$ maps points in camera space to points in world space. Initially, a large number of pose hypotheses (at most $N_{\max}$) are generated. We follow \cite{Cavallari2019} in generating each pose hypothesis by applying the Kabsch algorithm \cite{Kabsch1976} to $3$ point pairs\footnote{Applying PnP \cite{Hartley2004} to $4$ point pairs instead would make testing RGB-only, but since online training needs depth, Kabsch makes more sense here.} of the form $(\mathbf{x}_i^\mathcal{C},\mathbf{x}_i^\mathcal{W})$, in which $\mathbf{x}_i^\mathcal{C} = D(\mathbf{u}_i)K^{-1}\dot{\mathbf{u}}_i$ is the back-projection of a randomly chosen pixel $\mathbf{u}_i$ in the input image for which a reservoir index has been predicted, and $\mathbf{x}_i^\mathcal{W}$ is a corresponding world space point, randomly sampled from $M(\mathbf{u}_i)$, the modes of the point clusters in the predicted reservoir. We follow \cite{Cavallari2019} in subjecting each generated pose hypothesis to three geometric/colour-based checks, and if one of these checks fails, we try to replace the hypothesis with a new one as described therein.
\textbf{Preemptive RANSAC.} As per \cite{Cavallari2019}, the $\le N_{\max}$ pose hypotheses are first scored and pruned, so that at most $N_{\footnotesize\mbox{cull}}$ hypotheses are retained. The preemptive RANSAC process described in \cite{Cavallari2019} is then used to prune the remaining $\le N_{\footnotesize\mbox{cull}}$ hypotheses down to the best $16$, refining them using Levenberg-Marquardt optimisation \cite{Levenberg1944,Marquardt1963} in the process.
\textbf{Hypothesis Ranking.} Finally \cite{Cavallari2019}, the remaining $16$ hypotheses are refined using ICP \cite{Besl1992} with respect to the 3D scene model, and then scored and ranked by rendering synthetic depth images of the scene model from the ICP-refined poses and comparing these to the live depth image from the camera. The refined pose whose synthetic depth image is most similar to the live depth is then returned as the result.
\section{ScoreNet Architecture}
\lstset{
basicstyle=\scriptsize\ttfamily,
columns=flexible,
breaklines=true
}
\noindent As briefly mentioned in \S\ref{subsec:method-offlinetraining}, the architecture of our ScoreNets consists of a truncated VGG-16~\cite{Simonyan2015} feature extractor, followed by several $1{\times}1$ convolutional layers, to regress a 3D world space point for each relevant pixel (see Figure~\ref{fig:scorenet}).
A more detailed breakdown of the layer structure we use is shown in Figure~\ref{fig:layerstructure}. The feature extraction part of our networks broadly mirrors the structure of a standard VGG-16 feature extractor, but with a couple of modifications to suit our particular context:
\begin{enumerate}
\item Since (i) we want to be able to fill the reservoirs during adaptation at a reasonable rate to ensure that the camera can be relocalised even if camera tracking fails at a relatively early stage, and (ii) we want to predict correspondences for a sizeable subset of the pixels in the input image at test time to give RANSAC more potential correspondences to work with, it is important that the prediction images output by our network are not too small. For this reason, we truncate a conventional VGG-16 feature extractor after three rounds of downsampling, rather than using all of the usual layers.
\item Like \cite{Brachmann2018CVPR}, we use strided convolutions (with a stride of $2$), rather than using max-pooling as in VGG-16.
\end{enumerate}
Our modified feature extractor takes as input an RGB image of size $w \times h$, and produces as output a tensor of size $w/8 \times h/8 \times 512$. This is then fed through a series of $1{\times}1$ convolutional layers, as described in \S\ref{subsec:method-offlinetraining}, to produce a $w/8 \times h/8 \times 3$ tensor of 3D world space points.
To train a ScoreNet, we first initialise the feature extractor with weights learnt by pre-training on ImageNet \cite{Deng2009},\footnote{We adapted the \texttt{torchvision.models.vgg16\_bn} model from TorchVision \cite{TorchVision}, and used the ImageNet-trained weights supplied.} and then train the overall network for the task at hand on the RGB-D training sequence associated with a particular scene in one of our datasets. We use the Adam optimiser~\cite{Kingma2015} with an initial learning rate of $10^{-4}$ (which we reduce by a factor of $10$ whenever the validation loss has not improved over the last $10$ epochs), and batch normalisation~\cite{Ioffe2015}. Training a network (for $160$ epochs) takes from a few hours (for a short training sequence) to a few days (for a longer one).
\begin{figure}[!t]
\begin{lstlisting}
features: Sequential(
(0): Conv2d(3, 64, kernel_size=(3,3), stride=(1,1), padding=(1,1))
(1): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True)
(2): ReLU()
(3): Conv2d(64, 64, kernel_size=(3,3), stride=(1,1), padding=(1,1))
(4): BatchNorm2d(64, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True)
(5): ReLU()
(6): Conv2d(64, 128, kernel_size=(3,3), stride=(2,2), padding=(1,1))
(7): BatchNorm2d(128, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True)
(8): ReLU()
(9): Conv2d(128, 128, kernel_size=(3,3), stride=(1,1), padding=(1,1))
(10): BatchNorm2d(128, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True)
(11): ReLU()
(12): Conv2d(128, 256, kernel_size=(3,3), stride=(2,2), padding=(1,1))
(13): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True)
(14): ReLU()
(15): Conv2d(256, 256, kernel_size=(3,3), stride=(1,1), padding=(1,1))
(16): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True)
(17): ReLU()
(18): Conv2d(256, 256, kernel_size=(3,3), stride=(1,1), padding=(1,1))
(19): BatchNorm2d(256, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True)
(20): ReLU()
(21): Conv2d(256, 512, kernel_size=(3,3), stride=(2,2), padding=(1,1))
(22): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True)
(23): ReLU()
(24): Conv2d(512, 512, kernel_size=(3,3), stride=(1,1), padding=(1,1))
(25): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True)
(26): ReLU()
(27): Conv2d(512, 512, kernel_size=(3,3), stride=(1,1), padding=(1,1))
(28): BatchNorm2d(512, eps=1e-05, momentum=0.1, affine=True, track_running_stats=True)
(29): ReLU()
)
regression: Sequential(
(0): Conv2d(512, 4096, kernel_size=(1,1), stride=(1,1))
(1): ReLU()
(2): Conv2d(4096, 4096, kernel_size=(1,1), stride=(1,1))
(3): ReLU()
(4): Conv2d(4096, 3, kernel_size=(1,1), stride=(1,1))
)
\end{lstlisting}
\caption{A string representation of the layer structure of our PyTorch-based ScoreNet architecture.}
\label{fig:layerstructure}
\vspace{-\baselineskip}
\end{figure}
Note that it would have been possible to deploy an architecture other than VGG-16 (e.g.\ ResNet-50~\cite{He2016}) for feature extraction purposes. In our case, we chose to use a VGG-based architecture both for simplicity, and for consistency with similar architectures that have been used for this task by other authors \cite{Brachmann2017CVPR,Brachmann2018CVPR}, but there is nothing fundamentally important about this architecture per se: any network that can map pixels that have a similar appearance to similar areas in the pre-training scene should be viable in this context. Indeed, in \S\ref{subsec:trainingefficiency} of this supplementary material, we show that even after a few epochs of training, one of our ScoreNets is already able to predict points in the pre-training scene well enough to allow the camera to be relocalised.
\section{Full Results and Hyperparameter Values}
\label{sec:fullresults}
\begin{table*}[!t]
\centering
\scriptsize
\begin{tabular}{ccc}
\toprule
\textbf{Name} & \textbf{7-Scenes} & \textbf{Cambridge Landmarks} \\
\midrule
clustererSigma & 0.1 & 0.1 \\
clustererTau & 0.05 & 0.4 \\
maxClusterCount & 50 & 50 \\
minClusterSize & 20 & 5 \\
reservoirCapacity & 4096 & 4096 \\
\midrule
maxCandidateGenerationIterations & 6000 & 6000 \\
maxPoseCandidates ($N_{\mbox{max}}$) & 1024 & 2048 \\
maxPoseCandidatesAfterCull ($N_{\mbox{cull}}$) & 64 & 64 \\
maxTranslationErrorForCorrectPose & 0.05 & 0.1 \\
minSquaredDistanceBetweenSampledModes & 0.09 & 0.0225 \\
poseUpdate & True & True \\
ransacInliersPerIteration ($\eta$) & 512 & 512 \\
usePredictionCovarianceForPoseOptimization & True & False \\
\bottomrule
\end{tabular}
\caption{The hyperparameter values we used when testing on 7-Scenes \cite{Shotton2013} and Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017}.}
\label{tbl:hyperparameters}
\vspace{-\baselineskip}
\end{table*}
\noindent In \S\ref{sec:experiments}, we presented the performance (after ICP and hypothesis ranking) of two variants of our relocaliser: one based on a ScoreNet trained on \emph{Office} from 7-Scenes \cite{Shotton2013}, and another based on a ScoreNet trained on \emph{Great Court} from Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017}. For space reasons, the results for all other variants were necessarily deferred to this supplementary material, as were the values of the hyperparameters we use in each case. In this section, we present full results (before ICP, after ICP, and after ICP and hypothesis ranking) for variants of our relocaliser trained on every scene from 7-Scenes and Cambridge Landmarks (except \emph{Heads} from 7-Scenes, for which no validation sequence is available). We also provide the raw (i.e.\ \emph{without} grid-based adaptation) results of our ScoreNets on both datasets.
The hyperparameter values we used for each dataset are shown in Table~\ref{tbl:hyperparameters}. Most hyperparameters are the same in both cases, although we use a slightly higher value of \texttt{clustererTau} (the maximum distance there can be between two world space points that are part of the same cluster) and a slightly lower value of \texttt{minClusterSize} (the minimum number of points in a cluster) for the outdoor scenes to reflect the larger scale at which we are operating and the increased difficulty of collecting large enough clusters in that case. We found that these sets of hyperparameters gave good results, although further improvements in performance may be possible via more extensive tuning.
We based the grid cell size ($\ell$) we used for each ScoreNet on the size of the scene on which it was trained. For all the networks trained on sequences from 7-Scenes \cite{Shotton2013}, we used $\ell = 10$cm. For the networks trained on sequences from Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017}, we used $\ell = 1$m for most scenes, but $\ell = 4$m for \emph{Great Court}, in line with the greater size of that scene ($8000$m$^2$, vs.\ e.g.\ $2000$m$^2$ for \emph{Old Hospital}). In all cases, we set $C$ (the number of grid cells along each side of the grid) so as to ensure that $C\ell$, the side length of the grid, was equal to $1$km. In practice, $C$ can be set to any suitable value that will ensure that the grid covers the pre-training scene.
For the 7-Scenes dataset \cite{Shotton2013}, the raw (i.e.\ \emph{without} grid-based adaptation) results of our 7-Scenes ScoreNets are shown in Table~\ref{tbl:7scenes-raw}, whilst our results with grid-based adaptation enabled are shown in Table~\ref{tbl:7scenes-7scenes-adapted}. Several observations can be made. Firstly, as would be expected given the results reported by \cite{Cavallari2019}, both ICP and hypothesis ranking significantly improve the results in many cases. Secondly, our grid-based adaptation scheme consistently improves relocalisation performance on the scene on which a particular ScoreNet was trained (compare the numbers along the diagonal in Table~\ref{tbl:7scenes-7scenes-adapted} to the raw numbers in Table~\ref{tbl:7scenes-raw}). This is because by predicting a reservoir for each pixel rather than a single point, grid-based adaptation effectively allows us to predict multiple correspondences per pixel (based on the clusters in the reservoirs), which in turn gives RANSAC the opportunity to generate a more diverse range of candidate poses and thereby improve performance. Thirdly, whilst there is some slight variation in performance between ScoreNets trained on different scenes, average performance with adaptation and after ranking for all the networks is state-of-the-art.
For Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017}, the raw results of our Cambridge ScoreNets are shown in Table~\ref{tbl:cambridge-raw}, whilst our results with grid-based adaptation enabled are shown in Table~\ref{tbl:cambridge-cambridge-adapted}. For outdoor scenes, the 5cm limit on translation error is a much sterner test, which is reflected in the fact that in all cases, the pre-ICP percentages are relatively poor (indeed, most other methods, with the exception of \cite{Cavallari2019}, report only average median localisation errors for this reason). However, as with the 7-Scenes results, it is noticeable that ICP and hypothesis ranking significantly improve performance, leading to state-of-the-art results overall on this dataset. Notably, our grid-based adaptation scheme again improves performance compared to the raw predictions.
Finally, to test how well our grid-based adaptation scheme works between datasets, we evaluated our Cambridge ScoreNets on 7-Scenes (see Table~\ref{tbl:cambridge-7scenes-adapted}), and our 7-Scenes ScoreNets on Cambridge Landmarks (see Table~\ref{tbl:cambridge-7scenes-adapted}). Our Cambridge ScoreNets were able to adapt relatively well to indoor scenes, with each network achieving over $85$\% on average on the 7-Scenes dataset. Conversely, our 7-Scenes ScoreNets also performed well on almost all of the outdoor scenes. They did struggle to adapt well to \emph{Great Court}, which exhibits a much wider range of depth values than the other scenes (see \S\ref{subsec:datasetanalysis}). On the whole, though, these results clearly indicate the potential of our method to adapt well between scenes in different datasets.
\section{Additional Experiments}
\subsection{Generalisation to Novel Poses}
\noindent As noted by \cite{Cavallari2017,Cavallari2019}, an important facet of a relocaliser's performance is its ability to generalise to test images captured from poses that are quite far from the training trajectory.\footnote{Since we are in an online relocalisation context, that means the \emph{online} training trajectory, not the trajectory used during offline pre-training.} To examine how well our relocaliser is able to do this, we follow the methodology originally proposed in \cite{Cavallari2017} of grouping the test images into bins based on the novelty of their ground truth poses with respect to the training trajectory, and evaluating the percentage of frames from each bin that our relocaliser is able to successfully relocalise to within 5cm/5$^\circ$ of the ground truth.
We perform separate evaluations for 7-Scenes \cite{Shotton2013} and Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017}. For 7-Scenes, we follow \cite{Cavallari2017,Cavallari2019} in using bins specified in terms of a maximum translation and rotation difference with respect to the nearest training pose (a test pose must be within both thresholds to fall into a particular bin), and group all test poses that are more than either 50cm or 50$^\circ$ from the nearest training pose into a single bin.
For Cambridge Landmarks, we found (see \S\ref{subsec:datasetanalysis}) that a wider range of test poses were available: for this reason, we divided the final bin into several new bins that we denote as $< n$ metres, for various $n$. A test pose will fall into such a bin if and only if (i) it could not have fallen into a lower bin, and (ii) its translation distance from the nearest training pose is within the specified threshold.
Our results on 7-Scenes for our relocaliser pre-trained on \emph{Office} are shown in Figure~\ref{fig:generalisation-7scenes}. Although the performance of our relocaliser does gradually decrease as the test pose's distance from the training trajectory increases, it does so quite gracefully, and we remain able to relocalise more than 50\% of frames even at a significant distance ($> 50$cm or $> 50^\circ$) from the training trajectory, indicating that as expected, our approach generalises reasonably well to novel poses.
Similar results on Cambridge Landmarks for our relocaliser pre-trained on \emph{Great Court} can be seen in Figure~\ref{fig:generalisation-cambridge}. As with the indoor scenes, performance gradually decreases as the pose novelty increases, but we remain able to relocalise relatively well even at a distance of several metres from the training trajectory.
\stufig{width=\linewidth}{images/generalisation-7scenes-crop}{Evaluating the ability of our relocaliser pre-trained on \emph{Office} to generalise to novel poses (i.e.\ ones far from the online training trajectory) on 7-Scenes \cite{Shotton2013}.}{fig:generalisation-7scenes}{!t}
\stufig{width=\linewidth}{images/generalisation-cambridge-crop}{Evaluating the ability of our relocaliser pre-trained on \emph{Great Court} to generalise to novel poses (i.e.\ ones far from the online training trajectory) on Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017}. Note that the pre-ICP performance is not shown in this case, since very few images are relocalised to within 5cm/5$^\circ$ before ICP outdoors.}{fig:generalisation-cambridge}{!t}
\subsection{Dataset Analysis}
\label{subsec:datasetanalysis}
\stufig{width=\linewidth}{images/dataset-cambridge-crop}{The proportions of test frames from Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017} that are within certain distances of the training trajectories. Notably, many of the test poses in this dataset are quite novel, making this in practice a much harder dataset than some of the indoor datasets like 7-Scenes \cite{Shotton2013} and Stanford 4 Scenes \cite{Valentin2016}.}{fig:dataset-cambridge}{!t}
\begin{stusubfig*}{!t}
\begin{subfigure}{.45\linewidth}
\centering
\includegraphics[width=\linewidth]{images/depthvalues-7scenes-crop}
\end{subfigure}%
\hspace{8mm}%
\begin{subfigure}{.45\linewidth}
\centering
\includegraphics[width=\linewidth]{images/depthvalues-cambridge-crop}
\end{subfigure}%
\caption{The distributions of depth values in the training sequences of the different scenes in the 7-Scenes \cite{Shotton2013} and Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017} datasets. Notably, all of the depth values in 7-Scenes are less than 4m, which makes sense given the range of the Kinect sensor with which it was captured, whilst the depth values for the outdoor scenes are much greater.}
\label{fig:depthvalues}
\vspace{-\baselineskip}
\end{stusubfig*}
\noindent In \cite{Cavallari2019}, Cavallari et al.\ provided a detailed analysis of the 7-Scenes \cite{Shotton2013} and Stanford 4 Scenes \cite{Valentin2016} datasets,
to better understand the performance of their relocaliser in each case.
In this section, we provide an analysis of the Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017} dataset, which we use in this paper to evaluate our relocaliser's outdoor performance.
\textbf{Test Pose Novelty.} The key aspect of the datasets that \cite{Cavallari2019} examined was the percentages of the test poses they contained that were within various distances of the training trajectory. They showed in particular that for Stanford 4 Scenes, the novelty of the test poses is relatively low (most of them are within $30$cm/$30^\circ$ of the training trajectory), whereas for 7-Scenes, far more of the test poses exhibit some novelty (particular those from the \emph{Fire} scene). Nevertheless, the physical constraints of an indoor environment limit how novel the test poses in any such dataset can be in practice (only a small percentage of the test poses in 7-Scenes are more than $50$cm/$50^\circ$ from the training trajectory). For Cambridge Landmarks, we present a similar analysis in Figure~\ref{fig:dataset-cambridge}. Since this is an outdoor dataset, we would naturally expect the range of test poses to be significantly higher than for the indoor datasets, and indeed our results show that this is the case: for every scene (and especially \emph{Old Hospital} and \emph{Great Court}), a large proportion of the test poses are at a significant distance from the training trajectory (more than $50$cm or $50^\circ$), making this dataset quite challenging in practice.
\textbf{Range of Depth Values.} Above and beyond test pose novelty, it is also interesting to examine the range of depth values present in the test frames for the various scenes in the Cambridge Landmarks dataset. Clearly we would expect the depth range to be much larger for outdoor scenes, since on average there are fewer occluders outdoors, and indeed our results in Figure~\ref{fig:depthvalues} show this to be the case. More interestingly, however, it is noticeable that for \emph{Great Court}, the depth range is significantly broader than those for the other scenes. On the one hand, this explains why a relocaliser based on depth-adaptive features that were designed for use indoors with relatively small depth values \cite{Cavallari2017,Cavallari2019} might be expected to perform poorly on such a scene (since large depth values will cause the offsets used to be too small to be usable). On the other hand, it seems plausible that training a ScoreNet on a sequence exhibiting a wider range of depths might yield a network that can predict a more diverse range of points in the pre-training scene, potentially allowing it to generalise better to other scenes at test time (indeed, we observe that our \emph{Great Court} network does in fact have the greatest ability to generalise of all the networks we trained).
\subsection{Correspondence Quality}
\label{subsec:correspondencequality}
\stufig{width=\linewidth}{images/correspondencequality-crop}{Evaluating how the performance of our relocaliser trained on \emph{Office} \cite{Shotton2013} and tested on \emph{Office} changes as the percentage of `good' correspondences (those within $10$cm of the ground truth) changes. See \S\ref{subsec:correspondencequality} for a discussion.}{fig:correspondencequality}{!t}
\noindent We observed in \S\ref{subsec:experiments-correspondencevisualisation} that whilst our relocaliser relies on the prediction of correspondences between pixels in the input image and 3D world space points in the target scene to estimate the camera pose, it is not necessary to predict perfect correspondences for every pixel. Indeed, in principle we only need enough good correspondences to allow RANSAC to find and verify $3$ good ones that can be used to estimate the camera pose using the Kabsch algorithm \cite{Kabsch1976}. In this section, we explore the implications of this by evaluating how the relocalisation performance of our \emph{Office} network is affected as we vary the proportion of predicted 3D world space points that lie within $10$cm of the ground truth.
To perform this experiment, we classify, for each frame, all of the predicted 3D world space points\footnote{Note that when using grid-based adaptation, these points will be the modes of clusters in the reservoirs, not points predicted by the ScoreNet.} available as either `good' (within $10$cm of the ground truth) or `poor' (the converse). We then evaluate our relocaliser multiple times on the \emph{Office} test sequence, keeping/discarding different fractions of good/bad correspondences. We further repeat this whole process multiple times for: (i) the raw predictions from the ScoreNet; (ii) the adapted predictions with pose optimisation during RANSAC disabled, no ICP and no hypothesis ranking; (iii) the same as (ii), but with pose optimisation enabled; (iv) the same as (iii), but with ICP enabled; and (v) the same as (iv), but with hypothesis ranking enabled. The reason for exploring the effect that pose optimisation during RANSAC has on the performance is to explain why, with grid-based adaptation enabled, we are able to obtain poses within 5cm/5$^\circ$ of the ground truth for nearly $60$\% of the test frames, even with no `good' correspondences and before ICP or hypothesis ranking. This turns out to be a result of the way in which the third-party camera pose estimation backend we use \cite{Cavallari2019} optimises the pose hypotheses multiple times during the RANSAC process. Without this, our relocaliser performs poorly when there are no `good' correspondences present, as expected.
The results of our experiment are shown in Figure~\ref{fig:correspondencequality}. When grid-based adaptation is disabled, we find that around $50$\% of the correspondences need to be `good' before relocalisation performance starts to become acceptable. By contrast, with grid-based adaptation enabled, a much smaller proportion of `good' correspondences (around $20$\%) are needed. Performing pose optimisation during RANSAC significantly improves the performance when fewer than $20$\% of the correspondences are `good', but does not otherwise change the proportion needed for good performance. Notably, if we then also perform ICP and hypothesis ranking, this further improves the performance so much that excellent relocalisation performance ($> 98$\%) can be achieved even when all of the correspondences are more than $10$cm from the ground truth.
Intuitively, the explanation for this is that the correspondences we use, even though they are individually further than $10$cm from the ground truth, are still good enough that RANSAC with pose optimisation enabled can use them to generate a pose that is within the ICP convergence basin. With hypothesis ranking enabled, there is an even greater chance that one of the $16$ candidate poses generated will be within the ICP convergence basin, allowing relocalisation to succeed. This implies that the link between correspondence quality and relocalisation performance is actually quite weak when the full camera pose estimation backend is used, which in practice means there is a significant margin for error when predicting correspondences. In \S\ref{subsec:trainingefficiency}, we exploit this observation to show that it is possible to significantly reduce the time for which we train our ScoreNets offline without affecting relocalisation performance.
\subsection{Offline Training Efficiency}
\label{subsec:trainingefficiency}
\noindent In \S\ref{subsec:correspondencequality}, we saw that in practice, excellent relocalisation results can be obtained even with a very small proportion of `good' correspondences. Since arguably the only purpose of training a ScoreNet for many epochs is to improve the quality of the correspondences it predicts, and we have just seen that this is not really necessary to achieve excellent relocalisation performance, there are at least two questions that should be asked at this point. Firstly, can even a ScoreNet that has been trained for a much smaller number of epochs than the default (in our case, $160$ epochs) produce correspondences that are good enough for relocalisation to succeed? And secondly, if so, for how many epochs do we really need to train a ScoreNet? To answer these questions, we trained a ScoreNet on the \emph{Great Court} training sequence from Cambridge Landmarks \cite{Kendall2015,Kendall2016,Kendall2017}, and evaluated its post-ranking performance, both with and without grid-based adaptation, on the \emph{Great Court} testing sequence after every $5$ training epochs.
The results of this process are shown in Figure~\ref{fig:training-efficiency}. Notably, they show that it is not in fact necessary to train a ScoreNet for the full $160$ epochs, and that a much smaller number of epochs (around $70$) suffices to achieve maximum relocalisation performance. They also show that with grid-based adaptation enabled, very good results can already be achieved after training for only $20$ epochs, allowing significant time to be saved during the offline training process.
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const assert = require("assert");
const fs = require("fs");
exports.testGetAttributes = function () {
const jha = fs.path(java.lang.System.getProperty("java.home")).getAttributes();
assert.deepEqual(jha, fs.getAttributes(java.lang.System.getProperty("java.home")));
assert.isTrue(jha.creationTime instanceof Date);
assert.isTrue(jha.lastAccessTime instanceof Date);
assert.isTrue(jha.lastModifiedTime instanceof Date);
assert.isTrue(jha.isDirectory);
assert.isFalse(jha.isRegularFile);
assert.isFalse(jha.isOther);
assert.isTrue(typeof jha.isSymbolicLink === "boolean");
assert.isTrue(Number.isInteger(jha.size));
};
if (require.main === module) {
require('system').exit(require("test").run(module.id));
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,746 |
{"url":"https:\/\/madhavamathcompetition.com\/category\/iitjee-advanced\/","text":"Pre-RMO or RMO algebra practice problem: infinite\u00a0product\n\nFind the product of the following infinite number of terms:\n\n$\\frac{7}{9} \\times \\frac{26}{28} \\times \\frac{63}{65} \\times \\ldots = \\prod_{m=2}^{\\infty}\\frac{m^{3}-1}{m^{3}+1}$\n\n$m^{3}-1=(m-1)(m^{2}+m+1)$, and also, $m^{3}+1=(m+1)(m^{2}-m+1)=(m-1+2)((m-1)^{2}+(m-1)+1)$\n\nHence, we get $P_{m}=\\frac{7}{9} \\times \\frac{26}{28} \\times \\frac{63}{65} \\times \\ldots \\times \\frac{m^{3}-1}{m^{3}+1}$, which in turn, equals\n\n$(\\frac{1}{3} \\times \\frac{7}{3}) \\times (\\frac{2}{4} \\times \\frac{13}{7}) \\times (\\frac{3}{5} \\times \\frac{21}{13})\\times \\ldots (\\frac{m-1}{m+1} \\times \\frac{m^{2}+m+1}{m^{2}-m+1})$, that is, in turn equal to\n\n$\\frac{2}{3} \\times \\frac{m^{2}+m+1}{m(m+1)}$, that is, in turn equal to\n\n$\\frac{}{} \\times (1+ \\frac{1}{m(m+1)})$, so that when $m \\rightarrow \\infty$, and then $P_{m} \\rightarrow 2\/3$.\n\npersonal comment: I did not find this solution within my imagination !!! \ud83d\ude42 \ud83d\ude42 \ud83d\ude42\n\nThe credit for the solution goes to \u201cPopular Problems and Puzzles in Mathematics\u201d by Asok Kumar Mallik, IISc Press, Foundation Books. Thanks Prof. Mallik !!\n\nCheers,\n\nNalin Pithwa\n\nAlgebra Training for RMO and Pre-RMO: let\u2019s continue: Fibonacci Problem and\u00a0solution\n\nFibonacci Problem:\n\nLeonardo of Pisa (famous as Fibonacci) (1173) wrote a book \u201cLiber Abaci\u201d (1202), wherein he introduced Hindu-Arabic numerals in Europe. In 1225, Frederick II declared him as the greatest mathematician in Europe when he posed the following problem to defeat his opponents.\n\nQuestion:\n\nDetermine the rational numbers x, y and z to satisfy the following equations:\n\n$x^{2}+5=y^{2}$ and $x^{2}-5=z^{2}$.\n\nSolution:\n\nDefinition: Euler defined a congruent number to be a rational number that is the area of a right-angled triangle, which has rational sides. With p, q, and r as a Pythagorean triplet such that $r^{2}=p^{2}+q^{2}$, then $\\frac{pq}{2}$ is a congruent number.\n\nIt can be shown that square of a rational number cannot be a congruent number. In other words, there is no right-angled triangle with rational sides, which has an area as 1, or 4, or $\\frac{1}{4}$, and so on.\n\nCharacteristics of a congruent number: A positive rational number n is a congruent number, if and only if there exists a rational number u such that $u^{2}-n$ and $u^{2}+n$ are the squares of rational numbers. (Thus, the puzzle will be solved if we can show that 5 is a congruent number and we can determine the rational number $u(=x)$). First, let us prove the characterisitic mentioned above.\n\nNecessity: Suppose n is a congruent number. Then, for some rational number p, q, and r, we have $r^{2}=p^{2}+q^{2}$ and $\\frac{pq}{2}=n$. In that case,\n\n$\\frac{p+q}{2}, \\frac{p-q}{2}$ and n are rational numbers and we have\n\n$(\\frac{p+q}{2})^{2}=\\frac{p^{2}+q^{2}}{4}+\\frac{pq}{2}=(\\frac{r}{2})^{2}+n$ and similarly,\n\n$(\\frac{p-q}{2})^{2}=(\\frac{r}{2})^{2}-n$.\n\nSetting $u=\\frac{r}{2}$, we get $u^{2}-n$ and $u^{2}+n$ are squares of rational numbers.\n\nSufficiency:\n\nSuppose n and u are rational numbers such that $\\sqrt{u^{2}-n}$ and $\\sqrt{u^{2}+n}$ are rational, when\n\n$p=\\sqrt{u^{2}+n}+\\sqrt{u^{2}-n}$ and $q=\\sqrt{u^{2}+n}-\\sqrt{u^{2}-n}$\n\nand 2n are rational numbers satisfying $p^{2}+q^{2}=(2n)^{2}$ is a rational square and also $\\frac{pq}{2}=n$, a rational number which is a congruent number.\n\nSo, we see that the Pythagorean triplets can lead our search for a congruent number. Sometimes a Pythagorean triplet can lead to more than one congruent number as can be seen with $(9,40,41)$. This set obviously gives 180 as a congruent number. But, as $180=5 \\times 36=5 \\times 6^{2}$, we can also consider a rational Pythagorean triplet $(\\frac{9}{6}, \\frac{40}{6}, \\frac{41}{6})$, which gives a congruent number 5 (we were searching for this congruent number in this puzzle!). We also determine the corresponding $u=\\frac{r}{2}=\\frac{41}{12}$.\n\nThe puzzle\/problem is now solved with $x=\\frac{41}{12}$, which gives $y=\\frac{49}{12}$, and $z=\\frac{31}{12}$.\n\nOne can further show that if we take three rational squares in AP, $u^{2}-n$, and $u^{2}$, and $u^{2}+n$, with their product defined as a rational square $v^{2}$ and n as a congruent number, then $x=u^{2}$, $y=v$ is a rational point on the elliptic curve $y^{2}=x^{3}-n^{2}x$.\n\nReference:\n\n1) Popular Problems and Puzzles in Mathematics: Asok Kumar Mallik, IISc Press, Foundation Books, Amazon India link:\n\nhttps:\/\/www.amazon.in\/Popular-Problems-Puzzles-Mathematics-Mallik\/dp\/938299386X\/ref=sr_1_1?s=books&ie=UTF8&qid=1525099935&sr=1-1&keywords=popular+problems+and+puzzles+in+mathematics\n\n2) Use the internet, or just Wikipedia to explore more information on Fibonacci Numbers, Golden Section, Golden Angle, Golden Rectangle and Golden spiral. You will be overjoyed to find relationships amongst all the mentioned \u201cstuff\u201d.\n\nCheers,\n\nNalin Pithwa\n\nEuler Series question and\u00a0solution\n\nQuestion:\n\nMengoli had posed the following series to be evaluated:\n\n$1+ \\frac{1}{2^{2}} + \\frac{1}{3^{2}} + \\frac{1}{4^{2}} + \\ldots$.\n\nSome great mathematicians, including Liebnitz, John Bernoulli and D\u2019Alembert, failed to compute this infinite series. Euler established himself as the best mathematician of Europe (in fact, one of the greatest mathematicians in history) by evaluating this series initially by a not-so-rigorous method. Later on, he gave alternative and more rigorous ways of getting the same result.\n\nProve that the series converges and gets an upper limit. Then, try to evaluate the series.\n\nProof:\n\nDue Nicolas Oresine:\n\nConsider the following infinite series: $\\phi(s)=1 + \\frac{1}{2^{s}} + \\frac{1}{3^{s}} + \\frac{1}{4^{s}} + ldots$\n\nWe can re-write the preceding series as follows: $\\phi(s) = 1+ (\\frac{1}{2^{s}}+\\frac{1}{3^{s}}) + (\\frac{1}{4^{s}} + \\frac{1}{5^{s}} + \\frac{1}{6^{s}} + \\frac{1}{7^{s}}) + \\ldots$, which in turn is less than\n\n$1 + (\\frac{2}{2^{s}}) + (\\frac{4}{4^{s}}) + \\ldots$. Now, the RHS of this can be re-written as\n\n$1+(\\frac{2}{2^{s}}) + (\\frac{4}{4^{s}}) + \\ldots=1 + \\frac{1}{2^{(s-1)}}+ (\\frac{1}{2^{(s-1)}})^{2} + \\ldots$, which is a geometric series and it is given by\n\n$\\frac{1}{1-\\frac{1}{2^{(s-1)}}}$.\n\nNow, we can say that $\\phi(s)$ will converge if $\\frac{1}{2^{(s-1)}}<1 \\Longrightarrow s >1$.\n\nIn order to prove what is asked, we start with $\\phi(s)=1 + \\frac{1}{2^{s}}+ \\frac{1}{3^{s}}+ \\frac{1}{4^{s}}+\\ldots$\n\nAnd, then multiply both sides by $\\frac{1}{2^{s}}$ and then subtract the resulting equation from the preceding equation to get\n\n$(1-\\frac{1}{2^{2}})\\phi(s)=1+\\frac{1}{3^{s}}+\\frac{1}{5^{s}}+\\ldots$\n\nwhere all the terms containing the reciprocals of the sth power of even numbers vanished.\n\nRepeating this procedure with $\\frac{1}{3^{s}}$ gives\n\n$(1-\\frac{1}{2^{s}})(1-\\frac{1}{3^{s}})\\phi(s)=1+\\frac{1}{5^{s}}+ \\ldots$\n\nwhere all terms containing the reciprocals of the sth power of multiples of 3 vanished.\n\nBy continuing this with all prime numbers, we get\n\n$\\prod_{p}(1-\\frac{1}{p^{s}})\\phi(s)=1$, where p represents all prime numbers. Thus, we get\n\n$\\phi(s)=1 + \\frac{}{} + \\frac{}{} + \\frac{}{} + \\ldots =\\frac{1}{\\prod_{p}(1-\\frac{1}{p^{s}})}$\n\nThis is a remarkable result because the LHS is concerned with only positive integers, whereas the RHS is concerned with only primes. This result is known as the \u201cGolden Key of Euler\u201d.\n\nRiemann created his famous $\\zeta-$ function by extending the variable s to the entire complex plane, except $s=1$ with\n\n$\\zeta(s)=1+ \\frac{1}{2^{s}} + \\frac{1}{3^{s}} + \\ldots$.\n\nThis function is now very famous as the Riemann zeta function.\n\nHow can we apply the Golden Key of Euler to Mengoli\u2019s question that we started with?\n\nAns. In the Golden Key of Euler, substitute $s=2$.\n\nHence, we get the upper limit of the given series is 2.\n\nEuler\u2019s proof (1775):\n\nThe proof ran as follows:\n\nIt is a little roundabout way of arriving at the correct answer from a known result. Consider McLaurin\u2019s series expansion of sin x:\n\n$\\sin{(x)}=x - \\frac{x^{3}}{3!} + \\frac{x^{5}}{5!} -\\frac{x^{7}}{7!} + \\frac{x^{9}}{9!} + \\ldots$\n\nBy dividing both sides by x and then substituting $y=x^{2}$ on the right side, we get the following:\n\n$\\frac{\\sin{(x)}}{x} = 1-\\frac{y}{3!} + \\frac{y^{2}}{5!} - \\frac{y^{3}}{7!} + \\ldots$\n\nBy taking a special value of $x=n\\pi$ (and, hence $y=n^{2}\\pi^{2}$), we get the following:\n\n$\\frac{\\sin (n\\pi)}{(n\\pi)}=0=1-\\frac{y}{3!} + \\frac{y^{2}}{5!} - \\frac{y^{3}}{5!}+ \\ldots$\n\nNote\u00a0 that preceding equation is not a polynomial, but an infinite series.\u00a0But, Euler still treated it as a polynomial (that is why it was not accepted as a rigorous result) and observed that this \u201cinfinite\u201d polynomial has roots equal to $y_{n}=n^{2}x^{2}$. Then, Euler had used the fact that the sum of the reciprocals of the roots is determined by the coefficient of the linear term (here, the y-term) when the constant is made unity. (check this as homework quiz, for a quadratic to be convinced).\u00a0So, Euler had arrived at the following result:\n\n$1-\\sum_{i=1}^{\\infty}\\frac{6}{y_{n}}=0 \\Longrightarrow \\sum_{i=1}^{\\infty}\\frac{1}{y_{n}}=\\frac{1}{6}$. With $y_{n}=n^{2}(\\pi)^{2}$, we get the following:\n\n$\\sum_{i=1}^{\\infty}\\frac{1}{n^{2}(\\pi)^{2}}=\\frac{1}{6}$ or, $\\sum_{1}^{n^{2}}\\frac{1}{n^{2}}=\\frac{(\\pi)^{2}}{6}$.\n\nAnother proof also attributed to Euler that uses the series expansion of sin (x) goes as follows below:\n\n$\\sin {(x)}$ has roots given by 0, $\\pm \\pi$, $\\pm 2\\pi$, $\\pm 3\\pi$, \u2026So does this polynomial that Euler reportedly constructed:\n\n$x(1-\\frac{x^{2}}{(\\pi)^{2}})(1-\\frac{x^{2}}{(2\\pi)^{2}})(1-\\frac{x^{2}}{(3\\pi)^{2}})\\ldots=0$\n\nSo, Euler considered the preceding equation to be equivalent to:\n\n$\\sin{(x)}=x - \\frac{x^{3}}{3!} + \\frac{x^{5}}{5!} - \\frac{x^{7}}{7!} + \\ldots=0$\n\nThen, he had equated the coefficient of $x^{3}$ in both to get the result:\n\n$\\sum_{n=1}^{\\infty}\\frac{1}{n^{2}(\\pi)^{2}} = \\frac{1}{3!} = \\frac{1}{6}$.\n\nThus, $\\sum_{n=1}^{\\infty}\\frac{1}{n^{2}}=\\frac{(\\pi)^{2}}{6}$.\n\nLater on, Euler had provided a few more alternate and rigorous proofs of this result.\n\nReference: Popular Problems and Puzzles in Mathematics by Asok Kumar Mallik, IISc Press, Foundation Books.","date":"2019-07-17 12:47:02","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\": 84, \"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\": 0.9092519879341125, \"perplexity\": 474.56977678118017}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195525187.9\/warc\/CC-MAIN-20190717121559-20190717143559-00382.warc.gz\"}"} | null | null |
/* TEMPLATE GENERATED TESTCASE FILE
Filename: CWE122_Heap_Based_Buffer_Overflow__c_CWE805_wchar_t_loop_83_goodG2B.cpp
Label Definition File: CWE122_Heap_Based_Buffer_Overflow__c_CWE805.string.label.xml
Template File: sources-sink-83_goodG2B.tmpl.cpp
*/
/*
* @description
* CWE: 122 Heap Based Buffer Overflow
* BadSource: Allocate using malloc() and set data pointer to a small buffer
* GoodSource: Allocate using malloc() and set data pointer to a large buffer
* Sinks: loop
* BadSink : Copy string to data using a loop
* Flow Variant: 83 Data flow: data passed to class constructor and destructor by declaring the class object on the stack
*
* */
#ifndef OMITGOOD
#include "std_testcase.h"
#include "CWE122_Heap_Based_Buffer_Overflow__c_CWE805_wchar_t_loop_83.h"
namespace CWE122_Heap_Based_Buffer_Overflow__c_CWE805_wchar_t_loop_83
{
CWE122_Heap_Based_Buffer_Overflow__c_CWE805_wchar_t_loop_83_goodG2B::CWE122_Heap_Based_Buffer_Overflow__c_CWE805_wchar_t_loop_83_goodG2B(wchar_t * dataCopy)
{
data = dataCopy;
/* FIX: Allocate and point data to a large buffer that is at least as large as the large buffer used in the sink */
data = (wchar_t *)malloc(100*sizeof(wchar_t));
data[0] = L'\0'; /* null terminate */
}
CWE122_Heap_Based_Buffer_Overflow__c_CWE805_wchar_t_loop_83_goodG2B::~CWE122_Heap_Based_Buffer_Overflow__c_CWE805_wchar_t_loop_83_goodG2B()
{
{
size_t i;
wchar_t source[100];
wmemset(source, L'C', 100-1); /* fill with L'C's */
source[100-1] = L'\0'; /* null terminate */
/* POTENTIAL FLAW: Possible buffer overflow if source is larger than data */
for (i = 0; i < 100; i++)
{
data[i] = source[i];
}
data[100-1] = L'\0'; /* Ensure the destination buffer is null terminated */
printWLine(data);
free(data);
}
}
}
#endif /* OMITGOOD */
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,872 |
Richard Stilgoe's Orpheus - The Musical will be performed by students and alumni from the Arts Educational School at The Other Palace Theatre from 26th-31st March with special guests including Rob Brydon, Bertie Carvel, Patricia Hodge, Miles Jupp, Jim Carter and Jane Asher making one-off cameo appearances playing the role of the Greek Chorus on different nights.
Directed by Katie Beard, Richard Stilgoe's musical is a retelling of the Greek myth of Orpheus (whose voice was able to charm anything and anyone that heard it). To celebrate the Orpheus Trust's 20th birthday, they are presenting Orpheus' voyage to find the golden fleece and his doomed journey down to Hades to rescue his wife Euridice, while battling the efforts of the bored and wilful gods and an unreliable satnav.
Stilgoe founded the Orpheus Trust in 1998 in order to inspire young disabled people to have greater independence through performing arts experiences. The Orpheus Centre offers students the chance to stay for up to three years, with the students being aged between 18 and 25 and having learning and/or physical disabilities. He said: "The whole purpose of Orpheus is to mix young disabled people with the real world and not separate them. These are terrific and confident performers who concentrate on what they can do rather than what they can't. " | {
"redpajama_set_name": "RedPajamaC4"
} | 1,886 |
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<td class="colFirst"><a href="#org.springframework.web.reactive.result.method.annotation">org.springframework.web.reactive.result.method.annotation</a></td>
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<div class="block">Infrastructure for annotation-based handler method processing.</div>
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<h3>Uses of <a href="../../../../../../../../org/springframework/web/reactive/result/method/annotation/AbstractNamedValueSyncArgumentResolver.html" title="class in org.springframework.web.reactive.result.method.annotation">AbstractNamedValueSyncArgumentResolver</a> in <a href="../../../../../../../../org/springframework/web/reactive/result/method/annotation/package-summary.html">org.springframework.web.reactive.result.method.annotation</a></h3>
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<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../../../../org/springframework/web/reactive/result/method/annotation/CookieValueMethodArgumentResolver.html" title="class in org.springframework.web.reactive.result.method.annotation">CookieValueMethodArgumentResolver</a></span></code>
<div class="block">Resolve method arguments annotated with <code>@CookieValue</code>.</div>
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<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../../../../org/springframework/web/reactive/result/method/annotation/ExpressionValueMethodArgumentResolver.html" title="class in org.springframework.web.reactive.result.method.annotation">ExpressionValueMethodArgumentResolver</a></span></code>
<div class="block">Resolves method arguments annotated with <code>@Value</code>.</div>
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<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../../../../org/springframework/web/reactive/result/method/annotation/MatrixVariableMethodArgumentResolver.html" title="class in org.springframework.web.reactive.result.method.annotation">MatrixVariableMethodArgumentResolver</a></span></code>
<div class="block">Resolves arguments annotated with <a href="../../../../../../../../org/springframework/web/bind/annotation/MatrixVariable.html" title="annotation in org.springframework.web.bind.annotation"><code>@MatrixVariable</code></a>.</div>
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<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../../../../org/springframework/web/reactive/result/method/annotation/PathVariableMethodArgumentResolver.html" title="class in org.springframework.web.reactive.result.method.annotation">PathVariableMethodArgumentResolver</a></span></code>
<div class="block">Resolves method arguments annotated with @<a href="../../../../../../../../org/springframework/web/bind/annotation/PathVariable.html" title="annotation in org.springframework.web.bind.annotation"><code>PathVariable</code></a>.</div>
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<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../../../../org/springframework/web/reactive/result/method/annotation/RequestAttributeMethodArgumentResolver.html" title="class in org.springframework.web.reactive.result.method.annotation">RequestAttributeMethodArgumentResolver</a></span></code>
<div class="block">Resolves method arguments annotated with an @<a href="../../../../../../../../org/springframework/web/bind/annotation/RequestAttribute.html" title="annotation in org.springframework.web.bind.annotation"><code>RequestAttribute</code></a>.</div>
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<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../../../../org/springframework/web/reactive/result/method/annotation/RequestHeaderMethodArgumentResolver.html" title="class in org.springframework.web.reactive.result.method.annotation">RequestHeaderMethodArgumentResolver</a></span></code>
<div class="block">Resolves method arguments annotated with <code>@RequestHeader</code> except for
<a href="https://docs.oracle.com/javase/8/docs/api/java/util/Map.html?is-external=true" title="class or interface in java.util"><code>Map</code></a> arguments.</div>
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<td class="colFirst"><code>class </code></td>
<td class="colLast"><code><span class="memberNameLink"><a href="../../../../../../../../org/springframework/web/reactive/result/method/annotation/RequestParamMethodArgumentResolver.html" title="class in org.springframework.web.reactive.result.method.annotation">RequestParamMethodArgumentResolver</a></span></code>
<div class="block">Resolver for method arguments annotated with @<a href="../../../../../../../../org/springframework/web/bind/annotation/RequestParam.html" title="annotation in org.springframework.web.bind.annotation"><code>RequestParam</code></a> from URI
query string parameters.</div>
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,691 |
New Lagos-Ibadan Railway Will Be Ready In May – NRC Boss
NRC Gives Update On Railway Project
The 156-kilometre new Lagos-Ibadan railway will be ready in May this year, the Managing Director, Nigerian Railway Corporation, Mr Fidet Okheria, has said.
He also said commercial operation would commence on the 85km Iju-Abeokuta section.
This will be after the ongoing free train ride by passengers on the line expected to last for three months expires, Sunmence understands.
The Iju-Abeokuta section of the standard gauge new railway is currently operating a daily free train, which will last for three months.
Minister of Transportation, Hon. Rotimi Amaechi led several dignitaries including Governor Ibikunle Amosun of Ogun State and Alake of Egbaland, Oba Adedotun Gbadebo, on a train ride from Iju to Abeokuta on February 8 to signify the commitment of the Federal Government to the new railway project.
The entire stretch of the new line from Lagos to Ibadan would have been completed in May this year," he added.
Responding to a question by our correspondent on what was left to be done, the NRC managing director said the contractor handling the project, China Civil Eng, would need to lay rail track from Iju to Ebute Meta and extend it to the Lagos port.
Substantial work on the Ibadan-Abeokuta end of the project because it had approached the construction simultaneously from both the Lagos and Ibadan ends when the project started.
Okheria also said the CCECC was not expected to encounter much difficulty in the execution of the Lagos-Iju work having secured the cooperation of the Lagos State Government and other affected stakeholders and had jointly agreed on how to handle the various knotty issues.
About 30 coaches, including two sets of the diesel multiple unit, would be dedicated to the new line after the three months' free train ride, the NRC boss said.
The Deputy Director, Public Relations at the NRC, Mr Yakubu Mahmoud, said the three-month free train period was also to enable the corporation to effectively determine the number of trains that would be run on the route when full commercial operation commenced.
We are investigating Atiku's son-inlaw over money laundering allegation – EFCC
Mon Mar 4 , 2019
The Economic and Financial Crimes Commission said it had begun an investigation into alleged money laundering by the Finance Director of Atiku Abubakar's group of companies, Alhaji Babalele Abdullahi. Acting Spokesman of EFCC, Mr Tony Orilade, stated this in Abuja, on Monday, and confirmed to the News Agency of Nigeria that Abdullahi was […]
Fans React as Nigeria Artist Davido Trolls President Buhari on Social Media
Police to question Catholic mum-of-five Caroline Farrow for 'using wrong pronoun to describe transgender girl'
Piers Morgan BLASTS Kylie Jenner's self-made billionaire title: 'I thought she got it because her sister made a sex tape'
Oscar 2019 Full List Of Winners (All Categories)
FG withdraws corruption charges against Danjuma Goje
FG considers excise duty on soft drinks | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,902 |
Der Sächsische Landespokal 2021/22 war die 32. Austragung des Sächsischen Landespokals (Sponsorenname: Wernesgrüner Sachsenpokal) der Männer im Amateurfußball. Der Landespokalsieger ist für den DFB-Pokal 2022/23 qualifiziert.
Der Titelverteidiger 1. FC Lokomotive Leipzig schied überraschend bereits in der 3. Hauptrunde aus. Nachdem auch der als einziger Drittligist im Teilnehmerfeld favorisierte FSV Zwickau im Halbfinale ausgeschieden war, qualifizierten sich für das Endspiel zwei Regionalligisten: der Rekordpokalsieger Chemnitzer FC (bis dato elf Titel) und die BSG Chemie Leipzig, die den Landespokal bislang einmal (2017/18) gewonnen hatte. Die Chemnitzer gewannen das Finale, das am 21. Mai 2022 im Chemnitzer Stadion an der Gellertstraße ausgetragen wurde, mit 2:1 (1:1) und sicherten sich somit ihren zwölften Landespokalsieg.
Teilnehmende Mannschaften
Für den Sächsischen Landespokal 2021/22 qualifizierten sich alle sächsischen Mannschaften der 3. Liga 2021/22, der Regionalliga Nordost 2021/22, der Oberliga Nordost 2021/22, der Sachsenliga 2021/22, der Landesklasse Sachsen 2021/22, die dreizehn Kreis- bzw. Stadtpokalsieger der Saison 2020/21 (wobei zwei Kreispokale in der Vorsaison aufgrund der Spielbetriebseinschränkungen in Folge der COVID-19-Pandemie nicht beendet werden konnten) sowie die SG Weißig, der im Zuge der vom SFV getroffenen Regelungen für die Fortsetzung des pandemiebedingt unterbrochenen Pokalspielbetriebs in der Vorsaison ein zusätzlicher Pokalstartplatz zugesprochen worden war. Zweite Mannschaften höherklassiger Vereine waren von der Teilnahme ausgeschlossen.
Modus
Der Sächsische Landespokal wird im K.-o.-Modus ausgetragen; im Falle eines Unentschiedens nach 90 Minuten folgen Verlängerung und Elfmeterschießen. Die Paarungen werden vor jeder Runde ausgelost. Heimrecht hat stets die klassentiefere Mannschaft, sonst entscheidet die Reihenfolge der Ziehung bei der Auslosung (die Mannschaft mit Heimrecht kann allerdings zugunsten der anderen Mannschaft auf das Heimrecht verzichten). In der 1. und 2. Hauptrunde können die über die Kreispokale qualifizierten Mannschaften nicht gegeneinander spielen. Das Heimrecht im Finale wird für den Fall einer Paarung aus zwei gleichklassigen Mannschaften separat ausgelost.
Die qualifizierten Mannschaften steigen je nach Ligazugehörigkeit gestaffelt in den Wettbewerb ein: In der 1. Hauptrunde spielen zunächst nur die über die Sachsenliga, die Landesklasse und die Kreispokale qualifizierten Mannschaften sowie die SG Weißig, wobei hier neun Freilose vergeben werden. Die Regional- und Oberligisten sind für die 2. Hauptrunde, die Drittligisten für die 3. Hauptrunde gesetzt.
1. Hauptrunde
|}
Freilose:
Radebeuler BC 08 (VI)
Kickers 94 Markkleeberg (VI)
SV Edelweiß Rammenau (VII)
Roter Stern Leipzig (VII)
SV Sachsenwerk Dresden (VII)
SpVgg Reinsdorf-Vielau (VII)
ESV Delitzsch (VII)
SV Zeißig (VII)
FC Stollberg (VII)
2. Hauptrunde
|}
3. Hauptrunde
|}
Achtelfinale
|}
Viertelfinale
|}
Halbfinale
Im Rahmen der Auslosung zu den Halbfinalpaarungen wurde außerdem per Auslosung festgelegt, dass im Falle einer Finalpaarung aus zwei gleichklassigen Mannschaften der Sieger der Halbfinalpartie SG Motor Wilsdruff – Chemnitzer FC im Finale Heimrecht haben würde.
|}
Finale
Einzelnachweise
2022
Sachsischer Landespokal 2021 22 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,085 |
Stay updated with JFCS activities in the community.
Home » News & Impact » Endowment » Named Endowment Funds » Preisler Shorenstein Institute Offers Holocaust Education for Thousands
Preisler Shorenstein Institute Offers Holocaust Education for Thousands
Like many children who had a parent who survived the Holocaust, Lydia Preisler Shorenstein felt an obligation to find a way to honor her father's memories and experiences, as well as to educate others about the tragedy of the Holocaust so that history would not continue to repeat itself.
For Lydia the call to action came when she and her late husband, Doug Shorenstein, visited the United States Holocaust Memorial Museum in Washington D.C. along with her family, including Lydia's parents, Simon and Etelka.
Lydia Preisler Shorenstein and Doug Shorenstein with children Sandra, Brandon, and Danielle
Simon Preisler was born in Fancsicovo, Czechoslovakia to an observant Jewish family. A survivor of Auschwitz, he met and married his wife, Etelka, and they settled in Germany after the war, where he had a successful career in business.
Lydia learned the value of giving back through her father, who was a leader of the Frankfurt Jewish community and a well-known philanthropist. His courage and wisdom served as a reminder of the strength of the human spirit in the face of hate and indifference.
Douglas Shorenstein's father, Walter, was a successful real-estate developer who instilled in his children a deep commitment to civic causes. Following in his father's footsteps, Doug assumed leadership of Shorenstein Properties and served on boards and committees of hospitals and universities. Doug succumbed to cancer last year after a courageous battle. His profound impact on the San Francisco skyline and on our community cannot be measured, but it will be lasting.
Lydia Shorenstein in conversation with teachers at the annual Day of Learning. Over 750 educators and students come each year.
Lydia and Doug founded the Preisler Shorenstein Institute providing Holocaust education to thousands of youth, and they created an endowment to permanently support it.
They saw an opportunity in partnering with the JFCS Holocaust Center, and Lydia joined the Center's Council of Children of Survivors.
Susan Lowenberg, who serves with Lydia on the Council, has witnessed memory, which is very special to me to do," says Lydia of her involvement with the program.
This year Lydia also supported The Children of Willesden Lane BAY AREA BIG READ presented by the JFCS Holocaust Center. With 7,000 students and educators participating, it was the largest Holocaust education initiative Northern California has ever seen.
The BIG READ called upon students to read a true account of a musical prodigy who was rescued from the Nazis through the Kindertransport—and culminated in four days of musical performances at the Herbst Theatre in San Francisco. Lydia's support helped make this program a great success.
Don't miss another story! Subscribe to our newsletter.
Posted by Admin on November 12, 2016
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Stories & Testimonials (58) | {
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} | 8,455 |
Drupon Rinchen Dorje's last visit in 2016 is big!
Don't miss this outstanding and final opportunity in 2016 to receive the peerless instructions of Mahamudra. The quintessential instruction on meditation, the correct view and the nature of mind of the Drikung Kagyu. In an unbroken transmission, this rare and precious instruction was introduced to Tibet by the Buddhist Mahasiddha Tilopa, a master and sage from India, who passed the teaching to Naropa, from Naropa to Marpa Chokyi Lodro, from Marpa to Milarepa, to Gampopa the heart-son of Milarepa, to Phagmodrupa, and to Lord Jigten Sumgon, the progenitor of the Drikung Monastery and Drikung tradition of Tibet.
In an unbroken passage from guru to disciple, down to the present day - this teaching will be delivered by Drupon Rinchen Dorje, a great and accomplished meditation master in his own right.
Drupon Rinchen Dorje will lecture from the precious pith instruction and writings of the Drikung Kyopa (Lord) Jigten Sumgon. The "Introduction to Mahamudra, The Co-emergent Unification", is a short text, accessible for the beginner, but loaded with meanings for even the most advanced practitioner. Copies of the text will be available for purchase.
A favorite subject and instruction of H.E. Garchen Rinpoche, don't miss this rare and precious opportunity. The translator for this event is Lama Gyaltzen, a Tibetan lama residing in Santa Fe NM, with a great command of the English language.
__Anyone who would like to offer service or meals during Drupon's visit, please contact us! | {
"redpajama_set_name": "RedPajamaC4"
} | 1,071 |
Q: React JS - how to get google-map-react to work? I want to put two maps on my page but I'm not able to. This is what I got:
Can anyone help me?
This is the result
A: Your function Map is using the useState hook, this is not permitted and yields the error you are getting in the console.
You only need to use regular variable declarations:
Intead of:
const [status,setStatus] = useState(null)
Just declare it as
let status = null
And instead of using setStatus, just reassign the value using plain JS:
status = "new value"
EDIT: I see you got more errors in the console but this will at least fix one
Refer to https://reactjs.org/docs/hooks-rules.html to know more about said rules
| {
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} | 5,669 |
Q: List of plots generated in ggplot2 using scale_color_gradientn have wrong coloring I'm attempting to use library(scales) and scale_color_gradientn() to create a custom mapping of colors to a continuous variable, in an attempt to limit the effect of outliers on the coloring of my plot. This works for a single plot, but does not work when I use a loop to generate several plots and store them in a list.
Here is a minimal working example:
library(ggplot2)
library(scales)
data1 <- as.data.frame(cbind(x = rnorm(100),
y = rnorm(100),
v1 = rnorm(100, mean = 2, sd = 1),
v2 = rnorm(100, mean = -2, sd = 1)))
#add outliers
data1[1,"v1"] <- 200
data1[2,"v1"] <- -200
data1[1,"v2"] <- 50
data1[2,"v2"] <- -50
#define color palette
cols <- colorRampPalette(c("#3540FF","black","#FF3535"))(n = 100)
#simple color scale
col2 <- scale_color_gradient2(low = "#3540FF",
mid = "black",
high = "#FF3535"
)
#outlier-adjusted color scale
{
aa <- min(data1$v1)
bb <- quantile(data1$v1, 0.05)
cc <- quantile(data1$v1, 0.95)
dd <- max(data1$v1)
coln <- scale_color_gradientn(colors = cols[c(1,5,95,100)],
values = rescale(c(aa,bb,cc,dd),
limits = c(aa,dd))
)
}
Plots:
1. Plot with simple scales - outliers cause scales to stretch out.
ggplot(data1, aes(x = x, y = y, color = v1))+
geom_point()+
col2
2. Plot with outlier-adjusted scales - correct color scaling.
ggplot(data1, aes(x = x, y = y, color = v1))+
geom_point()+
coln
3. The scales for v1 do not work for v2 as the data is different.
ggplot(data1, aes(x = x, y = y, color = v2))+
geom_point()+
coln
#loop to produce list of plots each with own scale
{
plots <- list()
k <- 1
for (i in c("v1","v2")){
aa <- min(data1[,i])
bb <- quantile(data1[,i],0.05)
cc <- quantile(data1[,i], 0.95)
dd <- max(data1[,i])
colm <- scale_color_gradientn(colors = cols[c(1,5,95,100)],
values = rescale(c(aa,bb,cc,dd),
limits = c(aa,dd)))
plots[[k]] <- ggplot(data1, aes_string(x = "x",
y = "y",
color = i
))+
geom_point()+
colm
k <- k + 1
}
}
4. First plot has the wrong scales.
plots[[1]]
5. Second plot has the correct scales.
plots[[2]]
So I'm guessing this has something to do with the scale_color_gradientn() function being called when the plotting takes place, rather than within the loop.
If anyone can help with this, it'd be much appreciated. In base R I would bin the continuous data and assigning hex colors into a vector used for fill color, but I'm unsure how I can apply this within ggplot.
A: You need to use a closure (function with associated environment):
{
plots <- list()
k <- 1
for (i in c("v1", "v2")){
colm <- function() {
aa <- min(data1[, i])
bb <- quantile(data1[, i], 0.05)
cc <- quantile(data1[, i], 0.95)
dd <- max(data1[, i])
scale_color_gradientn(colors = cols[c(1, 5, 95, 100)],
values = rescale(c(aa, bb, cc, dd),
limits = c(aa, dd)))
}
plots[[k]] <- ggplot(data1, aes_string(x = "x",
y = "y",
color = i)) +
geom_point() +
colm()
k <- k + 1
}
}
plots[[1]]
plots[[2]]
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,360 |
SEX DRUGS HiV PARTY – PUNK ALL DAYER
The Sex Drugs And HIV project was started in 1996 to support and raise awareness for 4 charity's; Terrence Higgins Trust, Rape Crisis, Release, Cancer Research UK. It took 15 years to record a double album of 40 songs with over 200 musicians. The album will be issued as 3 disks, (a double CD and DVD of all the songs).
The Sex Drugs & HIV show at the New Cross Inn on the 26th May is raise funds for the project for pressing the album CD's. The event will start at 2pm with the 1st public showing of the Sex Drugs & HIV album/film followed by Max Splodge's Rock'n'Roll Bingo with 'prize's from the stars'.
The event will host punk, rock, ska & reggae bands, featuring musicians involved with the Sex Drugs & HIV project, including musicians from; The Revillos, The Godfathers, Angelic Upstarts, Sham 69, Chelsea, The Cure, R.D.F, P.A.I.N, The Astonaughts,The Duel, Electric Cocks, Tenpole Tudor, Splodgenessabounds. The night will end with a set of songs from the Sex Drugs & HIV album performed by the Rock'n'Roll Gypsies, an all stars band fronted by Mat Sargent who founded the project. Advance tickets £10 or £12 on the door. For more information on the Sex Drugs & HIV project visit www.sexdrugsandhiv.com
Published on: 22nd March 2012 at 12:42
Categories: Artists, Entertainment, News, charities, music and tagged 200, angelic upstarts, Cancer Research UK, Chelsea, Electric Cocks, mat sargent, max splodge, musicians, new cross inn, P.A.I.N, party, Project, punk, R.D.F, Rape Crisis, reggae, release, Rock, Rock'n'Roll Gypsies, sex drugs & HIV, Sham 69, Ska, ska & reggae, Splodgenessabounds, Tenpole Tudor, terrence higgins trust, The Astonaughts, the cure, the duel, The Godfathers, The Revillos | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,330 |
James Albert Cooper (born September 28, 1955) is a former American football offensive lineman in the National Football League (NFL) for the Dallas Cowboys. He played college football at Temple University.
Early years
Cooper attended Cardinal Dougherty High School in Philadelphia, not far from the Temple campus. He was a three-sport athlete, earning varsity letters in football, baseball, and track.
He was the Catholic League champ in the shot put. His teammates elected him captain in all three sports during his senior year.
College career
Cooper accepted a football scholarship from Temple University. He played both on the offensive and defensive lines, before settling at offensive tackle, where he was a three-year starter.
He was inducted into the Philadelphia Chapter of the Pennsylvania Sports Hall of Fame.
Professional career
Cooper was selected by the Dallas Cowboys in the sixth round (164th overall) of the 1977 NFL Draft. It proved difficult at first, almost being traded to the Denver Broncos along with fullback Jim Jensen and getting released at the end of training camp on September 9. On September 16, he was re-signed after Jim Eidson was lost for the year with a knee injury. He took advantage of this second opportunity, by learning 3 positions (Tackle, Guard, Center) and becoming the most versatile offensive lineman on the team. As a rookie, he mostly backed up both guard positions.
In 1978, he was listed as the backup at center. In 1979, he started the first 11 games at right tackle, while Rayfield Wright was recovering from a knee injury.
In 1980, he took over the starting position after Wright retired. center John Fitzgerald nicknamed the Cowboys offensive line as the "Four Irishmen and a Scott", when it was formed by him, Fitzgerald, Pat Donovan, Tom Rafferty and Herb Scott.
In the 1982 season, he helped blocked on Tony Dorsett's record 99-yard touchdown run against the Minnesota Vikings.
In 1984, he missed half of the season, after being injured on a bizarre accident, when he slipped while rising from a table at a night club, while watching Monday Night Football. He dislocated his right ankle and in the process broke a bone and ligaments. He was replaced by Phil Pozderac the rest of the season.
In 1985, he started 15 games, missing the season finale with a neck injury. Cooper retired in 1986, after being injured most of that year. He finished his career after playing in 133 games, including 2 Super Bowls.
Personal life
After retirement, he focused on managing his family owned company Industrial Container Services.
References
1955 births
Living people
Players of American football from Philadelphia
American football offensive tackles
Temple Owls football players
Dallas Cowboys players | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,362 |
\section{Introduction}
\label{sec:intro}
Since the first experimental realization of Bose-Einstein condensation in 1995 the expansion of quantum gases, after release of the confining trap, has systematically provided crucial information on the physical properties of such systems. In Ref.~\cite{Anderson1995, Davis1995} the observation of the bimodal expansion at finite temperature actually provided crucial evidence for the occurrence of Bose-Einstein condensation. Later experiments with larger numbers of atoms and stronger interaction effects have provided strong evidence for the hydrodynamic nature of the expansion~\cite{Ernst1998,Stenger1998}. The expansion of the condensate has also been employed to characterize the emergence of the superfluid to Mott insulator transition in the presence of a periodic optical lattice~\cite{Greiner2002}, the effects of localization in the presence of disorder~\cite{Roati2008}, the occurrence of Bloch oscillations~\cite{Roati2004} and a large variety of physical phenomena in both Bose and Fermi quantum gases~\cite{Pitaevskii2016}.
In many cases the condensate is in the so-called Thomas-Fermi limit, where the equation of state of uniform matter can be directly employed in the local density approximation and the mechanism of the expansion is well described using the hydrodynamic formalism of superfluids. In usual superfluids at low temperature, the hydrodynamic picture is based on the irrotationality constraint for the velocity field which takes the most famous expression ${\bf v} =(\hbar/m)\nabla \phi$ where $\phi$ is the phase of the order parameter, $\hbar$ is the reduced Plank constant and $m$ is the atomic mass. The hydrodynamic approach has proven quite useful to describe the macroscopic dynamic behavior of quantum gases, including the study of the low frequency collective oscillations~\cite{Stringari1996} and the expansion of the gas after release of the trap~\cite{Castin1996,Kagan1996,Dalfovo1997}, in excellent agreement with experiments.
The recent experimental investigation of spin-orbit coupled quantum gases has enlarged the horizon of the possible scenarios characterizing the dynamic behavior of quantum gases~\cite{Lin2011,Fu2011,Zhang2012,Ji2015,Wang2012,Cheuk2012,Qu2013,Olson2014} (for recent review articles, see~\cite{Zhai2015,Li2015}). In particular the presence of spin-orbit coupling in binary mixtures of Bose-Einstein condensates is known to affect the irrotationality constraint and the current-phase relation
${\bf j} =n (\hbar/m)\nabla \phi$ holding in usual superfluids. This is the consequence of the breaking of Galilean invariance and is at the origin of novel features exhibited by such systems, like the violation of Landau's criterion for superfluidity~\cite{Zhu2012,Zheng2013,Ozawa2012a}, the quenching of the sound velocity~\cite{Martone2012} and of the superfluid density~\cite{Zhang2016}, the emergence of diffused vorticity in the rotational flow~\cite{Stringari2017}. Experimentally a first direct evidence for the violation of the superfluid current-phase relation was given by the quenching of the frequency of the center-of-mass oscillation in harmonically trapped configurations~\cite{Zhang2012} and of the sound velocity~\cite{Ji2015}. In a very recent work \cite{Khamehchi2016} the expansion of a Bose-Einstein condensate in the presence of spin-orbit coupling was investigated with special focus on the plane-wave phase, where intriguing effects associated with the occurrence of negative values of the effective mass were observed.
In the present paper we investigate the expansion of a spin-orbit coupled Bose gas focusing on the single-minimum phase, where a simplified hydrodynamic description is formulated in terms of scaling variables and analytic solutions to the nonlinear equations characterizing the expansion can be obtained. We find that, for values of the Raman coupling close to phase transition between the single-minimum phase and the plane-wave phase, the expansion along the direction of spin-orbit coupling is dramatically quenched exhibiting an effective localization caused by the lowering of the superfluid flow. A typical example of quenched expansion caused by spin-orbit coupling is shown in Fig.~\ref{Fig1} where the integrated density $n(x, y)=\int n(\mathbf{r})dz$ obtained by numerically solving the Gross-Pitaevskii equations is reported for different expansion times, starting from an initially isotropic density distribution. At the phase transition, corresponding to the choice $\Omega=\Omega_\text{c}$ for the strength of the Raman coupling, the expansion is frozen along the direction of spin-orbit coupling (the vertical axis).
\begin{figure}[t!]
\centerline{
\includegraphics[width=8.6cm]{fig_2Ddensity.pdf}}
\caption{Contour plot of the total integrated density $n(x, y)$ of the expanding spin-orbit coupled condensate for (a) $\Omega=\Omega_\text{c}$ and (b) $\Omega=1.5\Omega_\text{c}$ (In each panel, the brighter the color, the higher the density). For each value of Raman coupling $\Omega$, the density profiles have been shown after an expansion time $\omega_\text{ho}t=0$, $4.7$ and $9.4$, respectively. In the top left panel we also show a length scale corresponding to $10\mu m$. The results are obtained by solving the Gross-Pitaevskii equations with the spin-orbit coupling along the vertical direction. See the main text for the detailed description of the simulation parameters.}
\label{Fig1}
\end{figure}
\section{Hydrodynamic formalism}
\label{sec:hydro}
We consider the following Raman induced spin-orbit coupled single-particle Hamiltonian~\cite{Ho2011,Li2012}
\begin{equation}
H_\text{sp}=\frac{1}{2m}(p_x-\hbar k_0\sigma_z)^2 + \frac{p_y^2}{2m} + \frac{p_z^2}{2m} - \frac{\Omega}{2}\sigma_x
\label{eq:sp}
\end{equation}
where $p_\mu=-i\hbar \nabla_\mu$ ($\mu=x,y,z$) is the canonical momentum, $\hbar k_0$ is the recoil momentum and $\Omega>0$ is the Raman coupling strength. The detuning term $(\delta/2)\sigma_z$ can be set to zero with a proper choice of the frequency of the two Raman lasers. Interaction effects will be taken into account through the interaction energy $V_\text{int}=(1/2)\sum_{\alpha\beta}\int d\mathbf{r}g_{\alpha\beta}n_\alpha n_\beta$ where $n_\alpha$ is the density distribution of the $\alpha$-th component, $g_{\alpha\beta}=4\pi\hbar^2a_{\alpha\beta}/m$ are the coupling constants in different spin channels and $a_{\alpha\beta}$ the corresponding scattering lengths. For simplicity, we assume that the interspin and intraspin interaction constants are the same, i.e., $g_{11}=g_{22}=g_{12}=g$. The spin-orbit coupled Bose-Einstein condensate is known to exhibit a second-order phase transition at the critical Raman coupling strength $\Omega_\text{c}=4E_\text{r}$ where $E_\text{r}=\hbar^2k_0^2/2m$ is the recoil energy. For $\Omega<\Omega_\text{c}$, the lower band of the single particle dispersion exhibits double minima at the wave vectors $\pm k_0\sqrt{1-(\Omega/\Omega_\text{c})^2}$. The condensate picks up one of them spontaneously and the resulting ground state then corresponds to a spin-polarized plane-wave phase. For $\Omega>\Omega_\text{c}$, the lower band has instead only a single minimum at zero momentum and the resulting ground state is named as single-minimum phase or zero-momentum phase which is instead spin balanced at equilibrium.
The spin-orbit Hamiltonian Eq.~(\ref{eq:sp}) is known to affect in a deep way the expression for the current operator along the $x$ direction which takes the form
\begin{equation}
{\hat j}_x=({\hat p}_x -\hbar k_0 {\hat \sigma_z})/m
\end{equation}
and, as we will explicitly discuss, causes the violation of the usual current-phase relation ${\bf j} =n (\hbar/m)\nabla \phi$.
As discussed in~\cite{Stringari2017}, the relative phase of the two components is locked for the ground state and for the low frequency dynamics. Therefore, the dynamic behavior of the spin-orbit coupled Bose-Einstein condensate is described by the densities of the two components $n_\alpha$ and by their common phase $\phi$. Starting from the coupled Gross-Pitaevskii equations in the presence of spin-orbit coupling and neglecting quantum pressure effects one can then derive the following equations:
\begin{eqnarray}
&& \frac{\partial n}{\partial t} + \frac{\hbar}{m}\nabla\cdot(n\nabla\phi)-\frac{\hbar k_0}{m}\nabla_x s_z=0 \label{eq:continuity} \\
&& \hbar\frac{\partial\phi}{\partial t} + \frac{\hbar^2}{2m}(\nabla\phi)^2 + gn + V_\text{ext}-\frac{\Omega}{2}\frac{n}{\sqrt{n^2-s_z^2}} = 0 \label{eq:Euler} \\
&& -\frac{\hbar^2 k_0}{m}\nabla_x \phi + \frac{\Omega}{2}\frac{s_z}{\sqrt{n^2-s_z^2}} = 0 \label{eq:spinphase}
\end{eqnarray}
where $V_\text{ext}$ is the confining harmonic potential, $n=n_1+n_2$ and $s_z=n_1-n_2$ are the total density and spin density. Note that these quantities are position- and time-dependent in the expansion dynamics. From the continuity equation~(\ref{eq:continuity}), it is natural to introduce the superfluid velocity $v_x=(\hbar/m)\nabla_x\phi-\hbar k_0s_z/(mn)$, $v_{y,z}=(\hbar/m)\nabla_{y,z}\phi$. By writing the spin density in terms of the local quasi-momentum $p_x= \hbar\nabla_x \phi$ with the help of Eq. (\ref{eq:spinphase}) one recovers the hydrodynamic formalism presented in the recent work~\cite{Khamehchi2016}, based on the inclusion of the lower branch of the single-particle spectrum.
In the following we will consider the single-minimum phase ($\Omega \ge \Omega_\text{c}$), where, at equilibrium, the system is unpolarized and its phase can be chosen equal to zero. To obtain simple analytic expressions, we assume that the spin density accumulated during the expansion is small compared to the total density, i.e., $s_z\ll n$. One can then expand the square roots in the above equations and include terms quadratic in the spin density, generalizing the linearized expressions derived in~\cite{Stringari2017}. Using Eq.~(\ref{eq:spinphase}) a useful relation between the spin density and the superfluid velocity is then obtained
\begin{equation}
\frac{s_z}{n}=\frac{2\hbar k_0}{\Omega}\frac{m^*}{m}v_x \label{eq:spindensity2}
\end{equation}
with the effective mass given by
\begin{equation}
m^*=m\left(1-\Omega_\text{c}/\Omega \right)^{-1}
\end{equation}
Using these relations, after eliminating the spin density, the above differential equations can be recast in the form
\begin{eqnarray}
&& \frac{\partial n}{\partial t} + \nabla\cdot(n\mathbf{v}) = 0 \label{eq:HD1} \\
&& m^*\frac{\partial v_x}{\partial t} + \nabla_x\left(\frac{m^*}{2}v_x^2+gn+V_\text{ext} \right) = 0 \label{eq:HD2} \\
&& m\frac{\partial v_{y,z}}{\partial t} + \nabla_{y,z}\left(\frac{m}{2}v_{y,z}^2+gn+V_\text{ext} \right) = 0
\label{eq:HD3}
\end{eqnarray}
hereafter called the equations of quadratic hydrodynamics, where we have retained only quadratic terms in the velocity field. Equations of the same form can also be derived in the plane-wave phase ($\Omega \le \Omega_\text{c}$), where the effective mass is given by $m^*=m(1-(\Omega/\Omega_\text{c})^2)^{-1}$. In the plane-wave phase the equations of quadratic hydrodynamics have however a more limited range of applicability because in this case high order terms in the velocity field become soon relevant and cause the appearance of asymmetric effects in the expansion dynamics along the direction of spin-orbit coupling, as shown in \cite{Khamehchi2016}.
\section{Expansion dynamics}
\label{sec:expansion}
Similar to the case in the absence of spin-orbit coupling, the equations~(\ref{eq:HD1})-(\ref{eq:HD3}) of quadratic hydrodynamics admit a solution of the inverted parabolic form $n(\mathbf{r}, t)=n_0(t)\left(1-x^2/R_x^2(t)-y^2/R_y^2(t)-z^2/R_z^2(t)\right)$ within the region where $n(\mathbf{r}, t)$ is positive and zero elsewhere~\cite{Castin1996,Kagan1996,Dalfovo1997}. The peak density $n_0(t)$ is determined by the normalization condition $n_0(t)=15N/[8\pi R_x(t)R_y(t)R_z(t)]$ where $N$ is the number of atoms. The Thomas-Fermi radii $R_\mu(t)$ can be written in the scaled form $R_\mu(t)=b_\mu(t)R_\mu^{(0)}$ ($\mu=x,y,z$) in terms of the dimensionless scaling parameters $b_\mu(t)$ and of the Thomas-Fermi radii $R_\mu^{(0)}$ before the expansion. The velocity field is parameterized as $v_\mu(x_\mu, t)=a_\mu(t)x_\mu$. Substituting these ansatz into the hydrodynamic equations, we find $a_\mu(t)=\dot{b}_\mu(t)/b_\mu(t)$ and, after switching off the trapping potential, the equations for the scaling parameters $b_{\mu}$ take the form
\begin{equation}
\ddot{b}_\mu(t)-\frac{\tilde{\omega}_\mu^2}{b_\mu(t)b_x(t)b_y(t)b_z(t)}=0 \; , \label{eq:Db}
\end{equation}
where $\tilde{\omega}_x=\omega_x\sqrt{m/m^*}$ and $\tilde{\omega}_{y,z}=\omega_{y,z}$. The initial conditions are determined by the properties of the system at equilibrium, thus $a_\mu(t=0)=0$ and $b_\mu(t=0)=1$. By solving the above differential equations for $b_\mu(t)$ and $a_\mu(t)$ with these initial conditions, one can obtain the density profile and the velocity field of the condensate as well as, according to Eq. (\ref{eq:spindensity2}), the spin density during the expansion. It is worth noticing that, different from the plane-wave phase, the
total density of the single-minimum phase is always symmetric in the spin-orbit coupling direction during the expansion.
Close to the critical transition point from the single-minimum phase to the plane-wave phase, $m/m^*\to 0$ and thus $\tilde{\omega}_x\to 0$. In this regime it is thus possible to write down the analytical expression~\cite{Castin1996}
\begin{eqnarray}
&& b_{y,z}(\tau) = \sqrt{1+\tau^2} \\
&& b_x(\tau) = 1+\lambda^2(\tau \arctan \tau - \ln\sqrt{1+\tau^2}) \label{bx}
\end{eqnarray}
for the solution of the hydrodynamic expansion differential Eq.~(\ref{eq:Db}),
where we have assumed initial isotropic trapping ($\omega_x=\omega_y=\omega_z\equiv \omega_\text{ho})$, yielding $\lambda = {\tilde \omega}_x/{\tilde \omega}_y=\sqrt{m/m^*}$, and introduced the dimensionless time $\tau=\omega_\text{ho}t$.
\begin{figure}[tbp]
\centerline{
\includegraphics[width=8.6cm]{fig_width.pdf}}
\caption{Scaled Thomas-Fermi radius $R_\mu(t)/R_0$ of the expanding spin-orbit coupled Bose-Einstein condensate for different Raman coupling strengths (a) $\Omega=\Omega_\text{c}$ and (b) $\Omega=1.5\Omega_\text{c}$. The dashed red lines and the solid red lines are the results for $R_x(t)$ and $R_{y,z}(t)$ obtained from the hydrodynamic theory. The circles and squares in blue are the corresponding results obtained by numerically solving the Gross-Pitaevskii equations.}
\label{Fig2}
\end{figure}
Using the result $v_x =\frac{\dot{b}_x}{b_x}x$ and Eq. (\ref{bx}) we finally find that, in the limit $m/m^* \to 0$, the spin density can be simplified to
\begin{equation}
\frac{s_z}{n}= \arctan \tau \sqrt{\frac{\mu_0}{E_\text{r}}} \frac{x}{R_0}
\label{eq:largetimesz}
\end{equation}
with $\arctan \tau \to \pi/2$ for $\tau\to\infty$. In Eq.~(\ref{eq:largetimesz}), we have introduced the initial Thomas-Fermi radius $R_0=R_\mu^{(0)}$ and $\mu_0=m\omega_\text{ho}^2R_0^2/2$. Result (\ref{eq:largetimesz}) shows that, when $\Omega=\Omega_\text{c}$ the above simplified hydrodynamic approach, based on the condition $s_z\ll n$, is no longer valid for large expansion times $\tau$, unless $\mu_0$ is sufficiently small compared to $E_\text{r}$.
The hydrodynamic results, based on the solution of the scaling equations for $b_\mu(t)=R_\mu(t)/R_\mu^{(0)}$, are shown as solid and dashed red lines in Fig.~\ref{Fig2} for two different values of the Raman coupling strength $\Omega=\Omega_\text{c} $ and $\Omega=1.5 \Omega_\text{c}$. For the sake of simplicity in these calculations we have assumed initial spherical trapping (similar features are predicted also in the case of anistropic trapping). We find that the expansion along the spin-orbit coupling direction (dashed red lines in Fig.~\ref{Fig2}) is slower than the one along the other directions (solid red lines in Fig.~\ref{Fig2}) and, in particular, it is frozen at the critical transition point between the single-minimum phase and plane-wave phase, exhibiting an effective localization. The corresponding integrated density $n(x)=\int dydz n(\mathbf{r})=(15N/16R_x(t))(1-x^2/R_x^2(t))^2$ and spin density profiles obtained with the help of Eq.~(\ref{eq:spindensity2}) are instead shown in Fig.~\ref{Fig3} for two different expansion times.
\begin{figure}[t!]
\centerline{
\includegraphics[width=8.6cm]{fig_density_column.pdf}}
\caption{Plot of the total integrated density $n(x)$ and the spin density $s_z(x)$ of the expanding spin-orbit coupled condensate for (a) $\Omega=\Omega_c$ and (b) $\Omega=1.5\Omega_c$. For each value of Raman coupling, the density profiles after an expansion time $\omega_{ho}t=0$, $4.7$, and $9.4$ are shown. The solid and dashed red lines are the results obtained from the hydrodynamic theory while the blue symbols are the results obtained by numerically solving the Gross-Pitaevskii equations, respectively. In the left-bottom panel, the solid green line is a fiting function of the form $f(x)= \beta \exp(-|x/R_0|^\alpha/\lambda)$ with $\beta=9750$, $\alpha=1.17$ and $\lambda=0.6$.
}
\label{Fig3}
\end{figure}
To check the validity of the hydrodynamic approach, a full numerical calculation of the coupled Gross-Pitaevskii equations has been carried out under the same conditions of spherical trapping. We consider a spin-orbit coupled Bose gas with $N=1\times 10^5$ of $^{87}$Rb atoms confined in a spherical harmonic trap with $\omega_\text{ho}=2\pi\times 50$Hz. The Raman laser wavelength $\lambda=782$nm which determines the recoil momentum $k_0=2\pi/\lambda$ and the recoil energy $E_\text{r}$. For our simulation parameters, we have the initial Thomas-Fermi radius $R_0=8.46\mu m$ and $\mu_0/E_\text{r}\approx 0.2$. After long enough imaginary-time evolution, we prepare the condensate at the ground state for different values of Raman coupling strengths $\Omega$. Then we release the harmonic trap and observe the expansion dynamics of the condensate. The numerical results for the radii are shown by the blue symbols in Fig.~\ref{Fig2} where, in order to compare quantitatively the simulation results with the hydrodynamic predictions, we have numerically calculated the mean square root radii $w_\mu(t)=\sqrt{\langle x_\mu^2\rangle}$ and then determined $R_\mu(t)$ through the Thomas-Fermi relation $R_\mu(t)=\sqrt{7}w_\mu(t)$. The corresponding density and spin density profiles are instead reported in Fig.~\ref{Fig3}.
The agreement between the simulation and the hydrodynamic theory results is excellent for Raman coupling $\Omega=1.5\Omega_\text{c}$ at all expansion times. At the critical transition point $\Omega=\Omega_\text{c}$, we observe deviations at large expansion times, reflecting the fact that spin density is not always small compared to the total density. In particular, as shown by the simulation results in Fig.~\ref{Fig2}(a), the expansion of the condensate along the spin-orbit coupling direction at the critical transition point is not completely frozen. Furthermore, as explicitly shown in Fig.~\ref{Fig3}(a), for larger expansion times the total density cannot be longer described as an inverted parabola. This is easy to understand since during the expansion the condensate accumulates a local phase (and hence a local quasi momentum) which changes the values of the effective mass, resulting in a more complicated behavior of the density distribution. For $\Omega=\Omega_c$, the density at $\omega_\text{ho}t=9.4$ can be approximated by a function of the form $n(x)\approx \beta \exp(-|x/R_0|^\alpha/\lambda)$, reflecting an analogy with the exponential behavior observed in~\cite{Roati2008} during the expansion in the presence of localization induced by disorder.
\section{Conclusion}
\label{sec:conclusion}
In conclusion, we have studied the expansion dynamics of the spin-orbit coupled Bose-Einstein condensate using a hydrodynamic formalism allowing for analytic scaling solutions. The expansion along the spin-orbit coupling direction is slowed down in a dramatic way near the critical transition point between the single-minimum phase and the plane-wave phase. The slowering is caused by the quenching of the superfluid flow corresponding to a huge increase of the effective mass. The comparison with the numerical solution of the coupled Gross-Pitaevskii equations indicates that our hydrodynamic approach, which takes into account up to quadratic terms in the velocity field, works very well for the calculation of the density and of the spin density for short expansion times. For a Rashba spin-orbit coupled Bose gas~\cite{Wang2010,Wu2011,Ozawa2012,ZhangYP2012}, the investigation of the expansion dynamics can be carried out similarly using the corresponding hydrodynamic formalism~\cite{Stringari2017}.
\begin{acknowledgements}
We would like to thank F. Dalfovo and P. Engels for discussions and comments. This work was supported by the QUIC grant of the Horizon2020 FET program and by Provincia Autonoma di Trento.
\end{acknowledgements}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,452 |
\section{Introduction}
Complex quantum trajectories were first obtained \cite{mvj1} by modifying the de Broglie's guiding wave approach to quantum mechanics. Here, trajectories were drawn for the cases of harmonic oscillator, potential step, wave packets etc. For getting this trajectory representation, first we substitute $\Psi=e^{i\hat{S}/\hbar}$ in the Schrodinger equation, which gives the quantum Hamilton-Jacobi equation (QHJE) \cite{oldpaps,goldstein}
\begin{equation}
\frac {\partial \hat{S}}{\partial t} + \left[ \frac{1}{2m}\left( \frac
{\partial \hat{S}}{\partial x}\right)^2 +V\right] =
\frac{i\hbar}{2m} \frac{\partial^2 \hat{S}}{\partial x^2}. \label{eq:qhje}
\end{equation}
Then we postulate an equation of motion for the particle, similar to that used by de Broglie:
\begin{equation}
m\dot{x} \equiv \frac {\partial \hat{S}}{\partial x}= \frac {\hbar
}{i} \frac {1}{\Psi}\frac {\partial \Psi}{\partial x}.
\label{eq:xdot} \end{equation}
The quantum trajectories $x(t)$ were found by integrating this equation with respect to time; in general, they lie in a complex $x$-plane, with $x=x_r+i x_i$. This results in a modified version of the de Broglie-Bohm (dBB) quantum mechanics \cite{dbpap,bohm,dBB,valentini}. Eq. (\ref{eq:xdot}) was used by Leacock and Padgett \cite{leacock} to obtain eigenvalues in many bound state problems, without having to solve the corresponding Schrodinger equation.
It shall be noted that the canonical momentum is not always the mechanical momentum, even in classical mechanics with Cartesian coordinates. For instance, when there are velocity-dependent potentials, the canonical and mechanical momenta are different. In the case of a charged particle in a magnetic field with the vector potential ${\bf A}$ , the mechanical momentum may be written as
\begin{equation}
m{\bf \dot{x}} = {\bf \nabla}{\hat S}-e{\bf A}/c.
\end{equation}
This equation in Cartesian coordinates shall thus be the equation of motion we adopt for charged particles in higher dimensions with electromagnetic field.
The Floyd-Faraggi-Matone (FFM) trajectory representation \cite{floyd,faraggi,carroll,floyd2,floyd3} is another modified dBB version but with real trajectories and is based on a generalised Hamilton-Jacobi equation equivalent to that used in dBB. But this representation differs from dBB quantum mechanics mainly in the use of the equation of motion. Here, for stationarity the equation of motion for the trajectory time $t$, relative to its constant coordinate $\tau$, is given as a function of $x$ by
\begin{equation}
t-\tau =\partial W/\partial E
\end{equation}
where $W$ is the Hamilton's characteristic function appearing in the generalised Hamilton-Jacobi equation and $E$ is the energy. Carroll \cite{carroll} finds that for stationarity the above Jacobi's theorem is valid, for $W$ is a Legendre transform of Hamilton's principal function. Floyd \cite{floyd2} notes that as Jacobi's theorem also determines the equation of motion in classical mechanics, it is universal transcending across the division between classical and quantum mechanics. In this way, FFM claims to be a deterministic theory.
The complex function $\hat{S}$, which may be called the complex Hamilton's principal function in the present modified de Broglian mechanics, and the complex QHJE (\ref{eq:qhje}) itself are quite different from the corresponding entities in dBB or FFM representations.
Moreover, probability is an integral part of the new formulation and one can also consider equation (\ref{eq:xdot}) as propagating probability densities. In fact, it was shown in a previous work \cite{mvj2} that the quantum trajectories obtained in this scheme are the same as the characteristic curves propagating information about a conserved probability density. Hence we continue to use (\ref{eq:xdot}), with appropriate modifications as mentioned above when necessary, as the equation of motion in this modified dBB formulation.
The complex trajectory approach gives the paths in the $n=1$ harmonic oscillator as shown in Fig. 1. In this case, the paths are given by $(\alpha^2x_r^2-\alpha^2x_i^2-1)^2+4\alpha^4x_r^2x_i^2 = \; A^2$, a constant for each path. Here $A$ is real and positive, and $\alpha^2$ has the usual definition, equal to $m\omega_0/\hbar$. It is interesting to note that these figures are the famous Cassinian ovals. The Jacobi lemniscate, which is the special case of these ovals, corresponds to $A=1$.
\begin{figure}[ht]
\centering{\resizebox {0.5 \textwidth} {0.3 \textheight }
{\includegraphics {shm11.eps}}
\caption{ The complex trajectories in the $n=1$ harmonic oscillator case, drawn for various values of $A$ and with axes $X_r\equiv \alpha x_r$ and $X_i\equiv \alpha x_i$. These are the famous Cassinian ovals. The lemniscate corresponds to $A=1$. } \label{fig:n1shmtraj}}
\end{figure}
Recently, the prospects of a dynamical explanation of quantum probability in this scenario was explored \cite{mvj2}. We have shown that in this modified de Broglian mechanics, the Born's probability along the real axis can be obtained as an exponential function of a line integral along the real axis. Similarly in the extended $x_rx_i$-plane, a probability density is obtained as the exponential of a trajectory integral. In both these cases, integrands involve the particle's complex velocities, indicating a dynamical origin of quantum probability. In the extended case, we obtain a conserved probability, which agrees with the boundary condition (Born's rule) along the real axis in most regions. However, for other regions (e.g., the subnests inside the lemniscate in the $n=1$ harmonic oscillator case, which do not enclose any poles of $\dot{x}$), such a definition could not be obtained. In fact, a probability density was not proposed for such regions in \cite{mvj2} and it was considered a missing link in the formalism.
Another important question faced by this trajectory representation is why the complex extended motion is not appreciable in the classical limit. Our everyday experience is that particle trajectories are in real space, obeying classical rules. In this letter, we study the probability distribution of particles in the complex plane in detail and adopt for those regions left out in \cite{mvj2} an alternative definition, first proposed in \cite{wyatt3,yang3} as the probability density for the entire plane. This density satisfies the boundary condition on the real axis, though it is not a conserved one. We show that the combined probability for the entire extended plane does not diverge and there is no difficulty in normalizing it, contrary to what happens in the above case. Nonconservation of probability points to the prospects of particle creation and destruction, a fundamental feature of quantum phenomena. Most importantly, the new scheme reveals that large part of the probability for a classical harmonic oscillator lies extremely close to the real line, thereby explaining why the complex extended motion does not leave any imprint in the classical limit.
We restrict ourselves to one particle stationary states in 1-dimension for simplicity. The paper is organized as follows. In the next section, we review the problem with probability density faced in \cite{mvj2}. The third section presents the alternative definition of probability density adopted for the subnests and obtains a form of continuity equation obeyed by it. In section 4, the classical limit of a harmonic oscillator in the light of our combined probability density and its implication for complex trajectories are discussed. Section 5 comprises our conclusion.
\section{Probability from velocity field}
As in the case of dBB quantum mechanics, the present modified de Broglie-Bohm mechanics too is constructed in such a way that it agrees with the results of standard quantum mechanics \cite{mvj1}. This is achieved by accepting the Born's probability axiom along the real line in these representations. However, we note that one of the challenges before such a quantum trajectory representation is to explain this probability axiom. In the standard dBB approach, there were several attempts to obtain the $\Psi^{\star}\Psi$ probability distribution from more fundamental assumptions \cite{dBB}. (The FFM trajectory representation does not involve probability and is considered a deterministic description.)
In addition to that defined along the real line, it is desirable to have a probability density defined everywhere in the extended complex $x$-plane. An earlier attempt made in \cite{poirier} to define such a density was to write $\rho (x)= \bar{\Psi (x)}\Psi (x)$, where $\bar{\Psi (x)} \equiv \Psi^{\star}(x^{\star})$, with $x$ complex. With the help of time-dependent Schrodinger equation, the author shows that, in general, ${\partial \rho}/{\partial t}\neq j^{\prime}(x,t)$.
This arguably leads to nonconservation of probability along trajectories. But it shall be noted that this negative result is based on the choices made in \cite{poirier} for the probability density and flux. Moreover, this definition leads to complex probability off the real axis, which is undesirable.
Another proposal is to define probability as $\Psi^{\star}(x)\Psi(x)$ itself \cite{wyatt3,yang3}. Though this has the advantage of being real everywhere, it is not conserved at any point in the extended plane. In addition, this diverges for large values of $x_i$ in many cases and is not generally a normalisable probability.
Using a different approach, it was shown in \cite{mvj2} that the complex trajectory representation is capable of explaining the quantum probability as originating from dynamics itself. Here, the Born's probability density to find the particle on the real axis around some point at $x=x_{r0}$ was obtained as
\begin{equation}
\Psi^{\star}\Psi (x_{r0},0,t) \equiv P(x_{r0},t) ={\cal N} \exp \left({-\frac{2m}{\hbar}\int^{{x_{r0}}} \dot{x}_i dx_r}\right) , \label{eq:psistarpsi}
\end{equation}
where the integral is taken along the real line. In addition, an extended probability density $\rho(x_r,x_i)$ in the $x_rx_i$-plane for stationary states was proposed in \cite{mvj2} and showed, with the aid of complex-extended Schrodinger equation, that the continuity equation follows from it. The proposal was that if $\rho_0$, the extended probability density at some point $(x_{r0},x_{i0})$ is given, then $\rho(x_r,x_i)$ at another point that lies on the trajectory which passes through $(x_{r0},x_{i0})$ is
\begin{equation}
\rho (x_r(t),x_i(t)) = \rho_0 \exp\left[ \frac{-4}{\hbar}\int_{t_0}^{t} Im\left(\frac{1}{2}m\dot{x}^2+V(x)\right)dt^{\prime} \right]. \label{eq:rho_def}
\end{equation}
Here, the integral is taken along the trajectory $[x_r(t^{\prime}),x_i(t^{\prime})]$ which passes through $(x_{r0},x_{i0})$. The continuity equation
was shown to follow from this axiom by using the extended version of the Schrodinger equation. While evaluating $\rho$ with the help of (\ref{eq:rho_def}) above, one should specify $\rho_0$ at $(x_{r0},x_{i0})$ and if we choose this point as $(x_{r0},0)$, the point of crossing of the trajectory on the real line, then $\rho_0$ can assume the Born's value $P(x_{r0})$.
However, we may note that in some regions of the complex plane, there can be disagreement between the values of $\rho$ and $P(x_r)$ at the points of crossing of the trajectories on the real line. This happens where the complex trajectories do not enclose any poles of $\dot{x}$, described as `subnests' in \cite{mvj1}. Stated more clearly, the problem here is that as a particle trajectory is traversed in these regions, even while the probability $\rho $ at one point of crossing $x_{r0}$ on the real axis agrees with $P(x_{r0})$, at the other point, say the point $x_{r1}$ where the trajectory again crosses the real line, the probability calculated according to (\ref{eq:rho_def}) happens to be different from that of $P(x_{r1})$.
On the other hand, if we directly solve the continuity equation for stationary states, it is easy to see that its solution and the probability density given by the trajectory integral (\ref{eq:rho_def}) give identical results for regions outside the subnests \cite{mvj2}. But for the subnests, the boundary condition overdetermines the problem and we are unable to find a solution that agrees with the Born's rule everywhere on the real line; i.e., there is complete agreement between the extended probability density $\rho$ and the Born's probability density only in the regions outside the subnests.
\section{Probability inside the subnests}
Given this situation, one can ask whether it is possible to find a probability density for the subnests that can agree with the Born's rule on the real line, even if it is not conserved in the region. A natural choice for such a definition is the extended probability $\rho^{\prime} =\Psi^{\star}(x)\Psi (x)$, suggested in \cite{wyatt3,yang3}. A trajectory integral form for $\rho^{\prime}$, similar to that in Eq. (\ref{eq:rho_def}) above, was proposed in \cite{wyatt4}. For stationary states, this can be written as
\begin{equation}
\rho^{\prime} (x_r,x_i) \propto \rho_0 \exp\left[ \frac{-4}{\hbar}\int_{t_0}^{t} Im\left(\frac{1}{2}m\dot{x}^2\right)dt^{\prime} \right]. \label{eq:rho_alt_ls1}
\end{equation}
We propose to adopt this as the probability density for such subnests. Here also, the integral is to be evaluated over the trajectory of the particle. The absence of the potential term $V(x)$ in the integrand marks this definition from that in (\ref{eq:rho_def}). Therefore this distribution will not be conserved. Instead of the continuity equation, we get
\begin{equation}
\frac{\partial \rho^{\prime}}{\partial t} + \frac{\partial (\rho^{\prime} \dot{x}_r)}{\partial x_r}+\frac{\partial (\rho^{\prime} \dot{x}_i)}{\partial x_i}=\frac{4}{\hbar} \rho^{\prime} (x_r,x_i)\; {\hbox {Im}}[V(x)] \label{eq:noncons_gen}
\end{equation}
Thus there are sources and sinks for probability in the subnests. However, as seen below, the net source of probability in this region for the $n=1$ harmonic oscillator is zero, indicating that the total probability for the subnested region can remain conserved.
Thus instead of being `nonviable', the trajectories in this region reveal an important feature of quantum phenomena, namely nonconservation of particles. This indicates creation and destruction of particles in such regions. Hence we can anticipate that the representation is capable of allowing a smooth transition to quantum field theory. It is interesting to note that the Floyd-Farraggi-Matone trajectory representation too allows creation and destruction of particles \cite{floyd3}.
\section{Probability and classical limit}
The extended probability density for the $n=1$ harmonic oscillator in the region inside the lemniscate (for $A<1$ and $x_r>0$), computed using the trajectory integral approach in (\ref{eq:rho_alt_ls1}) is shown overlapped with the `leaf-shaped' surface $\Psi^{\star}\Psi$ in this case, in Fig. 2.
\begin{figure}[ht]
\centering{\resizebox {0.8 \textwidth} {0.5 \textheight }
{\includegraphics {shm1in_traj_dirct.eps}}
\caption{ The extended probability density $\rho(x_r,x_i)$ for the $ A <1$, $x_r>0$ region of the $n=1$ harmonic oscillator, evaluated along the trajectories, overlapped with the extended $\Psi^{\star}\Psi$ probability density. This probability density does not obey the continuity equation, but the total probability will be conserved. Also it agrees with the Born rule along the real line.} \label{fig:n1shm_ls1}} \end{figure}
Using the above definition $\rho^{\prime} \equiv \Psi^{\star}\Psi$ and the expression for $\dot{x}$ in the $n=1$ case, we shall see that
\begin{equation}
\frac{\partial (\rho^{\prime} \dot{x}_r)}{\partial x_r}+\frac{\partial (\rho^{\prime} \dot{x}_i)}{\partial x_i} \propto e^{-\alpha^2 (x^2-y^2)}(x_rx_i^3+x_r^3x_i). \label{eq:noncons_n1}
\end{equation}
The quantity on the right hand side is the density distribution of the `probability source'. It is positive for the first and third quadrants and negative for the second and fourth ones. When the entire region inside the lemniscate is considered, the net source of probability is seen to be zero. Thus the total extended probability can be normalized for the $n=1$ harmonic oscillator.
It is found that the probability in the region inside the lemniscate is substantial; $43.25\%$ of the total probability lies inside this region for the $n=1$ harmonic oscillator. The maximum value of $x_i$ for the lemniscate (a measure of its width) in this case is $x_i^{max}= X_i^{max}/\alpha =0.4858/\alpha$. This explains how classical particles are confined close to the real line. For instance, consider a classical harmonic oscillator of mass $m\sim 1$ kg and frequency $\sim 1$ Hz in the $n=1$ state. The maximum value of $x_i$ for its lemniscate is $x_i^{max}\approx 10^{-17}/\sqrt{m\omega_0}$ in units of metres. Thus $43.25\%$ of the total probability in this case lies within the lemniscate of width $\sim 10^{-17}$ m. It may also be noted that since the extended probability for $A>1$ decreases fast, most of the probability outside also lies close to the real line.
For the higher energy eigenstates of the harmonic oscillator, the lemniscates are seen to be narrower than that of the low energy ones. For example, in the $n=2$ case, the complex paths are described by
\begin{equation}
\left[ (X_r^2+X_i^2)^2-5(X_r^2-X_i^2)+\frac{25}{4}\right]^2(X_r^2+X_i^2) = \hbox{constant.}
\end{equation}
These are shown in Fig. (\ref{fig:n2shmtraj}). The $X_i^{max}$ for the 3-fold lemniscate in this case is 0.4125 and hence its $x_i^{max}$ is less than that in the previous case. For $n=3$, the complex paths are described by a more detailed expression, and are shown in Fig. (\ref{fig:n3shmtraj}). The $X_i^{max}$ for this 4-fold lemniscate is found numerically as $\approx 0.39$, which is again smaller than that in the previous cases. Thus it can be assured that for a classical oscillator of any energy and having mass 1 kg., the width of the region, where the large part of probability lies, is less than or of the order of $10^{-17}/\sqrt{\omega_0}$ m. This indicates that the probable imaginary part of position for classical particles are of extremely small size. However, we may also see that an electron executing harmonic oscillation with frequency $\omega_0$ has to accommodate relatively large values for $x_i$, approximately equal to $ 0.01/\sqrt{\omega_0}$ m.
\begin{figure}[ht]
\centering{\resizebox {0.5 \textwidth} {0.3 \textheight }
{\includegraphics {shm22.eps}}
\caption{ The complex trajectories in the $n=2$ harmonic oscillator case.} \label{fig:n2shmtraj}}
\end{figure}
\begin{figure}[ht]
\centering{\resizebox {0.5 \textwidth} {0.3 \textheight }
{\includegraphics {shm33.eps}}
\caption{ The complex trajectories in the $n=3$ harmonic oscillator case. } \label{fig:n3shmtraj}}
\end{figure}
In summary, the probability axiom in the modified dBB quantum mechanics helps to distinguish the classical limit of quantum harmonic oscillator as one in which the oscillator is probable to be found only very close to the real axis. This result is very important for complex quantum trajectories, for it explains why the complex extension is not observable even indirectly in the classical limit. We anticipate that this property is true in other problems too.
\section{Conclusion}
The present work is a continuation of that in \cite{mvj2} to obtain a probability density for the entire complex $x$-plane, in the modified de Broglie-Bohm quantum mechanics. In the earlier work, a conserved probability (\ref{eq:rho_def}) was proposed for particles in 1-dimensional stationary states, but only for those regions where trajectories enclose poles of the velocity field. This continues to be so in the present framework. But for those regions where trajectories do not enclose any poles, described as subnests, a conserved probability could not be found in \cite{mvj2} and it was considered a missing link in the formalism. We here adopt $\Psi^{\star}(x)\Psi(x)$, which is equivalent to that in (\ref{eq:rho_alt_ls1}), as the extended probability density for those special regions and note that the combined probability density helps to find answers to many pertinent questions that arise in this context.
First of all, when compared to all other proposals for an extended probability density, the present combined probability has the important advantage that it is normalisable in the $x_rx_i$-plane. At the same time, along the real axis in all regions, it agrees with the Born's probability, as required. But we note that the proposed probability for subnests does not obey a continuity equation. However, it can be seen that for the entire subnested region, the net value of this probability is conserved. This property of local nonconservation of particles, also found in the FFM trajectory representation, is argued to be characteristic of quantum phenomena in general and is not to be dismissed as unphysical. In addition, it explains why the imaginary part of complex trajectories do not leave any observable imprints in the case of classical particles. The present scheme is such that for a classical harmonic oscillator, probability is considerable only for those trajectories which lie extremely close to the real line, whereas for an electron in harmonic motion, trajectories have substantial probability even when their imaginary part is appreciable. The observation that complex trajectories with large imaginary part are least probable in the classical limit is physically much relevant for the representation.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,435 |
Acrossocheilus paradoxus és una espècie de peix de la família dels ciprínids i de l'ordre dels cipriniformes.
Els adults poden assolir fins a 20 cm de longitud total. Es troba a Taiwan.
Referències
Bibliografia
Shen, S.C. (ed.), 1993. Fishes of Taiwan. Department of Zoology, National Taiwan University, Taipei. 960 p.
paradoxus | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,945 |
\section{Introduction}
In $1969$, M. Ribe \cite{Ribe} proved that two Banach spaces which are uniformly homeomorphic must be crudely finitely representable in each other. Since then, the Ribe program has attracted significant attention (see \cite{Naor} for a survey on the Ribe program), with the goal of providing purely metric characterizations of local properties of Banach spaces. An important result in this area is that of Bourgain, Milman, and Wolfson, who defined one notion of metric type $p$ and proved that any family of metric spaces with no non-trivial type must contain almost isometric copies of the Hamming cubes. Another goal within the Ribe program is to find, for a given class of important linear operators between Banach spaces, natural metric analogues within the class of Lipschitz maps between metric spaces (see \cite{ChavezDominguez}\cite{FJ}, \cite{JS}). One such class is the class of super-Rosenthal operators, for which Beauzamy \cite{Beauzamy} gave a linear characterization in terms of a sequence of subtype constants (we discuss the notion of subtype in Section $2$). The goal of this work is to undertake the process of proving the Lipschitz analogue of Beauzamy's linear result for the super-Rosenthal operators. We define different notions of subtype constants for a Lipschitz map (or more generally, a uniformly Lipschitz collection of maps) between Banach spaces, which are the analogues of linear subtype constants appearing in the literature in the aforementioned work of Beauzamy and the work of Hinrichs \cite{Hinrichs}. We prove the non-linear analogues of the results found in the work of Beauzamy and Hinrichs, in that if the subtype constants of a uniformly Lipschitz family of maps exhibit the asymptotically worst possible behavior, then the Lipschitz maps preserve copies of the Hamming cubes. Our subtype constants are based on the Bourgain, Milman, Wolfson notion of metric type. We next make these descriptions precise, and then state the main result.
We agree to the convention that $\frac{0}{0}=0$. Given a map $g:(U, d_U)\to (V, d_V)$ between metric spaces, we let $$\text{Lip}(g)=\sup_{x,y\in U}\frac{d_V(g(x), g(y))}{d_U(x,y)}.$$ For a map $g:(U, d_U)\to (V, d_V)$, we let $\text{dist}(g)=\infty$ if $g$ is not injective, and otherwise we let $\text{dist}(g)=\text{Lip}(g)\text{Lip}(g^{-1})$, where $g^{-1}$ is understood to be defined on $g(U)$.
We let $2^n=\{\pm 1\}^n$ be the (vertex set of the) Hamming cube. Given $\ee\in 2^n$, we denote the coordinates of $\ee$ by $\ee(1), \ee(2), \ldots$. We endow $2^n$ with the normalized graph metric $$\partial_n (\ee, \delta)=\frac{1}{n}|\{i: \ee(i)\neq \delta(i)\}|.$$ When no confusion can arise, we will suppress the subscript $n$ and just write $\partial$. We also endow $2^n$ with the uniform probability measure $\mathbb{P}_n$, also suppressing the subscript when no confusion can arise. Given $1\leqslant i\leqslant n$, we let $d_i$ denote the function on $2^n$ which changes the $i^{th}$ coordinate and leaves the other coordinates unchanged.
To avoid cumbersome notation, if $(X, d_X)$, $(Y, d_Y)$ are metric spaces and $f:2^n\to X$, $F:X\to Y$ are functions, we let $\varrho^f_X$, $\varrho^f_Y$, respectively, denote the pseudometrics on $2^n$ given by $$\varrho^f_X(\ee, \delta)= d_X(f(\ee), f(\delta))$$ and $$\varrho^f_Y(\ee, \delta)=d_Y(F\circ f(\ee), F\circ f(\delta)).$$
Now suppose we have $\lambda>0$ fixed and a collection $\mathcal{F}$ of $\lambda$-Lipschitz functions between (possibly different) metric spaces. For $n\in\nn$, we let $a_n(\mathcal{F})$ denote the infimum of those $a>0$ such that for each $F:X\to Y\in \mathcal{F}$ and $f:2^n\to X$, $$\mathbb{E}\varrho_Y^f(\ee, -\ee) \leqslant a\text{Lip}(f:2^n\to X).$$ Note that $a\leqslant \lambda$. These are the Lipschitz analogues of the linear quantities appearing in \cite{Beauzamy}. We note that there appears a factor of $n^{-1}$ on the expectation. The reason is because this constant $n^{-1}$ has been subsumed by $\text{Lip}(g)$ and our convention of using the normalized graph metric on $2^n$.
For $1<p<\infty$, let $b_{p,n}(\mathcal{F})$ denote the infimum of those $b>0$ such that for each $F:X\to Y\in \mathcal{F}$ and $f:2^n\to X$, $$\mathbb{E}\varrho_Y^f(\ee, -\ee)^p \leqslant b^p n^{p-1}\sum_{i=1}^n \mathbb{E}\varrho_X^f(\ee, d_i\ee)^p.$$ Let us note that by combining the triangle and H\"{o}lder inequalities, $b_{p,n}(\mathcal{F})\leqslant \lambda$. In the case $p=2$, these are the Lipschitz analogues of the linear quantities appearing in \cite{Hinrichs}, as well as the generalization to maps of the metric type $1$ constant as defined by Bourgain, Milman, and Wolfson.
Let us say that $\mathcal{F}$ \emph{crudely factors the Hamming cubes} provided that there exist constants $c,D>0$ such that for each $n\in\nn$, there exist $F:X\to Y\in \mathcal{F}$ and $f:2^n\to X$ and constants $a,b>0$ such that for each $\ee, \delta\in 2^n$, $$\frac{a}{D} \partial(\ee, \delta) \leqslant d_X(f(\ee), f(\delta)) \leqslant a D \partial(\ee, \delta),$$ $$\frac{b}{D}\partial (\ee, \delta) \leqslant d_Y(F\circ f(\ee), F\circ f(\delta)) \leqslant bD \partial (\ee, \delta),$$ and $b\geqslant ac$. An important feature of this definition is that the scaling factors $a,b$ be uniformly equivalent (that is, $ac\leqslant b\leqslant a\lambda D^2$). Let us say that $\mathcal{F}$ \emph{factors the Hamming cubes} provided that there exists a constant $c>0$ such that for each $D>1$ and each $n\in\nn$, there exist $F:X\to Y\in \mathcal{F}$ and $f:2^n\to X$ and constants $a,b$ such that for each $\ee, \delta\in 2^n$, $$\frac{a}{D} \partial(\ee, \delta) \leqslant d_X(f(\ee), f(\delta)) \leqslant a D \partial(\ee, \delta),$$ $$\frac{b}{D}\partial (\ee, \delta) \leqslant d_Y(F\circ f(\ee), F\circ f(\delta)) \leqslant bD \partial (\ee, \delta),$$ and $b\geqslant ac$.
We now present the main theorem.
\begin{theorem*} The following are equivalent: \begin{enumerate}[(i)]\item $\mathcal{F}$ factors the Hamming cubes. \item $\mathcal{F}$ crudely factors the Hamming cubes. \item $\lim\sup_n a_n(\mathcal{F})>0$. \item For each $1<p<\infty$, $\lim\sup_n b_{p,n}(\mathcal{F})>0$. \item For some $1<p<\infty$, $\lim\sup_n b_{p,n}(\mathcal{F})>0$. \end{enumerate}
\label{main theorem}
\end{theorem*}
It is obvious that $(i)\Rightarrow (ii)\Rightarrow (iii)\Rightarrow (iv)\Rightarrow (v)$. To see why $(iii)\Rightarrow (iv)$, note that for $F:X\to Y\in \mathcal{F}$ and $f:2^n\to X$, since $\varrho_X^f(\ee, d_i\ee)\leqslant \text{Lip}(f)/n$ for each $\ee\in 2^n$ and $1\leqslant i\leqslant n$, $$\mathbb{E}\varrho_Y^f(\ee, -\ee) \leqslant \Bigl[\mathbb{E}\varrho_Y^f(\ee, -\ee)^p\Bigr]^{1/p}\leqslant b_{p,n}(\mathcal{F})\Bigl[n^{p-1}\sum_{i=1}^n \varrho_X^f(\ee, d_i\ee)^p\Bigr]^{1/p} \leqslant b_{p,n}(\mathcal{F})\text{Lip}(f),$$ so $a_n(\mathcal{F})\leqslant b_{p,n}(\mathcal{F})$. Thus the main part of this work is concerned with proving the implication $(v)\Rightarrow (i)$.
We note that if $l/2\leqslant k\leqslant l$, then $b_{p,k}(\mathcal{F})\leqslant 2^{1-1/p}b_{p,l}(\mathcal{F})$. Indeed, for $F:X\to Y\in \mathcal{F}$ and $f:2^k\to X$, we can extend $f:2^k\to X$ to a function $g:2^l\to X$ by $g(\ee)=f(\ee(1) ,\ldots, \ee(k))$. Then $$\mathbb{E}_{2^k}\varrho^f_Y(\ee, -\ee)^p=\mathbb{E}_{2^l}\varrho_Y^g(\ee, -\ee)^p\leqslant b_{p,l}(\mathcal{F})^pl^{p-1}\sum_{i=1}^l \mathbb{E}_{2^l}\varrho_X^g(\ee, d_i\ee)^p\leqslant 2^{p-1}b_{p,l}(\mathcal{F})^pk^{p-1}\sum_{i=1}^k \mathbb{E}_{2^k}\varrho_X^f(\ee, d_i\ee)^p.$$
Applying this to $k\in\nn$ and $l=2^{\lceil \log_2(k)\rceil}$, we deduce that $\lim\sup_n b_{p,n}(\mathcal{F})>0$ if and only if $\lim\sup_n b_{p,2^n}(\mathcal{F})>0$. Thus our goal, completed in the fourth section of this work, will be to show that if for some $1<p<\infty$, $\lim\sup_n b_{p,2^n}(\mathcal{F})>0$, then $\mathcal{F}$ factors the Hamming cubes. In the fifth section, we use concentration of measure to provide a quantitatively sharp proof that $(iii)\Rightarrow (i)$.
We note that the definition of our quantities $b_{p,n}(\mathcal{F})$ are reminiscent of metric type as defined by Bourgain, Milman, and Wolfson in \cite{BMW}. One may also ask about Enflo's \cite{Enflo} definition of non-linear type. In the subtype regime, however, the two notions coincide. We give the details of this in the next section.
The author wishes to thank B. Randrianantoanina for making him aware of the coarse differentiation method of Eskin, Fisher, and Whyte.
\section{Spatial versus operator results; Subtype}
We first recall a result implicitly shown in \cite{BMW} in the particular case $p=2$. The general case $1<p<\infty$ follows by substituting their Fact $2.5$ with our Lemma \ref{flat}.
\begin{theorem}\cite[Theorem $2.6$]{BMW} For $1<p<\infty$, $l\in\nn$, and $D>1$, there exists a constant $0<a<1$ such that if $(Z, d_Z)$ is a metric space and $h:2^l\to Z$ is a function such that $$\mathbb{E}d_Z(h(\ee), h(-\ee))^p > (1-a) l^{p-1}\sum_{i=1}^l \mathbb{E}d_Z(h(\ee), h(d_i\ee))^p$$ and if $$t^p=\frac{1}{l}\sum_{i=1}^l \mathbb{E} d_Z(h(\ee), h(d_i\ee))^p,$$ then for any $\ee_1, \ee_2\in 2^l$, $$\frac{1}{D}\partial (\ee_1, \ee_2)\leqslant \frac{d_Z(h(\ee_1), h(\ee_2))}{t}\leqslant D\partial (\ee_1, \ee_2).$$
\label{BMW}
\end{theorem}
With the preceding remarkable result, in the case that $\mathcal{F}$ is a collection of identity operators (that is, in the spatial case), it is easy to complete the main theorem. This is because the function $n\mapsto b_{p,n}(\mathcal{F})$ is submultiplicative in the spatial case. From this it follows that either $b_{p,n}(\mathcal{F})=1$ (worst possible value) for all $n\in\nn$, in which case we immediately finish by Theorem \ref{BMW}, or $b_{p,n}(\mathcal{F})\underset{n}{\to}0$. But this method does not apply to the map case because of the lack of submultiplicativity of $n\mapsto b_{p,n}(\mathcal{F})$ in the non-spatial case.
More generally, one is often interested in a sequence of composition submultiplicative seminorms $(T_n)_{n=1}^\infty$ defined on the class of bounded, linear operators between Banach spaces (such as Rademacher or gaussian type $p$ \cite{Hinrichs}, Haar or marginale type $p$ \cite{Wenzel}, or asymptotic notions of Rademacher or basic type $p$ \cite{CDK}). By ``composition submultiplicative,'' we mean that for any pair of operators $A,B$ such that the composition $AB$ is defined, $T_{mn}(AB)\leqslant T_n(A)T_m(B)$ for any natural numbers $m,n$. In the case that $A=I_X$, we can apply this fact with $B=A=I_X$ to deduce that $T_{mn}(I_X)=T_{mn}(I_X^2)\leqslant T_m(I_X)T_n(I_X)$. The standard procedure in this case is to use these inequalities to prove that either $(T_n(I_X))_{n=1}^\infty$ exhibits the quantitatively worst possible behavior for each $n$ and use this to prove the presence of certain structures (such as $\ell_1^n$ subspaces in the Rademacher case), or to prove that $(T_n(I_X))_{n=1}^\infty$ is growing/shrinking rapidly enough to ensure some non-trivial power type behavior. This ``automatic power type'' phenomenon fails for all examples in the non-spatial case. One example, which is relevant to the subject of this work, is the diagonal operator $F:\ell_1\to \ell_1$ given by $F\sum_{n=1}^\infty a_ne_n= \sum_{n=1}^\infty \frac{a_n}{\log (n+1)}e_n$. This is compact, and cannot factor the Hamming cubes. But one can check that $F$ has no non-trivial Rademacher type, and therefore no non-trivial non-linear type in the sense of Bourgain, Milman, and Wolfson. This is because for $1<p<\infty$, $$\Bigl(\mathbb{E}\|F\sum_{i=1}^n \ee(i)e_i\|^p\Big)^{1/p}\geqslant \frac{n}{\log(n+1)}$$ and $$\Bigl(\sum_{i=1}^n \|e_i\|^p\Bigr)^{1/p}= n^{1/p}=o(n/\log(n+1)).$$ More generally, we can choose any $1<p<\infty$ and a sequence $(w_n)_{n=1}^\infty$ of positive numbers vanishing as slowly as we like and define the diagonal operator $F:\ell_1\to \ell_1$ by $F\sum_{n=1}^\infty a_ne_n=\sum_{n=1}^\infty a_nw_ne_n$. Then $b_{p,n}(F)$ is necessarily vanishing, but as slowly as we like. Examples such as this motivate the search for a characterization of when the worst possible behavior does not hold (in our case, worst possible behavior means factoring the Hamming cube, while in other cases it is crude finite representability/asymptotic crude finite representability of the identity operator of $\ell_1^n$, non-super weak compactness, or non-asymptotic uniform smoothability). A technique in this case is to define a sequence of \emph{subtype} constants of the map (or family of maps). One instance of this approach is due to Beauzamy \cite{Beauzamy}, who gave a characterization of when the identity on $\ell_1$ is crudely finitely representable in a linear operator using a sequence of constants which are the linear analogues of our $a_n(\mathcal{F})$. Hinrichs proved a similar result using constants which are the linear analogues of our $b_{2,n}(\mathcal{F})$. In \cite{CDK}, asymptotic analogues of the results of Beauzamy and Hinrichs were proven for both the asymptotic linear analogues of the $a_n(\mathcal{F})$ and $b_{p,n}(\mathcal{F})$ constants.
The general approach to subtype problems is as follows: Suppose we have a sequence $(T_n)_{n=1}^\infty$ as in the previous paragraph and positive numbers $(c_n)_{n=1}^\infty$ such that for each $\lambda>0$, $\lambda c_n$ is the supremum of $T_n(A)$ as $A$ ranges over all bounded, linear operators with $\|A\|\leqslant \lambda$. Then one may ask if, for a given class $\mathcal{A}$ of operators with norms not more than $\lambda$, does $(\sup_{A\in \mathcal{A}} T_n(A))_{n=1}^\infty$ exhibit the essentially worst possible behavior with respect to the sequence $(c_n)_{n=1}^\infty$ (that is, $\lim\sup_n \sup_{A\in \mathcal{A}} T_n(A)/c_n>0$)? We then say $A$ has \emph{subtype} if it does not exhibit the worst possible behavior (that is, $\lim_n \sup_{A\in \mathcal{A}} T_n(A)/c_n=0$). This has been applied when $T_n$ is the Rademacher/gaussian/Haar/martingale type $p$ norms. More generally, we may isolate non-linear subtype properties by replacing continuous, linear operators with Lipschitz functions, replacing the $(T_n)_{n=1}^\infty$ sequence with a sequence $(\tau_n)_{n=1}^\infty$ defined on the class of Lipschitz maps between metric spaces, and by replacing operator norm with Lipschitz constant. One then says that a class $\mathcal{F}$ has subtype if $\lim_n \sup_{F\in \mathcal{F}}\tau_n(F)/c_n=0$. This is the approach we take.
Now for a family $\mathcal{F}$ of $\lambda$-Lipschitz maps, let us define $e_{p,n}(\mathcal{F})$ to be the smallest constant $t>0$ such that for any $F:X\to Y\in \mathcal{F}$ and $f:2^n\to X$, $$\mathbb{E}\varrho^f_Y(\ee, -\ee)^p \leqslant t^p \sum_{i=1}^n \mathbb{E} \varrho_X^f(\ee, d_i\ee)^p.$$ Note that $e_{p,n}(F)\leqslant \lambda n^{1-1/p}$. Therefore with $c_n=n^{1-1/p}$, we can say $\mathcal{F}$ has \emph{Enflo subtype} if $\lim_n e_{p,n}(\mathcal{F})/c_n=0$. But $e_{p,n}(\mathcal{F})/n^{1-1/p}=b_{p,n}(\mathcal{F})$. Therefore the subtype approach applied to Enflo type recovers the same condition as the Bourgain, Milman, Wolfson approach.
\section{Rigidity results}
\begin{lemma} For $1<p<\infty$, $n\in\nn$, and $\Phi>1$, there exists $\phi=\phi(p,n,\Phi)\in (0,1)$ such that if $a=(a_i)_{i=1}^n\in \ell_p^n$ satisfies $\|a\|_{\ell_1^n}^p>\phi n^{p-1}\|a\|_{\ell_p^n}^p$, then $\max_i |a_i|\leqslant \Phi \min_i |a_i|$.
\label{flat}
\end{lemma}
\begin{proof} By the uniform convexity of $\ell_p^n$, there exists $0<\delta<1$ such that if $x,y\in B_{\ell_p^n}$ are such that $\|x+y\|_{\ell_p^n}>2(1-\delta)$, then $\|x-y\|_{\ell_p^n}<\frac{1}{2n^{1/p}}\cdot \frac{\Phi-1}{\Phi}$. Now let $\phi=(1-\delta)^p$. Suppose $a=(a_i)_{i=1}^n\in \ell_p^n$ satisfies $\|a\|_{\ell_1^n}^p>\phi n^{p-1} \|a\|_{\ell_p^n}^p$. Without loss of generality, let us assume that $a_i=|a_i|$ for all $1\leqslant i\leqslant n$. Let $x_i=a_i/\|a\|_{\ell_p^n}$ and $x=(x_i)_{i=1}^n\in S_{\ell_p^n}$. Let $y_i=n^{-1/p}$ for $1\leqslant i\leqslant n$ and $y=(y_i)_{i=1}^n\in S_{\ell_p^n}$. Note that $\|x\|_{\ell_1^n}>\phi^{1/p} n^{1-1/p}=(1-\delta)n^{1-1/p}$ and $\|y\|_{\ell_1^n}=n^{1-1/p}$. Then $$\|x+y\|_{\ell_p^n} \geqslant \|x+y\|_{\ell_1^n}/n^{1-1/p} = \frac{\|x\|_{\ell_1^n}+\|y\|_{\ell_1^n}}{n^{1-1/p}}\geqslant (1-\delta)+1>2(1-\delta).$$ Therefore $\|x-y\|_{\ell_p^n}\leqslant \frac{1}{2n^{1/p}}\cdot\frac{\Phi-1}{\Phi}$. Since $\max_i x_i\geqslant 1/n^{1/p}$, we deduce that $$\max_i x_i-\min_i x_i \leqslant |n^{-1/p}-\max_i x_i|+|n^{-1/p}-\min_i x_i| \leqslant 2\|x-y\|_{\ell_\infty^n}\leqslant \frac{1}{n^{1/p}}\cdot \frac{\Phi-1}{\Phi} \leqslant \Bigl(\frac{\Phi-1}{\Phi}\Bigr)\max_i x_i.$$ Rearranging yields that $$\max_i x_i\leqslant \Phi \min_i x_i,$$ and we deduce the result by homogeneity.
\end{proof}
\begin{lemma} Fix $1<p<\infty$. Let $\Omega$ be a probability space and let $D_Y, D_X, E_Y, E_X:\Omega\to \rr$, $\lambda, \Theta>0$, $a,b,\nu, \mu\in (0,1)$ be such that \begin{enumerate}[(i)]\item $D_Y, D_X, E_Y, E_X$ are non-negative, measurable functions on $\Omega$ such that $D_Y\leqslant E_Y$, $D_X\leqslant E_X$, $D_Y\leqslant \lambda^p D_X$, $E_Y\leqslant \lambda^p E_X$, $D_Y\leqslant (1+\nu)\Theta^p E_X$, \item $\mathbb{E}D_Y>(1-\nu)\Theta^p \mathbb{E}E_X$, $\mathbb{E}D_Y>(1-\mu) \mathbb{E}E_Y$, $\mathbb{E}D_X>(1-\mu)\mathbb{E}E_X$, \item $\lambda^p\Bigl(\frac{2\mu}{a}+\frac{2\nu}{b}\Bigr)<\Theta^p(1-\nu)$. \end{enumerate}
Then there exists $\ee\in \Omega$ such that $D_Y(\ee)>(1-a)E_Y(\ee)$, $D_X(\ee)>(1-a)E_Y(\ee)$, and $D_Y(\ee)>(1-b)\Theta^p E_X(\ee)$.
\label{expect}
\end{lemma}
\begin{proof} Let $A_Y=(D_Y\leqslant (1-a)E_Y),$ $A_X=(D_X\leqslant (1-a)E_X),$ and $B=(D_Y\leqslant (1-b)\Theta^p E_X).$ Then the conclusion of the lemma is equivalent to $A^c_Y\cap A_X^c\cap B^c\neq \varnothing$. We work by contradiction. Assume $A^c_Y\cap A^c_X\cap B^c=\varnothing$, so $\Omega=A_Y\cup A_X\cup B$. Let us first note that $$ (1-\mu)\mathbb{E}E_Y < \mathbb{E}D_Y = \mathbb{E}1_{A_Y}D_Y + \mathbb{E}1_{A_Y^c}D_Y \leqslant (1-a)\mathbb{E}1_{A_Y}E_Y + \mathbb{E}1_{A_Y^c}E_Y = \mathbb{E}E_Y- a \mathbb{E}1_{A_Y}E_Y.$$ From this it follows that $$ \mathbb{E}1_{A_Y}E_Y \leqslant \frac{\mu}{a}\mathbb{E}E_Y. $$ By replacing each $Y$ with $X$, we deduce that $$ \mathbb{E}1_{A_X}E_X \leqslant \frac{\mu}{a}\mathbb{E}E_X. $$
Also, \begin{align*} \Theta^p (1-\nu)\mathbb{E}E_X & < \mathbb{E}D_Y = \mathbb{E}1_BD_Y +\mathbb{E}1_{B^c}D_Y \leqslant \Theta^p(1-b)\mathbb{E}1_B E_X + \Theta^p(1+\nu)\mathbb{E}1_{B^c}E_X. \end{align*} Dividing by $\Theta^p$ and rearranging yields that $$\mathbb{E}1_B E_X \leqslant \frac{2\nu}{b}\mathbb{E}E_X.$$
Recalling that $A_Y\cup A_X\cup B=\Omega$, we deduce that \begin{align*} \Theta^p(1-\nu) \mathbb{E}E_X & < \mathbb{E}D_Y \leqslant \mathbb{E}1_{A_Y}D_Y+\mathbb{E}1_{A_X}D_Y+ \mathbb{E}1_BD_Y \\ & \leqslant \mathbb{E}1_{A_Y}E_Y + \lambda^p \mathbb{E}1_{A_X}E_X +\lambda^p \mathbb{E}1_BE_X \\ & \leqslant \frac{\mu}{a}\mathbb{E}E_Y + \frac{\mu\lambda^p}{a} \mathbb{E}E_X +\frac{2\nu\lambda^p}{b} \mathbb{E}E_X \\ & \leqslant \lambda^p\Bigl(\frac{2\mu}{a}+\frac{2\nu}{b}\Bigr)\mathbb{E}E_X. \end{align*}
Since $\mathbb{E}E_X>0$, this contradicts $(iii)$ and finishes the proof.
\end{proof}
For a natural number $n$, we let $[n]=\{1, \ldots, n\}$ denote the integer interval. Fix natural numbers $ l_1, \ldots, l_d$ and let $L=\prod_{j=1}^d l_j$. We define $T=\cup_{i=0}^d \Lambda_i$ as follows. We let $\Lambda_0=\{[L]\}$ consist of a single integer interval. Now suppose that for $i<d$, $\Lambda_i$ has been defined and consists of pairwise disjoint subintervals of $[L]$ each of which has cardinality $\prod_{j=i+1}^d l_j$. For each $I\in \Lambda_i$, let $\mathcal{I}_I=\{J^I_1, \ldots, J^I_{l_{i+1}}\}$ be a partition of $I$ into subintervals of equal cardinalilty (and therefore of cardinality $\prod_{j=i+2}^d l_j$). Now let $\Lambda_{i+1}=\cup_{I\in \Lambda_i} \mathcal{I}_I$. This completes the recursive definition of $\Lambda_0, \ldots, \Lambda_d$. Now let $T=\cup_{i=0}^d \Lambda_i$. We refer to $T$ as the $(l_1, \ldots, l_d)$ \emph{interval tree}. For $0<j\leqslant d$ and $J\in \Lambda_j$, let $J^-$ be the member $I$ of $\Lambda_{j-1}$ such that $J\subset I$. That is, $J^-$ is the interval $I\in \Lambda_j$ such that $J\in \mathcal{I}_I$.
\begin{rem}\upshape Suppose $l_1, \ldots, l_{d+1}$ are natural numbers and $T$ is the $(l_1, \ldots, l_{d+1})$ interval tree. Suppose that $(t_I)_{I\in T}$ is a collection of non-negative numbers such that for each $0\leqslant j\leqslant d$ and $I\in \Lambda_j$, $t_I\leqslant \sum_{J^-=I} t_J$. Then using this fact repeatedly yields that for any $0\leqslant i<j\leqslant d+1$ and $I\in \Lambda_i$, $$t_I\leqslant \sum_{I\supset J\in \Lambda_j} t_J.$$
Also, by H\"{o}lder's inequality, it follows that for any such $j$ and $I$, $$t_I^p\leqslant l_{j+1}^{p-1}\sum_{J^-=I} t^p_J,$$ and more generally, $$t^p_I \leqslant \Bigl(\prod_{m=i+1}^j l_m^{p-1}\Bigr)\sum_{I\supset J\in \Lambda_j} t^p_J$$ for any $0\leqslant i<j\leqslant d+1$ and $I\in \Lambda_i$. We will use this fact frequently in this section.
\label{vase}
\end{rem}
\begin{lemma} Fix $1<p<\infty$. Fix natural numbers $l_1, \ldots, l_d$, $0<\mu<1$, $\lambda, \Theta>0$, and $M>\lambda/\Theta$. Then for any $0<\eta_1<1$, there exists $0<\eta<\eta_1$ with the following property: Suppose $l_{d+1}$ is a natural number, $T$ is the $(l_1, \ldots, l_{d+1})$ interval tree, and $(r_I)_{I\in T}, (s_I)_{I\in T}$ are non-negative numbers such that \begin{enumerate}[(i)]\item for each $I\in T$, $r_I\leqslant \lambda s_I$, \item for each $I\in T\setminus \Lambda_{d+1}$, $r_I\leqslant \sum_{J^-=I}r_J$ and $s_I\leqslant \sum_{J^-=I}s_J$, \item for each $I\in \Lambda_d$, $r_I^p\leqslant (1+\eta)\Theta^p l_{d+1}^{p-1} \sum_{J^-=I}s^p_J$, \item $r_{[L]}>(1-\eta)\Theta^p\Bigl(\prod_{i=1}^{d+1} l_i^{p-1}\Bigr) \sum_{I\in \Lambda_{d+1}}s_I^p$. \end{enumerate} Then for any $0\leqslant j\leqslant d$, $0\leqslant i<d$, and $I_1\in \Lambda_i$, $\max_{I\in \Lambda_j} s_I\leqslant M\min_{I\in \Lambda_j} s_I$ and $r_{I_1}^p>(1-\mu)l_{j+1}^{p-1}\sum_{J^-=I_1} r_J^p$.
\label{eta}
\end{lemma}
\begin{proof} First fix $ \Phi>1$ such that $M>\Phi^3 \lambda/\Theta$. Now let $0<\phi<1$ be such that for any $1\leqslant n\leqslant \prod_{i=1}^d l_i$ and any $v=(v_i)_{i=1}^n\in \ell_p^n$ with $\max_i |v_i|>\Phi\min_i |v_i|$, $\|v\|_{\ell_1^n}^p<\phi n^{p-1}\|v\|_{\ell_p^n}^p$. Such a $\phi$ exists by Lemma \ref{flat}. Now fix $0<\eta<\eta_1$ so small that \begin{enumerate}[(a)]\item $\phi(1+\eta)<1-\eta$, \item $\frac{\Phi^p}{1+\eta}-\frac{1}{1-\eta}>\Bigl(\frac{1}{1-\eta}-\frac{1}{1+\eta}\Bigr)\Bigl(\prod_{i=1}^d l_i\Bigr)\Phi^p$, \item $M> \frac{ \Phi^3 \lambda}{(1-\eta)^{1/p}\Theta}$, \item $\Bigl(1-\frac{\mu}{\Phi^p \prod_{i=1}^d l_i}\Bigr)(1+\eta)<(1-\eta)$. \end{enumerate}
Now suppose that $l_{d+1}$, $(r_I)_{I\in T}$, $(s_I)_{I\in T}$ are as in the lemma.
Step $1$: For any $0\leqslant j\leqslant d$, $\max_I r_I\leqslant \Phi \min_I r_I$.
If it were not so, then by the choice of $\phi$ applied to the vector $(r_I)_{I\in \Lambda_j}\in \ell_p(\Lambda_j)$, $$\Bigl(\sum_{I\in \Lambda_j} r_I\Bigr)^p \leqslant \phi\Bigl(\prod_{i=1}^j l_i^{p-1}\Bigr)\sum_{I\in \Lambda_j}r_I^p.$$ Then \begin{align*} r_{[L]}^p & \leqslant \Bigl(\sum_{I\in \Lambda_j} r_I\Bigr)^p < \phi\Bigl(\prod_{i=1}^j l_i^{p-1}\Bigr)\sum_{I\in \Lambda_j}r_I^p \leqslant \phi\Bigl(\prod_{i=1}^d l_i^{p-1}\Bigr)\sum_{I\in \Lambda_d} r_I^p \\ & \leqslant \phi(1+\eta)\Theta^p\Bigl(\prod_{i=1}^{d+1} l_i^{p-1}\Bigr)\sum_{I\in \Lambda_{d+1}}s_I^p.\end{align*} But since $r_{[L]}^p>(1-\eta)\Theta^p (\prod_{i=1}^{d+1}l_i^{p-1})\sum_{I\in \Lambda_{d+1}}s_I^p$, we contradict item $(a)$ of our choice of $\eta$. This completes Step $1$. Note that this implies that for each $0\leqslant j\leqslant d$ and $I\in \Lambda_j$, $r_I>0$.
Step $2$: For any $0\leqslant j\leqslant d$, $$\max_{I\in \Lambda_j} \sum_{I\supset J\in \Lambda_{d+1}}s^p_J \leqslant \Phi^{2p} \min_{I\in \Lambda_j} \sum_{I\supset J\in \Lambda_{d+1}}s^p_J.$$
If it were not so, we could find $0\leqslant j\leqslant d$ and $I_1, I_2\in \Lambda_j$ such that $$\frac{1}{ \Phi^{2p}}\sum_{I_1\supset J\in \Lambda_{d+1}}s^p_J>\sum_{I_2\supset J\in \Lambda_{d+1}}s^p_J.$$ Then by Step $1$, \begin{align*} r_{I_1}^p & \leqslant \Phi^p r_{I_2}^p \leqslant \Phi^p \Bigl(\prod_{i=j+1}^d l_i^{p-1}\Bigr)\sum_{I_2\supset J\in \Lambda_d} r^p_J \leqslant \Phi^p(1+\eta)\Theta^p \Bigl(\prod_{i=j+1}^{d+1} l_i^{p-1}\Bigr)\sum_{I_2\supset J\in \Lambda_{d+1}}s^p_J \\ & \leqslant \frac{(1+\eta)\Theta^p}{\Phi^p}\Bigl(\prod_{i=j+1}^{d+1} l_i^{p-1}\Bigr)\sum_{I_1\supset J\in \Lambda_{d+1}}s_J^p. \end{align*} Note that for any $I\in \Lambda_j$, $$r_I^p\leqslant \Bigl(\prod_{i=j+1}^d l_i^{p-1}\Bigr)\sum_{I\supset J\in \Lambda_d} r_J^p \leqslant (1+\eta)\Theta^p\Bigl(\prod_{i=j+1}^{d+1} l_i^{p-1}\Bigr)\sum_{I\in \Lambda_{d+1}} s_I^p.$$ Since $$(1-\eta)\Theta^p\Bigl(\prod_{i=1}^{d+1} l_i^{p-1}\Bigr)\sum_{I\in \Lambda_{d+1}}s_I^p <r^p_{[L]} \leqslant \Bigl(\prod_{i=1}^j l_i^{p-1}\Bigr)\sum_{I\in \Lambda_j} r_I^p,$$ we see that \begin{align*} \frac{1}{(1+\eta)\Theta^p}\Bigl[ \Phi^p r_{I_1}^p +\sum_{I_1\neq I\in \Lambda_j}r_I^p\Bigr] \leqslant \Bigl(\prod_{i=j+1}^{d+1} l_i^{p-1}\Bigr)\sum_{I\in \Lambda_{d+1}}s^p_I \leqslant \frac{1}{(1-\eta)\Theta^p}\sum_{I\in \Lambda_j}r^p_I. \end{align*} Manipulating the first and last terms of this inequality, we deduce that \begin{align*} \Bigl(\frac{\Phi^p}{1+\eta}-\frac{1}{1-\eta}\Bigr)r_{I_1}^p & \leqslant \Bigl(\frac{1}{1-\eta}-\frac{1}{1+\eta}\Bigr)\sum_{I_1\neq I\in \Lambda_j} r_I^p \leqslant \Bigl(\frac{1}{1-\eta}-\frac{1}{1+\eta}\Bigr)\Phi^p |\Lambda_j|r_{I_1}^p \\ & \leqslant \Bigl(\frac{1}{1-\eta}-\frac{1}{1+\eta}\Bigr)\Phi^p \Bigl(\prod_{i=1}^d l_i\Bigr)r_{I_1}^p. \end{align*} Since $r_{I_1}>0$, we reach a contradiction of $(b)$ of our choice of $\eta$.
Step $3$: For any $0\leqslant j\leqslant d$, $\max_{I\in \Lambda_j} s_I\leqslant M \min_{I\in \Lambda_j} s_I$. Fix such a $j$ and let $$R=\max_{I\in \Lambda_j} r_I \hspace{15mm} S=\max_{I\in \Lambda_j} s_I \hspace{15mm} S_1=\max_{I\in \Lambda_j}\Bigl(\prod_{i=j+1}^{d+1}l_i^{p-1}\Bigr)\sum_{I\supset J\in \Lambda_{d+1}}s^p_J$$ and $$r=\min_{I\in \Lambda_j} r_I \hspace{15mm} s=\min_{I\in \Lambda_j} s_I \hspace{15mm} s_1=\min_{I\in \Lambda_j}\Bigl(\prod_{i=j+1}^{d+1}l_i^{p-1}\Bigr)\sum_{I\supset J\in \Lambda_{d+1}}s^p_J. $$ We know from Step $1$ that $R\leqslant \Phi r$. We know from Step $2$ that $S_1\leqslant \Phi^{2p} s_1$. We know from hypothesis that $R\leqslant \lambda S$, and we know from Remark \ref{vase} that $S^p\leqslant S_1$. Moreover, \begin{align*} \Bigl(\prod_{i=1}^j l_i^{p-1}\Bigr)s_1|\Lambda_j| & \leqslant \Bigl(\prod_{i=1}^{d+1}l_i^{p-1}\Bigr)\sum_{I\in \Lambda_j}\sum_{I\supset J\in \Lambda_{d+1}}s^p_J =\Bigl(\prod_{i=1}^{d+1}l_i^{p-1}\Bigr)\sum_{I\in \Lambda_{d+1}}s_I^p < [(1-\eta)\Theta^p]^{-1} r_{[L]}^p \\ & \leqslant [(1-\eta)\Theta^p]^{-1} \Bigl(\prod_{i=1}^j l_i^{p-1}\Bigr) \sum_{I\in \Lambda_j} r_I^p \leqslant [(1-\eta)\Theta^p]^{-1} \Bigl(\prod_{i=1}^j l_i^{p-1}\Bigr)R^p |\Lambda_j| \\ & \leqslant [(1-\eta)\Theta^p]^{-1}\Phi^p \Bigl(\prod_{i=1}^j l_i^{p-1}\Bigr)r^p |\Lambda_j| \leqslant [(1-\eta)\Theta^p]^{-1}\Phi^p \lambda^p \Bigl(\prod_{i=1}^j l_i^{p-1}\Bigr)s^p |\Lambda_j|.\end{align*} Therefore $ s_1\leqslant [(1-\eta)\Theta^p]^{-1}\Phi^p \lambda^p s^p$. Also, $S^p\leqslant S_1\leqslant \Phi^{2p} s_1,$ so $$S^p \leqslant \Phi^{2p} s_1 \leqslant [(1-\eta)\Theta^p]^{-1} \Phi^{3p} \lambda^p s^p.$$ Taking $p^{th}$ roots and appealing to $(c)$ finishes Step $3$.
Step $4$: For any $0\leqslant j<d$ and $I\in \Lambda_j$, $r^p_I>(1-\mu)l_{j+1}^{p-1}\sum_{J^-=I}r^p_J$. If it were not so, then for some $I_1\in \Lambda_j$, $r^p_I\leqslant (1-\mu)l_{j+1}^{p-1}\sum_{J^-=I_1}r_J^p$. Let us note that $$\sum_{J^-=I_1} r^p_J \geqslant \frac{1}{\Phi^p |\Lambda_{j+1}|}\sum_{J\in \Lambda_{j+1}}r^p_J \geqslant \frac{1}{\Phi^p \prod_{i=1}^d l_i}\sum_{J\in \Lambda_{j+1}}r^p_I,$$ so \begin{align*} \sum_{I\in \Lambda_j} r_I^p & \leqslant (1-\mu)l_{j+1}^{p-1}\sum_{J^-=I_1}r^p_J + l_{j+1}^{p-1}\sum_{I_1\neq I\in \Lambda_j}\sum_{ J^-=I}r^p_J \leqslant l_{j+1}^{p-1}\sum_{J\in \Lambda_{j+1}}r^p_J-l_{j+1}^{p-1}\mu \sum_{J^-=I_1}r_J^p \\ & \leqslant \Bigl(1-\frac{\mu}{\Phi^p\prod_{i=1}^d l_i}\Bigr)l_{j+1}^{p-1}\sum_{J\in \Lambda_{j+1}}r^p_J. \end{align*} Then \begin{align*} r_{[L]}^p & \leqslant \Bigl(\prod_{i=1}^j r_i^{p-1}\Bigr)\sum_{i\in \Lambda_j}r_I^p \leqslant \Bigl(1-\frac{\mu}{\Phi^p\prod_{i=1}^d l_i}\Bigr)\Bigl(\prod_{i=1}^{j+1}l_i^{p-1}\Bigr)\sum_{I\in \Lambda_{j+1}}r_I^p \\ & \leqslant \Bigl(1-\frac{\mu}{\Phi^p\prod_{i=1}^d l_i}\Bigr)\Bigl(\prod_{i=1}^d l_i^{p-1}\Bigr)\sum_{I\in \Lambda_d}r_I^p \\ & \leqslant \Bigl(1-\frac{\mu}{\Phi^p\prod_{i=1}^d l_i}\Bigr)(1+\eta)\Theta^p \Bigl(\prod_{i=1}^{d+1}l_i^{p-1}\Bigr)\sum_{I\in \Lambda_{d+1}}s_I^p.\end{align*} But since $r_{[L]}^p> (1-\eta)\Theta^p \Bigl(\prod_{i=1}^{d+1}l_i^{p-1}\Bigr)\sum_{I\in \Lambda_{d+1}}s^p_I$, this contradicts $(d)$ and finishes the proof.
\end{proof}
The following result is similar in spirit to the coarse differentiation result of Eskin, Fisher, and Whyte \cite{EFW}.
\begin{lemma} Fix $1< p<\infty$. Fix natural numbers $(l_1, \ldots, l_{d+1})$ and let $T$ be the $(l_1, \ldots, l_{d+1})$ interval tree. Suppose $0<\Delta,\mu<1$, $M>1$, $m\in\nn$, $\lambda>0$, and $(s_I)_{I\in T}\subset (0,\infty)$ are such that \begin{enumerate}[(i)]\item for each $I\in T\setminus \Lambda_{d+1}$, $s_I\leqslant \sum_{J^-=I}s_J$, \item for all $0\leqslant j\leqslant d$, $\max_{I\in \Lambda_j}s_I\leqslant M\min_{I\in \Lambda_j}s_I$, \item $s_{[L]}^p>(1-\nu/2)\frac{\Theta^p}{\lambda^p}\Bigl(\prod_{i=1}^{d+1}l_i^{p-1}\Bigr)\sum_{I\in \Lambda_{d+1}}s_I^p$, \item $(1-\mu\Delta/M^p)^m<(1-\nu/2)\frac{\Theta^p}{\lambda^p}$. \end{enumerate}
For each $0\leqslant j<d$, let $I_j=\{I\in \Lambda_j: s^p_I \leqslant (1-\mu)l_{j+1}^{p-1}\sum_{J^-=I} s^p_J\}$ and let $B=\{j<d: |I_j|\geqslant \Delta |\Lambda_j|\}$. Then $|B| \leqslant m$.
\label{goodX}
\end{lemma}
\begin{proof} First suppose $j\in B$. Let $s=\min_{I\in \Lambda_{j+1}} s_I$ and $S=\max_{I\in \Lambda_{j+1}}s_I\leqslant sM$. Let $A=\{J\in \Lambda_{j+1}:J^-\in I_j\}$ and note that $|A|/|\Lambda_{j+1}|=|I_j|/|\Lambda_j| \geqslant \Delta$. Then \begin{align*} \sum_{I\in \Lambda_j} s_I^p & = \sum_{I\in I_j}s^p_I+\sum_{I\in \Lambda_j\setminus I_j} s^p_I \leqslant (1-\mu)l_{j+1}^{p-1}\sum_{I\in I_j}\sum_{J^-=I}s_J^p + l_{j+1}^{p-1}\sum_{I\in \Lambda_j\setminus I_j}\sum_{J^-=I}s^p_J \\ & = l_{j+1}^{p-1}\sum_{I\in \Lambda_{j+1}}s_I^p -\mu l_{j+1}^{p-1} \sum_{I\in A} s^p_I \leqslant l_{j+1}^{p-1}\sum_{I\in \Lambda_{j+1}}s^p_I - \mu l_{j+1}^{p-1} s^p|A| \leqslant l_{j+1}^{p-1}\sum_{I\in \Lambda_{j+1}}s^p_I -l_{j+1}^{p-1} \frac{\mu S^p}{M^p}|A| \\ & \leqslant l_{j+1}^{p-1}\sum_{I\in \Lambda_{j+1}}s^p_I - \frac{\mu S^p}{M^p}l_{j+1}^{p-1} \Delta |\Lambda_{j+1}| \leqslant l_{j+1}^{p-1}\sum_{I\in \Lambda_{j+1}}s^p_I - \frac{\mu \Delta}{M^p}l_{j+1}^{p-1}\sum_{I\in \Lambda_{j+1}}s^p_I \\ & = (1-\mu\Delta/M^p)l_{j+1}^{p-1}\sum_{I\in \Lambda_{j+1}}s_I^p. \end{align*}
Now suppose that $|B|>m$ and fix $0\leqslant j_0<\ldots <j_m$, $j_i\in B$ for each $0\leqslant i\leqslant m$. Then \begin{align*} s^p_{[L]} & \leqslant\Bigl(\prod_{i=1}^{j_0} l_i^{p-1}\Bigr) \sum_{I\in \Lambda_{j_0}}s^p_I \leqslant \Bigl(1-\frac{\mu\Delta}{M^p}\Bigr)\Bigl(\prod_{i=1}^{j_0+1} l_i^{p-1}\Bigr) \sum_{I\in \Lambda_{j_0+1}}s^p_I \\ & \leqslant \Bigl(1-\frac{\mu\Delta}{M^p}\Bigr)\Bigl(\prod_{i=1}^{j_1} l_i^{p-1}\Bigr) \sum_{I\in \Lambda_{j_1}}s^p_I \leqslant \Bigl(1-\frac{\mu\Delta}{M^p}\Bigr)^2 \Bigl(\prod_{i=1}^{j_1+1} l_i^{p-1}\Bigr) \sum_{I\in \Lambda_{j_1+1}}s^p_I\\ & \leqslant \ldots \leqslant \Bigl(1-\frac{\mu\Delta}{M^p}\Bigr)^m \Bigl(\prod_{i=1}^{j_m+1}l_i^{p-1}\Bigr)\sum_{I\in \Lambda_{j_m+1}} s_I^p \\ & \leqslant \Bigl(1-\frac{\mu\Delta}{M^p}\Bigr)^m \Bigl(\prod_{i=1}^{d+1}l_i^{p-1}\Bigr)\sum_{I\in \Lambda_{d+1}} s_I^p. \end{align*} This contradicts $(iii)$ and $(iv)$ and finishes the proof.
\end{proof}
\begin{lemma} Fix $1< p<\infty$. Fix natural numbers $(l_1, \ldots, l_{d+1})$ and let $T$ be the $(l_1, \ldots, l_{d+1})$ interval tree. Suppose $0<\Delta, \nu<1$, $\lambda>0$, $M>1$, and $(r_I)_{I\in T}, (s_I)_{I\in T}\subset [0,\infty)$ are such that \begin{enumerate}[(i)]\item for each $I\in T$, $r_I\leqslant \lambda s_I$, \item for each $I\in T\setminus \Lambda_{d+1}$, $r_I\leqslant \sum_{J^-=I}r_J$, $s_I\leqslant \sum_{J^-=I}s_J$, \item for all $0\leqslant j\leqslant d$, $\max_{I\in \Lambda_j} s_I \leqslant M\min_{I\in \Lambda_j} s_I$, \item $\Delta M^p \leqslant \nu\Theta^p/2\lambda^p$, \item $r_{[L]}^p>(1-\nu/2)\Theta^p\Bigl(\prod_{i=1}^{d+1}l_i^{p-1}\Bigr)\sum_{I\in \Lambda_{d+1}}s^p_I$. \end{enumerate}
Then for any $0\leqslant j<d$, $|\{I\in \Lambda_j: r_I^p\leqslant (1-\nu)l_{j+1}^{p-1}\Theta^p \sum_{J^-=I}s^p_I\}|<(1-\Delta)|\Lambda_j|$.
\label{YtoX}
\end{lemma}
\begin{proof} Suppose not. Fix $0\leqslant j<d$ such that $B=\{I\in \Lambda_j: r_I^p\leqslant (1-\nu)\Theta^p\sum_{J^-=I}s^p_I\}$ has cardinality at least $(1-\Delta)|\Lambda_j|$. Let $s=\min_{I\in \Lambda_{j+1}}s_I$ and $S=\max_{I\in \Lambda_{j+1}}s_I\leqslant sM$. Let $A=\{J\in \Lambda_{j+1}: J^-\in B\}$ and note that $|A|/|\Lambda_{j+1}|=|B|/|\Lambda_j|\geqslant 1-\Delta$. Now \begin{align*} \sum_{J\in \Lambda_{j+1}\setminus A}s^p_J \leqslant |\Lambda_{j+1}\setminus A|S^p\leqslant \Delta |\Lambda_{j+1}|S^p \leqslant \Delta M^p\sum_{I\in \Lambda_{j+1}}s_I^p \leqslant \frac{\nu \Theta^p}{2\lambda^p}\sum_{I\in \Lambda_{j+1}}s_I^p. \end{align*} Then \begin{align*} r_{[L]}^p & \leqslant \Bigl(\sum_{i=1}^j l_i^{p-1}\Bigr)\sum_{I\in \Lambda_j} r^p_I = \Bigl(\sum_{i=1}^j l_i^{p-1}\Bigr)\Bigl[\sum_{I\in B} r^p_I+\sum_{I\in \Lambda_j\setminus B} r^p_I\Bigr] \\ & \leqslant \Bigl(\sum_{i=1}^{j+1} l_i^{p-1}\Bigr)\Bigl[(1-\nu)\Theta^p \sum_{I\in B}\sum_{J^-=I}s^p_J+\lambda^p\sum_{I\in \Lambda_j\setminus B}\sum_{J^-=I}s^p_J\Bigr] \\ & = \Bigl(\sum_{i=1}^{j+1} l_i^{p-1}\Bigr)\Bigl[(1-\nu)\Theta^p \sum_{I\in A}s^p_I+\lambda^p\sum_{I\in \Lambda_{j+1}\setminus A}s^p_I\Bigr] \\ & \leqslant \Bigl(\prod_{i=1}^{j+1}l_i^{p-1}\Bigr)\Bigl[(1-\nu)\Theta^p\sum_{I\in \Lambda_{j+1}}s_I^p+ \frac{\nu\Theta^p}{2}\sum_{I\in \Lambda_{j+1}}s_I^p \Bigr] \\ & = (1-\nu/2)\Theta^p \Bigl(\prod_{i=1}^{j+1}l_i^{p-1}\Bigr) \sum_{I\in \Lambda_{j+1}}s_I^p \\ & \leqslant (1-\nu/2)\Theta^p \Bigl(\prod_{i=1}^{d+1}l_i^{p-1}\Bigr)\sum_{I\in \Lambda_{d+1}}s^p_I. \end{align*} This contradiction finishes the proof.
\end{proof}
\section{Proof of $(v)\Rightarrow (i)$}
Fix $1<p<\infty$. Let $\lambda=\sup_{F\in \mathcal{F}}\text{Lip}(F)\in (0,\infty)$ and let $\Theta=\lim\sup_n b_{p, 2^n}(\mathcal{F})\in (0,\lambda]$. Fix $0<\vartheta<\Theta$.
Fix $0<b<1$ such that $(1-b)^{1/p}\Theta>\vartheta$. Next fix $0<\nu<b$ such that $\nu/b<(1-\nu)\Theta^p/4\lambda^p$. Fix $n_0\in \nn$ such that for all $n\geqslant n_0$, $b_{p, 2^n}(\mathcal{F})\leqslant (1+\nu)^{1/p}\Theta$. Fix $n\geqslant n_0$ and $D>1$, and let $l=2^n$. By Theorem \ref{BMW}, there exists $0<a<1$ such that if $(Z, d_Z)$ is any metric space and $h:2^l\to Z$ satisfies $$\mathbb{E}d_Z(h(\ee), h(-\ee))^p>(1-a)l^{p-1}\sum_{i=1}^l d_Z(h(\ee), h(d_i\ee))^p$$ and if $$t^p=\frac{1}{l}\sum_{i=1}^l d_Z(h(\ee), h(d_i\ee))^p,$$ then for any $\ee_1, \ee_2\in 2^l$, $$\frac{1}{D}\partial(\ee_1, \ee_2) \leqslant \frac{d_Z(h(\ee_1), h(\ee_2))}{t}\leqslant D\partial(\ee_1, \ee_2).$$ Now fix $0<\mu<a$ such that $\mu/a<(1-\nu)\Theta^p/4\lambda^p$. Fix $M>\lambda/\Theta$. Fix $0<\Delta<1$ such that $\Delta M^p<\nu\Theta^p/2\lambda^p$. Fix $m\in\nn$ such that $$\Bigl(1-\frac{\Delta \mu}{M^p}\Bigr)^m<(1-\nu/2)\frac{\Theta^p}{\lambda^p}.$$ Fix $d>m+1$ and let $l_1=\ldots =l_d=l$. Fix $0<\eta<\nu$ according to Lemma \ref{eta} with all of these choices of parameters. Fix $n_1\in\nn$ such that for all $n\geqslant n_1$, $b_{p, 2^n}(\mathcal{F}) \leqslant (1+\eta)^{1/p}\Theta$. Fix $n_2>n_1+nd$ such that $b_{p, 2^{n_2}}(\mathcal{F})>(1-\eta/2)^{1/p}\Theta$. Let $l_{d+1}=2^{n_2-nd}>2^{n_1}$ and let $L=\prod_{i=1}^{d+1}l_i=2^{n_2}$. Let $T$ be the $(l_1, \ldots, l_{d+1})$ interval tree. Fix $F:X\to Y\in \mathcal{F}$ and $f:2^L\to X$ such that $$\mathbb{E}\varrho^f_Y(\ee, -\ee)^p>(1-\eta/2)\Theta^p\Bigl(\prod_{i=1}^{d+1}l_i^{p-1}\Bigr)\sum_{i=1}^L \mathbb{E}\varrho_X^f(\ee, d_i\ee)^p.$$ For an interval $I\in T$ and $\ee\in 2^L$, let $I\ee\in 2^L$ be the member of $2^L$ given by
\begin{displaymath}
I\ee(i) = \left\{
\begin{array}{lr}
\ee(i) & : i\in [L]\setminus I\\
-\ee(i) & : i\in I.
\end{array}
\right.
\end{displaymath}
For each $I\in T$, let $$r_I=\Bigl[\mathbb{E}\varrho^f_Y(\ee, I\ee)^p\Bigr]^{1/p}$$ and $$s_I=\Bigl[\mathbb{E}\varrho^f_X(\ee, I\ee)^p\Bigr]^{1/p}.$$
\begin{claim} \begin{enumerate}[(i)]\item $r_{[L]}^p>(1-\eta/2)\Bigl(\prod_{i=1}^{d+1}l_i^{p-1}\Bigr)\Theta^p\sum_{I\in \Lambda_{d+1}} s_I^p$. \item For any $I\in T$, $r_I\leqslant \lambda s_I$. \item For any $0\leqslant j\leqslant d$ and $I\in \Lambda_j$, $r_I\leqslant \sum_{J^-=I}r_J$ and $s_I\leqslant \sum_{J^-=I}s_j$. \item For all $I\in T$, $r_I^p\leqslant (1+\nu)\Theta^p l_{j+1}^{p-1}\sum_{J^-=I}s_J^p$. \item For any $I\in \Lambda_d$, $r_I^p\leqslant (1+\eta)l_{d+1}^{p-1}\Theta^p \sum_{J^-=I}s^p_J$. \end{enumerate}
\label{claim1}
\end{claim}
\begin{proof}$(i)$ This follows from our choice of $F$, $f$, and the fact that $$r_{[L]}^p=\mathbb{E}\varrho_Y^f(\ee, -\ee)^p$$ and $$\Bigl(\prod_{i=1}^{d+1}l_i^{p-1}\Bigr)\sum_{J\in \Lambda_{d+1}}s^p_I=\Bigl(\prod_{i=1}^{d+1}l_i^{p-1}\Bigr)\sum_{i=1}^L \varrho_X^f(\ee, d_i\ee)^p.$$
$(ii)$ This follows from the fact that $\text{Lip}(F)\leqslant \lambda$.
$(iii)$ Fix $0\leqslant j\leqslant d$ and $I\in \Lambda_j$. Enumerate $\{J\in T: J^-=I\}$ as $(I_i)_{i=1}^{l_{j+1}}$. For $i=0, \ldots, l_{j+1}$, define $J_i:2^L\to 2^L$ by letting $J_0$ be the identity and $J_i=I_iJ_{i-1}$. For any $\ee\in 2^L$, $J_0\ee=\ee$ and $J_{l_{j+1}}\ee=I\ee$. Then for any $\ee\in 2^L$, $$\varrho_Y^f(\ee, I\ee)\leqslant \sum_{i=1}^{l_{j+1}}\varrho_Y^f(J_{i-1}\ee, J_i\ee).$$ Now the triangle inequality on $L_p(2^L)$ yields that $$r_I \leqslant \sum_{i=1}^{l_{j+1}} \Bigl[\mathbb{E}\varrho_Y^f(J_{i-1}\ee, J_i\ee)^p\Bigr]^{1/p}.$$ But $\varrho^f_Y(J_{i-1}\ee, J_i\ee)$ and $\varrho_Y^f(\ee, I_i\ee)$ have the same distribution, so $$r_I \leqslant \sum_{i=1}^{l_{j+1}} \Bigl[\mathbb{E}\varrho_Y^f(J_{i-1}\ee, J_i\ee)^p\Bigr]^{1/p} = \sum_{i=1}^{l_{j+1}} \Bigl[\mathbb{E}\varrho_Y^f(\ee, I_i\ee)^p\Bigr]^{1/p} = \sum_{J^-=I}r_J.$$ Replacing each $Y$ with $X$ yields that $s_I\leqslant \sum_{J^-=I}s_J$.
$(iv)$ and $(v)$ Let $I$ and $(I_i)_{i=1}^{l_{j+1}}$ be as in the proof of $(iii)$. Define $g:2^L\times 2^{l_{j+1}}\to 2^L$ by letting \begin{displaymath}
g(\ee, \delta)(i) = \left\{
\begin{array}{lr}
\ee(i) & : i\in [L]\setminus I\\
\delta(k)\ee(i) & : i\in I_k.
\end{array}
\right.
\end{displaymath} For fixed $\ee\in 2^L$, let $f_\ee:2^{l_{j+1}}\to X$ be given by $f_\ee(\delta)=f(g(\ee, \delta))$. Note that $g(\ee, -\delta)=Ig(\ee, \delta)$ and $g(\ee, d_i\delta)=I_ig(\ee, \delta)$ for all $\ee\in 2^L$, $\delta\in 2^{l_{j+1}}$, and $1\leqslant i\leqslant l_{j+1}$. From this it follows that $f_\ee(-\delta)=f(Ig(\ee, \delta))$ and for $1\leqslant i\leqslant l_{j+1}$, $f_\ee(d_i\ee)=f(I_ig(\ee, \delta))$. Then \begin{align*} r_I^p & = \mathbb{E}\varrho_Y^f(\ee, I\ee)^p = \mathbb{E}_\delta \mathbb{E}_\ee \varrho_Y^f(g(\ee, \delta), Ig(\ee, \delta))^p \\ & = \mathbb{E}_\ee \mathbb{E}_\delta \varrho_Y^{f_\ee}(\delta, -\delta)^p \leqslant \mathbb{E}_\ee b_{p, l_{j+1}}(\mathcal{F})^p l_{j+1}^{p-1} \sum_{i=1}^{l_{j+1}}\mathbb{E}_\delta \varrho_X^{f_\ee}(\delta, d_i\delta)^p \\ & = \mathbb{E}_\ee b_{p, l_{j+1}}(\mathcal{F})^p l_{j+1}^{p-1} \sum_{i=1}^{l_{j+1}}\mathbb{E}_\delta \varrho_X^f(g(\ee, \delta), I_ig(\ee, \delta))^p \\ & = b_{p, l_{j+1}}(\mathcal{F})^p l_{j+1}^{p-1} \sum_{i=1}^{l_{j+1}}\mathbb{E}_\delta \mathbb{E}_\ee\varrho_X^f(g(\ee, \delta), I_ig(\ee, \delta))^p \\ & = b_{p, l_{j+1}}(\mathcal{F})^p l_{j+1}^{p-1} \sum_{i=1}^{l_{j+1}} \mathbb{E}_\ee\varrho_X^f(\ee, I_i\ee)^p= b_{p, l_{j+1}}(\mathcal{F})^p l_{j+1}^{p-1}\sum_{J^-=I} s_J^p.\end{align*} The fact that $b_{p, l_{j+1}}(\mathcal{F})\leqslant (1+\nu)^{1/p}\Theta$ for all $j$ yields $(iv)$, and the fact that $b_{p, l_{d+1}}(\mathcal{F}) \leqslant (1+\eta)^{1/p}\Theta$ yields $(v)$.
\end{proof}
\begin{claim} There exist $0\leqslant j_0<d$ and $I\in \Lambda_{j_0}$ such that \begin{enumerate}[(i)]\item $r_I^p>(1-\mu)l^{p-1}\sum_{J^-=I}r_J^p$, \item $s_I^p>(1-\mu)l^{p-1}\sum_{J^-=I}s_J^p$, and \item $l^{p-1}\sum_{J^-=I}r^p_J\geqslant r_I>(1-\nu)\Theta^p l^{p-1} \sum_{J^-=I}s^p_J$. \end{enumerate}
\label{claim2}
\end{claim}
\begin{proof} For each $0\leqslant j<d$, let $$A_j=\Bigl\{I\in \Lambda_j: s_I^p\leqslant (1-\mu)l^{p-1}\sum_{J^-=I} s_J^p\Bigr\}$$ and $$B_j=\Bigl\{I\in \Lambda_j: r_I^p\leqslant (1-\nu)\Theta^pl^{p-1}\sum_{J^-=I}s_J^p\Bigr\}. $$ By Lemma \ref{goodX}, there are at most $m$ values of $j<d$ such that $|A_j|\geqslant \Delta|\Lambda_j|$. Here we note that $$s_{[L]}^p\geqslant \frac{r_{[L]}^p}{\lambda^p}> (1-\eta/2)\frac{\Theta^p}{\lambda^p}\Bigl(\prod_{i=1}^{d+1}l_i^{p-1}\Bigr)\sum_{I\in \Lambda_{d+1}}s_I^p.$$ Since $m+1<d$, there exists at least one value $j_0<d$ such that $|A_{j_0}|<\Delta |\Lambda_{j_0}|$. By Lemma \ref{YtoX}, for this $j_0$, $|B_{j_0}|< (1-\Delta)|\Lambda_{j_0}|$. Thus $$|A_{j_0}|+|B_{j_0}|<|\Lambda_{j_0}|,$$ whence there exists $I\in \Lambda_{j_0}\setminus (A_{j_0}\cup B_{j_0})$. Since $I\in \Lambda_{j_0}\setminus A_{j_0}$, $(ii)$ is satisfied for this $I$. Since $I\in \Lambda_{j_0}\setminus B_{j_0}$ and since $l^{p-1}\sum_{J^-=I}r_J^p \geqslant r_I^p$, $(iii)$ is satisfied for this $I$. Since $j_0<d$ and $I\in \Lambda_{j_0}$, $(i)$ is satisfied for this $I$ by Lemma \ref{eta}.
\end{proof}
For the remainder of the proof, $I$ is the fixed interval from Claim \ref{claim2}. Enumerate $\{J\in T: J^-=I\}$ as $(I_i)_{i=1}^l$ and define $g:2^L\times 2^l\to 2^L$ by letting \begin{displaymath}
g(\ee, \delta)(i) = \left\{
\begin{array}{lr}
\ee(i) & : i\in [L]\setminus I\\
\delta(j)\ee(i) & : i\in I_j.
\end{array}
\right.
\end{displaymath} We also define the functions $J_0, \ldots, J_l:2^L\to 2^L$ by letting $J_0$ be the identity function and $J_i=I_iJ_{i-1}$. Note that $\ee=J_0\ee$ and $J_l\ee=I\ee$.
Now define $D_Y, E_Y, D_X, E_X:2^L\to [0, \infty)$ by $$D_Y(\ee)= \mathbb{E}_\delta \varrho^f_Y(g(\ee, \delta), Ig(\ee, \delta))^p,$$ $$E_Y(\ee)=l^{p-1}\sum_{i=1}^l \mathbb{E}_\delta \varrho_Y^f(J_{i-1}g(\ee, \delta), J_i g(\ee, \delta))^p,$$ $$D_X(\ee)= \mathbb{E}_\delta \varrho^f_X(g(\ee, \delta), Ig(\ee, \delta))^p,$$ and $$E_X(\ee)=l^{p-1}\sum_{i=1}^l \mathbb{E}_\delta \varrho_X^f(J_{i-1}g(\ee, \delta), J_i g(\ee, \delta))^p.$$
\begin{claim} The functions $D_Y, E_Y, D_X, E_X$ satisfy the hypotheses of Lemma \ref{expect}.
\label{claim3}
\end{claim}
\begin{proof} $D_Y\leqslant \lambda^p D_X$ and $E_Y\leqslant \lambda^p E_X$ follow from the fact that $\text{Lip}(F)\leqslant \lambda$. For a fixed $\delta\in 2^l$ and $\ee\in 2^L$, $$\varrho_Y^f(g(\ee, \delta), Ig(\ee, \delta))^p\leqslant l^{p-1}\sum_{i=1}^l \varrho_Y^f(J_{i-1}g(\ee, \delta), J_ig(\ee, \delta))^p$$ follows from the triangle and H\"{o}lder inequalities. Taking expectations over $\delta$ with $\ee$ held fixed yields that $D_Y\leqslant E_Y$. Replacing $Y$ with $X$ yields that $D_X\leqslant E_X$. Now fix $\ee\in 2^L$ and define $f_\ee(\delta)=g(\ee, \delta)$. Define $d_{<i}, d_{\leqslant i}:2^l\to 2^l$ by letting $d_{<i}\delta$ be the member of $2^l$ by replacing $\delta(k)$ with $-\delta(k)$ for each $k<i$ and leaving the other coordinates of $\delta$ unchanged. Let $d_{\leqslant i}=d_id_{<i}$. Note that $\varrho_X^{f_\ee}(\delta, d_i\delta)^p$ and $\varrho_X^{f_\ee}(d_{<i}\delta, d_{\leqslant i\delta})$ have the same distribution as functions of $\delta\in 2^l$. Note also that $g(\ee, d_{<i}\delta)=J_{i-1}g(\ee, \delta)$ and $g(\ee, \delta_{\leqslant i}\delta)=J_i g(\ee, \delta)$. Then \begin{align*} D_Y(\ee) & = \mathbb{E}_\delta\varrho_Y^f(g(\ee, \delta), Ig(\ee, \delta))^p = \mathbb{E}_\delta \varrho_Y^{f_\ee}(\delta, -\delta)^p \\ & \leqslant b_{p,l}(\mathcal{F})^p l^{p-1} \sum_{i=1}^l \mathbb{E}_\delta \varrho_X^{f_\ee}(\delta, d_i\ee)^p = b_{p,l}(\mathcal{F})^pl^{p-1}\sum_{i=1}^l \mathbb{E}_\delta \varrho_X^{f_\ee}(d_{<i}\delta, d_{\leqslant i}\delta)^p \\ & \leqslant (1+\nu)\Theta^p l^{p-1} \sum_{i=1}^l \mathbb{E}_\delta \varrho_X^f(J_{i-1} g(\ee, \delta), J_i g(\ee, \delta))^p = (1+\nu)\Theta^p E_X(\ee). \end{align*} This yields that $D_Y, E_Y, D_X, E_X$ satisfy hypothesis $(i)$ of Lemma \ref{expect}.
For a fixed $\delta\in 2^l$ and $1\leqslant i\leqslant l$, $\varrho_Y^f(\ee, I\ee)$ and $\varrho_Y^f(g(\ee, \delta), Ig(\ee, \delta))$ have the same distribution as functions of $\ee\in 2^L$, as do $\varrho_Y^f(\ee, I_i\ee)$ and $\varrho_Y^f(J_{i-1}g(\ee, \delta), J_ig(\ee, \delta))$. The analogous statements hold with each $Y$ replaced by $X$. By exchanging order of integration of $\ee$ and $\delta$, we see that $$\mathbb{E}_\ee D_Y(\ee)=\mathbb{E}_\ee\mathbb{E}_\delta \varrho_Y^f(g(\ee, \delta), Ig(\ee, \delta))^p = \mathbb{E}_\delta \mathbb{E}_\ee \varrho^f_Y (g(\ee, \delta), Ig(\ee, \delta))^p = \mathbb{E}_\ee \varrho^f_Y (\ee, I\ee)^p=r_I^p.$$ We similarly deduce that $\mathbb{E}_\ee E_Y(\ee)=l^{p-1}\sum_{J^-=I} r_J^p$, $\mathbb{E}_\ee D_X(\ee)=s_I^p$, and $\mathbb{E}_\ee E_X(\ee)=l^{p-1}\sum_{J^-=I} s_J^p$. Thus hypothesis $(ii)$ of Lemma \ref{expect} is satisfied because $I$ satisfies the conclusions of Claim \ref{claim2}.
Hypothesis $(iii)$ of Lemma \ref{expect} is satisfied by our chocies of $\mu, a, \nu$, and $b$.
\end{proof}
Now by Lemma \ref{expect}, there exists $\ee_0\in 2^L$ such that $$D_Y(\ee_0)>(1-a)E_Y(\ee_0),$$ $$D_X(\ee_0)>(1-a)E_X(\ee_0),$$ and $$E_Y(\ee_0)\geqslant D_Y(\ee_0)>(1-b)\Theta^p E_X(\ee_0).$$ Now define $h:2^l\to X$ by letting $h(\delta)=f(g(\ee_0, \delta))$. Let $$r^p=\frac{1}{l}\sum_{i=1}^l \mathbb{E}_\delta \varrho_Y^h( \delta, d_i\delta)^p= E_Y(\ee_0)/l^p$$ and $$s^p=\frac{1}{l}\sum_{i=1}^l \mathbb{E}_\delta \varrho_X^h(\delta,d_i\delta)^p=E_X(\ee_0)/l^p \leqslant (1-b)\Theta^p r^p.$$ Then since $$\mathbb{E}_\delta \varrho^h_Y(\delta, -\delta)^p=D_Y(\ee_0)>(1-a)E_Y(\ee_0)=(1-a)l^{p-1}\sum_{i=1}^l \mathbb{E}_\delta \varrho_Y^h(\delta, d_i\delta)^p$$ and $$\mathbb{E}_\delta \varrho^h_X(\delta, -\delta)^p=D_X(\ee_0)>(1-a)E_X(\ee_0)=(1-a)l^{p-1}\sum_{i=1}^l \mathbb{E}_\delta \varrho_X^h(\delta, d_i\delta)^p,$$ it follows from our choice of $a$ and Theorem \ref{BMW} that for any $\delta_1, \delta_2\in 2^l$, $$\frac{1}{D}\partial (\delta_1, \delta_2) \leqslant \frac{d_Y(h(\delta_1), h(\delta_2))}{r} \leqslant D\partial (\delta_1, \delta_2)$$ and $$\frac{1}{D}\partial (\delta_1, \delta_2) \leqslant \frac{d_X(F\circ h(\delta_1), F\circ h(\delta_2))}{s} \leqslant D\partial (\delta_1, \delta_2).$$ Since $r\geqslant \vartheta s$, this finishes the proof.
\begin{rem}\upshape We observe the following quantitative consequence of the previous proof and our remark from the introduction. If we define $c(\mathcal{F})$ to be the supremum of those $c>0$ such that for each $D>1$ and $n\in \nn$, there exist $F:X\to Y\in \mathcal{F}$, $f:2^n\to X$, and $a,b>0$ such that $b\geqslant ac$ and for each $\ee_1, \ee_2\in 2^n$, $$\frac{a}{D} \partial (\ee_1, \ee_2) \leqslant d_X(f(\ee_1), f(\ee_2)) \leqslant aD \partial (\ee_1, \ee_2)$$ and $$\frac{b}{D}\partial (\ee_1, \ee_2) \leqslant d_Y(F\circ f(\ee_1), F\circ f(\ee_2)) \leqslant bD \partial (\ee_1, \ee_2),$$ then $$\underset{n}{\lim\sup} b_{p,n}(\mathcal{F}) \geqslant c(\mathcal{F}) \geqslant \underset{n}{\lim\sup} b_{p,2^n}(\mathcal{F}) \geqslant 2^{1/p-1}\underset{n}{\lim\sup} b_{p,n}(\mathcal{F}).$$
\label{gg}
\end{rem}
\section{The quantities $a_n(\mathcal{F})$}
The goal of this section is to prove the implication $(iii)\Rightarrow (i)$ with the additional quantitative information: If $(i)$ is satisfied and $c(\mathcal{F})$ is as defined in Remark \ref{gg}, then $c(\mathcal{F})=\lim\sup_n a_n(\mathcal{F})=\lim_n a_n(\mathcal{F})$. It is obvious that $c(\mathcal{F})\leqslant \lim\sup_n a_n(\mathcal{F})$, so we establish the following criterion for obtaining the reverse inequality. The basis of this criterion is to use standard self-improvement arguments for embeddings into $X$ without worsening the scaling factors between the embeddings of the cube into $X$ via some $f$ and the corresponding embedding of the cube into $Y$ via $F\circ f$.
\begin{lemma} Suppose $\lambda=\sup_{F\in \mathcal{F}} \text{\emph{Lip}}(F)\in (0,\infty)$. If $\Theta>0$ is such that for any $0<\vartheta<\Theta$, $D>1$, and $l\in\nn$, there exist $F:X\to Y\in \mathcal{F}$ and $h:2^l\to X$ such that $\text{\emph{dist}}(F\circ h)\leqslant D$ and $\varrho_Y^h(\ee_1, \ee_2)\geqslant \vartheta d_X^h(\ee_1, \ee_2)$ for all $\ee_1, \ee_2\in 2^l$, then $c(\mathcal{F})\geqslant \Theta$.
\label{taco bell}
\end{lemma}
\begin{proof} First fix $0<\vartheta<\Theta$. For each $l\in\nn$, let $\xi_n=\xi_n(\mathcal{F})$ be the supremum of those constants $\xi>0$ such that for all $D>1$, there exist $F:X\to Y\in \mathcal{F}$ and $f:2^n\to X$ such that $\text{dist}(F\circ f)\leqslant D$, for each $\ee_1, \ee_2\in 2^n$, $\varrho_Y^f(\ee_1, \ee_2) \geqslant \vartheta d_X^f(\ee_1, \ee_2)$, and $$\mathbb{E}\varrho_X^f(\ee, -\ee)^2>n \xi^2 \sum_{i=1}^n \mathbb{E} \varrho_X^f(\ee, d_i\ee)^2.$$
Let us observe the following facts: \begin{enumerate}[(i)]\item For all $n\in\nn$, $\vartheta/\lambda\leqslant \xi_n\leqslant 1$. \item For all $m,n\in\nn$, $\xi_{mn}\leqslant \xi_m\xi_n$. \end{enumerate}
For the first fact, $\xi_n\leqslant 1$ follows as usual from the triangle and H\"{o}lder inequalities. By hypothesis, for each $D_1,D_2>1$, there exist $F:X\to Y\in \mathcal{F}$ and $f:2^n\to X$ such that $\text{dist}(F\circ f)\leqslant D=\min\{D_1, D_2\}$ and $\vartheta \varrho_X^f(\ee_1, \ee_2) \leqslant \varrho_Y^f(\ee_1, \ee_2)$ for all $\ee_1, \ee_2\in 2^n$. Then \begin{align*} \lambda^2 \mathbb{E}\varrho_X^f(\ee,-\ee)^2 & \geqslant \mathbb{E}\varrho_Y^f(\ee_1, -\ee)^2 \geqslant D^{-4}n\sum_{i=1}^n \varrho_Y^f(\ee, d_i\ee)^2 \geqslant \vartheta^2 D^{-4}n\sum_{i=1}^n \varrho_X^f(\ee, d_i\ee)^2. \end{align*} Now applying this as $D_1\downarrow 1$ with $D_2>1$ held fixed, we deduce that $\xi_n\geqslant \vartheta/\lambda D^2_2$. Unfixing $D_2>1$ yields that $\xi_n\geqslant \vartheta/\lambda$.
For the second item, suppose $\xi_{mn}>\xi_m\xi_n$ for some $m,n\in\nn$. Fix $\alpha>\xi_m$ and $\beta>\xi_n$ such that $\alpha\beta<\xi_{mn}$. By definition of $\xi_m$, there exists $D_1>1$ such that for all $F:X\to Y\in \mathcal{F}$ and $f:2^m\to X$ with $\text{dist}(F\circ f)\leqslant D_1$ and $\vartheta \varrho_X^f(\ee_1, \ee_2)\leqslant \varrho_Y^f(\ee_1, \ee_2)$ for all $\ee_1, \ee_2\in 2^m$, $$\mathbb{E}\varrho_X^f(\ee, -\ee)^2 \leqslant m\alpha^2 \sum_{i=1}^m \mathbb{E}\varrho_X^f(\ee, d_i\ee)^2.$$ Similarly, there exists $D_2>1$ such that for any $F:X\to Y\in \mathcal{F}$ and $f:2^n\to X$ with $\text{dist}(F\circ f)\leqslant D_2$ and $\vartheta \varrho_X^f(\ee_1, \ee_2)\leqslant \varrho_Y^f(\ee_1, \ee_2)$ for all $\ee_1, \ee_2\in 2^n$, $$\mathbb{E}\varrho_X^f(\ee, -\ee)^2 \leqslant n\beta^2\sum_{i=1}^n \mathbb{E}\varrho_X^f(\ee, d_i\ee)^2.$$ Let $D=\min\{D_1, D_2\}>1$ and fix $\xi, F, f$ such that $\alpha\beta<\xi<\xi_{mn}$ and $F:X\to Y\in \mathcal{F}$, $f:2^{mn}\to X$ satisfy $\text{dist}(F\circ f)\leqslant D$, $\vartheta \varrho_X^f(\ee_1, \ee_2)\leqslant \varrho_Y^f(\ee_1, \ee_2)$ for all $\ee_1, \ee_2\in 2^{mn}$, and $$\mathbb{E}\varrho_X^f(\ee, -\ee)^2>mn\xi^2\sum_{i=1}^{mn} \varrho_X^f(\ee, d_i\ee)^2.$$ Now as usual, let $I_1, \ldots, I_m$ be a partition of $[mn]$ into intervals of cardinality $n$ and define \begin{displaymath}
I_j\ee(i) = \left\{
\begin{array}{lr}
\ee(i) & : i\in [mn]\setminus I_j\\
-\ee(i) & : i\in I_j
\end{array}
\right.
\end{displaymath}
Define $g:2^{mn}\times 2^m\to 2^{mn}$ be defined by $g(\ee, \delta)(i)=\delta(j)\ee(i)$, where $i\in I_j$. We identify $\ee$ with $(\ee_i)_{i=1}^m$, where $(\ee((i-1)n+1), \ldots, \ee(in))=\ee_i\in 2^n$. For $\ee\in 2^{mn}$, we let $\ee_{-i}\in 2^{(m-1)n}$ be defined by $\ee_{-i}=(\ee_1, \ldots, \ee_{i-1}, \ee_{i+1}, \ldots, \ee_m)$. Note that for a fixed $\ee_{-i}$ and a fixed $\ee'\in 2^{mn}$, if $h$ is the map from $2^n$ to $X$ given by $\ee\mapsto f((\ee_1, \ldots, \ee_{i-1}, \ee, \ee_{i+1}, \ldots, \ee_m))$ or if $h$ is the map from $2^m$ to $X$ given by $\delta\mapsto f(g(\ee', \delta))$, then $\text{dist}(F\circ h)\leqslant \text{dist}(F\circ f)\leqslant D$ and $\vartheta \varrho_X^h(\ee_1, \ee_2)\leqslant \varrho_Y^h(\ee_1, \ee_2)$ for all $\ee_1, \ee_2\in 2^n$ (resp. $\ee_1, \ee_2\in 2^m$). Then \begin{align*} mn\xi^2\sum_{i=1}^{mn} \mathbb{E} \varrho_X^f(\ee, d_i\ee)^2 & < \mathbb{E}\varrho_X^f(\ee, -\ee)^2 =\mathbb{E}_\ee \mathbb{E}_\delta \varrho_X^f(g(\ee, \delta), g(\ee, -\delta))^2 \\ & \leqslant m\alpha^2 \sum_{i=1}^m \mathbb{E}_\ee\mathbb{E}_\delta \varrho_X^f(g(\ee, \delta), g(\ee, d_i\delta))^2 = m\alpha^2 \sum_{i=1}^m \mathbb{E}_\ee\mathbb{E}_\delta \varrho_X^f(g(\ee, \delta), I_i g(\ee, \delta))^2 \\ & = m\alpha^2 \sum_{i=1}^m \mathbb{E} \varrho_X^f(\ee, I_i\ee)^2 = m \alpha^2\sum_{i=1}^m \mathbb{E}_{\ee_{-i}}\mathbb{E}_{\ee_i} \varrho_X^f(\ee, I_i\ee)^2 \\ & \leqslant mn\alpha^2\beta^2 \sum_{i=1}^m \mathbb{E}_{\ee_{-i}}\sum_{j=(i-1)+1}^{ij} \mathbb{E}_{\ee_i}\varrho_X^f(\ee, d_j\ee)^2 =mn\alpha^2\beta^2\sum_{i=1}^{mn}\mathbb{E}\varrho_X^f(\ee, d_i\ee)^2. \end{align*} This contradiction yields $(ii)$.
Now since $(\xi_n)_{n=1}^\infty$ is submultiplicative and lies in $[\vartheta/\lambda, 1]$, it must be that $\xi_n=1$ for all $n\in\nn$. Indeed, if $\xi_n<1$, then for large enough $t\in \nn$, $\xi_{n^t}\leqslant \xi_n^t<\vartheta/\lambda$. Now fix $n\in\nn$ and $D>1$. By Theorem \ref{BMW}, there exists $0<\mu<1$ such that if $f:2^n\to X$ satisfies $$\mathbb{E}\varrho_X^f(\ee, -\ee)^2>(1-\mu)n\sum_{i=1}^n \mathbb{E}_X^f(\ee, d_i\ee)^2,$$ then $\text{dist}(f)\leqslant D$. By the definition of $\xi_n$ and since $\xi_n=1>1-\mu$, there exist $F:X\to Y\in \mathcal{F}$ and $f:2^n\to X$ such that $\text{dist}(F\circ f)\leqslant D$, $\vartheta \varrho_X^f(\ee_1, \ee_2)\leqslant \varrho_Y^f(\ee_1, \ee_2)$ for all $\ee_1, \ee_2\in 2^n$, and $$\mathbb{E}\varrho_X^f(\ee, -\ee)^2>(1-\mu)n\sum_{i=1}^n \mathbb{E}\varrho_X^f(\ee, d_i\ee)^2.$$ From this it follows that $\text{dist}(f)\leqslant D$. Moreover, if $a,b>0$ are such that $$\frac{a}{D}\partial (\ee_1, \ee_2)\leqslant \varrho_X^f(\ee_1, \ee_2) \leqslant aD\partial (\ee_1, \ee_2)$$ and $$\frac{b}{D}\partial (\ee_1, \ee_2)\leqslant \varrho_Y^f(\ee_1, \ee_2) \leqslant bD\partial (\ee_1, \ee_2)$$ for all $\ee_1, \ee_2$, then $a\vartheta/D\leqslant bD$. Since $D>1$, $n\in\nn$ are arbitrary, $c(\mathcal{F})\geqslant \vartheta$. Now we unfix $0<\vartheta<\Theta$ and deduce that $c(\mathcal{F})\geqslant \Theta$.
\end{proof}
\begin{proposition} If $\mathcal{F}$ is a uniformly Lipschitz collection of maps, then $\lim_n a_n(\mathcal{F})=\inf_n a_n(\mathcal{F})$.
\end{proposition}
\begin{proof} Let $\lambda=\sup_{F\in \mathcal{F}}\text{Lip}(F)\in (0, \infty)$. Fix $k,l\in \nn$. Let $I_1, \ldots, I_l$ be a partition of $[kl]$ into subintervals of cardinality $k$. Let $I_j:2^{kl}\to 2^{kl}$ be such that \begin{displaymath}
I_j\ee(i) = \left\{
\begin{array}{lr}
\ee(i) & : i\in [kl]\setminus I_j\\
-\ee(i) & : i\in I_j.
\end{array}
\right.
\end{displaymath}
Define $g:2^{kl}\times 2^l\to 2^{kl}$ by $g(\ee, \delta)(i)=\delta(j)\ee(i)$, where $j$ is such that $i\in I_j$.
Now fix any $F:X\to Y\in \mathcal{F}$ and $f:2^{kl}\to X$. Note that for fixed $\ee\in 2^{kl}$, the function $\delta\mapsto g(\ee, \delta)$ is distance preserving. Therefore for each fixed $\ee\in 2^l$, the function $f_\ee:2^l\to X$ given by $f_\ee(\delta)=f(g(\ee, \delta))$ satisfies $\text{Lip}(f_\ee)\leqslant \text{Lip}(f)$. Then \begin{align*} \mathbb{E}_\ee \varrho_Y^f(\ee, -\ee) & = \mathbb{E}_\delta \mathbb{E}_\ee \varrho_Y^f(g(\ee, \delta), g(\ee, -\delta)) = \mathbb{E}_\ee \mathbb{E}_\delta \varrho^{f_\ee}_Y(\delta, -\delta) \\ & \leqslant a_l(\mathcal{F})\mathbb{E}_\ee \text{Lip}(f_\ee)\leqslant a_l(\mathcal{F}) \text{Lip}(f). \end{align*} From this it follows that $a_{kl}(\mathcal{F})\leqslant a_l(\mathcal{F})$.
Now fix $l\in\nn$. For $m>l$, write $m=k_ml+r_m$ where $k_m\in\nn$ and $0\leqslant r_m<k$. For the moment, we suppress the subscript $m$ and simply write $k_m=k$ and $r_m=r$. Now fix $F:X\to Y\in \mathcal{F}$ and $f:2^m\to X$. Let $\delta=(\delta(1), \ldots, \delta(r))\in 2^r$ be arbitrary and define $g:2^{lk}\to 2^m$ by $g(\ee)=(\ee(1), \ldots, \ee(lk), \delta(1), \ldots, \delta(r))$. Define $h:2^m\to 2^{lk}$ by $h(\ee)=(\ee(1), \ldots, \ee(lk))$. Note that for each $\ee_1, \ee_2\in 2^{lk}$, $$\frac{1}{lk}\partial(\ee_1, \ee_2)= \frac{1}{m}\partial (g(\ee_1), g(\ee_2)).$$ Therefore the map $G:2^{lk}\to X$ given by $G(\ee)=f(g(\ee))$ has $\text{Lip}(G)\leqslant \frac{m}{lk}\partial (\ee_1, \ee_2)$. Define $H:2^m\to X$ by $H(\ee)=G(h(\ee))=f(g(h(\ee)))$. Let us also note that $\mathbb{E}\varrho_Y^G(\ee, -\ee)=\mathbb{E}\varrho_Y^H(\ee, -\ee)$. For any $\ee\in 2^m$, since $\ee$ and $g(h(\ee))$ differ in at most $r$ coordinates, $$\varrho_Y^f(\ee, g(h(\ee))) \leqslant \lambda \text{Lip}(f)r/m.$$ Therefore \begin{align*} \mathbb{E}\varrho_Y^f(\ee, -\ee) & \leqslant \mathbb{E}\varrho_Y^f(g(h(\ee)), g(h(-\ee)) + \mathbb{E}\varrho_Y^f(\ee, g(h(\ee))+ \mathbb{E}\varrho_Y^f(g(h(-\ee)), -\ee) \\ & \leqslant \mathbb{E}\varrho_Y^f(g(h(\ee)), g(h(-\ee))+ 2\lambda \text{Lip}(f)r/m = \mathbb{E}\varrho_Y^H (\ee, -\ee)+2\lambda \text{Lip}(f)r/m \\ & = \mathbb{E}\varrho_Y^G(\ee, -\ee)+2\lambda \text{Lip}(f)r/m \leqslant a_{lk}(\mathcal{F})\text{Lip}(G)+2\lambda \text{Lip}(f)r/m \\ & \leqslant \Bigl[a_l(\mathcal{F})\frac{m}{lk}+\frac{2\lambda r}{m}\Bigr]\text{Lip}(f). \end{align*} From this it follows that $$a_m(\mathcal{F}) \leqslant a_l(\mathcal{F})\frac{m}{lk}+\frac{2\lambda r}{m}.$$ Now once more writing $k=k_m$ and $r=r_m$ and noting that $\frac{m}{lk_m}\to 1$ and $\frac{2\lambda r_m}{m}\to 0$ as $m\to \infty$, we deduce that $a_l(\mathcal{F})\geqslant \lim\sup_m a_m(\mathcal{F})$. Since $l\in \nn$ was arbitrary, we are done.
\end{proof}
\begin{rem}\upshape If $L,l\in\nn$ are two natural numbers, $g:2^L\times 2^l\to 2^L$ is a function such that for each $\delta\in 2^l$, $\ee\mapsto g(\ee, \delta)$ is a bijection, and $\Omega\subset 2^L$ is such that $\mathbb{P}(\Omega)<1/2^l$, then there exists $\ee\in 2^L$ such that $\{g(\ee, \delta):\delta\in 2^l\}\subset \Omega^c$. Indeed, for each $\ee_0\in 2^L$ and $\delta_0\in 2^l$, define $\Omega_{\ee_0}=\{\delta\in 2^l: g(\ee_0, \delta)\in \Omega\}$, $\Omega^{\delta_0}=\{\ee\in 2^L: g(\ee, \delta_0)\in \Omega\}$, and $\Omega_1=\{(\ee, \delta)\in 2^L\times 2^l: g(\ee, \delta)\in \Omega\}$. Then if for each $\ee$, there exists $\delta\in 2^l$ such that $g(\ee, \delta)\in \Omega^c$, \begin{align*} \mathbb{P}(\Omega)=\frac{1}{2^l}\sum_{\delta\in 2^l}\mathbb{P}(\Omega)=\frac{1}{2^l}\sum_{\delta\in 2^l} \mathbb{P}(\Omega^\delta) = \mathbb{P}(\Omega_1) = \frac{1}{2^L}\sum_{\ee\in 2^L}\mathbb{P}(\Omega_\ee) \geqslant \frac{1}{2^L}\sum_{\ee\in 2^L}1/2^l = 1/2^l.\end{align*}
\label{density}
\end{rem}
Let us also recall the following simple consequence of the reverse triangle inequality, which we use as a substitute for Theorem \ref{BMW} in this section.
\begin{proposition} For each $l\in\nn$ and $D>1$, there exists $0<a<1$ such that if $(Z, d_Z)$ is any metric space and $h:2^l\to Z$ is a map such that $(1-a)\text{\emph{Lip}}(h) < \min_{\delta\in 2^l} d_Z(h(\delta), h(-\delta))$, then $h$ is an embedding with distortion not more than $D$.
\label{sharp}
\end{proposition}
\begin{proof} Fix $0<a<1/l$ so small that $1-al>1/D$. Fix $\delta\neq \delta_1\in 2^l$ and let $m=l \partial (\delta, \delta_1)$. Then \begin{align*} d_Z(h(\delta), h(\delta_1)) & \geqslant d_Z(h(\delta), h(-\delta))-d_Z(h(-\delta), h(\delta_1)) \geqslant (1-a)\text{Lip}(h)-\frac{l-m}{l}\text{Lip}(h) \\ & = \Bigl(\frac{m}{l}-a\Bigr)\text{Lip}(h) \geqslant \frac{m}{l}(1-al)\text{Lip}(h) > \frac{\text{Lip}(h)}{D} \partial (\delta, \delta_1). \end{align*}
From this it follows that $\text{Lip}(h^{-1})\leqslant D/\text{Lip}(h)$, so $\text{Lip}(h)\text{Lip}(h^{-1})\leqslant D$.
\end{proof}
We next recall the concentration of measure for the Hamming cube.
\begin{lemma}\cite{Harper,AM} There exist constants $\alpha, \beta>0$ such that for any $n\in\nn$ and $\lambda_1>0$, if $\Phi:2^n\to \rr$ is $\lambda_1$-Lipschitz and if $\phi$ is a median of $\Phi$, then for any $t>0$, $$\mathbb{P}\Bigl(|\Phi-\phi|>t\lambda_1\Bigr)\leqslant \alpha \exp(-\beta t n).$$
\label{concentrate}
\end{lemma}
\begin{proof}[Proof of $(iii)\Rightarrow (i)$] Let $\lambda=\sup_{F\in\mathcal{F}}\text{Lip}(F)\in (0,\infty)$ and $\Theta=\lim_n a_n(\mathcal{F})\in (0, \lambda]$. Fix $0<\vartheta<\Theta$. Fix $l\in\nn$ and $D>1$ such that $\vartheta<\Theta/D$. Let $0<a<1$ be chosen according to Proposition \ref{sharp}. Now fix $0<\mu<1/2$ such that $\frac{1+\mu}{1-\mu}>1-a$ and $\frac{(1-2\mu)}{D}\Theta>\vartheta$. Fix $0<\eta<\mu$ such that $$\frac{1-\mu}{l}+(1+\eta)\bigl(\frac{l-1}{l}\bigr)<1-\eta.$$ Fix $t>0$ such that $t<\mu\Theta/l$ and $$(l+1)t< \mu(1-2\mu)\Theta.$$ Note that the second inequality implies that for any $M\geqslant (1-2\mu)\Theta$, $$(l+1)t+M\leqslant (1+\mu)M.$$ Fix $k\in\nn$ so large that for all $m\geqslant k$, $a_m(\mathcal{F})\in ((1-\eta)\Theta, (1+\eta)\Theta)$, $\lambda \alpha \exp(-\beta tm/8\lambda)<t/4\lambda$, and $(l+1)\alpha \exp(-\beta t m/8\lambda)<1/2^l$. Fix $F:X\to Y\in \mathcal{F}$ and $f:2^{lk}\to X$ such that $$\mathbb{E}\varrho_Y^f(\ee, -\ee)>(1-\eta)\Theta \text{Lip}(f).$$ Let $T$ be the $(l,k)$ interval tree and for each $I\in T$, define $\Phi_I:2^{lk}\to \rr$ by $\Phi_I(\ee)=\varrho_Y^f(\ee, I\ee)$, where $I\ee$ is obtained by changing the signs of the coordinates of $\ee$ which lie in $I$ and leaving the other coordinates unchanged. Let $\phi_I$ be a median of $\Phi_I$ and let $r_I=\mathbb{E}\Phi_I$. For the remainder of the proof, fix a partition of $[lk]$ into intervals $I_1, \ldots, I_l$, where $I_j=\{(j-1)k+1, \ldots, jk\}$.
We first claim that $$(1-\eta)\Theta \text{Lip}(f) <r_{[lk]}< (1+\eta)\Theta \text{Lip}(f)$$ and for each $I\in \Lambda_1$, $$(1-\mu)\Theta \text{Lip}(f)\leqslant l r_I \leqslant (1+\eta)\Theta \text{Lip}(f).$$ The first pair of inequalities follows from the fact that $r_{[lk]}=\mathbb{E}\varrho_Y^f(\ee, -\ee)\leqslant a_{lk}(\mathcal{F})\text{Lip}(f)< (1+\eta)\Theta \text{Lip}(f)$ and $F,f$ were chosen such that $(1-\eta)\Theta \text{Lip}(f)<\mathbb{E}\varrho_Y^f(\ee, -\ee)$. Now fix $1\leqslant j\leqslant l$ and define $g:2^{lk}\times 2^k\to 2^{lk}$ by letting \begin{displaymath}
g(\ee, \delta)(i) = \left\{
\begin{array}{lr}
\ee(i) & : i\in [lk]\setminus I_j\\
\delta(m)\ee(i) & : i=(j-1)k+m.
\end{array}
\right.
\end{displaymath} Note that for a fixed $\ee\in 2^{lk}$, the map $f_\ee:2^l\to 2^{lk}$ given by $f_\ee(\delta)\mapsto g(\ee, \delta)$ scales distances by a factor of $1/l$. From this it follows that for a fixed $\ee$, the map $\delta\mapsto f(g(\ee, \delta))$ has Lipschitz constant not more than $\text{Lip}(f)/l$. Therefore \begin{align*} r_{I_j} & = \mathbb{E}_\ee\varrho_Y^f(\ee, I_j\ee) = \mathbb{E}_\ee \mathbb{E}_\delta \varrho_Y^f(g(\ee, \delta), g(\ee, -\delta)) \leqslant a_k(\mathcal{F})\mathbb{E}_\ee\text{Lip}(f_\ee) \leqslant (1+\eta)\Theta \text{Lip}(f)/l. \end{align*} From this we deduce that $$\max_{I\in \Lambda_1} lr_I \leqslant (1+\eta)\Theta\text{Lip}(f).$$ To see that $(1-\mu)\Theta \text{Lip}(f) \leqslant lr_I$ for all $I\in \Lambda_1$, suppose that there exists $I_0\in \Lambda_1$ such that $r_{I_0}<(1-\mu)\Theta \text{Lip}(f)/l$. Then \begin{align*} (1-\eta)\Theta \text{Lip}(f) & < r_{[lk]} \leqslant \sum_{I\in \Lambda_1} r_I <(1-\mu)\Theta \text{Lip}(f)/l+(l-1)(1+\eta)\Theta \text{Lip}(f)/l \\ & = \Bigl[\frac{1-\mu}{l}+ (1+\eta)\bigl(\frac{l-1}{l}\bigr)\Bigr]\Theta \text{Lip}(f) < (1-\eta)\Theta \text{Lip}(f).\end{align*} This is a contradiction and yields the remaining inequality. Here we are using the fact that $r_{[lk]}\leqslant \sum_{I\in \Lambda_1} r_I$, which follows from the triangle inequality as in the proof from the previous section.
Let $\Upsilon_I=(|\Phi_I-\phi_I|>t\text{Lip}(f)/4)$ and $\Omega_I=(|r_I-\phi_I|>t\text{Lip}(f))$. We claim that $\Phi_I$ is $2\lambda\text{Lip}(f)$-Lipschitz taking values in $[0, \lambda\text{Lip}(f)]$, so $\mathbb{P}(\Upsilon_I)\leqslant t/4\lambda$, $|\phi_I-r_I|\leqslant t/2$, and $\mathbb{P}\Big(\bigcup_{I\in T\setminus \Lambda_2}\Omega_I\Bigr)<1/2^l$. Since $\text{diam}(2^{lk})=1$ and $\text{Lip}(F\circ f)\leqslant \lambda\text{Lip}(f)$, we deduce that $\Phi_I$ takes values in $[0,\lambda\text{Lip}(f)]$. Next let us show that $\Phi_I$ is $2\lambda\text{Lip}(f)$-Lipschitz. Fix $\ee_1, \ee_2\in 2^{lk}$ and note that \begin{align*} \varrho_Y^f(\ee_1, I\ee_1) - \varrho_Y^f(\ee_2, I\ee_2) & \leqslant \varrho_Y^f(\ee_1, \ee_2)+\varrho_Y^f(\ee_2, I\ee_2)+\varrho_Y^f(I\ee_2, I\ee_1)-\varrho_Y^f(\ee_2, I\ee_2) \\ & = \varrho_Y^f(\ee_1, \ee_2)+ \varrho_Y^f(I\ee_1, I\ee_2)= 2\varrho_Y^f(\ee_1, \ee_2) \leqslant 2\lambda\text{Lip}(f) \partial(\ee_1, \ee_2). \end{align*} By symmetry, we deduce that $\Phi_I$ is $2\lambda\text{Lip}(f)$-Lipschitz. From this it follows that $$\mathbb{P}(\Upsilon_I) = \mathbb{P}\Bigl(|\Phi_I-\phi_I|> \frac{t}{8\lambda}(2\lambda \text{Lip}(f))\Bigr) \leqslant \alpha \exp(-\beta t k/8\lambda)<t/4\lambda.$$ Therefore \begin{align*} |\phi_I-r_I| & \leqslant \mathbb{E}1_{\Upsilon_I}|\phi_I-\Phi_I| + \mathbb{E}1_{\Upsilon_I^c}|\phi_I-\Phi_I| \leqslant \lambda\text{Lip}(f)\mathbb{P}(\Upsilon_I)+ t\text{Lip}(f)/4 \leqslant t\text{Lip}(f)/2. \end{align*} Therefore $\Omega_I\subset \Upsilon_I$ and $$\mathbb{P}\Bigl(\bigcup_{I\in T\setminus \Lambda_2} \Omega_I\Bigr) \leqslant \sum_{I\in T\setminus \Lambda_2} \mathbb{P}(\Upsilon_I) \leqslant (l+1)\alpha \exp(-\beta tk/8\lambda)<1/2^l.$$ From this and Remark \ref{density}, we may define $g:2^{lk}\times 2^l\to 2^{lk}$ by $g(\ee, \delta)(i)=\delta(j)\ee(i)$ when $i\in I_j$ and choose $\ee_0\in 2^{lk}$ such that $\{g(\ee_0, \delta): \delta\in 2^l\}\subset \bigcap_{I\in T\setminus \Lambda_2} \Omega_I^c$.
Now define $h:2^l\to X$ by $h(\delta)=f(g(\ee_0, \delta))$. Note that \begin{align*} (1-\mu) l\max_{i\in [l], \delta\in 2^l} \varrho_Y^h(\delta, d_i\delta) & \leqslant \frac{1-\eta}{1+\eta}\cdot l\max_{I\in \Lambda_1, \delta\in 2^l} \Phi_I(g(\ee_0, \delta)) \leqslant lt\text{Lip}(f)+ \frac{1-\eta}{1+\eta}\cdot l r_I \\ & \leqslant lt\text{Lip}(f)+(1-\eta)\Theta \text{Lip}(f) \leqslant lt\text{Lip}(f)+r_{[lk]} \\ & \leqslant (l+1)t\text{Lip}(f)+\min_{\delta\in 2^l}\Phi_{[lk]}(g(\ee_0,\delta)) \\ & < (1+\mu)\min_{\delta\in 2^l} \Phi_{[lk]}(g(\ee_0,\delta)) = (1+\mu)\min_{\delta\in 2^l}\delta_Y^h(\delta, -\delta). \end{align*} Here we are using the fact that since $$M \text{Lip}(f):=\min_{\delta\in 2^l} \Phi_{[lk]}(g(\ee_0, \delta)) \geqslant r_{[lk]} -t\text{Lip}(f) \geqslant ((1-\mu)\Theta - t)\text{Lip}(f) \geqslant (1-2\mu)\Theta \text{Lip}(f),$$ it follows from our choice of $t$ that $$(l+1)t\text{Lip}(f) +M\text{Lip}(f) < (1+\mu)M \text{Lip}(f).$$ Now our choice of $a$ and $\mu$ combined with Proposition \ref{sharp} yield that $\text{dist}(F\circ h)\leqslant D$.
We next show that for any $\delta_1, \delta_2\in 2^l$, $\vartheta \varrho_X^h(\delta_1, \delta_2) \leqslant \varrho_Y^f(\delta_1, \delta_2)$. First observe that the map $\delta\mapsto g(\ee_0, \delta)$ is length preserving, so $\text{Lip}(h)\leqslant \text{Lip}(f)$. Therefore $$\varrho_X^h(\delta_1, \delta_2) \leqslant \text{Lip}(f) \partial(\delta_1, \delta_2).$$ Now let us observe that \begin{align*} \text{Lip}(F\circ h) & = \max_{i\in [l], \delta\in 2^l} l\varrho_Y^h(\delta, d_i\delta)\geqslant \min_{I\in \Lambda_1, \delta\in 2^l} l\Phi_I(g(\ee_0, \delta)) \\ & \geqslant \min_{I\in \Lambda_1} lr_I - lt\text{Lip}(f) \geqslant (1-\mu)\Theta \text{Lip}(f) -lt\text{Lip}(f)\geqslant (1-2\mu)\Theta \text{Lip}(f). \end{align*} Since $h$ has distortion at most $D$, for any $\delta_1, \delta_2\in 2^l$, $$\varrho_Y^h(\delta_1, \delta_2) \geqslant \frac{(1-2\mu)}{D}\cdot \Theta \text{Lip}(f) \partial(\delta_1, \delta_2) \geqslant \vartheta \text{Lip}(f) \partial(\delta_1, \delta_2)\geqslant \vartheta\varrho_X^h(\delta_1, \delta_2).$$ An appeal to Lemma \ref{taco bell} finishes the proof.
\end{proof}
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{"url":"https:\/\/www.mersenneforum.org\/showthread.php?s=28a95111c960b8222479c68083eb66e5&t=1909","text":"mersenneforum.org Error 2252 with 23.7\n Register FAQ Search Today's Posts Mark Forums Read\n\n 2004-01-13, 07:43 #1 bej \u00a0 Jan 2004 3\u00d75 Posts Error 2252 with 23.7 I've been having trouble reporting results to primenet with several prime95 clients that were running version 23.7.1. I noticed another client running 23.4.1 was successfully connecting to primenet, and after back-leveling the 23.7 clients to 23.4, they also connect successfully. Is anyone else seeing the same problem? A didn't see anything in a brief scan of the forum. I traced the connect attempt of 23.4 and 23.7. 23.4 sends the following (good) HTTP request: Code: GET http:\/\/mersenne.org:80\/cgi-bin\/pnHttp.exe?ps&32516&.&. HTTP\/1.0 while 23.7 is sending only: Code: GET \/cgi-bin\/pnHttp.exe?ps&32516&.&. HTTP\/1.0 with no Host: header in the request. Any ideas on why 23.7 (also tried 23.8) sends the invalid HTTP request? Systems this is running on are both Win XP SP1 and W2K Pro & Server SP4. Thanks, Brian\n 2004-02-08, 23:46 #2 bej \u00a0 Jan 2004 3\u00d75 Posts Found the primenet.ini UseFullURL=1 setting in whatsnew.txt and that works for me with prime95 23.7. I never had problems communicating with primenet prior to 23.4 though (which the doc says always used relative urls?). It still looks like the current version of prime95 with no UseFullURL setting should be (but isn't) sending a Host: header in the request.\n 2004-02-09, 23:11 #3 dsouza123 \u00a0 \u00a0 Sep 2002 10100101102 Posts How did you trace it ?\n 2004-02-10, 01:08 #4 bej \u00a0 Jan 2004 3\u00b75 Posts I traced it with CommView, a Windows-based network\/packet monitor, on the system I'm running Prime95 on. But I just tried it again and noticed something I hadn't noticed before -- when the request is sent, I don't receive any response whatsoever -- not even a TCP ack. That made me suspicious of my (external) firewall -- when I bypass the firewall, the short URL form now works OK. And I must have been reading the HTTP 1.1 RFC previously also -- HTTP 1.1 requires the Host: header, but HTTP 1.0 doesn't. So since Prime95 claims to be sending an HTTP\/1.0 request, it's not an invalid request as I originally thought. Thanks for forcing me to look at this again.\n 2004-02-10, 02:20 #5 Prime95 P90 years forever! \u00a0 \u00a0 Aug 2002 Yeehaw, FL 162678 Posts OK you HTTP experts, you just volunteered to help fix prime95! Prime95 (linux version) has always sent the relative URL. It was that way in the original code from Peter Hunter. Windows Prime95 started using the same code in version 22. To make some proxy servers happy, I tried sending full URLs. 99% of users were happy, but someone had a setup where that did not work. So where do we go from here? Are there more than 3 choices: 1) Full URL. 2) Relative URL. 3) Relative URL with host: Which should be the default? Should we try one and if we get an error (which errors?) try another? What else should prime95 be doing?\n 2004-02-10, 06:03 #6 garo \u00a0 \u00a0 Aug 2002 Termonfeckin, IE 3\u00d7919 Posts Here is the relevant snippet from the HTTP1.1 spec. Code: The Request-URI is a Uniform Resource Identifier (section 3.2) and identifies the resource upon which to apply the request. Request-URI = \"*\" | absoluteURI | abs_path | authority The four options for Request-URI are dependent on the nature of the request. The asterisk \"*\" means that the request does not apply to a particular resource, but to the server itself, and is only allowed when the method used does not necessarily apply to a resource. One example would be OPTIONS * HTTP\/1.1 The absoluteURI form is REQUIRED when the request is being made to a proxy. The proxy is requested to forward the request or service it from a valid cache, and return the response. Option 3, relative URI with host field is the preferred method for contacting web servers but option 1 is REQUIRED for proxies. Last fiddled with by garo on 2004-02-10 at 06:08\n2004-02-10, 12:12 \u00a0 #7\njinydu\n\nDec 2003\nHopefully Near M48\n\n2\u00d73\u00d7293 Posts\n\nQuote:\n Originally Posted by bej Any ideas on why 23.7 (also tried 23.8) ... Thanks, Brian\nVersion 23.8? I've never heard of it, and couldn't find it at www.mersenne.org.\n\n2004-02-10, 15:26 \u00a0 #8\nS80780\n\nJan 2003\nfar from M40\n\n53 Posts\n\nQuote:\n Originally Posted by jinydu Version 23.8? I've never heard of it, and couldn't find it at www.mersenne.org.\nTry ftp:\/\/mersenne.org\/gimps\/ . There you'll find statistics, new versions and stuff. E.g., ftp:\/\/mersenne.org\/gimps\/p95v238.zip.\n\nCheers,\n\nBenjamin\n\n Similar Threads Thread Thread Starter Forum Replies Last Post garo GPU Computing 10 2013-04-27 19:01 Unregistered Information & Answers 3 2008-05-05 05:40 Unregistered PrimeNet 37 2006-11-29 20:36 Old man PrimeNet PrimeNet 0 2006-02-05 02:27 nevillednz PrimeNet 15 2004-05-17 23:08\n\nAll times are UTC. 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Archive For SUMMER 10
Downtown's Will to Form
Image Conflict
Issue 6 Introduction
Empty Amusement
In Front of the Sea
Audience as Participants
de[FENCING]
Game Seeds
Delivering an Emotion
What Amuses You?
She Was Not Amused
San Francisco troublemakers
© Alex Lehnerer
Essay by architect and urban designer Alex Lehnerer
Architect Alex Lehnerer writes about the "Opposition Drawings", the tool that emerged in San Francisco in the 1960s and 1970s and was used by supporters and opponents of the proposed high-rises.
Every self-respecting North American city needs one — including San Francisco, of course. One simply can no longer do without a high-profile financial or central business district. Finally by the 1960s, the "inspiring drama of the free enterprise" [1] required a suitable and solid representation in every larger US American city.
When it came to the development of a typical American downtown, San Francisco in this period was among the most sensitive and hesitant communities. Tall buildings stood in direct competition with tall hills, with the outstanding landscape qualities of the Bay Area and its fine-grained development contradicting the image the citizenry had of their town. And so San Francisco took its sweet time before allowing its high-rises to go forward. Between 1930 and 1958, only a single tall office building was constructed, and only in 1959 did the city acquire its first modern high-rise, the Crown Zellerbach Building on Market Street.
Policemen and Trouble Makers
Without criminals, there would be no police — and the same is true of fires and the fire department. Both institutions work simultaneously preventively as well as via intervention. Precedential cases originated by these troublemakers are constantly necessary, as they lead to sensitization. Such villains then compel the introduction of preventive measures designed to eliminate the emergent problem for the wider community once and for all.
In the San Francisco of the 1960s, a quartet of such troublemakers suddenly stood together in close proximity, generating public annoyance in the financial and business district: the Holiday Inn and the Transamerica Building were regarded as "too bizarre," the Embarcadero Center as "too large and too bulky "and the Bank of America Building as "too big and too dark." Nonetheless, it seemed almost impossible to charge the Bank of America with ignorance of its context — contradicting such an argument were the 1500 bay windows on its façade.
The anti high-rise movement then active in San Francisco had definitively lost the battle against the four offenders. All that remained was public defamation. The Transamerica Building had an especially hard time of it. Its appearance was said to be incompatible with an attractive downtown. Regarded as embarrassing and uneconomical, it was soon nicknamed "The Egyptian Embassy," and was illustrated in newspapers and magazines wearing a dunce's cap.
In the late 1960s and early 1970s, the wrath of activists was directed principally against purportedly ugly, unwieldy, excessively tall and poorly sited office buildings. [2] One recipient of ill favor was US Steel, when it proposed erecting a 150 meter tower on the waters of the Bay. Newspaper articles caricatured the building as an ungainly "steel giraffe," one that palpably disturbed the site's sense of balance and proportion. [3]
But the controversy gave rise to a remarkable tool. Since the ranks of the high-rise's opponents contained a few architects proficient in drafting, so-called Opposition Drawings [4] could be prepared. On the basis of a bird's-eye view seen from the direction of Bay Bridge, architects and supporters of the US Steel Building attempted to demonstrate just how well the structure was adapted to the existing urban context. The building's opponents now provided a perspective from the opposite direction, from Telegraph Hill. These images showed how strongly the US Steel Building would plunge the proud Bay Bridge into shadow — that is to say, how a private office building would blight a public icon.
In the end, in 1971, the San Francisco Supervisors (SUPES) granted permission for the construction of the US Steel Building on this site, stipulating that it would have a height of only 53 m (175 ft) at most. This deployment of counter-perspectives functioned as well in the case of the 20-story Haas Towers project — against which no objections were raised until the lawyer had it immortalized from a number of different perspectives from Russian Hill. Virtual buildings were now regarded as having evidentiary status, and the discipline of urban simulation was launched. The more scientific and the more complex the methods used to generate these anamorphic images, the greater the public's preparedness to regard them as objective. The first setting for such activities was located directly behind San Francisco's Bay Bridge, at the Environmental Simulation Laboratory at UC Berkeley, founded by Donald Appleyard in 1972 and containing fabulous apparatuses, gigantic city models, and swiveling endoscopes by means of which one could navigate at eye level through models of the city.
Real Simulation
Even such an ingenious simulation cannot replace a personal or real interaction, that is to say, a 1:1 experience of the critical object in urban space. Switzerland and its cities are well aware of this fact. Ideally, one drives or moves through the city physically and in real time.
In 2007, it was possible to contemplate a 126 m (413 ft) tall steel scaffolding that had been erected in the former industrial quarter of the Swiss town of Zürich (Kreis 5) from close up. Next to the Hardbrücke, its quartet of steel profiles marked out the edges and complete height of the future Swiss Prime Tower. According to law, each Swiss building project must be rendered beforehand in its actual planned dimensions through a socalled Baugespann (structural mockup). The ordinance includes the construction of dormer windows, tool sheds, but also of high-rise buildings. For a specified period of time (2 months), the citizenry, assuming it is willing to exercise a minimum of imagination, has the opportunity to visualize the project within its three-dimensional boundaries and to discuss its urban integrity. Thereafter follows an act of participatory democracy, the voting in a referendum. In the case of the Swiss Prime Tower, however, the full 126 meter tall scaffolding would not have been necessary. For reasons of proportionality and safety, the city would have preferred a "preview" measuring just a fraction of that height. Since they were planning to erect the tallest building in Switzerland to date, the clients spared no expense and effort in erecting this spectacular simulation, spending more than 100,000 Swiss francs for purposes of prestige and advertising.
The virtual envelope becomes reality for immediate public review: 126 m tall "Baugespann" of the future Swiss Prime Tower, Zürich.
Little Big Plan
The desire not only to simulate visual qualities, but to actively and preventively guide them as well was manifested beginning in the 1970s in San Francisco's General Plan, and finally in the same city's 1983 Downtown Plan, which was the work of planning director Dean Macris. [5] The plan reduced building heights and building bulk on the basis of a floor area ratio and geometric bulk definition. It envisioned landmark protection status for 266 important buildings, and required the preparation of a shadow study in order to ensure that new buildings would permit adequate sun and light to reach the surrounding streets. Whole streets and their views acquired quasi-landmark status.
All of this is familiar already from New York. Of interest is this manifestation of San Francisco's collective taste and the yearning for visual attractiveness. This approach called for the architectural treatment of high-rise roofs with hat-style structures in order to avoid the so-called "refrigerator look" (i.e., a monotonous sequence of androgynous glassfronted crates). [6]
Cynical commentators soon joined the fray. Allan Temko, architecture critic with the San Francisco Chronicle, commented that the plan was more concerned with aesthetic matters than with the effective restriction of growth and density in San Francisco's downtown:
"Nor would I trust Macris' chief assistant on design matters… His contributions on the Plan would not require architects but milliners. So we'd put these party hats on buildings, as if we didn't have the most colossal dunce cap in the world on the Transamerica Building." [7]
It was calculated that by the year 2000, the plan would in principle permit the construction of more than 24 million sf (2.2 million square meters) of new office surface, in particular to the south of Market Street. This meant growth rates similar to those registered in the years prior to the plan's adoption. Such figures recall the reproach of "over-zoning" that had been leveled against New York's 1916 resolution. As far as San Francisco's citizenry were concerned, Dean Macris' plan was simply too weak, and represented an inadequate planning for exercising control over the "vertical earthquake" taking place in San Francisco's downtown. Resistance was so great that in 1985, the city's Board of Supervisors (the municipal governing body) endowed the plan with the force of law, but with an addendum, a limit on the maximum building volume per annum. To begin with, this was set at a maximum of 950,000 sf for the entire city — a "growth cap" lower than the square meters covered by certain individual high-rises in New York City.
San Francisco's specific bulk areas.
Method of measuring bulk (top); Guidelines for each bulk area: a tower's max. diagonal and lateral dimensions (bottom)
Map of streets with excellent views that deserve protection, San Francisco.
Anti-refrigerator look
What San Francisco was proposing here was definitively "the first quota system for city planning." [8] Alongside the "beauty contest," designed to determine whether the "hat" on the high-rise tower corresponded to the prevailing ideal of beauty, decisions would be yielded to the discretion of a review panel, which granted building permits to the numerous applicants and filled the annual quota. Administratively, it is only possible to get a handle on this process when a multiplicity of additional guidelines are introduced, for example those regarding the economic relevance of a specific project for the city. One year later, in 1987, Proposition M reduced the annual quota by 50% to 475,000 sf, making high-rise development in San Francisco virtually impossible. Even where this was not the case, it was charged that (to cite New York Times critic Paul Goldberger) "they have all turned out to be tame examples of the post-modern style, cautious little buildings that struggle not to offend." Goldberger goes on to criticize the high degree of determination exercised through the required review process:
"San Francisco no longer has planners, it has design czars. The city government through its planning department and the public through referendums have become the controlling forces in determining what happens in downtown San Francisco. They go far beyond the normal mandate of setting out the basic outlines of growth through zoning laws, as planners do everywhere else; here, they determine the specific design of individual projects, and ultimately decide whether or not projects can go ahead at all." [9]
The "hats" set on the towers, however, were intended to counteract another downtown tendency familiar under the term Skyline Wall Syndrome. As speculative projects controlled by similar marginal conditions and land costs, office high-rises tend to have similar numbers of stories, and hence similar heights. In many cases, such utilization does not approach the potential legal limit, but instead corresponds to a purely economic calculus: how many times must the building lot be multiplied into the heights in order to be profitable while avoiding excessive costs, for example, for vertical accesses? This Economic Height often lies well below the maximum utilization set by law, and by falling below it, disempowers the potential for architectonic shaping offered by such legislation. And the developer of a neighboring parcel determines the height of his tower on the basis of similar calculations. The height differences between buildings deemed desirable on the basis of aesthetic criteria, then, can be administered on the basis of general rules only with great difficulty. And if the maximum possible utilization is set somewhat below the economic height that prevails in a particular period, then that standard functions as a height limit, once again generating a uniform sequence of buildings.
A genuine predicament for the city. A successful, that is to say, interesting and dramatic skyline with its peaks, valleys, and jagged canyons, is very difficult to shape consciously. Lowered height limits, once again, generate uniformity, while also restraining inner-city growth in undesirable ways. Helpful instead is a constant radical revision of building laws and maximally turbulent pricing developments in the land and real estate markets. One indispensable ingredient, of course, is a local, egocentric corporate headquarters that functions like a mountaintop cross, so to speak; another is a collective architectonic expression on the part of office towers that shifts rapidly over the years.
[1] Earle Shultz and Walter Simmons (1959), "Offices in the Sky", 7.
[2] Chester W. Hartman (1984), "The Transformation of San Francisco," 269.
[3] Donald Appleyard and Lois Fishman (1977), "High Rise Buildings Versus San Francisco," 87.
[5] Hartman (1984), 274.
[6] Ibid, 273
[7] The City of San Francisco (1983), "The Downtown Plan—Proposal for Citizen Review."
[8] Paul Goldberger (1987), "When Planning Can Be Too Much of a Good Thing."
This essay has been republished with permission from the author from the book "Grand Urban Rules" by Alex Lehnerer published by 010 Publishers in 2009.
Alex Lehnerer is an architect and urban designer who received his PhD from the ETH in Zurich and is currently based in Chicago, where he holds a position as Assistant Professor at the University of Illinois, School of Architecture. He is also partner at Kaisersrot in Zurich, CH and ALSO Architekten.
www.alexlehnerer.com | {
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Оксопентахлорорениевая кислота — неорганическое соединение металла рения с формулой H2[ReCl5O]. При нормальных условиях представляет собой неустойчивую жидкость.
Получение
Восстановление рениевой кислоты в концентрированном растворе соляной кислоты.
Свойства
Оксопентахлорорениевая кислота образует неустойчивую жидкость. Соединение не получено в свободном состоянии, существует только в водном растворе.
Литература
Комплексные соединения рения
Неорганические кислородсодержащие кислоты
Комплексные хлориды | {
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{"url":"https:\/\/www.meritnation.com\/ask-answer\/question\/volume-of-gases-evolved-when-dil-h2so4-is-electrlysed-using\/electrochemistry\/10234999","text":"# volume of gases evolved when dil .H2SO4 is electrlysed using 2F at STP\n\n1. Electrolysis of H2SO4 gives following two reactions :\n\n2H+ + 2e-\u00a0$\\to$\u00a0H2\n2H2$\\to$ O2 + 2e\u200b- \u00a0+ 4H+\n\n2. 2 mole of electrons give 1 mole of O2 and H2 each.\nSo,2F of electricity gives 2 moles of total gases.\n\n3. Volume of 1 mole = 22.4 L\nVolume of 2 moles = 44.8 L\n\n\u2022 -221\nWhat are you looking for?","date":"2021-01-25 11:32:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 2, \"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\": 0.32547828555107117, \"perplexity\": 11513.038462860433}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-04\/segments\/1610703565541.79\/warc\/CC-MAIN-20210125092143-20210125122143-00585.warc.gz\"}"} | null | null |
El Colegio Oficial de Ingeniería Informática de la Comunidad Valenciana (COIICV) es una corporación de derecho público integrada por los ingenieros en informática de la Comunidad Valenciana, España. El COIICV pertenece al Consejo General de Colegios Profesionales de Ingeniería Informática (CCII), corporación colegial de ingeniería informática a nivel nacional, desde la constitución del mismo.
Historia
El COIICV se creó por Ley 6/2000 de las Cortes Valencianas, siendo uno de los primeros colegios autonómicos de ingeniería informática en crearse en España.
La Asamblea Constituyente se celebró el 28 de noviembre de 2000, aprobándose los Estatutos presentados y se celebraron las primeras elecciones en la que participaron todos los ingenieros en informática censados.
A las elecciones a la Junta de Gobierno concurre una única candidatura encabezada por Julián del Valle Fernández, en aquel momento Presidente de la Asociación de Licenciados e Ingenieros en Informática de la Comunidad Valenciana, resultando elegida. Lo mismo sucedió tras las elecciones de 2004 y 2008. En 2012 se celebran las cuartas elecciones, resultando elegida la candidatura encabezada por Juan Pablo Peñarrubia Carrión. Las quintas elecciones se celebran en 2016, resultando elegida la candidatura encabezada por F. Alejandro Blasco Iborra, actual Presidente del Colegio.
El COIICV es miembro de pleno derecho del CCII.
Fines
Los fines esenciales del Colegio Oficial de Ingeniería Informática de la Comunidad Valenciana se resumen en promover el progreso de la sociedad de la información y el conocimiento, velar por el cumplimiento de los preceptos constitucionales sobre el uso de la informática, ordenar la profesión, defender los intereses profesionales de los colegiados, representar el ejercicio de la profesión en el ámbito autonómico, hacer cumplir la ética profesional y normas deontológicas del colectivo, velar por el adecuado nivel de calidad de las prestaciones profesionales de los colegiados a través de la formación y perfeccionamiento de éstos, fomentar la solidaridad profesional y la investigación, el desarrollo e innovación (I+D+i) en el ámbito de la informática y de las tecnologías afines.
Premios Sapiens
Los Premios Sapiens son los premios que anualmente concede el Colegio en el marco de semanainformatica.com y del Congreso de la Ingeniería Informática de la Comunidad Valenciana, con el objetivo de distinguir aquellos profesionales, empresas y administraciones públicas que sobresalen en la valorización de la Profesión Informática.
'Sapiens' hace referencia a sabiduría, conocimiento, inteligencia,... Éstas son justamente las características que el Colegio desea destacar, premiar y dar a conocer a la Sociedad como elementos íntimamente ligados a la Informática.
Se conceden premios en las siguientes categorías:
Premio al Profesional
Premio a la Empresa
Premio a la Administración Pública
Premio Académico
Premio al Proyecto Fin de Carrera
Premio al Proyecto
Referencias
Enlaces externos
www.coiicv.org.
www.ccii.es.
Ingenieros en Informática
Asociaciones de ingenieros | {
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Q: Can´t connect to heroku PostgreSQL database with Node.js and Express, no error is showing So i recently created an Account at heroku.com and created a PostgreSQL database. First of all i wanted to test if i could store something into the database (One table was already created without node.js) but for some odd reason the callback function of my dbClient.query never executed. I then realized that the Database does not connect to my code. My password, host, etc. should all be right and i can connect from extern sites but not from my node.js code.
Here´s my code:
var express = require("express");
var pg = require("pg");
var bodyParser = require("body-parser");
var CON_STRING = process.env.DB_CON_STRING;
if (CON_STRING == undefined) {
console.log("Error: Environment variable DB_CON_STRING not set!");
process.exit(1);
}
pg.defaults.ssl = true;
var dbClient = new pg.Client(CON_STRING);
dbClient.connect();
var urlencodedParser = bodyParser.urlencoded({
extended: false
});
var PORT = 3000;
var app = express();
app.set("views", "views");
app.set("view engine", "pug");
app.get("/shoppingitems", function (req, res) {
res.render("shoppingitems")
});
app.post("/shoppingitems", urlencodedParser, function(req, res){
var shoppingItem = req.body.shoppingItem;
dbClient.query("INSERT INTO shoppinglist (title) VALUES ($1)", [shoppingItem], function(dbError, dbResponse){
console.log("function called!");
res.redirect("/shoppingitems");
})
console.log("test");
})
app.listen(PORT, function () {
console.log(`Shopping App listening on Port ${PORT}`);
});
It does not work with other configs of dbClient.query.
A: You need to set the environment variable on your localhost, depending on your OS.
*
*For Mac & Linux use 'export' keyword in Terminal for YOUR_ENV_VAR, example:
export DB_CON_STRING=postgres://ufohlgoihdgfalr:2fb913jhazsxd541469b972fcac862ddf664620998e87c96787569poiude4ab0b@ec2-34-238-26-109.compute-1.amazonaws.com:5432/d587yhytte32gj
You can check your variable using 'echo' and $ sign in Mac and Linux, for example
echo $DB_CON_STRING
It should give you the string if the variable is set or not.
Note that will be only work for the current session in the Terminal, if you want to keep it and use it for a long time or after closing/opening the terminal then it is better to add it to your .bash_profile. Same way, just make it saved in the /uerMacUser/.bash_profile
For windows you can do it using GUI, for quick access use in 'Run' rundll32.exe sysdm.cpl,EditEnvironmentVariables
Heroku Postgres uses different environment variable named 'DATABASE_URL' so you might want to consider this if you want to deploy on heroku later, otherwise your application will not start on heroku.
| {
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This wonderful selection of books shows characters appreciating others and the beauty of the world.
This picture book list highlights different communication methods in positive and effective ways.
These wonderful picture books feature female lead characters as smart, confident, and courageous.
Share these picture books to inspire children to be reflective about their own learning and behaviour.
These picture books present independent and effective thinkers who develop creative ideas, make connections and problem solve.
Have a discussion with your child about donating some of their old toys to charity. Take a look at some great picture books to support ideas of recycling and generosity. | {
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// -*- c++ -*-
// Generated by gmmproc 2.46.2 -- DO NOT MODIFY!
#ifndef _GIOMM_TLSPASSWORD_P_H
#define _GIOMM_TLSPASSWORD_P_H
#include <glibmm/private/object_p.h>
#include <glibmm/class.h>
namespace Gio
{
class TlsPassword_Class : public Glib::Class
{
public:
#ifndef DOXYGEN_SHOULD_SKIP_THIS
typedef TlsPassword CppObjectType;
typedef GTlsPassword BaseObjectType;
typedef GTlsPasswordClass BaseClassType;
typedef Glib::Object_Class CppClassParent;
typedef GObjectClass BaseClassParent;
friend class TlsPassword;
#endif /* DOXYGEN_SHOULD_SKIP_THIS */
const Glib::Class& init();
static void class_init_function(void* g_class, void* class_data);
static Glib::ObjectBase* wrap_new(GObject*);
protected:
//Callbacks (default signal handlers):
//These will call the *_impl member methods, which will then call the existing default signal callbacks, if any.
//You could prevent the original default signal handlers being called by overriding the *_impl method.
//Callbacks (virtual functions):
static const guchar* get_value_vfunc_callback(GTlsPassword* self, gsize* length);
static const gchar* get_default_warning_vfunc_callback(GTlsPassword* self);
};
} // namespace Gio
#endif /* _GIOMM_TLSPASSWORD_P_H */
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,094 |
Macrocoma aladina é uma espécie de escaravelho de folha de Arábia Saudita, descrito por Daccordi & Medvedev em 1996.
Referências
aeneonigra
Coleópteros da África Setentrional
Coleópteros descritos em 1996 | {
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Activist investor pushes McDonald's to make a shake-up
By Dave Carpenter
CHICAGO (AP) – An activist hedge-fund investor stepped up pressure Tuesday on McDonald's Corp., telling Wall Street the fast-food chain can boost its stock price dramatically if it spins off 65 percent of company-owned restaurants and focuses on more profitable real estate.
William Ackman, whose Pershing Square Capital Management LP owns a 4.9 percent stake in McDonald's, outlined publicly for the first time a strategy similar to one Pershing used to force divestitures _ and a higher share price _ this year at rival Wendy's International Inc.
McDonald's signaled its willingness to fight the Pershing proposal, dismissing it as "an exercise in financial engineering" that could pose serious strategic and financial risks and hurt its long-term health.
In a detailed presentation aimed at demonstrating that the company's stock is undervalued, Ackman told a New York investment conference he has urged McDonald's to spin off nearly two-thirds of the 8,000 restaurants it owns and borrow $14.7 billion against its real estate to buy back shares.
Those moves, he said, would help drive McDonald's stock as much as 50 percent to $45 per share to $50 per share and improve management focus and incentives at both the real estate-centered McDonald's and the new unit. The Oak Brook, Ill.-based chain owns 37 percent of the land beneath its more than 30,000 McDonald's restaurants worldwide.
Pershing bought a 4.9 percent stake in McDonald's in September, although Ackman acknowledged after the presentation that "almost our entire position" is in options, not shares.
The purchase stirred widespread speculation about a possible McDonald's spinoff in light of Pershing's move this year at Wendy's International Inc., where it bought a stake and nudged management to sell real estate and spin off its Tim Hortons coffee-and-doughnut chain.
McDonald's, however, issued a statement for at least the fourth time in two months saying it intends to continue with its existing strategy and is not interested in a restructuring involving its real estate. The company said two separate outside advisers had evaluated the Pershing idea negatively.
"The proposal is an exercise in financial engineering and does not take into account McDonald's unique business model," said the statement, attributed to Chief Financial Officer Matthew Paull. "While we remain open to ideas, we simply will not jeopardize the long-term health of our company, nor our relationships with customers, franchisees and suppliers for such a financial engineering exercise."
Despite their differences, Ackman praised the company's management.
"This is not a case at all where we think McDonald's is doing a bad job," Ackman said in an hour-long speech to the Value Investing Congress that was accompanied by slides and broadcast on the Internet. "Quite the opposite _ we like management."
He called McDonald's "a very shareholder-friendly company. … So far we just haven't gotten the right idea, but we're getting close."
Backing from other institutional investors would give Ackman's effort more of a chance, regardless of McDonald's resistance. Vornado Realty Trust is among those to have made purchases in the wake of Pershing's stake, and others could follow. Leon Cooperman, who runs the $4 billion-plus Omega Fund, attended the conference and called Ackman's proposal "very impressive."
Asked by reporters afterward if he might take the effort hostile if McDonald's doesn't cooperate, Ackman said: "I'm the most persistent person you will ever meet, particularly when I believe I'm right. That said, I don't think this will ever go to a proxy battle. I'm a huge believer in the power of our ideas."
He said there is a fundamental misperception that McDonald's is a restaurant company that owns a lot of real estate, yet nearly 90 percent of its economic earnings _ excluding rent, fees and other items _ come from real estate.
Getting McDonald's management out of the business of running restaurants would shed a low-margin, capital-intensive unit and enable it to focus on getting better results from franchisees, who already own a majority of the outlets anyway, he said.
Shares in the company fell 61 cents to $33.32 in afternoon trading on the New York Stock Exchange. The stock is up about 10 percent in the past year but _ as Ackman noted _ it is virtually unchanged from five years ago, closing at $33.31 on Nov. 15, 2000. | {
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Q: Correct SQL usage in Python Using PYMYSQL I am attempting to write a SQL in python using PYMYSQL, which searches a table for a certain record with a set value, however while this sounds simple I cannot seem to do it below is my query:
SELECT Series_ID FROM series_information WHERE Series_Name "'+data +'"'
where the data is the value that I am searching for however the following error occurs:
pymysql.err.ProgrammingError: (1064, 'You have an error in your SQL syntax; check the manual that corresponds to your MySQL server version for the right syntax to use near \'"Spice And Wolf"\' at line 1')
The problem I believe is that I am not sure how to properly escape the data value if it has spaces in it and therefore would require quotation marks in the SQL query.
A: You're missing a comparison (like, =, etc) between Series_Name and data, as well as a ';' on the end of the query.
`'SELECT Series_ID FROM series_information WHERE Series_Name = "'+data +'";'
A: `SELECT Series_ID FROM series_information WHERE Series_Name "'+data +'"'`
Is not a valid SQL query did you mean:
`'SELECT Series_ID FROM series_information WHERE Series_Name like "'+data +'"'`
| {
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Avertissement de l'éditeur
Les numéros de pages et de notes de bas de page apparaissant dans les renvois internes correspondent à ceux de l'édition papier. Dans cette édition numérique, des liens sont installés permettant d'accéder aux passages concernés, mais selon la taille de caractères sélectionnée, le numéro de page peut-être différent de celui de l'édition papier.
Introduction
Jaurès, un inconnu célèbre
Jean Jaurès est un des hommes politiques français les plus célèbres. Un grand nombre de Français connaissent son nom grâce à de multiples rues ou bâtiments publics, quand ce ne sont pas directement des boutiques sises « rue Jean-Jaurès », qui rappellent la mémoire du tribun socialiste. Début 2014, l'université de Toulouse a été rebaptisée du nom de Jean Jaurès à l'approche des commémorations du centenaire de son assassinat. Mais qui connaît son œuvre hors des cercles de passionnés d'histoire et de quelques militants aguerris ? Assurément peu de monde. À ceux qui ne le connaissent pas ou peu, la présente anthologie entend faire découvrir Jean Jaurès à travers un choix de ses textes les plus emblématiques, recoupant quelques grands thèmes politiques de son temps.
De la République au socialisme
Né en 1859 dans une famille bourgeoise de Castres, Jaurès n'était pas destiné à devenir le grand dirigeant socialiste que l'histoire a retenu. Ses premiers pas dans la vie publique ne le distinguent guère d'autres républicains « opportunistes », c'est-à-dire de centre gauche. Ce brillant élève, normalien agrégé de philosophie et enseignant à l'université, un temps maire-adjoint de la municipalité de Toulouse et jeune député en 1885, à l'origine hostile aux socialistes, aurait certainement pu, comme bien d'autres, s'assurer une carrière politique confortable dans le sud de la France.
Mais Jaurès opère progressivement un « passage au socialisme » dans les années 1889-18921. Deux éléments majeurs ont joué dans cette « conversion », longtemps discutée par les historiens : d'une part, l'étude de la philosophie allemande, à laquelle il a consacré sa thèse complémentaire sur Les Origines du socialisme allemand (1892), et qu'il a découverte notamment à travers les discussions avec le bibliothécaire de l'École normale supérieure, Lucien Herr ; d'autre part, l'importance de la question sociale, dont il prend conscience au contact des mineurs de Carmaux, ces hommes aux conditions de travail souvent effroyables, rencontrés à l'occasion d'une grande grève en 1892. Jaurès est bien désormais devenu socialiste en 1892. À cette époque, les socialistes constituent encore un ensemble de nébuleuses dispersées ; dix ans plus tard, Jaurès se fixera comme tâche urgente l'unification de ces différents courants. Pour le moment il est bien l'un des leurs, élu en 1893 député du Tarn sur la base du programme du Parti ouvrier de Jules Guesde et Paul Lafargue, alors proches des idées de Karl Marx. Mais par rapport à ces derniers, le socialisme jaurésien va conserver ses attaches républicaines, jugées suspectes par une bonne partie des socialistes influencés par le marxisme. La République n'est-elle pas fondamentalement « bourgeoise », comme tend à le montrer la violence de la répression anti-ouvrière ?
Jaurès pense qu'il existe un lien profond de continuité entre la République et le socialisme. Certes, la République actuelle est imparfaite, mais elle reste pour lui le meilleur régime politique existant et c'est en son sein que le socialisme peut être progressivement développé : la République sociale ne peut pas naître, selon lui, d'un grand cataclysme comme Marx et Engels le prévoyaient en 1848, mais d'une démarche gradualiste qui tienne compte des acquis républicains. Cette visée politique doit être mise en rapport avec la grande admiration que Jaurès porte à l'héritage de la Révolution française de 1789. Dès les années 1889-1890, il établit une continuité étroite entre les combats révolutionnaires de 1789 et le socialisme contemporain. Entre 1900 et 1904, il publie les premiers volumes (plusieurs milliers de pages) dédiés aux premières années de la Révolution française, 1789-1794, d'une vaste Histoire socialiste de la France contemporaine, lancée à son initiative. Lire l'introduction (1900) et la conclusion (1908) à ces volumes – reproduites dans le présent recueil – nous permet de comprendre admirablement bien qui était Jaurès, et la vision qu'il avait du socialisme. Un irréductible optimisme quant à l'avenir traverse ces pages, particulièrement sensible dans la conclusion de 1908.
Jaurès, les républicains, les socialistes, les syndicalistes
S'il faut lire Jaurès pour le comprendre, il est nécessaire également de le situer par rapport aux autres grandes figures et forces politiques de son temps. Jaurès n'était de son vivant pas plus connu que Jules Guesde ou d'autres personnalités républicaines, que la postérité a pourtant souvent moins retenues. Il n'a fondamentalement jamais rompu avec les républicains, notamment les radicaux, avec lesquels il eut, pour reprendre la formule de Rémy Pech, « une dispute sans rupture2 ». Ses rapports avec les différents courants socialistes (cinq à la fin du XIXe siècle !) ont été, quant à eux, divers selon les moments. Jaurès a eu des relations difficiles avec les « guesdistes », les partisans de Jules Guesde considérés comme les introducteurs du marxisme en France. Un clivage important le sépare des proches de Jules Guesde et d'Édouard Vaillant sur la question décisive de l'État, plus précisément sur la participation gouvernementale. En 1899, un ministre socialiste, Alexandre Millerand, entre pour la première fois dans un gouvernement, celui de Waldeck-Rousseau. Dans le contexte de l'affaire Dreyfus, c'est un gouvernement de « défense républicaine », face au péril menaçant le régime, qui justifie selon certains la participation à l'exercice du pouvoir ; Jaurès est de ceux-là, quand Guesde et Vaillant vouent aux gémonies ce qui leur apparaît comme une inadmissible compromission. Ligne de partage qui recoupe les divergences sur l'attitude à adopter par rapport au capitaine Dreyfus : alors que Guesde juge néfaste de lancer les ouvriers dans une bataille pour un militaire bourgeois, Jaurès, lui, après quelques hésitations, fait appel à un principe d'humanité, au-delà des clivages de classe. Il est nécessaire pour le tribun socialiste de se retrouver aux côtés de ceux qui, comme Zola, accusent la justice d'être partiale et complice de l'institution militaire pour en couvrir les erreurs. Les Preuves sont un des grands textes de Jaurès sur le sujet, dont on lira la préface dans le présent recueil. Sur quelques points, il peut néanmoins exister des ponts entre Jaurès et les guesdistes ; parfois ces derniers ont renoncé assez tôt à certains aspects les plus doctrinaires, par exemple sur la question de la petite propriété paysanne : alors que certains marxistes estimaient que la petite propriété disparaîtra avec la concentration capitaliste, la majorité des socialistes français considère qu'elle doit être défendue. Au congrès de Nantes du Parti ouvrier français (guesdiste), en 1894, s'exprime sur ce point un accord entre Jaurès et Paul Lafargue, gendre de Karl Marx et proche de Guesde3.
Mais à partir de 1899, les terrains d'entente se font rares. Il existe bien « deux méthodes » différentes, dont débattent Jules Guesde et Jean Jaurès à l'invitation du maire de Lille, lors d'un meeting mémorable en 19004. Deux ans plus tard, le soutien de Jaurès au nouveau gouvernement d'Émile Combes s'inscrit dans la même logique : soutenant l'action réformatrice des radicaux – mais cette fois-ci sans participation des socialistes au gouvernement – Jaurès est alors élu vice-président de la Chambre des députés et devient le principal homme politique, avec Aristide Briand, qui permet le vote, puis la promulgation, de la loi de séparation des Églises et de l'État, à la fin de l'année 1905. Certains socialistes se demandent pourquoi Jaurès va si loin dans l'alliance avec les radicaux sur la question laïque : l'essentiel n'est-il pas de conquérir de nouveaux droits sociaux et politiques pour les ouvriers et de se situer dans une perspective de renversement du capitalisme ? Pour Jaurès, qui ne délaisse pas l'horizon révolutionnaire, il est nécessaire de détacher les ouvriers de l'emprise du catholicisme. Comment penser qu'une société encore très encadrée par l'Église puisse être réceptive aux idées socialistes ? Tout ce qui permettra la laïcisation des esprits, dans le respect des croyances religieuses de chacun, est un pas en avant pour le socialisme. La différence de méthode est encore très nette. Pourtant, entre-temps, quelques mois avant la loi de séparation, la dispersion des socialistes a enfin cessé : tous se retrouvent, au-delà de leurs divergences, au congrès du Globe à Paris, en avril 1905, qui fonde la Section française de l'Internationale ouvrière (SFIO), le parti socialiste désormais unifié. Plus que tout, Jaurès craignait que se fissure l'unité et qu'ainsi le socialisme français retombe dans sa situation d'éclatement d'avant 1905. L'unité était une des grandes œuvres de sa vie, et il y tenait tout particulièrement. Les déchirements ultérieurs de la gauche française l'auraient probablement vivement affecté...
Formellement, la rhétorique des partisans de Guesde l'emporte dans les statuts du nouveau parti5. Mais au congrès de la SFIO à Toulouse en 1908, il ne fait aucun doute que le point de vue de Jaurès a marqué des points décisifs. Lorsque, deux ans plus tard, Jules Guesde souhaite s'opposer à la première loi sur les retraites « ouvrières et paysannes », parce que celle-ci n'offre à ses yeux que de faibles garanties pour les plus modestes, il est bien isolé face à un Jean Jaurès qui, malgré les nettes insuffisances de la loi, la vote parce qu'elle permet d'inscrire dans la République le principe même de la retraite6. Sur le long terme, c'est donc bien la perspective jaurésienne qui s'impose dans le socialisme français, même si elle est toujours contrebalancée par d'autres options politiques.
Quelle est au fond cette perspective ? Une attitude sagement gradualiste et réformiste, quand d'autres socialistes révolutionnaires souhaitent renverser le capitalisme ? Gardons-nous de plaquer les réalités d'une autre époque sur le socialisme de 1900 : les oppositions et ruptures postérieures à la révolution russe de 1917 ne s'imposent pas comme des évidences au temps de Jaurès. D'aucuns ont pu évoquer un « réformisme révolutionnaire7 » typiquement jaurésien, une « synthèse jaurésienne » prenant à rebours des oppositions trop simples qui ne sauraient refléter la pensée d'un socialisme français pluriel et complexe. En effet, il n'existe pas de modèle révolutionnaire clefs en main comme le proposera ultérieurement la révolution bolchevique, avec notamment la forme de parti centralisée théorisée par Lénine. Jaurès n'a pas de théorie du parti, et s'il s'oppose aux vues les plus centralisatrices des guesdistes inspirés par le modèle allemand, il ne propose pas de conception alternative, tout en restant attaché à l'existence d'un parti politique pluraliste mais uni sur l'essentiel. Il n'hésite certes pas à se référer à l'exemple de la Révolution française de 1789-1794, mais moins pour en répéter les phases que pour s'inspirer des grands principes dont elle a permis l'éclosion. C'est à elle d'ailleurs qu'il pense spontanément lorsque de grands bouleversements affectent d'autres pays où la révolution semble de nouveau à l'ordre du jour. Ses propos lyriques mobilisent spontanément la glorieuse tradition révolutionnaire française lorsque, en janvier 1905, sur les terres de l'autoritarisme honni du tsar, une révolution commence, soulevant les espoirs dans toute l'Europe. Le mot « révolution » fait bien partie du vocabulaire jaurésien, mais il n'apparaît pas antinomique avec les conquêtes partielles que sont les réformes. Jaurès utilise significativement la formule « évolution révolutionnaire » – qu'il reprend rapidement d'une formule de Marx – pour exposer sa vision politique8. Tout au plus, si l'on souhaitait lui attribuer une étiquette, pourrait-on qualifier Jaurès de « réformiste radical » : profondément pacifique, il se méfie des ruptures violentes et brutales et, en ce sens, se distingue bien d'autres socialistes davantage portés vers la rupture révolutionnaire avec l'ordre existant. Mais il se méfie également du conservatisme des républicains et n'hésite pas à monter au créneau pour défendre des réformes politiques et sociales que certains jugent dangereuses ou impossibles : là encore, son choix de défendre certaines lois (sur la laïcité ou les retraites) indique bien sa démarche.
Jaurès est par ailleurs un socialiste attentif aux autres sensibilités, convaincu que c'est du débat – même ferme et parfois virulent – que peut sortir la pertinence d'une ligne politique. C'est dans cet esprit que l'on doit concevoir la fondation du journal L'Humanité en avril 1904, un « journal socialiste » qu'il n'envisagera jamais comme un quotidien inféodé au parti, mais comme une tribune libre des mouvements ouvriers et socialistes de son temps. Au risque parfois de débats houleux et de déchirures avec certains de ses amis, il tiendra à cette option pluraliste jusqu'au bout. En témoigne un éditorial de L'Humanité de 1911 où il rend un hommage appuyé à Paul Lafargue, qui vient de se suicider avec sa femme. Quelques semaines plus tôt, Lafargue et Jaurès s'affrontaient pourtant sur des points majeurs avec une virulence manifeste ! L'activité de journaliste est essentielle chez Jaurès : avant (et parfois en parallèle à) L'Humanité, il fut un actif contributeur de La Dépêche et de La Petite République. Homme d'action, il a publié des centaines d'articles de journaux, souvent bien plus longs et touffus que ceux que l'on peut lire aujourd'hui ; il suffit de consulter certains de ses articles, denses et parfois même difficiles à saisir tant ils mobilisent de références historiques et philosophiques. Jaurès journaliste est un exemple remarquable d'homme politique suivant l'actualité la plus immédiate, sans perdre de vue une profondeur d'analyse liée à une immense culture9.
Un point particulier doit être mentionné au sujet du syndicalisme. La situation française est alors très spécifique. La Confédération générale du travail (CGT), fondée en 1895, est animée par une majorité syndicaliste révolutionnaire très hostile aux partis politiques et au parlementarisme, considérant la « grève générale » comme le principal moyen d'action pour la classe ouvrière, et qualifiant le suffrage universel de duperie empêchant la mobilisation active des masses populaires10. Lors du congrès d'Amiens en 1906, la CGT réaffirme son autonomie jalouse à l'égard de la SFIO dans un texte resté dans l'histoire sous l'intitulé de « charte d'Amiens ». Jaurès est évidemment opposé aux conceptions des membres de la CGT, lui qui demeure attaché aux formes parlementaires républicaines. Mais son évolution sur l'usage de la grève générale est sensible après 1905 : lucide sur la force que représente le syndicalisme de la CGT, et peut-être aussi conscient des limites d'une démocratie parlementaire qui ne peut se suffire à elle-même, il en vient à considérer l'usage ponctuel de la grève générale comme un moyen d'action utile. Un véritable dialogue fructueux s'installe ainsi entre Jaurès et la CGT syndicaliste révolutionnaire à partir des années 1906-1908, dans un contexte où le gouvernement de Clemenceau réprime les grèves ouvrières11. Jaurès se distingue là encore des partisans de Jules Guesde souhaitant soumettre le syndicat au parti, ne voyant le premier que comme l'appendice du second. Une même attitude peut être observée au sujet du mouvement coopératif : contre ceux qui tiennent qu'il doit être mis au service du parti, Jaurès y voit un des moyens de l'émancipation sociale, à côté de la SFIO et du syndicat. Il en arrive à considérer qu'un militant socialiste doit se retrouver autour des trois pôles que sont le parti, le syndicat, la coopérative, montrant sa réelle sensibilité aux diverses formes d'expression du mouvement ouvrier, là où d'autres voulaient faire primer impérativement l'une d'entre elles. Sensibilité qu'expriment ses hommages ponctuels à Pierre-Joseph Proudhon, figure influente des mouvements ouvriers des années 1840-1860 et père de l'anarchisme attaché à l'idée de coopération. Jaurès croit bien évidemment en l'action régulatrice de l'État contre les anarchistes, mais sait aussi parfois se montrer attentif aux arguments critiquant la bureaucratie et l'autoritarisme de certains courants républicains ou socialistes. L'empreinte proudhonienne, voire libertaire, existe chez lui, même si elle n'est pas dominante.
Jaurès et l'Allemagne
Jaurès comprend et lit l'allemand, bien mieux que l'anglais, ce qui est alors fréquent dans le socialisme international. Il a de nombreuses amitiés en Europe, notamment chez les socialistes belges, comme Émile Vandervelde, leader du Parti ouvrier belge et un temps président de l'Internationale socialiste. Mais le grand modèle de l'époque, c'est bien le socialisme allemand. Jaurès entretient un rapport particulier à ce socialisme et à son évolution. On l'a dit, sa thèse complémentaire en latin est consacrée à l'étude des origines de ce mouvement et, surtout, dans de nombreux articles et discours, il s'y réfère régulièrement. Plusieurs raisons expliquent cette attention. En premier lieu, Jaurès est philosophe de formation, et l'Allemagne de Kant et de Hegel est alors incontournable. Quand Jaurès se fait historien, avec son Histoire socialiste de la France contemporaine, il s'intéresse de près à l'impact de la Révolution française en Allemagne, et rédige en personne le volume, publié en 1907, dédié à la guerre franco-allemande de 1870. L'Allemagne est également le pays où a été fondé le premier parti ouvrier indépendant en Europe, en 1863 par Ferdinand Lassalle, une des branches de la future social-démocratie allemande, qui prend le nom de SPD en 1890. C'est bien évidemment aussi le pays d'origine de Marx et d'Engels, et du « marxisme » (le terme se généralise dans les années 1880) qui constitue la référence doctrinale du SPD, parti qui est alors le plus puissant dans le cadre de l'Internationale socialiste, fondée en 1889. Sans cette Internationale, la coordination et l'action collective des socialistes en Europe, face aux risques de conflits guerriers, ne sont pas possibles ; c'est la raison pour laquelle, en dépit de la distance qui sépare le républicanisme jaurésien du marxisme des sociaux-démocrates allemands, le tribun socialiste français reste toujours attentif aux évolutions de leur parti, et milite activement dans les instances de l'Internationale. Il publie plusieurs articles sur les débats internes du SPD, notamment sur la question de la « révision » du marxisme autour de la figure d'Eduard Bernstein.
Un des grands moments de cette amitié, teintée de débats intenses voire d'oppositions fermes, est le congrès de l'Internationale socialiste d'Amsterdam en 1904, où s'affrontent notamment les conceptions de Jaurès et d'August Bebel, le président du SPD. Amsterdam est un congrès décisif pour les Français : il va en sortir une motion incitant fortement les socialistes français à l'unité, qui facilitera la création de la SFIO en 1905. Mais c'est aussi de République dont il est question dans les débats. Le SPD réclamait formellement une République allemande, dans la continuité des idéaux de la révolution de 1848. Mais, de plus en plus, cette revendication était passée au second plan. Jaurès, lui, on l'a vu, reste fermement attaché à la forme républicaine, fondement imparfait mais nécessaire pour aller plus loin. Or, Bebel, et avec lui nombre de marxistes allemands comme Karl Kautsky – la grande autorité intellectuelle de l'époque –, considèrent que Jaurès est atteint de « superstition républicaine ». Pour eux – comme d'ailleurs, on l'a souligné, pour les marxistes français comme Guesde et Lafargue – la forme républicaine ouvre tout au plus la voie à une expression plus affirmée des luttes de classes, permettant de faire apparaître plus nettement l'affrontement entre la bourgeoisie et le prolétariat. Elle leur semble même devenir superflue au regard de l'évolution politique en France et en Allemagne : l'Empire allemand de Bismarck n'a-t-il pas octroyé des assurances sociales bien plus avantageuses que celles proposées par la République française ? À quoi bon valoriser une République qui semble en retard, sur le plan social, par rapport au Reich ? Jaurès conteste cette lecture, qu'il juge étroite, des luttes politiques en France : n'est-ce pas plutôt parce que la social-démocratie n'ose plus affronter le pouvoir impérial en place en Allemagne qu'elle ne demande plus l'établissement d'une République ? Autrement dit : derrière des formules marxistes très radicales, ne se cache-t-il pas des motifs moins avouables ? Paradoxalement, Rosa Luxemburg – pourtant très critique à l'égard du tribun socialiste français – lui donnera raison au moins sur ce point quelques années plus tard. En 1910, elle rompt avec son vieil ami Karl Kautsky, car celui-ci refuse de publier un article... qui demande notamment de faire de la propagande pour revendiquer la République allemande ! Il y a de ce point de vue, au-delà de l'enjeu précis de la question républicaine, une véritable distance de Jaurès à l'égard du marxisme tel que le comprennent ses homologues allemands. Jaurès considère bien qu'il existe des « luttes de classes » : on trouve sous sa plume régulièrement l'expression « classe ouvrière », et le matérialisme historique fait bien partie de son héritage intellectuel, notamment lorsqu'il s'agit d'interpréter un processus comme la Révolution française. Jaurès n'est par ailleurs pas si éloigné des dernières réflexions de Friedrich Engels (qui meurt en 1895) considérant l'État non pas seulement comme l'instrument d'une classe sociale (« l'État bourgeois »), mais comme un des lieux où s'expriment les contradictions entre les classes sociales ; il est donc nécessaire d'intervenir au niveau de l'État, d'assumer les compromis, ce qui n'exclut pas d'abandonner la perspective socialiste. Mais Jaurès accorde une importance bien supérieure à la forme politique des institutions et croit dans la vertu des valeurs républicaines.
Il existe chez Jaurès une dimension morale, manifeste dans son appréciation de la République qui le rend si singulier, même si cet aspect ne doit pas être surévalué, car le socialiste français pourrait certainement être rapproché sur ce point de ceux qui, en Allemagne, opèrent un « retour à Kant » pour mieux attaquer ce qu'ils considèrent comme les rigidités d'analyse issues de la méthode dialectique. La dimension religieuse de l'engagement jaurésien a été elle aussi l'objet de plusieurs études significatives, et faire de notre dirigeant socialiste un athée intransigeant n'aurait en effet aucun sens, tant une fibre mystique traverse sa trajectoire et ses écrits12. Qui plus est, Jaurès, à plusieurs reprises, entend se démarquer d'un matérialisme trop rigide, qu'il relève notamment dans les essais historiques de Paul Lafargue, et il se distingue ainsi assez nettement de Marx, même s'il est difficile de conclure sur le rapport qu'il entretient avec la grande figure théorique du socialisme allemand. Jaurès est un homme politique rédigeant des éditoriaux jour après jour, il lit énormément, mais cite aussi de mémoire, et parfois de façon approximative, des auteurs qu'il n'a pas toujours eu le loisir d'étudier en profondeur, Marx le premier. Distance à l'égard des marxistes affirmés de son temps, assurément ; son rapport à l'œuvre de Marx reste, lui, plus difficile à saisir.
Jaurès est bien le critique sévère de certaines prédictions « catastrophistes », de la part de ceux qui prennent au pied de la lettre le Manifeste du parti communiste de 1848 pour ne jurer que par l'effondrement à venir du capitalisme. On ne peut être par ailleurs que marqué par la faible présence, voire l'absence, chez lui d'une réflexion sur l'économie politique, absence qu'il partage au demeurant avec d'autres socialistes français comme Guesde. Par contraste, l'attention de ses amis sociaux-démocrates allemands – toutes tendances confondues – aux problèmes économiques et à la nature du capitalisme est saisissante ; en France, il faut aller chercher du côté des économistes universitaires pour trouver une discussion sérieuse sur ce terrain. Philosophe et historien, Jaurès s'intéresse bien évidemment aux évolutions économiques du monde, comme en atteste sa lecture d'un ouvrage très discuté à l'époque, Le Capital financier13 de Rudolf Hilferding (1910). Il se soucie du problème de la répartition des richesses, des questions douanières et fiscales. Mais l'on chercherait en vain une pensée théorique un peu originale sur l'économie. Sa formation intellectuelle et son action politique incessante ne le portent pas vers une analyse approfondie de l'évolution industrielle et agricole du pays. Aussi l'absence de telles questions dans cette anthologie, qui pourrait surprendre, ne tient pas à un choix délibéré, mais reflète la faible place objective qu'elles occupent dans le corpus jaurésien.
Jaurès, la paix et le monde
Malgré les divergences et les polémiques avec les sociaux-démocrates allemands, de grands moments de fraternité existent, à l'image de ce discours de 1905 que Jaurès devait proclamer à Berlin ; mais les autorités lui refusèrent l'entrée sur le territoire allemand. L'entente concertée avec le SPD reste de toute façon inévitable pour Jaurès jusqu'aux dernières heures de son existence, afin de sauver ce qui demeure au fond son principal dessein : la paix.
Jaurès, martyr de la paix : c'est assurément cette grande cause qui l'a rendu célèbre en France et dans le monde entier. Sur ce point, soulignons que Jaurès a toujours été un patriote convaincu, défenseur de son pays et de ses valeurs. À tel point que, pendant ses premières années d'engagement politique, et même une fois ses convictions socialistes établies, il reste fermement persuadé de la nécessité de la colonisation en Afrique du Nord pour apporter aux indigènes la civilisation française, cette colonisation devant assurer la première étape indispensable avant l'émancipation. Les brutalités de la conquête militaire le rendront de plus en plus circonspect et, à la Chambre, il devient un des plus farouches opposants à la politique coloniale, ce qui lui vaut les attaques virulentes du flanc droit de l'assemblée14. Jaurès n'ira jamais jusqu'à remettre en cause le principe même du colonialisme, mais son réformisme colonial va bien plus loin que celui de nombreux socialistes de l'époque, y compris ceux qui, sur d'autres terrains, étaient plus radicaux que lui. C'est là que l'on saisit à quel point son patriotisme se combine avec un vigoureux internationalisme. Car au-delà des amitiés européennes qu'il n'a cessé de tisser tout au long de son existence, et ce malgré de nombreuses tensions et affrontements idéologiques, Jaurès est également un des rares à avoir porté son attention sur les peuples non européens, certes selon une perspective qui peut nous paraître aujourd'hui étroite et limitée, mais incontestablement audacieuse dans le contexte de son époque. Nul hasard dans le fait qu'il soit un des premiers à s'intéresser aux mouvements anticoloniaux, lorsqu'il rédige son Histoire socialiste de la France contemporaine : là où la plupart des historiens de la Révolution française avaient superbement ignoré les luttes extra-métropolitaines, Jaurès offre un premier aperçu qui sera salué ultérieurement par les historiens de la colonisation.
« Un peu d'internationalisme éloigne de la patrie, beaucoup d'internationalisme y ramène » : on doit avoir toujours à l'esprit cette citation de L'Armée nouvelle pour comprendre Jaurès. L'un n'entend pas exclure l'autre. L'Armée nouvelle mérite à ce sujet une attention spécifique : c'est, avec l'Histoire socialiste, l'autre ouvrage majeur de Jaurès. À l'origine, il s'agit d'un projet de loi de 1910, transformé plus tard en un livre chez le même éditeur que pour l'Histoire socialiste, J. Rouff15. Sa lecture permet d'éviter les contresens sur le pacifisme de Jaurès, qui n'est ni abstrait, ni absolu, et s'insère dans un contexte précis. Jaurès ne méprise ni les frontières, les différentes contingences liées à la protection de la patrie ; il souhaite une réforme profonde de l'armée pour préserver la paix ; mais la condition du maintien de celle-ci ne peut pas être un antimilitarisme apatride. Bien évidemment, Jaurès s'oppose de toutes ses forces au bellicisme ambiant et à l'esprit de revanche contre l'Allemagne, que partage alors une grande partie de la classe politique française. Les projets de L'Armée nouvelle ne seront d'ailleurs pas retenus et la loi des trois ans de service militaire, contre laquelle Jaurès et les socialistes manifestent au Pré-Saint-Gervais, s'impose en 1913. Mais Jaurès se distingue également d'un socialiste comme Gustave Hervé16, partisan d'une action antimilitariste virulente et hostile à toute réforme de l'armée, ainsi que de tous ceux qui prônent un internationalisme radical souhaitant la disparition des frontières. Rosa Luxemburg polémique par exemple durement avec lui lorsqu'elle rend compte de L'Armée nouvelle : elle l'accuse d'abandonner les revendications traditionnelles du mouvement socialiste, notamment la dissolution de l'armée permanente au profit de milices ouvrières17.
Postérités jaurésiennes
Le lien entre patriotisme et internationalisme, le rapport complexe entre les deux, est un des legs les plus certains de Jaurès à la gauche française dans sa diversité. Mais Jaurès, assassiné le 31 juillet 1914 par un exalté nationaliste, à la veille du déclenchement de la guerre, n'a pas pu se prononcer sur l'attitude des socialistes à l'égard de la défense nationale, en août 1914. Se serait-il opposé à « l'union sacrée » ? Qu'aurait-il fait face à la révolution russe de 1917, notamment après la prise de pouvoir par les bolcheviks ? Ne faisons pas parler les morts. Il n'en demeure pas moins que Jaurès a été immédiatement récupéré pour justifier l'union sacrée, à l'occasion de son enterrement. Mais quelques années plus tard, sa figure allait au contraire devenir l'emblème de la paix. À elles seules, les utilisations politiques et historiques de Jaurès pourraient occuper des centaines de pages ; des contributions s'y sont d'ailleurs attachées, montrant l'incroyable mobilisation de la figure du tribun socialiste pour des causes... de natures très diverses18 ! Citons ici simplement quelques exemples significatifs, montrant combien la connaissance de l'œuvre de Jaurès est difficilement dissociable des usages qui en ont été faits au cours du XXe siècle. Après le congrès de Tours de 1920, qui signe la séparation définitive entre socialistes et communistes, ces derniers tentent dans un premier temps de récupérer la figure de Jaurès – démarche facilitée par la prise en main du journal L'Humanité qui va devenir l'organe central du PCF –, avant d'osciller entre le rejet de celui qui apparaît comme un plat « réformiste », par contraste avec l'audace révolutionnaire léniniste, et la mise en avant (notamment pendant les périodes d'unité de la gauche) du fondateur du journal L'Humanité, ardent combattant pacifique. Jaurès a une longue postérité dans la mémoire socialiste, même si elle n'est pas, là aussi, exempte de contradictions. Son ami Léon Blum, futur chef du gouvernement de Front populaire en 1936, s'y réfère avec passion. Mais la « popularité » de la figure ne s'arrête pas là : pendant l'occupation allemande, une partie de la collaboration « pacifiste », prônant une nouvelle amitié « franco-allemande » dans le cadre du nouvel ordre de 1940, n'a pas hésité à se situer dans le sillage de Jaurès, opération menée notamment par d'anciens socialistes passés à la collaboration19 !
Parallèlement à ces utilisations, l'étude scientifique de Jaurès commence avec la création, en 1959, d'une Société d'études jaurésiennes, présidée successivement par Ernest Labrousse, Madeleine Rebérioux, puis, encore aujourd'hui, Gilles Candar. Parmi les grands héritages de Jaurès, il faut souligner la place singulière qu'occupe son Histoire socialiste de la Révolution française, rééditée à plusieurs reprises car considérée comme fondatrice d'une tradition d'étude. Les plus grands historiens de la Révolution, d'Albert Mathiez à Michel Vovelle, s'en sont réclamés, et même François Furet, pourtant très critique de ces derniers, a vu dans Jaurès un des rares à donner une interprétation forte du processus révolutionnaire20. Marquant et durable aussi est l'investissement de Jaurès pour la publication de documents historiques : il est à l'origine d'une commission parlementaire pour l'étude de documents d'histoire économique pendant la Révolution, surnommée par la suite « Commission Jaurès », qui fonctionna jusqu'en 200021. Mais si les études jaurésiennes se développent, la figure de Jaurès ne devient pas pour autant un objet froid. Les récentes élections présidentielles de 2007 et 2012 l'ont montré avec un certain éclat : les candidats de toute tendance se sont réclamés de l'héritage jaurésien, notamment et de façon plus surprenante à droite, où l'on n'a pas hésité à souligner le défenseur du « travail » ou de la « patrie »22... En avril 2014, à vingt-quatre heures d'intervalle s'opposent le discours du président François Hollande rendant hommage, à Carmaux, à « l'optimisme » jaurésien23, et celui de Jean-Luc Mélenchon, à Castres, se saisissant de l'ardeur combattante de Jaurès pour prôner une « opposition de gauche » au président de la République, accusé d'avoir abandonné le terrain des classes populaires qui préoccupait tant le tribun assassiné en 1914.
Faut-il voir dans tout cela une trahison, une victoire posthume, ou plus simplement des usages multiples (et/ou abusifs ?) d'une figure incontournable de l'histoire de la vie politique française ? À partir d'une lecture attentive des textes replacés dans leur contexte, il est peut-être possible d'en décider : c'est ce qu'entend proposer modestement cette anthologie regroupant des textes célèbres, et d'autres inédits depuis l'époque de leur première parution. Chacun pourra alors se faire son opinion sur cette figure si souvent citée, mais finalement mal connue, qu'est Jean Jaurès.
Jean-Numa Ducange.
1 Jean Jaurès, Le Passage au socialisme (1889-1893), in Œuvres, t. II, Paris, Fayard, 2011.
2 Jean-Michel Ducomte et Rémy Pech, Jaurès et les radicaux : une dispute sans rupture, Toulouse, Privat, 2011.
3 Gilles Candar et Jean-Numa Ducange, « Paul Lafargue : la propriété paysanne et l'évolution économique », Cahiers Jaurès, no 195-196, p. 70-80.
4 Jean Jaurès, Jules Guesde, Rosa Luxemburg, Les Discours des deux méthodes, Paris, Le Passager clandestin, 2014 (2e éd.).
5 Claude Willard, Jules Guesde, l'apôtre et la loi, Paris, Éditions ouvrières, 1991.
6 Gilles Candar et Guy Dreux, Une loi pour les retraites. Débats socialistes et syndicalistes autour de la loi de 1910, Lormont, Le Bord de l'eau, 2010.
7 Voir notamment Bruno Antonini, État et socialisme chez Jean Jaurès, Paris, L'Harmattan, 2004.
8 Jean-Paul Scot, Jaurès et le réformisme révolutionnaire, Paris, Seuil, 2014.
9 Charles Silvestre, Jaurès, la passion du journaliste, Pantin, Le Temps des cerises, 2010.
10 Miguel Chueca, Déposséder les possédants. La grève générale aux « temps héroïques » du syndicalisme révolutionnaire (1895-1906), Marseille, Agone, 2008.
11 Alain Boscus, Jean Jaurès, la CGT et le syndicalisme révolutionnaire, Toulouse, Institut CGT d'histoire sociale, 2010.
12 Voir notamment les travaux de Jordi Blanc et son édition de textes philosophiques, Valence, Vent terral, 2014.
13 Rudolf Hilferding, Le Capital financier : étude sur le développement récent du capitalisme, Paris, Éditions de Minuit, 1979.
14 Voir Jean Jaurès, Le Pluralisme culturel, in Œuvres, t. XVII, Paris, Fayard, 2014.
15 Jean-François Chanet (dir.), « Lire L'Armée nouvelle », Cahiers Jaurès, no 207-208, janvier-juin 2013.
16 Gilles Heuré, Gustave Hervé. Itinéraire d'un provocateur, Paris, La Découverte, 1997.
17 Rosa Luxemburg, Le Socialisme en France, Marseille, Agone, 2013.
18 Vincent Duclert, Jaurès, 1859-1914 : la politique et la légende, Paris, Autrement, coll. « Vies parallèles », 2013.
19 Guillaume Pollack, « Une mémoire improbable : Jaurès sous l'Occupation (1940-1944) », Cahiers Jaurès, no 211, janvier-mars 2014, p. 95-114.
20 Christophe Prochasson, « Sur une réception de l'Histoire socialiste de la Révolution française : François Furet lecteur de Jean Jaurès », Cahiers Jaurès, no 200, avril-juin 2011, p. 49-67.
21 Michel Vovelle, « Un centenaire qui n'aura pas lieu », Annales historiques de la Révolution française, no 332, avril-juin 2003, p. 179-182.
22 Marion Fontaine, « Les usages politiques de Jaurès », Cahiers Jaurès, no 200, avril-juin 2011, p. 17-35.
23 Vincent Chambarlhac, « Une vidéo de l'Élysée... (Hollande dans la continuité de Jaurès ?) », 22 avril 2014, Comité de vigilance face aux usages publics de l'histoire. <http://cvuh.blogspot.fr/2014/04/une-video-de-lelysee-hollande-dans-la.html>
I
Le militant républicain et socialiste
Le socialisme de Jean Jaurès est profondément ancré dans l'héritage de la Révolution de 1789. À l'origine républicain modéré, Jaurès devient progressivement socialiste entre 1889 et 1892, sans jamais abandonner le cadre républicain, contrairement à d'autres socialistes qui s'obstinent à ne voir dans la République que l'instrument moderne de la domination bourgeoise.
Sa conviction profonde se retrouve dans ses multiples combats, depuis son engagement en faveur du capitaine Dreyfus en 1898 jusqu'à sa défense de la laïcité, en passant par la question de l'usage de la grève ouvrière, la définition de la propriété paysanne, la pertinence d'une nationalisation partielle de l'économie...
Le grand historien Ernest Labrousse disait du socialisme français qu'il était un « maximalisme de la République ». La formule s'applique merveilleusement bien aux textes de Jaurès que l'on va lire dans cette première partie.
« Le socialisme de la Révolution française »
Lorsqu'il rédige cet article pour La Dépêche, Jaurès n'est plus député (il a été battu l'année précédente), mais est en revanche maire-adjoint à l'instruction publique de la ville de Toulouse, d'où son intérêt précoce pour les questions éducatives. Il est progressivement convaincu par les idées socialistes : à travers ce texte, on voit combien l'héritage de la Révolution française, célébré par les républicains un an plus tôt lors du centenaire de 1789, lui permet d'opérer un passage progressif de la République au socialisme.
Quand on considère nos congrès socialistes si restreints et si agités, celui de Calais1, qui semble presque se confondre avec une grève locale, celui de Châtellerault2, où les membres du parti ouvrier se séparent et s'insultent misérablement, et qu'on les compare à cette magnifique assemblée des Trade Unions anglaises, adhérant définitivement au socialisme, et à ce congrès de Halle3, où tous les délégués de l'Allemagne socialiste, ouvriers aux mains calleuses, commerçants millionnaires, docteurs aux lunettes d'or, délibèrent fraternellement sur l'organisation et le programme du parti, on est un moment tenté de croire que le socialisme, possible en Angleterre, puissant en Allemagne, est voué en France à un lamentable avortement. Il n'en est rien et les partis rétrogrades et oligarchiques se réjouissent trop tôt.
Les écoles socialistes peuvent se transformer ; les sectes socialistes, après avoir rendu des services momentanés, peuvent périr par l'exclusivisme ; mais il y a en France un immense parti socialiste qui s'appelle tout simplement le parti républicain. Ni l'Angleterre, ni l'Allemagne n'ont dans leur passé une République démocratique comme celle qui fut proclamée en France en 1792. Dès lors, les espérances d'émancipation des travailleurs anglais et des travailleurs allemands ne prennent pas précisément la forme républicaine, et voilà pourquoi le parti des réformes populaires s'y appelle plus spécialement le parti socialiste. Au contraire, en France, le seul mot de République, tout plein des rêves grandioses des premières générations républicaines, contient à lui tout seul toutes les promesses d'égalité fraternelle.
Il se peut qu'à un jour prochain le vrai parti républicain français, celui qui ne se borne point à accepter la République comme un fait accompli et inévitable, mais qui l'aime comme la forme nécessaire du droit, soit amené à se déclarer tout entier parti socialiste-républicain ; il se peut que la République française s'appelle bientôt dans le monde une République socialiste. Pour ma part, je le désire et je le crois, et j'en dirai prochainement les raisons ; mais, en attendant, le parti républicain français qui se réclame de la Révolution française est, qu'il le dise ou non, un parti socialiste, car la Révolution contient le socialisme tout entier.
Je sais bien qu'on l'a contesté et que des doctrinaires exclusifs comme Louis Blanc, qui ne voyait jamais qu'un côté des choses, ont signalé dans la Révolution française le triomphe de l'individualisme4. Mais la Révolution française, en tout ce qu'elle a fait, en tout ce qu'elle a pensé, était manifestement socialiste.
Elle l'était en proclamant la République. Aujourd'hui, quand le socialisme réclame contre la détention exclusive des moyens de production par un nombre restreint de capitalistes, l'économie politique orthodoxe lui répond que les travailleurs manuels ne sont pour rien dans la création des grandes entreprises, source de la richesse. Sans doute, dit-elle, les entreprises industrielles auraient été impossibles sans les ouvriers ; mais ce sont ceux qui les ont conçues et dirigées jusqu'ici qui en sont les véritables créateurs ; les salariés sont la condition de la richesse, ils n'en sont pas la cause, et ils n'ont pas droit, par conséquent, à entrer en participation du capital industriel et de la puissance économique.
Or, en France, les défenseurs de la monarchie traditionnelle, de la légitimité, ont toujours raisonné ainsi. Sans doute, disent-ils, sans la foule des paysans et des ouvriers, sans le peuple de France, laborieux et brave, jamais la royauté n'aurait pu créer la France moderne une et forte ; mais enfin c'est une famille, la famille royale, qui, par son initiative, son esprit de suite, son habileté dans la guerre, les alliances, les mariages, a créé peu à peu la France. La foule obscure et imprévoyante a été aux mains de cette famille un instrument nécessaire, mais un instrument. Cette famille a donc à jamais la propriété légitime du pouvoir souverain et elle peut déléguer comme il lui plaît une partie de cette autorité aux grandes familles loyales qui l'ont aidée de près dans son œuvre d'agrandissement et d'unité.
À tous ces raisonnements, la Révolution a répondu que la tradition historique ne pouvait primer éternellement le droit humain, qu'il était impossible et inutile d'aller faire la part dans le passé de tous les éléments qui avaient concouru à la formation de la France, et qu'en fait, du jour où les hommes sentaient s'éveiller en eux le besoin de la liberté, ils avaient droit à la liberté. Qu'a donc fait la Convention5 en proclamant la République ? Elle a transféré à la nation tout entière la propriété politique de la France qu'une famille entendait se réserver indéfiniment, sous prétexte qu'elle en avait dirigé la formation séculaire. Appliquez ces maximes à l'ordre économique, et vous avez le socialisme absolu.
La Révolution a été socialiste dans l'organisation de la famille. Quand elle a supprimé ou presque supprimé le droit de tester du père de famille, elle a substitué la volonté de la nation à la volonté individuelle du créateur de la richesse et elle a réglé dans l'intérieur de la famille la distribution de la richesse, selon des principes d'égalité sociale. À l'heure de la mort, ce n'est pas le père, c'est-à-dire celui qui a créé la richesse, qui a droit sur cette richesse, ce sont les enfants, étrangers le plus souvent à sa formation ; c'est dans la famille le socialisme presque absolu.
La Révolution française a été socialiste dans l'organisation de l'enseignement public. La Convention avait institué non seulement des écoles primaires gratuites, mais encore, dans chaque chef-lieu de département, des écoles secondaires gratuites qui portaient le nom d'écoles centrales. Si nous suivions aujourd'hui ses principes, il y aurait gratuité complète dans nos collèges et dans nos lycées comme dans nos écoles, et, en même temps, ce seraient les meilleurs élèves de nos écoles primaires qui seraient appelés à bénéficier de l'enseignement secondaire.
Ainsi, dans la pensée de la Convention, le degré d'éducation que chaque enfant devait recevoir était déterminé, non point par la fortune de ses parents, mais par sa valeur personnelle. Or, il eût été absurde que l'enfant du pauvre, ainsi appelé par une éducation plus haute aux fonctions directrices de l'ordre social, eût été ensuite écarté de ces fonctions par l'absence de capital. Un pareil système d'éducation avait donc pour conséquence immédiate et forcée de subordonner le capital à l'homme, la propriété à la valeur personnelle. Le système d'éducation décrété par la Révolution était donc, en lui-même et dans ses conséquences immédiates, le socialisme le plus hardi qui ait été rêvé.
La Révolution a été socialiste dans l'administration du domaine public. Ce n'est pas elle qui eût consenti à démembrer la puissance de l'État au profit des compagnies de chemins de fer. Elle détestait et brisait tous les monopoles concédés à des particuliers, et elle eût exécuté elle-même, et par les ressources de l'État, les grands travaux publics, au lieu de les abandonner aux financiers, qui, depuis Louis-Philippe, ont rançonné la France. La Convention avait ordonné d'immenses travaux d'assainissement dont elle devait être remboursée par des annuités servies par les propriétaires, et, avec une audace qui nous confond aujourd'hui, elle avait chargé ses architectes de rebâtir les villages de France, composés alors presque partout de misérables huttes.
La Révolution était socialiste dans sa conception de la propriété. Avant la journée funeste du 31 mai6, girondins et montagnards firent un suprême effort pour se rapprocher et pour discuter ensemble la constitution nouvelle de la République. Vergniaud démontra, dans un discours admirable, que la République française ne devait pas comprimer l'essor de la richesse, du luxe, des lettres, de toutes les joies de la civilisation ; qu'elle devait être une nouvelle Athènes agrandie et sans esclaves ; et Robespierre acquiesça à ces vues en disant : « Il ne s'agit pas de proscrire l'opulence, mais de rendre la pauvreté honorable », c'est-à-dire de lui donner la sécurité et l'indépendance. Mais, en même temps, girondins et montagnards s'accordèrent à reconnaître qu'on ne pouvait pas abandonner les relations économiques des hommes entre eux aux seules lois du hasard et de la force, et ils approuvèrent ensemble des propositions décisives, qui, consacrées définitivement après le 31 mai, firent partie de la Constitution de 17937.
Article 7 : « La propriété est le droit qu'a chaque citoyen de jouir et de disposer de la portion de biens qui lui est garantie par la loi. » Voilà donc que, pour la Convention, la propriété est avant tout non pas un fait naturel, mais un fait social, soumis, par conséquent, au contrôle suprême de la société.
Article 9 : « Le droit de propriété ne peut préjudicier ni à la sûreté, ni à la liberté, ni à l'existence, ni à la propriété de nos semblables. » Or, avec le développement du machinisme et du capital anonyme absorbant peu à peu la petite industrie, c'est-à-dire la petite propriété, et pesant d'un poids énorme sur la conscience politique et religieuse des travailleurs, il est évident que l'état et le mode actuel de la propriété en France ne répondent plus aux conditions impératives posées par la Convention.
Article 11 : « La société est obligée de pourvoir à la subsistance de tous ses membres, soit en leur procurant du travail, soit en assurant les moyens d'exister à ceux qui sont hors d'état de travailler. » Voilà l'organisation de l'assistance et du travail.
Je n'ai pas même effleuré ce sujet immense ; je crois en avoir assez dit pour montrer que la Révolution, dans tout son développement libre, de 1789 à 1795, a été imprégnée de socialisme, et que, du jour où, rompant avec les incohérences de la première heure et le funeste essai de monarchie constitutionnelle, elle proclama la République, elle formula en même temps et d'une manière expresse les vérités socialistes.
Je tire de là deux conclusions. D'abord, c'est qu'il y a en France, malgré des apparences contraires, un immense parti socialiste qui est le parti de la Révolution, et, ensuite, c'est que, le socialisme étant contenu dès l'origine dans l'idée républicaine, les socialistes les plus absolus travaillent contre eux-mêmes lorsqu'ils s'isolent du grand parti républicain.
Pour moi, je me sens plus près, par la raison et par le cœur, d'un républicain, si modéré soit-il, qui verra dans la République non seulement le fait mais le droit, que des prétendus socialistes qui ne se réclameraient pas de la République ou qui se tiendraient à l'écart du grand parti républicain. Notre but doit être, non pas de fonder des sectes socialistes en dehors de la majorité républicaine, mais d'amener le parti de la révolution à reconnaître hardiment et explicitement ce qu'il est, c'est-à-dire un parti socialiste. Avant peu, il y sera contraint.
La Dépêche, 22 octobre 1890, texte intégral.
Pour Dreyfus
Ce texte, qui intervient en pleine affaire Dreyfus8, commente notamment la décision du gouvernement d'Henri Brisson de saisir le 27 septembre 1898 la Cour de cassation de l'arrêt du procès Dreyfus de décembre 1894. Si Jaurès doute des résultats, il considère la saisine comme un premier pas vers la vérité. Cette préface aux Preuves, ouvrage où il détaille son point de vue sur l'affaire Dreyfus, a été présentée lors d'un meeting en octobre 1898.
Je réunis en volume les articles publiés dans La Petite République sur l'affaire Dreyfus. Je tiens d'abord à remercier les lecteurs du journal qui m'ont permis d'entrer dans le détail d'une affaire compliquée, et qui ont bien voulu me suivre dans d'assez longues déductions.
Évidemment, le prolétariat ne veut plus se tenir à des formules générales. Il a, sur l'évolution de la société, une conception d'ensemble ; et l'idée socialiste éclaire devant lui le chemin. Mais il veut aussi connaître à fond, et jusque dans les moindres ressorts, le mécanisme des grands événements. Il sait que s'il ne démêle pas les intrigues compliquées de la réaction il est à la merci de tous les mensonges démagogiques : et il vient de donner la mesure de sa force intellectuelle en déjouant un complot où Rochefort était le répondant de l'abbé Garnier9.
Saisir la direction générale du mouvement économique qui va vers le socialisme, et pénétrer par l'analyse le détail de la réalité complexe et mouvante, voilà, pour le prolétariat, la pensée complète. Et désormais, dans toutes les grandes crises nationales, il faudra compter avec lui.
Un premier et grand résultat est atteint. La procédure de révision est engagée et la Cour de cassation est saisie du dossier de l'affaire. Mais la lutte n'est pas finie : et il y aurait un péril extrême à s'endormir. Les hommes qui ont machiné l'odieux procès contre Picquart10 pour empêcher l'ouverture de la révision recourront sans doute aux tentatives les plus audacieuses, les plus criminelles, pour troubler et fausser la révision commencée, pour affoler et égarer l'opinion. Désarmer pendant qu'ils se livreront aux plus louches manœuvres, ce serait une fois de plus trahir la vérité. Ce serait aussi trahir la classe ouvrière sur qui la haute réaction militaire épuiserait ses vengeances. Donc, la bataille continue.
Ce n'est pas que nous ayons aucune raison précise de mettre en doute pour l'affaire Dreyfus la bonne foi et le courage de la Cour de cassation. Il se peut très bien qu'elle comprenne la grandeur de son devoir et de son rôle, qu'elle veuille faire acte de vérité, produire au jour tous les crimes et toutes les hontes, corriger les erreurs et refouler les violences de la justice militaire. Mais il se peut aussi qu'elle se heurte à de rudes obstacles et que sa vigueur défaille. Elle trouvera devant elle deux difficultés principales. D'abord, le terrain de l'affaire Dreyfus est comme encombré de décisions judiciaires ineptes et iniques, qui peuvent arrêter ou gêner tout au moins la marche de l'enquête. Esterhazy11 est acquitté à la suite d'une véritable comédie judiciaire ; mais enfin il est acquitté et il est sans doute malaisé de l'appeler de nouveau à s'expliquer. La chambre des mises en accusation, malgré les charges écrasantes de l'information Bertulus, a mis hors de cause, pour les faux Speranza, Esterhazy, du Paty de Clam et Mme Pays ; la Cour de cassation a eu beau flétrir ces arrêts étranges ; elle a été obligée de les confirmer au fond et quoique pour le faux Blanche un sentier reste ouvert aux poursuites, un gros bloc obstrue le grand chemin.
Enfin l'autorité militaire s'est saisie du colonel Picquart par une procédure jésuitique, mais qui n'est peut-être pas littéralement illégale. Elle essaiera sans doute, par le petit bleu, de retenir à elle l'affaire Dreyfus, et d'opposer à la révision la condamnation criminelle, mais légale, du colonel Picquart, étranglé à huis clos.
Dans le terrain que doit fouiller la Cour de cassation il n'y a pas un seul fragment de vérité qui ne soit recouvert d'un mensonge judiciaire. La Cour de cassation aura-t-elle le courage de briser ces mensonges légaux pour chercher la vérité ? Pourra-t-elle concilier la fonction légale qui lui est assignée par le Code avec la fonction quasi révolutionnaire que lui assignent les événements ?
Elle est la gardienne de la loi : or, la loi, par une application monstrueuse, a travaillé jusqu'ici, dans toute cette affaire, contre la vérité.
La Cour de cassation pourra-t-elle rétablir la vérité sans froisser la loi ? Et comment délogera-t-elle Esterhazy et du Paty des abris légaux que la trahison gouvernementale a ménagés à l'espion et au faussaire ? Voilà la première difficulté.
Il en est une autre. La Cour de cassation découvrira certainement, dans son enquête, des vérités terribles. Il est impossible que la longue série des faux produits par les bureaux de la Guerre ait pu être fabriquée sans la complicité, ou du moins sans la complaisance des grands chefs. De plus, la forfaiture du général Mercier communiquant aux juges, en violation de la loi, des pièces inconnues de l'accusé et empruntant même ces pièces à un autre dossier que celui de l'affaire Dreyfus est certaine. Sur le général Mercier pèsent donc les responsabilités les plus lourdes.
La Cour de cassation aura-t-elle l'énergie d'attaquer les grands chefs, les grands coupables ? Et sachant que pour eux la lumière serait mortelle osera-t-elle faire toute la lumière ?
Encore une fois, il n'y a dans mes paroles aucune intention blessante pour la Cour de cassation. Il se peut qu'elle s'élève au-dessus de toute crainte, au-dessus de toute fausse prudence et qu'elle ait l'entier courage de l'entière vérité.
Je dis seulement que les crimes prolongés de la haute armée et la longue suite des mensonges judiciaires ont créé une situation si terrible que peut-être aucune force organisée de la société d'aujourd'hui ne peut résoudre le problème sans le concours passionné de l'opinion.
Quelle est l'institution qui reste debout ? Il est démontré que les conseils de guerre ont jugé avec la plus déplorable partialité ; il est démontré que l'État-Major a commis des faux abominables pour sauver le traître Esterhazy et que la haute armée a communié, sous les espèces du faux, avec la trahison.
Il est démontré que les pouvoirs publics, par ignorance ou lâcheté, ont été traînés pendant trois ans à la remorque du mensonge.
Il est démontré que les magistrats civils, du président Delegorgue au procureur Feuilloley12, se sont ingéniés, par des artifices de procédure, à couvrir les crimes militaires.
Et le suffrage universel lui-même, dans son expression légale et parlementaire n'a fait trop longtemps, jusqu'à l'éclair du coup de rasoir, que donner au mensonge et au faux l'investiture nationale.
Oui, quelle est l'institution qui reste debout ? Il n'en est plus qu'une : c'est la France elle-même. Un moment, elle a été surprise, mais elle se ressaisit et même si tous les flambeaux officiels s'éteignent, son clair bon sens peut encore dissiper la nuit.
C'est elle et elle seule qui fera la révision. J'entends par là que tous les organes légaux, la Cour de cassation, les conseils de guerre, sont incapables désormais de la vérité complète, si la conscience française n'exige pas chaque jour toute la vérité.
Voilà pourquoi, bien loin de désarmer aujourd'hui, les citoyens qui ont entrepris le combat contre les violences et les fraudes de la justice militaire doivent redoubler d'efforts pour éveiller et éclairer le pays. Voilà pourquoi aussi nous tenons à fournir au prolétariat les éléments de discussion et de preuve que nous avons recueillis.
Beaucoup même de nos adversaires de la première heure ont bien voulu nous dire qu'ils avaient été ébranlés par notre démonstration. Mais toujours un doute revient en eux : comment est-il possible, disent-ils, que sept officiers français aient condamné un autre officier sans des preuves décisives ? À vrai dire, un argument aussi général exclurait a priori toute erreur judiciaire. Mais il est faux que toujours et en tout cas il y ait entre officiers cette solidarité étroite.
Oui, quand ils ont à se défendre contre les civils ou contre les simples soldats, ils font bloc. Mais il existe entre eux de terribles rivalités de carrière, d'amour-propre et d'ambition. Que de fois sur le champ de bataille même les généraux se sont trahis les uns les autres, pour ne pas laisser à un rival tout l'éclat de la victoire !
Or, depuis quelques années, il y avait dans l'armée d'implacables luttes de clan. Le parti clérical, ayant perdu pendant la période républicaine de la République la direction des administrations publiques, des services civils, s'était réfugié dans l'armée. Là, les anciennes classes dirigeantes, les descendants de l'armée de Condé se groupaient en une caste hautaine et fermée. Là, l'influence des jésuites, recruteurs patients et subtils de la haute armée, s'exerçait souverainement. Fermer la porte à l'ennemi, au républicain, au dissident, protestant ou juif, était devenu le mot d'ordre.
Depuis des années, la presse catholique signalait le nombre croissant des juifs qui par l'École polytechnique ou l'École de Saint-Cyr entraient dans l'armée. Drumont avait allumé une sorte de guerre civile contre les officiers juifs13.
Or, voici qu'un juif pénètre, le premier de sa race, à l'État-Major, au cœur même de la place. Après lui sans doute d'autres vont venir : et dans l'antique domaine que se réservait l'aristocratie cléricale exclue un moment des autres fonctions, voici que l'intrus va s'installer.
Vite il faut couper court au scandale. Tout d'abord des rumeurs vagues, des théories générales sont propagées : par quelle imprudence la nation française accueille-t-elle, au centre même de son institution militaire, la race maudite, le peuple de trahison qui, ne pouvant plus crucifier Dieu retiré dans les hauteurs, va crucifier la Patrie ?
Et aussitôt qu'à l'État-Major des fuites de documents sont constatées, c'est vers le juif que se tournent secrètement les yeux : Ah ! quelle chance si c'était lui ! Ah ! quelle faveur de la Providence, quelle grâce divine si dans le premier juif qui viole de sa seule présence le sanctuaire de l'État-Major la trahison s'était logée ! Par lui et en lui tous les autres seraient à jamais discrédités.
Aussi, quand du Paty de Clam constate entre l'écriture du bordereau et l'écriture de Dreyfus quelques vagues analogies, toutes ces haines sournoises, ayant trouvé leur centre, se précipitent et s'organisent. C'est la soudaine cristallisation de la haine.
Dans quelle mesure du Paty de Clam et Henry, les deux meneurs du procès Dreyfus, furent-ils dupes eux-mêmes de cet entraînement ? Y eut-il de leur part complaisance fiévreuse au préjugé général ? Ou bien est-ce de parti pris, en pleine conscience, qu'ils frappèrent l'innocent ? Nous ne le saurons avec certitude que lorsque l'enquête aura été poussée à fond : il nous est impossible encore de savoir quelle fut la part de l'entraînement à demi volontaire, quelle fut la part du calcul scélérat.
Mais ce qui est sûr dès maintenant, c'est que, dans les bureaux de la Guerre, les cœurs et les cerveaux étaient prêts dès longtemps pour la condamnation du juif. Et voilà sans doute la cause maîtresse d'erreur.
Mais elle ne suffisait pas. Il y a fallu encore l'ambitieuse sottise d'un ministre médiocre et infatué. Le général Mercier, d'abord hésitant, fut peu à peu entraîné par un système combiné de flatteries et de menaces.
Ce pauvre esprit présomptueux, qui prétendait « de son seul flair d'artilleur » résoudre sans étude les problèmes techniques les plus ardus, avait été grisé à la Chambre par les applaudissements qui suivaient sa banale parole. Il crut qu'il pouvait, par l'affaire Dreyfus, jouer un grand rôle : mater les juifs, sauver la France des menées de trahison, conquérir les bonnes grâces de l'Église et l'appui de Rochefort, c'était bâtir à nouveau, sur une base plus solide, la fortune de Boulanger14. Quand son entourage clérical vit qu'il souriait à cette pensée, il le brusqua en communiquant aux journaux le nom de l'officier prévenu. Plus tard, L'Éclair s'est vanté qu'il a fallu enlever de vive force son assentiment. Mais quand il eut sauté le pas, quand il se fut livré à La Libre Parole, quand il eut mis toute sa fortune ministérielle sur cette carte, à tout prix il voulut gagner la partie.
Qu'on joigne à cela la sottise de tout le personnel judiciaire de l'armée, qu'on se rappelle la lamentable imbécillité de Besson d'Ormescheville et de Ravary15, on comprendra qu'en ces cerveaux fatigués, l'erreur la plus grossière ait pu germer.
Et par une sorte de fatalité, il se trouve qu'au conseil de guerre qui doit juger Dreyfus, il n'y a aucun officier d'artillerie. Peut-être un officier d'artillerie eût-il fait observer aux juges que le bordereau contenait des détails inapplicables à un artilleur. Il y a notamment à propos du frein hydraulique, substitué par l'auteur du bordereau au frein hydropneumatique, une erreur qu'un officier d'artillerie n'aurait pu commettre.
Personne, au conseil, n'a pu avertir les juges. Et ceux-ci, délibérant sous la communication impérative de pièces secrètes, ont condamné comme à la manœuvre.
Ainsi, bien loin qu'il faille s'étonner de la condamnation de Dreyfus innocent, tant de forces d'erreur et de crime concouraient à le perdre que c'eût été presque miracle qu'il échappât.
Comment ceux qui s'étonnent de la condamnation de Dreyfus ne trouvent-ils pas plus stupéfiant qu'en plein XIXe siècle, en pleine France républicaine, sous un régime d'opinion publique et de contrôle, l'État-Major ait pu accumuler en secret, pendant trois ans, les crimes que l'aveu d'Henry a fait éclater au jour ? Oui, pendant trois ans, comme en un antre profond et inaccessible à la lumière, la haute armée de la France a pu fabriquer des faux, se livrer à toutes les manœuvres de mensonge, peut-être même se débarrasser par le crime de Lemercier-Picard et d'Henry16, et il a fallu, si je puis dire, un accident, une surprise de clarté, pour que ce fonctionnement normal de scélératesse fût soupçonné du pays.
Sous la République française, avec le gouvernement parlementaire, avec la liberté de la presse et de la tribune, les crimes obscurs des républiques italiennes, assassines et empoisonneuses, ont pu être continués pendant trois ans. Cette guerre à coups de papier faux est comme la reproduction de la guerre sournoise avec des coupes empoisonnées que se livraient les Italiens du XVe et du XVIe siècle. Voilà l'étrange, voilà le surprenant, et non que Dreyfus innocent ait été condamné.
Il faut donc écarter cette sorte de préjugé, et regarder directement les faits. Or, à l'examen des faits, il est certain que Dreyfus est innocent. Les dirigeants ont pu affirmer sa culpabilité. Tant qu'ils l'ont fait en termes généraux, leur affirmation échappait à toute discussion. Mais dès qu'ils essaient de préciser et de produire une preuve, cette preuve tombe. Toutes les fois qu'ils puisent dans le fameux dossier, c'est pour remonter à la surface du puits mystérieux ou une sottise ou un faux.
Faut-il croire qu'un sort leur a été jeté ? Tous les bâtons sur lesquels ils s'appuient se brisent en leurs mains ; c'est du bois pourri. Et lorsque la révision se fera, lorsque le procès recommencera au grand jour, il sera difficile ou mieux il sera impossible à l'État-Major de dresser un acte d'accusation et il s'abîmera lui-même dans le néant.
Aussi, désespérant de trouver désormais des charges sérieuses contre Dreyfus, la haute armée, aidée par la faiblesse des gouvernants et la complicité sournoise de l'Élysée, tente une diversion suprême en essayant de déshonorer et de perdre le colonel Picquart.
De là, la monstrueuse accusation de faux dressée contre lui à propos du petit bleu.
D'avance, dans la série même des articles réunis aujourd'hui dans ce volume, nous avons répondu à cette accusation. J'ajoute seulement, dans cette courte préface, que cette machination scélérate est préparée dès longtemps. Évidemment, l'État-Major lui-même la trouve risquée. Tant qu'il a espéré qu'il pourrait se sauver et empêcher la révision sans recourir à cette scélératesse suprême, il l'a ajournée, et c'est seulement quand la révision menaçante était déjà sur lui, qu'il a frappé ce coup de désespoir.
Mais dès longtemps, il le méditait et le tenait en réserve. Dès longtemps, les deux faussaires, Henry et du Paty, préparaient contre Picquart l'accusation de faux.
Elle s'annonce tout d'abord dans la lettre qu'Henry écrit au colonel Picquart en juin 1897, et où il parle de « la tentative de suborner deux officiers du service pour leur faire dire qu'un document classé au service était de l'écriture d'une personnalité déterminée ». Henry qui avait déjà fabriqué la fausse lettre contre Dreyfus préparait en ce moment contre Picquart de faux témoignages.
Les dépositions de Lauth17, si perfides à la fois et si incohérentes, portent la marque d'un entraînement incomplet.
Puis, en novembre 1897, c'est la fausse dépêche Blanche où Esterhazy et du Paty disent au colonel Picquart : « On a des preuves que le bleu a été fabriqué par Georges. » Ainsi, c'est dans un faux que l'accusation de faux commence à s'essayer : c'est une pièce fausse qui sert de berceau au mensonge encore vagissant. Mais dès lors, contre les menteurs et faussaires se dresse cette question terrible : comment n'avez-vous pas, dès la première heure, dénoncé officiellement le colonel Picquart ?
Au procès Esterhazy, en janvier 1898, quand il faut à tout prix sauver le uhlan, l'illustre Ravary, dans son rapport, essaie de jeter le doute sur l'authenticité du petit bleu. Mais ici la question se fait plus pressante encore : Esterhazy est accusé de trahison. L'ancien chef du service des renseignements prétend avoir reçu de ses agents une pièce qui établit entre Esterhazy et M. de Schwartzkoppen18 des relations louches.
Si cette pièce est fausse, Esterhazy est victime de la plus abominable machination. Si elle est authentique, il y a contre lui une présomption grave. Le premier devoir des enquêteurs et des juges est donc de tirer au clair l'authenticité du petit bleu. Mais non, ils se contentent d'insinuations perfides. Ils n'osent pas dénoncer formellement comme fausse une pièce qu'ils savent authentique. Ils se bornent à la discréditer par des sous-entendus. Jamais machination plus scélérate ne s'étala plus cyniquement.
Aussi attendrons-nous, pour discuter de nouveau et plus à fond cette accusation misérable, de savoir si l'État-Major persiste dans cette manœuvre. Il est si répugnant d'engager une discussion sérieuse avec les organisateurs d'un guet-apens, que nous ajournons la nouvelle discussion de fond que nous pourrions produire.
Il nous serait aisé de démontrer par les paroles mêmes de M. Lauth la fausseté de plusieurs parties de son témoignage et l'authenticité du petit bleu. Mais il nous plaît d'attendre que l'État-Major produise les nouvelles pièces fausses qu'il a sans doute fabriquées pour cette tentative suprême.
À cette heure, il nous suffit d'avertir une fois de plus les citoyens pour qu'ils ne permettent pas que le colonel Picquart soit jugé dans l'ombre. Qu'on l'accuse en plein jour ; nous ne demandons pas autre chose et nous avons la certitude que l'infamie de ses accusateurs éclatera. Plus de huis clos ! Voilà le mot d'ordre des républicains, des honnêtes gens. Que ce soit notre cri de guerre ! Et par la seule force de la lumière, nous vaincrons. Et notre grande France généreuse, faisant face une fois de plus aux puissances de réaction et de ténèbres, aura bien mérité du genre humain.
Les Preuves, préface, 29 septembre 1898.
« Le collectivisme et les paysans »
Un des principaux arguments contre les socialistes consistait à les accuser de vouloir s'en prendre unilatéralement à toute propriété, y compris la petite propriété paysanne. Il est vrai que les socialistes ont longtemps considéré que le développement du capitalisme allait favoriser la concentration de la propriété, première étape nécessaire avant la socialisation, et qu'il était par conséquent rétrograde de vouloir s'opposer à ce processus. Néanmoins, Jaurès, comme Paul Lafargue, a défendu dès le congrès de Nantes du Parti ouvrier français de 1894 l'idée qu'il était nécessaire de défendre la petite propriété paysanne (« le petit champ est l'outil du paysan, comme la varlope est celui du menuisier et le bistouri celui du chirurgien »), provoquant d'ailleurs la colère de Friedrich Engels en Allemagne. Des années plus tard, Jaurès reprend cette idée et développe son point de vue.
Je crois que les adversaires du socialisme commencent à renoncer à nous traiter de « partageux19 ». Ils nous prêtaient une conception si absurde qu'ils ne rencontraient plus guère qu'incrédulité. D'abord, le partage serait tous les jours à recommencer. Puis, la tendance de la science et de la grande industrie est, non pas de morceler la production, mais, au contraire, de créer de vastes organismes, d'immenses usines, des réseaux de voies ferrées couvrant tout un pays : et l'idée d'un partage, d'un morcellement, va au rebours même de la civilisation. Enfin, les paysans, qu'on cherchait surtout à effrayer et à animer contre nous par ce mot de partage et de partageux, se sont fait bientôt un petit raisonnement bien simple. S'il s'agit des terres, entre qui les partagera-t-on ? Les travailleurs de la terre, les journaliers, les fils de métayers, même les fils de petits propriétaires, quittent les champs pour aller à la ville, dans les usines, où ils ont un plus haut salaire. Comment supposer que les socialistes veulent ramener les ouvriers à la campagne par l'appât d'un morceau de terre où ils ne récolteraient que misère et souffrance ? Non, c'est trop absurde et il faut renoncer à nous attribuer une pensée aussi folle. Mais, maintenant, c'est avec le collectivisme qu'on cherche à effrayer le paysan. Et je conviens qu'il y a progrès dans la discussion : car la question sera serrée de plus près. Donc, on dit aux cultivateurs : « Les collectivistes veulent que l'État s'empare de toutes les propriétés ; donc ils veulent que l'État s'empare des terres : le collectivisme prendra sa terre au paysan. »
Et je réponds tout de suite, d'une manière catégorique, absolue : « Non, les collectivistes ne veulent pas prendre leurs terres aux paysans. »
Je pourrais me borner à dire ceci : « S'il les leur prenait, qu'en ferait-il ? à qui les donnerait-il ? par qui les ferait-il travailler ? À peine aurait-il pris leurs petits domaines aux propriétaires paysans, il serait obligé de les leur rendre, trop heureux qu'ils continuent à les cultiver. »
Oui, je pourrais me borner à cette réponse de bon sens. Mais il faut s'expliquer davantage. D'abord, il est faux que le collectivisme veuille transférer à l'État toutes les propriétés, même industrielles. C'est de la grande propriété capitaliste qu'il s'agit. C'est de celle qui a concentré en de vastes entreprises les capitaux et les hommes et qui a séparé la propriété du travail. Personne aujourd'hui, au moins parmi les radicaux, ne se scandalise à l'idée que l'État reprenne la banque, les chemins de fer et les mines ; pourquoi ne pas accepter que cette transformation des grandes entreprises capitalistes en services publics s'étende à toutes les grandes entreprises capitalistes, verreries, hauts fourneaux, sucreries, grands tissages, raffineries, etc., etc., qui prennent de plus en plus, par leur étendue, un caractère social ? Je cherche en vain comment les radicaux, après avoir accepté la mainmise de l'État sur un certain nombre de grandes entreprises capitalistes, peuvent s'effaroucher à la pensée que la socialisation s'étendrait plus loin. Nous objectera-t-on que, dans le programme radical, il ne s'agit, pour les mines et les chemins de fer, que d'un rachat et que les socialistes parlent d'une expropriation sans indemnité ? D'abord, si c'est là une question très importante pour toute une catégorie de personnes, cela ne touche pas au fonctionnement définitif du système social. Même avec la charge plus ou moins lourde et plus ou moins longue d'une indemnité, le service public, le service social n'en sera pas moins substitué à l'entreprise capitaliste. En second lieu, il est tout à fait faux que le socialisme ait pris parti sur la question de l'indemnité. Marx disait souvent, au témoignage d'Engels, que la « Révolution sociale », c'est-à-dire la transformation de la propriété capitaliste en propriété sociale, s'accomplirait au meilleur marché possible si une indemnité était accordée aux détenteurs actuels du capital. Il entendait par là qu'une sorte d'arrangement amiable entre la force nouvelle et souveraine du prolétariat et le privilège capitaliste aboli épargnerait à la société bien des secousses et des misères. Quelle serait cette indemnité ? Il est puéril de chercher d'avance à en déterminer la forme. Mais on peut très bien concevoir que ce soit d'abord une rente d'État, convertie peu à peu, à mesure que se développera la production collectiviste, en assignations sur les produits de l'activité sociale. Il est infiniment désirable que les transitions soient ménagées, que les habitudes et les intérêts soient respectés le plus possible dans l'inévitable transformation que la justice exige, que la raison réclame et que le prolétariat, par son organisation croissante, saura obtenir.
Nous objectera-t-on encore que nous ne voulons pas nous arrêter à la forme d'un service public d'État ? Il est bien vrai que nous ne concevons pas du tout la grande production collectiviste sur le modèle des administrations publiques d'aujourd'hui, avec leur fonctionnarisme et leur hiérarchie. Ce seront les travailleurs organisés qui seront appelés à intervenir, à côté de la nature elle-même, dans la direction de la production, dans la marche de l'industrie socialisée. Leur influence, d'abord réduite, ira croissant peu à peu, à mesure que leur puissance dans l'État et sur l'État grandira. Le surcroît de bien-être qu'ils trouveront dans l'industrie transformée ira grandissant aussi à mesure que s'amortiront et s'éteindront les indemnités allouées aux anciens détenteurs du capital. L'initiative créatrice des groupes locaux ou régionaux ira croissant aussi, sous la seule réserve que jamais un homme ne pourra être exploité par d'autres hommes, que l'industrie socialisée ne comportera que des associés ayant titre égal et, pour un même travail, même rémunération. Ainsi la grande entreprise capitaliste sera transformée d'abord en service public, et ce service public, sous l'action de la démocratie industrielle, se transformera graduellement en propriété sociale des travailleurs organisés. Il n'y a pas, là, catastrophe ou cataclysme, ni brusque surgissement d'une cité nouvelle, mais une évolution profonde qui sera peut-être rapide, que nous hâterons de toutes nos forces, mais où toute forme nouvelle sera solidement appuyée sur la réalité préexistante.
Or, qu'on le remarque bien, cette Révolution sociale, soumise nécessairement, comme toute action humaine ou toute force naturelle, à la loi de la continuité, cette Révolution sociale développée en formes successives et en mouvements gradués n'aura même pas pour effet d'abolir la petite propriété individuelle, ce qui reste de l'artisan de la petite industrie. Celle-ci joue aujourd'hui un rôle subordonné, elle est, en bien des points, refoulée ou, en tout cas, subordonnée par la grande industrie capitaliste. Mais elle n'est point abolie. Elle subsiste encore, elle se défend tant bien que mal contre la concurrence effroyable du grand capital. Elle ramasse les miettes de la production ou elle se glisse dans les intervalles de la grande industrie. Or, puisqu'elle n'a pas été supprimée par la concurrence de la grande industrie capitaliste, puisqu'elle a gardé encore un rôle distinct, pourquoi serait-elle supprimée brusquement par la transformation de la grande industrie capitaliste en service public d'abord et bientôt en propriété sociale ? Elle ne s'y annexerait que peu à peu, en vertu d'arrangements amiables et d'équitables contrats, où les artisans, les petits industriels et les ouvriers travaillant avec eux trouveraient les plus larges et les plus fraternelles garanties.
Or, pour en revenir à l'objet immédiat de cet article, comment le collectivisme, qui n'abolira même pas la petite propriété industrielle, qui lui ménagera seulement des voies plus douces vers des formes plus élevées et plus sûres, comment le collectivisme pourrait-il menacer la petite propriété paysanne ? Je répète de la façon la plus nette, la plus formelle, qu'il est faux, absolument faux, que le socialisme collectiviste songe à exproprier les paysans propriétaires. Il n'y a jamais songé. Cela est contraire à tout son principe. Cela est contraire à toutes ses déclarations. Je le prouverai dans mon prochain article par des citations décisives. Et, en même temps, je ne me bornerai pas au côté négatif de la question, je ne me bornerai pas à dire ce que le collectivisme ne fera point aux paysans, je dirai ce qu'il fera, quelle action positive il exercera dans les campagnes au profit des travailleurs agricoles et de l'agriculture elle-même. Mais, dès maintenant, les propriétaires paysans peuvent être assurés que, bien loin de menacer leur droit, bien loin de vouloir leur arracher la parcelle d'indépendance qu'ils ont conquise par un labeur héroïque, le socialisme collectiviste veut garantir leur droit, accroître leur liberté, améliorer leur condition et les préparer, sans spoliation aucune, avec leur libre consentement tous les jours plus joyeux, à des formes supérieures de vie et de civilisation.
La Dépêche, 2 octobre 1901.
« Jaurès à ses électeurs »
Jaurès, battu aux élections en 1898, retrouve son siège de député en avril 1902 dans le cadre d'une victoire nationale du Bloc des gauches, qui va mener une politique offensive sur la question des rapports de l'État avec l'Église. C'est le début d'un des grands moments politiques de Jaurès : vice-président de la Chambre des députés début 1903, il soutient le gouvernement et va jouer un rôle majeur dans l'élaboration de la loi de séparation des Églises et de l'État. Voici la profession de foi qu'il présente à ses électeurs en 1902.
Citoyens,
Républicain et socialiste, je viens vous redemander le mandat de défendre la République, de fortifier la démocratie, d'organiser et d'émanciper les travailleurs industriels et agricoles.
Ce mandat nous fut arraché, il y a quatre ans, par la violence et la calomnie.
Vous vous souvenez des brutalités odieuses qui supprimèrent, en 1898, toute parole libre dans notre région20.
Et pendant que la réaction faisait ainsi le silence et la nuit, elle me calomniait odieusement à propos de mon zèle dans l'affaire Dreyfus. Elle me présentait comme un Judas, vendu aux ennemis de la patrie.
Maintenant toute la France éclairée et loyale sait la vérité.
Elle sait qu'un innocent avait été condamné par erreur et qu'il avait été maintenu au bagne par des manœuvres scélérates, par le mensonge, le parjure et le faux.
Elle connaît la trahison d'Esterhazy, qui est le vrai coupable, les aveux d'Henry, le faussaire, l'arrêt de la Cour de cassation21.
Je suis fier d'avoir contribué à sauver l'innocent et à démasquer les traîtres. Ce que des aveugles ou des infâmes m'ont reproché comme un crime, je le revendique comme l'honneur impérissable de ma vie d'homme et de citoyen.
À la faveur de l'obscurité qu'elle avait jetée dans les esprits, la réaction a tenté une fois de plus, il y a quatre ans, d'étrangler la République, de tuer en France l'esprit de la Révolution.
Par l'union de tous les républicains, l'assaut contre-révolutionnaire a été repoussé.
Cette union nécessaire et loyale qui va s'affirmer partout, sous des formes diverses, mais avec une égale force et un égal succès, aux élections générales, n'implique pour aucun républicain une confusion ou une abdication. Quand les républicains modérés, les radicaux, les socialistes sont conduits à voter tous ensemble contre la réaction, les modérés et les radicaux n'adhèrent point pour cela au socialisme, et les socialistes n'abandonnent pas une seule parcelle de leurs idées et de leurs revendications. Mais ils affirment tous ensemble que la liberté républicaine est le commun patrimoine et qu'elle est la condition absolue de l'évolution régulière de la démocratie.
Le Parti socialiste, dont je défendrai toujours avec une fidélité passionnée toute la doctrine et tout le programme, a le droit de faire appel à tous les républicains, parce qu'aux heures de crise il a toujours lutté au premier rang pour la liberté et parce qu'il s'est associé à toutes les réformes qui pouvaient faire aimer la République.
Jamais il n'abandonnera son idéal. Jamais il ne s'enfermera dans une intransigeance brouillonne et stérile. Il secondera les bonnes volontés, stimulera les hésitations, secouera les inerties et brisera de toute la puissance du prolétariat organisé les résistances égoïstes.
Citoyens, dans les réunions, toujours publiques, que j'ai multipliées pendant quarante-cinq jours, je me suis expliqué devant vous tous sur les réformes prochaines, sur la réforme du système d'impôts, sur la réduction vigoureuse de la durée du service et la transformation de toute l'institution militaire. Je me suis expliqué aussi sur les premières applications collectivistes, qui arracheront la nation et les travailleurs – travailleurs de la terre et travailleurs de l'industrie – à la domination et à l'exploitation du capitalisme.
Ces explications, je les renouvelle dans la circulaire plus étendue que j'adresse à chaque électeur.
Je veux seulement répondre ici aux calomnies les plus grossières et aux sophismes de la réaction.
On n'ose presque plus dire que nous sommes des partageux, que nous voulons prendre leur terre aux paysans. Leur bon sens a fait justice de ces sottises. J'y répondrai pour ma part en agissant. Je veux organiser les travailleurs ruraux de notre région, propriétaires cultivateurs et métayers, en mutualités de crédit agricole, en syndicats fédérés. Je veux apprendre aux cultivateurs, si défiants les uns des autres, si enfermés dans un individualisme étroit, quelle force incomparable sera pour eux l'association, quand les associations paysannes seront secondées par la commune démocratique, par l'État républicain et socialiste.
Mais si cette calomnie usée échappe à la réaction, elle ne renonce pas à calomnier.
La réaction ment quand elle dit que l'impôt général et progressif sur le revenu s'ajoutera aux autres et qu'il pèsera particulièrement sur les paysans. L'impôt général et progressif remplacera les impôts injustes d'aujourd'hui, notamment l'impôt foncier, et il atteindra la bourgeoisie capitaliste et rentière qui, aujourd'hui, ne paie pas sa part.
La réaction ment quand, pour exciter contre nous le fanatisme religieux et détourner le peuple de l'œuvre de réformes, elle dit que nous voulons détruire par la force les croyances religieuses, démolir ou fermer les églises. La liberté de toutes les croyances, la liberté de tous les cultes est un article essentiel du programme républicain et socialiste. La religion est, à nos yeux, chose privée qui ne relève et doit relever que de la conscience de chacun. Et l'État ne doit pas plus intervenir pour la détruire que pour l'imposer ou la propager.
Mais la nation républicaine, qui ne peut vivre que par la liberté et progresser que par la science, doit assurer à tous les enfants, dans toutes les écoles, un enseignement conforme aux principes de la liberté et aux données certaines de la science. Elle ne peut permettre que des moines financiers, politiciens et factieux préparent par un enseignement de contre-révolution des générations de guerre civile et de coup d'État. Et l'État républicain doit reprendre sur tout l'enseignement le droit de contrôle souverain et de direction effective dont il a été dépouillé en 1850 par les manœuvres jésuitiques préparatoires du 2 Décembre22.
La réaction ment quand elle dit que je suis un fomenteur de grèves. Pendant les cinq ans qu'a duré mon mandat, il n'y a pas eu à la mine un seul mouvement de grève, et si, au début du conflit, les grands patrons de la verrerie avaient fait preuve de l'esprit de conciliation dont les ouvriers étaient animés, ils auraient épargné à eux-mêmes, aux ouvriers et à la cité, une longue épreuve, d'où le droit prolétarien n'est sorti triomphant qu'après de trop cruelles souffrances. Les vrais agitateurs, les vrais fomenteurs de grèves, ce sont ceux qui, depuis 1885, ont essayé de faire de la mine un moyen de domination politique23.
Vous ne vous laisserez donc pas troubler, citoyens, par les clameurs des contre-révolutionnaires. Tous les socialistes, tous les travailleurs, tous les républicains, avec un élan admirable, renverseront l'ennemi.
La réaction, à la fois paresseuse et violente, incapable et tracassière, n'a su servir en rien vos intérêts et vos droits. Elle n'a su que semer des germes de discorde en introduisant la politique à la mine et à la verrerie.
La victoire républicaine et socialiste dans la circonscription sera le signal de la réconciliation définitive entre tous les ouvriers de la mine et de la verrerie.
Oubliant les divisions et les querelles d'hier, ils travailleront en paix, avec une sagesse toujours active, à la commune émancipation.
Vive la République sociale !
La Petite République, 28 avril 1902.
L'enseignement laïque
La laïcité est un des principaux combats de Jaurès pendant le gouvernement d'Émile Combes, entre 1902 et 1905. Dans sa circonscription, Jaurès expose sa conception de la laïcité, proche de celle de certains radicaux, mais distincte de celle d'une partie des socialistes qui considère cette question comme secondaire ; d'où le long argumentaire qu'il développe ici. On retrouve notamment la prégnance de la question scolaire, que Jaurès suivait de près depuis ses fonctions exercées à la municipalité de Toulouse en 1890.
Mesdames, messieurs, jeunes élèves,
Je remercie votre municipalité – laissez-moi dire la nôtre24 – de m'avoir donné, dans cette fête des écoles laïques, l'occasion de dire une fois de plus que l'éducation rationnelle et scientifique du peuple est un besoin essentiel, une nécessité vitale de la République. Cette éducation doit être l'objet d'une sollicitude constante, et la communication doit être incessante entre la vie de la nation et la vie de l'école. L'enseignement national dans une démocratie n'est pas une forme immobile et figée, ce n'est pas un mécanisme monté une fois pour toutes et qu'on abandonne ensuite à son fonctionnement. L'éducation est liée à toute l'évolution politique et sociale et il faut qu'elle se renouvelle et s'élargisse à mesure que s'élargissent et se renouvellent les problèmes.
Déjà, il y a trente-quatre ans, au lendemain des désastres effroyables que le despotisme avait déchaînés sur la patrie, un grand cri s'éleva de tout le parti républicain : « Il faut refaire la France ; il faut l'éclairer, il faut l'éduquer25. » La tyrannie est fille et mère de l'ignorance, ou plutôt elle est l'ignorance même ; car, en subordonnant toutes les volontés à une seule, en résumant toute la force active de la patrie dans une dynastie ou dans une caste, elle rend inutile, au moins dans la conduite de la chose publique, l'intelligence de tous, et c'est une loi de la vie qu'un organe inutile languisse et s'atrophie. Il se peut que, dans des sociétés compliquées où les intérêts privés sont si variés et si ardents, l'intelligence subsiste, appliquée au maniement de ces intérêts ; et un observateur superficiel ne constaterait point tout d'abord, dans une nation serve, une diminution de pensée. Mais l'intelligence de tous, exclue du gouvernement de la cité et de l'administration de la vie nationale, a perdu tout ensemble son plus haut objet et son plus vigoureux ressort ; et, dès que survient une crise, elle ne suffit plus à la force des événements.
La liberté républicaine, qui donne à tout citoyen le droit et qui lui crée le devoir d'intervenir dans la conduite des affaires publiques, qui l'oblige sans cesse à avoir une opinion et une volonté, est donc un incessant appel, en tous les hommes, à la force de la pensée, à la force du vouloir. Elle est donc la grande et universelle éducatrice.
Mais cette éducation par la liberté serait insuffisante, elle investirait les citoyens de droits et de devoirs supérieurs à leurs facultés, si la nation ne mettait pas tous les citoyens en état de se reconnaître dans la complication des événements, et de dégager de la contrariété des égoïsmes le droit de chacun et l'intérêt de tous.
C'est pourquoi l'éducation de tous par la liberté républicaine doit être soutenue de l'éducation de tous par l'école, mais par l'école de la nation et de la raison, par l'école civile et laïque. Oui, c'est là ce qu'à peine sortis du gouffre, criaient, il y a trente-quatre ans, les républicains et les patriotes : « Refaire la France par l'éducation de tous, et éduquer tous les citoyens par la République et par l'école, la souveraineté agissante pour tous, la lumière pour tous, la responsabilité pour tous. » C'est là, pour appliquer à ces jours tourmentés et tragiques la grande image du Dante, le premier appel jeté par les naufragés dès que, roulés par la vague, ils abordaient au rivage, tout haletants encore et presque suffoqués. C'est là ce que, dès 1871 et 1872, sous le double fardeau de l'occupation étrangère et de la réaction versaillaise, Gambetta proposait au pays en ses discours de Bordeaux, du Havre, d'Angers, de Saint-Quentin, de La Ferté-sous-Jouarre, partout où il portait son admirable apostolat républicain26.
Et pourquoi ne se bornait-il pas à demander l'instruction pour tous, l'éducation pour tous ? Pourquoi voulait-il qu'elle fût civile et laïque, exclusivement civile et laïque ? On ne l'accusera point, j'imagine, d'être un sectaire ; car ceux-là mêmes qui aujourd'hui nous outragent de ce mot, nous tous républicains dévoués à l'œuvre de laïcité, invoquent volontiers contre nous la largeur de sa pensée et de sa politique. Non, il n'était pas un sectaire. Certes, il savait bien qu'un gouvernement n'est fort, qu'il ne peut être agissant que s'il agit conformément à ses principes, c'est-à-dire avec un parti. Il a dit bien souvent : « On ne gouverne qu'avec son parti. » Il a même tiré de cette formule un jugement historique singulièrement grave. Il a déclaré, en septembre 1871, que si le gouvernement de la Défense nationale à Paris n'avait pas su utiliser pleinement les énergies dont la grande ville abondait, si, au lieu de les susciter et de les organiser, il les avait laissées se perdre à demi en une sorte de flottement mou, c'était faute d'une direction politique assez nette, d'une action politique assez ferme, c'était « parce qu'il n'y avait pas de parti qui régnât à Paris, et que le gouvernement n'était pas un gouvernement de parti ». Il signifiait ainsi d'une façon plus générale que, même pour l'organisation de la défense nationale, même pour l'effort qui tend au salut de tous, au bien-être et à la grandeur de tous, le centre d'impulsion, le ressort d'action doit être dans un parti, c'est-à-dire dans un système d'idées politiques et sociales très défini et très clair, sans lequel l'apparente conciliation des forces n'est que confusion stagnante et impuissance.
Mais ce parti, centre nécessaire et ardent de l'action, même nationale, il ne le concevait pas de façon étroite. Il cherchait à y attirer le plus possible toutes les forces divergentes ou jadis hostiles. L'homme qui disait avoir l'âme assez large pour être dévot tout ensemble à Voltaire et à Jeanne la bonne Lorraine, qui rendait témoignage de la gloire hautaine de la vieille monarchie comme de la noble fierté de la Révolution, qui appelait contre l'envahisseur toutes les forces anciennes et nouvelles, des ouvriers révolutionnaires aux chrétiens de l'Ouest, qui glorifiait Hoche d'avoir vaincu, mais surtout d'avoir pacifié, qui jetait d'ardentes paroles aux paysans de France, à ces fils de la terre, élevés au-dessus du limon par la Révolution libératrice, et qui invitait les survivants des anciennes classes et des anciens partis à entrer dans la République avec leur politesse d'esprit et de mœurs et à devenir une des parures de la France nouvelle, cet homme n'avait point, en effet, l'étroitesse de l'esprit de secte. Et si, au lendemain même de l'orage qui avait bouleversé le sol, il proposait ce programme laïque qui devait soulever nécessairement les controverses les plus violentes et les résistances les plus passionnées, si lui, l'homme de la conciliation nationale, il jetait à la France, encore toute meurtrie et déchirée, cette formule de laïcité qui allait irriter les divisions anciennes et provoquer des divisions nouvelles, c'est bien que la laïcité de l'éducation était à ses yeux une nécessité nationale, une nécessité utile, la condition même du relèvement de la patrie et de l'institution de la liberté, l'âme, le souffle, la respiration même de la République.
Même une société aristocratique, selon lui, si elle ne veut pas languir dans une routine superstitieuse ou se laisser fasciner par des rêves mystiques, si elle veut vivre d'une vie naturelle, moderne, active, doit faire appel à une éducation de laïcité et justifier le privilège de son autocratie, non par l'investiture du pouvoir clérical, mais par l'activité sociale exceptionnelle de cette aristocratie. Et pour une société démocratique, à moins qu'elle ne se soit laissé envahir et corrompre jusqu'aux moelles, sous prétexte de libéralisme, par les principes de servitude, la question ne se pose même pas.
Démocratie et laïcité sont deux termes identiques. Qu'est-ce que la démocratie ? Royer-Collard27, qui a restreint arbitrairement l'application du principe, mais qui a vu excellemment le principe même, en a donné la définition décisive : « La démocratie n'est autre chose que l'égalité des droits. » Or, il n'y a pas égalité des droits si l'attachement de tel ou tel citoyen à telle ou telle croyance, à telle ou telle religion, est pour lui une cause de privilège ou une cause de disgrâce. Dans aucun des actes de la vie civile, politique ou sociale, la démocratie ne fait intervenir, légalement, la question religieuse. Elle respecte, elle assure l'entière et nécessaire liberté de toutes les consciences, de toutes les croyances, de tous les cultes, mais elle ne fait d'aucun dogme la règle et le fondement de la vie sociale. Elle ne demande pas à l'enfant qui vient de naître, et pour reconnaître son droit à la vie, à quelle confession il appartient, et elle ne l'inscrit d'office dans aucune Église. Elle ne demande pas aux citoyens, quand ils veulent fonder une famille, et pour leur reconnaître et leur garantir tous les droits qui se rattachent à la famille, quelle religion ils mettent à la base de leur foyer, ni s'ils y en mettent une. Elle ne demande pas au citoyen, quand il veut faire, pour sa part, acte de souveraineté et déposer son bulletin dans l'urne, quel est son culte et s'il en a un. Elle n'exige pas des justiciables qui viennent demander à ses juges d'arbitrer entre eux, qu'ils reconnaissent, outre le Code civil, un code religieux et confessionnel. Elle n'interdit point l'accès à la propriété, la pratique de tel ou tel métier, à ceux qui refusent de signer tel ou tel formulaire et d'avouer telle ou telle orthodoxie. Elle protège également la dignité de toutes les funérailles, sans rechercher si ceux qui passent ont attesté avant de mourir leur espérance immortelle, ou si, satisfaits de la tâche accomplie, ils ont accepté la mort comme le suprême et légitime repos. Et quand sonne le tocsin de la patrie en danger, la démocratie envoie tous ses fils, tous ses citoyens, affronter sur les mêmes champs de bataille le même péril, sans se demander si, contre l'angoisse de la mort qui plane, ils chercheront au fond de leur cœur un recours dans les promesses d'immortalité chrétienne, ou s'ils ne feront appel qu'à cette magnanimité sociale par où l'individu se subordonne et se sacrifie à un idéal supérieur, et à cette magnanimité naturelle qui méprise la peur de la mort comme la plus dégradante servitude.
Mais qu'est-ce à dire ? Et si la démocratie fonde en dehors de tout système religieux toutes ses institutions, tout son droit politique et social, famille, patrie, propriété, souveraineté, si elle ne s'appuie que sur l'égale dignité des personnes humaines appelées aux mêmes droits et invitées à un respect réciproque, si elle se dirige sans aucune intervention dogmatique et surnaturelle, par les seules lumières de la conscience et de la science, si elle n'attend le progrès que du progrès de la conscience et de la science, c'est-à-dire d'une interprétation plus hardie du droit des personnes et d'une plus efficace domination de l'esprit sur la nature, j'ai bien le droit de dire qu'elle est foncièrement laïque, laïque dans son essence comme dans ses formes, dans son principe comme dans ses institutions, et dans sa morale comme dans son économie. Ou, plutôt, j'ai le droit de répéter que démocratie et laïcité sont identiques.
Mais, si laïcité et démocratie sont indivisibles, et si la démocratie ne peut réaliser son essence et remplir son office, qui est d'assurer l'égalité des droits, que dans la laïcité, par quelle contradiction mortelle, par quel abandon de son droit et de tout droit, la démocratie renoncerait-elle à faire pénétrer la laïcité dans l'éducation, c'est-à-dire dans l'institution la plus essentielle, dans celle qui domine toutes les autres, et en qui les autres prennent conscience d'elles-mêmes et de leur principe ? Comment la démocratie, qui fait circuler le principe de laïcité dans tout l'organisme politique et social, permettrait-elle au principe contraire de s'installer dans l'éducation, c'est-à-dire au cœur même de l'organisme ?
Que les citoyens complètent, individuellement, par telle ou telle croyance, par tel ou tel acte rituel, les fonctions laïques, l'état civil, le mariage, les contrats, c'est leur droit, c'est le droit de la liberté. Mais, de même qu'elle a constitué sur des bases laïques l'état civil, le mariage, la propriété, la souveraineté politique, c'est sur des bases laïques que la démocratie doit constituer l'éducation.
La démocratie a le devoir d'éduquer l'enfance ; et l'enfance a le droit d'être éduquée selon les principes mêmes qui assureront plus tard la liberté de l'homme. Il n'appartient à personne, ou particulier, ou famille, ou congrégation, de s'interposer entre ce devoir de la nation et ce droit de l'enfant.
Comment l'enfant pourra-t-il être préparé à exercer sans crainte les droits que la démocratie laïque reconnaît à l'homme, si lui-même n'a pas été admis à exercer sous forme laïque le droit essentiel que lui reconnaît la loi, le droit à l'éducation ? Comment plus tard prendra-t-il au sérieux la distinction nécessaire entre l'ordre religieux qui ne relève que de la conscience individuelle, et l'ordre social et légal qui est essentiellement laïque, si lui-même, dans l'exercice du premier droit qui lui est reconnu et dans l'accomplissement du premier devoir qui lui est imposé par la loi, il est livré à une entreprise confessionnelle, trompé par la confusion de l'ordre religieux et de l'ordre légal ? Qui dit obligation, qui dit loi, dit nécessairement laïcité. Pas plus que le moine ou le prêtre ne sont admis à se substituer aux officiers de l'état civil dans la tenue des registres, dans la constatation sociale des mariages, pas plus qu'ils ne peuvent se substituer aux magistrats civils dans l'administration de la justice et l'application du Code, ils ne peuvent, dans l'accomplissement du devoir social d'éducation, se substituer aux délégués civils de la nation, représentants de la démocratie laïque.
Voilà pourquoi, dès 1871, le parti républicain demandait indivisiblement la République et la laïcité de l'éducation. Voilà pourquoi, depuis trente-cinq ans, tout recul et toute somnolence de la République a été une diminution ou une langueur de la laïcité ; et tout progrès, tout réveil de la République, un progrès et un réveil de la laïcité.
Mais pourquoi ceux qu'on appelle les croyants, ceux qui proposent à l'homme des fins mystérieuses et transcendantes, une fervente et éternelle vie dans la vérité et la lumière, pourquoi refuseraient-ils d'accepter jusque dans son fond cette civilisation moderne, qui est, par le droit proclamé de la personne humaine et par la foi en la science, l'affirmation souveraine de l'esprit ? Quelque divine que soit pour le croyant la religion qu'il professe, c'est dans une société naturelle et humaine qu'elle évolue. Cette force mystique ne sera qu'une force abstraite et vaine, sans prise et sans vertu, si elle n'est pas en communication avec la réalité sociale ; et ses espérances les plus hautaines se dessécheront si elles ne plongent point, par leur racine, dans cette réalité, si elles n'appellent point à elles toutes les sèves de la vie.
Quand le christianisme s'est insinué d'abord et installé ensuite dans le monde antique, certes, il s'élevait avec passion contre le polythéisme païen et contre la fureur énorme des appétits débridés. Mais quelque impérieux que fût son dogme, il ne pouvait pas répudier toute la vie de la pensée antique ; il était obligé de compter avec les philosophies et les systèmes, avec tout l'effort de sagesse et de raison, avec toute l'audace intelligente de l'hellénisme ; et, consciemment ou inconsciemment, il incorporait à sa doctrine la substance même de la libre pensée des Grecs. Il ne recruta point ses adeptes par artifices, en les isolant, en les cloîtrant, sous une discipline conventionnelle. Il les prenait en pleine vie, en pleine pensée, en pleine nature, et il les captait, non par je ne sais quelle éducation automatique et exclusive, mais par une prodigieuse ivresse d'espoir qui transfigurait sans les abolir les énergies de leur âme inquiète.
Et, plus tard, au XVIe siècle, quand des réformateurs chrétiens prétendirent régénérer le christianisme et briser, comme ils disaient, l'idolâtrie de l'Église, qui avait substitué l'adoration d'une hiérarchie humaine à l'adoration du Christ, est-ce qu'ils répudièrent l'esprit de science et de raison, qui se manifestait alors dans la Renaissance ? De la Réforme à la Renaissance, il y a certes bien des antagonismes et des contradictions. Les sévères réformateurs reprochaient aux humanistes, aux libres et flottants esprits de la Renaissance, leur demi-scepticisme et une sorte de frivolité. Ils leur faisaient grief, d'abord de ne lutter contre le papisme que par des ironies et des critiques légères, et de n'avoir point le courage de rompre révolutionnairement avec une institution ecclésiastique viciée que n'amenderaient point les railleries les plus aiguës. Ils leur faisaient grief ensuite de si bien se délecter et s'attarder à la beauté retrouvée des lettres antiques qu'ils retournaient presque au naturalisme païen, et qu'ils s'éblouissaient, en curieux et en artistes, d'une lumière qui aurait dû servir surtout, suivant la Réforme, au renouvellement de la vie religieuse et à l'épuration de la croyance chrétienne.
Mais, malgré tout, malgré ces réserves et ces dissentiments, c'est l'esprit de la Renaissance que respiraient les réformateurs. C'étaient des humanistes, c'étaient des hellénistes, qui se passionnaient pour la Réforme ; il leur semblait que pendant les siècles du Moyen Âge, une même barbarie, faite d'ignorance et de superstition, avait obscurci la beauté du génie antique et la vérité de la religion chrétienne. Ils voulaient, en toutes choses divines et humaines, se débarrasser d'intermédiaires ignorants ou sordides, nettoyer de la rouille scolastique et ecclésiastique les effigies du génie humain et de la charité divine, répudier pour tous les livres, pour les livres de l'homme et pour les livres de Dieu, les commentaires frauduleux ou ignorés, retourner tout droit au texte d'Homère, de Platon ou de Virgile, comme au texte de la Bible et de l'Évangile, et retrouver le chemin de toutes les sources, les sources sacrées de la beauté ancienne, les sources divines de l'espérance nouvelle, qui confondraient leur double vertu dans l'unité vivante de l'esprit renouvelé.
Qu'est-ce à dire ? C'est que jusqu'ici, ni dans les premiers siècles, ni au XVIe, ni dans la crise des origines, ni dans la crise de la Réforme, le christianisme, quelque transcendante que fût son affirmation, quelque puissance d'anathème que recelât sa doctrine contre la nature et la raison, n'a pu couper ses communications avec la vie, ni se refuser au mouvement des sèves, au libre et profond travail de l'esprit.
Mais maintenant, pour le grand effort qui va de la Réforme à la Révolution, l'homme a fait deux conquêtes décisives ; il a reconnu et affirmé le droit de la personne humaine, indépendant de toute croyance, supérieur à toute formule ; et il a organisé la science méthodique, expérimentale et inductive, qui tous les jours étend ses prises sur l'univers.
Oui, le droit de la personne humaine à choisir et à affirmer librement sa croyance, quelle qu'elle soit, l'autonomie inviolable de la conscience et de l'esprit, et en même temps la puissance de la science organisée qui, par l'hypothèse vérifiée et vérifiable, par l'observation, l'expérimentation et le calcul, interroge la nature et nous transmet ses réponses, sans les mutiler ou les déformer à la convenance d'une autorité, d'un dogme ou d'un livre, voilà les deux nouveautés décisives qui résument toute la Révolution ; voilà les deux principes essentiels, voilà les deux forces du monde moderne.
Ces principes sont si bien, aujourd'hui, la condition même, le fond et le ressort de la vie, qu'il n'y a pas une seule croyance qui puisse survivre si elle ne s'y accommode, ou si même elle ne s'en inspire.
Maintenant il s'agit de savoir si les tenants du dogme sont disposés enfin à accepter nettement, et jusqu'en leur fond, ces principes vitaux. Que gagneraient-ils à s'insurger contre eux ? Ils ne le peuvent pas sans s'exposer eux-mêmes à une incessante défaite, à un incessant désaveu.
À quoi leur a servi, au siècle dernier, de lancer l'anathème, en un document retentissant, aux libertés et aux droits modernes, à la liberté de conscience et de pensée, à tout le droit de la Révolution ? Devant le scandale qu'il a provoqué, même dans l'immense majorité des croyants en qui un commencement d'esprit moderne a pénétré, ils ont dû si bien l'expliquer, l'atténuer, le déguiser, que ce fut presque comme une rétractation.
À quoi leur a servi de dénoncer si longtemps et de nier comme impie le nouveau système du monde entrevu par Copernic et Galilée ? Longtemps, ils ont prolongé leur résistance, puisque c'est seulement en 1855 qu'ils ont levé l'index sur les œuvres de Copernic. Mais cette résistance a fini comme elle devait finir, par une capitulation. Et maintenant les proscripteurs se glorifient d'avoir des astronomes revêtus de la robe du moine qui interrogent et calculent le mouvement des astres selon le système qu'ils avaient proscrit. Maintenant ils commentent le Coeli enarrant gloriam Dei28 au moyen de ces grandes découvertes de l'esprit vouées par eux durant des siècles à l'anathème et au bûcher. Ils font servir à la gloire de Dieu ces vérités de la science qu'au nom de ce même dieu ils tentèrent d'abolir. Et l'on ne sait qu'admirer le plus dans ce long effort contre la science, si c'est son atrocité ou si c'est sa vanité.
Impuissante aussi sera la résistance des tenants du dogme contre l'application scientifique des règles de la critique à l'étude des textes de l'Ancien et du Nouveau Testament. Si les prêtres comme l'abbé Loisy, qui reconnaissent la nécessité de cette méthode, sont inquiétés et frappés, on voit cependant dans la main qui frappe un commencement d'hésitation ; et ils peuvent se rassurer à demi et se consoler à la vue du mouvement qui s'est produit dans l'Église même, pour la doctrine du transformisme.
Il y a un peu plus de trente ans, un évêque véhément et illustre refusait de siéger à l'Académie française à côté du noble et sage Littré, coupable d'avoir accueilli l'hypothèse de l'évolution vitale et du transformisme des espèces29. Le même évêque, à la tribune de l'Assemblée nationale, dans le débat sur la liberté de l'enseignement, s'écriait qu'il y aurait scandale à laisser pénétrer dans l'enseignement, et même dans le haut enseignement, ces doctrines impies et dégradantes. Et ce perpétuel anathème contre l'effort de l'esprit et la vérité croissante suffit à juger l'enseignement confessionnel. Or, quelques années après, des représentants de l'orthodoxie catholique, et qui n'ont pas encore été désavoués, M. de Vogüé, M. Brunetière30, tentaient d'adapter à la tradition religieuse cette conception nouvelle de la science, et ils interprétaient l'évolution comme le symbole visible par où la force créatrice se manifestait.
Mais si les tenants du dogme sont ainsi obligés de céder en détail aux progrès de la conscience et de la science, et de concilier successivement avec leur doctrine des vérités qu'ils dénoncent d'abord comme incompatibles avec leur foi, s'ils sont contraints de se traîner à la suite du droit humain victorieux et de la science humaine victorieuse, s'ils entrent enfin, balbutiants et trébuchants, dans les voies mêmes que longtemps ils ensanglantèrent de leurs persécutions et obstruèrent de leurs anathèmes, pourquoi n'ont-ils pas la sagesse et le courage d'aller d'emblée jusqu'au bout ? Pourquoi n'acceptent-ils pas jusqu'au fond et dans toutes leurs conséquences possibles ces deux grands principes du monde moderne qu'ils ne peuvent plus abolir, qui sont l'élément vital de toute pensée, et avec lesquels il faudra bien qu'ils accordent leur espérance transcendante, s'ils ne veulent pas que comme une flamme que rien ne nourrit plus, elle s'éteigne lamentablement ?
Mais, s'ils acceptent ces deux principes, ils acceptent par là même l'école laïque qui n'en est que l'application à l'enseignement. Car, d'un côté, en éveillant dans les esprits le besoin de la réflexion et du contrôle, en écartant de l'éducation toute contrainte intellectuelle, en soumettant aux esprits les objets sur lesquels la conscience et la raison s'exercent librement, elle donne à la personne humaine le sentiment de son droit et de sa valeur. Et, d'un autre côté, elle ne limite par aucun dogmatisme, par aucun parti pris confessionnel, la puissance de la science ; elle ne se livre à aucune agression systématique contre aucune croyance, mais elle ne subordonne par aucune complaisance servile les vérités de la science aux intérêts du dogme.
Ainsi se dissiperaient les préjugés ; ainsi s'apaiseraient les fanatismes ; ainsi le jour viendra où tous les citoyens, quelle que soit leur conception du monde, catholiques, protestants, libres penseurs, reconnaîtront le principe supérieur de laïcité. Et la conscience de tous ratifiera les lois nécessaires et bienfaisantes dont l'effet prochain sera, je l'espère, de rassembler dans les écoles laïques, dans les écoles de la République et de la nation, tous les fils de la République, tous les citoyens de la nation.
Et n'est-ce point pitié de voir les enfants d'un même peuple, de ce peuple ouvrier si souffrant encore et si opprimé et qui aurait besoin, pour sa libération entière, de grouper toutes ses énergies et toutes ses lumières, n'est-ce pas pitié de les voir divisés en deux systèmes d'enseignement comme entre deux camps ennemis ?
Et à quel moment se divisent-ils ? À quel moment des prolétaires refusent-ils leurs enfants à l'école laïque, à l'école de lumière et de raison ? C'est lorsque les plus vastes problèmes sollicitent l'effort ouvrier : réconcilier l'Europe avec elle-même, l'humanité avec elle-même, abolir la vieille barbarie des haines, des guerres, des grands meurtres collectifs, et, en même temps, préparer la fraternelle justice sociale, émanciper et organiser le travail.
Ceux-là vont contre cette grande œuvre, ceux-là sont impies au droit humain et au progrès humain, qui se refusent à l'éducation de laïcité. Ouvriers de cette cité, ouvriers de la France républicaine, vous ne préparerez l'avenir, vous n'affranchirez votre classe que par l'école laïque, par l'école de la République et de la raison.
Discours de Castres, 30 juillet 1904
(L'Humanité, 2 août 1904).
Grève générale et suffrage universel ouvrier
Depuis 1906 et le congrès d'Amiens, la CGT se revendique indépendante des partis politiques (par la « charte d'Amiens »), et tout particulièrement de la SFIO récemment créée. Un des grands points de divergence entre la CGT et la SFIO concerne les moyens d'action. La première revendique l'usage de la grève générale contre le suffrage universel qu'il juge un instrument de duperie ; les socialistes français, eux, sont divisés sur la question, mais Jaurès fait partie de ceux pour qui le suffrage universel est sacré. Très hostile à la grève générale, il la considérera progressivement comme un moyen d'action possible et complémentaire.
Je persiste à penser, malgré les objections qui me sont faites et qui me paraissent bien vaines, que le référendum, l'appel au suffrage universel ouvrier, est la condition d'une action très forte des syndicats. Ce n'est pas en s'isolant que la minorité syndicale prouvera sa force et développera son action, mais en mettant tout en œuvre, toutes les énergies ouvrières, en assurant l'unité du mouvement par la consultation préalable de toutes les forces, par l'éveil de toutes les responsabilités.
De même que les grèves partielles, et plus encore, c'est par le suffrage universel ouvrier que la grève générale, aux heures critiques de l'évolution sociale, aura toute son efficacité.
D'abord, c'est la condition même du succès réel du mouvement. Plus le champ sur lequel il doit se produire est vaste, plus il risque d'avorter par l'incertitude où sont les ouvriers du sentiment vrai de leurs camarades. Ils hésitent à se risquer, ne sachant pas si les autres, au loin, marcheront.
C'est par là surtout que s'explique le formidable échec de la grève générale essayée il y a quelques années par le syndicat des chemins de fer. Le syndicat était puissant. Il croyait avoir pris toutes ses mesures. Il espérait être suivi par les sections. Personne ne bougea. Pourquoi ? Parce qu'à la minute critique, quand il s'agit pour les travailleurs de jouer leur pain, leur vie, la vie de leur famille, chacun d'eux se demande avec angoisse si les autres se décideront ; une effroyable sensation d'isolement pèse alors sur chacun des ouvriers.
La certitude d'une action collective ne peut leur être donnée que dans une délibération sérieuse et publique, aboutissant à un vote. C'est ainsi qu'ont procédé en Angleterre les ouvriers et employés des chemins de fer, il y a deux ans. Le syndicat proposa la grève générale au suffrage universel des ouvriers, et une majorité imposante se prononça pour la grève, après discussion dans les meetings et devant la presse, et au scrutin secret. Devant cette menace sérieuse, devant cette ferme résolution, les Compagnies entrèrent en composition. On a beaucoup discuté en Angleterre, dans le monde ouvrier pour savoir si le syndicat a tiré un suffisant parti de cette démonstration de force, et s'il n'eût pas mieux valu pour lui déclarer en effet la grève générale qu'accepter les combinaisons compliquées proposées par le ministre anglais Lloyd31. C'est un débat où je n'ai pas à entrer en ce moment. Mais une chose est certaine ; c'est qu'après le vote des ouvriers, personne, en Angleterre, ne douta un instant que la grève générale, si elle avait été déclarée, aurait été effective. Les ouvriers des chemins de fer en avaient l'assurance ; et cette assurance même rendait la grève générale possible. Ils n'étaient pas réduits à se demander anxieusement : que se passera-t-il au-delà de l'horizon immédiat que mesure notre regard ? Ils se sentaient portés par une grande force de solidarité certaine.
Le vote délibéré, réfléchi, de tous les travailleurs intéressés dans le conflit, constitue déjà un effet de masse. Il constitue aussi pour tous les ouvriers un engagement d'honneur. Et comment sans cela ne redouteraient-ils point, avant de se jeter dans un combat qui peut soudain se creuser comme un gouffre, les hésitations, les défections, les ajournements de groupes lointains, dont la volonté n'aurait pas été interrogée et manifestée ? Et quand il s'agira de la grève générale d'un grand nombre de corporations diverses, quand il faudra faire de la grève vraiment générale non plus un mot stérilement agité, mais une réalité d'action, quand il faudra par exemple associer à un même mouvement de protestation ou de revendication, par le refus collectif et simultané du travail, les salariés des chemins de fer, des mines, du textile, de la métallurgie, de la verrerie, des services publics, de la typographie, des transports urbains, de l'éclairage, de la grande propriété foncière, s'imagine-t-on vraiment que pour mobiliser une masse aussi énorme et aussi diverse, il suffira d'un mot d'ordre donné de haut par quelques syndicats ou même par la Confédération générale du travail en qui les syndicats seront groupés ?
Le doute paralysant croîtra en chaque individu en proportion même de la masse à émouvoir. Non seulement chaque corporation, chaque salarié, se demandera si ses camarades inconnus, dispersés sur toute la surface du pays, obéiront au signal donné, mais chaque corporation se demandera si les autres corporations marcheront. Ce sera un flottement mortel ; et bientôt, entre ceux qui se seront risqués tout seuls ou par petits groupes et ceux qui se seront abstenus, les récriminations, les âpres défiances ; ce sera, pour la plus grande joie de la bourgeoisie ironique, la guerre intestine dans le prolétariat, l'effort ambitieux des salariés retombant sur eux et les écrasant, comme une masse maladroitement soulevée par des bras incohérents et débiles.
Il n'y a qu'un moyen de mettre dans ce mouvement si ample d'éléments si multiples la clarté, la confiance, la certitude, c'est d'appeler le suffrage universel ouvrier, dans toutes les corporations, où il y a un commencement sérieux d'organisation, à se prononcer sur la grève générale. J'admire ceux qui disent qu'il y aurait là une abdication pour lui au contraire ! Ce sont les syndicats qui poseront les problèmes ; ce sont eux qui dégageront des plaintes, des griefs, des revendications confuses de la masse ouvrière, les questions précises qui deviendront le programme de la grève générale, questions assez diversifiées pour émouvoir directement en quelques points précis chacune des corporations, assez vastes pour les animer et les passionner toutes d'une ardeur commune, d'une même et grandiose espérance. Ce sont les militants du syndicalisme qui dirigeront et coordonneront la propagande. Et jamais ils ne trouveront le peuple ouvrier plus disposé à écouter, à réfléchir, à penser, puisqu'au bout de toutes ces libres controverses il y aura un vote, c'est-à-dire un acte qui pourra être le prélude de toute une série d'actes émouvants. Ce sera pour eux aussi le moyen de connaître à fond le prolétariat, d'éclairer à ses diverses profondeurs cet océan obscur et dormant. Ils sauront quels sont les problèmes qui réussissent à émouvoir la masse, où doit se porter l'effort d'éducation et la propagande. Si la majorité des salariés, dans l'ensemble des corporations, s'est prononcée pour la grève, si au contraire, dans telle ou telle des corporations, la majorité s'est prononcée contre, c'est auprès de celle-ci qu'ils agiront, forts de la résolution authentique et manifestée de la majorité des prolétaires.
Dira-t-on que ces mouvements de masse seront nécessairement rares ? Sans doute ! Mais c'est la condition absolue de leur efficacité. Ce n'est pas impunément qu'on épuiserait la classe ouvrière par des tentatives étourdiment répétées. La grève générale elle-même dégénérant peu à peu en une série d'escarmouches qui n'en auraient que le nom, perdrait tout prestige et toute vertu. L'essentiel n'est pas que la grève générale se produise fréquemment ; c'est une impossibilité, c'est une contradiction. L'essentiel est que quand elle se produit, aux moments décisifs de l'évolution et de la bataille sociale, pour des motifs graves et des objets de première importance, elle se produise en effet avec le maximum d'efficacité. Une grève générale sérieuse, conduite vraiment par l'ensemble du prolétariat, le couvrirait pour de longues années contre toute tentative systématique de réaction sociale, contre toute atteinte et toute menace à ses libertés syndicales, aux lois de protection ouvrière déjà conquises. Elle obligerait la démocratie, pour de longues années, à appliquer plus loyalement les lois de protection ouvrière, à hâter le travail de la protection sociale et la graduelle émancipation du prolétariat. Ce sont de mauvaises montres, celles qu'il faut remonter sans cesse. Ce serait de mauvaises grèves générales, celles qu'il faudrait recommencer à chaque minute. Elles doivent être un ébranlement énorme, capable de se propager au loin, comme des ondulations de l'eau sous la chute d'un énorme bloc. Il est plus facile, à coup sûr, de jeter au vent des poignées de sable.
Dira-t-on que, par cette préparation nécessairement lente, et par cette consultation publique du suffrage universel ouvrier, le prolétariat donnera au pouvoir et à la bourgeoisie tout le loisir d'organiser la résistance ou l'attaque ? Mais il ne faut pas être dupe des mots. En fait, le pouvoir est toujours préparé au maximum. Il est servi par une centralisation administrative formidable, par une concentration militaire permanente. Et la bourgeoisie sera tous les jours mieux organisée. Elle se syndique, elle se fédère, elle s'essaie aux lock-out. S'imaginer qu'elle sera surprise à l'état de sommeil par une grève vraiment générale serait puérilité, et quand elle serait décrétée par un Comité central dictatorial auquel la classe ouvrière aurait laissé toute initiative, elle ne peut éclater comme un coup de foudre. Il y faudra toujours des délibérations, une préparation. Elle ne jaillira pas comme d'une nuée diffuse subitement concentrée et sous prétexte de surprendre le patronat et l'État qui seront en réalité préparés, c'est la classe ouvrière qui se retirera à elle-même la profonde et vaste préparation qui est la condition absolue de la victoire ; c'est elle qui se retirera la puissance prodigieuse de l'effet de masse produit par la grève générale, libre et éclatante décision du suffrage universel ouvrier.
L'Humanité, 9 septembre 1908.
« Pas d'équivoque »
Jaurès a peu écrit sur l'économie et n'a guère développé d'analyse originale du capitalisme, voire de « l'impérialisme », à l'heure où cette question est âprement débattue dans le socialisme international, notamment autour de l'ouvrage de Rudolf Hilferding Le Capital financier (1910). Dans certains textes, dont celui que nous reproduisons ici, il a néanmoins avancé sur quelques points majeurs, notamment sur la question des nationalisations, qui fut au cœur des revendications programmatiques de la gauche socialiste française depuis cette époque, et au moins jusqu'aux années 1980.
Quand le Parti socialiste demande que, dès maintenant, les monopoles capitalistes qui dominent la vie politique et économique du pays soient nationalisés ou communalisés, quand il demande, par exemple, que les voies ferrées, les mines, la banque, la fabrication et la raffinerie du sucre, les assurances, la production des engrais chimiques, les grands services d'alimentation, fassent retour à la nation, quand il demande l'extension du domaine municipal et de la gestion municipale aux transports locaux, omnibus et tramways, à l'éclairage, gaz et électricité, au logement, il ne confond pas du tout ces nationalisations ou municipalisations accomplies dans le cadre et sous les lois générales du système capitaliste toujours dominant, avec ce que sera le régime de la propriété socialisée dans un monde d'où le capitalisme et le salariat auront disparu.
Il sait très bien que sur les services ainsi nationalisés ou municipalisés en régime capitaliste pèsera encore, au moins pour une part, le prélèvement capitaliste, par l'intérêt et l'amortissement des emprunts contractés pour le rachat, ou pour le développement de la production. Il sait que les conditions de travail et de salaire, tout en pouvant être haussées au point où elles serviront de modèle et constitueront une force d'entraînement, resteront cependant dominées par les conditions d'ensemble de la production et soumises aux lois générales de l'équilibre capitaliste. Il sait tout cela, et il le rappelle fortement aux prolétaires, pour les mettre en garde contre toute confusion et toute illusion, à l'heure même où il lutte le plus énergiquement pour accroître le domaine économique de la nation et des communes. Il sait que ce n'est point là la réalisation, même partielle, de son idéal suprême, et il n'oublie pas non plus que ce n'est qu'une part, une faible part de l'œuvre immédiate qu'il doit accomplir pour la protection et l'extension des libertés ouvrières, pour la protection, l'organisation et l'émancipation progressive du travail.
Mais il sait, en même temps, d'accord avec tout l'instinct et toute la pensée du socialisme et de la classe ouvrière militante depuis plus d'un siècle, qu'il y a le plus grand intérêt pour le prolétariat en marche à affaiblir le capitalisme privé, à diminuer son influence morale et sociale en lui arrachant les domaines où son activité s'exerce avec une concentration souveraine. C'est une grande force pour le socialisme de démontrer à la démocratie routinière et réfractaire que la gestion du capitalisme privé, l'appât du profit et du dividende, la direction et l'initiative des oligarchies du capital ne sont pas nécessaires au fonctionnement, au progrès des grandes entreprises. Il y a pour lui un intérêt de premier ordre à prouver que dans une démocratie où grandit la puissance organisée du prolétariat, la collectivité nationale ou communale peut exercer de grandes fonctions économiques dans des conditions nouvelles, avec la participation des consommateurs groupés et des travailleurs organisés à l'administration, à l'impulsion et au contrôle. Voilà l'objet à atteindre. Voilà l'effort à accomplir. Voilà la bataille à livrer. Il s'agit tout à la fois d'accroître aux dépens du capitalisme privé, le domaine économique de l'État, et de faire pénétrer dans l'État des mœurs nouvelles, la force de contrôle de la démocratie et des consommateurs, la force de contrôle, la liberté organisée et le droit des salariés.
Voilà pourquoi, dans cet effort, dans cette bataille qui commence à peine, et qui se marque en ce moment même par des actes émouvants, par la lutte des instituteurs syndiqués et confédérés, par la protestation et la démission de délégués de cheminots de l'État, par l'affirmation de solidarité avec les maîtres laïques de la Fédération des salariés des services publics, voilà pourquoi dans le grand combat profond, intérieur, le socialisme, le syndicalisme, la coopération prolétarienne sont trois forces nécessairement unies.
Le Parti socialiste, parti de la démocratie sociale, portant en lui l'idée de la complète émancipation ouvrière, de la plus haute civilisation humaine et de l'entière démocratie ; le syndicalisme, groupant les salariés de toute corporation dans une vaste unité où ils prennent conscience de leur force immédiate et de leur force d'avenir ; le coopératisme, représentant et appelant à l'organisation la force immense, mais trop longtemps diffuse et stérilisée des consommateurs, ces trois puissances distinctes et unies, transformeront l'idée du service public, bureaucratique, oligarchique et fiscal en une vaste organisation de démocratie et de travail, en un magnifique apprentissage des forces et des vertus par lesquelles, un jour, l'ordre nouveau fonctionnera dans sa plénitude. Une partie des profits, accaparés à l'heure actuelle par le capital, une partie de l'influence sociale de direction concentrée par des états-majors de grandes compagnies et sociétés capitalistes passeront à la collectivité nationale et municipale, pénétrée tous les jours davantage d'esprit démocratique et d'influences prolétariennes.
Ainsi l'État sera peu à peu économiquement agrandi et socialement transformé. Ainsi dans l'État de classe et de privilège se formera le rudiment d'un État de travail, d'administration populaire et d'égalité, et cette organisation nouvelle, en se développant, amorcera, contribuera à préparer l'ordre vraiment socialiste où l'État ne sera plus qu'une société libérée des classes et des violences de classes, que la forme et la garantie du respect réciproque des droits des individus, le lien de la grande et noble discipline consentie par les citoyens et les producteurs égaux pour l'œuvre commune de production et de civilisation, le point de ralliement et l'organe central d'exécution des volontés manifestées par les groupes professionnels associés par les communes fédérées dans la grande unité nationale.
C'est bien là ce qu'ont pressenti depuis la Révolution française, depuis le plein avènement du monde moderne, tous les socialistes, tous les communistes32, tous les prolétaires luttant pour la liberté et la justice. C'est là une des premières pensées des ouvriers de France, quand s'éveilla d'abord en eux, en face de la bourgeoisie dont ils étaient à la fois les alliés contre l'ancien régime et les adversaires dans le régime nouveau, un puissant instinct de classe. C'est dans cet esprit, c'est dans ce pressentiment que, dès 1792, les ouvriers mineurs et les démocrates révolutionnaires d'Anzin, constatant que la tyrannie des seigneurs d'ancien régime se continuait dans la tyrannie, dans les manœuvres contre-révolutionnaires des seigneurs du capital, demandent aux Assemblées de la Révolution la nationalisation des mines, dans une pétition d'une haute valeur historique que M. Schmidt a retrouvée et publiée dans les documents sur la vie économique de cette grande époque33. C'est dans cet esprit, c'est pour accroître les garanties et la puissance des ouvriers s'éveillant sous l'orage à la vie de classe et aux plus sublimes ambitions d'avenir, qu'en 1794 les travailleurs de la Nièvre demandent la nationalisation des forges, des mines et de l'industrie des forêts à leur compatriote Chaumette, qui transmet leur revendication à la Commune, et, par elle, à la Convention. J'en ai trouvé dans les journaux du temps et publié le texte. C'est sous la poussée des forces ouvrières qu'au lendemain de la rupture avec la Gironde, les délégués de la Montagne, qui ne pouvaient se sauver et sauver la Révolution que par l'appui du prolétariat, menaçaient les patrons de nationaliser les fabriques, s'ils ne payaient pas aux ouvriers un salaire suffisant. J'ai publié la photographie du procès-verbal de la municipalité de Castres, où est reproduit le discours impérieux du conventionnel Baudot. Les délégués de la Convention menaçaient partout les boulangers, s'ils ne vendaient pas à des prix raisonnables, de créer des boulangeries municipales. Et la nationalisation du commerce des grains était, aux jours de révolte, une des revendications les plus véhémentes du prolétariat souffrant.
C'est aussi la création de grands services nationaux et communaux qui constitue pour Babeuf, puis pour Buonarroti34 le programme de transition, de préparation qu'on proposait à la démocratie pour l'orienter vers le communisme. C'est par de grandes nationalisations progressives que Marx et Engels, dans Le Manifeste communiste, préludent à l'œuvre de transformation totale, dans la période où le prolétariat sera assez fort pour imprimer sa direction au monde social, pas assez pour réaliser le communisme entier. C'est cette formule de nationalisation qui éclate dans les revendications ouvrières de 1848.
Et par une suite invincible de tradition, c'est Bebel et Liebknecht35 affirmant, dans les congrès de la démocratie socialiste allemande, que les nationalisations seraient plus utiles et plus bienfaisantes à mesure que se développerait la démocratie et que par elle le contrôle du peuple et du prolétariat sur les services nationaux serait plus efficace. Elle est donc pour eux une partie essentielle du programme d'action de la classe ouvrière dans la période où celle-ci sera assez forte pour exercer un contrôle réel mais où elle ne pourra encore réaliser toute sa formule révolutionnaire.
Mais en France même, aux dernières élections municipales de Paris, quelle est la parole qui faisait le plus vibrer la classe ouvrière parisienne ? C'est le cri d'indignation et de révolte de nos militants contre la trahison de radicaux de l'Hôtel de Ville, qui livrèrent aux compagnies capitalistes les services des transports et de l'éclairage, tout ce que la commune devait garder dans son domaine. Et quelle est la partie du programme qui portait le plus ? C'est celle où nos amis revendiquaient pour la Ville de Paris le droit de posséder et de construire des logements pour faire échec aux prétentions abusives et aux loyers usuraires des propriétaires bourgeois.
Mais quel est le syndicaliste, parmi les signataires de la lettre ouverte, qui s'est levé alors pour dire que le Parti socialiste faisait fausse route, et que ce sont les radicaux de l'Hôtel de Ville, qui en laissant au capitalisme privé les services les plus essentiels avaient le mieux respecté les conditions d'organisation de la classe ouvrière ? Quel est, parmi les syndicalistes, quel est, parmi mes contradicteurs immédiats, celui qui a protesté quand le Congrès général des cheminots a voté la nationalisation des chemins de fer, quand le Congrès général des mineurs a voté la nationalisation des mines ?
Il y a quelques jours à peine, le puissant Congrès anglais des Trade Unions votait à l'unanimité une résolution demandant la nationalisation immédiate de la terre, des chemins de fer et des usines. Les organisations ouvrières anglaises, qui sont de plus en plus pénétrées de la pensée socialiste et de l'esprit de classe, et qui n'ont jamais été plus militantes, seront-elles accusées par les citoyens Griffuelhes et Jouhaux36 de travailler contre l'action d'ensemble de la classe ouvrière, et de développer un corporatisme étroit pour rendre plus difficile l'organisation prolétarienne ? Et si cet argument ne vaut ni contre les cheminots et les mineurs de France, éléments si importants de la Confédération, ni contre le Congrès international des mineurs, ni contre l'unanimité des Trade Unions anglaises, de quel droit les citoyens Jouhaux, Griffuelhes, et leurs amis l'invoquent-ils contre nous ? De quel droit tentent-ils de diviser par là, en France, le syndicalisme et le socialisme ? Ce n'est pas entre eux et nous qu'ils creuseraient un fossé, mais entre eux et toute la tradition française et européenne du prolétariat militant, entre eux et tout le prolétariat de l'Europe.
Mais j'ai là-dessus des questions précises à poser.
L'Humanité, 17 septembre 1912.
1 Congrès de la Fédération nationale des syndicats ouvriers en octobre 1890, dominé par les partisans de Jules Guesde (1845-1922).
2 Congrès de la Fédération des travailleurs socialistes de France en octobre 1890 ; les partisans de Jean Allemane (1843-1935) quittent le congrès et créent le Parti ouvrier socialiste révolutionnaire dit « allemaniste ».
3 Le congrès de Halle, en 1890, est le premier congrès de la social-démocratie allemande après douze ans de clandestinité.
4 Louis Blanc (1811-1882) considérait en effet que les premières étapes de la Révolution de 1789 étaient avant tout individualistes et bourgeoises.
5 Convention : assemblée révolutionnaire qui gouverna la France de septembre 1789 à octobre 1795.
6 La journée du 31 mai 1793 fut marquée (comme celle du 2 juin) par la chute des girondins au profit des montagnards (Robespierre et Danton).
7 La Constitution de 1793 n'entrera jamais en application, mais fut une référence importante pour le mouvement socialiste au XIXe siècle.
8 Voir É. Zola, L'Affaire Dreyfus, Paris, Le Livre de Poche, coll. « Classiques », 2010.
9 Henri Rochefort (1831-1913), ancien communard un temps proche des socialistes, est désormais une des personnalités les plus opposées à Dreyfus. L'abbé Garnier (1850-1920) était une figure de l'hostilité au socialisme et de l'antisémitisme.
10 À la tête de la section statistique de l'armée et antisémite, Picquart est néanmoins convaincu de l'innocence de Dreyfus. Envoyé à l'étranger, il est arrêté en juillet 1898 pour avoir transmis des documents secrets et devient ainsi un héros des dreyfusards.
11 Esterhazy (1847-1923), officier d'origine austro-hongroise auteur du fameux bordereau qui permit l'accusation à l'encontre de Dreyfus.
12 Deux figures qui firent tout pour éviter la révision du procès.
13 Édouard Drumont (1844-1917) est un des représentants les plus célèbres de l'antisémitisme. Son livre La France juive (1886) eut un impact considérable, tout comme son journal La Libre Parole.
14 Boulanger (1837-1891) : officier français ayant provoqué, dans la seconde moitié des années 1880, une grave crise politique de la Troisième République. À partir de thèmes démagogiques, il parvint à réunir autour de lui nombre de déçus du bonapartisme, de la monarchie et même quelques socialistes. Son ascension fut stoppée en 1889.
15 Le premier est l'officier chargé de l'instruction judiciaire lors du premier procès de Dreyfus ; le second était rapporteur au premier conseil de guerre de 1897 et chargé de l'instruction contre Esterhazy, qu'il couvrit totalement.
16 Le colonel Henry, lié à Esterhazy, s'est suicidé en août 1898.
17 Adjoint d'Henry.
18 Attaché militaire à l'ambassade d'Allemagne, qui s'était lié avec Esterhazy en 1894, avant de rompre avec lui deux ans plus tard.
19 « Partageux » renvoyait à l'idée que les socialistes voulaient tout partager, notamment les propriétés ; le mot était employé par les conservateurs afin d'effrayer les paysans et de les détourner du socialisme.
20 Jaurès fut battu aux élections législatives, en 1898, à la suite d'une campagne particulièrement haineuse à son encontre.
21 Voir la préface aux Preuves, p. 38-51, et partie I, notes 8 à 18.
22 Allusion à la loi Falloux (liberté de l'enseignement secondaire) du 15 mars 1850, et au 2 décembre 1851 (coup d'État de Louis-Napoléon Bonaparte, qui signe la disparition de la Deuxième République).
23 Allusion à son adversaire le marquis de Solages ; 1885 correspond à la première campagne électorale victorieuse de Jaurès dans le Tarn.
24 En 1904, le radical Louis Vieu est élu à la tête de la municipalité de Castres sur une liste d'union de la gauche.
25 On considérait alors que les succès des Prussiens, lors de la guerre de 1870, étaient le résultat d'une éducation plus poussée des soldats.
26 Malgré des divergences notables, Jaurès conserva toujours une certaine admiration pour Gambetta ; il souhaitait même écrire une biographie de l'homme politique.
27 Pierre-Paul Royer-Collard (1763-1845), avocat et philosophe qui contribua à former les élites intellectuelles de la monarchie de Juillet.
28 « Les cieux racontent la gloire de Dieu » (Psaume 19).
29 Allusion à Mgr Dupanloup (1802-1878), évêque d'Orléans, académicien, député, puis sénateur. L'élection à l'Académie française d'Émile Littré (1801-1881), l'auteur du Dictionnaire, eut lieu en 1871.
30 Le vicomte Eugène de Vogüé (1848-1910), écrivain, académicien, et un temps député, et Ferdinand Brunetière (1849-1906), académicien et directeur de la Revue des Deux Mondes, sont deux personnalités importantes du monde catholique.
31 David Lloyd George (1863-1945), chancelier de l'Échiquier (ministre des Finances) du gouvernement britannique, développa un programme social (création de retraites ouvrières et lois d'assurances sociales).
32 Dans les années 1880-1914, il n'existe pas de distinction forte entre « socialistes » et « communistes » ; il ne faut donc pas plaquer ici les clivages ultérieurs nés de la révolution russe de 1917.
33 Sur Jaurès et la Révolution française, voir p. 31.
34 Gracchus Babeuf et Filippo Buonarroti sont deux des principaux acteurs de la « Conspiration pour l'égalité » de 1796-1797, considérée par Karl Marx comme l'embryon du premier « parti communiste agissant » et très respectée à ce titre par les socialistes.
35 Wilhelm Liebknecht (1826-1900) et August Bebel (1840-1913), principaux dirigeants de la social-démocratie allemande. Ils s'opposèrent tous les deux au vote des crédits pour la guerre de 1870.
36 Victor Griffuelhes (1874-1923) et Léon Jouhaux (1879-1954) sont deux des principaux dirigeants de la CGT.
II
Le défenseur de l'amitié franco-allemande
Cette deuxième partie se concentre sur les rapports de Jaurès avec l'Allemagne, pays de première importance à son époque tant la « revanche » anime les débats politiques. Les textes qu'on va lire permettent par ailleurs d'éclairer sous un angle original l'histoire des relations entre Allemands et Français, en montrant les prémisses d'un projet européen jaurésien, fondé sur la paix et le socialisme.
Jaurès tente en effet de développer les bases d'une amitié entre les deux nations pour éviter la guerre. Il revient par ailleurs, tout au long de sa trajectoire, sur l'histoire du pays voisin, tandis que ses controverses avec ses amis sociaux-démocrates allemands montrent toute la difficulté qu'il y a à fonder une solidarité franco-allemande au-delà des singularités des mouvements socialistes de chaque pays.
Le socialisme allemand
Jaurès soutient sa thèse complémentaire (à l'époque en latin) sur les origines du socialisme allemand. Sa formation intellectuelle tient beaucoup aux grands philosophes comme Kant, Hegel, mais aussi Marx. Il explique sa démarche dans les premières pages de cette thèse, où il relie la philosophie allemande aux combats politiques du XIXe siècle.
Nous savons à la vérité que le socialisme allemand n'est pas une philosophie pure et contemplative ; il lutte et combat pour abattre les fondements de la société civile actuelle. Ce n'est pas seulement une doctrine, mais encore un parti dans l'État. Mais la philosophie elle-même revêt parfois des dehors belliqueux, fourbit ses armes et se mêle au combat politique ; elle ne regarde pas seulement le ciel, mais aussi la terre. – Si Socrate a fait descendre la philosophie du ciel, le socialisme en a fait descendre la justice ; c'est-à-dire qu'il cherche dans la région « des idées » des vues pratiques pour l'arrangement de cette vie terrestre. – Fichte, Lassalle, Marx, Schäffle1 furent à la fois des précurseurs et des maîtres.
En pénétrant le socialisme allemand, on y trouve incluse une philosophie. Celle-ci prétend qu'il y a dans l'histoire et l'économie politique une certaine dialectique2 qui change les formes des choses et les relations des hommes. – Elle définit la liberté, non pas comme une abstraite faculté de pouvoir choisir entre des contraires, comme une hypothétique indépendance de chaque citoyen pris individuellement, mais comme la véritable base de l'égalité des hommes et de leur communion. Enfin, cette philosophie ne poursuit pas un fantôme céleste, une vaine image de justice séparée du monde et de l'ordre naturel des choses ; elle exige une justice matérielle mêlée aux choses elles-mêmes et s'appuyant sur elles. Au socialisme allemand tient donc une solide doctrine dialectique du devenir universel, de la liberté humaine, de la Nature et de Dieu.
Il n'est pas nécessaire, pour saisir exactement le lien du socialisme et de la philosophie allemande, que nous embrassions toute l'histoire de cette philosophie ; il suffit que nous interrogions ces hommes qui ont pour ainsi dire façonné le génie et la pensée allemandes. Ce ne sont pas, en effet, les talents médiocres et inférieurs qui influent sur les événements et les cours de l'histoire, mais bien les esprits les plus élevés. Or, qui niera que Luther, Kant, Fichte et Hegel ne tiennent le premier rang parmi les théologiens ou les philosophes de l'Allemagne ?
Il n'y a pas lieu de s'étonner de ce que nous laissions de côté cette doctrine matérialiste qui procède de l'une des faces de la philosophie hégélienne, bien qu'en Économique elle se rapprochât du socialisme. Par exemple, Feuerbach ne fut pas le maître du célèbre Marx, mais son condisciple. Tous deux ont donné la même interprétation de l'hégélianisme : l'un dans la philosophie, l'autre dans l'économie politique. Marx lui-même déclare avoir embrassé la dialectique hégélienne, pour la convertir en matérialisme et transformer ses vaines futilités en un métal terrestre, fer ou or. Au surplus, comme je ne rattache pas le socialisme allemand au matérialisme de « l'extrême gauche hégélienne », mais à ces idéalistes qui s'appellent Luther, Kant, Fichte et Hegel, je veux, non seulement atteindre les vraies sources profondes du socialisme allemand, mais encore découvrir la future évolution de ce socialisme. En effet, si aujourd'hui le socialisme allemand combat sous des apparences matérialistes, derrière le bouclier du matérialisme, c'est là l'aspect, non pas de la paix future, mais uniquement de la lutte présente. Les socialistes s'affirment et se croient matérialistes, pour les facilités de leur démonstration, afin que cette terre, délivrée cependant de tous les fantômes de la superstition, apparût sous une lumière dure et crue, d'autant plus hérissée de rudes misères ; mais, dans les replis profonds du socialisme, survit le souffle allemand de l'idéalisme.
Cette constatation sera manifeste lorsque nous aurons examiné la contribution apportée au socialisme par Luther, Kant, Fichte et Hegel ; il deviendra non moins évident que les socialistes ont été les vrais disciples de la philosophie allemande et comme du génie allemand lui-même.
Tout d'abord, cela indiquera plus clairement combien les événements découlent des idées, combien l'histoire dépend de la philosophie. À première vue, l'on pourrait croire que le socialisme a surtout fleuri en Angleterre, puisque c'est particulièrement en Angleterre qu'a insolemment grandi le nouvel ordre économique, qui a pour base fondamentale l'Argent. En Angleterre, de toutes façons, il était facile de saisir le procès économique. Mais qui l'a vu et décrit ? Ce n'est pas un philosophe anglais, c'est un Allemand habitant l'Angleterre, Karl Marx. Si Marx n'eût pas eu gravée dans son esprit la dialectique hégélienne, il n'eût pas rattaché tout le mouvement économique de l'Angleterre à cette dialectique socialiste. L'Angleterre a fourni les faits, mais la philosophie allemande les a interprétés. Le socialisme était né dans l'esprit allemand bien avant l'accroissement anormal de la grande industrie, et l'apparition des conditions constitutives du socialisme économique.
Les Origines du socialisme allemand (1892), extrait.
« Le problème subsiste »
Jaurès salue dans ce texte la grande œuvre de Karl Marx et de Ferdinand Lassalle qui, au-delà de leurs divergences, œuvrèrent tous les deux à la création de forces politiques qui allaient devenir, après leur fusion, le Parti social-démocrate allemand (qui prend le nom de SPD en 1890). Mais Jaurès critique également leur intransigeance et leur mépris des traditions républicaines françaises, dont héritent les sociaux-démocrates allemands. La confrontation a un relief tout particulier au congrès de l'Internationale socialiste d'Amsterdam en 1904, qui incite notamment les socialistes français à l'unité : ce sera chose faite l'année suivante, avec la création de la SFIO.
Marx et Lassalle n'ont pas été seulement les théoriciens et les organisateurs du socialisme allemand ; ils ont été les grands consolateurs de la pensée allemande. Quelle détresse en effet pour celle-ci de constater ou la débilité ou l'inanité des efforts tentés par la démocratie pour fonder un régime de liberté et de dignité humaine ! Quel contraste entre l'audace de l'esprit allemand saisissant et façonnant l'univers, et la langueur ou l'impuissance de l'action allemande ! Quoi ! même la grande commotion de la Révolution française n'a pas suscité en Allemagne des énergies de liberté et un mouvement efficace ! Et la fière autonomie de la volonté revendiquée par Kant est réduite à s'exercer dans le secret de la conscience, ou dans l'humble sphère des relations privées, et elle ne réussit pas à s'exprimer dans un ordre politique et social qui lui soit conforme ! Les paroles de feu de Fichte s'éteignent en tombant sur l'inerte et lourde servitude allemande comme des étincelles sur le pavé.
Quoi ! même quand le régime napoléonien, de violence, de conquête, de pillerie et d'insolence, a enfin secoué toute l'Allemagne, quand elle s'est dressée tout entière, peuples et princes, contre la tyrannie étrangère, et quand de cette collaboration nationale des rois et des peuples on peut attendre, au lendemain de la victoire, une collaboration de liberté et des garanties constitutionnelles, les peuples sont joués par les rois, et ils se laissent jouer ! La liberté bafouée et trahie est réduite aux conspirations impuissantes du Tugendbund3 ! Quelle faiblesse, et quelle humiliation ! Quelle souffrance pour les esprits fiers !
Mais quoi ! même la nouvelle secousse révolutionnaire de 1830, dont toute l'Europe est ébranlée, n'aboutit en Allemagne qu'à des tentatives incohérentes et débiles4. L'air allemand reste irrespirable aux hommes libres ; Heine5 s'exile, et lance de loin ses flèches éblouissantes au colosse allemand endormi. Börne6 s'exile aussi ; et de Paris, en une de ses lettres stridentes, il chante l'hymne ironique : « Ô patience, tourne vers nous, Allemands, ta face lunaire ! »
Dans la révolution de 1848, la tristesse et l'humiliation s'aggravent encore de toutes les espérances trompées. Elle tombe, vaincue par la contre-révolution, sans laisser même au peuple la force d'un grand souvenir ; et on dirait qu'elle n'a servi qu'à discréditer le suffrage universel, qui a masqué un moment comme un décor de papier la réalité subsistante de la servitude. Enfin, comme pour achever la déroute de l'idéalisme allemand, voici que le grand rêve de l'unité allemande s'accomplit, mais non point comme les plus hautes consciences l'avaient rêvée. Voici que le fer et le feu réalisent, sous la direction du militarisme prussien, ce que n'a pu accomplir la force de la pensée et de la démocratie7. Voici que l'Allemagne devient une nation puissante entre les nations sans devenir une nation libre. Et l'histoire inflige à l'idéalisme allemand, depuis un siècle, cette double et croissante défaite, cette double et croissante humiliation, d'abord de lui refuser l'accomplissement de son rêve, et ensuite de l'accomplir sans lui et contre lui.
C'est de ce désespoir, c'est de ce destin qui semblait irrévocable que Marx et Lassalle ont sauvé l'idéalisme allemand et la démocratie allemande. Malgré des variations superficielles de tactique et de vifs dissentiments, leur trouvaille commune, trouvaille de génie, fut de dégager et de réserver une force neuve, qui ne fût pas compromise dans la triste solidarité des humiliations et des défaites, et de lui confier l'avenir. Cette force, pressentie par eux avant même qu'elle fût capable de jouer un rôle historique efficace, ce sera le prolétariat allemand. Le premier souci de Marx, dès qu'il pense, c'est de concentrer sur la bourgeoisie allemande tout le poids des défaites de la démocratie, toute la responsabilité de l'impuissance révolutionnaire de l'Allemagne. Sentence exclusive et partiale ? Oui, peut-être, en un sens : car d'abord la bourgeoisie allemande, dans son effort morcelé d'émancipation, se heurtait à des obstacles que même la vigueur révolutionnaire de la bourgeoisie française n'eût pas sans doute brisés ; et ce n'était pas la seule bourgeoisie, c'était toute la démocratie allemande, avec ses pauvres éléments de prolétariat et d'artisanerie comme avec ses forces bourgeoises, qui avait été ou inerte ou impuissante en 1789, en 1830, en 1848. Mais il était vrai que la démocratie allemande échouait misérablement dans cette période de l'histoire où c'est la bourgeoisie qui suscitait et dirigeait le mouvement révolutionnaire.
Ainsi Marx avait le droit de faire porter à la bourgeoisie allemande la responsabilité du désastre de la démocratie. Surtout il avait le droit de sauver l'avenir en dégageant de la solidarité de la défaite, en protégeant contre les humiliations de l'histoire cet humble prolétariat dont la croissance pouvait seule, un jour, libérer l'Allemagne. Et ce prolétariat, débile encore et obscur, comme il en pressent la grandeur ! Comme il la prépare, comme il la crée en la définissant !
C'est lui, c'est ce prolétariat démuni qui continuera, et qui réalisera dans les faits le haut idéalisme allemand. C'est en lui, et en lui seul, que prendra corps l'infinie liberté de l'esprit, affirmée par la philosophie de l'Allemagne ; car n'ayant pas d'intérêts constitués et compacts, il n'a pas de préjugés ou d'égoïsmes qui s'opposent au développement de l'esprit ; et l'ardente vie intérieure de Fichte, flamme repliée et refoulée qui ne pouvait se communiquer que par échappées à un monde solide et compact, se développera comme une grande force sociale dans ce prolétariat nécessairement généreux dont aucune parcelle d'âme n'est réfractaire à la chaleur de l'idée. Précisément parce qu'il est tout misère, servitude, ignorance, parce qu'il est l'absolu de la privation, le prolétariat est l'absolu de la revendication, l'absolu de l'aspiration, c'est-à-dire l'absolu de l'humanité et de l'esprit. C'est donc par lui que la grande force de la pensée allemande passera dans la réalité.
Le mouvement économique qui suscite et développe la grande industrie prépare, par l'expropriation capitaliste de la petite industrie, l'expropriation socialiste du capital. Que le prolétariat prenne seulement conscience de cette dialectique ; que cette révolution économique, qui n'est encore que nature, soit élevée à l'Idée, à l'idée consciente, à l'idée « pour soi », et la libération du travail et des hommes va s'accomplir. Ainsi Marx faisait du prolétariat l'héritier ou plutôt le représentant historique de la plus haute pensée allemande. Ainsi il construisait le pont sublime par où la conscience allemande passerait sans déchoir des plus audacieuses spéculations sur l'univers à l'audace de la révolution sociale, forme suprême de la pensée. Ainsi, à l'heure même où l'idéalisme allemand semblait le plus accablé et le plus bafoué par l'histoire, il le relevait soudain et l'exaltait en l'identifiant à ce prolétariat qui devait être la grande force historique de l'Allemagne. La période de dépression et de découragement était donc close, et sous l'abondance marxiste de l'ironie et du sarcasme une grande espérance s'affirmait.
C'est là ce qui justifie l'attitude politique de Lassalle en 1863, dans le conflit entre Bismarck et les libéraux de la Chambre prussienne. Bien loin d'appeler la classe ouvrière au secours des libéraux, il l'invita à se constituer contre eux, au risque de faire le jeu de Bismarck, au moins momentanément. Coupable eût été cette tactique si Lassalle avait pu croire que l'informe et impuissante démocratie allemande, qui n'avait été que néant ou désastre, en 1789, en 1830, en 1848, était maintenant en état, même avec le concours d'une élite prolétarienne plus excitée, d'accomplir la grande œuvre nationale et démocratique que l'Allemagne attendait. Mais Lassalle crut démêler dans l'agitation libérale les mêmes symptômes de débilité, les mêmes signes d'avortement qu'offrait depuis un siècle la confuse démocratie bourgeoise et artisane de l'Allemagne. Et il jugea funeste de noyer dans cette impuissance et cette défaite chroniques la force neuve du prolétariat. Cette tactique d'isolement et d'intransigeance prolétarienne lui était suggérée, et sans doute imposée, par toute l'histoire défaillante de la Révolution allemande8, comme la tactique inverse de la classe ouvrière française soutenant, malgré Proudhon, l'opposition républicaine réveillée contre l'Empire lui était suggérée et imposée par toute l'histoire tourmentée et brisée, mais rebondissante et victorieuse de la Révolution française.
C'est donc en conformité avec les exigences de l'histoire allemande comme avec la pensée de Marx et de Lassalle que la démocratie socialiste allemande s'est constituée et développée. Il semblait qu'en dédaignant et négligeant ce libéralisme bourgeois qui pouvait paraître encore la seule force immédiate, la seule chance prochaine de démocratie, et en confiant tout l'avenir à un prolétariat qui n'était encore qu'un germe débile, Marx et Lassalle avaient pris un détour bien long et bien hasardeux. En fait, la démocratie socialiste allemande, depuis qu'elle a pu exercer son action dans une Allemagne relativement unifiée, a grandi avec une force et une rapidité singulières ; elle a été, pour le prolétariat allemand, une puissance admirable d'organisation et d'éducation. Et des éléments merveilleux ont été réunis pour la solution du problème auquel depuis un siècle la démocratie allemande a succombé.
Mais le problème n'est pas résolu. Il reste toujours à savoir comment se réalisera en Allemagne une démocratie politique permettant l'évolution légale du prolétariat et la réalisation graduelle du plan communiste. Il reste à savoir comment le suffrage universel allemand s'élèvera de son rôle subalterne d'aujourd'hui à l'état de puissance dirigeante et souveraine. C'est ici que je touche au vif des grands débats d'Amsterdam9. C'est ici que commence à apparaître l'aberration de ceux des socialistes allemands qui, pour faire le jeu de quelques socialistes français étrangement déviés de la tradition socialiste française, ont jeté la défaveur sur la démocratie politique, juste à l'heure où le socialisme allemand a besoin d'instituer à son tour la démocratie politique et l'effective souveraineté du peuple, sous peine de continuer l'avortement bourgeois de la Révolution allemande par un avortement prolétarien.
L'Humanité, 16 septembre 1904, texte intégral.
« Controverses allemandes »
Dans cet article, Jaurès polémique frontalement avec ses camarades allemands : l'enjeu central est la question de la revendication républicaine. L'Allemagne était alors un empire et, formellement, le SPD demandait une république allemande ; en réalité, cette revendication est de moins en moins avancée. Les socialistes français comme Jaurès sont critiqués pour leur attachement à la tradition républicaine, que beaucoup d'Allemands considèrent comme secondaire, voire nuisible, car trop rattachée à l'histoire de la bourgeoisie. Jaurès retranscrit ici le débat qu'il mène sur cette question avec plusieurs responsables politiques allemands ayant publié des articles dans le Vorwärts, journal quotidien berlinois du SPD. Il les cite largement pour montrer leurs contradictions sur le sujet.
La tendance démocratique dans le socialisme allemand. – Les articles du Vorwärts. – Kurt Eisner et la République. Bebel contre Bebel.
S'il était entendu, pour tous les socialistes allemands, que la monarchie a plus de fécondité sociale que la République, et qu'elle est préférable à bien des égards pour le prolétariat ; si, au lieu de regarder en face la réalité, et de se préparer, par un effort énergique et méthodique, à instituer dans l'Empire la démocratie politique, ils s'obstinaient à couvrir l'impuissance où les réduit encore une Constitution d'autorité, en affectant le dédain de la République et en rabaissant de parti pris l'action du socialisme dans une démocratie républicaine, tout dialogue entre socialistes allemands et socialistes français serait inutile ou au moins prématuré. Nous n'aurions qu'à attendre, sans explication aucune, sans controverse aucune, que la leçon des événements, l'instinct de la vie et la croissance même de leur parti induisent les socialistes allemands à un effort délibéré vers la démocratie politique, instrument nécessaire de l'œuvre sociale.
La véritable action de Bebel
Mais il n'en est rien. Je suis sûr que Bebel10 lui-même a cédé à un entraînement de polémique : il a été pris au piège de la motion de Dresde11 ; et Guesde lui a joué le méchant tour de faire appel aux formules allemandes pour justifier sa propre tactique. Bebel est un politique trop avisé pour s'immobiliser dans une théorie d'indifférence politique et d'impuissance. Il constatera bientôt qu'en Allemagne même le besoin s'accroît de donner à la puissance électorale croissante du prolétariat une forme et une sanction politiques et d'entrer dans un effort de démocratie. Ce qui doit le frapper, c'est que ce n'est pas seulement chez ceux qu'on appelle les révisionnistes que cette tendance s'affirme. Le fait qu'à Amsterdam même et malgré son intervention vigoureuse, 15 voix contre 30, dans la délégation allemande, se sont prononcées pour la motion beaucoup plus large et beaucoup plus souple d'Adler et de Vandervelde12, a été sans doute un avertissement pour cette intelligence si vive et si attentive. Un des délégués les plus considérables de la délégation allemande (et un radical) disait : « Bebel a donné, dans son discours, l'impression que le parti est le but du parti. Or, c'est pour obtenir des réformes et pour développer ses moyens d'action que le prolétariat constitue le parti. »
Les articles d'Eisner
Mais surtout Bebel a dû être frappé de la force, de l'énergie et de la véhémence de conviction avec lesquelles Kurt Eisner13, un des principaux rédacteurs du Vorwärts, a pris parti pour la République, pour la démocratie politique, et a tenté de rompre la fausse et trompeuse solidarité que la complaisance doctrinaire de Kautsky a laissé établir entre le socialisme allemand et la conception de Guesde14.
Je ne connais pas personnellement Kurt Eisner : je ne l'ai jamais rencontré et je n'ai jamais correspondu avec lui. C'est seulement par ses articles du Vorwärts que je puis démêler ses tendances. Je me hâte de dire qu'il est loin d'approuver sans restriction notre politique ; et il formule contre elle de sérieuses réserves que je discuterai. Mais il a l'esprit libre ; et du centre même de cette Allemagne semi-absolutiste, où la lutte des classes a une forme compacte, il a le sens de la vie parlementaire et de la démocratie. Il a, dans un article de la Neue Zeit15, qui fit sensation, indiqué, du point de vue théorique et sans entrer dans l'examen de la politique socialiste italienne et française, que le socialisme pouvait être conduit, dans les pays parlementaires, à participer à une action gouvernementale et à une action de majorité. Mais sa préoccupation obsédante (et c'est pourquoi il désire passionnément l'unité du socialisme français), c'est que le parti socialiste soit si fortement constitué, et en communication si constante avec le prolétariat, que les périls de cette action parlementaire ou même gouvernementale s'atténuent, et que seuls subsistent les effets utiles.
Il n'est pas « révisionniste » et « bernsteinien ». Il me semble, si je comprends bien sa pensée, que ce qu'il reproche à Bernstein16 et à quelques-uns de ses amis, c'est d'avoir dispersé leur critique théorique sur trop de points, c'est d'avoir dépensé un trop grand effort à écarter des théories surannées, comme celle de la « catastrophe », qui, en fait, n'avaient plus de racine dans la pensée du parti, et de n'avoir pas suffisamment concentré leur propagande sur la nécessité d'une action politique plus vigoureuse et plus efficace. Je ne discute pas en ce moment, j'expose seulement quelles me paraissent être les idées directrices de Kurt Eisner pour qu'il n'y ait pas de malentendu sur le sens de ses déclarations. Au demeurant, il paraît supporter impatiemment qu'on se tire d'affaire avec des mots et qu'on se donne l'illusion de résoudre les problèmes qui sollicitent le parti en groupant sous un vocable tendancieux, celui de révisionnisme ou un autre, des idées très diverses.
J'ajoute que c'est Liebknecht, si je ne me trompe, qui a discerné le vif et libre esprit de Kurt Eisner et qui l'a introduit au Vorwärts. En tout cas, je me souviens qu'au congrès de Dresde Bebel a rendu hommage à la haute valeur philosophique et politique de ses articles ; et Kautsky17, dans la polémique toute récente qui s'est émue entre Kurt Eisner et lui au sujet du congrès d'Amsterdam, se dépite que « la tête dirigeante » du Vorwärts se permette cette controverse.
Je ne ferai pas à Bebel et à Kautsky l'injure de supposer qu'en reproduisant les articles d'Eisner je leur fournis contre lui un grief complémentaire. Ces articles sont maintenant dans le domaine international, et j'ai marqué assez nettement la pensée propre de Kurt Eisner, j'insisterai assez, dans quelques jours, sur les critiques qu'il nous adresse, pour que nul ne m'accuse de l'avoir tiré arbitrairement à nous et ne le soupçonne de s'y être prêté. Si je ne cite pas en même temps les articles de Kautsky auxquels il répond, ce n'est pas pour n'éclairer qu'un côté du débat, c'est d'abord parce que les thèses de Bebel, de Kautsky et de Guesde ont été déjà formulées à Amsterdam, c'est ensuite parce que je me propose de citer bientôt Kautsky pour discuter des sophismes avec lesquels il est temps d'en finir, si on ne veut pas que le mouvement socialiste soit paralysé par les abstractions les plus funestes et les plus enfantines, aussi mortelles à la pensée qu'à l'action.
République et monarchie sociale
Kurt Eisner dit donc dans une première réponse à Kautsky et à ses formules complaisantes pour la monarchie sociale : « On pourrait croire que Bebel a pensé que dans la monarchie les classes ne sont pas aussi directement et aussi brutalement en lutte que dans la République. Une pareille opinion, que naturellement Bebel n'a pas et ne peut pas avoir, serait une concession à cette théorie foncièrement fausse des professeurs domestiques du droit monarchique, que la monarchie a la force merveilleuse d'être au-dessus des partis et de réaliser l'équilibre de justice entre les classes. L'expérience historique apprend en fait, et la logique interne de l'organisation sociale de l'État démontre qu'aucune classe dominante ne supporte une monarchie forte qui ne se livre point à elle corps et âme. C'est pourquoi une monarchie, quand elle n'est point réduite à n'être qu'une forme vide et un décor, mais quand elle est plus ou moins profondément engagée dans l'absolutisme, ne cherche pas, même le plus modestement du monde, à équilibrer les intérêts entre les classes dominantes et les classes dominées ; elle cherche seulement à instituer des compromis entre les différents intérêts des classes dominantes elles-mêmes. Elle doit sans cesse veiller à garder les bonnes grâces des "fidèles soutiens du trône" ; et ainsi elle doit leur dispenser tour à tour – et toujours aux dépens du prolétariat – tous les avantages imaginables.
« La nécessité de s'appuyer sur les classes dominantes et de concilier sans cesse les multiples antagonismes d'intérêts qui éclatent dans l'évolution économique entre les diverses fractions de la société possédante, pour n'être point déchirée elle-même dans ce conflit, contraint la monarchie à renforcer sans cesse les privilèges économiques et sociaux.
« D'un autre côté, les classes dominantes, dans leur rivalité empressée autour de la faveur de la couronne, deviennent toujours plus réactionnaires ; et tandis qu'elles cherchent à s'adapter à la nature des cours et à se prêter aux conditions d'influence sous les régimes dynastiques, leur égoïsme de classe se manifeste non seulement sous la forme la plus grossière, mais encore sous la forme la plus insensée et la plus contraire même à leurs propres intérêts. Par suite, dans aucune République la lutte de classe n'est plus brutale et en même temps plus absurde que dans la Prusse monarchique et dans la Saxe monarchique.
« Pendant qu'ainsi, d'un côté, la monarchie absolue ou semi-absolue fait une politique de compromis entre les divers intérêts des classes possédantes et aux dépens du prolétariat, d'un autre côté les classes dominantes se sentent si sûres derrière le rempart d'une monarchie forte, qu'elles ne voient presque aucune nécessité de s'entendre avec le prolétariat, de l'apaiser par des concessions. Et si la monarchie éprouve le besoin, pour faire accepter au prolétariat les compromis qu'elle institue sur son dos entre les divers intérêts des classes possédantes, de lui faire quelques semblants de concession, les classes dominantes s'opposent obstinément à ces sortes de tentatives. Ainsi, il n'y a de "réforme sociale monarchique" que celle que permettent les classes possédantes. Et c'est pourquoi encore nulle part il n'y a une domination de classe aussi réactionnaire et aussi bornée que celle que les hobereaux et les féodaux bourgeois de fabrique confondus avec eux exercent dans la monarchique Prusse et la monarchique Saxe.
« Au contraire, dans les Républiques démocratiques ou dans ces États constitutionnels qui n'ont de la monarchie que l'apparence, comme l'Angleterre, les classes dominantes sont obligées, dans leur propre lutte d'intérêts, d'attirer le prolétariat de leur côté par des concessions. Les concessions sociales dans ces États peuvent parfois n'avoir pas autant d'éclat extérieur que la réforme sociale bureaucratique, d'ailleurs si contente de soi, des États monarchiques, mais leur valeur interne est bien supérieure. Et, avant tout, les classes dominantes doivent chercher à gagner, par des libertés politiques, la sympathie et l'aide des masses dont le concours est indispensable à leur domination même... »
La « superstition républicaine »
« Ce que Kautsky18 appelle "la superstition républicaine" est donc une conception très révolutionnaire, très radicale et, dans tous les cas, absolument nécessaire. »
Kurt Eisner constate, au reste, que ceux qui, comme Kautsky, ont essayé de systématiser contre la République les paroles de Bebel à Amsterdam, sont en contradiction avec Bebel lui-même, qui a dit au congrès de Dresde : « Si l'Allemagne prussienne avait une République, si pâle fût-elle, nous aurions très probablement le suffrage universel égal, direct et secret pour tous les corps représentatifs, une indemnité pour les députés, une plus juste répartition des circonscriptions électorales, dans la mesure du possible la représentation proportionnelle, une législation beaucoup plus libérale sur les associations et les réunions, une plus grande liberté de la presse, une réforme sociale plus parfaite, et en faveur de laquelle notre puissance pèserait bien plus qu'aujourd'hui dans la balance, un système militaire plus démocratique, un gouvernement responsable devant le Parlement ; bref, toute une série des articles les plus immédiats de notre programme serait réalisée, pour lesquels nous devons maintenant encore livrer de longs et sans doute très durs combats et supporter bien des sacrifices.
« Et si maintenant Vollmar et Göhre19 veulent prétendre que la forme de gouvernement est chose secondaire, que penseront, en lisant de pareilles assertions, nos camarades belges, français, autrichiens, italiens ? »
Je ne veux pas, aujourd'hui, ajouter de longs commentaires à ce que j'ai cité de Kurt Eisner, et à la citation que lui-même fait de Bebel. Je ne dirai qu'un mot : c'est qu'il est prodigieux que ce soit pour plaire à des socialistes français et pour faire leur jeu que le Bebel républicain du congrès de Dresde se soit livré, à Amsterdam, à l'apologie de la monarchie sociale !
À supposer qu'il n'y ait pas, entre le langage de Bebel à Dresde et son langage à Amsterdam, une contradiction directe, brutale, grossière, en tout cas le langage d'Amsterdam est d'une tout autre inspiration, beaucoup moins démocratique, beaucoup moins républicaine. Bebel, obligé de se faire beaucoup moins républicain qu'il ne l'était pour entrer dans le jeu de Guesde et de Vaillant, c'est un des paradoxes les plus extraordinaires qu'ait vus l'histoire des partis. Le blanquisme aboutissant, en la personne de Vaillant20, et pour la passagère et illusoire satisfaction d'un intérêt de secte, à obtenir du chef socialiste allemand l'atténuation et presque le désaveu de ses déclarations républicaines antérieures : c'est le signe des déformations prodigieuses que l'esprit de coterie fait subir au sens révolutionnaire.
Oui, en vérité, Bebel entraîné, pour défendre contre nous la politique de Guesde et de Vaillant, à relever à Amsterdam la monarchie qu'il accablait à Dresde, à déprécier à Amsterdam la République qu'il glorifiait à Dresde, c'est, pour la politique de Guesde et de Vaillant, la condamnation la plus ironique, la plus effroyable qui se puisse imaginer. La présence de Guesde et de Vaillant à un congrès socialiste international se traduisant par un relèvement du niveau européen de la monarchie, et par un abaissement du niveau européen de la République, c'est un spectacle d'un comique profond et où la pensée même éclate de rire.
Mais il me reste à traduire l'article si pénétrant et si vif où Kurt Eisner touche directement aux choses de France.
L'Humanité, 18 septembre 1904, texte intégral.
L'Allemagne révolutionnaire
Dans son Histoire socialiste de la France contemporaine, Jean Jaurès consacre des centaines de pages à l'étude de la Révolution française à l'étranger. Un volume spécifique est consacré à l'Europe pendant la période révolutionnaire, notamment à l'Allemagne. Ce qui peut nous paraître évident aujourd'hui ne l'était pas à l'époque : très peu d'historiens français sortaient alors de France pour s'intéresser aux conséquences internationales de cette révolution ! De ce point de vue, l'Histoire socialiste de Jaurès est un moment charnière – trop méconnu – de l'établissement d'un récit historique commun entre la France et l'Allemagne, que les deux guerres mondiales allaient temporairement briser.
I. La condition politique et économique de l'Allemagne
L'influence de l'Encyclopédie
L'Allemagne était toute préparée à s'intéresser à la Révolution française. L'action intellectuelle de la France sur l'Allemagne au XVIIIe siècle avait été immense. Voltaire, Diderot, Rousseau, l'Encyclopédie, l'Académie des sciences avaient au-delà du Rhin suscité les idées, passionné les esprits. Et même quand l'esprit allemand prit conscience de son originalité, quand il s'affranchit, dans l'ordre de l'art et de la pensée, de l'influence exclusive de la France et se créa sa littérature, son théâtre, sa philosophie, il resta en communication vivante avec l'esprit français. C'est Klopstock qui donne le premier au génie allemand une expression épique et lyrique vraiment nationale et c'est Klopstock qui vibrera d'enthousiasme aux premiers événements de la Révolution française, aux premières affirmations de la liberté21. En Lessing22, qui libère le théâtre allemand de l'imitation servile du théâtre français et qui donne à la critique religieuse une profondeur inconnue en France, la marque de l'esprit critique français, si nette et si aiguë, est toujours visible. Lorsque Kant résout le problème des rapports de la pensée et de l'être par une solution d'une hardiesse incomparable, lorsqu'il fait l'accord de la pensée et du monde sur la primauté de la pensée créant elle-même les lois selon lesquelles le monde se manifeste, que fait-il sinon justifier la science, glorifier la pensée, affermir les fondements de la connaissance et de l'expérience, c'est-à- dire continuer à sa manière la grande tradition du XVIIIe siècle français ? Il intervient en réalité pour protéger contre l'offensive possible du doute les magnifiques audaces de la science expérimentale. Il consolide la voie où marchèrent les encyclopédistes, et il en fait la voie royale de la pensée, législatrice des choses.
En tous les esprits allemands de la seconde moitié du XVIIIe siècle, chez les plus modestes comme chez les plus grands, se marquent les traits décisifs de la culture française. C'est un libre souci de la vérité universelle, c'est la haine ou le dédain du préjugé, c'est l'incessant appel à la raison, c'est la large sympathie humaine qui va à tous les peuples et à toutes les races, surtout à tous les efforts de civilisation et de pensée, sous quelque forme et en quelque nation qu'ils se produisent ; c'est le besoin de tout comprendre et de tout harmoniser, de briser l'unité factice de la tradition pour créer l'unité vivante de la science et de l'esprit ; c'est l'inspiration encyclopédique et cosmopolite, la passion de la science et de l'humanité ; c'est le grand mouvement que les Allemands ont appelé l'Aufklärung, reflet du mot que le XVIIIe siècle français aimait tant et qui avait alors un éclat tout jeune et tout vif : les lumières.
En même temps, et par un lien plus particulier, par une influence plus singulière et plus pénétrante, le Genevois protestant Rousseau, avec son rationalisme religieux, avec son sens douloureux des problèmes moraux, mettait en communication profonde la pensée de la France et la conscience de l'Allemagne. Quelle fut son action sur toute la pensée allemande, je n'ai pas à le dire.
Comment une Allemagne ainsi façonnée par notre XVIIIe siècle, ainsi pénétrée d'esprit français, ne se serait-elle point émue au grand événement de liberté qui, en 1789, ébranlait toute la France ? Comment n'aurait-elle pas été attentive à cette affirmation des Droits de l'Homme qui semblait donner à un fait historique l'ampleur de la pensée et à l'action particulière d'un peuple une valeur symbolique et universelle ?
Mais si l'Allemagne, au moins l'Allemagne pensante, était ainsi disposée d'abord à la sympathie envers la Révolution, il ne pouvait y avoir entre l'Allemagne et la France cette communauté d'action que fonde seule l'union durable des esprits. L'Allemagne, malgré la hardiesse de ses penseurs, n'était pas à l'état révolutionnaire : elle n'était pas prête à accomplir chez elle la révolution de liberté et de démocratie bourgeoise que la France, à ses risques et périls, essayait glorieusement.
Les obstacles à l'action révolutionnaire
Le morcellement politique
Quatre obstacles principaux s'opposaient en Allemagne à l'action révolutionnaire. D'abord le morcellement politique de l'Allemagne empêchait les mouvements d'ensemble. Elle était divisée en plusieurs centaines de petits États. Dans la France centralisée et à peu près unifiée, même avant 1789, le terrain large et uni se prêtait, si l'on peut dire, à des opérations de masses. Les Français des diverses régions, des diverses provinces, malgré certaines diversités de législation et de coutumes, vivaient sous le même pouvoir et à peu près sous la même loi. Dès lors, les bourgeois et les prolétaires de la Bretagne, de l'Île-de-France, du Languedoc, de la Provence, du Dauphiné, n'étant pas animés les uns contre les autres par de violentes rivalités provinciales, disposaient de toute leur énergie contre les privilèges des nobles et du clergé, contre l'arbitraire du roi et des bureaux : ils avaient des intérêts communs évidents, d'où procédait bientôt une action commune.
Au contraire, l'extrême division politique de l'Allemagne en 1789 dispersait la pensée des classes exploitées et l'égarait. Les bourgeois et prolétaires allemands se demandaient, non pas ce qu'ils deviendraient eux-mêmes dans une grande transformation révolutionnaire, mais ce que deviendrait l'État particulier auquel des liens multiples d'habitude, d'intérêt et de vanité les attachaient encore.
L'autonomie relative de chacun de ces États, si dommageable qu'elle fût à la vie générale de l'Allemagne, à son activité économique, à sa force nationale et à sa liberté, offrait cependant aux esprits superficiels des avantages immédiats. Chacune de ces petites cours avait sa clientèle de fonctionnaires, de fournisseurs et de marchands. Elle apparaissait comme un centre de vie, comme un foyer de richesse et, tandis que l'élan de la production et des échanges qui résulterait d'un mouvement d'unification démocratique paraissait lointain ou incertain, la perte que pouvait entraîner pour toutes ces petites capitales et ces petits États une vaste commotion sociale pouvait être prochaine.
À ces inquiétudes de l'égoïsme routinier se joignaient parfois des préoccupations d'un ordre plus élevé. Par sa diversité même et son morcellement, l'Allemagne offrait çà et là un refuge aux libres esprits : c'était une coquetterie ou une gloire pour quelques-uns de ces petits princes d'accueillir les hauts génies qui agrandissaient la pensée allemande. Goethe avec Wieland, avec les frères Humboldt et les frères Schlegel, avec Voss, Jean-Paul, plusieurs autres, avait trouvé à Weimar une noble liberté ; qui sait ce que réserverait à la pensée une Allemagne unifiée par une secousse violente ? Ainsi le souci de la libre culture confirmait, chez les intelligences d'élite, cette politique particulariste où abondait déjà le bourgeois de petite ville, « le philistin allemand ».
La rivalité austro-prussienne
En outre, les intrigues rivales de l'Autriche et de la Prusse qui cherchaient à dominer l'Allemagne éveillaient de justes défiances. Lorsqu'en 1785 se forma la « Ligue des princes allemands », dirigée par la Prusse, elle fut plutôt un moyen de combat imaginé par celle-ci contre l'Autriche qu'un moyen d'émancipation pour l'Allemagne. Ainsi la conscience nationale n'avait aucun centre politique où elle pût s'attacher et le Reichstag, l'Assemblée d'Empire où se réunissaient les représentants des princes et des villes, n'avait qu'un semblant de vie. On n'y discutait même plus ; les princes ne prenaient plus la peine d'y venir en personne : ils y faisaient connaître leur volonté par des mémoires que lisaient leurs secrétaires et, naturellement, de cet échange protocolaire de pensées diverses et confuses, qui se refusaient à toute délibération et à toute adaptation, aucun mouvement ne pouvait naître.
Les Allemands cherchaient à se consoler de leur impuissance à se créer une vie nationale en se disant que par là ils vivaient plus librement d'une vie humaine. Goethe, en deux vers qui constataient cette radicale incapacité, disait aux Allemands :
« C'est en vain que vous espérez, vous, Allemands, former une nation. Mais c'est une raison de plus pour vous de devenir des hommes libres : et cela, vous le pouvez. »
Illusion puérile et mensonge des mots ! Car comment séparer l'homme du citoyen, du producteur ?
Comment l'homme peut-il être libre, si le citoyen est opprimé, si le producteur est chargé d'entraves ? Pour libérer « l'homme », il fallait à l'Allemagne comme à la France une révolution ; or, cette révolution n'était possible que par un mouvement concerté et vaste, et ce mouvement même supposait une vie nationale puissante et une.
[...]
V. Les Allemands révolutionnaires
Les réformes
Est-ce à dire que la faillite de la Révolution française en Allemagne est complète ? Non, certes. D'abord, ce n'était pas en vain que depuis trois ans se déployait le spectacle prodigieux de la France révolutionnaire. Si obtus, si endormis que fussent encore les paysans d'Allemagne, ils apprenaient l'abolition des corvées et des dîmes, et ils s'étonnaient. Les hommes d'État les plus avisés comprenaient bien en Allemagne que, pour prévenir un soulèvement analogue à celui de la France, il faudrait réaliser quelques réformes, alléger le fardeau du peuple. Quelques souverains de petits États, notamment le fantasque et despote margrave de Hesse, eurent bien la pensée qu'il suffirait de mesures répressives pour écraser même les germes de la Révolution. Et en quelques points, la liberté de la presse dont s'enorgueillissait depuis un tiers de siècle l'Allemagne de l'Aufklärung, parut menacée. Il fut interdit de parler politique dans les cabarets et les auberges. « Dans les hôtelleries, il n'y a plus maintenant, disait une revue satirique, qu'une différence entre les hommes et les bêtes : c'est que les hommes paient. » Le secret de la correspondance fut parfois violé. Mais l'Allemagne tenait à la liberté de la pensée et la réaction s'arrêta.
Ainsi, peu à peu, même par les journaux et les revues qui combattaient la Révolution, les idées de celle-ci se répandaient. Et les gouvernements sentaient approcher l'heure des concessions nécessaires. Dans le Nouveau Museum allemand, Schlosser, le serviteur et conseiller du margrave Frédéric de Bade, invitait les souverains à la prudence, à la prévoyance :
« Espérons, écrivait-il, qu'en Allemagne on sera plus sage qu'en France. Il est impossible d'empêcher le peuple de constater, par l'exemple même des Français, que les choses pourraient aller autrement qu'elles ne vont et il faut que le penchant à l'obéissance reste assez fort pour neutraliser les impulsions contraires. Or, pour fortifier l'habitude de l'obéissance, il faut que les princes fassent à temps les réformes indispensables : juste diminution des impôts, limitation des ravages du gibier, adoucissement des corvées, assistance pour les pauvres, facilités plus grandes données au travail, ferme surveillance des employés de l'État, justice plus rapide, voilà maintenant la seule éloquence qui puisse détourner les sujets de la révolte. »
Ainsi, malgré tout, les idées cheminaient, et d'innombrables semences tombaient dans les sillons ouverts.
Même, à l'épreuve de l'action, la haute pensée allemande devenait plus virile... Bien des esprits sans doute se repliaient, se retiraient. Mais d'autres prenaient leur parti de l'inévitable brutalité des grands mouvements humains. Ils maintenaient et élevaient toujours plus haut, contre les fureurs et les menaces croissantes de la réaction, l'idéal du droit et de la liberté, et ils faisaient ainsi, dans l'ordre de la pensée, l'apprentissage du combat.
Histoire socialiste de la France contemporaine (1902),
« La Révolution en Europe », chap. I et V, extraits.
« La paix et le socialisme »
(discours de Berlin)
Jaurès devait initialement prononcer ce discours à Berlin, mais le Reich ne l'autorisa pas à entrer en territoire allemand. La situation internationale avait été, peu de temps avant ce discours, extrêmement tendue, notamment en raison du conflit entre la France et l'Allemagne pour la mainmise sur le Maroc, opération que Jaurès combat et qui pouvait faire craindre la guerre entre les deux pays. Nous sommes également quelques mois après le déclenchement de la première révolution russe (voir p. 211), ce qui explique en partie le grand enthousiasme dont fait preuve ici Jaurès.
Citoyens, je suis heureux d'être ici, comme délégué du groupe socialiste du Parlement français, pour affirmer avec vous la solidarité, l'unité du prolétariat français et du prolétariat allemand, leur commune et ferme volonté d'assurer la paix, de conquérir la paix par l'organisation et l'émancipation de tous les travailleurs. Je me réjouis aussi qu'en ce moment même il y ait une détente dans les relations gouvernementales de la France et de l'Allemagne, et que le conflit marocain soit en voie de solution. Jusqu'à quel point ce conflit a-t-il menacé la paix ? Et les deux peuples ont-ils vraiment couru le risque monstrueux d'être jetés l'un sur l'autre ? On ne peut le dire avec précision : car une des beautés de la diplomatie, c'est qu'on ne peut jamais savoir avec certitude à quel point elle met en péril les peuples qu'elle a mission de sauver. Le chancelier de l'Empire allemand a dit à un grand journal français qu'il était heureux qu'on fût sorti enfin d'une situation qui avait été « tendue et périlleuse ». Ce sont là des paroles graves. Des sceptiques insinuent, il est vrai, qu'il y a eu comme un bluff des deux gouvernements, qui a été pris trop au sérieux par les peuples. Les diplomaties cherchaient, dit-on, à se tâter ; elles essayaient l'une sur l'autre la puissance magnétique de leurs attitudes et de leurs regards. Je ne sais s'il y a eu, dans le conflit, cette part de jeu et de parade. Ce serait, en tout cas, un jeu plein de péril. Quand deux mécaniciens lancent leur trains l'un contre l'autre sur la même voie et qu'on ne sait rien d'ailleurs de leurs intentions, on a beau dire qu'ils ne veulent qu'éprouver réciproquement la solidité de leurs nerfs, nul ne peut savoir comment les choses tourneront. Il se peut qu'un des mécaniciens au moins perde la tête. Il se peut qu'entraînés par la parade, ils lancent si bien leurs machines qu'ils ne puissent plus les arrêter à temps, et que la collision se produise. Si c'est un jeu, les voyageurs aimeraient mieux autre chose. Cette fois, le heurt a pu être évité : les deux mécaniciens se font des politesses ; on s'apprête même à orner de fleurs les locomotives et à les atteler toutes deux à l'interminable train de la conférence internationale. C'est bien ; mais cette alerte redoutable, survenue subitement en pleine tranquillité, en pleine sécurité, rappelle aux peuples et aux prolétaires combien la paix est fragile et précaire dans la société d'aujourd'hui avec les gouvernements d'aujourd'hui. Elle rappelle à toute la classe ouvrière d'Europe, à toute la classe ouvrière du monde son devoir d'union internationale, de vigilance internationale. Il ne faut pas que le prolétariat international soit un mot magnifique et vain. Il ne faut pas qu'il soit une force intermittente et superficielle, se manifestant à intervalles par ses congrès internationaux ou par les circulaires du Bureau socialiste international23. Il faut qu'il soit une force constante, toujours avertie, toujours éveillée, toujours en état de contrôler les événements à leur naissance, de surveiller dans leur germe les premiers conflits qui, en se développant, pourraient produire la guerre.
Et il n'y a dans ces paroles, croyez-le bien, aucune fanfaronnade socialiste. Nous ne sommes pas réunis pour échanger des illusions. Nous savons très bien, les uns et les autres, qu'il y a dans le monde capitaliste des forces formidables de conflit, d'anarchie violente, d'antagonismes exaspérés que le prolétariat universel, au degré insuffisant d'organisation et de puissance politique où il est parvenu, ne peut se flatter encore de maîtriser avec certitude. La concurrence économique de peuple à peuple et d'individu à individu, l'appétit du gain, le besoin d'ouvrir à tout prix, même à coups de canon, des débouchés nouveaux pour dégager la production capitaliste, encombrée et comme étouffée sous son propre désordre, tout cela entretient l'humanité d'aujourd'hui à l'état de guerre permanente et latente ; ce qu'on appelle la guerre n'est que l'explosion de ce feu souterrain qui circule dans toutes les veines de la planète et qui est la fièvre chronique et profonde de toute vie. Il faut bien chercher des clientèles lointaines, des clientèles exotiques et serviles, puisque tout le système, en retirant aux ouvriers une large part du produit de leur travail, restreint la libre consommation nationale. Oui, nous savons cela, et nous savons aussi que la force ouvrière n'est pas encore assez organisée, assez consciente, assez efficace, pour refouler et neutraliser ces forces mauvaises. Ou bien le prolétariat, séduit par une fausse apparence de grandeur nationale, et corrompu par une part dérisoire du butin capitaliste et colonial, ne s'oppose que mollement aux entreprises de la force. Ou bien les classes dirigeantes embrouillent si habilement la querelle née de l'antagonisme économique que les prolétaires n'en démêlent point l'origine. Ou bien, quand leur conscience est mieux avertie, ils ne disposent pas d'une action suffisante sur le mécanisme politique et gouvernemental, et leur opposition est submergée par tous les éléments flottants et inorganisés que le capitalisme met en mouvement aux heures de crise. Ou encore, les travailleurs socialistes de chaque nation, trop séparés encore les uns des autres, s'ignorant les uns les autres, désespèrent de l'utilité d'une action qui, pour être efficace, devrait être internationale ; et n'étant pas sûrs d'être soutenus de l'autre côté des frontières, ils s'abandonnent tristement à la fatalité. Oui, la protestation de la classe ouvrière ne suffit pas encore à dissiper tous les orages. La voix du prolétariat universel, qui commence à s'élever pourtant vibrante et forte au-dessus des nations agitées par une éternelle rumeur d'inquiétude et de guerre, ne peut pas répéter tout ce que dit la cloche de Schiller24. Elle peut bien dire : Vivos voco, mortuos plango, j'appelle les vivants, et je pleure sur les morts. Elle ne peut pas dire encore : Fulgura frango, je brise la foudre. Il nous reste encore une œuvre immense d'éducation et d'organisation à accomplir. Mais, malgré tout, dès maintenant, il est permis d'espérer, il est permis d'agir. Ni optimisme aveugle ni pessimisme paralysant. Il y a un commencement d'organisation ouvrière et socialiste, il y a un commencement de conscience internationale. Dès maintenant, si nous le voulons bien, nous pouvons réagir contre les fatalités de guerre que contient le régime capitaliste. Marx, quand il parle des premières lois anglaises qui ont réglementé la durée du travail, dit que c'est le premier réflexe conscient de la classe ouvrière contre l'oppression du capital. La guerre est, comme l'exploitation directe du travail ouvrier, une des formes du capitalisme, et le prolétariat peut engager une lutte systématique et efficace contre la guerre, comme il a entrepris une lutte systématique et efficace contre l'exploitation de la force ouvrière. Pas plus qu'il n'y a une loi d'airain du salaire25 qu'aucune action prolétarienne ne pourrait assouplir, pas plus qu'il n'y a un mètre d'airain de la journée ouvrière qu'aucune action prolétarienne ne pourrait réduire, il n'y a une loi d'airain de la guerre qu'aucune action prolétarienne ne pourrait fléchir. Le monde présent est ambigu et mêlé. Il n'y a en lui aucune fatalité, aucune certitude. Ni le prolétariat n'est assez fort pour qu'il y ait certitude de paix, ni il n'est assez faible pour qu'il y ait fatalité de guerre. Dans cette indécision des choses et cet équilibre instable des forces, l'action humaine peut beaucoup. La formidable part d'inconnu n'est pas redoutable seulement pour nous, socialistes. Elle l'est aussi pour ceux qui déchaîneraient témérairement des guerres dont nul aujourd'hui ne peut prévoir les conséquences politiques et sociales et les contrecoups intérieurs.
Donc, nous pouvons agir dès aujourd'hui, à quelque degré, sur la marche des événements, et comme nul ne peut déterminer d'avance le degré d'efficacité de notre action, nous devons donner tout notre effort comme si elle était en effet assurée du succès.
[...]
Et qu'on ne se méprenne pas sur notre pensée. Nous n'avons pas, nous, socialistes, la peur de la guerre. Si elle éclate, nous saurons regarder les événements en face, pour les faire tourner de notre mieux à l'indépendance des nations, à la liberté des peuples, à l'affranchissement des prolétaires. Si nous avons horreur de la guerre, ce n'est point par une sentimentalité débile et énervée. Le révolutionnaire se résigne aux souffrances des hommes, quand elles sont la condition nécessaire d'un grand progrès humain, quand, par elles, les opprimés et les exploités se relèvent et se libèrent. Mais maintenant, mais dans l'Europe d'aujourd'hui, ce n'est pas par les voies de la guerre internationale que l'œuvre de la liberté et de justice s'accomplira et que les griefs de peuple à peuple seront redressés. Certes, depuis cent cinquante ans, bien des violences internationales ont été commises en Europe, dont les meurtrissures subsistent encore en des millions de consciences, dont les conséquences pèsent lourdement sur l'Europe et sur le monde. Mais c'est par la croissance de la démocratie et du socialisme, et par là seulement, que ces souffrances seront apaisées, que ces problèmes douloureux seront résolus. La démocratie fait du consentement des personnes humaines la règle du droit national et international. Le socialisme veut organiser la collectivité humaine, mais ce n'est pas une organisation de contrainte et sous la loi générale de justice et d'harmonie qui préviendra toute tentative d'exploitation, il laissera aux nations la libre disposition d'elles-mêmes dans l'humanité, comme aux individus la libre disposition d'eux-mêmes dans la nation. Or, dans la paix, la croissance de la démocratie et du socialisme est certaine.
D'une guerre européenne peut jaillir la révolution, et les classes dirigeantes feront bien d'y songer ; mais il en peut sortir aussi, pour une longue période, des crises de contre-révolution, de réaction furieuse, de nationalisme exaspéré, de dictature étouffante, de militarisme monstrueux, une longue chaîne de violences rétrogrades et de haines basses, de représailles et de servitudes. Et nous, nous ne voulons pas jouer à ce jeu de hasard barbare, nous ne voulons pas exposer, sur ce coup de dé sanglant, la certitude d'émancipation progressive des prolétaires, la certitude de juste autonomie que réserve à tous les peuples, à tous les fragments de peuples, au-dessus des partages et des démembrements, la pleine victoire de la démocratie socialiste européenne.
C'est pourquoi, nous, socialistes français, sans qu'aucune personne humaine puisse nous accuser d'abaisser le droit, nous répudions à fond, aujourd'hui et à jamais, et quelles que puissent être les conjectures de la fortune changeante, toute pensée de revanche militaire contre l'Allemagne, toute guerre de revanche. Car cette guerre irait contre la démocratie, elle irait contre le prolétariat, elle irait donc contre le droit des nations, qui ne sera pleinement garanti que par le prolétariat et la démocratie. Aujourd'hui, la paix de l'Europe est nécessaire au progrès humain ; et la paix, la paix assurée, la paix durable, la paix confiante entre l'Allemagne et la France, qui a beaucoup fait en Europe pour le mouvement de la démocratie et l'éveil de la classe ouvrière, ne peut pas être à contresens de leur développement. C'est pourquoi, nous, socialistes français, en répudiant ici devant vous toute pensée de guerre, toute revendication armée, en adjurant la France et l'Allemagne de renoncer à tout antagonisme latent, à toute suspicion réciproque, et de concerter leur action pour l'affermissement de la paix, nous croyons servir, avec l'intérêt du prolétariat international, le plus haut intérêt de notre nation comme de la vôtre. Je puis, sans contradiction et sans embarras, parler ici tout à la fois en socialiste international et en fils de cette France qui a, sans doute, dans sa longue histoire, commis bien des fautes, qui, de Charles VIII à Louis XIV et de celui-ci à Napoléon, a trop souvent abusé de son unité nationale constituée avant les autres, pour brutaliser et offenser les nations morcelées encore et inorganisées ; qui, même sous la Révolution, a mêlé trop vite une ivresse de domination et d'orgueil au pur enthousiasme de la liberté universelle et de l'Humanité ; qui a laissé déflorer, comme dit votre poète Herwegh26, par la brutalité conquérante de ses soldats, la liberté qu'elle offrait au monde comme une fiancée ; qui, s'étant portée d'abord d'un mouvement héroïque à l'extrémité de la révolution et de la démocratie, n'a pu s'y maintenir, et a subi des vicissitudes de liberté et de réaction, parfois même un horrible mélange césarien de démagogie et d'esclavage ; qui a porté dans sa politique extérieure les contradictions ou les ambiguïtés de sa politique intérieure, aidant de sa sympathie, ou même de son effort, les nationalités à naître, et les arrêtant aussitôt à moitié développement ou les contrariant d'une jalousie secrète ; qui a payé d'un morceau de son âme et de sa chair les imprudences et les incohérences de ce despotisme napoléonien, dont elle fut tout ensemble la complice et la victime ; mais qui, à travers toutes ses étourderies, tous ses enivrements et toutes ses défaillances, a donné sans compter le meilleur de son sang pour les plus grandes causes, qui, la première, a ébranlé le vieux monde féodal et absolutiste, et, la première, a combattu le nouvel égoïsme bourgeois ; qui met au service de l'humanité des dons admirables : une fine et profonde culture, un instinct démocratique et républicain, la clarté de l'esprit et de la volonté, la rapidité de la décision, l'élan de la sympathie, et qui, aujourd'hui, ramenée par la dureté des événements et la salutaire croissance des autres peuples à une appréciation plus exacte du rôle de tous et de chacun, reste une des grandes forces de progrès humain et de libération ouvrière, une force nécessaire et inviolable, résolue, dans la limite de son droit, à ne se laisser ni violenter ni humilier. Oui, je parle ainsi de la France, sans embarras aucun, devant vous, socialistes d'Allemagne, parce que je sais que dans votre conscience vous vous efforcez de juger impartialement votre pays, comme nous nous efforçons de juger impartialement le nôtre. Ce fut pour nos deux nations, il y a trente-cinq ans, une grande faillite d'idéalisme, que nous ne soyons arrivés que par le chemin de la guerre, nous à la République, vous à l'unité. Ainsi, nous paraîtrons les uns devant les autres sans prétention exclusive et arrogante, et nous ne nous souviendrons du passé que pour faire tous ensemble le serment d'abjurer tout orgueil, toute haine, toute défiance, de travailler tous ensemble, d'un même cœur, à fonder la paix définitive de l'Allemagne et de la France, pour que les deux prolétariats puissent se donner tout entiers à l'œuvre d'émancipation, pour que les deux peuples puissent se donner tout entiers à l'œuvre de civilisation. Dans cette capitale de Berlin, où nos soldats sont entrés avant que les vôtres n'entrent à Paris, nous voulons resserrer, nous voulons proclamer devant le monde le pacte d'union de la classe ouvrière française et de la classe ouvrière allemande. Nous voulons rejeter ensemble toute pensée de violence internationale, nous voulons détester et dénoncer ensemble tous ceux, quels qu'ils soient, qui chercheraient à mettre aux prises les deux nations. Nous voulons opposer la diplomatie pacifique, ouverte, loyale du prolétariat international, à la diplomatie imprudente, avide ou cauteleuse des gouvernants capitalistes et des gouvernants féodaux. Et nous vous devons compte des efforts que nous faisons dans notre pays pour déjouer les manœuvres suspectes, et prévenir les entraînements funestes, comme vous nous devez compte des efforts faits par vous dans votre propre pays contre le chauvinisme arrogant et agressif.
L'Humanité, 9 juillet 1905, extraits.
1 Albert Schäffle (1831-1903), homme politique conservateur mais souhaitant réconcilier le socialisme avec l'État prussien, est considéré à ce titre comme un « socialiste de la chaire ».
2 La dialectique est un mode de raisonnement spécifique depuis l'Antiquité qui entend aboutir à une synthèse en dépassant les contradictions. Jaurès discute ici notamment de la dialectique marxiste, qui considère comme fondamentales les contradictions entre les classes sociales.
3 Association prussienne qui avait pour objectif d'exalter les valeurs germaniques pour les retourner contre la domination napoléonienne, entre 1808 et 1815.
4 Le sens premier de « débile » renvoie à un individu ou à quelque chose qui manque de vigueur ou de force. Voir, au début du texte, « la débilité ou l'inanité des efforts ».
5 Heinrich Heine (1797-1856), un des grands écrivains allemands du XIXe siècle, admirateur de la Révolution française et ami de Karl Marx.
6 Ludwig Börne (1786-1837), écrivain allemand contestataire, un des animateurs du courant dit « Jeune-Allemagne ».
7 Allusion à l'unité allemande réalisée en 1871 sous l'impulsion autoritaire du chancelier Otto von Bismarck.
8 Lassalle considérait les autres forces politiques allemandes comme une « masse réactionnaire » et voulait ainsi préserver l'indépendance du mouvement ouvrier.
9 Le congrès de l'Internationale socialiste a eu lieu à Amsterdam en août 1904.
10 August Bebel (1840-1913), président du SPD, disposait d'une autorité morale et politique considérable.
11 Au congrès de Dresde du SPD (1903) fut condamnée toute remise en question du marxisme.
12 Victor Adler (1852-1918) était un des principaux responsables de la social-démocratie autrichienne ; Émile Vandervelde (1866-1938), de la social-démocratie belge.
13 Kurt Eisner (1867-1919), figure nationale du SPD, membre de la rédaction du Vorwärts, et un des rares soutiens de Jaurès dans ce journal.
14 Voir Introduction, p. 11.
15 Revue théorique de la social-démocratie allemande, parue de 1883 à 1923.
16 Eduard Bernstein (1850-1932), dirigeant social-démocrate de premier plan, remet en cause le marxisme dans une série d'articles regroupés dans l'ouvrage Les Présupposés du socialisme (1899). Ses thèses provoquèrent une discussion dans tout le socialisme international.
17 Karl Kautsky (1854-1938), principal théoricien de la social-démocratie allemande, disposait alors d'une grande influence.
18 Karl Kautsky (voir partie II, note 17) critiquait régulièrement les conceptions de Jaurès.
19 Georg von Vollmar (1850-1922), un des premiers sociaux-démocrates à avoir remis en question la lecture marxiste ; Paul Göhre (1864-1928), pasteur devenu socialiste, critique du marxisme, comme Vollmar.
20 Édouard Vaillant (1840-1915) incarnait alors la sensibilité blanquiste (du nom de Blanqui, révolutionnaire célèbre du XIXe s.) du mouvement socialiste, alliée à Guesde et souvent hostile aux conceptions jaurésiennes.
21 Friedrich Klopstock (1724-1803), poète allemand célèbre notamment pour son poème de 1789 qui salue la Révolution française.
22 Gotthold Lessing (1729-1781) est un des écrivains allemands du mouvement des Lumières (Aufklärung).
23 Bureau chargé de diriger et de coordonner les actions de l'Internationale socialiste.
24 Le poète allemand Friedrich von Schiller (1759-1805) n'est pas cité ici par hasard : il était particulièrement apprécié dans les rangs de la social-démocratie allemande.
25 Allusion aux thèses de Ferdinand Lassalle, fondateur du premier parti ouvrier allemand en 1863, qui défendait la loi d'airain, selon laquelle le salarié ne pouvait pas toucher un salaire supérieur à ce qui lui permettait d'assurer sa subsistance minimale.
26 Georg Herwegh (1817-1875), poète allemand impliqué notamment dans la révolution de 1848.
III
Le philosophe et l'historien
Grand admirateur des Lumières, Jaurès se vivait comme un continuateur de leur esprit : aussi n'est-il pas strictement « historien » ou « philosophe », mais un socialiste intéressé par la pensée et l'action politique, que n'effraie aucune barrière disciplinaire.
Jaurès était néanmoins avant tout normalien, agrégé de philosophie, et fut un temps enseignant à l'université de Toulouse. Avant, puis pendant son action politique, il a publié plusieurs textes philosophiques qui montrent l'ancrage de son action dans une démarche originale, dont l'objectif est de puiser dans les différentes sources de la philosophie européenne, de Rousseau à Marx.
Mais Jaurès est aussi reconnu comme un historien majeur et singulier de la Révolution française et de son impact international : il a notamment mis l'accent sur l'importance des aspects économiques et sociaux pour comprendre le processus commencé en 1789, contribuant ainsi à une longue tradition d'étude qui inspirera de nombreux historiens au XXe siècle. Nous reproduisons ici notamment l'introduction de son ouvrage le plus volumineux, l'Histoire socialiste de la France contemporaine, dont il rédigea les premiers chapitres sur 1789-1794, ainsi que la conclusion de l'ensemble, vif et beau plaidoyer pour un monde débarrassé de l'emprise du capital.
« Idéalisme et matérialisme
dans la conception de l'histoire »
Élu député socialiste en 1893 sur la base du programme des proches de Jules Guesde et de Paul Lafargue, les introducteurs du marxisme en France, Jaurès se distingue néanmoins rapidement d'eux, notamment sur le plan théorique. Il refuse ce qu'il estime être un matérialisme mécaniste pour comprendre l'histoire et la société françaises ; il expose devant les étudiants « collectivistes » (le mot était alors fréquent pour désigner les marxistes français) sa perspective combinant matérialisme et idéalisme, dans le cadre d'une controverse avec Paul Lafargue1.
Citoyennes et citoyens,
Je vous demande d'abord toute votre patience, parce que c'est à une déduction purement doctrinale que j'entends me livrer ce soir devant vous.
Je veux aussi, tout d'abord, vous prémunir contre une erreur qui pourrait résulter de ce fait que le sujet que je vais traiter devant vous, j'en ai déjà parlé il y a quelques mois2. J'ai, alors, exposé la thèse du matérialisme économique, l'interprétation de l'histoire, de son mouvement selon Marx ; et je me suis appliqué à ce moment à justifier la doctrine de Marx, de telle sorte qu'il pouvait apparaître que j'y adhérais sans restriction aucune.
Cette fois-ci, au contraire, je veux montrer que la conception matérialiste de l'histoire n'empêche pas son interprétation idéaliste. Et, comme dans cette deuxième partie de ma démonstration, on pourrait perdre de vue la force des raisons que j'ai données en faveur de la thèse de Marx, je vous prie donc, pour qu'il n'y ait pas de méprise sur l'ensemble de ma pensée, de corriger l'une par l'autre, de compléter l'une par l'autre, les deux parties de l'exposé que nous avons été obligés de scinder.
J'ai montré, il y a quelques mois, que l'on pouvait interpréter tous les phénomènes de l'histoire du point de vue du matérialisme économique, qui, je le rappelle seulement, n'est pas du tout le matérialisme physiologique. Marx n'entend pas dire, en effet, le moins du monde, que tout phénomène de conscience ou de pensée s'explique par de simples groupements de molécules matérielles ; c'est là même une hypothèse que Marx et plus récemment Engels traitent de métaphysique et qui est écartée aussi bien par l'école scientifique que par l'école spiritualiste.
Ce n'est pas non plus ce que l'on appelle parfois le matérialisme moral, c'est-à-dire la subordination de toute l'activité de l'homme à la satisfaction des appétits physiques et à la recherche du bien-être individuel. Au contraire, si vous vous rappelez comment, dans son livre Le Capital, Marx traite la conception utilitaire anglaise, si vous vous rappelez comment il parle avec dédain, avec mépris, de ces théoriciens de l'utilitarisme comme Jeremy Bentham, qui prétendent que l'homme n'agit toujours qu'en vue d'un intérêt personnel consciemment recherché par lui, vous verrez qu'il n'y a rien de commun entre ces deux doctrines. Bien mieux, c'est l'inverse ; car précisément parce que Marx estime que les modes même du sentiment et de la pensée sont déterminés dans l'homme par la forme essentielle des rapports économiques de la société où il vit, par là, Marx fait intervenir dans la conduite de l'individu des forces sociales, des forces collectives, des forces historiques dont la puissance dépasse celle des mobiles individuels et égoïstes. Ce qu'il entend, c'est que ce qu'il y a d'essentiel dans l'histoire, ce sont les rapports économiques, les rapports de production des hommes entre eux.
C'est selon que les hommes sont rattachés les uns aux autres par telle ou telle forme de la société économique, qu'une société a tel ou tel caractère, qu'elle a telle ou telle conception de la vie, telle ou telle morale, et qu'elle donne telle ou telle direction générale à ses entreprises. De plus, suivant Marx, ce n'est pas selon une idée abstraite de justice, ce n'est pas selon une idée abstraite du droit, que les hommes se meuvent : ils se meuvent parce que le système social formé entre eux, à un moment donné de l'histoire, par les relations économiques de production, est un système instable qui est obligé de se transformer pour faire place à d'autres systèmes ; et c'est la substitution d'un système économique à un autre, par exemple de l'esclavage à l'anthropophagie, c'est cette substitution qui entraîne une correspondance naturelle, une transformation équivalente dans les conceptions politiques, scientifiques et religieuses : en sorte que, selon Marx, le ressort le plus intime et le plus profond de l'histoire, c'est le mode d'organisation des intérêts économiques.
Le nom du matérialisme économique s'explique donc en ce que l'homme ne tire pas de son cerveau une idée toute faite de justice, mais qu'il se borne à réfléchir en lui, à réfléchir dans sa substance cérébrale, les rapports économiques de production.
En regard de la conception matérialiste, il y a la conception idéaliste sous des formes multiples. Je la résumerai ainsi : c'est la conception selon laquelle l'humanité, dès son point de départ, a pour ainsi dire une idée obscure, un pressentiment premier de sa destinée, de son développement.
Avant l'expérience de l'histoire, avant la constitution de tel ou tel système économique, l'humanité porte en elle-même une idée préalable de la justice et du droit, et c'est cet idéal préconçu qu'elle poursuit, de forme de civilisation en forme supérieure de civilisation ; et quand elle se meut, ce n'est pas par la transformation mécanique et automatique des modes de la production, mais sous l'influence obscurément ou clairement sentie de cet idéal.
En sorte que c'est l'idée elle-même qui devient le principe du mouvement et de l'action, et que bien loin que ce soient les conceptions intellectuelles qui dérivent des faits économiques, ce sont les faits économiques qui traduisent peu à peu, qui incorporent peu à peu, dans la réalité et dans l'histoire, l'idéal de l'humanité.
Telle est, indépendamment des innombrables formules que la diversité des systèmes philosophiques ou religieux a données à cette thèse, la conception de l'idéalisme dans l'histoire. Or, remarquez, citoyens, en fait, ces deux conceptions qui semblent s'opposer l'une à l'autre, qui semblent être exclusives l'une de l'autre, je dirai presque que dans la conscience contemporaine, elles sont à peu près confondues et réconciliées. Il n'y a pas, en fait, un seul idéaliste qui ne convienne qu'on ne pourrait réaliser un idéal supérieur de l'homme sans une transformation préalable de l'organisme économique ; et, en revanche, il y a bien peu d'adeptes du matérialisme économique qui ne se laissent aller à l'idée de la justice et du droit, il y en a bien peu qui se bornent à prévoir dans la société communiste de demain une réalisation plus haute de la justice et du droit.
Y a-t-il là contradiction ? Marx a toujours voulu maintenir l'intégrité un peu âpre de sa formule et il n'a eu que railleries pour ceux qui croient ajouter à la force de l'évolution économique et du mouvement socialiste, en faisant appel à l'idée pure de justice ; il n'a eu que railleries pour ceux qui, selon sa parole, « veulent jeter sur la réalité de l'histoire, sur le corps même des faits, une sorte de voile tissé des fils les plus immatériels de la dialectique, bordé de fleurs de rhétorique et trempé de rosée sentimentale ».
Il s'agit pour nous de savoir si cette conciliation entre la conception matérialiste et la conception idéaliste de l'histoire, qui est réalisée en fait dans notre pays par l'instinct, peut-être aveugle, de la conscience socialiste, il s'agit de savoir si elle est théoriquement et doctrinalement possible ou s'il y a là une insoluble contradiction, si nous sommes obligés de faire un choix décisif entre les deux conceptions ou si nous pouvons logiquement et raisonnablement les considérer l'une et l'autre comme les deux aspects différents d'une même vérité.
Il m'est impossible de résoudre cette question particulière sans la rattacher à un problème plus général, sans dire comment, à mon sens, se pose aujourd'hui, devant l'esprit humain, le problème même de la connaissance. Au point de vue où, pour ma part, je suis placé, je dis et je crois constater que l'effort de la pensée humaine depuis quatre siècles, depuis la Renaissance, c'est la conciliation, la synthèse des contraires et même des contradictoires : là est la marque, la caractéristique de tout le mouvement philosophique et intellectuel.
La Renaissance se trouvait devant une sorte de contradiction en apparence insoluble : l'esprit chrétien persistant et l'esprit de l'Antiquité réveillé. Or l'esprit de l'Antiquité, c'était le culte, mieux que l'acceptation, l'adoration de la nature ; l'esprit chrétien, c'était la condamnation, la négation de la nature.
En sorte que les hommes pensants, au sortir du Moyen Âge, se sont trouvés en face d'un héritage intellectuel contradictoire, d'un dualisme à concilier, à ramener à l'unité.
Le problème s'aggravait particulièrement par le développement même de l'esprit scientifique et de la science expérimentale, car par l'étude rigoureuse, positive, des phénomènes naturels, par l'application de la mécanique, de la mathématique à l'étude des forces naturelles, la nature perdait ce prestige de beauté, cette apparence de vie intérieure et divine qu'elle avait eus pour les hommes antiques.
Il fallait, d'une part, concilier la nature telle que l'avait conçue l'Antiquité avec la conception chrétienne ; il fallait, d'autre part, concilier la nature telle que la faisait la science nouvelle, la nature, simple enchaînement de phénomènes déterminés par des nécessités purement mécaniques, avec la libre aspiration de l'esprit humain.
C'est d'abord Descartes qui, par un singulier artifice de méthode, commence par s'enfermer, comme le chrétien, dans l'intérieur de sa conscience, par rejeter la vie extérieure et écarter la nature comme un fantôme problématique.
Réduit à la constatation de sa pensée, il retrouve l'idée de Dieu, et réalise ainsi cette sorte d'isolement de la conscience et de Dieu, qui avec la répudiation de la nature est la marque du christianisme.
Puis, lorsqu'il s'est créé ainsi une première méthode, au lieu d'organiser simplement sa vie intérieure, comme le chrétien, il veut connaître avec certitude la nature elle-même ; en sorte qu'après avoir traversé l'état d'esprit chrétien, il ne s'en sert que pour fonder la science positive.
Dans Leibniz, vous voyez la même tentative pour ramener à l'unité l'homme et la nature, en montrant partout, jusque dans les forces purement matérielles, jusque dans cette table, jusque dans ce sol sur lequel nous marchons, quelque chose d'analogue à l'esprit, le désir, le sens même de la beauté, des rapports harmonieux, mathématiques et certains, dans les lois de la physique et les combinaisons de la chimie. C'est la même conciliation entre l'universel déterminisme et l'universelle liberté.
D'un côté, il affirme qu'il n'y a pas dans le monde un seul mouvement qui ne soit lié à l'infini à d'autres mouvements. Le mouvement que je détermine en ce moment dans l'atmosphère par l'émission de la voix est la suite d'innombrables mouvements antérieurs, et ce mouvement lui-même sera répercuté à l'infini, ébranlant imperceptiblement la muraille même de cette enceinte, par elle l'atmosphère extérieure, et par là se transmettant sous des formes obscures ; nous ne pouvons produire un seul mouvement, déplacer un seul grain de sable, sans modifier l'équilibre de l'univers tout entier.
Mais en même temps que cette liaison des mouvements, des phénomènes, des faits, est universelle et illimitée, il n'y a pas une seule force qui procède par la contrainte ; lorsqu'une boule de billard en choque une autre, cette dernière se met en mouvement ; mais elle ne le fait que suivant certaines lois d'élasticité qui lui sont propres, qui résultent de sa contexture, et ce mouvement qui a l'air de venir du dehors, jaillit du dedans : il y a tout ensemble continuité et spontanéité absolue.
Pour Spinoza, c'est la même conciliation entre la nature et Dieu, entre le fait et l'idée, entre la force et le droit.
Pour Kant, vous le savez tous, le problème philosophique consiste expressément à trouver la synthèse des affirmations contradictoires qui s'offrent à l'esprit de l'homme : l'univers est-il limité ou infini ? Le temps est-il limité ou infini ? La série des causes est-elle limitée ou infinie ? Tout est-il soumis à l'universelle et inflexible nécessité, ou y a-t-il une part pour la liberté des actions ?
Autant de thèses et d'antithèses, de négations et d'affirmations, entre lesquelles hésite l'esprit.
L'effort de la philosophie kantienne est tout entier dans la solution de ces contradictions, de ces antinomies fondamentales.
Enfin, c'est Hegel qui vient donner la formule même de ce long travail en disant que la vérité est dans la contradiction : ceux-là se trompent, ceux-là sont les jouets d'une logique étroite, illusoire, qui affirment une thèse sans lui opposer la thèse inverse. En fait, dans la nature, dans la réalité, les contraires se pénètrent, le fini par exemple et l'infini se pénètrent : ce plateau est limité, c'est une surface restreinte et pourtant dans la limite de cette surface, je puis tracer indéfiniment figures et figures ; en sorte que si vous vous bornez à affirmer la limitation du carré, vous ne dites qu'une part de la vérité, vous êtes dans l'erreur ; il est tout ensemble fini et infini.
De même vous vous trompez en séparant ce qui est rationnel de ce qui est réel, et ce qui est réel de ce qui est rationnel.
D'habitude, on s'imagine qu'une chose, parce qu'elle est, est une dérogation à l'idéal, qu'elle ne peut pas être, par exemple, la beauté, la vérité absolue ; on s'imagine que l'idéal ne peut être qu'une conception, que dès qu'il se réalise, il diminue. Ce sont là des idées arbitraires et fausses ; tout ce qui est rationnel rentre nécessairement dans la vie ; il n'y a pas une idée rationnelle qui ne soit traduite dans la réalité et il n'est pas une seule réalité qui ne puisse se ramener à une idée et recevoir une explication rationnelle.
Cette grande formule de la synthèse des contraires, de la conciliation des contradictions par l'identité du rationnel et de l'idéal, a eu une influence profonde.
Nous ne disons plus de telle ou telle période de l'histoire qu'elle n'est qu'une période de barbarie, nous disons : tout ce qui est, par cela seul qu'il est, tout ce qui a été, par cela seul qu'il a été, avait sa raison et sa racine dans la raison, mais ce n'était pas la raison complète.
Je crois inutile de rappeler aux adeptes de la doctrine de Marx, que Marx a été le disciple intellectuel de Hegel ; il le déclare, il le proclame lui-même dans son introduction du Capital (et Engels3, depuis quelques années, semble, par cette pente qui porte l'homme qui a longtemps vécu à revenir vers ses origines, s'appliquer à l'étude approfondie de Hegel lui-même). Il y a une application saisissante de cette formule des contraires, lorsque Marx constate aujourd'hui l'antagonisme des classes, l'état de guerre économique, opposant la classe capitaliste à la classe prolétarienne ; parce que cet antagonisme est né sous le régime capitaliste, sous un régime de guerre et de division, il prépare un régime nouveau de paix et d'harmonie. Selon la vieille formule d'Héraclite, que Marx se plaît à citer, « la paix n'est qu'une forme, un aspect de la guerre ; la guerre n'est qu'une forme, un aspect de la paix. Il ne faut pas opposer l'une à l'autre ; ce qui est lutte aujourd'hui est le commencement de la réconciliation de demain ».
La pensée moderne de l'identité des contraires se retrouve encore dans cette autre conception admirable du marxisme : l'humanité a été jusqu'ici conduite, pour ainsi dire, par la force inconsciente de l'histoire, jusqu'ici ce ne sont pas les hommes qui se meuvent eux-mêmes ; ils s'agitent et l'évolution économique les mène ; ils croient produire les événements ou s'imaginent végéter et rester toujours à la même place, mais les transformations économiques s'opèrent à leur insu même, et à leur insu elles agissent sur eux. L'humanité a été, en quelque sorte, comme un passager endormi qui serait porté par le cours d'un fleuve sans contribuer au mouvement, ou du moins sans se rendre compte de la direction, se réveillant d'intervalles en intervalles et s'apercevant que le paysage a changé.
Eh bien ! Lorsque sera réalisée la révolution socialiste, lorsque l'antagonisme des classes aura cessé, lorsque la communauté humaine sera maîtresse des grands moyens de production selon les besoins connus et constatés des hommes, alors, l'humanité aura été arrachée à la longue période d'inconscience où elle marche depuis des siècles, poussée par la force aveugle des événements, et elle sera entrée dans l'ère nouvelle où l'homme, au lieu d'être soumis aux choses, règlera la marche des choses. Mais cette ère prochaine de pleine conscience et de pleine clarté, elle n'a été rendue possible que par une longue période d'inconscience et d'obscurité.
Si les hommes, à l'origine incertaine de l'histoire, avaient voulu délibérément régler la marche des événements et des choses, ils auraient contrarié simplement le cours de ces événements, ils auraient gaspillé les ressources de l'avenir, et pour avoir voulu agir trop tôt avec pleine conscience, ils se seraient retiré le moyen d'agir jamais avec pleine conscience ; comme l'enfant que l'on aurait appelé trop tôt à la vie pleinement consciente de la raison réfléchie, et en qui l'on n'aurait pas laissé se produire l'évolution inconsciente de la vie organique et des premières manifestations de la vie morale, pour avoir été un penseur à la première heure de la vie, aurait été incapable de penser ensuite.
Pour Marx, cette vie inconsciente était la condition même et la préparation de la vie consciente de demain, et ainsi encore l'histoire se charge de résoudre une contradiction essentielle. Eh bien ! je demande si l'on ne peut pas, si l'on ne doit pas, sans manquer à l'esprit même du marxisme, pousser plus loin cette méthode de conciliation des contraires, de synthèse des contradictoires, et chercher la conciliation fondamentale du matérialisme économique et de l'idéalisme appliqué au développement de l'histoire.
Remarquez dans quel esprit – et je vous demande pardon de ces longs préliminaires, mais il n'y a pas de question particulière qui puisse être résolue si l'on ne s'est entendu sur une philosophie générale, – remarquez dans quel esprit je cherche cette conciliation du matérialisme économique et de l'idéalisme historique et moral.
Je ne veux pas faire à chacun sa part, je ne veux pas dire : il y a une partie de l'histoire qui est gouvernée par les nécessités économiques et il y en a une autre dirigée par une idée pure, par un concept, par l'idée, par exemple, de l'humanité ou de la justice ou du droit ; je ne veux pas mettre la conception matérialiste d'un côté d'une cloison, et la conception idéaliste de l'autre. Je prétends qu'elles doivent se pénétrer l'une l'autre, comme se pénètrent dans la vie organique de l'homme, la mécanique cérébrale et la spontanéité consciente.
Je dis qu'il n'y a pas dans le cerveau un seul mouvement qui ne corresponde d'une façon claire ou obscure à un état de conscience, et qu'il n'y a pas un seul état de conscience qui ne corresponde à un mouvement cérébral.
Et si l'on pouvait ouvrir le cerveau et suivre l'infinie délicatesse des mouvements qui s'y produisent, s'y déterminent et s'y enchaînent, on pourrait suivre par l'envers physiologique tout le travail psychologique de nos pensées, de nos conceptions, de nos volitions ; et pourtant il y a là une singulière antinomie résolue par la vie sans que nous paraissions nous en douter.
Oui, au moment même où je parle, mon idée, par quoi est-elle déterminée ?
Elle est déterminée par une idée antérieure avec laquelle elle a des rapports logiques, et toutes nos idées s'enchaînent les unes aux autres selon certains rapports logiques, intelligibles, ou de ressemblance, ou d'opposition, ou de causalité.
En sorte que dans la trame de nos pensées, n'interviennent, en apparence, que des forces logiques, et que, de plus, il semble que toute l'activité présente dans mon esprit conscient soit déterminée par une idée de l'avenir.
Si je prononce en ce moment des paroles, c'est bien parce que l'idée que j'exprime en cette minute a été longuement amenée par une idée antérieure et par toute la suite des idées antérieures ; mais c'est aussi parce que je veux réaliser dans l'avenir que je vois devant moi, un but, une intention, une fin ; c'est parce que je veux aboutir à une démonstration complète que je conduis en ce moment mes pensées dans la direction qu'elles suivent, en sorte que ma pensée présente, en même temps qu'elle est déterminée par la série des pensées antérieures, semble provoquée par une idée d'avenir.
Au contraire, dans le développement physiologique, mécanique des mouvements cérébraux, celui qui accompagne en ce moment la pensée que j'exprime n'est déterminé que par un mouvement antérieur ; en sorte, citoyens, que notre vie est à la fois physiologique et consciente, à la fois mécanique et spontanée. Dans l'enchaînement des mouvements cérébraux, le présent n'est déterminé que par le passé, tandis que dans l'enchaînement des idées, des concepts conscients, le présent semble déterminé par l'avenir.
On dirait donc qu'il y a contradiction entre le mode selon lequel fonctionne ma vie cérébrale et le mode selon lequel fonctionne le développement conscient de nos idées et de nos pensées.
Et pourtant, quoiqu'il y ait antinomie apparente entre ces deux modes, entre ces deux points de vue, la synthèse est faite, la conciliation est faite, et il n'y a pas une seule de mes pensées qui ne corresponde à un mouvement cérébral comme il n'y a pas un seul mouvement cérébral qui ne corresponde au moins à un commencement de pensée.
Or, il en est de même dans l'histoire et en même temps que vous pouvez expliquer tous les phénomènes historiques par la pure évolution économique, vous pouvez aussi les expliquer par le désir inquiet, permanent, que l'humanité a d'une forme supérieure d'existence, et pour préciser la question, voici, citoyens, comment, à mon sens, le problème se pose, voici les explications complémentaires que je suis obligé de demander aux théoriciens du marxisme.
Marx dit : « Le cerveau humain ne crée pas de lui-même une idée du droit qui serait vaine et creuse ; il n'y a dans toute la vie, même intellectuelle et morale, de l'humanité, qu'un reflet des phénomènes économiques dans le cerveau humain4. »
Eh bien ! je l'accepte. Oui, il n'y a dans tout le développement de la vie intellectuelle, morale, religieuse de l'humanité que le reflet des phénomènes économiques dans le cerveau humain ; oui, mais il y a en même temps le cerveau humain, il y a par conséquent la préformation cérébrale de l'humanité.
L'humanité est le produit d'une longue évolution physiologique qui a précédé l'évolution historique, et lorsque l'homme, selon cette évolution physiologique, a émergé de l'animalité, immédiatement inférieure, il y avait déjà dans le premier cerveau de l'humanité naissante des prédispositions, des tendances.
Quelles étaient-elles ?
Il y avait d'abord l'aptitude à ce que j'appellerai les sensations désintéressées. À mesure que l'on s'élève dans l'échelle de la vie animale, on constate que les sens purement égoïstes se subordonnent peu à peu aux sens esthétiques et désintéressés. Aux rangs inférieurs de l'animalité, la vue est peu développée, l'ouïe l'est peu, ce qui est développé c'est l'odorat, c'est la faculté de préhension ; c'est le goût, c'est-à-dire tous les sens qui sont mis surtout en mouvement par la proie, tous les sens qui mettent surtout en mouvement l'appétit physique et égoïste. Au contraire, à mesure qu'on s'élève dans l'animalité, voyez se développer le sens de l'ouïe, et le sens de la vue. Et c'est par l'œil qu'arrive à l'animal l'image de la proie, qu'il devrait saisir, il est vrai, mais en même temps bien d'autres images lui arrivent qui ne peuvent mettre en mouvement son appétit animal ; par l'ouïe, si l'animal recueille bien des bruits, des rumeurs qui peuvent le mettre sur la trace de la proie, ou qui peuvent l'avertir du danger, il lui arrive aussi bien des harmonies qui n'ont aucun rapport immédiat avec son appétit physique et les conditions positives de sa sécurité. En sorte que, par la vue inondée d'images qui dépassent la sensibilité immédiate de l'animal, par l'ouïe pénétrée de sonorités qui dépassent le besoin immédiat de l'animal, l'univers pénètre dans l'animalité, sous une autre forme que celle de la lutte pour la vie. C'est déjà dans l'animal le besoin, la joie, l'éblouissement de la lumière, c'est déjà le besoin, la joie, l'enchantement de la mélodie et de l'harmonie ; du fond de la vie organique purement égoïste va éclore peu à peu le sens esthétique et désintéressé, et dans la forêt profonde, toute frissonnante de rumeurs et de clarté, l'univers entre dans l'animal comme un roi.
Outre cette prédisposition première que l'homme-animal apportait au début de la longue évolution économique, il y avait de plus la faculté, déjà éveillée chez les animaux eux-mêmes, de saisir le général dans le particulier, le type de l'espèce dans l'individu, de démêler la ressemblance générique à travers les diversités individuelles.
Dans les autres individus qui vont passer devant lui, avec lesquels les lois du développement économique le mettront en contact, l'individu homme, et l'animal homme, ne verra pas seulement des formes associées ou ennemies, il verra des forces semblables et alors il y a en lui un premier instinct de sympathie imaginative qui, par la ressemblance saisie et constatée, lui permettra de deviner et de sentir les joies des autres, de deviner et d'éprouver leurs douleurs. Dès le début de la vie, à côté de l'égoïsme brutal, on trouve ce sentiment préparant la réconciliation fraternelle de tous les hommes après les séculaires combats.
Enfin, dès le début de sa vie, avant la première manifestation de sa pensée, l'homme a ce que l'on peut appeler le sens de l'unité, la première manifestation de son mouvement intellectuel, c'est la réduction de tous les êtres, de toutes les formes et de toutes les forces, à une unité vaguement entrevue ; voilà comment on peut dire que l'homme est, dès sa première heure, un animal métaphysicien, puisque l'essence même de la métaphysique, c'est la recherche de l'unité totale dans laquelle seraient compris tous les phénomènes et enveloppées toutes les lois.
La preuve de ce sens premier de l'unité est dans la création spontanée du langage, avec ses hiérarchies de mots, qui ne font que représenter des hiérarchies d'idées s'enveloppant les unes dans les autres, avec ses catégories verbales traduisant les catégories intellectuelles.
En résumé, j'accorde à Marx que tout le développement ultérieur ne sera que le réfléchissement de phénomènes économiques dans le cerveau, mais à condition que nous disions qu'il y a déjà dans ce cerveau, par le sens esthétique, par la sympathie imaginative et par le besoin d'unité, des forces fondamentales qui interviennent dans la vie économique.
Remarquez, encore une fois, que je ne juxtapose pas les facultés intellectuelles aux forces économiques, que je ne veux pas reconstituer ce syndicat de facteurs historiques que notre éminent ami Gabriel Deville5 a dispersé avec tant de vigueur, il y a quelques mois. Non, je ne veux pas de cette juxtaposition, mais je dis qu'il est impossible que les phénomènes économiques constatés pénètrent dans le cerveau humain, sans y mettre en jeu ces ressorts primitifs que j'analysais tout à l'heure. Et voilà pourquoi je n'accorde pas à Marx que les conceptions religieuses, politiques, morales, ne sont qu'un reflet des phénomènes économiques6. Il y a dans l'homme une telle pénétration de l'homme même et du milieu économique qu'il est impossible de dissocier la vie économique et la vie morale ; pour les subordonner l'une à l'autre, il faudrait d'abord les abstraire l'une de l'autre ; or, cette abstraction est impossible : pas plus qu'on ne peut couper l'homme en deux et dissocier en lui la vie organique et la vie consciente, on ne peut couper l'humanité historique en deux et dissocier en elle la vie idéale et la vie économique. Telle est ma thèse, dont je trouve la confirmation partielle dans la philosophie grecque.
Les Grecs n'ont pas commencé à constater les antinomies économiques, les lois qui établissent l'ordre dans la cité, l'opposition et la conciliation des pauvres et des riches, pour projeter ensuite leurs observations économiques sur l'univers ; non, ils ont d'un même coup d'œil et dans une même conception réuni les phénomènes économiques et les phénomènes naturels. Voyez Héraclite, Empédocle, Anaximandre ; ils constatent dans des formules uniques les liens et les contradictions des éléments, que ces éléments appartiennent à la nature, le chaud et le froid, le lumineux et le ténébreux, ou à l'organisme physiologique, les sains et les malades, ou à la vie intellectuelle, le parfait et l'imparfait, l'égal et l'inégal, ils font une table unique de ces oppositions empruntées soit à la nature, soit à la société, et c'est dans Héraclite le même mot Cosmos qui formule tout à la fois l'ordre du monde résultant de la conciliation des contraires, et l'ordre dans la cité résultant de la conciliation des factions. C'est d'une seule vue que les penseurs grecs saisissent l'ordre du monde se débrouillant du chaos social.
Ne pouvant en quelques paroles, à la fois trop longues et trop brèves, qu'effleurer la question, je me borne à adresser aux théoriciens marxistes une autre demande d'explications, et je leur dis ceci :
Quel jugement portez-vous, si vous en portez un (et de cela je suis certain), sur la direction du mouvement économique et du mouvement humain ?
Il ne suffit pas de dire qu'une forme de la production succède à une autre forme de la production ; il ne suffit pas de dire que l'esclavage a succédé à l'anthropophagie, que le servage a succédé à l'esclavage, que le salariat a succédé au servage, et que le régime collectiviste ou communiste succédera au salariat. Non, il faut encore se prononcer. Y a-t-il évolution ou progrès ? Et s'il y a progrès, quelle est l'idée décisive et dernière à laquelle on mesure les diverses formes du développement humain ? Et encore, si l'on veut écarter comme trop métaphysique cette idée de progrès, pourquoi le mouvement de l'histoire a-t-il ainsi été réglé de forme en forme, d'étape économique en étape économique, de l'anthropophagie à l'esclavage, de l'esclavage au servage, du servage au salariat, du salariat au régime socialiste, et non pas d'une autre façon ? Pourquoi, en vertu de quel ressort, je ne dis pas en vertu de quel décret providentiel, puisque je reste dans la conception matérialiste et positive de l'histoire, mais pourquoi, de forme en forme, le développement humain a-t-il suivi telle direction et non pas telle autre ?
Pour moi la raison en est simple, si l'on veut admettre l'action de l'homme comme homme, l'action de ces forces humaines initiales dont j'ai parlé.
C'est que, précisément parce que les rapports économiques de production s'adressent à des hommes, il n'y a pas une seule forme de production qui ne renferme une contradiction essentielle, tant que la pleine liberté et la pleine solidarité des hommes n'aura pas été réalisée.
C'est Spinoza qui a démontré admirablement la contradiction intime de tout régime tyrannique, de toute exploitation politique ou sociale de l'homme par l'homme, non pas en se plaçant au point de vue du droit abstrait, mais en montrant qu'on se trouvait là en présence d'une contradiction de fait. Ou bien la tyrannie fera à ceux qu'elle opprime tant de mal qu'ils cesseront de redouter les suites que pourrait avoir pour eux une insurrection, et alors les opprimés se soulèveront contre l'oppresseur, ou bien celui-ci, pour prévenir les soulèvements, ménagera dans une certaine mesure les besoins, les instincts de ses sujets, et il les préparera ainsi à la liberté. Ainsi, de toute façon, la tyrannie doit disparaître en vertu du jeu des forces, parce que ces forces sont des hommes.
Il en sera de même tant que l'exploitation de l'homme par l'homme n'aura pas pris fin. C'est Hegel encore qui a dit, avec une précision souveraine : « La contradiction essentielle de toute tyrannie politique ou économique, c'est qu'elle est obligée de traiter comme des instruments inertes des hommes qui, quels qu'ils soient, ne pensent jamais descendre à l'inertie des machines matérielles. »
Et remarquez que cette contradiction est, tout à la fois, une contradiction logique et une contradiction de fait.
C'est une contradiction logique, puisqu'il y a opposition entre l'idée même de l'homme, c'est-à-dire d'un être doué de sensibilité, de spontanéité et de réflexion, et l'idée de machine. C'est une contradiction de fait puisqu'en se servant de l'homme, outil vivant, comme d'un outil mort, on violente la force même dont on veut se servir et on aboutit ainsi à un mécanisme social discordant et précaire. C'est parce que cette contradiction viole à la fois l'idée de l'homme et la loi même de mécanique, selon laquelle la force homme peut être utilisée, que le mouvement de l'histoire est tout à la fois une protestation idéaliste de la conscience contre les régimes qui abaissent l'homme, et une réaction automatique des forces humaines contre tout arrangement instable et violent. Qu'était l'anthropophagie ? Elle était doublement contradictoire, car en obligeant l'homme à égorger l'homme en dehors même de l'excitation du combat, elle faisait violence à ce premier instinct de sympathie dont j'ai parlé : contradiction morale ; et, de plus, elle faisait de l'homme, qui a une aptitude certaine au travail réglé, à la production, une sorte de bête de proie dont on ne peut utiliser que la chair : contradiction économique. Dès lors l'esclavage devait naître, parce que la domestication de l'homme blessait moins l'instinct de sympathie et ménageait mieux l'intérêt du maître en tirant de l'homme, par le travail, beaucoup plus qu'il ne donnait par sa substance.
Et l'on ferait sans peine la même démonstration pour l'esclavage, pour le servage, pour le salariat. Dès lors, on comprend, puisque tout le mouvement de l'histoire résulte de la contradiction essentielle entre l'homme et l'usage qui est fait de l'homme, que ce mouvement tende, comme à sa limite, à un ordre économique où il sera fait de l'homme un usage conforme à l'homme. C'est l'humanité qui, à travers des formes économiques qui répugnent de moins en moins à son idée, se réalise elle-même. Et il y a dans l'histoire humaine non seulement une évolution nécessaire, mais une direction intelligible et un sens idéal. Donc, tout le long des siècles, l'homme n'a pu aspirer à la justice qu'en aspirant à un ordre social moins contradictoire à l'homme que l'ordre présent, et préparé par cet ordre présent, et ainsi l'évolution de ses idées morales est bien réglée par l'évolution des formes économiques, mais en même temps, à travers tous ces arrangements successifs, l'humanité se cherche et s'affirme elle-même, et quelle que soit la diversité des milieux, des temps, des revendications économiques, c'est un même souffle de plainte et d'espérance qui sort de la bouche de l'esclave, du serf et du prolétaire ; c'est ce souffle immortel d'humanité qui est l'âme de ce qu'on appelle le droit. Il ne faut donc pas opposer la conception matérialiste et la conception idéaliste de l'histoire. Elles se confondent en un développement unique et insoluble, parce que si on ne peut abstraire l'homme des rapports économiques, on ne peut abstraire les rapports économiques de l'homme, et l'histoire, en même temps qu'elle est un phénomène qui se déroule selon une loi mécanique, est une aspiration qui se réalise selon une loi idéale.
Et, après tout, n'en est-il pas de l'évolution de la vie comme de l'évolution de l'histoire ? Sans doute, la vie n'est passée d'une forme à une autre, d'une espèce à une autre, que sous la double action du milieu et des conditions biologiques immédiatement préexistantes et tout le développement de la vie est susceptible d'une explication matérialiste, mais en même temps on peut dire que la force initiale de vie concentrée dans les premières granulations vivantes et les conditions générales de l'existence planétaire déterminaient d'avance la marche générale et comme le plan de la vie sur notre planète. Ainsi, les êtres sans nombre qui ont évolué, en même temps qu'ils ont subi une loi, ont collaboré par une aspiration secrète à la réalisation d'un plan de vie. Le développement de la vie physiologique comme de la vie historique a donc été fait ensemble idéaliste et matérialiste. Et la synthèse que je vous propose se rattache à une synthèse plus générale que je ne puis indiquer sans la fortifier.
Mais pour revenir à la question économique, est-ce que Marx lui-même ne réintroduit pas dans sa conception historique l'idée, la notion de l'idéal du progrès, du droit ? Il n'annonce pas seulement la société communiste comme la conséquence nécessaire de l'ordre capitaliste : il montre qu'en elle cessera enfin cet antagonisme des classes qui épuise l'humanité ; il montre aussi que pour la première fois la vie pleine et libre sera réalisée par l'homme, que les travailleurs auront tout ensemble la délicatesse nerveuse de l'ouvrier et la vigueur tranquille du paysan, et que l'humanité se redressera, plus heureuse et plus noble, sur la terre renouvelée.
N'est-ce pas reconnaître que le mot justice a un sens, même dans la conception matérialiste de l'histoire, et la conciliation que je vous propose n'est-elle pas, dès lors, acceptée de vous ?
Conférence devant les étudiants collectivistes, salle
d'Arras (Paris), décembre 1894, texte intégral.
Les idées politiques et sociales
de Jean-Jacques Rousseau
Ce texte de Jaurès sur Rousseau a été publié en 1912, mais est issu en réalité d'une conférence prononcée en 1889, lorsque ses convictions républicaines commencent à se rapprocher du socialisme. Jaurès expose la thèse d'un Rousseau « révolutionnaire malgré lui », c'est-à-dire n'ayant jamais souhaité de bouleversement révolutionnaire tout en proposant des conceptions philosophiques – notamment dans le Contrat social – qui ont largement inspiré de grands révolutionnaires comme Robespierre. À l'image de Jaurès, de nombreux socialistes se considèrent alors comme les héritiers de Rousseau.
Avertissement
Lors de sa publication, dans la Revue de métaphysique et de morale (longtemps après avoir été prononcé), ce texte était accompagné de la note suivante : « Conférence prononcée le 19 décembre 1889, à la Faculté des lettres de Toulouse, par M. Jaurès, chargé de cours de philosophie, recueillie par M. Gheusi, professeur à l'Université de Toulouse, député de la Haute-Garonne. M. Jaurès, qui n'a pu revoir cette conférence, a bien voulu nous autoriser à la publier telle quelle. »
Je ne pourrai pas, en une heure, entrer dans le détail cependant si intéressant des idées politiques et sociales de J.-J. Rousseau. Je serai même obligé, faute de temps, de laisser de côté une partie essentielle de son œuvre, celle que E. Quinet considérait comme vitale, celle qui a trait aux rapports de la religion et de l'État, l'institution d'une religion civile. Je dois me borner à indiquer les grands traits des conceptions politiques et sociales de Rousseau, à marquer surtout quel était son état d'esprit et d'âme dans cet ordre de questions. Ce n'est peut-être pas entièrement inutile à notre époque, car l'homme extraordinaire que L. Blanc et G. Sand admiraient, n'est pas très pratiqué par la nouvelle génération politique et littéraire. Il n'en reste guère dans les esprits qu'une vague idée, une notion confuse de républicanisme théorique, de brillants mais dangereux paradoxes, d'incurable misanthropie.
D'abord, si l'on veut bien comprendre le sens profond de l'œuvre politique et sociale de J.-J. Rousseau, il faut se rappeler qu'il n'a commencé que fort tard à écrire, vers la cinquantaine, qu'il s'était nourri jusque-là, sans aucune préoccupation d'écrivain, de musique, de fortes et graves lectures et surtout de toutes les images familières ou grandioses de la nature, qu'il aimait les champs, les bois, le ciel changeant. Il avait, sous les arbres, étudié et rêvé, classé des herbes et songé à Dieu. – Puis un beau jour, ou plutôt une belle nuit, il s'endort au bord d'un lac et il se réveille avec la lumière. Rien ne pouvait remplacer dans son cœur sensible la contemplation muette de la nature. Lorsqu'il avait pu échapper aux importuns, qu'il avait tourné un certain angle de mur et qu'il se voyait brusquement en face de la campagne solitaire, il avait des pétillements de joie. Devant un certain site où il se trouvait seul, il s'imaginait être devant un coin de la nature où personne n'avait jamais pénétré, il s'imaginait qu'il l'avait découvert, et que l'innocence intacte des choses souriait à l'innocence retrouvée de la pensée. Cheminant à pied en France, en Suisse, en Italie, il composait dans son imagination des tableaux, il écrivait dans sa tête des paysages qu'il oubliait devant un horizon nouveau. Qui saura jamais le nombre de chefs-d'œuvre perdus de cet écrivain avide de vie et dédaigneux de gloire ? Eh bien ! Rousseau retenait, probablement à son insu, ces impressions diverses ; et un demi-siècle après, telle circonstance, tel son, telle couleur, tel parfum, tel paysage se traduisaient harmonieusement sous sa plume... Un pareil homme, vivant avec la nature, y cherchant l'activité de l'esprit, le pain du cœur, l'oubli des misères sociales, ne peut être le réformateur outré et fiévreux qu'on s'imagine. À vouloir réformer le monde, refaire les gouvernements, bouleverser la société, il aurait fallu y penser sans cesse, et il les fuyait. – Ah ! certes, il y avait pourtant dans une pareille existence, continuée cinquante ans en plein XVIIIe siècle, un germe, un commencement de réforme politique et sociale. Il était impossible à Rousseau vivant en communion de cœur avec la nature et Dieu, la liberté et la joie, de ne pas protester contre l'existence misérable, factice et servile que les gouvernements faisaient aux hommes, privés de tout par la folie des uns et la frivolité des autres, et succombant sous l'excès d'un travail malsain. Il était impossible à Jean-Jacques, lorsqu'il observait les gouvernements et les sociétés avec son esprit de vie libre, de ne pas constater qu'ils ne reposaient plus sur leurs bases. Les joies, même les plus naïves, soulevaient dans son esprit de terribles questions politiques et sociales. Mais il semblait en redouter, en contenir l'explosion. Il sentait que le combat qu'il fallait livrer, allait bouleverser toute sa vie, et il hésitait, ou tout au moins il attendait. Et si je veux me figurer à cette époque cette pensée faite pour révolutionner le monde, qui le révolutionnera en effet, qui en a sans doute l'inquiet pressentiment, qui tâche de s'arrêter, de se fixer dans sa sérénité première, je la comparerai à ces beaux lacs de la Suisse que Rousseau a tant aimés : on dirait qu'ignorant l'issue et la pente par où ils se précipitent en fleuves, ils s'enferment en eux-mêmes et que leur joie est de réfléchir les rivages verts et les nuées roses...
C'est ainsi que Jean-Jacques a été réformateur, révolutionnaire malgré lui, et que sa pensée a eu toute sa puissance. En effet, il n'apportait pas au monde les combinaisons arbitraires d'un cerveau inquiet, mais des conclusions naturelles, pleines de vie intérieure, très riches, interprétées par un esprit puissant. Lorsque les esprits entraient dans ses doctrines, qu'ils étaient entraînés par lui, au moment où ils pouvaient hésiter, résister, ils sentaient tout à coup que ses doctrines avaient pour arrière-fond la nature immense, joyeuse et libre. Le point de départ des idées sociales de Rousseau était l'amour du monde naturel ; il arrivait à une source délicieuse, cachée sous bois. En communiquant aux hommes ses joies, il communiquait sa doctrine. Il semblait qu'on ne pût revenir à la nature que par ses études. Danton disait dans sa prison, après les agitations furieuses de sa vie révolutionnaire : « Que je voudrais voir des arbres ! » Il y a là une contradiction bizarre que les épris des œuvres de Jean-Jacques n'avaient pas à redouter. Partout dans la doctrine du maître, circule la sève, pénètrent les senteurs des grands bois. Et les hommes qui retrouvaient à la fois la nature et la liberté, s'éprenaient pour l'âme que leur donnait cette révélation, de cette sorte d'adoration qui fut, dans la société vieillie, une grande force de transformation.
Rousseau a encore donné beaucoup d'autorité à ses idées, et notamment au commencement d'idée socialiste qui était en lui, par son désintéressement, son détachement personnel. Certes, il a eu de grands défauts, il a eu peut-être des vices ; mais dans sa longue vie de travaux, de pauvreté et de rêveries, s'il a connu l'orgueil, il n'a jamais connu l'envie. Or, s'il y avait eu seulement un peu d'envie dans le premier germe du socialisme français, ce socialisme eût été diminué et discrédité. – C'est Rousseau qui a dit, par une belle application d'une loi physique au monde moral : « l'eau n'agit jamais qu'au niveau de sa source » ; et si ses idées avaient eu leur source dans les bas-fonds de l'envie, elles se fussent depuis longtemps englouties dans la fange de leur origine.
Mais Rousseau, suivant le mot d'un homme d'esprit, n'était pas « un de ces philanthropes à pied pour qui la circulation des voitures était une injure personnelle ». Il plaignait beaucoup de riches, n'en jalousait aucun. Lui-même, dans un de ses Dialogues qui sont, avec ses Rêveries, comme son testament moral, dit qu'il a vécu dans un monde idéal où la lumière est plus belle, les sons plus éclatants et plus doux, les couleurs plus vives, les parfums plus exquis. Ce monde est le monde réel, savouré par des sens d'artiste. Pourquoi Rousseau veut-il apparaître, lui aussi, comme un privilégié du bonheur ? C'est pour écarter tout soupçon d'amertume, toute accusation d'envie et de méchanceté. « J'ai été heureux à ma façon, peut-il s'écrier ; ce n'est donc pas pour moi que je réclamais. » – Messieurs, il serait sacrilège de notre part de repousser l'appel de la justice, même lorsqu'il s'y mêlerait parfois l'âpre appel de la souffrance ; mais j'aime qu'il ne se soit pas glissé dans l'œuvre de Rousseau une seule goutte de fiel, un seul ferment de haine, et je voulais d'abord rétablir au profit de celui qui a lutté pour la justice ce premier titre d'honneur.
Mais ce désintéressement même contribuait à empêcher Rousseau de devenir je ne dis pas un homme d'action, mais un penseur d'action. Il regardait l'humanité, non avec sa passion, mais avec sa raison, et il ne croyait guère à la possibilité d'obtenir les transformations profondes exigées par le droit. Chose étrange ! Cet homme, qui a agi si puissamment sur la Révolution, ne croyait pas au succès possible de cette Révolution. Dans la dédicace aux magistrats suisses, dans son Discours sur l'inégalité, il écrit : « Les peuples, une fois accoutumés à des maîtres, ne sont plus en état de s'en passer. S'ils tentent de secouer le joug, ils s'éloignent d'autant plus de la liberté que, prenant pour elle une licence effrénée qui lui est opposée, leurs révolutions les livrent presque toujours à des séductions qui ne font qu'aggraver leurs chaînes. » Bonaparte, messieurs, est au bout de ces lignes. – À un autre point de vue, il reprend la même idée dans le Contrat social : « La plupart des peuples ainsi que des hommes ne sont dociles que dans leur jeunesse ; ils deviennent incorrigibles en vieillissant. Quand une fois les coutumes sont établies et les préjugés enracinés, c'est une entreprise dangereuse et vaine de vouloir les réformer. » – Mais dans le même chapitre du Contrat social il ajoute ces paroles, qui sont une prévision nette, quoique farouche, de la Révolution : « Ce n'est pas que, comme quelques maladies bouleversent la tête des hommes et leur ôtent le souvenir du passé, il ne se trouve quelquefois dans la durée des États des époques violentes où les révolutions font sur les peuples ce que certaines crises font sur les individus, où l'horreur du passé tient lieu d'oubli, et où l'État, embrasé par les guerres civiles, renaît pour ainsi dire de sa cendre et reprend la vigueur de la jeunesse en sortant des bras de la mort. »
Dans ces grandes commotions nationales, même si elles sont des commotions de liberté et de justice, Rousseau redoutait les dérèglements des passions mauvaises. Il déplorait dans les temps calmes les usurpations des riches, dans les temps troublés les brigandages des pauvres. Je ne suis pas sûr que pour cet homme concentré, fermé à certaines légèretés d'enthousiasme, la Révolution française n'eût pas été une nouvelle cause de désespoir. Il a, en termes catégoriques, condamné d'avance le régicide : « le sang d'un homme a plus de prix que la liberté du genre humain. » – Vous voyez que Robespierre a bien fait, pour accomplir en paix son voyage à Ermenonville, d'attendre qu'il n'y eût plus qu'un tombeau... – Et cependant, un siècle après ces paroles de renonciation et de doute, non seulement notre pays a traversé sans sombrer les terribles secousses de la Révolution française ; non seulement il paraît être entré définitivement – et c'est ma foi profonde – dans la liberté ordonnée ; non seulement cette liberté républicaine, que Rousseau ne croyait possible que pour de petits États, est devenue le patrimoine du grand pays de France, mais la France libre, avant d'être entièrement sortie de cette terrible crise, se retourne vers son passé, et par ses historiens, ses poètes, ses orateurs politiques, Quinet, Michelet, Hugo, Gambetta, au lieu de rejeter sa longue histoire d'avant la liberté, cherche à renouer par une brillante chaîne ses glorieuses annales et cette liberté qu'elle a conquise, et fait entrer le passé mieux connu que jamais dans sa conscience agrandie.
Si bien que si Rousseau s'est trompé, c'est pour n'avoir pas eu assez de confiance. – On traite volontiers cet homme de chimérique : toutes ses erreurs sont de n'avoir pas cru assez à sa chimère. Dans ce manque de foi est la clef de ces apparents paradoxes sur les progrès funestes de la civilisation. S'il avait cru possible de rectifier la société actuelle dans le sens de l'égalité et du bonheur, il aurait beaucoup moins célébré la condition primitive de l'homme. Mais il ne voulait pas que l'humanité eût tout à fait manqué sa vie et voyant le présent mauvais, ne croyant pas l'avenir meilleur, il attribuait l'innocence et la joie à certaines périodes lointaines du développement humain. Jamais il n'a célébré, sous le nom d'état de nature, la grossièreté première des hommes voisins de l'animalité, et lorsqu'il veut condamner la guerre il la donne comme un prolongement faux de l'état de nature7. Il montre que les hommes, en aliénant leur liberté naturelle, ont retrouvé plus qu'ils n'ont donné ; qu'ils ont échangé une manière d'être précaire contre la sécurité, l'indépendance contre la liberté, la force contre le droit. Même dans le Discours sur l'inégalité des conditions, Rousseau ne propose pas comme idéal la période la plus primitive du développement humain. L'état le meilleur dans le passé, ce n'est pas, suivant ses explications, le pur état de nature, c'est la société naissante. C'est lorsque les hommes ont déjà inventé le langage, fondé la famille, assuré la stabilité du foyer, lorsque l'agriculture, les travaux de la terre ont nécessité la constitution de la propriété individuelle, que les hommes, engagés dans les lieux de la société, déjà robustes mais encore souples, jouissent le plus pleinement des biens naturels et sociaux. La fameuse phrase de Rousseau sur la propriété n'a pas été très bien comprise. Oui, Rousseau a écrit : « Le premier qui ayant enclos un terrain s'avisa de dire : Ceci est à moi, et trouva des gens assez simples pour le croire, fut le vrai fondateur de la société civile. Que de crimes, de guerres, de meurtres, que de misères et d'horreurs n'eût point épargnés au genre humain celui qui arrachant les pieux ou comblant les fossés eût crié à ses semblables : Gardez-vous d'écouter cet imposteur ; vous êtes perdus si vous oubliez que les fruits sont à tous et que la terre n'est à personne ! » – Eh bien ! messieurs, malgré tout cela, Jean-Jacques ne conteste pas la nécessité de la propriété individuelle à un moment des révolutions humaines. Il n'en conteste pas non plus la légitimité. Aussitôt à la suite de ce passage, il déclare que la constitution de la propriété individuelle était inévitable et qu'elle est légitime lorsqu'elle est fondée sur le travail. Ce qu'il a voulu dire, c'est que d'abord dans l'institution, dans l'origine de cette propriété individuelle, il y a un mélange de droit et d'usurpation, de force, de hasard et de travail, et ensuite que les hommes n'étaient pas capables d'entourer cette propriété de telles garanties qu'elle ne dégénérât pas en instrument de tyrannie et de spoliation. C'est une force bonne et salutaire en soi, mais qui, insuffisamment maîtrisée, se déchaînera et aboutira aux plus monstrueuses inégalités.
C'est là le sens, la clef de toutes les théories de Rousseau sur le développement de la société. Elles peuvent se résumer ainsi : la faiblesse humaine est disproportionnée au progrès humain.
Oui, des hommes ont raison d'élever leur âme à Dieu et d'instituer un culte, mais ils ne restent pas maîtres de ces mouvements : les castes privilégiées, les superstitions dégradantes, le fanatisme horrible viennent leur dérober les douceurs premières de la religion. – Oui, les hommes ont raison d'instituer les lois, l'ordre, la paix ; mais les lois, gardiennes du droit, deviennent les gardiennes de l'illégalité, et la paix sociale n'est plus que la dérision de la paix, alors que ce que les hommes avaient voulu, c'est la paix dans la justice. – Oui, les hommes font bien de s'élever de l'instinct à la raison, mais des hardiesses téméraires et des sophismes dangereux font descendre cette raison au-dessous de l'instinct. L'histoire du progrès humain est ainsi une série de déceptions. L'homme est un artiste étrange, sublime et inconstant : il crée, il sculpte des statues vivantes, il les anime de son souffle libre, mais ce souffle se refroidit, les puissances de la matière prévalent, le sourire se change en grimace, les lèvres raillent, l'œuvre blesse l'artiste, la chose créée enchaîne le créateur, et Jean-Jacques crie à l'âme humaine : « Garde-toi de créer ! » Mais, messieurs, le vigoureux esprit de Rousseau lui retraçait pourtant avec tant de relief l'idéal de la liberté politique et de l'égalité sociale que, peu à peu, à son insu même, cet idéal lui apparaissait bientôt comme réel. Aussi dans le Contrat social il se laisse aller beaucoup plus à l'espérance que dans le Discours sur l'inégalité. Il y fait œuvre involontairement mais manifestement utile. Lorsqu'il montrait que le pacte social de la liberté était sans cesse violé par les gouvernements, lorsqu'il montrait que les hommes peuvent reprendre cette liberté de nature, il sapait la base sur laquelle reposait le pouvoir arbitraire, il ouvrait les portes du Droit.
Rousseau était républicain. Il l'était de patrie, étant né à Genève, il l'était d'âme, car nul n'a mieux compris que lui les joies de la liberté, de l'auguste liberté, comme il disait ; il l'était de raison, car selon lui, ou plutôt selon le droit, les hommes ne peuvent aliéner dans l'ordre social leur liberté naturelle qu'à la condition de la retrouver confirmée, élevée par ce même ordre social. Il se peut que les clauses de ce contrat n'aient jamais été exposées, mais partout elles ont été facilement adoptées et reconnues. Tout homme entrant dans l'ordre social doit y trouver l'égalité, en échange de la liberté dont il fait abandon. Il doit y retrouver une part de souveraineté égale à la part de souveraineté d'un autre ; chaque homme est une partie égale du souverain, et l'assemblée des volontés libres constitue effectivement ce souverain. Rousseau a ainsi proclamé à l'avance, sans le nommer, le suffrage universel ; et si les assemblées nationales de la Révolution avaient été plus imprégnées des doctrines de Jean-Jacques, elles n'auraient pas commis la faute mortelle, de diviser la France nouvelle en citoyens actifs et citoyens passifs. Rousseau a d'ailleurs le sentiment que le peuple doit avoir une certaine culture d'esprit. Il félicite les magistrats suisses d'avoir su former un peuple qui, par ses lumières et sa raison, est au-dessus des autres, et, citant à ce propos l'exemple de son père, il dit : « Qu'il me soit permis de citer un exemple dont il devrait rester de meilleures traces et qui sera toujours présent à mon cœur. Je ne me rappelle point sans la plus douce émotion la mémoire du vertueux citoyen de qui j'ai reçu le jour et qui souvent entretint mon enfance du respect qui vous était dû. Je le vois encore vivant du travail de ses mains et nourrissant son âme des vérités les plus sublimes. Je vois Tacite, Plutarque et Grotius8 mêlés devant lui avec les instruments de son métier. » Et plus loin : « Tels sont ces hommes instruits et sensés dont, sous le nom d'ouvriers et de peuple, on a chez les autres nations des idées si basses et si fausses. » Pour permettre à tous les citoyens de mêler ces deux vies politique et individuelle, Rousseau aurait réclamé pour eux non seulement les lumières, mais aussi les loisirs que le travail manuel ne laisse pas à beaucoup.
Un homme n'a pas le droit d'aliéner sa propre liberté et à plus forte raison celle de ses descendants. L'exercice de la souveraineté est indivisible. On ne peut pas opposer à la légitimité du souverain celle d'un autre pouvoir. Il n'y a pas de compromission possible entre le pouvoir légitime et un autre, pas de collaboration entre l'arbitraire et le droit.
En établissant ces principes, Rousseau condamnait les tentatives hybrides de monarchie constitutionnelle et affirmait la légalité de la démocratie républicaine. Ce n'est pas à dire qu'il veuille aboutir à la confusion des pouvoirs, à l'anarchie. Précisément parce que la souveraineté est indivisible, une fraction du peuple, isolée du reste, ne peut rien. Le règne des factions est une usurpation comme une tyrannie de caste. – Puis, Rousseau distingue profondément, quoiqu'on ait prétendu le contraire, trois pouvoirs : judiciaire, législatif, exécutif. Il n'admet pas que le droit qu'a le souverain de faire les lois lui confère celui de juger : « Sitôt qu'il s'agit d'un fait ou d'un droit particulier sur un point qui n'a pas été réglé par une convention générale et antérieure, c'est un procès où les particuliers intéressés sont une des parties et le public l'autre... Il serait ridicule de vouloir alors s'en rapporter à une expresse décision de la volonté générale qui ne peut être que la conclusion de l'une des parties et qui par conséquent n'est pour l'autre qu'une volonté étrangère, particulière, portée en cette occasion à l'injustice et à l'erreur. » Il faut donc que le pouvoir judiciaire soit un pouvoir distinct.
Quant au pouvoir exécutif, qui est si contesté aujourd'hui dans la démocratie française, le théoricien absolu de la démocratie a reconnu non seulement que ce pouvoir devait exister, mais encore qu'il était nécessaire qu'il eût une force propre lorsqu'il a dit : « Il faut à la force publique un agent propre qui la réunisse et la mette en œuvre selon la direction de la volonté générale, qui serve à la communication de l'État et du souverain, qui fasse en quelque sorte dans la personne publique ce que fait dans l'homme l'union de l'âme et du corps. Le gouvernement est une personne morale douée de certaines facultés. » Et encore : « Il lui faut un moi particulier, une sensibilité commune à ses membres, une force, une volonté propre qui tende à sa conservation. »
Dans les petits États, où l'écart entre la volonté du souverain et celle des particuliers ne saurait être très grand, le pouvoir exécutif peut être faible sans danger. Mais dans un grand État, il faut au pouvoir exécutif une très grande force ; et comme, suivant la pensée de Rousseau, le gouvernement se relâche à mesure que le nombre des magistrats s'accroît, il faut qu'une grande démocratie constitue une certaine unité de magistrature. Les Constituantes des États-Unis, quand elles ont assuré au Président de la République une initiative considérable, se sont visiblement inspirées des vues de Jean-Jacques, et je crois qu'aujourd'hui il conseillerait formellement à la démocratie française de donner toute force à son pouvoir exécutif.
C'est ici que se place la théorie de Rousseau sur le gouvernement parlementaire. Jean-Jacques est beaucoup trop sévère pour ce régime qui est l'essai au moins de la liberté. Il faut donc se rendre compte de la force avec laquelle il le montre comme l'organisation raisonnée de l'asservissement populaire. Il fallait être démocrate comme il l'était pour dénoncer hautement le parlementarisme anglais du XVIIIe siècle, ce jeu d'oligarchies rivales qui donne au peuple seulement la comédie de la souveraineté. J'ajoute que l'école des constitutionnels anglais a été une grande déviation pour notre pays.
Rousseau se rendait compte que la pratique de la souveraineté populaire, comme il l'entendait, avec la ratification constante des lois, était difficile. Il admettait qu'elle n'était pleinement praticable que dans un État très petit, et où les citoyens pouvaient disposer d'une grande somme de loisirs. Telles étaient les Républiques antiques : « Tout ce que le peuple avait à faire, il le faisait par lui-même ; il était sans cesse assemblé sur la place ; il habitait un climat doux ; il n'était pas avide ; des esclaves faisaient ses travaux. Sa grande affaire était sa liberté. »
Messieurs, je ne crois pas, pour ma part, que le problème dans ce qu'il a d'essentiel, c'est-à-dire la participation effective du véritable souverain à l'exercice de la souveraineté, soit insoluble. Ce n'est pas le lieu de discuter sur les moyens de le réaliser. Je voulais seulement vous montrer que Rousseau avait prévu les deux périls qui menacent actuellement notre démocratie : la nation française gouverne trop et ne légifère pas assez9. Ce peuple intervient constamment par ses représentants dans la pratique du pouvoir exécutif ; il pèse sur les administrations publiques, faussant leur droiture ; et il est incapable de suivre, de diriger l'élaboration des lois. Dans ces conditions, l'exercice intermittent du pouvoir pourrait livrer la démocratie française à toutes les oligarchies, surtout à l'oligarchie d'argent.
Messieurs, je m'étais proposé, et je vois que cette tâche dépasse mon temps, d'entrer aussi dans le détail des idées sociales de Jean-Jacques. Je vois, à mon grand regret, qu'il m'y faut renoncer pour ce soir ; je me bornerai à vous dire que si Jean-Jacques a formulé un système précis d'organisation politique qui contient, rigoureusement exprimés, tous les traits d'une constitution immédiatement applicable, il n'a pas indiqué par quel mécanisme, par quelles mesures pratiques, il comptait faire pénétrer dans les rapports des citoyens plus d'équité, plus de justice. Le problème qu'il s'est posé n'était pas avant tout un problème de propriété, mais bien de souveraineté. Nous pouvons dire qu'aujourd'hui la question de souveraineté est résolue selon le droit ; c'est celle de propriété qui reste à résoudre.
Mais si l'on ne trouve pas dans Jean-Jacques de solution précise, on peut y chercher une inspiration supérieure de justice. Lorsqu'on dit à ceux qui se contentent de prêcher la justice qu'ils devraient indiquer les moyens de la faire entrer dans l'ordre social, on raisonne juste. Mais le premier moyen de réaliser cette réforme, c'est de tomber d'accord sur la nécessité de la faire, et les hommes qui établissent cette convention commencent par là même à la réaliser.
Ah ! je ne me dissimule pas les difficultés qui attendent l'accomplissement de cette deuxième partie de l'œuvre sociale. Je sais qu'aux hommes de bon vouloir il faudra lutter contre l'ignorance dormante et routinière des foules assujetties, contre le discrédit que les démagogues et les charlatans de la démocratie jettent sur les meilleures causes, contre les basses habiletés des bas politiciens. Il faudra lutter contre les pharisiens de la démocratie, qui acceptent tout du droit, à l'exception des sacrifices qu'il commande, qui prétendent ouvrir leur âme à certaines espérances, pourvu qu'elles restent toujours de vagues chansons sous le ciel étoilé, qui veulent bien saluer la Justice lorsqu'elle passe dans les nuées, mais qui l'oppriment dès qu'elle descend sur la terre. Mais nous qui allons chercher dans Jean-Jacques l'inspiration de la justice, nous savons par une expérience qu'il n'avait pas, et qui s'appelle la Révolution française, qu'il ne faut jamais désespérer, et qu'un jour ou l'autre, dans notre pays de France, la grandeur des événements répond à la grandeur de la pensée10.
Revue de métaphysique et de morale, 1912,
texte intégral.
Une histoire socialiste
Dans cette introduction à la grande Histoire socialiste de la France contemporaine qu'il dirige, Jaurès expose sa conception de l'histoire de France, qui le fascine et le passionne. Il se réfère à Michelet, à qui il emprunte le lyrisme, à Plutarque, dont les Vies parallèles l'inspirent pour dresser des portraits de grands révolutionnaires, et surtout à Marx, dont il retient l'importance décisive des faits économiques et sociaux pour comprendre la complexité du processus révolutionnaire. C'est à la suite de cette publication que fut créée une commission parlementaire spécifique visant à publier de nombreux documents de l'époque révolutionnaire, et qui sera active pendant près d'un siècle (surnommée « la Commission Jaurès »).
C'est du point de vue socialiste que nous voulons raconter au peuple, aux ouvriers, aux paysans, les événements qui se développent de 1789 à la fin du XIXe siècle. Nous considérons la Révolution française comme un fait immense et d'une admirable fécondité ; mais elle n'est pas, à nos yeux, un fait définitif dont l'histoire n'aurait ensuite qu'à dérouler sans fin les conséquences. La Révolution française a préparé indirectement l'avènement du prolétariat. Elle a réalisé les deux conditions essentielles du socialisme : la démocratie et le capitalisme. Mais elle a été, en son fond, l'avènement politique de la classe bourgeoise.
Peu à peu, le mouvement économique et politique, la grande industrie, la croissance de la classe ouvrière qui grandit en nombre et en ambition, le malaise des paysans écrasés par la concurrence et investis par la féodalité industrielle et marchande, le trouble moral de la bourgeoisie intellectuelle qu'une société mercantile et brutale offense en toutes ses délicatesses, tout prépare une nouvelle crise sociale, une nouvelle et plus profonde révolution où les prolétaires saisiront le pouvoir pour transformer la propriété et la moralité. C'est donc la marche et le jeu des classes sociales depuis 1789 que nous voudrions retracer à grands traits. Il est toujours un peu arbitraire de marquer des limites, des divisions tranchantes dans le progrès ininterrompu et nuancé de la vie. Pourtant, on peut, avec une suffisante exactitude, distinguer trois périodes dans l'histoire de la classe bourgeoise et de la classe prolétarienne depuis un siècle.
D'abord, de 1789 à 1848, la bourgeoisie révolutionnaire triomphe et s'installe. Elle utilise contre l'absolutisme royal et contre les nobles la force des prolétaires, mais ceux-ci, malgré leur prodigieuse activité, malgré le rôle décisif qu'ils jouent en certaines journées, ne sont qu'une puissance subordonnée, une sorte d'appoint historique. Ils inspirent parfois aux possédants bourgeois une véritable terreur ; mais au fond ils travaillent pour eux ; ils n'ont pas une conception de la société radicalement différente : le communisme de Babeuf11 et de ses rares disciples ne fut qu'une convulsion sublime, le spasme suprême de la crise révolutionnaire avant l'apaisement du Consulat et du Premier Empire. Même en 1793 et en 1794 les prolétaires étaient confondus dans le tiers état : ils n'avaient ni une claire conscience de la classe ni le désir ou la notion d'une autre forme de propriété. Ils n'allaient guère au-delà de la pauvre pensée de Robespierre : une démocratie politiquement souveraine, mais économiquement stationnaire, faite de petits propriétaires paysans et de petite bourgeoisie artisane. La merveilleuse sève de vie du socialisme, créateur de richesse, de beauté et de joie, n'était point en eux : aux jours terribles, ils brûlaient d'une flamme sèche, flamme de colère et d'envie. Ils ignoraient la séduction, la puissante douceur d'un idéal nouveau.
Pourtant la société bourgeoise commence à peine à s'apaiser et à se fixer, et déjà la pensée socialiste s'essaie. Après Babeuf, voici de 1800 à 1848 Fourier, Saint-Simon, Proudhon, Louis Blanc. Voici sous Louis-Philippe les soulèvements ouvriers de Lyon et de Paris. À peine la révolution bourgeoise est-elle définitivement victorieuse, les prolétaires se demandent : d'où vient notre souffrance et quelle révolution nouvelle faudra-t-il accomplir ? Dans le flot de la révolution bourgeoise, d'abord bouillonnant et trouble, plus calme maintenant et plus clair, ils mirent leur pauvre visage exténué, et ils sont pris d'épouvante. Mais, avant 1848, malgré la multiplicité des systèmes socialistes et des révoltes ouvrières, la domination bourgeoise est encore intacte.
La bourgeoisie ne croit pas possible que le pouvoir lui échappe et que la propriété se transforme. Elle a, sous Louis-Philippe, la force de lutter à la fois contre les nobles et les prêtres, et contre les ouvriers. Elle écrase les soulèvements légitimistes de l'Ouest, comme les révoltes prolétariennes des grandes villes affamées. Elle croit naïvement, avec l'orgueil de Guizot12, qu'elle est l'aboutissement de l'histoire, qu'elle a des titres historiques et philosophiques au pouvoir définitif, qu'elle résume l'effort séculaire de la France et qu'elle est l'expression sociale de la raison. Les prolétaires de leur côté, malgré les soubresauts de la misère et de la faim, ne sont pas des révolutionnaires conscients. Ils entrevoient à peine la possibilité d'un ordre nouveau. C'est surtout dans la classe « intellectuelle » que les « utopies » socialistes recrutent d'abord des adeptes. Et d'ailleurs les systèmes socialistes sont très fortement imprégnés ou de pensée capitaliste, comme celui de Saint-Simon, ou de pensée petite bourgeoise, comme celui de Proudhon. Il a fallu la crise révolutionnaire de 1848 pour que la classe ouvrière prît conscience d'elle-même, pour qu'elle opérât, suivant le mot de Proudhon, sa scission définitive avec les autres éléments sociaux.
Et encore la deuxième période, celle qui va de février 1848 à mai 1871, du Gouvernement provisoire à la répression sanglante de la Commune, est-elle trouble et incertaine. Déjà, il est vrai, le socialisme s'affirme comme une force et comme une idée ; le prolétariat s'affirme comme une classe. La révolution ouvrière se dresse si menaçante contre l'ordre bourgeois que les classes dirigeantes coalisent contre elle toutes les puissances de la bourgeoisie et les propriétaires paysans affolés par le spectre rouge13. Mais il y a encore indécision et confusion dans les doctrines socialistes : en 1848, le communisme de Cabet, le mutuellisme de Proudhon, l'étatisme de Louis Blanc se heurtent désespérément, et le moule de pensée où doit prendre forme la classe ouvrière est inconsistant et inachevé ; les théoriciens se disputent le métal en fusion qui sort de la fournaise, et pendant qu'ils se querellent, la réaction, conduite par l'homme de Décembre14, brise tous les moules ébauchés et refroidit le métal. Sous la Commune même, blanquistes, marxistes, proudhoniens impriment à la pensée ouvrière des directions divergentes : nul ne peut dire quel idéal socialiste eût appliqué la Commune victorieuse.
En outre, il y a trouble et mélange dans le mouvement même comme dans la pensée. En 1848, la Révolution est préparée par la démocratie radicale des petits bourgeois autant et plus peut-être que par le socialisme ouvrier, et aux journées de Juin la démocratie bourgeoise couche sur le pavé ardent de Paris les prolétaires. En 1871 aussi, c'est d'un soulèvement de la bourgeoisie commerçante irritée par la loi des échéances et par la dureté des hobereaux de Versailles, c'est aussi de l'exaspération patriotique et des défiances républicaines de Paris que le mouvement de la Commune est sorti.
Le prolétariat socialiste n'a pas tardé à mettre sa marque révolutionnaire sur cette confusion et Marx a eu raison de dire, en ce sens, dans sa forte et systématique étude sur la Commune, que, par elle, la classe ouvrière a pour la première fois pris possession du pouvoir. C'est un fait nouveau et d'une incalculable portée ; mais le prolétariat a profité d'une sorte de surprise ; il était, dans la capitale isolée et surexcitée, la force la mieux organisée et la plus aiguë ; mais il n'était pas encore en état d'entraîner et d'assimiler la France ; celle-ci appartenait aux prêtres, aux grands propriétaires fonciers et à la bourgeoisie dont M. Thiers était le chef. La Commune a été comme une pointe rougie au feu qui se brise contre un gros bloc réfractaire. Mais, de 1848 à 1871, le progrès prolétarien est immense. En 1848, la participation du prolétariat au pouvoir est presque fictive : Louis Blanc et l'ouvrier Albert sont paralysés au Gouvernement provisoire ; et une bourgeoisie perfide organise contre eux la tricherie des ateliers nationaux. Les socialistes discutent platoniquement au Luxembourg, ils abdiquent et se résignent à n'être qu'une impuissante Académie. N'ayant pas la force d'agir, ils dissertent. Puis, quand la classe ouvrière trompée se soulève en juin, elle est écrasée avant d'avoir pu une minute toucher au pouvoir. En 1871, les fils des combattants de juin ont tenu le pouvoir ; ils l'ont exercé ; ils n'ont pas été l'émeute, ils ont été la Révolution.
Les prolétaires ainsi haussés au gouvernement ont pu en être précipités ; ils n'en ont pas moins donné aux nouvelles générations ouvrières un haut signal d'espérance qui a été compris. La Commune clôt la seconde période où le socialisme s'affirme comme une force de premier ordre, confuse encore et convulsive ; mais c'est bien elle, aussi, c'est bien la Commune qui a rendu possible la période nouvelle, celle où nous sommes tous engagés et où le socialisme procède méthodiquement à l'organisation totale de la classe ouvrière, à la conquête morale des paysans rassurés, au ralliement de la bourgeoisie intellectuelle désenchantée du pouvoir bourgeois, et à la prise de possession complète du pouvoir pour des formes nouvelles de propriété et d'idéal.
Maintenant la confusion n'est plus à craindre. Il y a dans la classe ouvrière et le parti socialiste unité de pensée. Malgré les chocs des groupes et les rivalités superficielles, toutes les forces prolétariennes sont unies, au fond, par une même doctrine et pour une même action. Si demain le prolétariat s'emparait du pouvoir tout entier, il en pourrait d'emblée faire un usage défini et décisif. Il y aurait à coup sûr des conflits de tendances. Les uns voudraient fortifier et pousser au plus haut l'action centrale de la communauté, les autres voudraient assurer aux groupes locaux des travailleurs la plus large autonomie possible. Pour régler les rapports nouveaux de la nation, des fédérations professionnelles, des communes, des groupes locaux, des individus, pour fonder à la fois la parfaite liberté individuelle et la solidarité sociale, pour donner forme juridique aux innombrables combinaisons de la propriété sociale et de l'action des individus, un immense effort de pensée sera nécessaire ; et dans cette complexité il y aura des désaccords. Mais, malgré tout, c'est un commun esprit qui meut aujourd'hui les socialistes, les prolétaires ; le socialisme n'est plus dispersé en sectes hostiles et impuissantes. Il est de plus en plus une grande unité vivante et qui multiplie les prises sur la vie. C'est de lui maintenant que toutes les grandes forces humaines, le travail, la pensée, la science, l'art, la religion même, entendue comme prise de possession de l'univers par l'humanité, attendent leur renouvellement et leur essor.
Comment, à travers quelles crises, par quels efforts des hommes et quelle évolution des choses le prolétariat a-t-il grandi jusqu'au rôle décisif qu'il va jouer demain ? C'est ce que nous tous, militants socialistes, nous nous proposons de raconter. Nous savons que les conditions économiques, la forme de la production et de la propriété sont le fond même de l'histoire. De même que pour la plupart des individus humains l'essentiel de la vie, c'est le métier, de même que le métier, qui est la forme économique de l'activité individuelle, détermine le plus souvent les habitudes, les pensées, les douleurs, les joies, les rêves même des hommes, de même, à chaque période de l'histoire, la structure économique de la société détermine les formes politiques, les mœurs sociales, et même la direction générale de la pensée. Aussi nous appliquerons-nous, à chaque période de ce récit, à découvrir les fondements économiques de la vie humaine. Nous tâcherons de suivre le mouvement de la propriété, et l'évolution même de la technique industrielle et agricole. Et, à grands traits, comme il convient dans un tableau forcément sommaire, nous marquerons l'influence de l'état économique sur les gouvernements, les littératures, les systèmes.
Mais nous n'oublions pas, Marx lui-même, trop souvent rapetissé par des interprètes étroits15, n'a jamais oublié que c'est sur des hommes qu'agissent les forces économiques. Or les hommes ont une diversité prodigieuse de passions et d'idées ; et la complication presque infinie de la vie humaine ne se laisse pas réduire brutalement, mécaniquement, à une formule économique. De plus, bien que l'homme vive avant tout de l'humanité, bien qu'il subisse surtout l'influence enveloppante et continue du milieu social, il vit aussi, par les sens et par l'esprit, dans un milieu plus vaste, qui est l'univers même.
Sans doute, la lumière même des étoiles les plus lointaines et les plus étrangères au système humain n'éveille, dans l'imagination du poète, que des rêves conformes à la sensibilité générale de son temps et au secret profond de la vie sociale, comme c'est de l'humidité cachée de la terre que le rayon de lune forme le brouillard léger qui flotte sur la prairie. En ce sens, même les vibrations stellaires, si hautes et si indifférentes qu'elles paraissent, sont harmonisées et appropriées par le système social et par les forces économiques qui le déterminent. Goethe, entrant un jour dans une manufacture, fut pris de dégoût pour ses vêtements qui exigeaient un si formidable appareil de production. Et pourtant, sans ce premier essor industriel de la bourgeoisie allemande, le vieux monde germanique, somnolent et morcelé, n'aurait pu ni éprouver ni comprendre ces magnifiques impatiences de vie qui font éclater l'âme de Faust.
Mais quel que soit le rapport de l'âme humaine, en ses rêves même les plus audacieux ou les plus subtils, avec le système économique et social, elle va au-delà du milieu humain, dans l'immense milieu cosmique. Et le contact de l'univers fait vibrer en elle des forces mystérieuses et profondes, forces de l'éternelle vie mouvante qui précéda les sociétés humaines et qui les dépassera. Donc, autant il serait vain et faux de nier la dépendance de la pensée et du rêve même à l'égard du système économique et des formes précises de la production, autant il serait puéril et grossier d'expliquer sommairement le mouvement de la pensée humaine par la seule évolution des forces économiques. Très souvent l'esprit de l'homme s'appuie sur le système social pour le dépasser et lui résister ; entre l'esprit individuel et le pouvoir social il y a ainsi tout à la fois solidarité et conflit. C'est le système des nations et des monarchies modernes, à demi émancipées de l'Église, qui a permis la libre science des Kepler et des Galilée ; mais une fois en possession de la vérité, l'esprit ne relève plus ni du prince, ni de la société, ni de l'humanité ; c'est la vérité elle-même, avec son ordonnance et son enchaînement, qui devient, si je puis dire, le milieu immédiat de l'esprit, et, bien que Kepler et Galilée aient appuyé leurs observations et leurs travaux d'astronomes aux fondements de l'État moderne, ils ne relevaient plus, après leurs observations ou leurs calculs, que d'eux-mêmes et de l'univers. Le monde social, où ils avaient pris leur point d'appui et leur élan, s'ouvrait, et leur pensée ne connaissait plus d'autres lois que les lois mêmes de l'immensité sidérale.
Il nous plaira, à travers l'évolution à demi mécanique des formes économiques et sociales, de faire sentir toujours cette haute dignité de l'esprit libre, affranchi de l'humanité elle-même par l'éternel univers. Les plus intransigeants des théoriciens marxistes ne sauraient nous le reprocher. Marx, en une page admirable, a déclaré que jusqu'ici les sociétés humaines n'avaient été gouvernées que par la fatalité, par l'aveugle mouvement des formes économiques ; les institutions, les idées n'ont pas été l'œuvre consciente de l'homme libre, mais le reflet de l'inconsciente vie sociale dans le cerveau humain. Nous ne sommes encore, selon Marx, que dans la préhistoire. L'histoire humaine ne commencera véritablement que lorsque l'homme, échappant enfin à la tyrannie des forces inconscientes, gouvernera par sa raison et sa volonté la production elle-même. Alors, son esprit ne subira plus le despotisme des formes économiques créées et dirigées par lui, et c'est d'un regard libre et immédiat qu'il contemplera l'univers. Marx entrevoit donc une période de pleine liberté intellectuelle où la pensée humaine, n'étant plus déformée par les servitudes économiques, ne déformera pas le monde. Mais, à coup sûr, Marx ne conteste pas que déjà, dans les ténèbres de la période inconsciente, de hauts esprits se soient élevés à la liberté ; par eux l'humanité se prépare et s'annonce. C'est à nous de recueillir ces premières manifestations de la vie de l'esprit : elles nous permettent de pressentir la grande vie ardente et libre de l'humanité communiste qui, affranchie de tout servage, s'appropriera l'univers par la science, l'action et le rêve. C'est comme le premier frisson qui dans la forêt humaine n'émeut encore que quelques feuilles, mais qui annonce les grands souffles prochains et les vastes ébranlements.
Aussi notre interprétation de l'histoire sera-t-elle à la fois matérialiste avec Marx et mystique avec Michelet. C'est bien la vie économique qui a été le fond et le ressort de l'histoire humaine, mais, à travers la succession des formes sociales, l'homme, force pensante, aspire à la pleine vie de la pensée, à la communion ardente de l'esprit inquiet, avide d'unité, et du mystérieux univers. Le grand mystique d'Alexandrie16 disait : « Les hautes vagues de la mer ont soulevé ma barque et j'ai pu voir le soleil à l'instant même où il sortait des flots. » De même, les vastes flots montants de la révolution économique soulèveront la barque humaine pour que l'homme, pauvre pêcheur lassé d'un long travail nocturne, salue de plus haut la première lueur de l'esprit grandissant qui va se lever sur nous.
Et nous ne dédaignerons pas non plus, malgré notre interprétation économique des grands phénomènes humains, la valeur morale de l'histoire. Certes, nous savons que les beaux mots de liberté et d'humanité ont trop souvent couvert, depuis un siècle, un régime d'exploitation et d'oppression. La Révolution française a proclamé les Droits de l'Homme ; mais les classes possédantes ont compris sous ce mot les droits de la bourgeoisie et du capital.
Elles ont proclamé que les hommes étaient libres quand les possédants n'avaient sur les non-possédants d'autre moyen de domination que la propriété elle-même, mais la propriété, c'est la force souveraine qui dispose de toutes les autres. Le fond de la société bourgeoise est donc un monstrueux égoïsme de classe compliqué d'hypocrisie. Mais il y a eu des heures où la Révolution naissante confondait avec l'intérêt de la bourgeoisie révolutionnaire l'intérêt de l'humanité, et un enthousiasme humain vraiment admirable a plus d'une fois empli les cœurs. De même dans les innombrables conflits déchaînés par l'anarchie bourgeoise, dans les luttes des partis et des classes, ont abondé les exemples de fierté, de vaillance et de courage. Nous saluerons toujours, avec un égal respect, les héros de la volonté, en nous élevant au-dessus des mêlées sanglantes, nous glorifierons à la fois les républicains bourgeois proscrits en 1851 par le coup d'État triomphant et les admirables combattants prolétariens tombés en juin 1848.
Mais qui nous en voudra d'être surtout attentifs aux vertus militantes de ce prolétariat accablé qui, depuis un siècle, a si souvent donné sa vie pour un idéal encore obscur ? Ce n'est pas seulement par la force des choses que s'accomplira la révolution sociale ; c'est par la force des hommes, par l'énergie des consciences et des volontés. L'histoire ne dispensera jamais les hommes de la vaillance et de la noblesse individuelles. Et le niveau moral de la société communiste de demain sera marqué par la hauteur morale des consciences individuelles dans la classe militante d'aujourd'hui. Proposer en exemple tous les combattants héroïques, qui depuis un siècle ont eu la passion de l'idée et le sublime mépris de la mort, c'est donc faire œuvre révolutionnaire. Nous ne sourions pas des hommes de la Révolution qui lisaient les Vies de Plutarque ; à coup sûr les beaux élans d'énergie intérieure qu'ils suscitaient ainsi en eux changeaient peu de choses à la marche des événements. Mais, du moins, ils restaient debout dans la tempête, ils ne montraient pas sous l'éclair des grands orages des figures décomposées par la peur. Et si la passion de la gloire animait en eux la passion de la liberté, ou le courage du combat, nul n'osera leur en faire grief.
Ainsi nous essaierons dans cette histoire socialiste, qui va de la Révolution bourgeoise à la période préparatoire de la Révolution prolétarienne, de ne rien retrancher de ce qui fait la vie humaine. Nous tâcherons de comprendre et de traduire l'évolution économique fondamentale qui gouverne les sociétés, l'ardente aspiration de l'esprit vers la vérité totale, et la noble exaltation de la conscience individuelle défiant la souffrance, la tyrannie et la mort. C'est en poussant à bout le mouvement économique que le prolétariat s'affranchira et deviendra l'humanité. Il faut donc qu'il prenne une conscience nette, dans l'histoire, et du mouvement économique et de la grandeur humaine. Au risque de surprendre un moment nos lecteurs par le disparate de ces grands noms, c'est sous la triple inspiration de Marx, de Michelet et de Plutarque que nous voudrions écrire cette modeste histoire, où chacun des militants qui y collaborent mettra sa nuance de pensée, où tous mettront la même doctrine essentielle et la même foi.
Histoire socialiste de la France contemporaine,
Introduction (1900), texte intégral.
« Le bilan social du XIXe siècle »
Dans ce texte de conclusion de l'Histoire socialiste de la France contemporaine, Jaurès fait preuve d'un très grand optimisme, en relevant les progrès réalisés depuis un siècle. L'unification de la SFIO en 1905, son influence grandissante dans le parti, notamment à l'occasion du congrès de Toulouse en 1908, le regain des mobilisations ouvrières (malgré la grande fermeté répressive de Clemenceau), expliquent cet entrain, que l'on retrouve dans d'autres textes de personnalités du mouvement ouvrier international de la même époque. Nombreux en effet étaient alors les militants convaincus de l'imminence d'un bouleversement politique qui mènerait au socialisme...
Arrivés au terme de notre long et sérieux effort, nous ne nous dissimulons pas, mes collaborateurs et moi, ce que notre œuvre a nécessairement d'insuffisant et d'incomplet : mais elle marque une direction nouvelle dans les recherches d'histoire. Le souci de démêler la trame économique des faits, l'origine profonde des événements politiques et sociaux, y éclate à chaque ligne. Pour pouvoir atteindre avec quelque certitude le mouvement même de la réalité sociale, pour surprendre dans la continuité, dans la familiarité quotidienne de la vie des hommes le secret des grandes crises qui se produisent par intervalles à la surface, il faudrait, comme Andler l'a indiqué dans sa substantielle introduction à l'étude d'Albert Thomas sur le Second Empire, un immense effort de recherche collective17. Il faudrait notamment, pour comprendre à fond le mouvement, l'action, la vie, de la classe ouvrière dans la société issue de la Révolution française, pouvoir suivre à la trace, année par année, presque jour par jour, les événements de tout ordre, surtout d'ordre économique, qui ont retenti dans l'existence des prolétaires et dans leur pensée : découvertes de la science, variations de la technique, transformations de l'outillage industriel, élargissement des marchés, fluctuation des prix ; prix des marchandises, prix du travail. Or, cette enquête suppose l'effort concerté de plusieurs générations de chercheurs. Nous n'avons pu que l'amorcer et faire apparaître, par quelques résultats partiels, l'importance de la méthode fondamentale qui soutient tout notre travail. Déjà l'immense recherche collective qui seule désormais peut éclairer l'histoire commence à s'organiser. La Société pour l'étude des documents économiques de la Révolution française a publié quelques volumes considérables, et j'ai constaté avec joie que ces publications confirmaient les vues générales que j'ai émises18. Mais que de précisions pourront être apportées ! Que de filons nouveaux seront découverts par ces forages ! Pour la révolution de 1848, des recherches du même ordre ont été ébauchées19. Ce sera une des fonctions des Bourses du travail20, quand elles seront devenues vraiment les organes de la conscience ouvrière, de tenir à jour la statistique vraie, réelle, vivante, de l'existence prolétarienne et de provoquer des recherches qui, des faits présents, sérieusement analysés et constatés, remonteront au passé et permettront ensuite, par un mouvement inverse, de redescendre le cours de l'évolution. Alors, vraiment, l'histoire sera bien la conscience des grands groupements humains. Elle ne sera plus une sorte de clarté partielle et partiale concentrée sur quelques personnages privilégiés ; c'est toute l'immense multitude des hommes qui entrera enfin dans la lumière : et le vrai dieu de l'histoire, le travail, sombre forgeron qui a forgé dans sa caverne obscure les destinées humaines, pareil à Vulcain bafoué et enseveli qui forgeait les armes des dieux d'en haut, montera au jour et manifestera sa force créatrice dans le rayonnement de la science et la gloire de l'esprit. Notre œuvre ne sera pas vaine si elle peut, par un commencement de lumière, donner le goût et le besoin d'une plus vaste clarté.
Il serait contradictoire, après l'aveu que j'ai fait, de prétendre déduire de cette première enquête d'histoire socialiste que nous soumettons au prolétariat, des conclusions trop formelles et trop impérieuses. On peut dire pourtant que c'est une grande leçon d'action et d'espérance qui se dégage pour la classe ouvrière des faits que nous venons d'exposer. Oui, d'espérance.
Certes le prolétariat est bien loin du but qu'il se propose. L'injustice essentielle n'est point abolie. Le monopole de fait de la propriété subsiste, et la domination économique de la classe capitaliste a pour effet d'abaisser et d'exploiter l'immense multitude des hommes qui ne possèdent que leur force de travail. La Bruyère disait : « Devant certaines misères on éprouve de la honte à être heureux. » Devant les iniquités, les souffrances qui tourmentent la société d'aujourd'hui et accablent la classe ouvrière, il y aurait une sorte d'impudence à étaler, dans le jugement d'ensemble porté sur l'évolution française depuis la Révolution, une sorte d'optimisme béat et satisfait. Mais il y a un optimisme vaillant et âpre qui ne se dissimule rien de l'effort qui reste à accomplir, mais qui trouve dans les premiers résultats péniblement et douloureusement conquis de nouvelles raisons d'agir, de combattre, de porter plus haut et plus loin la bataille.
En fait, la Révolution française a abouti. Ce qu'il y avait en elle de plus hardi et de plus généreux a triomphé. Deux traits caractérisent le mouvement politique et social de la France depuis 1789 jusqu'au commencement du XXe siècle. C'est d'abord l'avènement de la pleine démocratie politique. Tous les compromis monarchiques ont été balayés ; toutes les combinaisons de monarchie traditionnelle et de souveraineté ont été écartées ; toutes les contrefaçons césariennes ont été rejetées. La Constitution mixte de 1791 a sombré dans l'imbécillité et dans la trahison royales. La monarchie restaurée de 1815 s'est perdue dans son étroitesse d'esprit. La monarchie censitaire de 1830 a révélé l'incapacité de la bourgeoisie française à gouverner seule, parce qu'elle ne peut se défendre contre les forces subsistantes du passé sans faire appel aux forces de l'avenir. Deux fois la démocratie napoléonienne a été engloutie dans le désastre, et maintenant, sous la forme républicaine, c'est bien le peuple qui gouverne par le suffrage universel. Il dépend de lui de conquérir le pouvoir. Ou plutôt il l'a déjà conquis, puisque aucune force ne peut faire échec à sa volonté légalement exprimée. Mais il ne sait pas encore en faire usage. Il ne sait pas l'employer vigoureusement à sa pleine émancipation économique. Les millions de travailleurs, ouvriers ou paysans, ne sont plus théoriquement des citoyens passifs. Ils le sont restés trop souvent encore par la résignation aux vieilles servitudes, par l'indifférence à l'idée nouvelle qui les affranchira. Mais c'est déjà chose immense qu'il suffise d'un progrès d'éducation du prolétariat pour que sa souveraineté formelle devienne une souveraineté substantielle.
Aussi bien, dans l'ordre de l'enseignement aussi, le progrès est grand depuis un siècle. Tous les enfants de la nation sont appelés à l'école : le grand idéal de Condorcet est réalisé ou en voie de réalisation. Et ce n'est plus l'Église, complice des tyrannies sociales, qui domine l'éducation et façonne le peuple. Elle a été réduite à n'être plus qu'une association privée ; et c'est la science, c'est la raison qui animent l'enseignement public. C'est la grande lumière de l'Encyclopédie, mais plus large et plus ardente, qui emplit l'horizon. La pensée socialiste, héritière des audaces extrêmes du XVIIIe siècle, commence à pénétrer les instituteurs de la nation.
De même, au point de vue social et dans cette portion même de la démocratie française qui n'a pas encore adhéré au socialisme explicite, c'est une conception bourgeoise encore, mais déjà sociale, de la propriété qui a prévalu. Elle n'est pas, comme les Constituantes, la condition de la souveraineté politique : l'homme le plus pauvre, le plus dénué, est politiquement l'égal du plus riche. Elle n'est pas non plus intangible. En demandant au Capital, par un impôt progressif sur les successions, une part croissante des ressources publiques, en proclamant que l'État a le droit et le devoir d'imposer aux possédants des contributions pour assurer les non-possédants contre les risques naturels et sociaux, le radicalisme français subordonne théoriquement le droit de propriété au droit supérieur de la nation ; il reprend à son compte le mot de Robespierre définissant la propriété : la portion de ses biens garantie au citoyen par la loi. Et il se peut que le radicalisme, après avoir accepté cette formule, hésite à l'appliquer hardiment et pleinement. Il se peut qu'il redoute que cette formule, maniée par un prolétariat vigoureux et fort, et appliquée à une société où la puissance économique est concentrée à nouveau dans une oligarchie, ne conduise par degrés à la socialisation générale de la société capitaliste. Cette défaillance du radicalisme gouvernemental, si elle se produit, n'empêchera point l'effet de l'idée qui s'est développée dans la démocratie française.
C'est le socialisme lui-même qui se substituera alors au radicalisme dans la mise en œuvre de cette idée sociale de la propriété, et qui la poussera jusqu'à ses conséquences nécessaires. La brèche est ouverte par où il passera. C'est donc bien la formule la plus extrême, la plus logique, la plus démocratique de la Révolution française, qui, après un siècle de tâtonnements, de réactions, de rêves d'abord impuissants, de révolutions à demi manquées, est enfin entrée dans les faits. Ce que le génie révolutionnaire avait entrevu, affirmé, essayé, dans la fièvre et l'exaltation du combat est devenu la réalité normale et solide. On dirait une cime volcanique qui, après une série d'explosions, d'affaissements, de redressements, s'est enfin fixée à son niveau le plus élevé : elle est consolidée maintenant et élargie en un vaste plateau qui peut porter les assises de la grande cité nouvelle. Non, tous ceux qui ont lutté, souffert, espéré depuis un siècle, n'ont pas perdu leur effort ; leur souffrance n'a pas été vaine ; leur espérance n'a pas été décevante, et si le prolétariat peut se réjouir de cette victoire de la démocratie révolutionnaire, ce n'est pas seulement parce qu'elle lui permet d'espérer et de préparer une victoire plus décisive, mais parce que c'est lui, débile21 encore pourtant et incertain, qui a assuré le triomphe de la Révolution. C'est par lui qu'elle a été portée d'abord, comme en un jet de flammes, à ce niveau de 1793, d'où elle ne tarda pas à retomber, mais où sans cesse elle tendait à revenir. C'est lui qui a aidé, qui a obligé la bourgeoisie à en finir avec les prétentions renaissantes de l'Ancien Régime. C'est lui qui a arraché à la bourgeoisie son privilège étroit pour créer enfin une vaste démocratie politique qui évoluera en démocratie sociale. Qu'auraient fait durant tout le siècle les républicains sans les ouvriers ? À tous les moments de la lutte qui a préparé ou réalisé la démocratie politique, l'action du prolétariat est visible ; et ce sera, je crois, un des mérites de l'œuvre historique dont j'écris en ce moment les dernières lignes d'avoir éclairé ces traces.
C'est donc avec confiance que la classe ouvrière, qui a déjà fait dans le passé l'épreuve de sa force, peut aborder des luttes nouvelles. Entre l'oligarchie capitaliste et la démocratie socialiste, forme achevée de la démocratie républicaine, le combat à fond est engagé. C'est le privilège de la propriété qui sera vaincu. Mais pour réussir, il faut que le prolétariat comprenne bien les leçons de sa propre vie depuis un siècle. C'est par l'effort continu quotidien, c'est par la propagande incessante, qu'il a eu ses premiers succès. Si quelque chose ressort du récit que nous avons fait, c'est bien cette continuité profonde de la pensée et de l'action prolétariennes. Quand les hautes cimes ardentes et éclatantes s'éteignent, quand la révolution démocratique et populaire de 1793 et 1794 pâlit et s'éclipse, quand la généreuse révolution de 1848 est brutalement supprimée, on peut croire que la nuit est complète ; mais ceux qui regardent au fond des esprits, au fond des âmes, s'aperçoivent que dans la conscience ouvrière l'idéal survit secrètement, et à la moindre ouverture des événements, la lumière jaillit de nouveau. Grande leçon pour tous les gouvernements de répression, quels qu'ils soient, et de quelque nom qu'ils s'appellent. Grande leçon aussi et grand réconfort pour les combattants socialistes, car ils apprennent que l'effort obscur et constant de chacun se retrouve tout entier au jour des grandes crises. La persistance du communisme babouviste à travers toutes les persécutions et toutes les réactions, la persistance de la foi républicaine et de l'espérance ouvrière jusque sous le triomphe insolent du Second Empire, sont parmi les faits les plus remarquables de toute notre histoire. Quand l'énergie passionnée des consciences ouvrières a pu, sans aucun droit légal, sans aucune ressource d'organisation publique, sauver de l'oubli mortel et du désastre définitif la liberté et le socialisme, comment ne serait-elle point assurée de la victoire, disposant maintenant des moyens d'action multiples que le prolétariat a conquis ?
Le socialisme a grandi depuis un siècle, il est devenu une puissance par l'emploi simultané ou alternatif de deux méthodes en apparence contradictoires et que le libre génie des ouvriers a conciliées. Tantôt il s'est mêlé, avec le babouvisme, avec le blanquisme22, à tous les mouvements de la démocratie, à toutes les agitations du peuple. Tantôt, comme avec Fourier, avec Saint-Simon, avec quelques-uns des premiers ouvriers de l'Internationale, il a voulu isoler ou la pensée ou l'action du socialisme. Tantôt il considère que la conquête des libertés politiques est la condition préalable de l'avènement social des ouvriers ; et il concentre sur cette première tâche tout leur effort. Tantôt il les avertit de ne jamais détourner leurs vues et leur action de leur objet suprême et de leur idéal. À toutes les pages de l'Histoire socialiste, se retrouve ce conflit des tendances et des méthodes. Mais en fait, le prolétariat ne sacrifie jamais l'une à l'autre. Jamais il ne se désintéresse des événements confus et vastes où il peut essayer sa force et développer son action. Mais jusque dans cette impétuosité de mouvement, qui le jette dans toutes les batailles politiques et intellectuelles, il ne perd pas son intransigeance foncière ; il a le sens très vif que toute action ne vaut que comme un acheminement, comme un entraînement à la révolution de propriété ; que toute réforme ne vaut que comme un degré vers le but supérieur. Le grand problème tactique des jours présents, c'est de concilier en effet, non pas seulement d'instinct, mais délibérément ces deux méthodes également nécessaires. On peut dire que le marxisme fut à l'origine un essai de synthèse des deux tendances, puisqu'il invitait le prolétariat à participer à tous les mouvements de la démocratie, mais pour les faire tourner immédiatement à la victoire du communisme. Le même problème s'impose aujourd'hui à nous, mais en des circonstances différentes. Au temps du Manifeste communiste23, la révolution démocratique n'était pas accomplie en France ; elle n'était même pas ébauchée en Europe, et Marx pouvait croire que le prolétariat serait assez fort pour faire servir à ses propres fins les agitations prévues de la révolution démocratique bourgeoise.
Maintenant, c'est dans une démocratie puissamment constituée et qui évolue sous la loi suprême du suffrage universel, que la classe ouvrière se développe et agit. Les conditions de l'action ne sont plus celles que le marxisme avait prévues, mais la méthode essentielle du prolétariat doit bien être celle qu'il avait esquissée : la méthode complexe d'une classe à la fois très vivante et très âpre qui se mêle à tous les mouvements pour les ramener sans cesse à sa propre fin. Comment pourra-t-il se passionner à l'œuvre de la réforme, et la rattacher sans cesse à son idéal révolutionnaire ? Comment pourra-t-il contribuer au développement de la production et intervenir comme classe dans le fonctionnement de la vie capitaliste sans s'immobiliser dans ses cadres ? Il n'entre pas dans le dessein de l'Histoire socialiste de résoudre ces difficultés qui presseront demain et le Parti socialiste et la Confédération générale du travail24. Mais ce que cette histoire démontre, c'est que le socialisme a grandi dans la société née de la Révolution, parce qu'il a su tour à tour ou en même temps se répandre et se concentrer. Il a été tout ensemble dans la démocratie et au-dessus d'elle, et c'est la marque de la puissance vitale du prolétariat français qu'il n'ait pas succombé à la difficulté de cette tâche en apparence contradictoire mais qu'au contraire, il s'y soit fortifié.
Créer la démocratie en la dépassant a été, durant un grand siècle tourmenté et fécond, l'œuvre de la classe ouvrière. Diriger la démocratie en la dépassant et l'obliger enfin à se hausser au socialisme, ce sera sa grande œuvre de demain.
Histoire socialiste de la France contemporaine,
Conclusion (1908), texte intégral.
1 Pour lire le texte de son adversaire, on consultera Paul Lafargue, Paresse et révolution, Paris, Tallandier, 2009, p. 211-239.
2 Le 9 juillet 1894, Jaurès avait fait une conférence sur le matérialisme économique de Karl Marx devant les étudiants collectivistes, à l'hôtel des Sociétés savantes, à Paris.
3 Friedrich Engels est encore en vie au moment où Jaurès présente sa conférence. Il mourra l'année suivante, en 1895.
4 Jaurès cite souvent de mémoire les auteurs, ce qui rend très difficile la restitution originale des propos cités ; ici, il ne s'agit pas en réalité d'une citation de Marx, mais d'un résumé d'un chapitre de l'ouvrage d'Engels L'Anti-Dühring (1878).
5 Gabriel Deville (1854-1940), socialiste français, alors connu notamment pour son œuvre de vulgarisation du Capital de Marx (Abrégé du « Capital », 1884).
6 Vision incontestablement réductrice de l'œuvre de Marx, qui tient notamment à l'absence de disponibilité de nombreux textes du philosophe allemand.
7 Propos en écho aux convictions pacifistes de Jaurès ; d'ailleurs, dans L'Armée nouvelle (1911) il évoque Rousseau en termes proches.
8 Grotius (1583-1645), juriste et diplomate hollandais théoricien du droit naturel.
9 Jaurès fait allusion ici à sa propre expérience parlementaire et aux difficultés qu'il rencontra sur les questions législatives.
10 Jaurès relie nettement l'œuvre de Rousseau à la Révolution française ; les socialistes français, dans leur majorité, se considèrent alors comme des héritiers fidèles de l'un et de l'autre.
11 La « conjuration des Égaux » de Gracchus Babeuf (1796-1797) est alors considérée par les socialistes comme le « premier parti communiste agissant », pour reprendre l'expression de Marx.
12 François Guizot (1787-1874), grande figure libérale de la monarchie de Juillet (1830-1848).
13 L'insurrection ouvrière de juin 1848 et la Commune de Paris de mai-juin 1871 déclenchèrent en effet des réactions d'affolement dans les classes dominantes.
14 Allusion à Louis-Napoléon Bonaparte et à son coup d'État du 2 décembre 1851.
15 Allusion aux socialistes « marxistes » comme Jules Guesde.
16 Allusion à Philon d'Alexandrie (Ier siècle av. J.-C.), interprète juif de la Bible confrontant son interprétation à la philosophie grecque.
17 Charles Andler, socialiste français et germaniste à qui l'on doit notamment des traductions de Nietzsche, a publié une préface à la partie de l'Histoire socialiste consacrée au Second Empire, partie dont Albert Thomas (1878-1932) avait la charge.
18 À partir de 1903, et pendant près d'un siècle, une commission parlementaire, surnommée la « Commission Jaurès », publia de nombreux documents de l'époque révolutionnaire.
19 La Société d'histoire de la Révolution de 1848 est fondée en 1903.
20 À l'origine bureaux de placement pour les ouvriers, les Bourses du travail devinrent des lieux clefs du mouvement syndical en offrant aux travailleurs de nombreuses activités, notamment d'éducation populaire.
21 Voir partie I, note 4.
22 Voir partie II, note 20.
23 Allusion au Manifeste du parti communiste de Marx et Engels (1848).
24 Jaurès tente alors d'établir un dialogue entre la CGT à majorité syndicaliste révolutionnaire et le Parti socialiste (SFIO).
IV
L'inlassable internationaliste
et l'apôtre de la paix
Jaurès reste associé à son tragique assassinat du 31 juillet 1914, par un nationaliste exalté : il paya de sa vie son combat pour la paix. Ce dernier chapitre regroupe une série d'interventions et de textes relatifs à la situation internationale, qui illustre les multiples facettes du combat jaurésien pour la paix : la conviction que le système capitaliste amène la guerre, ou encore la dénonciation de la politique coloniale (notamment au Maroc, alors objet de toutes les attentions), mais non du colonialisme en tant que tel. D'autres textes relatifs à la situation internationale, notamment sur la révolution russe de 1905 qui avait ébloui Jaurès, permettent de saisir l'internationalisme vigoureux qui l'animait. Nous intégrons également un extrait de son principal ouvrage de la période, L'Armée nouvelle, où il propose une réforme en profondeur de l'armée française peu avant le déclenchement de la Grande Guerre.
Le capitalisme et la guerre
En 1893, Jaurès a été élu député sur la base du programme socialiste du Parti ouvrier. S'exprimant deux ans plus tard, à l'occasion du vote du budget, sur la question de l'armement de la nation, il prononce une célèbre intervention où il dénonce la société capitaliste qui « porte en elle la guerre, comme la nuée dormante porte l'orage ».
Jean Jaurès. – Messieurs, je remercie d'abord la Chambre d'avoir bien voulu, malgré la fatigue de ce très important mais long débat, nous permettre, à mes amis et à moi, de nous expliquer à cette tribune, et j'ai encore besoin de faire appel, non pas à une indulgence à laquelle nous n'avons aucun droit, mais à toute votre équité ; car lorsque nous venons ici, au nom de l'idée socialiste, discuter avec vous le budget présent de la guerre, nous nous heurtons à de graves et particulières difficultés.
Il y a en effet dissentiment entre nous, non seulement sur le mode de répartition de détail de notre budget, non seulement sur tel ou tel point particulier de l'organisation militaire, mais sur les conceptions sociales qui déterminent le principe même de cette organisation.
C'est dire, messieurs, qu'il y a entre nous – et cela crée une grande difficulté de discussion – un désaccord fondamental et irréductible.
De plus je rencontre d'emblée – et je ne peux pas ne pas les rencontrer – des problèmes troublants, poignants même, à propos desquels, depuis deux années, de vives polémiques sont dirigées contre nous : la guerre, le militarisme, l'idée que le prolétariat socialiste se fait de l'armée, de la patrie ; les rapports du socialisme français avec le socialisme international ; l'aspect que prennent pour nous, pour notre parti, ces questions territoriales dont notre ami Vaillant parlait tout à l'heure ; ces questions, si redoutables qu'elles soient, nous ne pouvons, nous ne devons pas les éluder. Elles sont posées devant le pays, elles doivent être posées devant le Parlement.
Et, messieurs, si vous suivez de près, comme vous l'avez fait assurément, les grandes discussions qui se produisent dans les parlements étrangers, vous verrez que partout, à Londres, à Rome, à Berlin, à Vienne, à Budapest, il est parlé de toutes choses, et, en particulier, des choses de la France, avec une entière liberté ; il y est parlé de notre pays, de nos gouvernants, de nos chefs d'État, de nos institutions, du mouvement de notre politique intérieure, de la répercussion que cette politique intérieure de la France peut avoir sur l'ensemble des affaires européennes. Il me semble que la même liberté de discussion fondamentale doit être assurée ici. (Très bien ! très bien ! à l'extrême gauche.)
Et si quelques-uns nous opposent, dans un scrupule de prudence patriotique, qui ne nous est pas étranger, croyez-le bien, que ce sont les douloureux souvenirs d'il y a vingt-cinq ans qui nous conseillent à tous ou cet excès de réserve ou cet excès de susceptibilité, je répondrai très nettement que notre pays, dans les épreuves qu'il a traversées, a peut-être perdu pour un moment quelque chose de sa substance, mais qu'il n'a rien perdu, rien laissé de sa puissance réelle, de sa fierté, de son droit plein à la liberté et à la vie. (Applaudissements.)
Messieurs, vous voulez la paix ; vous la voulez profondément. Toutes les classes dirigeantes de l'Europe, les gouvernements et les peuples la veulent aussi visiblement avec une égale sincérité.
Un membre à gauche. – Ou inégale ! (Mouvements divers.)
Jean Jaurès. – Et pourtant, dans cet immense et commun amour de la paix, les budgets de la guerre s'enflent et montent partout d'année en année, et la guerre, maudite de tous, redoutée de tous, réprouvée de tous, peut, à tout moment, éclater sur tous.
D'où vient cela ? Au risque de vous paraître affligé de la plus cruelle monotonie, je dois dire ici tout d'abord quelle est, selon nous, la raison profonde de cette contradiction, de ce perpétuel péril de guerre au milieu de l'universel désir de la paix. (Mouvements divers.) Tant que, dans chaque nation, une classe restreinte d'hommes possédera les grands moyens de production et d'échange, tant qu'elle possédera ainsi et gouvernera les autres hommes, tant que cette classe pourra imposer aux sociétés qu'elle domine sa propre loi, qui est la concurrence illimitée, la lutte incessante pour la vie, le combat quotidien pour la fortune et pour le pouvoir ; tant que cette classe privilégiée, pour se préserver contre tous les sursauts possibles de la masse, s'appuiera ou sur les grandes dynasties militaires ou sur certaines armées de métier des républiques oligarchiques, tant que le césarisme pourra profiter de cette rivalité profonde des classes pour les duper et les dominer l'une par l'autre (applaudissements à l'extrême gauche), écrasant au moyen du peuple aigri les libertés parlementaires de la bourgeoisie, écrasant au moyen de la bourgeoisie gorgée d'affaires le réveil républicain du peuple ; tant que cela sera, toujours cette guerre politique, économique et sociale des classes entre elles, des individus entre eux, dans chaque nation, suscitera les guerres armées entre les peuples. (Très bien ! très bien ! sur divers bancs.) C'est de la division profonde des classes et des intérêts dans chaque pays que sortent les conflits entre les nations.
Depuis un siècle, c'est la Pologne aboutissant par la guerre de ses paysans et de ses nobles à la guerre étrangère1. C'est la France révolutionnaire jetant un défi à l'Europe, d'abord pour répondre aux trahisons sourdes des privilégiés, puis pour mieux se débarrasser, dans une crise exaspérée, de ses nobles et de ses rois ; c'est l'aristocratie anglaise armant contre nous, pour sauver ses privilèges et ses grands fiefs coloniaux, toutes les forces monarchiques et féodales de l'Europe. C'est Napoléon, continuant et amplifiant la guerre au-delà du nécessaire et du juste par instinct, je le veux bien, par habitude peut-être et par ennui, mais aussi pour continuer son absolutisme qui se serait forcément détendu dans la paix, et pour faire du peuple idéaliste et violent de la Révolution une démocratie héroïque et subalterne.
Plus tard, ce sont les aventuriers de Décembre2, jouant d'une main tremblante et vieillie la France sur une suprême partie de dés ; et c'est la noblesse militaire de Prusse conduite par les Hohenzollern, imposant à l'Allemagne, par sa victoire sur la France, sa domination politique et sociale et imprégnant de militarisme féodal l'unité allemande qui aurait pu s'accomplir par d'autres voies.
Partout ce sont ces grandes compétitions coloniales où apparaît à nu le principe même des grandes guerres entre les peuples européens, puisqu'il suffit incessamment de la rivalité déréglée de deux comptoirs ou de deux groupes de marchands pour menacer peut-être la paix de l'Europe.
Et alors, comment voulez-vous que la guerre entre les peuples ne soit pas tous les jours sur le point d'éclater ? Comment voulez-vous qu'elle ne soit pas toujours possible, lorsque dans nos sociétés livrées au désordre infini de la concurrence, aux antagonismes de classes et à ces luttes politiques, qui ne sont bien souvent que le déguisement des luttes sociales, la vie humaine elle-même en son fond n'est que guerre et combat ?
Ceux qui de bonne foi s'imaginent vouloir la paix, lorsqu'ils défendent contre nous la société présente, lorsqu'ils la glorifient contre nous, ce qu'ils défendent en réalité, sans le vouloir et sans le savoir, c'est la possibilité permanente de la guerre ; c'est en même temps le militarisme lui-même, qu'ils veulent prolonger.
Car cette société tourmentée, pour se défendre contre les inquiétudes qui lui viennent sans cesse de son propre fonds, est obligée perpétuellement d'épaissir la cuirasse contre la cuirasse ; dans ce siècle de concurrence sans limite et de surproduction, il y a aussi concurrence entre les armées et surproduction militaire. L'industrie elle-même étant un combat, la guerre devient la première, la plus excitée, la plus fiévreuse des industries. (Très bien ! très bien ! à l'extrême gauche.)
Et il ne suffit pas aux nations de s'épuiser ainsi à entretenir les unes contre les autres des forces armées ; il faut encore – et ici je demande la permission de dire nettement ma pensée – que les classes privilégiées, possédantes de tous les pays, isolent le plus possible cette armée, par l'encasernement et par la discipline de l'obéissance passive, de la libre vie des démocraties. (Applaudissements à l'extrême gauche. – Interruptions au centre.)
On ne nous a pas caché depuis vingt ans que c'était là aujourd'hui, en Europe, la conception des armées de métier. L'Assemblée nationale acclamait l'illustre rapporteur de la loi militaire3 disant : « Quand on parle d'armée, il ne faut plus parler de démocratie » ; et elle couvrait de huées le défenseur de Belfort, Denfert-Rochereau, réclamant contre le dogme de l'obéissance passive. (Très bien ! très bien ! à l'extrême gauche.)
Et au moment même où, de l'autre côté de la frontière, un empereur d'armée disait récemment à ses soldats qu'il avait désormais besoin surtout de leur fidélité contre l'ennemi intérieur et qu'ils devaient être prêts à tirer, sans hésitation et sans faiblesse, sur leurs pères et sur leurs frères enrôlés par la démocratie sociale4, à ce moment même ou quelques jours après, dans cette discussion, on nous signifiait – et je remercie l'orateur du parti conservateur, comme je l'en ai remercié l'autre jour, de sa sincérité et de sa franchise5 – on nous signifiait que l'armée était la grande sauvegarde au-dehors et au-dedans. Nous avons compris ce que cela voulait dire.
Et voilà comment, messieurs, vous aboutissez à cette double contradiction : d'une part, tandis que tous les peuples et tous les gouvernements veulent la paix, malgré tous les congrès de philanthropie internationale, la guerre peut naître toujours d'un hasard toujours possible ; et d'autre part, alors que s'est développé partout l'esprit de démocratie et de liberté, se développent aussi les grands organismes militaires, qui, au jugement des penseurs républicains qui ont fait notre doctrine, sont toujours un péril chronique pour la liberté des démocraties. Toujours votre société violente et chaotique, même quand elle veut la paix, même quand elle est à l'état d'apparent repos, porte en elle la guerre, comme la nuée dormante porte l'orage. (Très bien ! très bien ! à l'extrême gauche.)
Intervention à la Chambre des députés, 7 mars 1895,
extrait.
Contre la politique coloniale
Au moment où l'intervention française au Maroc prend une tournure militaire de plus en plus affirmée, un débat s'engage à la Chambre des députés. À ce stade, Jaurès critique les excès des militaires et la politique coloniale, mais sans aller jusqu'à remettre en question les intérêts de la France en Afrique du Nord. Conviction qui peut apparaître très modérée au regard de l'histoire ultérieure, mais qui, à l'époque, était un engagement fort face à nombre de républicains convaincus du bien-fondé de toute la politique coloniale.
Jean Jaurès. – Messieurs, je ne veux pas toucher aux questions générales de politique extérieure qui ont déjà été si amplement développées ici ; je veux seulement parler en quelques mots de la question du Maroc.
Il y a quelques semaines, j'ai sonné l'alarme au sujet des tentatives ou des prétentions de quelques groupements coloniaux et militaires qui me paraissent impatients de déterminer une expédition au Maroc. Je crois qu'il y avait en effet péril.
Ce n'est pas, comme l'a dit tout à l'heure M. de Castellane, qu'une résolution gouvernementale très ferme eût été conçue dans ce sens. Ce qui me paraissait grave, ce qui me paraissait inquiétant, c'est, au contraire, que quelques chefs militaires peut-être médiocrement disciplinés... (Réclamations au centre et à droite.)
Vraiment il est extraordinaire que nous ne puissions même pas ici poser les problèmes avec clarté. (Très bien ! très bien ! à l'extrême gauche.)
Il y a eu en ces questions, dans la politique africaine, un précédent à mon avis redoutable et qui était de nature à encourager certaines imprudences et certaines tentatives d'initiative indisciplinée. Je ne porte aucun jugement sur l'expédition même qui a été faite dans les oasis sahariennes, sur la prise de possession du Touat6. Je n'en juge point les effets et les résultats, mais je suis bien obligé de constater qu'une expédition qui a coûté 50 millions, qui aujourd'hui même impose annuellement 10 millions de charges à notre budget, qui a mis la France en contact avec des populations nomades ou anarchiques, qui a créé entre nous et le Maroc des causes nouvelles de froissements et de rencontres, je suis bien obligé de constater que cette expédition n'a pas été délibérée ici, qu'elle n'a été ni voulue ni votée par le Parlement, que nous avons été réduits à consacrer et à légitimer après coup ce qui avait été le résultat peut-être accidentel, peut-être prémédité, d'entraînements militaires qui s'étaient transformés en expéditions définitives. (Applaudissements à l'extrême gauche et à gauche.)
Et pourquoi donc, messieurs, ne voulez-vous pas qu'une pareille tentation, qui avait abouti dans la question des oasis algériennes et du Sud-Oranais, n'ait pas pu renaître ? J'ai eu peur précisément que l'émotion causée par l'incident d'El-Moungar, par le massacre d'une de nos colonnes fût exploitée – et elle commençait à l'être – par ceux qui, au lieu de se borner aux mesures nécessaires de répression et de police voulaient, sous prétexte de poursuivre jusque dans son foyer marocain l'anarchie dont nous avions à souffrir, entamer, amorcer une expédition nouvelle.
Oh ! on ne l'aurait pas annoncé nettement et à grand fracas. Ce n'eût été d'abord qu'une opération de police, mais qui nous aurait conduits sur le territoire du sultan ; il y aurait eu rencontre et alors on aurait dit au Parlement : le drapeau est engagé ; nous ne pouvons plus reculer ! L'honneur national est en jeu, il exige que l'expédition continue ! (Applaudissements à l'extrême gauche et à gauche.)
Et c'est tout simplement pour prévenir ce danger que je l'ai signalé à temps. J'imagine avoir rendu par là, dans la mesure de mes forces, service à ceux qui veulent précisément d'une politique de sage pénétration au Maroc.
Si vous voulez que l'influence française y pénètre, qu'elle s'y étende, qu'elle s'y établisse, il faut que vous restiez, vous parlement, vous gouvernement responsable, maîtres de cette politique. Il faut que vous restiez libres d'en choisir le moment, d'en choisir le mode, d'en choisir la forme. (Nouveaux applaudissements.)
Ce n'est pas, j'ai hâte de le dire, une politique de pure négation et de pure défiance que j'apporte à cette tribune dans la question marocaine, ni une politique d'attente incertaine et inerte, qui s'en remettrait au hasard, à la merci des événements. Je suis convaincu que la France a, au Maroc, des intérêts de premier ordre ; je suis convaincu que ces intérêts mêmes lui créent une sorte de droit. Ce n'est pas impunément que les populations musulmanes ou marocaines pourraient être tournées, animées, groupées contre nous. Il y a entre toutes les parties du monde musulman une étroite solidarité qui s'est affirmée à bien des reprises et qui nous oblige à prolonger l'action morale de notre pays sur les parties mêmes des populations musulmanes qui ne sont pas directement incorporées à l'empire de la France.
Le danger apparaissait il y a vingt ans, au moment même où en Égypte, à une distance pourtant assez sérieuse, les progrès du mahdi7 éveillaient dans toute l'Afrique musulmane les ambitions et les convoitises. C'est de Foucauld, notre explorateur au Maroc, qui constatait à ce moment que les progrès du mahdi étaient amplifiés, que la légende le représentait déjà maître de Tunis, maître d'Alger. Les convoitises s'éveillaient, s'organisaient. Nous avons donc le droit de surveiller ces événements et de prendre des précautions contre cet ordre de dangers.
M. Gayraud. – Nous en avons aussi le devoir.
Jean Jaurès. – J'ajoute que la France a d'autant plus le droit de prolonger sur le Maroc son action économique et morale, qu'en dehors de toute surprise et de toute violence militaire, la civilisation qu'elle représente en Afrique auprès des indigènes est certainement supérieure à l'état présent du régime marocain.
Mes amis et moi ne sommes pas de ceux qui ont approuvé toujours la politique suivie à l'égard des indigènes ; nous en avons dénoncé les lacunes, ou les vices, ou les violences, ou les injustices ; mais du moins il y a un contrôle possible qui permet peu à peu de réparer, de redresser les iniquités et de réaliser au profit des populations musulmanes un régime de justice croissante. Et je suis obligé de constater – je le fais avec joie – que, malgré ses infirmités et ses vices, le régime français, en Algérie et en Tunisie, donne aux indigènes, aux musulmans, des garanties de sécurité, des garanties de bien-être, des moyens de développement infiniment supérieurs à ceux de ce régime marocain spoliateur, anarchique, violent, mauvais, qui absorbe et dévore toutes les ressources du pays, qui est secoué par les soubresauts d'un fanatisme morbide et bestial. Oui, il est à désirer, dans l'intérêt même des indigènes du Maroc comme dans l'intérêt de la France, que l'action économique et morale de notre pays s'y prolonge et s'y établisse. (Applaudissements sur un grand nombre de bancs.)
Mais comment le ferez-vous, messieurs ? Est-ce par la guerre ? Nul de vous ne la désire, nul de vous n'en veut. D'abord il faut bien reconnaître qu'elle serait une lourde et difficile entreprise. Les tribus sont désunies, mais elles sont armées, et il se pourrait bien que l'apparition de l'étranger en armes provoquât brusquement contre nous la réconciliation de ces tribus divisées.
Et puis, nous ne pourrions pas mener loin cette entreprise militaire sans être conduits peut-être à des opérations cruelles, sanglantes, dont l'effet se prolongerait en rancunes pendant des générations et des générations auprès des hommes mêmes que nous voudrions assimiler à notre nation. Et ce serait là le péril le plus redoutable qui pourrait naître pour nous de l'emploi de la force.
Enfin, messieurs, quel aveu d'impuissance si nous en étions réduits là ! On comprend que lorsque la France n'avait pas le pied en Algérie, on comprend que sous la Restauration, lorsque la France est allée châtier les pirates d'Alger, elle ait été obligée de recourir à la force pour pénétrer dans l'intérieur et pour s'installer ; mais maintenant que depuis trois quarts de siècle vous êtes établis là-bas, maintenant que vous y possédez une Algérie organisée, maintenant que vous pouvez agir sur le Maroc voisin par toutes les ressources morales, politiques et économiques dont dispose un grand pays comme la France, ce serait véritablement une faillite morale que d'être obligé d'avouer que vous n'avez que les armes comme moyen de pénétration. (Applaudissements à l'extrême gauche et à gauche.)
Intervention à la Chambre des députés,
20 novembre 1903.
La révolution européenne
Le 9 janvier 1905, une manifestation pacifique à Saint-Pétersbourg est réprimée par la troupe. C'est le début de la première révolution russe, souvent occultée au profit de celle de 1917, mais qui en son temps déclencha un immense enthousiasme dans toute l'Europe, notamment dans les rangs socialistes. Toute la Russie est touchée par une importante vague de grèves, que commente ici Jaurès.
Ce que demandent les ouvriers russes soulevés d'un bout à l'autre de l'Empire, c'est l'institution de la souveraineté nationale ; c'est une Constitution qui procède du droit égal de tous les citoyens. Exclus du droit de suffrage, ils se révoltent contre un régime qui les oblige à prouver par la force qu'ils ne sont pas quantité négligeable et néant. Déjà, ils bloquent Pétersbourg et Moscou, coupés presque de toute communication avec le dehors, par une grève générale des chemins de fer qui est le prélude d'une grève totale, appliquée à toute l'industrie. Là-dessus, nos journaux conservateurs et modérés, de L'Écho de Paris au Temps et aux Débats, se scandalisent et s'indignent. Que les ouvriers usent de la grève pour des revendications économiques, on daigne le leur concéder, sauf à contrarier le plus possible, par des violences de police, l'exercice de ce droit. Mais décréter la grève pour conquérir le suffrage universel, pour imposer la convocation d'une constituante, c'est un scandale et un attentat.
Mais quoi ! et par quels moyens, ces messieurs veulent-ils donc que les ouvriers russes protestent ? Est-ce le bulletin de vote ? Le tsarisme vient de le refuser à l'heure même où, pour la création d'une Douma d'Empire, il faisait semblant d'accorder à la nation quelques garanties. L'exclusion dont la classe ouvrière est frappée est d'autant plus outrageante que le pouvoir ne peut plus se tenir au vieux régime de bureaucratie incontrôlée. Il n'est rien de plus logique que leur réponse. À un pouvoir qui les considère comme néant, ils répliquent en se réduisant, en effet, à rien. Puisqu'on ne daigne pas les admettre, pour leur part, à l'exercice de la souveraineté nationale, c'est sans doute qu'ils sont inutiles et inexistants ; pourquoi donc continueraient-ils à travailler et à produire ? Puisque leur travail, qui assure la vie sociale, n'est pas un titre suffisant à la liberté, au pouvoir politique, ils cessent le travail. Que la société fonctionne sans eux, puisqu'il paraît qu'on peut, en effet, se passer d'eux. Et la grève générale, par laquelle ils anéantissent eux-mêmes leur puissance économique, est la réplique admirable et appropriée à l'acte du tsarisme insensé qui leur refuse toute puissance politique. Le Temps et les Débats ont-ils donc oublié l'histoire révolutionnaire de la France ? Dans la tournée oratoire du Mirabeau en Provence, quand il lançait des écoles des foudres contre l'orgueil des nobles et l'arbitraire des rois, il s'écriait : « Prenez garde de ne pas réduire au désespoir ce peuple qui produit tout et qui, pour être formidable, n'aurait qu'à être immobile. » C'était la menace de la grève générale, faite au nom du tiers état. Et si la Révolution n'avait pas trouvé sa voie, si la monarchie n'avait pas appelé toute la nation à former l'Assemblée constituante, si elle n'avait pas accordé au tiers ce doublement qui lui assurait le pouvoir, c'est peut-être par l'arrêt général de la production et des échanges que le tiers aurait brisé l'obstination royale, et l'orgueil des privilégiés.
Visiblement, c'est un plan révolutionnaire très délibéré qui s'accomplit en Russie. Les socialistes, longtemps divisés sur la tactique8, se sont réunis, ils ont abouti à des décisions communes. Sans doute, ils ont résolu d'employer toutes les formes de l'action, c'est-à-dire d'essayer de porter quelques-uns des leurs à la Douma, quelle qu'elle soit, afin d'y faire entendre la protestation du peuple, mais de peser d'abord sur le pouvoir par un grand mouvement révolutionnaire pour lui arracher une Constitution démocratique et la mise en œuvre du suffrage universel. Le mot d'ordre évident a été de ne compliquer la grève générale d'aucune violence inutile. C'est seulement la force de travail qui se retire et toute la vie sociale s'effondre ou fléchit.
Le mouvement ouvrier a d'autant plus de chances d'aboutir que toutes les sympathies secrètes ou déclarées de ceux qui veulent la liberté vont à lui. Ce n'est plus seulement un mouvement de classe : c'est le vaste mouvement national de tous ceux qui ont besoin d'être libres et qui ne trouvent qu'en la force de la classe ouvrière un moyen décisif de revendication. Les ingénieurs se solidarisent avec eux, et le patronat lui-même est obligé de souhaiter que la victoire des prolétaires sur un régime déplorable permette aux inévitables conflits du capital et du travail, de prendre une forme moins chaotique et moins trouble. L'imbécillité du tsarisme, refusant au peuple russe un régime de garantie universelle, dont la bourgeoisie industrielle a besoin comme le prolétariat, a coalisé contre la bureaucratie toutes les forces vives du monde moderne. Les faux sages peuvent railler ce qu'ils appellent l'inexpérience politique du peuple russe. Le pouvoir s'est chargé de faire rapidement son éducation et jamais le mot de Proudhon ne fut plus applicable : « la sottise des gouvernements fait la science des révolutionnaires. » Il n'y eut jamais, à ce compte, science supérieure à celle de la Révolution russe.
Ce nouveau soulèvement, plus vaste, plus méthodique que les soulèvements antérieurs, semble calculé pour démoraliser toutes les forces de résistance. À l'heure où le tsar, épuisé, à bout de souffle, espérant se remettre un peu des émotions et des fatigues morales de la guerre de Mandchourie9, quand il s'imaginait, par la piètre invention de sa contrefaçon de Douma, avoir amorti tout l'élan révolutionnaire, quand il espérait que la flamme des colères allait s'éteindre sous une vaste indifférence, comme un feu perdu sous la neige qui tombe, la Révolution se redresse, plus impérieuse et plus ardente que jamais, et elle lui demande compte des hypocrisies qu'il a ajoutées à ses violences. Où se réfugiera-t-il ? et l'intolérable charlatan qui donnait des airs de victoire à la définitive défaite rapportée par lui de Portsmouth suffira-t-il avec ses hâbleries à prévenir le désastre ?
Or, pendant qu'une lutte révolutionnaire pour le suffrage universel est engagée en Russie, une lutte révolutionnaire s'ouvre pour le suffrage universel en Autriche-Hongrie. L'empereur a compris enfin qu'il ne pouvait résister au nationalisme hongrois qu'en proposant à tous les peuples de Hongrie, aux groupes dominés par les Magyars comme aux Magyars eux-mêmes, le suffrage universel. Mais il voudrait bien le refuser aux peuples de l'Autriche. Calcul enfantin ! Car, dans cet ébranlement, le prolétariat de l'Autriche tout entière, Bohême et Autriche allemande, exige la plénitude du droit politique. Une agitation est commencée, la plus véhémente, la plus puissante qu'ait entreprise jusqu'ici la classe ouvrière autrichienne. Elle sera poussée, s'il le faut, jusqu'à la grève de masse, jusqu'à la révolution. Et il se trouve, par une admirable rencontre historique qui est la bonne fortune des classes ascendantes, qu'en Autriche, comme en Russie, le prolétariat, en se sauvant lui-même, sauve la nation. Car il n'y a que le suffrage universel qui puisse maintenir l'État autrichien sollicité par tant de forces divergentes. Lui seul sera le lien et le ciment.
Mais l'avènement de la démocratie en Russie, comme en Autriche, aura en Allemagne un contrecoup nécessaire. Comment l'Empire allemand pourra-t-il maintenir ce qui reste en lui d'absolutisme et de féodalisme, quand les grands peuples de l'Europe orientale, et celui-là même qui était plongé au plus profond de la servitude, auront affirmé enfin leur volonté libre ? C'est alors que l'Allemagne du Kaiser sera isolée, non pas par un plan d'agression des puissances hostiles, mais par la contradiction de son régime d'arbitraire et de caste à toute la démocratie européenne émancipée. Le peuple allemand, après la Révolution russe, après l'émancipation autrichienne, consentira-t-il à être le seul peuple de l'Europe livré à la volonté irresponsable d'un homme, qui ne semble pas très soucieux de ménager les nerfs de l'Europe ? Voici qu'au moment où semblait se produire une détente des esprits, son toast à la poudre sèche et à l'épée excite de nouveaux commentaires inquiets, les paniques vraies ou simulées. Le Temps fait écho à ces propos de moderne sauvagerie, et il les transpose de l'Allemagne à la France : « Nous connaissons la position de la France et de l'Europe. Par conséquent, hourrah pour la poudre sèche et l'épée aiguisée, pour le but toujours clair et les muscles toujours tendus. » Peut-être, la suite des événements de Russie inspirera-t-elle à l'empereur allemand des réflexions d'un autre ordre. Mais voilà où en est l'Europe. Voilà quelle vie inquiète et agitée lui sera faite. Voilà sous quel cauchemar de guerre et de violence elle vivra tant que la démocratie et la classe ouvrière n'y seront pas souveraines. C'est pourquoi nous suivons d'un cœur passionné la lutte héroïque et féconde du prolétariat russe.
L'Humanité, 28 octobre 1905, texte intégral.
« Discours du citoyen Jaurès »
Le 24 janvier 1908, Jean Jaurès intervient à la Chambre dans un débat sur le Maroc ; voici deux extraits de son discours publié dans L'Humanité, où l'on voit Jaurès prendre encore davantage de distance par rapport à la politique coloniale : les dénonciations se font plus vives et plus régulières que les années précédentes.
[...] Voilà bien longtemps que, mes amis et moi, nous nous obstinons à dire, à écrire, à répéter que votre intervention au Maroc, sous la forme où elle se produit, ne pouvait avoir d'autre effet que d'exciter, que de grouper contre la France, et contre la France seule, toutes les haines marocaines, d'ameuter contre vous, contre vous seuls, la passion du fanatisme religieux et de l'indépendance nationale et d'achever de perdre le sultan déjà ébranlé et dont vous faites tous les jours aux yeux des Marocains l'instrument détesté de l'étranger, de l'envahisseur, des usuriers lointains. (Très bien ! très bien ! à l'extrême gauche.) Voilà ce que nous disions et les choses se sont réalisées. Le sultan avec lequel, après bien des incertitudes, la France a témérairement lié partie est aujourd'hui à peu près ruiné dans son autorité. Il n'a plus qu'une seule des deux capitales marocaines. Il a été exclu, le 19 août, de Marrakech, de la capitale du Sud. Il a été exclu, le 3 janvier, de Fez, de la capitale du Nord. Lorsque son agent, El Mokri est venu en France, il a été frappé par la nouvelle de la révolution de Fez, mais il a essayé de reprendre contenance et il a dit : Oui, mais le sultan n'est pas perdu, il a encore Medinez, il a encore Ouezzan.
Et voilà que Medinez a proclamé l'autre sultan, voilà que les autorités religieuses d'Ouezzan ont proclamé à leur tour Moulaï Hafid ; il ne reste plus au sultan avec lequel la France a partie liée que quelques ports de la côte, et probablement parce que la France l'impose par la force. Je n'ai pas à rechercher quelles sont les circonstances du mouvement qui s'est produit à Fez. On va prétendre que c'est la foule qui a forcé la main aux autorités religieuses, qu'il y a eu là une sorte d'émeute populaire.
Mais plus vous direz cela, plus vous constaterez qu'en ce Maroc, où les autorités religieuses ont une telle prise sur le peuple, la foule a forcé la main aux autorités religieuses elles-mêmes, parce qu'elle les jugerait trop circonspectes, plus vous direz cela, et plus vous constaterez que nous sommes au Maroc en face d'un mouvement vaste et redoutable. [...]
Ah ! messieurs, prenons-y garde et n'allons pas dans cette aventure. Ne renonçons pas au seul rôle qui puisse aujourd'hui convenir à la France. Depuis qu'après avoir connu les ivresses et les ébouissements de la force elle en a connu les déceptions et les meurtrissures, son idéal, c'est d'être dans le monde la grande ouvrière de la paix et du droit. Son devoir, son rôle, c'est de veiller à écarter, à prévenir tous les conflits qui peuvent menacer la paix du monde. Et si un jour les rivalités économiques exaspérées risquaient de mettre aux prises Angleterre et Allemagne, votre devoir est de porter devant l'Europe une parole de paix, de prévenir et d'écarter l'orage. (Très bien ! très bien ! à l'extrême gauche.)
Comment le pourrez-vous, si vous êtes engagés dans cette ténébreuse intrigue du Maroc, où tous les intérêts se heurtent, où toutes les convoitises se froissent ? Comment aurez-vous l'esprit libre et les mains libres si vous êtes engagés à fond et si vous n'avez pas pour vous l'évidence du droit ?
J'ajouterai : quelle figure, si vous êtes maintenant pour les Marocains le peuple de l'Europe qui exerce le plus violemment la force des armes, quelle figure ferez-vous devant ce monde de l'islam qui commence à s'éveiller et où il y avait un si grand intérêt pour la France de conquérir et d'élargir des sympathies ?
Vous savez bien que ce monde musulman meurtri, tyrannisé tantôt par le despotisme de ses maîtres, tantôt par la force de l'Européen envahisseur, se recueille et prend conscience de son unité et de sa dignité. Deux mouvements, deux tendances inverses le disputent : il y a les fanatiques qui veulent en finir par la haine, le fer et le feu, avec la civilisation européenne et chrétienne, et il y a les hommes modernes, les hommes nouveaux, aujourd'hui Mustapha Kameh, comme est l'élite des musulmans de l'Inde unis aux hindous, comme est l'élite des musulmans de la Turquie qui viennent de tendre fraternellement la main aux Arméniens égorgés. Il y a toute une élite qui dit : l'islam ne se sauvera qu'en se renouvelant, qu'en interprétant son vieux livre religieux selon un esprit nouveau de liberté, de fraternité, de paix. (Applaudissements à l'extrême gauche.) Et cette élite, elle n'aspire pas à briser le cadre de la civilisation et de l'administration européennes ; elle reconnaît, par exemple, les admirables services que l'Angleterre a pu rendre à des castes opprimées, mais elle dit en même temps qu'il faut élever le peuple musulman au point où il pourra participer avec dignité à la civilisation européenne.
Et c'est à l'heure où ce mouvement se dessine que vous fournissez aux fanatiques de l'islam le prétexte, l'occasion de dire : « Comment se réconcilier avec cette Europe brutale ? Voilà la France, la France de justice et de liberté, qui n'a contre le Maroc d'autre geste que les obus, les canons, les fusils ! »
Vous faites, messieurs, contre la France, une politique détestable. (Applaudissements à l'extrême gauche.)
L'Humanité, 25 janvier 1908, extraits.
L'Armée nouvelle
Avec l'Histoire socialiste de la Révolution française, L'Armée nouvelle est l'autre ouvrage important de Jean Jaurès ; il s'agit en réalité, à l'origine, d'un projet de loi pour réformer l'armée française, transformé en un livre qui va être largement lu et discuté à l'époque, notamment dans les rangs du socialisme international. Plan de paix, L'Armée nouvelle vise à assurer à la France une sécurité durable à ses frontières. Jaurès mobilise l'histoire pour justifier sa vision de la nation armée, qui ne peut entrer dans un conflit qu'en cas de guerre défensive. L'adoption de la loi qui porte le service militaire à trois ans en 1913 signe l'échec du projet jaurésien.
C'est par les questions relatives à la défense nationale et à la paix internationale que j'aborde l'exposé du plan d'organisation socialiste de la France que je veux soumettre au Parlement en formules législatives. Il y a urgence et pour le socialisme et pour la nation, à définir ce que doivent être, dans la pensée socialiste, l'institution militaire et la politique extérieure de la France républicaine. Pour hâter, pour accomplir son évolution vers l'entière justice sociale, pour instaurer et même préparer librement un ordre nouveau où le travail sera organisé et souverain, la France a besoin avant tout de paix et de sécurité. Il faut écarter d'elle la sinistre diversion des aventures extérieures ; il faut écarter d'elle la menace des violences étrangères. Le premier problème qui s'impose donc à un grand parti de transformation sociale qui est résolu à aboutir, c'est celui-ci :
Comment porter au plus haut, pour la France et pour le monde incertain dont elle est enveloppée, les chances de paix ? Et si, malgré son effort et sa volonté de paix, elle est attaquée, comment porter au plus haut les chances de salut, les moyens de victoire ?
Il serait enfantin et dérisoire de proposer un vaste programme de travail, un long et grand effort systématique de réforme à un pays qui ne disposerait pas de lui-même, qui serait sans cesse à la merci ou des aventuriers du dedans en quête de conflits, ou des agresseurs du dehors, et toujours sous la menace ou dans le déchaînement de la guerre. Assurer la paix par une politique évidente de sagesse, de modération et de droiture, par la répudiation définitive des entreprises de force, par l'acceptation loyale et la pratique des moyens juridiques nouveaux qui peuvent résoudre les conflits sans violence ; assurer aussi la paix, vaillamment, par la constitution d'un appareil défensif si formidable que toute pensée d'agression soit découragée chez les plus insolents et les plus rapaces : il n'y a pas de plus haut objet pour le parti socialiste.
Ou plutôt c'est la condition même de son action et de sa vie. Il ne suffit pas qu'il ait cette double et indivisible volonté de paix internationale et d'autonomie nationale. Il faut qu'il persuade le pays tout entier, la démocratie tout entière, de la sincérité et de la force de son dessein : car comment pourrait-il inviter et entraîner la nation à des transformations sociales hardies si elle se croyait menacée par lui dans son existence même ? La France a besoin, pour l'œuvre de justice supérieure que le socialisme lui propose, de toute sa vie, c'est-à-dire de toute sa liberté, et comment faire monter la sève vers les fruits si l'on blesse la racine ? Comment surtout le parti socialiste pourrait-il proposer avec autorité les formes de défense nationale qui lui paraissent les plus efficaces, s'il pouvait être suspect aux yeux d'un seul homme de se désintéresser de la défense nationale elle-même ? Or, c'est par l'action et par l'action seule qu'il dissipera les malentendus créés par l'ignorance ou la perfidie, ou par les paradoxes inséparables des grands mouvements d'idées. Il ne désarmera certes pas la calomnie des charlatans de patriotisme qui couvrent d'un prétexte d'intérêt national les convoitises et les violences de l'esprit de classe ; mais il ralliera peu à peu les bons citoyens qui veulent épargner à la France les convulsions de la guerre et l'humiliation de la servitude.
Ce qui importe au socialisme, c'est donc de traduire en actes, par une application quotidienne, les résolutions essentielles de ses congrès nationaux et internationaux10 ; c'est de rendre visible et tangible sa pensée tout entière, sans mutilation, sans dénaturation. Que le socialisme lie sans cesse la libération des prolétaires à la paix de l'humanité et à la liberté des patries ; que le prolétariat organisé et pensant, éduquant peu à peu la masse, à demi inconsciente encore ou inerte, la guérisse des illusions chauvines et des entraînements belliqueux ; qu'il dénonce l'odieux et le ridicule de la guerre, dont le rôle dans l'histoire humaine fut terriblement ambigu, à la fois funeste et fécond, mais qui aujourd'hui, dans le monde de la démocratie et du travail, est pleinement surannée, absurde et criminelle ; qu'il menace sérieusement d'un acte de désespoir révolutionnaire tout gouvernement assez insensé et coupable pour déchaîner un conflit sans avoir épuisé manifestement toutes les chances de paix, tous les moyens de conciliation arbitrale et sans avoir couvert la patrie du verdict d'approbation de la conscience universelle ; qu'il mette au service de cette salutaire menace, à la fois patriotique et humaine, la force accrue de ses organisations développées et fédérées ; qu'il noue avec les prolétaires de tous les pays des relations internationales toujours plus étroites et institue ainsi pratiquement un commencement d'humanité ouvrière capable de mettre un peu d'ordre et d'équité dans le chaos des rivalités nationales ; qu'il intervienne en toute occasion pour donner vérité et plénitude aux premières garanties de paix, si timidement, parfois si hypocritement ébauchées à La Haye11 par les gouvernements eux-mêmes ; qu'il demande dans tous les traités de pays à pays l'insertion de clauses d'arbitrage universel et aussi de clauses de réciprocité protégeant partout les salariés et créant, par le libre consentement des patries historiques, une patrie sociale du travail ; mais que, dans cette grande œuvre même et pour la conduire à sa fin, il veille constamment sur l'indépendance de la patrie et sur ses moyens de défense ; qu'il ne s'en tienne pas à la formule générale des milices, mais qu'il précise pour le pays le mode d'organisation très robuste qu'il entend sous ce mot ; qu'il en démontre l'excellence et l'efficacité ; qu'il prouve par la conduite même de ses militants et par leur propagande dans le peuple ouvrier, par leur assiduité et leur zèle aux œuvres vivantes d'éducation militaire, aux sociétés de gymnastique et de tir, aux manœuvres de plein air et aux exercices en terrain varié dont l'efficacité se substituera à la stérile mécanique de l'enseignement de la caserne, qu'il démontre donc par son activité allègre que s'il combat le militarisme et la guerre, ce n'est point par égoïsme peureux, lâcheté servile ou paresse bourgeoise, mais qu'il est aussi résolu et prêt à assurer le plein fonctionnement d'un système d'armée vraiment populaire et défensif qu'à abattre les fauteurs de conflit : alors, il pourra défier les calomnies, car il portera en lui, avec la force accumulée de la patrie historique, la force idéale de la patrie nouvelle, l'humanité du travail et du droit.
C'est dans cet esprit, c'est pour concourir, selon mes forces, à cette action et éducation nécessaires, que j'essaie de définir au Parlement et les conditions dans lesquelles doit être organisée la défense de la patrie, et les garanties qui doivent être instituées pour le maintien de la paix.
Pour la nation aussi, il y a un intérêt vital à ce que tout malentendu entre elle et le socialisme soit dissipé. Un pays qui ne pourrait pas compter, aux jours de crise où sa vie même serait en jeu, sur le dévouement national de la classe ouvrière, ne serait qu'un misérable haillon. Par quelle aberration la démocratie française s'obstinerait-elle à calomnier la pensée de ce prolétariat de France qui proclame sans cesse le devoir et la volonté des prolétaires de défendre l'autonomie de la nation ? Ce qu'il demande à la nation, il a le droit et le devoir de le lui demander.
Il lui demande simplement de lui épargner un crime aussi funeste à la France qu'à la classe ouvrière elle-même, et ce crime, c'est de jeter les ouvriers dans une guerre contre leurs frères des autres pays sans avoir mis du côté de la France l'évidence du droit. Or, au moment où la France prend à la conférence de La Haye, très mollement, la direction de la politique d'arbitrage, au moment où elle propose que tout pays, à la veille d'un conflit, saisisse du conflit la conférence de La Haye, la France prend l'engagement moral de pratiquer cette politique. Elle n'y saurait manquer sans scandale. Et les travailleurs de France n'abusent pas vraiment quand ils adjurent la France de ne pas déchirer étourdiment et cruellement le lien de solidarité qui les lie aux ouvriers de tous les pays. Ils l'adjurent de rester fidèle à ce qu'il y a de meilleur en elle, et en protégeant son idéal le plus haut contre la surprise des passions ou la manœuvre d'intérêts subalternes, ils servent la patrie comme le prolétariat.
Ce qu'ils demandent encore, ce qu'ils ont le droit et le devoir de demander, c'est que la nation organise sa force militaire sans aucune préoccupation de classe ou de caste, sans autre souci que celui de la défense nationale elle-même.
J'ose prier les officiers eux-mêmes de méditer sans passion et sans préjugé l'idée générale du socialisme et l'application qu'il en fait à l'institution militaire. Il serait funeste pour eux, j'entends pour la vie de leur esprit, de céder aveuglément à des préventions haineuses contre des hommes qu'ils ne connaissent pas, contre des systèmes de pensée qui heurtent d'abord leurs habitudes. Ils ont fait l'épreuve, dans le drame de l'affaire Dreyfus, du péril de ces malentendus. Ils ont cru, sur la foi de quelques chefs compromis, sauver l'armée en luttant contre le vrai et le juste. Je ne sais quelle est la conclusion secrète qu'ils ont tirée de cette tragédie. Le pire serait que par une gageure insensée, ils aient persévéré silencieusement dans l'erreur première, ou que par dégoût d'un drame où leur conscience et leur raison furent engagées si à faux, ils en aient détourné leur pensée sans avoir déduit les conséquences nécessaires. Mais s'ils ont ce haut courage qui cherche dans les plus douloureuses épreuves et les plus mortifiantes erreurs une leçon et un renouvellement, ils ont certainement conclu qu'ils devaient garder l'intelligence ouverte et la conscience libre. Le jour où l'existence nationale serait en jeu, ils auraient à conduire à la bataille des millions de prolétaires ; quelle faiblesse et quelle tristesse si entre eux et ces hommes il y a comme un divorce moral, un irréparable malentendu de la conscience et de la pensée !
[...]
Tout ce que la France fera pour ajouter à sa puissance défensive accroîtra les chances de paix dans le monde. Tout ce que la France fera dans le monde pour organiser juridiquement la paix et la fonder immuablement sur l'arbitrage et le droit ajoutera à sa puissance défensive. C'est pourquoi je présente indivisément des projets organisant la défense et organisant la paix ; je ne veux pas seulement travailler de la sorte à propager des idées, à créer des tendances d'esprit. Je ne me livre pas simplement à une manifestation doctrinale, et mon unique objet n'est pas de dissiper des malentendus dont souffriraient également la noble patrie que j'aime et le grand parti que je sers. J'entends faire œuvre pratique, d'un intérêt immédiat et d'un effet prochain. Je suis convaincu que l'œuvre ébauchée à La Haye peut, si nous le voulons, se préciser et s'approfondir. Je suis convaincu que l'institution militaire de la France ne peut s'arrêter longtemps à l'état ambigu et contradictoire créé ou constaté par la loi de deux ans et qu'elle devra sans délai ou rétrograder vers des formes anciennes ou aboutir à un système fortement conçu de milices nationales. Et qu'on ne nous oppose pas dédaigneusement une fin de non-recevoir préalable. Qu'on ne se livre pas au jeu facile et puéril de railler mon inexpérience militaire et mon incompétence technique. Je ne décide pas, je propose. Il me serait d'ailleurs trop aisé de répondre qu'en un pays où c'est le Parlement, lequel n'est pas composé en majorité de techniciens militaires, qui décide de tout, il serait au moins étrange qu'une exception d'incompétence frappât aucun citoyen et discréditât aucune idée. Je pourrais ajouter qu'il m'a été loisible de m'informer et de m'instruire auprès d'officiers d'élite et que je n'y ai point manqué. Mais surtout j'ose dire que ce qui fait le plus défaut maintenant dans le travail de réorganisation militaire qui se poursuit au jour le jour dans notre pays, ce sont les vues d'ensemble ; c'est l'audace et la fermeté des constructions logiques.
L'Armée nouvelle (1911), extrait.
« Sang-froid nécessaire »
Il s'agit du dernier article paru du vivant de Jaurès, éditorial de L'Humanité du 31 juillet 1914, qui a valeur de testament. L'auteur y expose son analyse de la crise internationale en cours, mais il ne pourra jamais davantage expliquer ce qu'il aurait fait s'il avait assisté au déclenchement de la guerre... Il est assassiné au soir du 31 juillet 1914. Comme nombre de socialistes, il espérait beaucoup du congrès de l'Internationale qui devait se tenir à Vienne en août 1914, et qui ne se réunira jamais.
Que l'on mette si l'on veut les choses au pire, qu'on prenne en vue des plus formidables hypothèses les précautions nécessaires, mais de grâce qu'on garde partout la lucidité de l'esprit et la fermeté de la raison. À en juger par tous les éléments connus, il ne semble pas que la situation internationale soit désespérée. Elle est grave à coup sûr, mais toute chance d'arrangement pacifique n'a pas disparu. D'une part, il est évident que si l'Allemagne avait eu le dessein de nous attaquer, elle aurait procédé selon la fameuse attaque brusquée. Elle a au contraire laissé passer les jours, et la France comme la Russie ont pu mettre à profit ce délai, l'une, la Russie, pour procéder à une mobilisation partielle, l'autre, la France, pour prendre toutes les précautions comptables avec le maintien de la paix.
D'autre part, l'Autriche et la Russie sont entrées en négociations directes. La Russie demande à l'Autriche quel traitement elle réserve à la Serbie. L'Autriche a répondu qu'elle respecterait « son intégrité territoriale ». La Russie estime que ce n'est pas assez, qu'il faut en outre que « les droits de souveraineté de la Serbie soient garantis ».
La conversation est engagée. Même si un désaccord se précise entre les vues de l'Autriche et celles de la Russie, on pourra mesurer l'écart des idées et s'employer à la solution d'un problème dont les données seront déterminées. C'est alors, semble-t-il, que pourra intervenir cette pensée médiatrice de l'Angleterre qui cherche sa forme, ses moyens d'expression, mais qui finira bien par prévaloir, car elle répond au sentiment profond des peuples, et sans doute au désir même des gouvernements qui sentent monter vers eux, comme un châtiment, ce péril de guerre avec lequel un moment ils avaient cru jouer comme avec un instrument diplomatique.
Et si l'on juge de ce que serait la guerre elle-même et des effets qu'elle produirait par la panique, les sinistres rumeurs, les embarras économiques, les difficultés monétaires et les désastres financiers que déchaîne la seule possibilité du conflit, si l'on songe que dès maintenant il faut ajourner les règlements d'échéances et se préparer à décréter le cours forcé de petites coupures de billets de banque, on se demande si les plus fous ou les plus scélérats des hommes sont capables d'ouvrir une pareille crise.
Le plus grand danger à l'heure actuelle n'est pas, si je puis dire, dans les événements eux-mêmes. Il n'est même pas dans les dispositions réelles des chancelleries, si coupables qu'elles puissent être, il n'est pas dans la volonté réelle des peuples ; il est dans l'énervement qui gagne, dans l'inquiétude qui se propage, dans les impulsions subites qui naissent de la peur, de l'incertitude aiguë, de l'anxiété prolongée. À ces paniques folles les foules peuvent céder et il n'est pas sûr que les gouvernements n'y cèdent pas. Ils passent leur temps (délicieux emploi) à s'effrayer les uns et les autres et à se rassurer les uns et les autres. Et cela, qu'on ne s'y trompe pas, peut durer des semaines. Ceux qui s'imaginent que la crise diplomatique peut être et doit être résolue en quelques jours se trompent. De même que les batailles de la guerre moderne, se développant sur un front immense, durent sept ou huit jours, de même les batailles diplomatiques, mettant maintenant en jeu toute une Europe et un appareil formidable et multiple de nations puissantes, s'étendent nécessairement sur plusieurs semaines12. Pour résister à l'épreuve, il faut aux hommes des nerfs d'acier ou plutôt il leur faut une raison ferme, claire et calme. C'est à l'intelligence du peuple, c'est à sa pensée que nous devons aujourd'hui faire appel si nous voulons qu'il puisse rester maître de soi, refouler les paniques, dominer les énervements et surveiller la marche des hommes et des choses, pour écarter de la race humaine l'horreur de la guerre.
Le péril est grand, mais il n'est pas invincible si nous gardons la clarté de l'esprit, la fermeté du vouloir, si nous savons avoir à la fois l'héroïsme de la patience et l'héroïsme de l'action. La vue nette du devoir nous donnera la force de le remplir.
Tous les militants socialistes inscrits à la Fédération de la Seine sont convoqués dimanche matin, salle Wagram, à une réunion où sera exposée la situation internationale, où sera définie l'action que l'Internationale attend de nous. Des réunions multipliées tiendront en action la pensée et la volonté du prolétariat et prépareront la manifestation assurément magnifique qui préludera aux travaux du Congrès international. Ce qui importe avant tout, c'est la continuité de l'action, c'est le perpétuel éveil de la pensée et de la conscience ouvrières. Là est la vraie sauvegarde. Là est la garantie de l'avenir.
L'Humanité, 31 juillet 1914.
1 En 1795, la Pologne avait été partagée entre la Russie, la Prusse et l'Autriche.
2 Allusion à Napoléon III et à son entourage.
3 Loi militaire rapportée devant l'Assemblée par le général Chanzy en mai 1872 et qui portait le service à cinq ans pour ceux qui tiraient un mauvais numéro.
4 Allusion à une déclaration de l'empereur allemand Guillaume II.
5 Allusion à l'intervention de Jules Delafosse qui avait déclaré la veille que l'armée était « la meilleure sauvegarde de notre sécurité au-dehors et au-dedans ».
6 Les oasis du Touat furent occupées par les troupes françaises en 1900-1901, ce qui déclencha une révolte.
7 Mahdi (1844-1885), chef politique religieux qui a mené une révolte et fondé un État théocratique entre 1885 et 1898.
8 Depuis 1903, les sociaux-démocrates russes étaient divisés entre bolcheviks (« majoritaires ») et mencheviks (« minoritaires »).
9 Allusion à la guerre russo-japonaise de 1904-1905 qui s'est terminée par une défaite de la Russie.
10 Allusion à la motion du congrès de Stuttgart de l'Internationale socialiste de 1907, qui déclare notamment : « si une guerre menace d'éclater, c'est un devoir de la classe ouvrière dans les pays concernés [...] de faire tous les efforts pour empêcher la guerre par tous les moyens qui leur paraissent les mieux appropriés [...] ».
11 Il y eut deux conférences internationales à La Haye, en 1899 et en 1907.
12 Jaurès sous-estime incontestablement la gravité de la crise qui fera basculer le monde dans la Grande Guerre en un peu plus d'une semaine...
Repères biographiques
1859. 3 septembre : Naissance à Castres de Jean Jaurès.
1878. Entré premier à l'École normale supérieure.
1885. Député républicain du Tarn.
1889. Maître de conférences à la faculté des lettres de Toulouse. Battu aux élections législatives.
1890. Maire-adjoint à l'instruction publique à la mairie de Toulouse (jusqu'en 1893).
1891. Fusillade de Fourmies (1er mai).
1892. Jaurès reçu docteur. Grève des mineurs de Carmaux.
1893. Jaurès élu député socialiste du Tarn.
1896. Inauguration de la verrerie ouvrière d'Albi.
1898. « J'accuse...! » de Zola. Jaurès s'engage dans l'affaire Dreyfus. Il est battu aux élections législatives.
1899. Première participation d'un ministre socialiste au gouvernement Waldeck-Rousseau soutenu par Jaurès.
1900. Début de la publication de l'Histoire socialiste de la Révolution française.
1902. Jaurès réélu député.
1903. Jaurès élu vice-président de la Chambre des députés. Soutien au Bloc des gauches d'Émile Combes.
1904. Fondation du journal L'Humanité. Congrès de l'Internationale socialiste à Amsterdam.
1905. Fondation de la Section française de l'Internationale ouvrière. Révolution russe. Séparation des Églises et de l'État.
1906. Controverse avec Clemenceau, ministre de l'Intérieur. Jaurès réélu député. Mille morts lors de la catastrophe minière de Courrières.
1907. Révolte des vignerons du Midi. Congrès de l'Internationale à Stuttgart.
1908. Congrès à Toulouse de la SFIO ; discours de Jaurès sur l'« évolution révolutionnaire ».
1910. Proposition de loi de Jaurès sur « l'Armée nouvelle ». Il est réélu député.
1911. Voyage en Amérique latine.
1912. Congrès de l'Internationale socialiste à Bâle.
1913. Grand meeting du Pré-Saint-Gervais contre les trois ans de service militaire.
1914. Jaurès réélu député. 31 juillet : Jaurès assassiné. 1er août : mobilisation générale en France.
Bibliographie sélective
Biographies
Harvey GOLDBERG, Jean Jaurès : la biographie du fondateur du Parti socialiste, Paris, Fayard, 1970.
Max GALLO, Le Grand Jaurès, Paris, Robert Laffont, 1984.
Madeleine REBÉRIOUX, Jaurès, la parole et l'acte, Paris, Gallimard, « Découvertes », 1994.
Jean-Pierre RIOUX, Jaurès, Paris, Perrin, 2005 ; rééd. « Tempus », 2008.
Gilles CANDAR et Vincent DUCLERT, Jean Jaurès, Paris, Fayard, 2014.
Études
Alain BOSCUS et Rémy CAZALS (dir.), Sur les pas de Jaurès. La France de 1900, Toulouse, Privat, 2004 ; rééd. 2008.
Paul MARCUS, Jaurès l'humaniste, Paris, La Documentation française, « Tribuns », 2009.
Rémy PECH, Jaurès paysan, Toulouse, Privat, 2009.
Jean-Michel DUCOMTE, Quand Jaurès administrait Toulouse, Toulouse, Privat, 2009.
Charles SILVESTRE, Jaurès, la passion du journaliste, Pantin, Le Temps des cerises, 2010.
Jean-Michel DUCOMTE et Rémy PECH, Jaurès et les radicaux : une dispute sans rupture, Toulouse, Privat, 2011.
Charles SILVESTRE, La Victoire de Jaurès, Toulouse, Privat, 2013.
Vincent DUCLERT, Jaurès, 1859-1914 : la politique et la légende, Paris, Autrement, coll. « Vies parallèles », 2013.
Jacqueline LALOUETTE, Jean Jaurès. L'assassinat, la gloire et le souvenir, Paris, Perrin, 2014.
De Jaurès
Œuvres de Jean Jaurès, édition coordonnée par Gilles Candar, Paris, Fayard, 17 volumes (parution en cours).
Parus :
I. Les Années de jeunesse (1859-1889), édition établie par Madeleine Rebérioux et Gilles Candar (2009).
II. Le Passage au socialisme (1889-1893), édition établie par Madeleine Rebérioux et Gilles Candar (2011).
III. Philosopher à trente ans (les thèses et autres textes), édition établie par Annick Wajngart-Taburet (2000).
VI et VII. L'Affaire Dreyfus (1897-1899), édition établie par Éric Cahm (2001).
VIII. La Défense républicaine et la participation ministérielle (1899-1902), édition établie par Maurice Agulhon et Jean-François Chanet (2013).
XIII. L'Armée nouvelle, édition établie par Jean-Jacques Becker (2012).
XVI. Critique littéraire et critique d'art, édition établie par Michel Launay, Camille Grousselas et Françoise Laurent-Prigent (2000).
XVII. Le Pluralisme culturel, édition établie par Jean-Numa Ducange et Marion Fontaine (2014).
Anthologies et textes
Rallumer tous les soleils, édition établie par Jean-Pierre RIOUX, Paris, Omnibus, 2005.
Jaurès. Ce que dit un philosophe à la cité, édition établie par Claude DUPONT, Paris, Les Belles Lettres, 2010.
Gilles CANDAR et Guy DREUX, Une loi pour les retraites. Débats socialistes et syndicalistes autour de la loi de 1910, Lormont, Le Bord de l'eau, 2010.
Discours en Amérique latine 1911, préface de Jean-Luc MÉLENCHON, Paris, Bruno Leprince, 2010.
Le Socialisme et la Vie. Idéalisme et matérialisme, édition établie par Frédéric WORMS, Paris, Rivages poche/Petite bibliothèque, 2011.
Discours et conférences, édition établie par Thomas HIRSCH, Paris, Flammarion, « Champs classiques », 2011.
De l'éducation (anthologie), édition établie par Guy DREUX, Christian LAVAL, Madeleine REBÉRIOUX, Catherine MOULIN et Gilles CANDAR, Paris, Syllepse, 2005 ; rééd. Le Seuil, « Points/Essais », 2012.
Jaurès actuel, préface de Patrick LE HYARIC, Paris, L'Humanité, 2013.
Justice d'abord !, anthologie par Gilles CANDAR avec le concours de Grégoire KAUFFMANN, Paris, Garnier/Le Monde, « Les rebelles », collection dirigée par Jean-Noël Jeanneney, 2013.
Histoire socialiste de la Révolution française, Paris, Éditions sociales, 2014 (réimpression de l'édition d'Albert SOBOUL avec préfaces de Michel BIARD et de Jean-Numa DUCANGE).
Ainsi nous parle Jean Jaurès, anthologie de la Fondation Jean-Jaurès préparée par Marion FONTAINE, Paris, Fayard, « Pluriel », 2014.
Jaurès : controverses avec Jules Guesde et Rosa Luxemburg, Paris, Le Passager clandestin, 2014.
Pour en savoir plus sur Jaurès :
la Société d'études jaurésiennes
et la revue Cahiers Jaurès :
<http://www.jaures.info>. 4 rue Toussaint-Feron 75013 Paris.
Le Livre de Poche
Jean-Numa Ducange est historien, enseignant-chercheur et maître de conférences à l'université de Rouen. Il est l'auteur de nombreux articles et ouvrages sur l'histoire du socialisme. Il a notamment présenté, pour Le Livre de poche, Le 18 Brumaire de Louis Bonaparte, de Karl marx (coll. « Classiques de la philosophie »).
Couverture : Studio LGF. © Roger-Viollet/ © Leemage/ © Bridgeman.
© Librairie Générale Française, 2014, pour la présente édition.
ISBN : 978-2-253-17884-2
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{"url":"https:\/\/aptitude.gateoverflow.in\/6594\/nielit-2019-feb-scientist-c-section-a-2","text":"878 views\n\nStudy the following series \u2160 and\u00a0\u2161\u00a0comprising letters\/digits\/symbols and answer the questions given below.\n\nSeries I\u00a0:\u00a0 N P Q * 5 4 B Z \\$# 3 7 D 9 A ? F 8 G H 1 + D K 6 Series II: 7 4 \u20ac J M 1 \u00a9 Y P 8 Q R 5 T N 6 2 ? U A 3 9 @ I S What is the total number of symbols in series I each of which is immediately followed by a digit and immediately preceded by a letter together with the number of letters in Series II which are immediately preceded by a symbol and immediately followed by a letter? 1.$5$2.$4$3.$1$4.$3\\$\n\n210 points\n\n1\n1 vote","date":"2022-12-03 22:53:29","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 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\": 0.4680316150188446, \"perplexity\": 991.890675721233}, \"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\/1669446710941.43\/warc\/CC-MAIN-20221203212026-20221204002026-00394.warc.gz\"}"} | null | null |
"Let me start by saying I live about a mile from this place and work in Saint barnabas hospital and to be honest I never knew that it was a diner. Happened to go there with my co-worker and man that was the best thing we ever did. The waiter recommended panini with turkey and spinach and summer salad which we both shared and they both were fantastic and the service was exceptional. Good job." -Connie S. | {
"redpajama_set_name": "RedPajamaC4"
} | 2,691 |
RAILROAD.NET • View topic - Getting Re-Hired at NS?
I previously worked for NS in the Philly area as a CO for about a year, got furloughed. NS did attempt to call me back, but the problem is my fault. I attempted to change my mailing address within the NS Mainframe and it didn't change and I wasn't aware of it. So when the recall letter went to my old address and I didn't know about it until a month after the letter was sent, I couldn't go back. My union reps worked really hard and tried to get me back, but were unsuccessful. If you're thinking this is BS, I swear it's not.
In the year that I was at NS, I marked off only as needed which was 2 times sick and any other times I marked off it was with permission from a boss for Doctor's appointments, e.g. things that would need me to mark off of work that I had to take care of. I was never taken OOS for any rule violations, etc. The guys who I worked with enjoyed working with me and always got along with me. Bottom line is I wasn't a trouble maker and did what I was supposed to do how I was taught.
A friend who I actually trained during my time at NS, who made a move to Allentown during furloughs put a referral in for me as a CO for Allentown. He also gave me the number of the Terminal Super of Allentown so I could give him a call and tell him what happened to me and that I would like to come back. I did call him and he did seem interested.
Today I got an invite to a hiring session. Should I pass the Entry Exam, interview, and get rehired, my CO Card is still valid until February.
I'm curious to see what members of RR.net think on this question. Would NS send me back to Georgia for another 3 weeks of CT school, or since my card is still valid would they send me out to qualify and go?? I did explain this to the terminal super that if offered employment I'm comfortable being sent out on road trains and qualifying without going to Georgia again.
Thanks for any insight that anyone provides.
Re: Getting Re-Hired at NS?
I hope this works out for you. I think you'll just have to accept whatever they offer/require. It would be nice to be able to reclaim your original seniority date but I wouldn't hold my breath on that. Wishing you a long and safe career! Pay into railroad retirement for many years on my behalf!
Thanks. I definitely would take the offer either way. As far as my seniority date. That'd be awesome. But wouldn't happen I'd say. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,580 |
{"url":"https:\/\/webwork.maa.org\/moodle\/mod\/forum\/discuss.php?d=2879","text":"## WeBWorK Problems\n\n### Force reduced fractions in lists\n\nby Jason Aubrey -\nNumber of replies: 10\nHi All,\n\nWe have questions whose answer is a list of fractions. (E.g. below.) The Fraction-NoDecimals context works to force students to enter fractions instead of decimals, but we can't seem to force reduced fractions in lists.\n\nAlso, the author of the problem defined a \"no solution\" string. But, if a student enters \"no solution\" to a question which in fact has a solution the reply is \"numerators must be integers\" rather than \"incorrect.\"\n\nThanks,\nJason\n\n# DESCRIPTION\n# Solve an abs val eqn of the form k - |ax + b| = k, one solution.\n# ENDDESCRIPTION\n## DBsubject('Algebra')\n## DBchapter('Equations and Inequalities')\n## DBsection('Absolute Value')\n## KEYWORDS('linear', 'absolute value', 'equation')\n## Author('Rick Lynch')\n## Institution('University of Missouri-Columbia')\n\n###########################################################################\n# initialization\n###########################################################################\n\nDOCUMENT();\n\"PGstandard.pl\",\n\"MathObjects.pl\",\n\"PeriodicRerandomization.pl\",\n\"contextFraction.pl\"\n);\n\nTEXT(beginproblem());\n$showPartialCorrectAnswers = 1; PeriodicRerandomization(\"3\"); ########################################################################### # setup contexts and variables ########################################################################### Context(\"Fraction-NoDecimals\"); Context()->strings->add( \"no solution\"=>{}, \"none\"=>{alias=>\"no solution\"} );$var = list_random('x','y','z','w','p','q','r','s','t','u','v');\n\n$a = random(2,20)*random(-1,1,2);$b = -sgn($a)*random(1,20);$k = non_zero_random(-15,15);\n\n$eqn = \"$k - \\left| $a$var + $b\\right| =$k\";\n$answer = List(Fraction(-$b,$a)); ########################################################################### # state the problem ########################################################################### Context()->texStrings; BEGIN_TEXT Solve the following equation for $$var$$:$PAR\n$eqn$\n$PAR $$var =$$ \\{ ans_rule(20) \\} END_TEXT ########################################################################### # check the answer ########################################################################### ANS($answer->cmp(studentMustReduceFractions=>1));\n\n###########################################################################\n# use PeriodicRerandomization to write the answer and generate a new\n# version of the problem\n###########################################################################\n\nif ($attempts_modp == 0 &&$actualAttempts != 0) {\nBEGIN_TEXT\n$PAR$BBOLD Answer: $EBOLD $$var = answer$$$PAR\nEND_TEXT\n} else {\nBEGIN_TEXT\n$PAR$BBOLD Help: $EBOLD Enter your answers as a comma separated list if there is more than one correct answer. Write \"no solution\" if the equation has no solution.$BR\nEND_TEXT\n}\nContext()->normalStrings;\n\nPeriodicStatus();\n\nCOMMENT('Features Periodic Rerandomization. Edited and updated in 2012\/2013. $BR Desc: Solve an abs val eqn of the form k - |ax + b| = k, one solution.'); ENDDOCUMENT(); In reply to Jason Aubrey ### Re: Force reduced fractions in lists by Paul Pearson - Hi Jason, The bogus error message can be replaced by \"incorrect\" using error message customization: http:\/\/webwork.maa.org\/wiki\/ErrorMessageCustomization I don't know how to fix the other problem (forcing reduced fractions). Good luck! Paul Pearson In reply to Jason Aubrey ### Re: Force reduced fractions in lists by Robin Cruz - Jason, You can change the error message and you can do it without loading the answerHints.pl. Here's an example where the correct answer could be a string or a list. I took a substring of the original error message I wanted to change to use in the checker--You could just put \" \" for the error message if you didn't want a message to show up. I put the code at the end of this note. --rac ------------------------------------------------- #SET-UP Context()->strings->add( \"No zeros\"=>{alias=>'None'}, \"No solution\"=>{alias=>'None'} );$a1 = list_random(3,5);\n$b1 = list_random(7,4,2,1,-1,-2,-4,-7);$c1 = list_random(-2,-1,1,2);\n$p[0] = Compute(\"$a1 x^2 + ($a1*$c1+$b1) x +$b1*$c1\")->reduce;$temp = round(-1000*$b1\/$a1)\/1000;\n$z11 = Real(\"$temp\");\n$z12 = Real(\"-1*$c1\");\n$a[0] = List($z11,$z12);$a2 = random(2,5,1);\n$b2 = non_zero_random(-4,4,1);$c2 = random(5,20,1);\n$p[1] = Compute(\"$a2 x^2 + $b2 x +$c2\")->reduce;\n$a[1] = List(\"No zeros\");$n1 = random(0,1,1);\n\n$ans =$a[$n1]; ANS($ans->cmp->withPostFilter(sub {\nmy $ans = shift;$ans->{ans_message} = \"Enter the zeros of the function or $BITALIC No zeros$EITALIC. \"\nif $ans->{ans_message} =~ m\/not defined\/; return$ans;\n}));\n\n\n### Re: Force reduced fractions in lists\n\nby Davide Cervone -\nJason:\n\nThe problem has to do with the way lists do their checking, and how (or actually when) the Fraction object does the reduction check. The reduction check is done within the Fraction object's post-processing of the answer checker, and but the list checker doesn't actually call that (because it is not part of the equality check that it is doing to compare student entries to correct ones). This has come up before, and probably means that the pos-processing code should be refactored, but such changes are delicate with such a complicated package like MathObjects.\n\nAnyway, the solution (to both problems, as it turns out) is to use a custom checker for the list check. Note that for a list object, checker refers to the checker used to compare individual entries in the list to correct ones, not to a checker for the entire list (which is list_checker. I give a modified version of the key parts of the code below:\n\n DOCUMENT();\n\n\"PGstandard.pl\",\n\"MathObjects.pl\",\n\"contextFraction.pl\"\n);\n\nTEXT(beginproblem());\n$showPartialCorrectAnswers = 1; Context(\"Fraction-NoDecimals\"); Context()->strings->add( \"no solution\"=>{}, \"none\"=>{alias=>\"no solution\"} ); Context()->flags->set(reduceFractions=>0);$var = list_random('x','y','z','w','p','q','r','s','t','u','v');\nContext()->variables->are($var => 'Real');$a = random(2,20)*random(-1,1,2);\n$b = -sgn($a)*random(1,20);\n$k = non_zero_random(-15,15);$eqn = \"$k - \\left|$a $var +$b\\right| = $k\";$answer = List(\"-$b\/$a,1\/2\");\n\nBEGIN_TEXT\nSolve the following equation for $$var$$:\n$PAR $eqn$$PAR\n$$var =$$ \\{ ans_rule(20) \\}\nEND_TEXT\n\nANS($answer->cmp( entry_type=>\"a fraction\", checker => sub { my ($correct,$student,$ans,$nth,$value) = @_;\nreturn 0 unless Value::classMatch($student,'Fraction'); return$correct == $student if$student->isReduced;\n$correct->context->setError(\"Your$nth $value is not reduced\",\"\",undef,undef,$Value::CMP_WARNING)\nunless $ans->{isPreview}; return 0; } )); ENDDOCUMENT(); Note that I've added the reduceFractions=>0 flag to prevent student answers from being reduced automatically. I also add the variable used for the problem to the context (since the student might type that, and I'd prefer not to see the \"variable not defined\" message, since the variable IS defined. The main change is in the custom checker. We set the entry_type to \"a fraction\" so that the error messages will be a bit better (otherwise it would be \"fraction of integers\" which is awkward in most of the messages). The checker itself gets that correct and student answers, plus the answer hash and strings that can be used for error messages to indicate which list entry ($nth) we are, and what type of object ($value). We return 0 (no correct) if the student answer isn't a fraction (among other things, this will take care of the case where they enter \"no solution\" without causing an error message). Then, if the student's answer is reduced, return whether it equals the correct answer or not. Othewise (not reduced), print a warning message (unless we are previewing the asnwers), and return 0 (no correct). That should do what you want, I hope. Davide In reply to Davide Cervone ### Re: Force reduced fractions in lists by Jason Aubrey - Thanks! We had to tweak this a bit since we wanted integer answers to be accepted too. The answer checker was accepting 2\/1 but rejecting 2. We also added some checking so that it would gracefully handle string answers when fractions were expected and handle fractions when string answers were expected... ANS($answer->cmp( entry_type => \"a fraction\",\nchecker => sub {\nmy ($correct,$student, $ans,$nth, $value) = @_; if (Value::classMatch($correct,'String')) {\nreturn $correct ==$student;\n} else {\nreturn 0 if Value::classMatch($student,'String'); if (Value::classMatch($student, 'Real')) {\n$student = Fraction($student) if floor($student)==$student;\n}\nreturn 0 unless Value::classMatch($student,'Fraction'); return$correct == $student if$student->isReduced;\n$correct->context->setError(\"Your$nth $value is not reduced\", \"\", undef, undef,$Value::CMP_WARNING)\nunless $ans->{isPreview}; return 0; } } )); In reply to Jason Aubrey ### Re: Force reduced fractions in lists by Davide Cervone - Here is an simpler version: ANS($answer->cmp( entry_type => \"a fraction\",\nchecker => sub {\nmy ($correct,$student, $ans,$nth, $value) = @_; return$correct == $student unless Value::classMatch($student,'Fraction') && !$student->isReduced;$correct->context->setError(\"Your $nth$value is not reduced\",\"\",undef,undef,$Value::CMP_WARNING) unless$ans->{isPreview};\nreturn 0;\n}\n));\n\nIt should work for all the situations you indicate as well. The $correct ==$student check will work whether the student and correct answers are Strings, Reals, or Fractions, so really the only thing we need to check is if the warning message needs to be given (student entered a fraction that is not reduced).\n\nHope that works for you.\n\nDavide\n\n### Re: Force reduced fractions in lists\n\nby Richard Lynch -\nHi Davide, My name is Rick and I am the one working with Jason on this at Mizzou. I greatly appreciate your help thus far on this issue! I have a further question, though, if you don't mind?\n\nThe code:\n\nsub checkFracList {\nreturn (\nentry_type => \"a fraction\",\nchecker => sub {\nmy ($correct,$student, $ans,$nth, $value) = @_; if (Value::classMatch($correct,'String')) {\nreturn $correct ==$student;\n} else {\nreturn 0 if Value::classMatch($student,'String'); if (Value::classMatch($student, 'Real')) {\n$student = Fraction($student) if (floor($student)==$student));\n}\nreturn 0 unless Value::classMatch($student,'Fraction'); return$correct == $student if ($student->isReduced);\n$correct->context->setError(\"Your$nth $value is not reduced\", \"\", undef, undef,$Value::CMP_WARNING) unless $ans->{isPreview}; return 0; } } ); } works, but I realized that if the answer is not correct, it will still tell the students that it is not reduced. For instance, say that the answers are 1\/4, -5\/8 and a student enter 6\/8, -5\/8. Now 6\/8 is clearly wrong and it will count it as incorrect, however, it still displays the message that 6\/8 is not reduced. I think this could give the wrong impression to the student that they are correct, but just need to reduce their answer. Is there anyway to fix this? I tried to ignorantly replace$correct->context->setError(\"Your $nth$value is not reduced\", \"\", undef, undef, $Value::CMP_WARNING) unless$ans->{isPreview};\n\nwith the if statement:\n\nif ($correct ==$student) {\n$correct->context->setError(\"Your$nth $value is not reduced\", \"\", undef, undef,$Value::CMP_WARNING) unless $ans->{isPreview}; } but I believe the way that it is looping through the answers to check them makes this approach fail. Thanks! Rick In reply to Richard Lynch ### Re: Force reduced fractions in lists by Davide Cervone - You are right, the list checking is a little more complicated than you might think. For unordered lists, the checker may be called for the same student answer multiple times (as it looks for a correct answer that it matches). So the error messages from this phase of the checking are not retained (as otherwise spurious messages might be produced from comparing the student answer to the wrong correct ones). After it finds all the student answers that are correct, the list checker does a second round of comparisons on all the incorrect answers (comparing them to the first entry in the list) just to get syntax errors or other error messages. The reason my original code worked is that an unreduced fraction always produced an error, no matter what it was compared to, so the second phase produces the error message that is needed. But your fails since in order to produce the error message it has to be compared to the reduced answer that it matches (and the second phase just compares to the first entry in the list). There are two ways around this. The first is that you are allowed to supply a second checker that is used during the second phase (just to do syntax checking on the answers that don't match anything in the list). We could use the first answer checker to mark a fraction as not reduced when it matches a correct answer but isn't reduced, and then use the second to report that error. E.g., ANS($answer->cmp(\nentry_type => \"a fraction\",\nchecker => sub {\nmy ($correct,$student, $ans,$nth, $value) = @_; return 0 unless$correct == $student; return 1 unless Value::classMatch($student,'Fraction') && !$student->isReduced;$student->{notReduced} = 1 unless $ans->{isPreview}; return 0; }, extra => sub { my ($student, $ans,$nth, $value) = @_;$student->context->setError(\"Your $nth$value is not reduced\",\"\",undef,undef,$Value::CMP_WARNING) if$student->{notReduced};\n}\n));\n\nHere, we only check it the fraction is reduced when it is actually equal to the correct answer. In that case, we set a flag on the student answer that we can use later. Then the extra subroutine is used to check the extra answers (ones that didn't match anything in the list, i.e., are incorrect), and there we look for the flag and produce the warning if it is needed.\n\nFor this situation, it is not actually necessary to use the two separate checkers. The two functions can be combined into a single checker routine, as follows:\n\n ANS($answer->cmp( entry_type => \"a fraction\", checker => sub { my ($correct, $student,$ans, $nth,$value) = @_;\n$correct->context->setError(\"Your$nth $value is not reduced\",\"\",undef,undef,$Value::CMP_WARNING)\nif $student->{notReduced}; return 0 unless$correct == $student; return 1 unless Value::classMatch($student,'Fraction') && !$student->isReduced;$student->{notReduced} = 1 unless $ans->{isPreview}; return 0; } )); Here, we produce the error message whenever the student answer has been marked as not reduced. If this is still in the first phase, the error message will be ignored, so it doesn't matter if we generate it (and we go on to do the checking to see if the answer is correct and reduced or not, marking the student answer as not reduced for later). But if we are in the second phase, we produce the error message and it is displayed. So this latter version might be the easiest to use. Davide In reply to Davide Cervone ### Re: Force reduced fractions in lists by Richard Lynch - I ended up using the second form and it works. I really appreciate it! Thanks a ton!! Here is the current code I am using: ANS($answer->cmp(\nentry_type => \"a fraction\",\nchecker => sub {\nmy ($correct,$student, $ans,$nth, $value) = @_; return$correct == $student if Value::classMatch($correct,'String');\nreturn 0 if Value::classMatch($student,'String');$student = Fraction($student)->reduce if Value::classMatch($student,'Real') && floor($student) ==$student;\nreturn abs($student -$correct) < 1\/1000000 if Value::classMatch($student,'Real');$correct->context->setError(\"Your $nth$value is not reduced\", \"\", undef, undef, $Value::CMP_WARNING) if$student->{notReduced};\nreturn 0 unless $correct ==$student;\nreturn 1 unless Value::classMatch($student,'Fraction') && !$student->isReduced;\n$student->{notReduced} = 1 unless$ans->{isPreview};\nreturn 0;\n}\n));\n\nNow it looks a little different, so first let me explain what more I am trying to do (maybe I'm going a little too crazy with reduced fractions??)\n\nFirst off, I added the lines\n\nreturn $correct ==$student if Value::classMatch($correct,'String'); return 0 if Value::classMatch($student,'String');\n\nbecause it kept saying something along the line numerators of fractions must be integers and the fix you mentioned two posts ago didn't work for some reason. The next two lines\n\n$student = Fraction($student)->reduce if Value::classMatch($student,'Real') && floor($student) == $student; return abs($student - $correct) < 1\/1000000 if Value::classMatch($student,'Real');\n\nare what I'm trying to do next. My goal is to allow them also enter functions (only when necessary) in their answers as well as reduced fractions. For instance, maybe the problem has as answer sqrt(3)\/2 and -1\/2 and I'd like to still force no decimals and reduced fractions. Actually, the code I just posted above works for this given I do\n\nsince it will just check if the student entered an integer and turn it into a fraction otherwise (the only way a student can enter something that isn't an integer and is real is if they do it with a function since they can't enter decimals).\n\nNow here's the issue. I also want to force reduced functions. Say the answers are sqrt(3)\/2, 4. In this case (let's just worry about sqrts for now), I want to make sure they enter sqrt(3)\/2, 4 rather than sqrt(3)\/2, sqrt(16) in addition to forcing the reduced fractions. Is there a way to do this? I have seen your SimplifiedSqrt example on the wiki, but my attempts at trying to extend it have not worked. In fact, I wasn't even able to extend it to lists (without forcing reduced fractions) as I believe it is the same issue as before with the way that a lists loops through the answers.\n\nAny help would be greatly appreciated. Thanks in advance!","date":"2021-10-20 00:49:32","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.6236637234687805, \"perplexity\": 4723.166087763732}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585290.83\/warc\/CC-MAIN-20211019233130-20211020023130-00459.warc.gz\"}"} | null | null |
\section{Introduction}
The nucleus of female mammals embryo cells contain two X
chromosomes; one of these has to be inactivated in order to
equalize the dosage of X genes product with respect to males, and
this inactivation is crucial to survival \cite{CYZB,LKP,CTTB}.
Moreover, the X chromosome is conjectured to play a key role in
the arising of certain types of cancer \cite{JNCI1,JNCI2}.
The mechanism by which one of the two X chromosomes is inactivated
is poorly understood, despite extensive work on this problem
\cite{CYZB,LKP,CTTB}; it is known that co-localization of
chromosomes at the time X-chromosome inactivation (XCI) is
established plays a key role \cite{NTL,NCB,Xclose}.
Very recently, Nicodemi and Prisco \cite{NP} proposed a model to
explain XCI on the basis of a spontaneous symmetry-breaking
mechanism, related to the classical Ising model of Statistical
Mechanics, in a dynamical stochastic model of the single-agent
type.
Their idea is that blocking factor (BF) molecules diffuse in the
cell and can bind randomly to one or the other of the
X-chromosomes; however, by a collective effect, the affinity to a
chromosome increases when this already binds to other BF
molecules. The net effect is that once one of the two chromosomes,
by the effect of a fluctuation, binds to a larger number of BF
molecules, it will bind to more and more of these -- at a rate
higher than the other chromosome (note that the BF molecules could
have a greater probability to escape when more of them bind to the
chromosome; this effect should be dominated by the growth in
affinity in order for the collective effect devised by Nicodemi
and Prisco to set in). In the whole, as a result of diffusive
behavior, fluctuation and the affinity-enhancing collective
effect, a net current will be established leading to BF
concentration on one of the two X-chromosomes.
The model can also account for the relevance of co-localization at
the establishment of XCI; in facts, if the two X-chromosomes are
too far away, the diffusive behavior will prevail, and the BF
molecules escaping from the less populated X-chromosome will have
a very low probability to reach the other one.
The purpose of this short note is to provide a deterministic
(simplified) version of the Nicodemi-Prisco symmetry-breaking XCI
model, considering average exchanges of blocking factors between
each of the two X-chromosomes and the cell fluid environment. Thus
our model will amount to a mean-field version of the
Nicodemi-Prisco model; we will formulate it first in terms of
transition probabilities between states for a single BF molecule,
and then pass to a formulation in terms of a first order system of
ODEs, i.e. a Dynamical System \cite{Gle,GuH,Ver}, following a
procedure which is standard in Physics or in Quantitative Biology
\cite{Mur,ElG}.
Here we will consider the fluid environment as a homogeneous
reservoir, i.e. overlook the effects connected with the spatial
localization of the X-chromosomes; thus the model will be able to
predict the symmetry breaking leading to inactivation of one of
the X-chromosomes, but (in its simplest form, presented here) not
its dependence on co-localization of the chromosomes. An extension
of the model aiming at including co-localization effects will be
developed in forthcoming work.
Similarly, here we will only deal with average quantities and
flows; fluctuations can be included within this kind of modeling
by promoting the ODEs we obtain to {\it stochastic differential
equations} \cite{Gue,Oks,vKam}, as briefly discussed below.
\section{The model}
\label{sec:model}
We will consider a fixed total amount ($n$ units) of BF molecules;
these can be in three mutually exclusive states: they can bind to
one (state 1) or the other (state 2) of the two X-chromosomes, or
be diffusing in the cell environment (state 3). We denote the
number of units binding to the first X-chromosome at time $t$ as
$\^X(t)$, that of units binding to the second X-chromosome at time
$t$ as $\^Y(t)$, and that of units diffusing in the cell fluid at
time $t$ as $\^Z(t)$. We will refer to $x(t)$ and $y(t)$ as the
{\it occupation numbers} for the two X-chromosomes (or more
precisely, for the Xic of the chromosomes). Obviously,
\begin{equation}\label{totnumber} \^X(t) + \^Y(t) + \^Z(t) \ = \ N \ . \end{equation}
\subsection{Transition probabilities for a single BF molecule}
\label{sec:mod:TP}
Let us consider the probabilities $P(i \to j, \de t)$ for
transitions between these states -- in particular, transition from
state $i$ to state $j$ -- for each BF molecule (referred to as a
``particle'' in the following) in a time interval of length $\de
t$. In the following, we will use a simplified notation, and write
\begin{equation}\label{probshort} p_{ij} \ := \ P(i \to j , \de t) \ . \end{equation}
We assume that for $\de t$ sufficiently small, the probability a
particle initially in state 3 will end up being in state 1 or 2
(i.e. bound to one of the two X-chromosomes) is \begin{equation}\label{prob1}
p_{31} \simeq \alpha (\^X) \de t \, + \, o (\de t) \ ; \ \ p_{32}
\simeq \alpha (\^Y) \de t \, + \, o (\de t) \ . \end{equation} That is, it will
be proportional to the length of the time interval (but
independent of time $t$ itself) via a function $\alpha (x)$ or $\alpha
(y)$ representing the affinity with the concerned X-chromosome and
depending on the number of particles already binding to it.
Similarly, the probability that a particle initially binding to
one of the X-chromosomes will escape from it in a time interval
$\de t$ will be given by \begin{equation}\label{prob2} p_{13} \simeq
\beta(\^X,\^Z) \de t \, + \, o (\de t) \ ; \ \ p_{23} \simeq
\beta(\^Y,\^Z) \de t \, + \, o (\de t) \ . \end{equation}
That is, the escape probability will be
proportional to the length of the time interval via a function of
the number of particles binding to the concerned X-chromosome and
of the number of particles fluctuating in the fluid environment
(it is natural to assume these should enter only through their
density $z = Z/N$).
As for direct transitions from one to the other of the two
X-chromosomes, these will be impossible on account of their
non-zero space separation (the particle will have to diffuse
through the fluid to do that), so we will set \begin{equation}\label{prob3}
p_{12} \ = \ o (\de t) \ ; \ \ p_{21} \ = \ o (\de t) \ . \end{equation}
Needless to say, the probabilities of permanence in a given state
are obtained simply requiring the conservation of particles, i.e.
that the sum of probabilities for all transitions from a given
state to all possible state is one (Markov property); thus
\begin{equation}\label{prob4}
\begin{array}{rl}
p_{11} \ =& 1 - p_{12} - p_{13} = 1 - \beta(\^X,\^Z) \de t + o (\de t) \ , \\
p_{22} \ =& 1 - p_{21} - p_{23} = 1 - \beta(\^Y,\^Z) \de t + o (\de t) \ , \\
p_{33} \ =& 1 - p_{31} - p_{32} = 1 - [\alpha(\^X) + \alpha(\^Y)] \de t
+ o (\de t) \ .
\end{array} \end{equation}
\subsection{Evolution equations for occupation numbers}
Let now $\{ X(t),Y(t),Z(t)\}$ be the average occupation numbers
for states $\{1,2,3\}$ at time $t$; we look at the average
occupation numbers at time $t + \de t$. Note $Z(t)$ can be
expressed using \eqref{totnumber}, hence we need only two
equations. These will be given by \begin{equation}\label{eveq1a}
\begin{array}{rl} X (t+ \de t) =& X(t) - p_{13} X(t) + p_{31} Z(t) \ , \\
Y (t+ \de t) =& Y(t) - p_{23} Y(t) + p_{32} Z(t)
\ .
\end{array} \end{equation}
Using the expressions given above for $p_{ij}$ -- and omitting $o
(\de t)$ terms -- the \eqref{eveq1a} yield \begin{equation}\label{eveq1}
\begin{array}{rl}
X (t+ \de t) =& X(t) + [ \alpha(X(t)) \, Z(t) \, - \, \beta(X(t),Z(t)) \, X(t) ] \de t \\
Y (t+ \de t) =& Y(t) + [ \alpha(Y(t)) \, Z(t) \, - \, \beta(Y(t),Z(t)) \, Y(t) ] \de t
\ .
\end{array} \end{equation}
Dividing by $\de t$ and letting $\de t \to 0$, eq.\eqref{eveq1}
yields a system of ODEs describing the evolution of the average
occupation numbers for the three states (in which we make explicit
the expression of $Z(t)$ implied by the conservation law
\eqref{totnumber}): \begin{equation}\label{eveq3}
\begin{array}{rl}
d X / dt =& \alpha(X) \, (N - X - Y) \ - \ \beta(X,N-X-Y) \, X \ , \\
d Y / dt =& \alpha(Y) \, (N - X - Y) \ - \ \beta(Y,N-X-Y) \, Y \ .
\end{array} \end{equation}
Passing to consider the densities $x = X/N$ and $y = Y/N$, and
introducing the functions $a$ and $b$ defined through \begin{equation} a (w)
:= \alpha (N w) \ , \ \ b(w_1,w_2) := \beta (N w_1,N w_2) \ , \end{equation} we get
in the end the equations (symmetric under the exchange of $x$ and
$y$) \begin{equation}\label{eveq4}
\begin{array}{l}
d x / dt \ = \ a(x) \, (1-x-y) \ - \ b(x,1-x-y) \, x \ , \\
d y / dt \ = \ a(y) \, (1-x-y) \ - \ b(y,1-x-y) \, y \ .
\end{array} \end{equation}
\subsection{Minimal model}
So far we have worked in completely general terms within the
three-states description of the system. In order to have a
specific model, we should choose concrete forms for the functions
$\alpha(W)$ and $\beta (W)$ -- and hence of the $a(w)$ and $b(w)$ --
describing the probability of capture by and escape from
chromosomes in function of the associated occupation numbers $X,Y$
or densities $x,y$; this will yield a specific expression for the
evolution equations \eqref{eveq3} and \eqref{eveq4}.
In their work, Nicodemi and Prisco \cite{NP} argued that the
essential feature of their model for XCI is the collective
phenomenon enhancing affinity and hence the probability of capture
by X-chromosomes with a substantial number of binding particles.
Thus the function $\alpha (W)$ should go to a finite limit $\alpha_0$ for
$W \to 0$ (the BF have a non-zero affinity with the chromosomes
even when no BF molecules are binding to chromosomes, and more
generally when the collective behavior has not set in), and grow
with $W$, i.e. $\alpha' (W) > 0$.
As for $\beta(W,Z)$, we should similarly have a function with a
finite limit $\beta_0$ for $W \to 0$; as for its dependence on $W$
and $Z$, we assume it is only through the ratio $X/Z$ of BF
molecules binding to the chromosome and BF molecules in the fluid
environment.
In the following, we choose to deal with a ''minimal'' model, i.e.
consider a linear dependence on $w$ for both these quantities; we
set \begin{equation}\label{AB} \alpha (W) = a_0 + a_1 W \ , \ \ \beta (W) = b_0
+ b_1 (W/Z) \ ; \end{equation} this implies of course \begin{equation} a (w) = a_0 +
a_1 N w \ , \ \ b (w) = b_0 + b_1 (w/z) \ . \end{equation}
The parameters $(a_0,a_1,b_0,b_1)$ are all positive (we could
always set one of these parameters equal to one by rescaling the
time unit.) We also stress that in view of the discussion by
Nicodemi and Prisco \cite{NP}, we should expect the cooperative
effects enhancing affinity should be predominant over those
depressing the escape rate; this means we should expect $a_1 \gg
|b_1|$.
With these choices, the evolution equations for densities
\eqref{eveq4} read \begin{equation}\label{eveq4AB}
\begin{array}{rl}
d x / d t =& (a_0 + a_1 N x ) \, (1 - x - y)
\ - \ [b_0 + (b_1/(1 - N(x+y))) N x] \, x \ , \\
d y / d t =& (a_0 + a_1 N y ) \, (1 - x - y)
\ - \ [b_0 + (b_1/(1
- N(x+y))) N y] \, y \ .
\end{array} \end{equation}
We are specially interested in the large $N$ regime. In order to
study this, it is convenient to rescale time (note that
eqs.\eqref{eveq4AB} become singular for $N \to \infty$) and pass
to consider as independent variable \begin{equation} \tau \ := \ N t \ . \end{equation}
We also set \begin{equation} \varepsilon \ := \ 1/N \end{equation} (hence $t = \varepsilon \tau$); the
equations \eqref{eveq4} read then \begin{equation}\label{eqseps}
\begin{array}{rl}
d x / d \tau =& a_1 (1 - x - y) x + \varepsilon \, [a_0 (1 - x - y) -
b_0 x] \ + \ \varepsilon \, b_1 x^2 (x+y - \varepsilon)^{-1} \ , \\
d y / d \tau =& a_1 (1 - x - y) y + \varepsilon \, [a_0 (1 - x - y) -
b_0 y] \ + \ \varepsilon \, b_1 y^2 (x+y - \varepsilon)^{-1} \ ; \end{array}
\end{equation} at first order in $\varepsilon$, we have \begin{equation}\label{eqslim}
\begin{array}{rl}
d x / d \tau =& a_1 (1 - x - y) x \ + \ \varepsilon \, [a_0 (1 - x - y)
\ - \ (b_0 - b_1 x /(x+y)) x ] \ , \\
d y / d \tau =& a_1 (1 - x - y) y \ + \ \varepsilon \, [a_0 (1 - x - y)
\ - \ (b_0 - b_1 y /(x+y)) y ] \ . \end{array} \end{equation}
\section{Different time-scales for the dynamics}
\label{sec:timescales}
We want now to discuss some point concerning the qualitative
behavior of the dynamical system defined by \eqref{eveq4AB}; we
should consider it in the region (invariant under the dynamics) of
${\bf R}^2$ delimited by the coordinate axes and by the line $x+y=1$.
\begin{figure}
\begin{tabular}{|ll|}
\hline
(a) & \includegraphics[width=250pt]{fig1a.eps} \\
(b) & \includegraphics[width=250pt]{fig1b.eps} \\
(c) & \includegraphics[width=250pt]{fig1c.eps} \\
\hline \end{tabular}
\caption{Evolution of $x(t)$ (upper curves) and $y(t)$ (lower curves)
as a function of the rescaled time $\tau$, obtained numerically
integrating the equations \eqref{eqseps}.
The parameters defining the model have been set as follows:
$a_0 = 1$, $a_1 = 0.01$, $b_0 = 0.5$, $b_1 = 0.1$; while for $N$ we have
used $N=10000$, giving $\varepsilon = 0.0001$. Initial conditions are $x(0)= 0.011$,
$y(0)=0.01$.
(a): the initial expansion (this plot shows the evolution
for $0 \le \tau \le 10^3$); note the two densities grow together.
(b): on a longer timescale one observes how
the slow phase sets in
and the densities remain nearly constant over a long time
(this plot shows the evolution for $\tau \le 5 \cdot 10^4$).
(c): on a still longer timescale, the transition from the
saddle point $E_3$ to the stable equilibrium -- in this case,
$E_2$ -- is clearly visible and happens at an intermediate speed
(this plot shows the evolution for $\tau \le 2 \cdot 10^6$).}\label{fig1}
\end{figure}
\begin{figure}
\begin{tabular}{|ll|}
\hline
(a) & \includegraphics[width=250pt]{fig2a.eps} \\
(b) & \includegraphics[width=250pt]{fig2b.eps} \\
(c) & \includegraphics[width=250pt]{fig2c.eps} \\
\hline \end{tabular}
\caption{Evolution of $\xi(t) = x(t)+y(t)$ (dotted curves) and
$\eta(t)= x(t) - y(t)$ (solid curves) as a function of the
rescaled time $\tau$, obtained numerically
integrating the equations \eqref{eqseps}.
Parameters and initial conditions as in Figure \ref{fig1}.
(a): in the initial expansion $\xi$ grow abruptly, while $\eta$
remains near to zero (this plot shows the evolution
for $0 \le \tau \le 10^3$).
(b): in the slow phase both $\xi$ and $\eta$
remain nearly constant over a long time
(this plot shows the evolution for $\tau \le 5 \cdot 10^4$).
(c): in the intermediate speed phase, $\xi$ remains nearly constant, while
$\eta$ undergoes a relatively fast growth
(this plot shows the evolution for $\tau \le 2 \cdot 10^6$).}\label{fig2}
\end{figure}
\begin{figure}
\begin{tabular}{|ll|}
\hline
(a) & \includegraphics[width=250pt]{fig3a.eps} \\
(b) & \includegraphics[width=250pt]{fig3b.eps} \\
(c) & \includegraphics[width=250pt]{fig3c.eps} \\
\hline \end{tabular}
\caption{Evolution of $\xi'(\tau)$ (dotted curves) and
$\eta'(\tau)$ (solid curves) resulting from the numerical
integration of the equations \eqref{eqseps}.
Parameters and initial conditions as in Figure \ref{fig1}.
(a): in the initial expansion $\xi' (\tau)$ reaches values of order
$2.5 \cdot 10^{-3}$, while $\eta' (\tau)$ remains smaller than $7 \cdot
10^{-5}$.
(b): in the slow phase both speeds are very small, but $\xi' < 10^{-8}$
while $\eta' \simeq 10^{-7}$.
(c): in the intermediate speed phase, $\xi'$ remains extremely small (in this
case we actually have $\xi' \simeq 10^{-9}$), while
$\eta'$ slightly grows again; in this case $\eta' \simeq 10^{-6}$.}\label{fig3}
\end{figure}
As noted above, the symmetric fixed point $E_3$ is a saddle point
for the dynamics. It is easy to see that its (invariant) stable
manifold \cite{Gle,GuH,Ver} is just the line $y=x$.
Let us consider the situation where at first very few BF molecules
are binding to either one of the X-chromosomes, i.e. $x(0) \approx
0 \approx y(0)$; note that as $x(0) \approx y(0)$, the initial
data are close to the stable manifold for the saddle point $E_3$.
In a first stage, both $x(t)$ and $x(t)$ will grow exponentially
in $\tau$, with exponent $a_1$; this is easily seen from
\eqref{eveq4AB} (assuming $x \ll 1$, $y \ll 1$). More precisely,
near the origin one has $dx/d \tau \approx a_1 x$, $d y / d \tau
\approx a_1 y$; it follows that in this first phase, $x(t) \approx
y(t)$, i.e. the dynamics remains in a neighborhood of the
symmetric line $x=y$.
This fast growth will go on until the system gets near to $E_3$
(note this also means $1 - (x + y) = O (\varepsilon)$); at this point, a
slower evolution gets in. Once it reaches the vicinity of $E_3$,
the dynamics is dominated by the expansion rate near $E_3$, i.e.
by the positive eigenvalue $\lambda_2 (E_3) \approx (b_1/2) \varepsilon$, and
evolution takes place at a speed of order $\varepsilon$.
However, at some point (after a time of order $1/\varepsilon$) the system
gets far enough from $E_3$ to have again a dynamics non fully
described by the linearized system identified by $M_3$, and a
somewhat faster -- albeit not as fast as in the first phase --
evolution can take place.
This will lead the system near one of the non-symmetric equilibria
(depending on the initial data), where again the dynamics is
dominated by the eigenvalue which is smaller in modulus; this is
$\lambda_2 (E_{1,2}) = \varepsilon b_1$, i.e. the system gets again a speed
of order $\varepsilon$, and thus approaches the final equilibrium, as
already discussed, with a rate of order $\varepsilon$.
Summarizing, the evolution of our system will have (assuming
initial conditions are near to zero) four distinct phases:
\par\medskip\noindent
(1) A phase of rapid growth, exponential with expansion rate of
order $a_1$;
\par\noindent
(2) A slow phase (speed of order $\varepsilon$) spent in the vicinity of
the saddle point $E_3$;
\par\noindent
(3) An intermediate (moderate) speed phase, in which the system
travels from a neighborhood of $E_3$ to a neighborhood of either
ones of the asymmetric stable equilibria $E_{1,2}$ with a speed
substantially higher than $\varepsilon$;
\par\noindent
(4) A new slow phase, in which the system approaches exponentially
the stable equilibrium at a rate of order $\varepsilon$.
\medskip
This behavior is clearly shown in Figure \ref{fig1}, where we plot
the numerical solution to the full equations for $x(\tau)$ and
$y(\tau)$ (that is, without truncation to first order in $\varepsilon$)
on different time scales.
In Figure \ref{fig2}, we look at the evolution of the quantities
$\xi (t) := x(t) + y(t)$ and $\eta (t) := x(t) - y(t)$,
representing the total fraction of BF molecules binding to either
one of the X-chromosomes and the symmetry breaking measure
respectively. This again shows clearly that the overall evolution
of the system goes through the different phases enumerated above.
Confirmation is also obtained by Figure \ref{fig3}, where we plot
$d \xi / d \tau$ and $d \eta / d \tau$ on different timescales.
\section{Fluctuations}
\label{sec:noise}
The model equations \eqref{eqseps} or the limit ones
\eqref{eqslim} only deal with evolution of average quantities,
i.e. do not take into account in any way fluctuations. Our present
task is to go beyond this level of description, and take into
account {\it fluctuations} around the average dynamics.
As a first approximation, these can be taken into account by
transforming the equations \eqref{eqseps} or \eqref{eqslim} into
stochastic differential equations \cite{Gue,Oks,vKam} by adding a
noise term (see the Appendix for details). In other words, if the
deterministic equations are $dx/d\tau = f(x,y)$, $d y / d \tau =
g(x,y)$, we will write \begin{equation}\label{eqsepsnoise}
\begin{array}{rl}
d x =& f(x,y) \, d \tau \ + \ \gamma \, d \omega (\tau) \ , \\
d y =& g(x,y) \, d \tau \ + \ \gamma \, d \omega (\tau ) \ ,
\end{array} \end{equation}
with $\omega (\tau)$ a standard Wiener process and $\gamma$ a
parameter describing the size of fluctuations. Note that (as these
are fluctuations for an average) $\gamma$ will depend on $N$ and
hence on $\varepsilon$; more precisely, $\gamma \simeq \gamma_0 \sqrt{\varepsilon}$.
This dependence -- and the estimate $\gamma_0 \le \gamma_0 =
\sqrt{a_1}/2$ -- can be obtained by a standard procedure.
At first order in $\varepsilon$, we have \begin{equation}\label{eqslimnoise}
\begin{array}{rl}
d x =& \[ a_1 (1 - x - y) x + \varepsilon [a_0 (1 - x - y) \right. \\
& \left. - (b_0 - b_1 x /(x+y)) x ] \] d \tau + \sqrt{\varepsilon} \gamma_0 d \omega (\tau) \ , \\
d y =& \[ a_1 (1 - x - y) y + \varepsilon [a_0 (1 - x - y) \right. \\
& \left. - (b_0 - b_1 y /(x+y)) y ] \] d \tau + \sqrt{\varepsilon} \gamma_0 d \omega (\tau)
\ . \end{array} \end{equation}
When $1-x-y = O(\varepsilon)$, the noise term could
-- depending on the relation between $\varepsilon$, $a_1$ and $\gamma_0$ --
dominate the dynamics. That is, in the slow phase near the saddle
point (see the discussion in section \ref{sec:timescales}) random
motion due to fluctuations could be dominant: if $\gamma$ is large
enough, the time spent in a neighborhood $\mathcal{B}$ of $E_3$
will be determined by the escape time needed to exit $\mathcal{B}$
due to fluctuations. This will produce an acceleration of the
dynamics with respect to the purely deterministic one -- and
possibly could cause the system to cross the line $x=y$ and end up
near the other stable equilibrium (this happens in the numerical
simulation plotted in figure \ref{fig4}). On the other hand, if
$\gamma$ is small enough then the fluctuations will cause just a
twiggling of trajectories around deterministic ones.
\begin{figure}
\begin{tabular}{|c|}
\hline
\includegraphics[width=250pt]{fig4.eps} \\
\hline
\end{tabular}
\caption{Simulation of the stochastic process defined by \eqref{eqslimnoise};
the parameters of the model have been chosen as in the previous Figures (see
caption to Figure \ref{fig1} for their values), and we have set $\gamma = 10^{-4}$.
In this case the fluctuation do not seriously modify the dynamics, and the
discussion based on the deterministic model confirms its validity.}\label{fig4}
\end{figure}
Obviously dominance of fluctuation in the dynamics will take place
once the system gets near -- or very near if $\gamma$ is small -- to
the stable equilibrium point $E_1$ or $E_2$. There will be a
metastable equilibrium distribution around $E_{1,2}$ and -- over
extremely long times -- large fluctuations could drive the system
near to the other stable equilibrium; the equilibrium distribution
and the time needed for large fluctuations to appear can be
estimated via standard stochastic processes techniques
\cite{vKam}.
In Figure \ref{fig4} we show a numerical simulation of (the
stochastic process defined by) the stochastic differential
equations \eqref{eqslimnoise}.
\section{Conclusions}
We have provided a mean-field formulation of the model recently
proposed by Nicodemi and Prisco \cite{NP} for the X-chromosome
inactivation in mammals \cite{LKP,CYZB,CTTB,NTL,NCB,Xclose}.
Like their model, the version proposed here explains
$X$-chromosome inactivation as the result of a {\it spontaneous
symmetry breaking} in the dynamics of blocking factors molecules
binding to the X-chromosome inactivation centers (Xic). While
their approach was based on a single-agent model, in our version
the considered equations deal directly with average quantities.
After discussing a general formulation of the model, we considered
a minimal version, in which the affinity grows linearly with the
number of BF molecules already binding to the Xic and the rate of
escape of BF molecules increases linearly on the ratio between
density of BF molecules binding to the X-chromosomes and of BF
molecules floating in the fluid environment with the same number.
Our model contains four positive parameters, related to affinity
and escape probability for the BF molecules.
We showed that when $b_1 < b_0$, non-symmetric equilibria exist;
under the same condition they are stable (while symmetric
equilibria are unstable). Thus in such case we have a {\it
spontaneous symmetry breaking}, i.e. non-symmetric equilibria
describe the asymptotic state of the system. We were also able to
estimate the rate of approach to equilibrium as a function of the
parameters describing the system and the total BF population.
A qualitative discussion showed that when the initial situation
corresponds to most of the BF molecules floating in the fluid, one
should expect the dynamics to undergo different phases. We ran
some numerical simulations (one of these is shown in the Figures)
confirming the conclusions reached by the qualitative discussion
mentioned above. We also briefly discussed how fluctuations can be
taken into account within our scheme.
Finally, we would like to stress that our model differs from that
of Nicodemi and Prisco only in the mathematical description it
considers, while the physical basis is the same. Thus, we are just
providing an alternative description -- in terms of dynamical
systems and ODEs rather than of stochastic processes -- for the
Physics described by Nicodemi and Prisco in their paper \cite{NP}.
This mathematically simpler description allows to make more
detailed predictions, in particular about different timescales in
the dynamics; these were fully confirmed by numerical simulations.
Our model, like the one of Nicodemi and Prisco \cite{NP}, did not
take into account co-localization; work is in progress to take
this into account within the present description of
X-inactivation.
\section*{Appendix}
\label{sec:noise:technical}
In this Appendix we will briefly discuss the passage from the
physical model of section \ref{sec:model} to the stochastic
differential equations \eqref{eqsepsnoise} and
\eqref{eqslimnoise}.
In order to take into account full detail of fluctuations in our
model, we should consider the flows $\Phi_{ij}$ of particles from
state $i$ to state $j$. Denoting by $\nu_{ij} (\de t)$ the number
of particles passing from state $i$ to state $j$ in a time
interval of length $de t$, we have of course \begin{equation} \Phi_{ij} =
\nu_{ij} - \nu_{ji} \ . \end{equation} The transition probabilities $p_{ij}$
for a single particle are given in section \ref{sec:mod:TP}; the
$\nu_{ij}$ will thus be Poisson distributed with parameter
$\lambda_{ij} = p_{ij} n_i$, where $n_i$ is the population of state
$i$ at the beginning of the time interval.
Needless to say, if we look just at expectation values we obtain
again \eqref{eveq1}; however, our present task is to go beyond
this level of description, and take into account first and higher
order momenta of the Poisson distribution -- that is, in physical
terms, {\it fluctuations} around the average dynamics.
Let us fix our attention, for the sake of concreteness, on the
population $X$ (or density $x = X/N$) of the state 1 and its
variation $\de X$ (or $\de x$) in a time interval of length $\de
t$; we also write $Z$ and $z$ rather than expressing these
quantities in terms of $(X,Y)$ and $(x,y)$, for ease of notation.
We obviously have $de X = \nu_{31} - \nu_{13}$. Recall that
$\nu_{31}$ is Poisson distributed with parameter $\lambda = \alpha (X) Z
\de t$ (i.e. $\mathcal{P} (\nu_{31} = k) = \lambda^k e^{-\lambda}/k!$),
and $\nu_{13}$ is also Poisson distributed but with parameter $\mu
= \beta (X,Z) X$. Thus $\Phi_{31}$ is Poisson with mean $\lambda - \mu$
and variance $\sigma^2_{31} = (\lambda + \mu )$. Thus for $X$ we have the
stochastic differential equation $ d X = (\lambda - \mu) d t + \sigma d
\omega(t)$; in the case of our minimal model this reads
\begin{equation}
d X = [(a_0 + a_1 X) Z - (b_0 + b_1 X/Z)X] d t
+ \sqrt{(a_0 + a_1 X) Z - (b_0 + b_1 X/Z) X} d \omega(t) \ . \end{equation}
Passing to density variables this reads $d x = N^{-1} [(a_0 + a_1
N x) N z - (b_0 + b_1 x/z) N x] d t + N^{-1}[(a_0 + a_1 N x) -
(b_0 + b_1 x/z) N x ]^{1/2} d \omega(t)$; we should also pass to the
rescaled time variable $\tau = N t$, so that $dt = (1/N)(d \tau )$
and $d w (t) = (1 / \sqrt{N}) d \omega (\tau )$.
In this way we get, using again $N^{-1} = \varepsilon$,
\begin{equation}
d x = [a_1 x z + \varepsilon (a_0 z - (b_0 + b_1 x/z) x ] d \tau
+ \sqrt{a_1 x z + \varepsilon (a_0 z - (b_0 + b_1 x/z) x}
\sqrt{\varepsilon} \, d \omega(\tau) \ . \end{equation}
We thus have a noise of intensity $\gamma = \^\gamma_0 \sqrt{\varepsilon}$; note
that at order zero in $\varepsilon$, $\^\gamma_0 = \sqrt{a_1 x z}$; as the
sum of $x$ and $z$ cannot be higher than one, this can be
estimated by $\^\gamma_0 \le \gamma_0 = \sqrt{a_1}/2$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,692 |
Zapora Hardap (niem. Hardap-Damm) – betonowa zapora wodna, zbudowana na rzece Fish River w Namibii w Afryce, niedaleko miasta Mariental w regionie Hardap.
Poprzez budowę długiej na 860 metrów i wysokiej na 40 metrów zapory, rzeka Fish River została spiętrzona na długości około 30 km, tworząc zbiornik wodny o powierzchni ok. 28,7 km². Budowa tamy została ukończona w 1962 roku. Powstały zbiornik zawiera około 294,5 milionów metrów sześciennych wody i zaopatruje okoliczne farmy oraz Mariental w słodką wodę. Zapora ma wielkie znaczenie dla rolnictwa. Zmagazynowana woda pozwala na lepsze zaopatrzenie w wodę farm, co umożliwia uprawę warzyw i owoców na większą skalę. Ponadto utworzono tutaj strefę rekreacyjną Hardap, z kilkoma ośrodkami wypoczynkowymi.
Galeria
Przypisy
Hardap | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,261 |
I'm a Maryland native and could never imagine living anywhere else. I married my high school sweetheart for time and all eternity in the Washington D.C. Temple after I fell in love with his endless sense of humor. I enjoyed a short stint as a labor and delivery nurse before becoming a stay-at-home mom for my two beautiful daughters. In addition to my new-found love of writing, my passions are: my faith in Jesus Christ, gardening, and yoga. My favorite delights are decadent homemade cakes, cookies, or brownies—never store-bought.
I started in February of 2013 and have one completed novel published on Createspace and two manuscripts in the revisions stages for a chronicle of three books. It is about immortal beings, dragons, and the fight between light and dark inside each of us and around us.
My latest novel is Master of Lies (Chronicles of the Half-Emrys, #1).
I started writing this book back when I was doing a Lord of the Rings movie marathon. I even based some of my characters off of Tolkien's. I imagined the rolling hills of Middle Earth for my realm of Terrin and the murky swamps of the Dead Marshes for Rolant. The torturous snow peaks of the Misty Mountains could be my Eirwen Mountains, where the entrance to Gorlassar is hidden. And when you enter Gorlassar and fly to the capital Emryn city called Mared, I imagined the majestic music from when the Fellowship entered the great hall in the Mines of Moria.
I would have to say music is what carried the feeling in my book. I mentioned the music from the mines of Moria scene. I did listen to the LOTR's sound track quite a bit in the early phases of my free writing. But I took any music that uplifted or motivated me, from Evanescence's Fallen CD to the Piano Guys. I think Twilight's various sound tracks for all of their movies was a major inspiration for me. Evenstar (LOTR) was my inspiration for the final scene before the epilogue in my book.
Every hour I jump up and dance to a song in my kitchen. I read somewhere that exercising five minutes per hour can prevent the detrimental effects of sitting. Plus it revs me up, and I enjoy my day more. You'd be surprised how fast an hour flies by when you are elbow deep in a novel.
My favorite childhood author was Lois Duncan. But more recently, The Shatter Me series by Tahereh Mafi showed me that writing is an art. The way she even used her formatting to evoke the character's emotion opened my eyes. I was blown away. Brandon Mull is my ultimate favorite author. He speaks to the kid in me.
I am currently working on Book Two in my half-Emrys Chronicles. The Two Masters is in the revision stages, and I hope to release it in the summer of 2015. The Chronicle picks up after my first book, The Master of Lies, when Ahnalyn's son, Einion, is about to be crowned king of Terrin.
Still learning this. I just stuck my foot in the door. I have been looking into free sites to list my book and filling out author interviews.
Get a support group. Join a critique group. Free write without stopping until the book is done. Forget the grammar at first. Just get it down. Find people to critique, beta-read, and proofread. But have fun. And don't let anyone tell you how to write your book. Take the advice and use what works for you.
And remember, only write the parts people want to read!
The protagonist should have a terrifically hard time getting what she wants. The more obstacles thrown in the way of her progress, the better, but the best obstacles are ones that come from her trying one thing or another that goes terribly wrong, despite her best efforts. The reader wants her to triumph in the end (or else fail miserably if she's actually evil, and her goals are simply self-serving).
Once I complete my Half-Emrys series, I have several spin-offs planned. Lost Emrys. Fallen Emrys. I'm stuck in this world. It's become my life.
The Book of Mormon. The Shatter Me Series. The Fablehaven Series. Okay, I cheated. So that's more like ten books. | {
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} | 8,682 |
Поганий хлопчик Пека () — американська кінокомедія режисера Сема Вуда 1921 року.
Сюжет
У ролях
Джекі Куган — Генрі Пек
Вілер Окман — доктор Мартін
Доріс Мей — Летті (сестра Генрі)
Реймонд Гаттон — батько Бадді
Джеймс Корріген — Джордж Пек (батько Генрі)
Лілліан Лейтон — Ма Пек
Чарльз Гаттон — Бадді (друг Генрі)
Кейті Стівенс — молода дівчина з морозивом
Собака Квіні
Дін Райснер
Посилання
Кінокомедії США
Фільми англійською мовою | {
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} | 14 |
This centrally located house is warm & welcoming, spacious & stylish. With 2002 asf, highlights: 4 bedrooms + a bonus rooms, a generous living room with over-sized vinyl windows, hardwood floors, & wood stove, & 2nd family room downstairs! More? An open kitchen with ample storage, dining room, & access to the great deck overlooking the level, fenced yard complete with sprinkler system. Add in efficient gas heat, on-demand hot water, work-shop/garage, & RV parking...& you've just unlocked your home for the holidays! | {
"redpajama_set_name": "RedPajamaC4"
} | 9,363 |
Dierks Bentley has a deep connection to the mountains out West. He grew up in Arizona, climbing those mountains and exploring everything the outdoors had to offer. Mountain living was the impetus that eventually formulated Dierks' latest album, The Mountain, so it was a very natural thing.
Dierks is currently scaling the charts with his latest single, "Burning Man," which features his tour mates Brothers Osborne. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,964 |
Journal of Fluid Mechanics (2)
Highlights of Astronomy (1)
Ryan Test (2)
Axial creeping flow in the gap between a rigid cylinder and a concentric elastic tube
S. B. Elbaz, A. D. Gat
Journal: Journal of Fluid Mechanics / Volume 806 / 10 November 2016
We examine transient axial creeping flow in the annular gap between a rigid cylinder and a concentric elastic tube. The gap is initially filled with a thin fluid layer. We employ an elastic shell model and the lubrication approximation to obtain governing equations for the elastohydrodynamic interaction. At long axial length scales viscous forces are balanced by elastic tension, while at shorter length scales the viscous–elastic balance is achieved by means of an interplay between elastic bending, tension and shear stresses. Based on a viscous gravity current analogy in the tensile–viscous regime, we devise propagation laws for displacement flows which are induced by a variety of boundary conditions and examine different limits of the prewetting thickness. Next we focus on the moving elastohydrodynamic contact line at the edge of a penetrating film. A uniform matched asymptotic solution connecting the interior tension-based region with a boundary layer region near the propagation front is presented. Finally, a constructive example is shown in which isolated moving deformation patterns are created and superimposed to form a travelling wave displacement field. The presented interaction between viscosity and elasticity may be applied to fields such as soft robotics and micro-scale or larger swimmers by allowing for the time-dependent control of an axisymmetric compliant boundary.
Dynamics of viscous liquid within a closed elastic cylinder subject to external forces with application to soft robotics
Viscous flows in contact with elastic structures apply both pressure and shear stress at the solid–liquid interface and thus create internal stress and deformation fields within the solid structure. We study the interaction between the deformation of elastic structures, subject to external forces, and an internal viscous liquid. We neglect inertia in the liquid and solid and focus on viscous flow through a thin-walled slender elastic cylindrical shell as a basic model of a soft robot. Our analysis yields an inhomogeneous linear diffusion equation governing the coupled viscous–elastic system. Solutions for the flow and deformation fields are obtained in closed analytical form. The functionality of the viscous–elastic diffusion process is explored within the context of soft-robotic applications, through analysis of selected solutions to the governing equation. Shell material compressibility is shown to have a unique effect in inducing different flow and deformation regimes. This research may prove valuable to applications such as micro-swimmers, micro-autonomous systems and soft robotics by allowing for the design and control of complex time-varying deformation fields.
Crystallization Behavior of Hf-rich Aluminates and Influence on Film Dielectric Properties
M. Climent, B. Crivelli, G. Righini, S. Alberici, M. Alessandri, A. C. Elbaz, G. Pavia, C. Wiemer
Published online by Cambridge University Press: 28 July 2011, D10.7
In this study, the investigation of crystallization behaviour of Hf-rich aluminate is presented. Different alloys were deposited by ALCVD™ with composition ranging between 16 and 47 Al2O3 mol%. Post-deposition annealings were carried out in single or sequential mode using purified N2 at atmospheric pressure. Process temperature and time were varied from 700°C to 900°C and from 1' to 30' respectively. Upon these conditions, film thermal evolution was observed without any relevant increasing of in interface layer and any change in material composition. Measurements on 20 Al2O3 mol% films evidenced that thermal treatments up to 800°C promoted initial shrinking in thickness and material densification. Above 900°C, all considered aluminates were found to crystallize in orthorhombic phase maintaining original alloy composition. The higher the alumina content, the lower the grain size, the higher the crystallites density. Stability of orthorhombic crystalline structure was demonstrated upon single prolonged annealing up to 30' and upon sequential processes. In correspondence with film crystallization, enhancement of dielectric constant was detected with an increasing trend upon hafnia content. For 20 Al2O3 mol% aluminate, change in k form 19 to 40-45 was observed together with limited degradation in conduction and breakdown characteristics.
Physical-Chemical Evolution of Hf-aluminates upon Thermal Treatments
B. Crivelli, M. Alessandri, S. Alberici, D. Brazzelli, A. C. Elbaz, S. Frabboni, G. Ghidini, J. W. Maes, G. Ottaviani, G. Pavia, C. Wiemer
Published online by Cambridge University Press: 01 February 2011, D2.10
This study presents an investigation on physical-chemical stability of (HfO2)x(Al2O3 )1-x alloys upon prolonged post-deposition annealings. Two different Hf-aluminates were deposited by ALCVDTM, containing 34% and 74% Al2O3 mol% respectively. Post-deposition annealings (PDA) were carried out in O2 or N2 atmosphere, at 850°C and 900°C for 30 minutes. Interfacial layer (IL) increase after PDA was detected on all the samples, but with small differences between N2 and O2 treatments. Stack composition was characterized by means of XRR, XRF, RBS and TOF-SIMS. Growth of interface layer was justified by limited oxygen incorporation from external ambient. Silicon diffusion from the substrate into high-k material and aluminum/hafnium redistribution were observed and associated to annealing temperature. XRD and planar TEM analysis evidenced first grain formation and then, in the case of Hf-rich samples, almost complete crystallization. Overall, Hf-aluminates were found to remain XRD amorphous during high temperature prolonged treatments up to 900°C for 74% and 850°C for 34% alloys respectively. Differently from HfO2, (HfO2)0.66(Al2O3 )0.34 alloy was observed to crystallized in orthorhombic phase. Hf-aluminates were also electrically characterized by means of C(V) and I(V) measurements on basic capacitors. Variations in material electrical properties were found consistent with change in physical-chemical film structure. Increase in k value up to 30 was observed on Hf-rich samples crystallized in orthorhombic phase.
Deep Surveys and Cosmology
S. J. Oliver, S. Sergeant, P. Goldschmidt, R.G. Mann, M. Rowan-Robinson, N. Eaton, A. Efsathiou, C. Gruppioni, T.J. Sumner, B. Mobasher, A. Verma, L. Danese, E. Egami, D. Elbaz, A. Franceschini, I. Gonzalez-Serrano, M. Kontizas, A. Lawrence, R. Mcmahon, H.U. Nørgaard-Nielsen, I. Pérez-Fournon
Journal: Highlights of Astronomy / Volume 11 / Issue 2 / 1998
Published online by Cambridge University Press: 14 August 2015, pp. 1122-1124
Surveys with ISO (Kessler et al 1996), in particular with the CAM (Cesarsky et al 1996) and PHOT (Lemke et al 1996) instruments, will greatly extend our understanding of extra-galactic populations and their cosmological evolution. The main advantages that ISO surveys have over e.g IRAS are increased sensitivity/depth and wavelength coverage. Within the Guaranteed and Open Time programmes there are many field surveys which will efficiently map the limits in these parameters. In this talk I will briefly overview those surveys before concentrating in more detail on one survey in particular, the ISO survey of the Hubble Deep Field (HDF), to illustrate the kind of results that can be expected. | {
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} | 8,593 |
{"url":"https:\/\/solvedlib.com\/n\/ul-heel-jenes-1-u6-cangalata-0l-l-12-5,10593757","text":"# UL Heel Jenes 1 U6 cangalata 0l L 12.5\n\n###### Question:\n\nUL Heel Jenes 1 U6 cangalata 0l L 12.5\n\n#### Similar Solved Questions\n\n##### HzSO4 \/ methanolOCH; OHOCH;OCH;\nHzSO4 \/ methanol OCH; OH OCH; OCH;...\n##### 2F2FF47 1 N\n2F 2F F 4 7 1 N...\n##### Please help with both problems ! Question 19 (5 points) Given that the mass of the...\nPlease help with both problems ! Question 19 (5 points) Given that the mass of the Earth is 5.972 * 10^24 kg and the radius of the Earth is 6.371 * 10^6 m and the gravitational acceleration at the surface of the Earth is 9.81 m\/s^2 what is the gravitational acceleration at the surface of an alie...\nHow do I switch from LIFO to FIFO method on the balance sheet if the income for the last three years would have been $45,000 higher using the FIFO method. Use FIFO when calculating the revised stockholder's equity figure. 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For each equation, write short sentence explaining what IS wrong: (r) =x' Tk excxuras Crc Syfuse 40 b2 mltiphte Tosny 4+ Mo 10 onul es pouer _ Nov 9 4c corec+ Onsu\u00e2\u201a\u00ac X&#...\n##### Experiment 24 Advance Study Assignment State how and explain why the equilibrium 2X + Y 2 Z will shift if X is removed.b extra Y is added,some Z is added:\nExperiment 24 Advance Study Assignment State how and explain why the equilibrium 2X + Y 2 Z will shift if X is removed. b extra Y is added, some Z is added:...\n##### TABLE IISensilizalion of Intracellulor Phage to DNase by Freezing, Thawing, and Firalion toith FormaldehydeUnadorbed Infected cclls [rozen , thawed, Iniccted [rozen, dtcells aEred fxed fred fredLow speced sediment fraction +Total pe Acid-soluble_ Acid-soluble after DNex86 0.5 28Low spced supernatant fractionTotal pe Acid-solubk Acid-soluble after DNE ,0.85.5\nTABLE II Sensilizalion of Intracellulor Phage to DNase by Freezing, Thawing, and Firalion toith Formaldehyde Unadorbed Infected cclls [rozen , thawed, Iniccted [rozen, dtcells aEred fxed fred fred Low speced sediment fraction + Total pe Acid-soluble_ Acid-soluble after DNex 86 0.5 28 Low spced super...\n##### The random variables X and Y havc joint probability density function given by J3\/4 for - 1<I<1lyl <r, f(z,y) (0 otherwise.The support for (X,Y) is shaded in the following diagramCalculate the marginal probability density function fx of X. Calculate the marginal cumulative distribution function Fy of Y. Calculate P(X +Y > 3\/4)\nThe random variables X and Y havc joint probability density function given by J3\/4 for - 1<I<1lyl <r, f(z,y) (0 otherwise. The support for (X,Y) is shaded in the following diagram Calculate the marginal probability density function fx of X. Calculate the marginal cumulative distribution fun...\n##### Write MATLAB function that can calculate the determinant of a n x n matrix Your program should be carry out the following functions: Shows the given matrix in the command window Determination of the given size of the matrices Notification of the user when the matrix is not square in size Performing the proper calculation of minors, cofactors etc. into each for loop Check the correctness of determinant found by the code by comparing the values calculated via MATLAB command, deto_ Display the valu\nWrite MATLAB function that can calculate the determinant of a n x n matrix Your program should be carry out the following functions: Shows the given matrix in the command window Determination of the given size of the matrices Notification of the user when the matrix is not square in size Performing ...\n##### Urr .Iruvavmly inuu dulu IVUIL) Chapter 3: Probability (Add and Mult) Score: 24.229999999999997\/38 31\/38 answered ments...\nUrr .Iruvavmly inuu dulu IVUIL) Chapter 3: Probability (Add and Mult) Score: 24.229999999999997\/38 31\/38 answered ments \u2022 Question 17 The table summarizes results from pedestrian deaths that were caused by automobile accidents. Pedestrian Deaths Driver Pedestrian Intoxicated? Intoxicated? Yes N...\n##### Questions What functional group is present in triglyceride?2. What functional group is present in fatty acid?3. Draw the structure of oleic acid.4. Draw the structure of glyceryl triolein.What do lipids have in common?What type of solvent would be needed to remove an oil spot? Why?The melting point of stearic acid is 708C,and the melting point of oleic acid is 4*C. Explain in detail why their melting points are s0 different:Based on pan 3 of the experiment; which oil is more unsaturated, safflo\nQuestions What functional group is present in triglyceride? 2. What functional group is present in fatty acid? 3. Draw the structure of oleic acid. 4. Draw the structure of glyceryl triolein. What do lipids have in common? What type of solvent would be needed to remove an oil spot? Why? The melting ...\n##### FoddmnmECulrm ChrrabrCintni HociouajCHEM 231 @cdone @cionat Columin Chrenuiodrupy Poat -Lubcruin Cuoalion:RhlcaldordleenannnAtor nnnna Emn 4two comoonant Tal; ccllctng 5 tuctkns tra tncton: Noto tpottd ecnd GTn TC pat jeaulng ILC &tha Iructona Iren ia cdurra l pelued bolaw Vihd ottra \"dllovirg rascnscould have led ta ta rrauit?Seececnlco ILu udboibr:WnedrIFa uanrntonpoT Lotunt nom 415used a8ntint JonpO & sovcnt syeter Nas Usad 43 the obulnt Githa ccmpounos eired Ironi clmmn anmplu w\nFoddmnmE Culrm Chrrabr Cintni Hociouaj CHEM 231 @cdone @cionat Columin Chrenuiodrupy Poat -Lubcruin Cuoalion: Rhlcaldordleenannn Ator nnnna Emn 4two comoonant Tal; ccllctng 5 tuctkns tra tncton: Noto tpottd ecnd GTn TC pat jeaulng ILC &tha Iructona Iren ia cdurra l pelued bolaw Vihd ottra \"...\n\n-- 0.025037--","date":"2022-07-05 10:33:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.597349226474762, \"perplexity\": 14465.388137761965}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-27\/segments\/1656104542759.82\/warc\/CC-MAIN-20220705083545-20220705113545-00479.warc.gz\"}"} | null | null |
Council accused of "cultural vandalism" after destroying historic sculpture
The sculpture, created by town artist David Harding in 1972, has been removed and destroyed by Fife Council.
Fife Council has been accused of "cultural vandalism" after destroying one of Glenrothes' public sculptures rather than seeking to repair it.
A brick spiral, one of two created by Glenrothes' first appointed town artist David Harding in 1972, which was situated on land adjacent to Muirfield Drive in Newcastle, was removed by council workmen two weeks ago and destroyed.
Andrew Demetrius and Jeremy Howard from St Andrews Universary were responsible for the successful exhibition celebrating Goenrothes town art recently.
The work had been fenced off for the past three years after being damaged during the construction of new houses.
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Commenting on the decision to remove the art work, Lesley-Jane Robertson, lead officer – area housing management, said: "Roundell has been fenced off for a few years and was unable to be repaired.
"Following a request made from the chair of Glenrothes West Tenants and Residents Association on behalf of the local community, the decision to remove the Roundell was made by councillors at their recent ward 13 meeting."
However, the residents' group has moved to distance itself from the controversy claiming it alerted the council in a bid to make the sculpture safe.
"At no time did we indicate to the council that we wanted the historic sculpture removed or destroyed," said Lesley Bain, chairman.
"We were only acting on the information from residents that children had been able to get past the safety fencing and were playing on the damaged spiral and that we wanted the council to make it safe.
"We are as dismayed as everyone else that a long-standing and much enjoyed example of the town's public art has now been removed for good."
Ironically, the removal comes as the town plays host to an exhibition celebrating the unique collection of public art and sculpture and the 50th anniversary of Mr Harding's appointment as the first town artist in the UK.
With a growing appreciation of the town's public art collection and its cultural significance, Linda Ballingall, chairman of Glenrothes Heritage Centre, said the move was an act of "cultural vandalism" and "ill judged".
She said: "It's most disappointing. The council needs to re-evaluate its stance on the town's art, not just take the easy option to remove the problem."
Jeremy Howard from St Andrews University, who along with colleague Andrew Demetrius, were responsible of the successful town art exhibition which took place in October, said he has also been left disappointed by the council's actions.
He said: "Of course I don't know how decisions to demolish or maintain are made, but I would have thought that such a work would have provided a great opportunity for potential construction-work apprentices to learn a trade; and that consideration should be given to how the artwork belongs to something bigger than itself and its particular location, ie, to the wider community and the uniquely imaginative sense of identity-shaping that makes Glenrothes a very special place.
"I would think, and many others I have spoken to are of like mind, not least David Harding, who has suggested the idea himself, that a preservation group should be formed, and that its role should be to press for sensitive care and timely maintenance of the town's works by those in charge.
"Consultation should be paramount and restoration shouldn't be piecemeal or partial. It should be regarded as an opportunity. We named our exhibition 'The Enduring Town Art of Glenrothes'.
"Unfortunately many visitors pointed out that the case of the vanished Brick Spiral is far from isolated.
"My hope would be that officials, townsfolk and advisors could work together to ensure that the material and integrity of the works really endures." | {
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Safari to the Suburbs 1: Luxuriant Refuse at the Pearl Fincher Museum
by Robert Boyd
I live in the suburbs. I know there are art-lovers here, but we are fairly uncommon--compared to those who live in hipper inner city neighborhoods. We are less concentrated in the population for the simple reason that the suburbs themselves are less concentrated. The suburbs place a high premium on outward conformity, so it's hard to tell who's artsy. Deeper inside the city, people are more likely to wear their artsiness outwardly--in their lawn art, their art cars, their fashions, their tattoos and hairstyles. But we suburbanites keep that stuff inside the walls of our homes. There are exceptions--the suburbs are too vast to be homogenous. But the problem with the suburbs is that there isn't enough of a concentration of artistic types to encourage arts institutions to form. Few people start galleries in the 'burbs. We hardly have any arts institutions.
But even so, there are exceptions. The Pearl Fincher Museum of Fine Art is one I've written about before. Located in Spring, it is near Steubner-Airline just north of FM 1960 on Cypresswood Dr. It's in a neighborhood that appears to be fairly upper-middle-class. The houses are the brick generica favored by Houston developers. Just to give you an idea of how spread out things are up there, the Woodlands is about 14 miles from the Pearl Fincher. Nonetheless, the Pearl Fincher has community support--that can be seen by their well-attended openings, and their website claims that they have over 2000 donors. (They are not a 501(c)3 organization, so their finances are not reported publicly. Correction: They are a 501(c)3, but the name of the non-profit is the Cypress Creek Fine Art Association, not the Pearl Fincher Museum of Fine Art.)
Most of their previous exhibits have been exhibits by local artists (including student shows) or exhibits culled from the collections of local collectors. This seems like a good exhibition strategy for a young (established in 2008) suburban museum. It's an approach that promotes buy-in from the local artistic community. (Indeed, the Pearl Fincher has had a series of shows with the subtitle "Northwest Houston Collects".) But it's not a viable long-term strategy, because once you have that buy-in, you need to occasionally show things that are not from your own backyard. You're nurturing a local scene, and that means paying attention to that local scene, which the Pearl Fincher is doing well, but it also means injecting new ideas into that scene.
So I was excited when I heard about Luxuriant Refuse, a contemporary group show curated by Melissa Grobmyer and featuring the work of Adela Andea, Johnston Foster, Alison Foshee, Sarah Frost, Gwyneth Leech, Shawne Major, Aurora Robson, Betsabeé Romero, and Paul Villinski.
Aurora Robson, Belch (aka Tarball), 2009, discarded PET bottles, tinted polycrylic, rivets, steel armature, mica powder
Aurora Robson's work may be familiar to Houstonians from The Great Indoors, her installation at the Rice Gallery, or her installation at the Rice Recreation and Wellness Center. Those colorful pieces contrast to this large black encrusted sphere, floating slightly above eye-level like a menacing alien spacecraft.
Adela Andea, Bioluminescence, 2012, pool noodles and cold cathode fluorescent lights mounted on wire mesh
Equally alien, but much more colorful is Bioluminescence by Adela Andea. I question the categorization of pool noodles as refuse--is she actually making sculpture out of thrown-away materials, or did she buy a bunch of pool noodles new? But who cares? It's a pretty work and it's something I haven't really seen before at the Pearl Fincher--a site specific installation. (I know Andea has presented this work elsewhere, but I suspect it varies depending on the place of installation.)
Johnston Foster, River Euphrates, 2004, traffic cones, wood, plastic trash cans, duct tape, Venetian blinds
Johnston Foster's work--animals assembled from castaway plastic detritus--fits the theme of the show better. River Euphrates, a rhino made out of traffic cones, was a crowd-pleaser. The inclusion of the oxpeckers on the rhino's back was a nice touch.
The thing with pieces like River Euphrates, Bioluminescence and Belch (aka Tarball) is that they look cool. One could imagine someone curating a show of art made of trash and debris that would have been much more austere, more intellectual, more challenging. The choice here was to be accessible. Is the audience being condescended to? Is there a calculation that this suburban audience might not appreciate more difficult work? Or to put it another way, is there a deliberate choice not to alienate the audience?
(Years ago, when I worked for The Comics Journal, we were about to run an interview with cartoonist Paul Chadwick, the creator of a comic called Concrete. I mentioned to one of the editors that to me, Chadwick seemed totally mainstream and therefore should not be included in our magazine. This editor wisely responded, "To you and me he may seem mainstream, but there are many readers for whom Chadwick is quite radical." This has stuck with me and seems appropos here.)
Alison Foshee, Ivanhoe, 2012, labels on canvas
The artists in this show veer between those who create work that is pretty and those whose work makes you think, "Wow, that's amazing." (Or sometimes both.) Alison Foshee uses product labels to create her beautiful, somewhat abstract flowers.
Betsabeé Romero, 2007, Tire, carved rubber truck tire
Betsabeé Romero's Tire belongs more in the "wow" category. Houstonians may remember his work which was included in the show Cosmopolitan Routes: Houston Collects Latin American Art at the MFAH.
Sarah Frost, Sign Off, 2011, discarded computer keyboard keys
Sarah Frost is also one of the "wow" artists. Sign-Off, like many of her pieces, is composed of old computer keyboard keys. These keys are not only trash (and therefore appropriate for inclusion in this exhibit), but they're obsolete as well. Sign-Off is a formal work of light and dark, but it is also a reminder of how we produce millions of these little things knowing that most will be thrown away after at most a few years.
Frost was one of the stand-out artists at the Houston Fine Art Fair last year, and her piece there was one of the ones that sold. (I will return to the subject of the HFAF below.)
Paul Villinski, Consonance, 1993-2006, gold leaf on found work gloves
Paul Villinski's Consonance is one work that is neither "pretty" nor "wow," and to me is perhaps the most intriguing work in the show. It combines the grungy (old work gloves) with the glamorous (gold leaf). The seven gloves are arranged horizontally so that the boundary between glove and gold leaf forms a kind of horizon line. He has, in a sense, created a landscape. This piece is from a series he did with gloves where he combined the sweaty, dirty work glove with some elegant addition, like gold leaf or embroidery.
I mentioned the Houston Fine Art Fair earlier. They sponsored this exhibit and had promotional material available for attendees on opening night. In addition, curator Melissa Grobmyer is a partner with M.K.G. Art Management, LLC., a private company that provides art acquisition and divestment services, appraisals, art inventory management, art leases and corporate archival services. Now it's not unusual for a museum show to be sponsored by a private company. It's the cost we have to pay to see exhibits in a country where the government doesn't typically pay the bills for art museums. But Luxuriant Refuse is a little different--both Hamptons Expo Group (parent company of HFAF) and M.K.G. are companies in the art business. And doing a show like this is a way to drum up business for themselves.
In fact, I think it's kind of brilliant (if ethically shaky). We know there are a lot of well-off people in North Houston. Some of them are into art, and the Pearl Fincher Museum has been cultivating those art-lovers for the past four years. M.K.G. would no doubt like to serve as art consultant for energy company executives in the Woodlands. HFAF would like the collectors who loan work to the Pearl Fincher--as well as aspirational collectors--to visit the art fair this September. I have no idea what deal was made between Hamptons Expo Group and the Pearl Fincher or between M.K.G. and the Pearl Fincher. But on the face of it, each party benefits. M.K.G. and HFAF get access to potential clients while the Pearl Fincher gets a very nice art exhibit. However, the exhibit is also risky for the Pearl Fincher--it's possible that people will see the museum as a shill for the two commercial enterprises. (Tyler Green would have a meltdown over this.)
I'm not sure how I feel. The blatant combination of art business and art museum feels a little skeevy. But I suspect that without the help of M.K.G. and Hamptons Expo Group, the Pearl Fincher would have found it difficult to mount a show like Luxurient Refuse. And that would have been a shame. It's the kind of show that can ease skeptical people into contemporary art. And I think this is an important task for museums--particularly those like the Pearl Fincher Museum located in the vast suburban plain far from the art center.
Posted by Robert Boyd at 1:46 PM
Labels: Adela Andea, Alison Foshee, assemblage, Aurora Robson, Betsabeé Romero, installation, Johnston Foster, Melissa Grobmyer, Paul Villinski, Sarah Frost, sculpture
I really enjoyed your post. Thanks for coming out to the Pearl again. You obviously spent a lot of time with the artwork and I appreciate your thoughtful evaluation.
I wanted to clarify one point. The Pearl Fincher Museum of Fine Arts is a 501(c)(3) organization, though you would not be able to find it under that name. Cypress Creek Fine Art Association was the original organization that raised funds for the building renovation before the Fincher family gave the funds and their mother's name to the facility. Our 501(c)(3) is still registered under that original name and there is a official dba filed with Harris County as Pearl Fincher Museum of Fine Arts. As I look back, this is something that is definitely not clear on our website.
Thanks again for your visit,
Rosemary Hickman
Education Director
Pearl Fincher Museum of Fine Arts
Robert Boyd June 22, 2012 at 3:39 PM
No wonder I couldn't find you guys in GuideStar. Thanks for the clarification.
Melissa Grobmyer July 14, 2012 at 11:18 AM
Dear Robert:
Thank you for your review. I want to address a couple of points you made in your assessment of Luxuriant Refuse.
My whole career, I have resisted the idea that art has to be austere, opaque, or challenging (whatever that means) to engage the viewer in a serious thought process or to qualify as "good" art. If something is beautiful or "cool," as you say, should that somehow disqualify the work as "good" and automatically become prosaic or pandering? Can't a work of art that is "cool" also invite the viewer to think more deeply about the nuance of a subject matter?
Furthermore, as Abstract Expressionism, Minimalism, Conceptualism, and Pop came to the foreground in the American art establishment in the mid 20th century, a decided sea-change in artist training and development encouraged "idea" and discouraged draftsmanship, or "craft".
The result is a contemporary art culture that undervalues skill and overvalues "concept." Add in an exploded art market and lack of traditional idea of connoisseurship, and guess what results: a lot of very high concept art whose opacity is its only virtue, and which is actually as deep as a puddle. Talk about a shill.
In my opinion, art should engage us on many levels. Beauty can be a virtue, as can "wow" factor. I have been thinking about the theme of the show for years now, and actually chose art that was decidedly not didactic or pedagogical, because I don't like being preached to. I touch on this in the exhibition brochure. The placement of the show in the suburbs had everything to do with finding an art space to mount the show in and nothing to do with somehow bringing suburbanites into a dialog about contemporary art. If that has happened, fantastic! But that was not the point.
Finally regarding your concerns about the ethics of mixing the art business, the museum business, and corporate money: I am sure you know that museums are supported by corporate and private money without which they would not exist. Museums have large development departments that specialize in raising money for the institution.
In exchange for their generosity, the corporation or private individual then gets to have their name on the building, exhibition brochure, or party invitation. They get to publicly communicate that they are embedded in the cultural landscape of a place. This buying and selling of influence is very much a part of the art world in the US, as you know. Ethics come in when the benefactor gains too much influence with the institution, unbalancing decision-making etc. This is admittedly a fine line that I will not argue here.
Furthermore, art dealers regularly donate or heavily discount works of art by artists they are promoting, in order to get the work into the museum and in order to increase the salability of the artist's work. This symbiosis is firmly in place, ethically suspect or not.
But specifically, what is "skeevy" about HFAF supporting an exhibition at the Pearl Fincher? I am not sure how it is any different from any other corporate support of any other exhibition at any other institution. Of course HFAF wants people in the Woodlands to know about the art fair. To pretend that corporations give money to cultural institutions out of the pure generosity of their corporate hearts is silly. But I am not sure what is unethical about this particular arrangement.
Finally, just to clarify my involvement, I proposed the exhibition to the Pearl Fincher months before my firm took on HFAF as a client. If my company garners a client out of the show, great, but that was not the point of mounting the show. It was just an idea I wanted to explore, having been thinking and writing about art in various capacities for 20 years. It worked out that the timing was perfect. We needed money to mount the show, and HFAF, as well as Crateworks and ANR, stepped up.
I am glad you enjoyed the show. It was fun to put together, and I hope to have the opportunity to do it again in the future.
Melissa Grobmyer
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The Gift Voucher couldn't be easier or more convenient, simply purchase, print it or email it as the perfect present for any occasion!
To redeem your recipient just simply need to email fleckybennett@hotmail.co.uk to be offered a choice of a public Flecky Bennett Ghost Walk, and once confirmed the gift voucher will be applied to the performance.
Once Gift Vouchers have been redeemed they become guaranteed tickets on the selected performance and therefore are strictly non exchangeable or refundable. | {
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Ping Chau ( ) es una pequeña isla de la Región Administrativa Especial de Hong Kong, en el sur de China. También es conocida como Tung Ping Chau (en chino: 東平洲). Tung (chino: 東, es decir, este) se antepone al nombre para evitar posibles confusiones con Peng Chau (坪洲), otra isla en Hong Kong con un nombre pronunciado idéntico en el idioma chino cantonés. Administrativamente, la isla forma parte del distrito de Tai Po, en los Nuevos Territorios.
Geográficamente, Ping Chau es una isla en alta mar situada en la esquina noreste de Hong Kong en la bahía de Mirs y que está cerca de la frontera con la provincia de Guangdong en China continental. La isla tiene una superficie de 1,16 km² y está formada de roca pizarra. La isla está mucho más cerca de China continental (4 km) que la masa principal de Hong Kong. Se encuentra cerca de Nan'ao de Dapeng. La isla además es el punto más oriental de Hong Kong.
El Parque Marino Tung Ping Chau fue declarado en 2001 como el cuarto parque marino de Hong Kong. Ocupa una zona marítima de alrededor de 270 hectáreas que encierra la isla de Ping Chau.
Véase también
Geografía de Hong Kong
Geografía de China
Referencias
Archipiélagos e islas de Hong Kong | {
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Horse rescue organization to hold 12th annual Benefit Horse Show
SAFE's event to be held Aug. 4-5
Redmond-based Save a Forgotten Equine (SAFE) will hold its 12th annual Benefit Horse Show Aug. 4-5 at Donida Farm Equestrian Center in Auburn.
Presented by F5 Networks, the proceeds from this two-day horse show will benefit SAFE's horse rescue program. SAFE is a nonprofit organization dedicated to the rescue, rehabilitation, retraining and adoption of neglected and abused horses.
There is something fun for every one — and every horse — at the SAFE Benefit Horse Show. Competitors of all ages will be vying for blue ribbons in dressage, hunters, English and western pleasure, halter, showmanship and trails. Special classes for rescued and senior horses celebrate the continued usefulness of horses that have been saved from at-risk situations.
Spectators are welcome to join in on the family-friendly fun. Entry to the show grounds and parking are free all weekend. Food and beverages are available for sale. No dogs will be allowed on the showgrounds.
Members of the public are also invited to a Saturday night party, sponsored by F5 Networks, 14 Hands Winery, and Sparkling Ice. Dinner will be woodfire pizza and fresh salad catered by Veraci Pizza and a summer favorite — Sub Zero Ice Cream. Diners will be entertained by their second annual Hobby Horse Challenge, where creative riders mount stick horses and perform dressage and trail maneuvers with a dash of humor sure to be enjoyed by all. The party will also include a performance by Broken Arrow Mounted Archers, of Arlington — a group of talented riders showcasing outstanding horsemanship paired with exciting archery skills.
Dinner tickets are $20 per person and are available online at http://show.safehorses.org.
SAFE provides its horses with needed veterinary care, hoof care and additional training to help ensure them a safe and productive future.
Once these horses are ready to go to permanent homes, SAFE undertakes a rigorous screening process to find ideal placements for them — carefully reviewing applications, checking references and conducting site visits. Horses are adopted out under contract, and SAFE maintains a responsibility to each horse's well-being throughout its lifetime.
SAFE works to support Animal Control divisions in King, Snohomish, Pierce, Whatcom and Skagit counties. SAFE also helps horse owners in the community who can no longer care for their animals.
Since 2005, SAFE has rescued and carefully re-homed more than 220 horses. SAFE is a grassroots organization that relies almost entirely on public donations to fund its programs. SAFE is a 501(c)(3) exempt organization, and all donations made to SAFE are tax-deductible.
For information about SAFE or about the SAFE Benefit Horse show, visit http://www.safehorses.org, or contact Bonnie Hammond, executive director, via email: bonnie@safehorses.org, or phone: 206-331-0006.
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\section{Introduction}
In this paper, we provide minimal network realizations for the class of impedance functions realized by series-parallel networks containing three energy storage elements and a finite number of resistive elements. Using the force-current analogy, the results in the paper are directly applicable to the design of both electric circuits without transformers, and mechanical controllers containing the inerter \cite{mcs02}. From a practical perspective, the mechanical applications are of particular interest, as space and cost constraints motivate the design of mechanical systems which have a simple configuration (e.g., series-parallel) and contain the least possible number of elements. Indeed, low complexity networks of the type considered in the paper are relevant to vibration control in a diverse range of application areas, including automotive vehicles \cite{chen_14}; railway vehicles \cite{fucheng_10, fucheng_12, jiang_vsd_12}; buildings \cite{fucheng_07, LNW_SVS, ZJN_LVSMS}; motorcycle steering systems \cite{Limebeer_steering2, Limebeer_Steering}; and aircraft landing systems \cite{YJN_MLGSS}. The use of passive components is particularly beneficial when there are safety or regulatory requirements, or when access to a reliable energy source cannot be guaranteed.
This paper is inspired by the modern control-based framework for the design of mechanical systems pioneered in \cite{mcs02}, which is reminscent of the behavioral notion of control by interconnection \cite{JWBAOIS}. In this framework, the design problem is to synthesize a dynamical controller taken from a broadly defined class (in this case, the class of series-parallel networks), to interconnect with a given environment (e.g., the connection of a vibration absorber between consecutive floors of a building). Familiar system-theoretic notions such as realizability and minimality take on a different perspective in this framework, resulting in a number of interesting open questions \cite{kalman2010, HugJSmOp, camwb, JW_LTIRLC}. Indeed, despite the relative simplicity of series-parallel networks, it is not known how to obtain a minimal series-parallel realization for an arbitrary impedance function.
To date, much of the literature on this topic has focussed on impedance functions of McMillan degree two (biquadratics). The paper \cite{LIN} provided minimal series-parallel realizations for the class of biquadratic impedances realized by series-parallel networks containing two energy storage elements and a finite number of resistors. An algebraic characterisation of this class of impedances was subsequently provided in \cite{JiangSmith11}. Somewhat surprisingly, it was shown in \cite{HugSmSP} that six energy storage elements are required in order to realize a so-called \emph{biquadratic minimum function} with a series-parallel network. Analogous results to those in \cite{LIN, JiangSmith11, HugSmSP} have also been obtained for general resistor-inductor-capacitor (RLC) networks (i.e., without the series-parallel restriction) in \cite{Reichert69, JZJ_TR, HugGMF}. We also note the papers \cite{ChenSmith09, Chen13, Chen15}, which consider the class of impedances realized by mechanical (or RLC) networks containing one inerter (capacitor), one damper (resistor) and a finite number of springs (inductors).
In contrast, there have been relatively few systematic studies of impedance functions of McMillan degree three (bicubics). A notable exception is the recent paper \cite{ZJ_ERBI}, which considered the realization of a special class of bicubic functions termed \emph{essential regular}, and provided a minimal series-parallel realization for each impedance in this class. The benefits of bicubic impedance functions were illustrated in that paper using an example from \cite{fucheng_9b}, which demonstrated a reduction in the wheel load index of a railway vehicle suspension by $36\%$ when using a suspension with a bicubic impedance as opposed to biquadratic. However, not all series-parallel networks with three energy storage elements have essential regular impedances. Thus, the purpose of the present paper is to solve the minimal realization problem for the class of impedance functions realized by series-parallel networks containing three energy storage elements and a finite number of resistors.
In addition to the 24 network structures presented in \cite{ZJ_ERBI}, we present a further 40 network structures required to realize the entire class. In contrast to existing studies, we introduce a novel continuity-based technique to eliminate resistors from non-minimal networks in order to obtain the aforementioned minimal series-parallel network realizations.
For comparison with existing literature, we state our results in terms of electrical series-parallel networks. However, as described earlier, the results have direct practical relevance to the design of mechanical controllers using the force-current analogy. Thus, in the following section, we describe this analogy and elaborate on the motivation for the paper in the context of passive mechanical control. The rest of the paper is then structured as follows. In Section \ref{sec:ncm}, we present a formal approach to network classification that serves to simplify the statement and proof of our main results. Sections \ref{sec:ac} and \ref{sec:blbqi} summarise relevant established results in the passive network synthesis literature.
Our main results then follow in Sections \ref{sec:nmgs} and \ref{sec:mgsbc}. In Lemmas \ref{lem:z30mgs}--\ref{lem:z03mgs} and Theorems \ref{thm:z12nmgs}--\ref{thm:z21nmgs}, we provide network realizations for the class of impedances realized by series-parallel networks containing at most three energy storage elements. The realizations are non-minimal (in the number of elements used) except in cases when all energy storage elements are of the same type. Minimal series-parallel realizations are then provided in Theorems \ref{thm:z12mgs}--\ref{thm:z21mgs}.
Finally, our notation is as follows. The real numbers are denoted $\mathbb{R}$; $\mathbb{R}^{n}$ denotes $n$-dimensional Euclidean space; $\Re{(z)}$ denotes the real part of the complex number $z$; and $\mathbb{R}[s]$ (resp., $\mathbb{R}(s)$) denotes the polynomials (resp., rational functions) in the indeterminate $s$ with coefficients from $\mathbb{R}$.
\section{Passive mechanical control}
\label{sec:pmc}
The relevance of this paper to passive mechanical control is due to an analogy between electrical and mechanical networks. This so-called force-current analogy is outlined in Fig.\ \ref{fig:ema}. Indeed, it was the absence of a mechanical equivalent to a capacitor which inspired the invention of the inerter \cite{mcs02}, a two-terminal passive device for which the force transmitted through the device is proportional to the relative acceleration of the two terminals (in contrast, a mass has the property that the force applied is proportional to its acceleration \emph{relative to a fixed point}, which is analogous to a \emph{grounded} capacitor). The force-current analogy establishes a one-to-one correspondence between electrical and mechanical networks, whereby resistors are replaced with dampers, inductors with springs, and capacitors with inerters. With these substitutions, the impedances of the corresponding networks are equal.\footnote{Here, we define the impedance as the transfer function from driving-point current (resp., transmitted force) to driving-point voltage (resp., relative velocity of the two terminals). Note that, in the mechanical case, this is often referred to as the admittance.}
Thus, the results in this paper
provide statements about minimal mechanical network realizations.
\begin{figure}[!t]
\scriptsize
\begin{center}
\leavevmode
\begin{comment}
\begin{psfrags}
\psfrag{el}[c][c]{\normalsize Electrical}
\psfrag{me}[c][c]{\normalsize Mechanical}
\psfrag{f2}[l][r]{spring}
\psfrag{f4}[l][l]{inerter}
\psfrag{f6}[l][l]{damper}
\psfrag{f8}[l][l]{inductor}
\psfrag{g1}[l][l]{capacitor}
\psfrag{g3}[l][l]{resistor}
\psfrag{i}[c][c]{$\displaystyle i$}
\psfrag{vm}[c][c]{$\displaystyle v^{-}$}
\psfrag{vp}[c][c]{$\displaystyle v^{+}$}
\psfrag{F}[c][c]{$\displaystyle F$}
\psfrag{sp}[l][l]{$\displaystyle \frac{dF}{dt} = k (v^{+}{-}v^{-})$}
\psfrag{Zs}[c][c]{$\displaystyle \frac{s}{k}$}
\psfrag{kg0}[l][l]{$\displaystyle k > 0$}
\psfrag{ma}[l][l]{$\displaystyle F=b \frac{d(v^{+}{-}v^{-})}{dt}$}
\psfrag{Zma}[c][c]{$\displaystyle \frac{1}{bs}$}
\psfrag{bg0}[l][l]{$\displaystyle b > 0$}
\psfrag{f5}[r][r]{$\displaystyle Z(s)\;=\frac{1}{c}$}
\psfrag{da}[l][l]{$\displaystyle F=c(v^{+}{-}v^{-})$}
\psfrag{Zd}[c][c]{$\displaystyle \frac{1}{c}$}
\psfrag{cg0}[l][l]{$\displaystyle c > 0$}
\psfrag{f7}[r][r]{$\displaystyle Z(s)\;=Ls$}
\psfrag{in}[l][l]{$\displaystyle L\frac{di}{dt} = (v^{+}{-}v^{-})$}
\psfrag{Zl}[c][c]{$\displaystyle Ls$}
\psfrag{Lg0}[l][l]{$\displaystyle L > 0$}
\psfrag{f9}[r][r]{$\displaystyle Z(s)\;=\frac{1}{C}s$}
\psfrag{ca}[l][l]{$\displaystyle i=C \frac{d(v^{+}{-}v^{-})}{dt}$}
\psfrag{Zc}[c][c]{$\displaystyle \frac{1}{Cs}$}
\psfrag{Cg0}[l][l]{$\displaystyle C > 0$}
\psfrag{g2}[r][r]{$\displaystyle Z(s)\;=R$}
\psfrag{re}[l][l]{$\displaystyle Ri= (v^{+}{-}v^{-})$}
\psfrag{Zr}[c][c]{$\displaystyle R$}
\psfrag{Rg0}[l][l]{$\displaystyle R > 0$}
\includegraphics[page=2,width=0.88\hsize]{wmrte_pics}
\end{psfrags}
\end{comment}
\includegraphics[page=2, width=0.88\hsize]{bicubic_pics}
\end{center}
\caption{Passive electrical and mechanical elements.}
\label{fig:ema}
\end{figure}
To illustrate the practical relevance of our results, we now consider a specific example. The paper \cite{ZJN_LVSMS} considered the problem of controlling vertical vibrations in a three-storey building by installing a passive mechanical vibration suppressor between the ground and first floor (see Fig.\ \ref{fig:mnc}). This design problem can be posed generally as a constrained optimization problem. For example,
minimize the $\|H\|_{\infty}$ norm of the transfer function from the base displacement to the maximum inter-storey displacement subject to the constraint that the controller can be realized by a series-parallel damper-spring-inerter network with a bound on the number of elements. Similar optimization problems can be posed for a wide range of other applications (see, e.g., \cite{chen_14, fucheng_10, fucheng_12, jiang_vsd_12, fucheng_07, LNW_SVS, Limebeer_steering2, Limebeer_Steering, YJN_MLGSS}), but are often very computationally demanding. As we argue in the following two paragraphs, the computational complexity of these design problems can be substantially reduced by fundamental studies of the type presented in this paper.
The paper \cite{ZJ_SIA} noted two approaches to an optimization problem of this type---the \emph{immitance-based} and \emph{structure-based} approaches---and also proposed a third hybrid method termed the \emph{structure-immitance approach}. In the immitance-based approach, the optimization is performed over all controller impedances taken from a certain class. Here, a key challenge is to identify an appropriate class of impedances. If this is defined too broadly, then there will be no guarantee that the impedance can be realized by a passive mechanical system. In many studies (e.g., \cite{smith_06}), the optimization is performed over the class of positive-real impedances (with an upper bound on the McMillan degree), in which case it is known that a passive mechanical realization necessarily exists. But the number of elements required to realize the optimal impedance cannot be predicted in advance, and is potentially large. For example, to realize a biquadratic \emph{minimum} function with a series-parallel network, six energy storage elements are required \cite{HugSmSP}; yet to realize a biquadratic \emph{regular} function requires only two energy storage elements \cite{JiangSmith11}. Thus, instead of optimizing over the class of positive-real impedances, it is preferable to optimize over the class of impedances which can be realized by networks of a given complexity. One contribution of this paper is to describe one such class of impedances, namely the class of impedances which are realized by series-parallel networks containing three energy storage elements (springs and inerters) and a finite number of resistive elements (dampers). We also provide explicit network realizations for every impedance in this class, and these networks contain the least possible number of elements in almost all cases.
\tikzset{ mn/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short] ++(4.0,0)
($(O)+(0.5,0)$) to[spring] ++(0,3)
($(O)+(3.5,0)$) to[twoport]
++(0,2.0)
to[spring] ++(0,1.0)
($(O)+(0,3.0)$) to[short] ++(4.0,0)
to[short] ++(0,0.3)
to[short] ++(-4.0,0)
to[short] ++(0,-0.3)
($(O)+(0.5,3.3)$) to[spring] ++(0,1.0)
($(O)+(0.5,5.3)$) to[spring] ++(0,1.0)
($(O)+(0,6.3)$) to[short] ++(4.0,0)
to[short] ++(0,0.3)
to[short] ++(-4.0,0)
to[short] ++(0,-0.3)
($(O)+(0.5,6.6)$) to[spring] ++(0,1.0)
($(O)+(0.5,8.6)$) to[spring] ++(0,1.0)
($(O)+(0,9.6)$) to[short] ++(4.0,0)
to[short] ++(0,0.3)
to[short] ++(-4.0,0)
to[short] ++(0,-0.3);}}
\begin{figure}[!b]
\centering
\normalsize
\begin{tikzpicture}[scale=0.4, every node/.style={transform shape}]
\pic at (0,0) [] {mn};
\end{tikzpicture}
\caption{Vertical vibration suppression in three storey building (see \cite{ZJN_LVSMS}).}
\label{fig:mnc}
\end{figure}
In the structure-based approach, the controller's configuration is fixed, and the element parameters are varied in order to optimize performance. This approach has the advantage of providing a guarantee on the controller's complexity. However, the approach fails to identify alternative (potentially simpler) controller configurations with improved performance. To overcome this drawback, it is possible to perform structure-based optimization on a number of different configurations. Indeed, the structure-immitance approach of \cite{ZJ_SIA} outlines a systematic way of considering all possible configurations of series-parallel networks which have a given fixed number of each type of element. However, except in particularly simple cases, it is unlikely to be computationally feasible to cover all configurations, since the number of series-parallel networks grows significantly as the number of elements is increased \cite{Lomnicki_SP}. Another contribution of this paper is to show that the impedance of any series-parallel network containing three energy storage elements and a finite number of resistors (dampers) can always be realized by one of a small number of series-parallel network configurations, each of which contains at most seven elements. With this result, the structure-immitance approach becomes computationally tractable when applied to networks containing at most three energy storage elements, irrespective of the number of resistors (dampers).
\section{Series-parallel networks: classification and minimality}
\label{sec:ncm}
We begin this section with some technical preliminaries on network classification, which simplify the statement and proof of our results. This formalism is similar to \cite{JiangSmith11, HugSmSP, MS_LCGA}. We then formally state the problem considered in this paper.
We define a series-parallel network in the manner of \cite{Riordan_42}. Specifically, an individual resistor, inductor or capacitor is a series-parallel network, and a network is series-parallel if it is either a series or parallel connection of two series-parallel networks. The impedance $Z$ of a series (resp., parallel) connection of two networks $N_{1}$ and $N_{2}$ with impedances $Z_{1}$ and $Z_{2}$ satisfies $Z = Z_{1} + Z_{2}$ (resp., $1/Z = 1/Z_{1} + 1/Z_{2}$). For a given series-parallel network $N$ with impedance $Z(s)$, there exists a series-parallel network whose impedance is $Z(1/s)$ (denoted $N^{i}$), and a series-parallel network whose impedance is $1/Z(s)$ (denoted $N^{d}$). The network $N^{i}$ is obtained by replacing inductors (with impedance $Xs$) with capacitors (with impedance $X/s$) and vice-versa; and $N^{d}$ is obtained by interchanging series and parallel connections and inverting the impedances of each element (which again replaces inductors with capacitors and vice-versa). In particular, $(N^{i})^{d} = (N^{d})^{i}$, and we denote this network by $N^{p}$. We will also define network classes (denoted $\mathcal{N}_{1}, \mathcal{N}_{2}$, etc) as sets which contain all networks of a given fixed structure, in addition to networks obtained by replacing certain resistors in this structure with open or short circuits (see, e.g., Fig.\ \ref{fig:bq1c1l}). Finally, for a given network class $\mathcal{N}$, we let $\mathcal{N}^{i} = \lbrace N \mid \exists N_{b} \in \mathcal{N} \text{ with } N = N_{b}^{i}\rbrace$, and the network classes $\mathcal{N}^{d}$ and $\mathcal{N}^{p}$ are defined similarly.
\begin{comment}
Purpose of this paper is to show the following result:
\begin{theorem}
\label{thm:hlts}
Let $Z$ be the impedance of a series-parallel network containing at most three energy storage elements. Then $Z$ is the impedance of a series-parallel network containing at most three energy storage elements and at most four resistors.
\end{theorem}
In addition, for any given $Z$ as in Theorem \ref{thm:hlts}, we provide a series-parallel network containing at most three energy storage elements and four resistors whose impedance is equal to $Z$.
\end{comment}
To formally state our results, and compare them with the existing literature, we next introduce the concepts of generic network classes and minimal generating sets
The impedance of a given RLC network always takes the form of a ratio of two polynomials
\begin{align}
p(s) &= p_{n}s^{n} + p_{n-1}s^{n-1} + \cdots + p_{1}s + p_{0}, \text{ and} \nonumber\\
q(s) &= q_{n}s^{n} + q_{n-1}s^{n-1} + \cdots + q_{1}s + q_{0}. \label{eq:pqd}
\end{align}
Here, the coefficients $p_{0}, \ldots , p_{n}, q_{0}, \ldots , q_{n}$ are all polynomial functions in the network's element values (inductances, capacitances, etc.) \cite{HMS_GEN}. Following \cite{HMS_GEN}, we call the set of impedances realised by a given network class $\mathcal{N}$ the \emph{realizability set} of $\mathcal{N}$, which can be characterised by the vector of coefficients $(p_{0}, \ldots , p_{n}, q_{0}, \ldots , q_{n})$ and viewed as a (semi-algebraic) subset of $\mathbb{R}^{2n+2}$. For any given $N \in \mathcal{N}$, the dimension of the realizability set of $\mathcal{N}$ ($m$) is no greater than one plus the number of elements in $N$ \cite[Lemma 2]{HMS_GEN}, and $\mathcal{N}$ is called generic if there exists $N \in \mathcal{N}$ which contains exactly $m-1$ elements \cite[Definition 1]{HMS_GEN}. It follows from \cite[Lemma 2]{HMS_GEN} that almost all networks from a given generic network class are \emph{minimal} in the sense that their impedance cannot be realized by a network containing strictly fewer elements.\footnote{More precisely, the set of impedances in the realizability set that can be realized with strictly fewer elements is a subset of the realizability set whose codimension is at least one.}
\begin{definition}[Generating/ minimal generating sets]
\label{defn:gs}
Let $\mathcal{Z}_{m,n}$ be the set of impedances realized by series-parallel networks containing at most $m$ capacitors and $n$ inductors; and let $\mathcal{Z}_{M} = \cup_{m,n \mid m + n = M} \mathcal{Z}_{m,n}$ be the set of impedances realized by series-parallel networks containing at most $M$ energy storage elements.
\begin{comment}
\begin{align*}
\hspace*{-0.3cm} \mathcal{Z}_{M} &\coloneqq \lbrace Z \mid Z \text{ is the impedance of a series-parallel network}\\
&\hspace{0.6cm}\text{containing at most $M$ energy storage elements}\rbrace,\text{ and} \\
\hspace*{-0.3cm} \mathcal{Z}_{m,n} &\coloneqq \lbrace Z \mid Z \text{ is the impedance of a series-parallel network}\\
&\hspace{0.6cm} \text{containing at most $m$ capacitors and $n$ inductors}\rbrace,
\end{align*}
\end{comment}
We call $\mathcal{N}$ a \emph{generating set} for $\mathcal{Z}_{m,n}$ (resp., $\mathcal{Z}$) if (i) $\mathcal{N}$ is a set of series-parallel networks, each containing at most $m+n$ (resp., $M$) energy storage elements; and (ii) for every single $Z \in \mathcal{Z}_{m,n}$ (resp., $Z \in \mathcal{Z}_{M}$), there exists a network $N \in \mathcal{N}$ whose impedance is $Z$. We call $\mathcal{N}$ a \emph{minimal generating set} for $\mathcal{Z}_{m,n}$ (resp., $\mathcal{Z}$) if (i) $\mathcal{N}$ is a generating set; and (ii) $\mathcal{N}$ is the union of generic network classes.
\end{definition}
\begin{problem}
\label{prob:fgs}
Given $\mathcal{Z}_{m,n}$ (resp., $\mathcal{Z}_{M}$) as in Definition \ref{defn:gs}:
\begin{enumerate}[label=(\alph*)]
\item Find a generating set for $\mathcal{Z}_{m,n}$ (resp., $\mathcal{Z}_{M}$).\label{nl:fgsa}
\item Find a minimal generating set for $\mathcal{Z}_{m,n}$ (resp., $\mathcal{Z}_{M}$).\label{nl:fgsb}
\end{enumerate}
\end{problem}
Clearly, any solution to problem \ref{prob:fgs}\!\ref{nl:fgsb} also solves \ref{prob:fgs}\!\ref{nl:fgsa}. Solutions to problem \ref{prob:fgs}\!\ref{nl:fgsb} for the cases $m=0$ and $n=0$ are provided by the so-called \emph{Foster} and \emph{Cauer} forms; and problem \ref{prob:fgs}\!\ref{nl:fgsb} has also been solved for the case $M = 2$.
As discussed in \cite{HugJSmOp}, both the increased complexity of algebraic manipulations, and the growth in the number of candidate network structures, present
considerable barriers to extending these
results to cases with $M > 2$. Thus, in contrast with existing approaches, we present a novel continuity-based argument to solve problem \ref{prob:fgs}\!\ref{nl:fgsb} in the case $M = 3$. This results in an implicit description of the sets $\mathcal{Z}_{1,2}$ and $\mathcal{Z}_{2,1}$ in terms of a quantifier (see, e.g., Theorem \ref{thm:bcmrac}). Remark \ref{rem:qe} then indicates how to compute an explicit description of $\mathcal{Z}_{3}$, and illustrates the algebraic complexity of this problem.
\section{Algebraic criteria for network realizations}
\label{sec:ac}
The impedance of a given RLC network always takes the form $Z = p/q$ for some polynomials $p, q$ as in (\ref{eq:pqd}). Here, no generality is lost by assuming that at least one of $p_{n}, q_{n}$ is non-zero, and $p, q$ are coprime (equivalently, the McMillan degree of $p/q$ is $n$).
In \cite{HugSmAI}, several necessary algebraic conditions were presented for a function $Z = p/q$ to be the impedance of an RLC network. These relate to the Sylvester matrices and their determinants:
\def\matriximg{%
\begin{matrix}
q_{n}& q_{n-1}& q_{n-2}& \cdots\\
p_{n}& p_{n-1}& p_{n-2}& \cdots\\
0& q_{n}& q_{n-1}& \cdots\\
0& p_{n}& p_{n-1}& \cdots\\
0& 0& q_{n}& \cdots\\
\vdots& \vdots& \vdots& \ddots
\end{matrix}
}%
\begin{align*} \mathcal{S}_{k}(p,q) &:=
\left. \left[\vphantom{\matriximg} \kern-\nulldelimiterspace \right. \overbrace{\matriximg}^{\text{$2(n-k)$ columns}} \right.
\left. \left.\vphantom{\matriximg}\right] \kern-2\nulldelimiterspace \right\} \text{\scriptsize $2(n-k)$ rows}, \\
\text{and } R_{k}(p,q) &:= \abs{\mathcal{S}_{k}(p,q)}, \hspace{0.15cm} (k = 0, 1, \ldots, n-1).
\end{align*}
Here, $R_{0}(p,q)$ is proportional to the resultant of $p$ and $q$ (and is equal in magnitude if $p_{n} \neq 0$ and $q_{n} \neq 0$), and $p$ and $q$ have at least $r$ roots in common (counting according to multiplicity) if and only if $R_{0}(p, q) = \cdots = R_{r-1}(p, q) = 0$ \cite{HugSmAI}. The following result is then shown in \cite[Theorem 8]{HugSmAI}:
\begin{lemma}
\label{lem:ai}
Let $p, q$ in (\ref{eq:pqd}) be coprime, and let $Z = p/q$ be the impedance of an RLC network $N$ containing at most $n$ energy storage elements. Then $R_{0}(p,q) \neq 0$, and the number of capacitors (resp., inductors) in $N$ is equal to the number of permanences (resp., variations) in sign in the sequence:
\begin{equation*}
1, R_{n-1}(p,q), \ldots , R_{1}(p,q), R_{0}(p,q).
\end{equation*}
In any subsequence of zero values, $R_{k}(p,q) \neq 0$, $R_{k-1}(p,q) = R_{k - 2}(p,q) = \ldots = 0$, signs are assigned to the zero values as follows: $\text{sign}(R_{k - j}) = (-1)^{j(j-1)/2}\text{sign}(R_{k})$
\end{lemma}
\begin{remark}
Let $m_{1}, m_{2}, n_{1}, n_{2}$ be integers with $M = m_{1} + n_{1} = m_{2} + n_{2}$ and $m_{1} \neq m_{2}$. From Lemma \ref{lem:ai}, if $Z \in \mathcal{Z}_{m_{1}, n_{1}}$ has McMillan degree $M$, then $Z \not\in \mathcal{Z}_{m_{2}, n_{2}}$.
\end{remark}
Lemma \ref{lem:ai} can also be stated in terms of the \emph{Bezoutian matrix} associated with the polynomials $p$ and $q$:
\begin{definition}
\label{def:bezd}
Let $p, q$ be as in (\ref{eq:pqd}). Then $\mathcal{B}(q,p)$ is the matrix whose entries $\mathcal{B}_{ij}$ satisfy
\begin{equation*}
\frac{q(z)p(w) - p(z)q(w)}{z - w} = \sum_{i=1}^{n}{\sum_{j=1}^{n}{\mathcal{B}_{ij}z^{i-1}w^{j-1}}}.
\end{equation*}
\end{definition}
From \cite[Section 6]{HugSmAI}, $R_{k}(p,q)$ is equal to the determinant formed from the final $n{-}k$ rows and columns of $\mathcal{B}(q,p)$. In particular, we note the following
\begin{corollary}
\label{cor:dbei}
Let $p,q$ in (\ref{eq:pqd}) be coprime, and let $Z = p/q$ be the impedance of an RLC network $N$ containing at most $n$ energy storage elements. Then $\abs{\mathcal{B}(q,p)} = R_{0}(p,q) \neq 0$, and $N$ contains an even number of inductors if and only if $\abs{\mathcal{B}(q,p)} = R_{0}(p,q) > 0$.
\end{corollary}
\begin{comment}
Finally in this section, we provide a lower bound on the number of elements required to realize the set $\mathcal{Z}_{n}$.
\begin{lemma}
\label{lem:mgsnr}
If $\mathcal{N}$ is a minimal generating set for $\mathcal{Z}_{n}$, then there exists a network $N \in \mathcal{N}$ which contains at least $2n+1$ elements.
\end{lemma}
\begin{IEEEproof}
Let $p$ and $q$ be as in (\ref{eq:pqd}). It suffices to show that
\begin{enumerate}
\item The set of $p_{i}, q_{i}$ ($i = 0, \ldots , n$) such that there exists $Z \in \mathcal{Z}_{n}$ with $Z = p/q$ is a (semi-algebraic) set of dimension $2(n+1)$.\label{nl:dsas}
\item
Let $\mathcal{N}$ be a network class, and let all networks in $\mathcal{N}$ have the same structure and contain at most $n$ energy storage elements and at most $m$ elements in total. Then the set of $p_{i}, q_{i}$ ($i = 0, \ldots , n$) such that $Z = p/q$ is the impedance of a network from $\mathcal{N}$ is a (semi-algebraic) set of dimension less than or equal to $2m+1$. \label{nl:updsas}
\end{enumerate}
Let $\mathcal{N}$ be the network class described in condition \ref{nl:updsas}, and suppose there exists $N \in \mathcal{N}$ whose impedance $Z$ takes the form $Z = p/q$. Denote the element values (i.e., the resistances of the resistors, inductances of the inductors, etc.) by $E_{i}$ ($i = 1, \ldots , m$), so $E_{i} > 0$ for $i = 1, \ldots , m$. Then, by Kirchhoff's tree formula, there exist $2(n+1)$ polynomial functions in the element values ($f_{0}, \ldots , f_{n}$, $g_{0}, \ldots , g_{n}$) such that the set described in condition \ref{nl:updsas} is the projection of the semi-algebraic set
\begin{align*}
p_{0} = kf_{0}(E_{1}, \ldots , E_{m}), \ldots , p_{n} = kf_{n}(E_{1}, \ldots , E_{m}), \\
q_{0} = kg_{0}(E_{1}, \ldots , E_{m}), \ldots , q_{n} = kg_{n}(E_{1}, \ldots , E_{m}), \\
E_{i} > 0 \hspace{0.1cm} (i = 1, \ldots , m), \hspace{0.1cm} k \neq 0,
\end{align*}
onto the parameters $p_{0} \ldots p_{n}$ and $q_{0} \ldots q_{n}$ (see \cite[Section 2.4.1]{MS_LAD}). Condition \ref{nl:updsas} then follows from \cite[Lemma 5.30]{ARAG}. Then, to prove \ref{nl:dsas}, it suffices to show that there exists a network class $\mathcal{N}$ which (i) comprises networks with $n$ energy storage elements; and (ii) has the property that the set of $p_{i}, q_{i}$ ($i = 0, \ldots , n$) such that there exists $N \in \mathcal{N}$ with impedance $Z = p/q$ is a semi-algebraic set of dimension $2(n+1)$. This can be shown to be the case for the Cauer canonical networks \cite{HugCF} (and also for the networks in \cite{ZJ_ERBI}). For these networks, it can be shown from results in \cite{HugCF} that there exists a one-to-one semi-algebraic function from the coefficient values to the element values and the scaling parameter $k$ (i.e., from $p_{0}, \ldots , p_{n}, q_{0}, \ldots , q_{n}$ to $E_{1}, \ldots , E_{2n+1}, k$).
\end{IEEEproof}
\end{comment}
\section{Bilinear and biquadratic impedances}
\label{sec:blbqi}
In this section, we summarise several known results on those impedances
which are realized by series-parallel networks containing at most two energy storage elements. These results provide minimal generating sets for $\mathcal{Z}_{0}, \mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$. We also prove a couple of lemmas on the properties of the sets $\mathcal{Z}_{1}$ and $\mathcal{Z}_{2}$. These lemmas will be used in Section \ref{sec:mgsbc} to obtain a minimal generating set for $\mathcal{Z}_{3}$.
A classical result in passive network synthesis is that the impedance of any network which contains only one type of energy storage element can always be realized by the so-called Cauer canonical networks. These networks provide minimal generating sets for $\mathcal{Z}_{0,k}$ and $\mathcal{Z}_{k,0}$ for $k = 0, 1, 2, \ldots$.\footnote{To see that the Cauer canonical network classes are generic, it suffices to show that there is a one to one function from the impedance parameters to the element values and scaling parameters (denoted $E_{1}, \ldots , E_{m}, c$ in \cite[equation (3)]{HMS_GEN}). This is easily shown from results in \cite{HugCF}.} In the case of $\mathcal{Z}_{0,0}$, a minimal generating set is given by the set of all resistors (see Fig.\ \ref{fig:c0r}). The cases $\mathcal{Z}_{0,1}$ and $\mathcal{Z}_{1,0}$ are covered in the following lemma:
\begin{lemma}
\label{lem:blgs}
Consider the network classes in Figs.\ \ref{fig:c0r}--\ref{fig:bl1inags}. The following hold.
\begin{enumerate}
\item $\mathcal{N}_{1} \cup \mathcal{N}_{2}$ is a minimal generating set for $\mathcal{Z}_{1,0}$.
\item $\mathcal{N}_{1} \cup \mathcal{N}_{2a}$ is a minimal generating set for $\mathcal{Z}_{1,0}$.
\item $\mathcal{N}_{1} \cup \mathcal{N}_{3}$ is a minimal generating set for $\mathcal{Z}_{0,1}$.
\item $\mathcal{N}_{1} \cup \mathcal{N}_{3a}$ is a minimal generating set for $\mathcal{Z}_{0,1}$.
\end{enumerate}
\end{lemma}
\tikzset{ r1/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R=$R_{1}$] ++(1.7,0)
to[short,-*] ++(0.3,0);}}
\begin{figure}[!b]
\centering
\normalsize
\begin{tikzpicture}[scale=0.57, every node/.style={transform shape}]
\pic at (0,0) [] {r1};
\node at (-1.6,-0.0) {$\text{(i) } R_{1} \geq 0$};
\end{tikzpicture}
\caption{Network $N_{1}$.
We define $\mathcal{N}_{1}$ as the set of all networks of the form of $N_{1}$ which satisfy condition (i).}
\label{fig:c0r}
\end{figure}
\tikzset{ cbl1/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R=$R_{1}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{2}}$] ++(1.7,0)
to[short] ++(0,-0.7)
to[short,-*] ++(0.3,0)
($(O)+(2.0,0)$) to[short] ++(0,-0.7)
to[C=$\frac{1}{C_{1}s}$] ++(1.7,0)
to[short] ++(0,0.7);}}
\tikzset{ lbl1/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R=$R_{1}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{2}}$] ++(1.7,0)
to[short] ++(0,-0.7)
to[short,-*] ++(0.3,0)
($(O)+(2.0,0)$) to[short] ++(0,-0.7)
to[L=$L_{1}s$] ++(1.7,0)
to[short] ++(0,0.7);}}
\begin{figure}[!b]
\centering
\normalsize
\begin{tikzpicture}[scale=0.57, every node/.style={transform shape}]
\node at (-0.7,-0.2) {$\begin{array}{rl}\text{(i)}& R_{1} \geq 0\\
\text{(ii)}& G_{2} \geq 0\\
\text{(iii)}& C_{1} > 0 \\
\text{(iv)}& L_{1} > 0
\end{array}$};
\node at (1.3,1.2) {$N_{2}$}
\pic at (0,0) [] {cbl1};
\node at (7.0,1.2) {$N_{3}$}
\pic at (5.7,0) [] {lbl1};
\end{tikzpicture}
\caption{Networks $N_{2}$ and $N_{3}$.
We define $\mathcal{N}_{2}$ (resp., $\mathcal{N}_{3}$) as the set of all networks of the form of $N_{2}$ (resp., $N_{3}$) which satisfy conditions (i)--(iii) (resp., conditions (i), (ii) and (iv)).}
\label{fig:bl1in}
\end{figure}
\tikzset{ lbl2/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{1}}$] ++(3.4,0)
to[short] ++(0,-0.7)
to[short,-*] ++(0.3,0)
($(O)+(0.3,0)$) to[short] ++(0,-0.7)
to[R=$R_{2}$] ++(1.7,0)
to[L=$L_{1}s$] ++(1.7,0)
to[short] ++(0,0.7);}}
\tikzset{ cbl2/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{1}}$] ++(3.4,0)
to[short] ++(0,-0.7)
to[short,-*] ++(0.3,0)
($(O)+(0.3,0)$) to[short] ++(0,-0.7)
to[R=$R_{2}$] ++(1.7,0)
to[C=$\frac{1}{C_{1}s}$] ++(1.7,0)
to[short] ++(0,0.7);}}
\begin{figure}[!b]
\centering
\normalsize
\begin{tikzpicture}[scale=0.57, every node/.style={transform shape}]
\node at (-0.7,-0.2) {$\begin{array}{rl}\text{(i)}& G_{1} \geq 0\\
\text{(ii)}& R_{2} \geq 0\\
\text{(iii)}& C_{1} > 0 \\
\text{(iv)}& L_{1} > 0
\end{array}$};
\node at (1.3,1.2) {$N_{2a}$}
\node at (7.0,1.2) {$N_{3a}$};
\pic at (0,0) [] {cbl2};
\pic at (5.7,0) [] {lbl2};
\end{tikzpicture}
\caption{Networks $N_{2a}$ and $N_{3a}$.
We define $\mathcal{N}_{2a}$ (resp., $\mathcal{N}_{3a}$) as the set of all networks of the form of $N_{2a}$ (resp., $N_{3a}$) which satisfy conditions (i)--(iii) (resp., conditions (i), (ii) and (iv)).}
\label{fig:bl1inags}
\end{figure}
In fact, any bilinear impedance realized by an RLC (not necessarily series-parallel) network can also be realized by one of the networks in Figs.\ \ref{fig:c0r}--\ref{fig:bl1inags}, in accordance with the following well known result:
\begin{comment}
An algebraic description of the set of bilinear impedances in $\mathcal{Z}_{1}$ can then be given in terms of polynomials
\begin{align}
f(s) &:= f_{1}s + f_{0}, \text{ and}\\
g(s) &:= g_{1}s + g_{0},\label{eq:fgd}
\end{align}
in a coprime factorisation $Z = f/g$ of the impedance $Z$. Specifically, we have the following result.
\end{comment}
\begin{lemma}
\label{lem:blec}
Let the McMillan degree of $Z \in \mathbb{R}(s)$ be less than or equal to one. The following are equivalent:
\begin{enumerate}
\item $Z$ is the impedance of an RLC network.
\item $Z$ is positive-real (i.e., (i) $Z$ is analytic in the open right half plane, and (ii) $\Re{(Z(s_{0}))} \geq 0$ whenever $\Re{(s_{0})} > 0$).\label{nl:blprc}
\item $Z \in \mathcal{Z}_{1}$.
\end{enumerate}
\end{lemma}
\begin{comment}
In addition, we have the following well known result on constant impedances:
\begin{lemma}
\label{lem:cec}
Let $Z \in \mathbb{R}$. Then the following are equivalent:
\begin{enumerate}
\item $Z$ is the impedance of an RLC network.
\item $Z \geq 0$.\label{nl:cprc}
\item $Z \in \mathcal{Z}_{0}$.
\end{enumerate}
\end{lemma}
\begin{remark}
Note that condition \ref{nl:blprc} in Lemmas \ref{lem:blec} and \ref{lem:cec} are equivalent to $Z$ being \emph{positive-real}, i.e., (i) $Z$ is analytic in $\mathbb{C}_{+}$, and (ii) $\Re{(Z(s_{0}))} \geq 0$ for all $s_{0} \in \mathbb{C}_{+}$.
\end{remark}
\end{comment}
A minimal generating set for $\mathcal{Z}_{2}$, which first appeared in \cite{LIN}, is described in the following lemma.
\begin{lemma}
\label{lem:bqmgs}
Let $\mathcal{N}_{1}$--$\mathcal{N}_{9}$ be as in Figs.\ \ref{fig:c0r}--\ref{fig:bq1c1l}. The following hold.
\begin{enumerate}
\item $\mathcal{N}_{1} \cup \mathcal{N}_{2} \cup \mathcal{N}_{4}$ is a minimal generating set for $\mathcal{Z}_{2,0}$.
\item $\mathcal{N}_{1} \cup \mathcal{N}_{3} \cup \mathcal{N}_{5}$ is a minimal generating set for $\mathcal{Z}_{0,2}$.
\item The union of $\mathcal{N}_{1}$--$\mathcal{N}_{3}$ and $\mathcal{N}_{6}$--$\mathcal{N}_{9}$ is a minimal generating set for $\mathcal{Z}_{1,1}$.
\end{enumerate}
\end{lemma}
\begin{IEEEproof}
See Lemma \ref{lem:ai}, \cite{LIN} and \cite[Theorem 1]{JiangSmith11}. That the network classes $\mathcal{N}_{1}$--$\mathcal{N}_{9}$ are generic was established in \cite{MS_LC}.
\end{IEEEproof}
\tikzset{ ccbq1/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R=$R_{1}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$R_{2}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{3}}$] ++(1.7,0)to[short] ++(0,-1.4)
($(O)+(3.7,0.7)$) to[short] ++(0,-0.7)
to[C=$\frac{1}{C_{1}s}$] ++(1.7,0)
to[short,-*] ++(0.3,0)
($(O)+(2.0,0)$) to[short] ++(0,-1.4)
to[C=$\frac{1}{C_{2}s}$] ++(3.4,0)
to[short] ++(0,1.4);}}
\tikzset{ llbq1/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R=$R_{1}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$R_{2}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{3}}$] ++(1.7,0)
to[short] ++(0,-1.4)
($(O)+(3.7,0.7)$) to[short] ++(0,-0.7)
to[L=$L_{1}s$] ++(1.7,0)
to[short,-*] ++(0.3,0)
($(O)+(2.0,0)$) to[short] ++(0,-1.4)
to[L=$L_{2}s$] ++(3.4,0)
to[short] ++(0,1.4);}}
\begin{figure}[!b]
\centering
\normalsize
\begin{tikzpicture}[scale=0.57, every node/.style={transform shape}]
\node at (1.3,1.8) {$N_{4}$};
\pic at (0,0) [] {ccbq1};
\node at (7.8,1.8) {$N_{5}$};
\pic at (6.5,0) [] {llbq1};
\end{tikzpicture}
\caption{Networks $N_{4}$ and $N_{5}$. In $N_{4}$, (i) $R_{1} \geq 0$, (ii) $R_{2} > 0$, (iii) $G_{3} \geq 0$, and (iv) $C_{1}, C_{2} > 0$. In $N_{5}$, conditions (i)--(iii) hold, and (v) $L_{1}, L_{2} > 0$. We define $\mathcal{N}_{4}$ (resp., $\mathcal{N}_{5}$) as the set of all networks of the form of $N_{4}$ (resp., $N_{5}$) which satisfy conditions (i)--(iv) (resp., conditions (i)--(iii) and (v)).
\label{fig:bq2co2i}
\end{figure}
\tikzset{ clbq1/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R=$R_{1}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$R_{2}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{3}}$] ++(1.7,0)
to[short] ++(0,-1.4)
($(O)+(3.7,0.7)$) to[short] ++(0,-0.7)
to[C=$\frac{1}{C_{1}s}$] ++(1.7,0)
to[short,-*] ++(0.3,0)
($(O)+(2.0,0)$) to[short] ++(0,-1.4)
to[L=$L_{1}s$] ++(3.4,0)
to[short] ++(0,1.4);}}
\tikzset{ clbq2/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R=$R_{1}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$R_{2}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{3}}$] ++(1.7,0)to[short] ++(0,-1.4)
($(O)+(3.7,0.7)$) to[short] ++(0,-0.7)
to[L=$L_{1}s$] ++(1.7,0)
to[short,-*] ++(0.3,0)
($(O)+(2.0,0)$) to[short] ++(0,-1.4)
to[C=$\frac{1}{C_{1}s}$] ++(3.4,0)
to[short] ++(0,1.4);}}
\tikzset{ clbq3/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[short] ++(0,-0.7)
to[C=$\frac{1}{C_{1}s}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{2}}$] ++(3.4,0)
to[short,-*] ++(0.3,0)
($(O)+(2.0,-0.7)$) to[short] ++(0,-0.7)
to[R=$R_{3}$] ++(1.7,0)
to[L=$L_{1}s$] ++(1.7,0)
to[short] ++(0,1.4)
($(O)+(0.3,0)$) to[short] ++(0,1.4)
to[R=$\frac{1}{G_{1}}$] ++(5.1,0)
to[short] ++(0,-1.4);}}
\tikzset{ clbq4/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[short] ++(0,-0.7)
to[L=$L_{1}s$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{2}}$] ++(3.4,0)
to[short,-*] ++(0.3,0)
($(O)+(2.0,-0.7)$) to[short] ++(0,-0.7)
to[R=$R_{3}$] ++(1.7,0)
to[C=$\frac{1}{C_{1}s}$] ++(1.7,0)
to[short] ++(0,1.4)
($(O)+(0.3,0)$) to[short] ++(0,1.4)
to[R=$\frac{1}{G_{1}}$] ++(5.1,0)
to[short] ++(0,-1.4);}}
\begin{figure}[!b]
\centering
\normalsize
\begin{tikzpicture}[scale=0.57, every node/.style={transform shape}]
\node at (1.3,1.8) {$N_{6}$};
\pic at (0,0) [] {clbq1};
\node at (7.8,1.8) {$N_{7}$};
\pic at (6.5,0) [] {clbq2};
\node at (1.3,-2.2) {$N_{8}$};
\pic at (0,-4.0) [] {clbq3};
\node at (7.8,-2.2) {$N_{9}$};
\pic at (6.5,-4.0) [] {clbq4};
\end{tikzpicture}
\caption{Networks $N_{6}$--$N_{9}$. In $N_{6}$ and $N_{7}$, (i) $R_{1}, R_{2} \geq 0$, (ii) $G_{3} \geq 0$, and (iii) $C_{1}, L_{1} > 0$. We define $\mathcal{N}_{6}$ (resp., $\mathcal{N}_{7}$) as the set of all networks of the form of $N_{6}$ (resp., $N_{7}$) which satisfy conditions (i)--(iii). In $N_{8}$ and $N_{9}$, condition (iii) holds, (iv) $R_{3} \geq 0$, and (v) $G_{1}, G_{2} \geq 0$. We define $\mathcal{N}_{8}$ (resp., $\mathcal{N}_{9}$) as the set of all networks of the form of $N_{8}$ (resp., $N_{9}$) which satisfy conditions (iii)--(v).
\label{fig:bq1c1l}
\end{figure}
In \cite{JiangSmith11}, an algebraic description of the biquadratic impedances in $\mathcal{Z}_{2}$ was provided in terms of the polynomials
\begin{align}
c(s) &:= c_{2}s^{2} + c_{1}s + c_{0}, \text{ and} \nonumber\\
d(s) &:= d_{2}s^{2} + d_{1}s + d_{0}, \label{eq:cdd}
\end{align}
in a coprime factorisation $Z = c/d$ of the impedance $Z$. We summarise these results in the following lemma:
\begin{lemma}
\label{lem:bqrrc}
Let $Z = c/d$ where $c, d$ are as in (\ref{eq:cdd}); let $R_{0}(c,d) \neq 0$; and let
\begin{align}
\alpha_{1} &:= c_{1}d_{0} - c_{0}d_{1}, \hspace{0.1cm} \alpha_{2} := c_{2}d_{0} - c_{0}d_{2}, \hspace{0.1cm} \alpha_{3} := c_{2}d_{1} - c_{1}d_{2}, \nonumber \\
\lambda_{1} &:= d_{1}\alpha_{1} - d_{0}\alpha_{2}, \hspace{0.1cm} \lambda_{2} := c_{1}\alpha_{3} - c_{2}\alpha_{2}, \nonumber \\
\lambda_{3} &:= d_{2}\alpha_{2} - d_{1}\alpha_{3}, \text{ and } \lambda_{4} := c_{0}\alpha_{2} - c_{1}\alpha_{1}.\label{eq:bqid1}
\end{align}
\begin{comment}
\begin{align}
\alpha_{1} &:= c_{1}d_{0} - c_{0}d_{1}, \label{eq:bqid1}\\
\alpha_{2} &:= c_{2}d_{0} - c_{0}d_{2}, \\
\alpha_{3} &:= c_{2}d_{1} - c_{1}d_{2}, \\
\lambda_{1} &:= d_{1}\alpha_{1} - d_{0}\alpha_{2}, \\
\lambda_{2} &:= c_{1}\alpha_{3} - c_{2}\alpha_{2}, \\
\lambda_{3} &:= d_{2}\alpha_{2} - d_{1}\alpha_{3}, \text{ and} \\
\lambda_{4} &:= c_{0}\alpha_{2} - c_{1}\alpha_{1}, \label{eq:bqid7}
\end{align}
\end{comment}
Then $Z \in \mathcal{Z}_{2}$ if and only if at least one of the following conditions holds:
\begin{remunerate}
\labitem{(Q\arabic{muni})}{nl:bqcb1} $c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}, \lambda_{1}$ have the same sign;\footnote{Here, and throughout, we say a set of real numbers have the same sign if either all are non-negative or all are non-positive} and $\alpha_{2} \geq 0$.
\labitem{(Q\arabic{muni})}{nl:bqcb2} $c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}, \lambda_{2}$ have the same sign; and $\alpha_{2} \geq 0$.
\labitem{(Q\arabic{muni})}{nl:bqcb3} $c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}, \lambda_{3}$ have the same sign; and $\alpha_{2} \leq 0$.
\labitem{(Q\arabic{muni})}{nl:bqcb4} $c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}, \lambda_{4}$ have the same sign; and $\alpha_{2} \leq 0$.
\end{remunerate}
In particular, if $R_{0}(c,d) > 0$, then \ref{nl:bqcb1} (resp., \ref{nl:bqcb3}) holds if and only if \ref{nl:bqcb2} (resp., \ref{nl:bqcb4}) holds.
Moreover, with $N_{4}$--$N_{9}$ as in Figs.\ \ref{fig:bq2co2i}--\ref{fig:bq1c1l}, then
\begin{enumerate}
\item If condition \ref{nl:bqcb1} holds and $R_{0}(c,d) < 0$, then $\alpha_{1} > 0$, $d_{0}, d_{2} \neq 0$, and $Z$ is the impedance of $N_{6}$ with
\begin{align*}
&R_{1} = \tfrac{c_{0}}{d_{0}}, \hspace{0.1cm} R_{2} = \tfrac{\alpha_{2}}{d_{0}d_{2}}, \hspace{0.1cm} G_{3} = \tfrac{-d_{2}\lambda_{1}}{R_{0}(c,d)}, \\
&C_{1} = \tfrac{-d_{2}^{2}\alpha_{1}}{R_{0}(c,d)} \text{ and } L_{1} = \tfrac{\alpha_{1}}{d_{0}^{2}}.
\end{align*}
\item If condition \ref{nl:bqcb3} holds and $R_{0}(c,d) < 0$, then $\alpha_{3} < 0$, $d_{0}, d_{2} \neq 0$, and $Z$ is the impedance of $N_{7}$ with
\begin{align*}
&R_{1} = \tfrac{c_{2}}{d_{2}}, \hspace{0.1cm} R_{2} = \tfrac{-\alpha_{2}}{d_{0}d_{2}}, \hspace{0.1cm} G_{3} = \tfrac{-d_{0}\lambda_{3}}{R_{0}(c,d)}, \\
&C_{1} = \tfrac{-d_{2}^{2}}{\alpha_{3}} \text{ and } L_{1} = \tfrac{R_{0}(c,d)}{d_{0}^{2}\alpha_{3}}.
\end{align*}
\item If condition \ref{nl:bqcb4} holds and $R_{0}(c,d) < 0$, then $\alpha_{1} < 0$, $c_{0}, c_{2} \neq 0$, and $Z$ is the impedance of $N_{8}$ with
\begin{align*}
&G_{1} = \tfrac{d_{0}}{c_{0}}, \hspace{0.1cm} G_{2} = \tfrac{-\alpha_{2}}{c_{0}c_{2}}, \hspace{0.1cm} R_{3} = \tfrac{-c_{2}\lambda_{4}}{R_{0}(c,d)}, \\
&C_{1} = \tfrac{-\alpha_{1}}{c_{0}^{2}} \text{ and } L_{1} = \tfrac{c_{2}^{2}\alpha_{1}}{R_{0}(c,d)}.
\end{align*}
\item If condition \ref{nl:bqcb2} holds and $R_{0}(c,d) < 0$, then $\alpha_{3} > 0$, $c_{0}, c_{2} \neq 0$, and $Z$ is the impedance of $N_{9}$ with
\begin{align*}
&G_{1} = \tfrac{d_{2}}{c_{2}}, \hspace{0.1cm} G_{2} = \tfrac{\alpha_{2}}{c_{0}c_{2}}, \hspace{0.1cm} R_{3} = \tfrac{-c_{0}\lambda_{2}}{R_{0}(c,d)}, \\
&C_{1} = \tfrac{-R_{0}(c,d)}{c_{0}^{2}\alpha_{3}} \text{ and } L_{1} = \tfrac{c_{2}^{2}}{\alpha_{3}}.
\end{align*}
\item If condition \ref{nl:bqcb1} holds and $R_{0}(c,d) > 0$, then $\alpha_{1} > 0$, $d_{0}, \lambda_{1} \neq 0$, and $Z$ is the impedance of $N_{5}$ with
\begin{align*}
&R_{1} = \tfrac{c_{0}}{d_{0}}, \hspace{0.1cm} R_{2} = \tfrac{\alpha_{1}^{2}}{d_{0}\lambda_{1}},\hspace{0,1cm} G_{3} = \tfrac{d_{2}\lambda_{1}}{R_{0}(c,d)}\\
&L_{1} = \tfrac{\alpha_{1}R_{0}(c,d)}{\lambda_{1}^{2}} \text{ and } L_{2} = \tfrac{\alpha_{1}}{d_{0}^{2}}
\end{align*}
\item If condition \ref{nl:bqcb3} holds and $R_{0}(c,d) > 0$, then $\alpha_{3} < 0$, $d_{2}, \lambda_{3} \neq 0$, and $Z$ is the impedance of $N_{4}$ with
\begin{align*}
&R_{1} = \tfrac{c_{2}}{d_{2}}, \hspace{0.1cm} R_{2} = \tfrac{\alpha_{3}^{2}}{d_{2}\lambda_{3}},\hspace{0,1cm} G_{3} = \tfrac{d_{0}\lambda_{3}}{R_{0}(c,d)}\\
&C_{1} = \tfrac{-\lambda_{3}^{2}}{\alpha_{3}R_{0}(c,d)} \text{ and } C_{2} = \tfrac{-d_{2}^{2}}{\alpha_{3}}
\end{align*}
\end{enumerate}
\end{lemma}
\begin{remark}
Note that conditions \ref{nl:bqcb1}--\ref{nl:bqcb4} in the above lemma are equivalent to $Z$ being \emph{regular} (i.e., $Z$ is positive-real and the least value of the real part of either $Z(j\omega)$ or $1/Z(j\omega)$ occurs at either $\omega = 0$ or $\omega = \infty$) \cite{JiangSmith11}.
\end{remark}
\begin{remark}
\label{rem:z2lo2c}
If $Z$ is the impedance of one of the networks $N_{4}$--$N_{9}$ in Figs.\ \ref{fig:bq2co2i}--\ref{fig:bq1c1l}, then it is straightforward to show that $Z$ is biquadratic. Also, if $c, d$ as in (\ref{eq:cdd}) are a coprime factorization for $Z$ (i.e., $Z = c/d$ and $R_{0}(c,d) \neq 0$), then the element values in these networks are uniquely determined by the coefficients $c_{0}$--$c_{2}$ and $d_{0}$--$d_{2}$ in accordance with the expressions in Lemma \ref{lem:bqrrc}
\end{remark}
From Lemmas \ref{lem:bqmgs} and \ref{lem:bqrrc}, we obtain the following two lemmas, which provide an alternative characterisation for the set $\mathcal{Z}_{2}$. This characterisation will be used in Section \ref{sec:mgsbc} to obtain a minimal generating set for $\mathcal{Z}_{3}$. The papers \cite{LIN, JiangSmith11} provide an indirect proof of these two lemmas. For completeness, we present a more direct algebraic proof.
\begin{lemma}
\label{lem:bqec}
Let $c, d$ be as in (\ref{eq:cdd}) with $R_{0}(c,d) < 0$, let $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as in (\ref{eq:bqid1}), and consider the following set of inequalities:
\begin{remunerate}
\labitem{(Q\Alph{muni})}{nl:bqca1} $c_{0}d_{0}, d_{2}\lambda_{1}, \alpha_{2}d_{0}d_{2} \geq 0$; $d_{0}, d_{2} \neq 0$; and $\alpha_{1} > 0$.
\labitem{(Q\Alph{muni})}{nl:bqca2} $c_{2}d_{2}, d_{0}\lambda_{3}, -\alpha_{2}d_{0}d_{2} \geq 0$; $d_{0}, d_{2} \neq 0$; and $\alpha_{3} < 0$.
\labitem{(Q\Alph{muni})}{nl:bqca3} $c_{0}d_{0}, c_{2}\lambda_{4}, -\alpha_{2}c_{0}c_{2} \geq 0$; $c_{0}, c_{2} \neq 0$; and $\alpha_{1} < 0$.
\labitem{(Q\Alph{muni})}{nl:bqca4} $c_{2}d_{2}, c_{0}\lambda_{2}, \alpha_{2}c_{0}c_{2} \geq 0$; $c_{0}, c_{2} \neq 0$; and $\alpha_{3} > 0$.
\end{remunerate}
With \ref{nl:bqcb1}--\ref{nl:bqcb4} as in Lemma \ref{lem:bqrrc}, then \ref{nl:bqcb1} (resp., \ref{nl:bqcb2}, \ref{nl:bqcb3}, \ref{nl:bqcb4}) is satisfied if and only if \ref{nl:bqca1} (resp., \ref{nl:bqca4}, \ref{nl:bqca3}, \ref{nl:bqca2}) is satisfied.
\begin{comment}
\begin{enumerate}
\item \ref{nl:bqcb1} holds if and only if \ref{nl:bqca1} holds.\label{nl:z11ac1}
\item \ref{nl:bqcb2} holds if and only if \ref{nl:bqca4} holds.\label{nl:z11ac2}
\item \ref{nl:bqcb3} holds if and only if \ref{nl:bqca3} holds.\label{nl:z11ac3}
\item \ref{nl:bqcb4} holds if and only if \ref{nl:bqca2} holds.\label{nl:z11ac4}
\end{enumerate}
\end{comment}
\end{lemma}
\begin{IEEEproof}
To show that \ref{nl:bqcb1} $\Rightarrow$ \ref{nl:bqca1}, it suffices to show that condition \ref{nl:bqcb1} and $R_{0}(c,d) < 0$ together imply that $d_{0}, d_{2} \neq 0$ and $\alpha_{1} > 0$. To see this, we note the following relationships:
\begin{align}
\lambda_{1} + d_{0}\alpha_{2} &= d_{1}\alpha_{1}, \label{eq:bql1c1} \\
\alpha_{2} + c_{0}d_{2} &= c_{2}d_{0}, \text{ and} \label{eq:bql1c2} \\
c_{2}\lambda_{1} - R_{0}(c,d) &= -d_{2}\lambda_{4}. \label{eq:bql1c3}
\end{align}
If $d_{2} = 0$, then (\ref{eq:bql1c3}) implies that $R_{0}(c,d) = 0$. But $R_{0}(c,d) < 0$, so we conclude that $d_{2} \neq 0$. If $d_{0} = 0$, then it follows from (\ref{eq:bql1c2}) that $c_{0} = 0$ (since $d_{2} \neq 0$), but this again implies that $R_{0}(c,d) = 0$. We conclude that $d_{0}, d_{2} \neq 0$. Furthermore, if $\alpha_{1} \leq 0$, then it follows from (\ref{eq:bql1c1}) that $d_{1} = \alpha_{2} = \lambda_{1} = 0$ (since $d_{0} \neq 0$).
But this implies that $R_{0}(c,d) = -c_{1}d_{2}\alpha_{1}$. Since $R_{0}(c,d) < 0$, then we conclude that $\alpha_{1}$ must be positive. It follows that $d_{0}, d_{2} \neq 0$ and $\alpha_{1} > 0$, which completes the proof of \ref{nl:bqcb1} $\Rightarrow$ \ref{nl:bqca1}.
To show that \ref{nl:bqca1} $\Rightarrow$ \ref{nl:bqcb1}, we recall that (\ref{eq:bql1c1})--(\ref{eq:bql1c3}) hold, and we note the following additional relationships:
\begin{align}
\alpha_{2}d_{2}\lambda_{1} - R_{0}(c,d)d_{0}d_{2} &= d_{2}^{2}\alpha_{1}, \text{ and} \label{eq:bql1c4}\\
\alpha_{1} + c_{0}d_{1} &= c_{1}d_{0}. \label{eq:bql1c5}
\end{align}
Here, (\ref{eq:bql1c4}) implies that $d_{0}$ and $d_{2}$ have the same sign. Since, in addition, $d_{0}, d_{2} \neq 0$ and $\alpha_{2}d_{0}d_{2} \geq 0$, then we conclude that $\alpha_{2} \geq 0$. Then, (\ref{eq:bql1c2}) implies that $c_{2}$ and $d_{0}$ have the same sign, and so $c_{0}, c_{2}, d_{0}, d_{2}$ and $\lambda_{1}$ have the same sign, and $\alpha_{2} \geq 0$. Next, note that (\ref{eq:bql1c1}) implies that $d_{1}$ and $\lambda_{1}$ have the same sign. Finally, (\ref{eq:bql1c5}) implies that $c_{1}$ and $d_{0}$ have the same sign, which completes the proof of \ref{nl:bqca1} $\Rightarrow$ \ref{nl:bqcb1}.
The proof of the remaining conditions are analogous. Specifically, in the above argument, we swap $c_{0}$ with $c_2$ and $d_0$ with $d_2$ to prove that \ref{nl:bqcb4} $\iff$ \ref{nl:bqca2}; we swap $c$ and $d$ to prove that \ref{nl:bqcb3} $\iff$ \ref{nl:bqca3}; and we swap $c_0$ with $d_2$, $c_1$ with $d_1$, and $c_2$ with $d_0$ to prove that \ref{nl:bqcb2} $\iff$ \ref{nl:bqca4}.
\end{IEEEproof}
\begin{lemma}
Let $c, d$ be as in (\ref{eq:cdd}) with $R_{0}(c,d) > 0$, and consider the following sets of inequalities:
\begin{remunerate}
\setcounter{muni}{4}
\labitem{(Q\Alph{muni})}{nl:bqca5} $c_{2}, d_{0}, d_{2}, \lambda_{3}$ have the same sign; $d_{2}, \lambda_{3} {\neq} 0$; and $\alpha_{3} {<} 0$
\labitem{(Q\Alph{muni})}{nl:bqca6} $c_{0}, d_{0}, d_{2}, \lambda_{1}$ have the same sign; $d_{0}, \lambda_{1} {\neq} 0$; and $\alpha_{1} {>} 0$
\end{remunerate}
The following hold:
\begin{enumerate}
\item Conditions \ref{nl:bqcb3}, \ref{nl:bqcb4} and \ref{nl:bqca5} are all equivalent.\label{nl:bq1retc1}
\item Conditions \ref{nl:bqcb1}, \ref{nl:bqcb2} and \ref{nl:bqca6} are all equivalent.\label{nl:bq1retc2}
\end{enumerate}
\end{lemma}
\begin{IEEEproof}
We first prove condition \ref{nl:bq1retc1}.
To show that \ref{nl:bqcb3} $\Rightarrow$ \ref{nl:bqca5}, it suffices to show that if \ref{nl:bqcb3} holds and $R_{0}(c,d) > 0$, then $d_{2}, \lambda_{3} \neq 0$ and $\alpha_{3} < 0$. To show this, we note the following relationships
\begin{align}
R_{0}(c,d)d_{2} + d_{0}\alpha_{3}^{2} &= -\alpha_{2}\lambda_{3}, \label{eq:bql1ca1} \\
\lambda_{3} + c_{2}d_{1}^{2} &= d_{2}(c_{2}d_{0} - c_{0}d_{2} + c_{1}d_{1}), \label{eq:bql1ca2} \\
\alpha_{2}^{2} + R_{0}(c,d) &= \alpha_{1}\alpha_{3}, \text{ and} \label{eq:bql1ca3} \\
\lambda_{3} - d_{2}\alpha_{2} &= -d_{1}\alpha_{3}. \label{eq:bql1ca4}
\end{align}
From (\ref{eq:bql1ca3}), $\alpha_{3} = 0$ implies $R_{0}(c,d) = 0$. But $R_{0}(c,d) > 0$, so we conclude that $\alpha_{3} \neq 0$. Then, from (\ref{eq:bql1ca2}), $d_{2} = 0$ implies $c_{2} = 0$ or $d_{1} = 0$, and in either case we have $\alpha_{3} = 0$, so we conclude that $d_{2}, \alpha_{3} \neq 0$. Next, from (\ref{eq:bql1ca1}), we find that $\lambda_{3} = 0$ implies that either $R_{0}(c,d) = 0$ or $d_{2} = 0$, neither of which is possible, and so $d_{2}, \alpha_{3}, \lambda_{3} \neq 0$. Finally, since $d_{1}, d_{2}$ and $\lambda_{3}$ have the same sign and $\alpha_{2} \leq 0$, then (\ref{eq:bql1ca4}) implies that $\alpha_{3} < 0$.
To see that \ref{nl:bqca5} $\Rightarrow$ \ref{nl:bqcb3}, we recall that (\ref{eq:bql1ca1})--(\ref{eq:bql1ca4}) hold, and we note the following additional relationships:
\begin{align}
\alpha_{3}^{2}d_{0}d_{2} + \lambda_{3}^{2} + R_{0}(c,d)d_{2}^{2} &= -d_{1}\alpha_{3}\lambda_{3}, \label{eq:bql1ca5} \\
-\alpha_{3} + c_{2}d_{1} &= c_{1}d_{2}, \text{ and} \label{eq:bql1ca6} \\
-\alpha_{1} + c_{1}d_{0} &= c_{0}d_{1}. \label{eq:bql1ca7}
\end{align}
Then (\ref{eq:bql1ca5}) implies that $d_{1}$ has the same sign as $\lambda_{3}$, whereupon (\ref{eq:bql1ca6}) implies that $c_{1}$ has the same sign as $d_{2}$. Also, (\ref{eq:bql1ca3}) implies that $\alpha_{1} \leq 0$, whereupon (\ref{eq:bql1ca7}) implies that $c_{0}$ has the same sign as $d_{1}$. Finally, (\ref{eq:bql1ca1}) implies that $\alpha_{2} \leq 0$.
An analogous argument proves that \ref{nl:bqcb4} is equivalent to:
\begin{remunerate}
\setcounter{muni}{6}
\labitem{(Q\Alph{muni})}{nl:bqca7} $c_{0}, c_{2}, d_{0}, \lambda_{4}$ have the same sign; $c_{0}, \lambda_{4} {\neq} 0$; and $\alpha_{1} {<} 0$
\end{remunerate}
Accordingly, to complete the proof of the present lemma, we will show that \ref{nl:bqca5} and \ref{nl:bqca7} are equivalent. To see this, assume initially that \ref{nl:bqca5} holds and $d_{2} > 0$. Then $c_{2}, d_{0} \geq 0$ and the Sylvester matrix $\mathcal{S}_{0}(a,b)$ is positive definite. It follows that all of the principal minors of $\mathcal{S}_{0}(a,b)$ are positive, whence $c_{0} > 0$, $\alpha_{1} < 0$ and $\lambda_{4} > 0$. If, on the other hand, $d_{2} < 0$, then $\mathcal{S}_{0}(a,b)$ is negative definite, in which case $c_{0}, \alpha_{1}, \lambda_{4} < 0$. In either case, condition \ref{nl:bqca7} holds. A similar argument proves that, if \ref{nl:bqca7} holds and $R_{0}(c,d) > 0$, then \ref{nl:bqca5} holds.
The proof of condition \ref{nl:bq1retc2} is analogous to the above.
\end{IEEEproof}
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\section{A non-minimal generating set for $\mathcal{Z}_{3}$}
\label{sec:nmgs}
In this section, we derive a generating set for $\mathcal{Z}_{3}$, the set of impedances realized by series-parallel networks containing at most three energy storage elements. The case in which the energy storage elements are all of the same type is solved by the Cauer canonical networks in accordance with the following two lemmas:
\begin{lemma}
\label{lem:z30mgs}
The union of $\mathcal{N}_{1}$, $\mathcal{N}_{2}$, $\mathcal{N}_{4}$ and $\mathcal{N}_{10}$ is a minimal generating set for $\mathcal{Z}_{3,0}$ (see Figs.\ \ref{fig:c0r}, \ref{fig:bl1in}, \ref{fig:bq2co2i} and \ref{fig:bc3c}).
\end{lemma}
\begin{lemma}
\label{lem:z03mgs}
The union of $\mathcal{N}_{1}$, $\mathcal{N}_{3}$, $\mathcal{N}_{5}$ and $\mathcal{N}_{10}^{i}$ is a minimal generating set for $\mathcal{Z}_{0,3}$.
\end{lemma}
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\begin{figure}[!b]
\centering
\normalsize
\begin{tikzpicture}[scale=0.57, every node/.style={transform shape}]
\node at (-1.6,2.5) {$N_{10}$};
\node at (-1.6,0.0) {$\begin{array}{rl}\text{(i)}& R_{1} \geq 0\\
\text{(ii)}& R_{2}, R_{3} > 0\\
\text{(iii)}& G_{4} \geq 0\\
\text{(iv)}& C_{1}, C_{2}, C_{3} > 0
\end{array}$};
\pic at (0,0) [] {ccbc1};
\end{tikzpicture}
\caption{Network $N_{10}$.
We define $\mathcal{N}_{10}$ as the set of all networks of the form of $N_{10}$ which satisfy conditions (i)--(iv).
\label{fig:bc3c}
\end{figure}
The main contribution of this section is to derive generating sets for the cases in which both types of energy storage elements are present (i.e., for the sets $\mathcal{Z}_{1,2}$ and $\mathcal{Z}_{2,1}$). These are described in the following two theorems:
\begin{theorem}
\label{thm:z12nmgs}
The union of $\mathcal{N}_{1}$--$\mathcal{N}_{3}$, $\mathcal{N}_{5}$--$\mathcal{N}_{9}$, $\mathcal{N}_{11}$--$\mathcal{N}_{15}$ and $\mathcal{N}_{11}^{p}$--$\mathcal{N}_{15}^{p}$ is a generating set for $\mathcal{Z}_{1,2}$ (see Figs.\ \ref{fig:c0r}, \ref{fig:bl1in}, and \ref{fig:bq2co2i}--\ref{fig:bcr6}).
\end{theorem}
\begin{theorem}
\label{thm:z21nmgs}
The union of $\mathcal{N}_{1}$--$\mathcal{N}_{4}$, $\mathcal{N}_{6}$--$\mathcal{N}_{9}$, $\mathcal{N}_{11}^{i}$--$\mathcal{N}_{15}^{i}$ and $\mathcal{N}_{11}^{d}$--$\mathcal{N}_{15}^{d}$ is a generating set for $\mathcal{Z}_{2,1}$.
\end{theorem}
Note from \cite[Corollary 2]{HMS_GEN} that the network classes in Theorems \ref{thm:z12nmgs} and \ref{thm:z21nmgs} are not generic.
To prove Theorems \ref{thm:z12nmgs} and \ref{thm:z21nmgs}, we will use the following well known network transformation:
\begin{lemma}
\label{lem:eqr}
For an arbitrary given impedance $H \in \mathbb{R}(s)$, the impedance of the two networks in Fig.\ \ref{fig:eqn} are equivalent under the transformations $R_{2} = R_{1}(1+R_{1}G_{1})$, $G_{2} = G_{1}/(1+R_{1}G_{1})$, and $\alpha_{2} = \alpha_{1}(1+R_{1}G_{1})^{2}$ (equivalently, $R_{1} = R_{2}/(1+R_{2}G_{2})$, $G_{1} = G_{2}(1+R_{2}G_{2})$, and $\alpha_{1} = \alpha_{2}/(1+R_{2}G_{2})^{2}$).
\begin{comment}
Let $Z$ be the impedance of the RLC network on the left of Fig.\ \ref{fig:eqn}, for some $0 \leq R_{1} \in \mathbb{R}$, $0 \leq G_{1} \in \mathbb{R}$, and some RLC network $N_{1}$ with impedance $Z_{1}$. Then $Z$ is also realized by the RLC network on the right of Fig.\ \ref{fig:eqn}, where (i) $R_{2} = R_{1}(1+R_{1}G_{1})$; (ii) $G_{2} = G_{1}/(1+R_{1}G_{1})$; and (iii) $N_{2}$ is the network obtained by scaling the impedance of each element in $N_{1}$ by $(1+R_{1}G_{1})^{2}$, and has impedance $Z_{2} = Z_{1}(1+R_{1}G_{1})^{2}$.
Conversely, let $Z$ be the impedance of the RLC network on the right of Fig.\ \ref{fig:eqn}, for some $0 \leq R_{2} \in \mathbb{R}$, $0 \leq G_{2} \in \mathbb{R}$, and some RLC network $N_{2}$ with impedance $Z_{2}$. Then $Z$ is also realized by the RLC network on the left of Fig.\ \ref{fig:eqn}, where (i) $R_{1} = R_{2}(1+R_{2}G_{2})$; (ii) $G_{1} = G_{2}(1+R_{2}G_{2})$; and (iii) $N_{2}$ is the network obtained by scaling the impedance of each element in $N_{1}$ by $1/(1+R_{1}G_{1})^{2}$, and has impedance $Z_{1} = Z_{2}/(1+R_{2}G_{2})^{2}$.
\end{comment}
\end{lemma}
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\begin{figure}[!h]
\centering
\normalsize
\begingroup
\renewcommand*{\arraystretch}{1.7}
\begin{tikzpicture}[scale=0.57, every node/.style={transform shape}]
\pic at (0,0) [] {eqn1};
\pic at (4.5,0) [] {eqn2};
\end{tikzpicture}
\endgroup
\caption{Two networks with equivalent impedance.}
\label{fig:eqn}
\end{figure}
\begin{IEEEproof}[Proof of Theorem \ref{thm:z12nmgs}]
The proof of this theorem is similar to the method proposed in \cite[Section 2(b)]{ZJ_SIA}. In contrast to that paper, we will use the network transformation in Lemma \ref{lem:eqr} to eliminate several redundant elements.
If $Z$ is the impedance of a series-parallel network containing two or fewer energy storage elements, then $Z$ is the impedance of a network from one of the classes $\mathcal{N}_{1}$--$\mathcal{N}_{3}$ or $\mathcal{N}_{5}$--$\mathcal{N}_{9}$. Accordingly, it remains to consider the case in which $Z \in \mathcal{Z}_{1,2}$ is the impedance of a series-parallel network $N$ which contains exactly three energy storage elements. Then, at some stage in the construction of $N$, a network $N_{a}$ containing one energy storage element is connected either in series or parallel with a network $N_{b}$ containing two energy storage elements, and all subsequent stages in the construction of $N$ involve the addition of resistors in series or in parallel.
Consider first the case in which $N_{a}$ and $N_{b}$ are connected in parallel. Since a series or parallel connection of two resistors can always be realized by a single resistor, then it is easily shown from Lemma \ref{lem:eqr} that $Z$ is realized by a network of the form of $N_{u}$ in Fig.\ \ref{fig:gsz12}. Since $N_{a}$ contains one energy storage element, then its impedance is realized by a network from $\mathcal{N}_{2}$ or $\mathcal{N}_{3}$, and by a network from $\mathcal{N}_{2a}$ or $\mathcal{N}_{3a}$, by Lemma \ref{lem:blgs}. It follows that the impedance of a network of the form of $N_{u}$ in Fig.\ \ref{fig:gsz12} can be realized by the impedance of a network of the form of $N_{v}$ or $N_{w}$ in that figure, where $N_{c}$ is a series-parallel network containing at most two energy storage elements. Furthermore, in network $N_{v}$ (resp., $N_{w}$), the network $N_{c}$ necessarily contains at most one inductor and one capacitor (resp., at most two inductors and no capacitors). Thus, from Section \ref{sec:blbqi}, it follows that $Z$ is the impedance of a network from one of the classes $\mathcal{N}_{2}$--$\mathcal{N}_{3}$, $\mathcal{N}_{5}$--$\mathcal{N}_{9}$, or $\mathcal{N}_{11}$--$\mathcal{N}_{15}$.
The case with $N_{a}$ and $N_{b}$ connected in series is similar. In this case, we find that $Z$ is the impedance of a network from one of the classes $\mathcal{N}_{2}$--$\mathcal{N}_{3}$, $\mathcal{N}_{5}$--$\mathcal{N}_{9}$, or $\mathcal{N}_{11}^{p}$--$\mathcal{N}_{15}^{p}$
\end{IEEEproof}
\tikzset{ gn1/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R] ++(1.7,0)
to[short] ++(0,1.4)
to[twoport, t=$N_{a}$] ++(1.7,0)
to[short] ++(0,-1.4)
($(O)+(2.0,0)$) to[twoport, t=$N_{b}$] ++(1.7,0)
to[short,-*] ++(0.3,0)
($(O)+(2.0,0)$) to[short] ++(0,-1.4)
to[R] ++(1.7,0)
to[short] ++(0,1.4);}}
\tikzset{ gn2/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R] ++(1.7,0)
to[short] ++(0,0.7)
to[R] ++(1.7,0)
to[L] ++(1.7,0)
to[short] ++(0,-0.7)
to[short,-*] ++(0.3,0)
($(O)+(2.0,0)$) to[short] ++(0,-0.7)
to[twoport, t=$N_{c}$] ++(3.4,0)
to[short] ++(0,0.7);}}
\tikzset{ gn3/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R] ++(1.7,0)
to[short] ++(0,0.7)
to[R] ++(1.7,0)
to[C] ++(1.7,0)
to[short] ++(0,-0.7)
to[short,-*] ++(0.3,0)
($(O)+(2.0,0)$) to[short] ++(0,-0.7)
to[twoport, t=$N_{c}$] ++(3.4,0)
to[short] ++(0,0.7);}}
\begin{figure}[!h]
\centering
\normalsize
\begingroup
\renewcommand*{\arraystretch}{1.7}
\begin{tikzpicture}[scale=0.57, every node/.style={transform shape}]
\node at (1.0,1.7) {$N_{u}$};
\pic at (0,0) [] {gn1};
\node at (5.5,2.1) {$N_{v}$};
\pic at (4.5,1.5) [] {gn2};
\node at (5.5,-0.9) {$N_{w}$};
\pic at (4.5,-1.5) [] {gn3};
\end{tikzpicture}
\endgroup
\caption{Structure of the generating sets for $\mathcal{Z}_{1,2}$ in Theorem \ref{thm:z12nmgs}.}
\label{fig:gsz12}
\end{figure}
\tikzset{ clbc1a/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R=$R_{4}$] ++(1.7,0)
to[short] ++(0,-1.4)
to[R=$R_{1}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$R_{2}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{3}}$] ++(1.7,0)
to[short,-*] ++(0.3,0)
($(O)+(5.4,-0.7)$) to[short] ++(0,-0.7)
to[C=$\frac{1}{C_{1}s}$] ++(1.7,0)
to[short] ++(0,1.4)
($(O)+(3.7,-1.4)$) to[short] ++(0,-1.4)
to[L=$L_{1}s$] ++(3.4,0)
to[short] ++(0,1.4)
($(O)+(2.0,0)$) to[short] ++(0,1.4)
to[R=$R_{5}$] ++(2.55,0)
to[L=$L_{2}s$] ++(2.55,0)
to[short] ++(0,-1.4);}}
\tikzset{ clbc1b/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R=$R_{4}$] ++(1.7,0)
to[short] ++(0,-1.4)
to[R=$R_{1}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$R_{2}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{3}}$] ++(1.7,0)
to[short,-*] ++(0.3,0)
($(O)+(5.4,-0.7)$) to[short] ++(0,-0.7)
to[L=$L_{1}s$] ++(1.7,0)
to[short] ++(0,1.4)
($(O)+(3.7,-1.4)$) to[short] ++(0,-1.4)
to[C=$\frac{1}{C_{1}s}$] ++(3.4,0)
to[short] ++(0,1.4)
($(O)+(2.0,0)$) to[short] ++(0,1.4)
to[R=$R_{5}$] ++(2.55,0)
to[L=$L_{2}s$] ++(2.55,0)
to[short] ++(0,-1.4);}}
\begin{comment}
\begin{figure}[!t]
\centering
\normalsize
\begin{tikzpicture}[scale=0.57, every node/.style={transform shape}]
\node at (-1.6,6.5) {$N_{11}$};
\node at (-1.6,1.0) {$N_{12}$};
\node at (-1.6,-2.0) {$\begin{array}{rl}\text{(i)}& R_{1}, R_{2}, R_{4}, R_{5} \geq 0\\
\text{(ii)}& G_{3} \geq 0\\
\text{(iii)}& C_{1}, L_{1}, L_{2} > 0
\end{array}$}
\pic at (0,5.5) [] {clbc1a};
\pic at (0,0) [] {clbc1b};
\end{tikzpicture}
\caption{Networks $N_{11}$ and $N_{12}$.
We define $\mathcal{N}_{11}$ (resp., $\mathcal{N}_{12}$) as the set of all networks of the form of $N_{11}$ (resp., $N_{12}$) which satisfy conditions (i)--(iii).}
\label{fig:bcr2}
\end{figure}
\end{comment}
\begin{figure}[!t]
\centering
\normalsize
\begin{tikzpicture}[scale=0.57, every node/.style={transform shape}]
\pic at (0,0) [] {clbc1a};
\pic at (8.0,0) [] {clbc1b};
\node at (1.5,1.5) {$N_{11}$};
\node at (9.5,1.5) {$N_{12}$};
\end{tikzpicture}
\caption{Networks $N_{11}$ and $N_{12}$. Here, (i) $R_{1}, R_{2}, R_{4}, R_{5} \geq 0$; (ii) $G_{3} \geq 0$; and (iii) $C_{1}, L_{1}, L_{2} > 0$.
We define $\mathcal{N}_{11}$ (resp., $\mathcal{N}_{12}$) as the set of all networks of the form of $N_{11}$ (resp., $N_{12}$) which satisfy conditions (i)--(iii).}
\label{fig:bcr2}
\end{figure}
\tikzset{ clbc1c/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R=$R_{4}$] ++(1.7,0)
to[short] ++(0,-2.1)
to[C=$\frac{1}{C_{1}s}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{2}}$] ++(3.4,0)
to[short] ++(0,1.4)
($(O)+(3.7,-2.1)$) to[short] ++(0,-0.7)
to[R=$R_{3}$] ++(1.7,0)
to[L=$L_{1}s$] ++(1.7,0)
to[short] ++(0,1.4)
($(O)+(2.0,0)$) to[R=$\frac{1}{G_{1}}$] ++(5.1,0)
to[short,-*] ++(0.3,0)
($(O)+(2.0,0)$) to[short] ++(0,1.4)
to[R=$R_{5}$] ++(2.55,0)
to[L=$L_{2}s$] ++(2.55,0)
to[short] ++(0,-1.4);}}
\tikzset{ clbc1d/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R=$R_{4}$] ++(1.7,0)
to[short] ++(0,-2.1)
to[L=$L_{1}s$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{2}}$] ++(3.4,0)
to[short] ++(0,1.4)
($(O)+(3.7,-2.1)$) to[short] ++(0,-0.7)
to[R=$R_{3}$] ++(1.7,0)
to[C=$\frac{1}{C_{1}s}$] ++(1.7,0)
to[short] ++(0,1.4)
($(O)+(2.0,0)$) to[R=$\frac{1}{G_{1}}$] ++(5.1,0)
to[short,-*] ++(0.3,0)
($(O)+(2.0,0)$) to[short] ++(0,1.4)
to[R=$R_{5}$] ++(2.55,0)
to[L=$L_{2}s$] ++(2.55,0)
to[short] ++(0,-1.4);}}
\begin{comment}
\begin{figure}[!t]
\centering
\normalsize
\begin{tikzpicture}[scale=0.57, every node/.style={transform shape}]
\node at (-1.6,6.5) {$N_{13}$};
\node at (-1.6,1.0) {$N_{14}$};
\node at (-1.6,-2.0) {$\begin{array}{rl}\text{(i)}& R_{3}, R_{4}, R_{5} \geq 0\\
\text{(ii)}& G_{1}, G_{2} \geq 0\\
\text{(iii)}& C_{1}, L_{1}, L_{2} > 0
\end{array}$}
\pic at (0,5.5) [] {clbc1c};
\pic at (0,0) [] {clbc1d};
\end{tikzpicture}
\caption{Networks $N_{13}$ and $N_{14}$.
We define $\mathcal{N}_{13}$ (resp., $\mathcal{N}_{14}$) as the set of all networks of the form of $N_{13}$ (resp., $N_{14}$) which satisfy conditions (i)--(iii).}
\label{fig:bcr5}
\end{figure}
\end{comment}
\begin{figure}[!t]
\centering
\normalsize
\begin{tikzpicture}[scale=0.57, every node/.style={transform shape}]
\pic at (0,0) [] {clbc1c};
\pic at (8.0,0) [] {clbc1d};
\node at (1.5,1.5) {$N_{13}$};
\node at (9.5,1.5) {$N_{14}$};
\end{tikzpicture}
\caption{Networks $N_{13}$ and $N_{14}$. Here, (i) $R_{3}, R_{4}, R_{5} \geq 0$; (ii) $G_{1}, G_{2} \geq 0$; and (iii) $C_{1}, L_{1}, L_{2} > 0$.
We define $\mathcal{N}_{13}$ (resp., $\mathcal{N}_{14}$) as the set of all networks of the form of $N_{13}$ (resp., $N_{14}$) which satisfy conditions (i)--(iii).}
\label{fig:bcr5}
\end{figure}
\tikzset{ clbc2a/.pic={
\ctikzset{bipoles/thickness=1}
\node[anchor=center] at (1,0) (O) {};
\draw (O) to[short,*-] ++(0.3,0)
to[R=$R_{4}$] ++(1.7,0)
to[short] ++(0,-1.4)
to[R=$R_{1}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$R_{2}$] ++(1.7,0)
to[short] ++(0,0.7)
to[R=$\frac{1}{G_{3}}$] ++(1.7,0)
to[short,-*] ++(0.3,0)
($(O)+(5.4,-0.7)$) to[short] ++(0,-0.7)
to[L=$L_{1}s$] ++(1.7,0)
to[short] ++(0,1.4)
($(O)+(3.7,-1.4)$) to[short] ++(0,-1.4)
to[L=$L_{2}s$] ++(3.4,0)
to[short] ++(0,1.4)
($(O)+(2.0,0)$) to[short] ++(0,1.4)
to[R=$R_{5}$] ++(2.55,0)
to[C=$\frac{1}{C_{1}s}$] ++(2.55,0)
to[short] ++(0,-1.4);}}
\begin{figure}[!t]
\centering
\normalsize
\begin{tikzpicture}[scale=0.57, every node/.style={transform shape}]
\node at (-1.6,1.0) {$N_{15}$};
\node at (-1.6,-2.0) {$\begin{array}{rl}\text{(i)}& R_{1}, R_{4}, R_{5} \geq 0\\
\text{(ii)}& R_{2} > 0\\
\text{(iii)}& G_{3} \geq 0\\
\text{(iv)}& C_{1}, L_{1}, L_{2} > 0
\end{array}$}
\pic at (0,0) [] {clbc2a};
\end{tikzpicture}
\caption{Network $N_{15}$.
We define $\mathcal{N}_{15}$ as the set of all networks of the form of $N_{15}$ which satisfy conditions (i)--(iv).}
\label{fig:bcr6}
\end{figure}
\section{A minimal generating set for $\mathcal{Z}_{3}$}
\label{sec:mgsbc}
The contribution of this section is to derive a minimal generating set for $\mathcal{Z}_{3}$, the set of impedances realized by series-parallel networks containing at most three energy storage elements. From Lemmas \ref{lem:z30mgs} and \ref{lem:z03mgs}, it suffices to find generating sets for $\mathcal{Z}_{1,2}$ and $\mathcal{Z}_{2,1}$ that comprise generic network classes. These are provided in Theorems \ref{thm:z12mgs} and \ref{thm:z21mgs}, which refer to the network classes in Table \ref{tab:n2047d}. That the network classes in these theorems are generic can be routinely checked using the condition in \cite[Theorem 1]{HMS_GEN}.
\begin{table}[h]
\caption{Definition of network classes $\mathcal{N}_{16}$--$\mathcal{N}_{30}$. Refer to Figs.\ \ref{fig:bcr2}--\ref{fig:bcr6} for definitions of network classes $\mathcal{N}_{11}$--$\mathcal{N}_{15}$.}
\centering
\begin{tabular}{p{8cm}}
\hline
$\mathcal{N}_{16}$ (resp., $\mathcal{N}_{17}, \mathcal{N}_{18}$) is the set of all networks from $\mathcal{N}_{11}$ for which $R_{1} = 0$ (resp., $G_{3} = 0$, $R_{5} = 0$).\\
$\mathcal{N}_{19}$ (resp., $\mathcal{N}_{20}, \mathcal{N}_{21}$) is the set of all networks from $\mathcal{N}_{12}$ for which $R_{1} = 0$ (resp., $G_{3} = 0$, $R_{5} = 0$).\\
$\mathcal{N}_{22}$ (resp., $\mathcal{N}_{23}, \mathcal{N}_{24}$) is the set of all networks from $\mathcal{N}_{13}$ for which $G_{1} = 0$ (resp., $R_{3} = 0$, $R_{5} = 0$).\\
$\mathcal{N}_{25}$ (resp., $\mathcal{N}_{26}, \mathcal{N}_{27}$) is the set of all networks from $\mathcal{N}_{14}$ for which $G_{1} = 0$ (resp., $R_{3} = 0$, $R_{5} = 0$).\\
$\mathcal{N}_{28}$ (resp., $\mathcal{N}_{29}, \mathcal{N}_{30}$) is the set of all networks from $\mathcal{N}_{15}$ for which $R_{1} = 0$ (resp., $G_{3} = 0$, $R_{5} = 0$).\\
\hline
\end{tabular}
\label{tab:n2047d}
\end{table}
\begin{theorem}
\label{thm:z12mgs}
The union of $\mathcal{N}_{1}$--$\mathcal{N}_{3}$, $\mathcal{N}_{5}$--$\mathcal{N}_{9}$, $\mathcal{N}_{16}$--$\mathcal{N}_{30}$, and $\mathcal{N}_{16}^{p}$--$\mathcal{N}_{30}^{p}$ is a minimal generating set for $\mathcal{Z}_{1,2}$ (see Figs.\ \ref{fig:c0r}, \ref{fig:bl1in}, \ref{fig:bq2co2i} and \ref{fig:bq1c1l} and Table \ref{tab:n2047d}).
\end{theorem}
\begin{theorem}
\label{thm:z21mgs}
The union of $\mathcal{N}_{1}$--$\mathcal{N}_{4}$, $\mathcal{N}_{6}$--$\mathcal{N}_{9}$, $\mathcal{N}_{16}^{i}$--$\mathcal{N}_{30}^{i}$, and $\mathcal{N}_{16}^{d}$--$\mathcal{N}_{30}^{d}$ is a minimal generating set for $\mathcal{Z}_{2,1}$.
\end{theorem}
\begin{comment}
\begin{remark}
The paper \cite{ZJ_ERBI} considered a special class of bicubic positive-real functions which they called \emph{essential-regular}. In that paper, it was shown that a bicubic impedance can be realized by a so-called \emph{simple series-parallel network} if and only if it is essential-regular.
It is then easily shown from results in \cite{ZJ_ERBI} and Lemmas \ref{lem:ai} and \ref{lem:eqr} that $Z$ is the impedance of a simple series-parallel network containing 1 capacitor and 2 inductors if and only if $Z$ is the impedance of a network from $\mathcal{N}_{18}, \mathcal{N}_{21}, \mathcal{N}_{24}, \mathcal{N}_{27}, \mathcal{N}_{30}, \mathcal{N}_{18}^{p}, \mathcal{N}_{21}^{p}, \mathcal{N}_{24}^{p}, \mathcal{N}_{27}^{p}$, or $\mathcal{N}_{30}^{p}$. It will be shown in remark \ref{rem:ercc} that there exist $Z \in \mathcal{Z}_{1,2}$ which cannot be realized by these classes. Theorem \ref{thm:z12mgs} shows that these classes can be augmented by a further 20 network classes to obtain a minimal generating set for $\mathcal{Z}_{1,2}$.
\end{remark}
\end{comment}
To prove Theorems \ref{thm:z12mgs} and \ref{thm:z21mgs}, we first obtain an algebraic description of the impedances of the networks in the generating sets for $\mathcal{Z}_{1,2}$ and $\mathcal{Z}_{2,1}$ described in Theorems \ref{thm:z12nmgs} and \ref{thm:z21nmgs}. Then, given any such network, we show that it is possible to vary the element values continuously without changing the network's impedance until one of the resistors can be replaced with either an open or a short circuit, thereby arriving at the network classes $\mathcal{N}_{16}$--$\mathcal{N}_{30}$. The proof considers separately the case in which the impedance is bicubic and the case in which the impedance is biquadratic. The case in which the impedance is bilinear or constant valued is covered by Lemmas \ref{lem:blgs}--\ref{lem:blec}.
\subsection{Bicubic impedances in $\mathcal{Z}_{3}$}
We first consider the bicubic impedances realized by the networks described in Theorem \ref{thm:z12nmgs}. In particular, we obtain explicit descriptions for the element values in these networks in terms of the polynomials
\begin{align}
a(s) &= a_{3}s^{3} + a_{2}s^{2} + a_{1}s + a_{0}, \text{ and} \nonumber \\
b(s) &= b_{3}s^{3} + b_{2}s^{2} + b_{1}s + b_{0}, \label{eq:abd}
\end{align}
in a coprime factorization $Z = a/b$ for the impedance $Z$.
\begin{theorem}
\label{thm:bcn1013}
Let $a,b$ in (\ref{eq:abd}) be coprime; let $b_{i} > 0$ for at least one value of $i \in 0, \ldots , 3$; let
\begin{comment}
\begin{align*}
&\begin{bmatrix}c_{0}& c_{1}& c_{2}\end{bmatrix} {=} \begin{bmatrix}1& -x& x^{2}\end{bmatrix}\mathcal{B}(a,b), \hspace{0.1cm} c(s) {=} c_{2}s^{2} + c_{1}s+ c_{0}, \\
&\tilde{a} {=} a(-x), \hspace{0.1cm} \tilde{b} {=} b(-x), \hspace{0.1cm} \tilde{c} {=} c(-x), \\
&\begin{bmatrix}d_{0}& d_{1}& d_{2}\end{bmatrix} {=} \tilde{b}\begin{bmatrix}1& -x& x^{2}\end{bmatrix}\mathcal{B}(b,c), \hspace{0.1cm} d(s) {=} d_{2}s^{2} + d_{1}s + d_{0};
\end{align*}
and let $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as in (\ref{eq:bqid1});
\end{comment}
\begin{comment}
\begin{enumerate}[label=(\roman*)]
\item $\begin{bmatrix}c_{0}& c_{1}& c_{2}\end{bmatrix} = \begin{bmatrix}1& -x& x^{2}\end{bmatrix}\mathcal{B}(a,b)$, $c(s) = c_{2}s^{2} + c_{1}s+ c_{0}$,
\item $\tilde{a} = a(-x)$, $\tilde{b} = b(-x)$, $\tilde{c} = c(-x)$,
\item $\begin{bmatrix}d_{0}& d_{1}& d_{2}\end{bmatrix} = \tilde{b}\begin{bmatrix}1& -x& x^{2}\end{bmatrix}\mathcal{B}(b,c)$, $d(s) = d_{2}s^{2} + d_{1}s + d_{0}$,
\item $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as in (\ref{eq:bqid1}),
\end{enumerate}
\end{comment}
\begin{itemize}[leftmargin=0cm]
\item[] $\begin{bmatrix}c_{0}& c_{1}& c_{2}\end{bmatrix} = \begin{bmatrix}1& -x& x^{2}\end{bmatrix}\mathcal{B}(a,b)$,
\item[] $c(s) = c_{2}s^{2} + c_{1}s+ c_{0}$,
\item[] $\tilde{a} = a(-x)$, $\tilde{b} = b(-x)$, $\tilde{c} = c(-x)$,
\item[] $\begin{bmatrix}d_{0}& d_{1}& d_{2}\end{bmatrix} = \tilde{b}\begin{bmatrix}1& -x& x^{2}\end{bmatrix}\mathcal{B}(b,c)$,
\item[] $d(s) = d_{2}s^{2} + d_{1}s + d_{0}$, and
\item[] $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as in (\ref{eq:bqid1});
\end{itemize}
and consider the following sets of inequalities:
\begin{remunerate}
\labitem{(C\arabic{muni})}{nl:bcc1a} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2,3$; $\tilde{a}, \tilde{b}, c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}$ and $\lambda_{1}$ have the same sign; $\tilde{c}, \alpha_{2} \geq 0$; and $R_{0}(a,b) > 0$.
\labitem{(C\arabic{muni})}{nl:bcc1b} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2,3$; $\tilde{a}, \tilde{b}, c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}$ and $\lambda_{3}$ have the same sign; $\tilde{c}, -\alpha_{2} \geq 0$; and $R_{0}(a,b) > 0$.
\labitem{(C\arabic{muni})}{nl:bcc1c} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2,3$; $\tilde{a}, \tilde{b}, c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}$ and $\lambda_{4}$ have the same sign; $\tilde{c}, -\alpha_{2} \geq 0$; and $R_{0}(a,b) > 0$.
\labitem{(C\arabic{muni})}{nl:bcc1d} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2,3$; $\tilde{a}, \tilde{b}, c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}$ and $\lambda_{2}$ have the same sign; $\tilde{c}, \alpha_{2} \geq 0$; and $R_{0}(a,b) > 0$.
\end{remunerate}
Then $Z$ is the impedance of a network from $\mathcal{N}_{11}$ (resp., $\mathcal{N}_{12}$, $\mathcal{N}_{13}$, $\mathcal{N}_{14}$) if and only if there exists $x \geq 0$ such that $Z = a/b$ and condition \ref{nl:bcc1a} (resp., \ref{nl:bcc1b}, \ref{nl:bcc1c}, \ref{nl:bcc1d}) is satisfied.
Moreover,
\begin{enumerate}
\item If $x \geq 0$ is such that \ref{nl:bcc1a} holds, then $\tilde{b}, d_{0}, d_{2}, \tilde{c} \neq 0$, $\alpha_{1} > 0$, and $Z$ is the impedance of $N_{11}$ with \label{nl:bcev1}
\begin{align*}
&R_{1} {=} \tfrac{\tilde{c}c_{0}}{d_{0}}, \hspace{0.1cm} R_{2} {=} \tfrac{\tilde{c}\alpha_{2}}{d_{0}d_{2}}, \hspace{0.1cm} G_{3} {=} \tfrac{d_{2}\lambda_{1}}{\tilde{b}^{4}\tilde{c}^{2}R_{0}(a,b)}, \hspace{0.1cm} R_{4} {=} \tfrac{\tilde{a}}{\tilde{b}},\\
&R_{5} {=} \tfrac{\tilde{c}x}{\tilde{b}^{2}}, \hspace{0.1cm} C_{1} {=} \tfrac{d_{2}^{2}\alpha_{1}}{\tilde{b}^{4}\tilde{c}^{2}R_{0}(a,b)}, \hspace{0.1cm} L_{1} {=} \tfrac{\tilde{c}\alpha_{1}}{d_{0}^{2}} \text{ and } L_{2} {=} \tfrac{\tilde{c}}{\tilde{b}^{2}}.
\end{align*}
\item If $x \geq 0$ is such that \ref{nl:bcc1b} holds, then $\tilde{b}, d_{0}, d_{2}, \tilde{c} \neq 0$, $\alpha_{3} < 0$, and $Z$ is the impedance of $N_{12}$ with\label{nl:bcev2}
\begin{align*}
&R_{1} {=} \tfrac{\tilde{c}c_{2}}{d_{2}}, \hspace{0.1cm} R_{2} {=} \tfrac{-\tilde{c}\alpha_{2}}{d_{0}d_{2}}, \hspace{0.1cm} G_{3} {=} \tfrac{d_{0}\lambda_{3}}{\tilde{b}^{4}\tilde{c}^{2}R_{0}(a,b)}, \hspace{0.1cm} R_{4} {=} \tfrac{\tilde{a}}{\tilde{b}},\\
&R_{5} {=} \tfrac{\tilde{c}x}{\tilde{b}^{2}}, \hspace{0.1cm} C_{1} {=} \tfrac{-d_{2}^{2}}{\tilde{c}\alpha_{3}}, \hspace{0.1cm} L_{1} {=} \tfrac{-\tilde{b}^{4}\tilde{c}^{2}R_{0}(a,b)}{d_{0}^{2}\alpha_{3}} \text{ and } L_{2} {=} \tfrac{\tilde{c}}{\tilde{b}^{2}}.
\end{align*}
\item If $x \geq 0$ is such that \ref{nl:bcc1c} holds, then $\tilde{b}, c_{0}, c_{2}, \tilde{c} \neq 0$, $\alpha_{1} < 0$, and $Z$ is the impedance of $N_{13}$ with\label{nl:bcev3}
\begin{align*}
&G_{1} {=} \tfrac{d_{0}}{\tilde{c}c_{0}}, \hspace{0.1cm} G_{2} {=} \tfrac{-\alpha_{2}}{\tilde{c}c_{0}c_{2}}, \hspace{0.1cm} R_{3} {=} \tfrac{c_{2}\lambda_{4}}{\tilde{b}^{4}R_{0}(a,b)}, \hspace{0.1cm} R_{4} {=} \tfrac{\tilde{a}}{\tilde{b}}, \\
&R_{5} {=} \tfrac{\tilde{c}x}{\tilde{b}^{2}}, \hspace{0.1cm} C_{1} {=} \tfrac{-\alpha_{1}}{\tilde{c}c_{0}^{2}}, \hspace{0.1cm} L_{1} {=} \tfrac{-c_{2}^{2}\alpha_{1}}{\tilde{b}^{4}R_{0}(a,b)} \text{ and } L_{2} {=} \tfrac{\tilde{c}}{\tilde{b}^{2}}.
\end{align*}
\item If $x \geq 0$ is such that \ref{nl:bcc1d} holds, then $\tilde{b}, c_{0}, c_{2}, \tilde{c} \neq 0$, $\alpha_{3} > 0$, and $Z$ is the impedance of $N_{14}$ with\label{nl:bcev4}
\begin{align*}
&G_{1} {=} \tfrac{d_{2}}{\tilde{c}c_{2}}, \hspace{0.1cm} G_{2} {=} \tfrac{\alpha_{2}}{\tilde{c}c_{0}c_{2}}, \hspace{0.1cm} R_{3} {=} \tfrac{c_{0}\lambda_{2}}{\tilde{b}^{4}R_{0}(a,b)}, \hspace{0.1cm} R_{4} {=} \tfrac{\tilde{a}}{\tilde{b}}, \\
&R_{5} {=} \tfrac{\tilde{c}x}{\tilde{b}^{2}}, \hspace{0.1cm} C_{1} {=} \tfrac{\tilde{b}^{4}R_{0}(a,b)}{c_{0}^{2}\alpha_{3}}, \hspace{0.1cm} L_{1} {=} \tfrac{\tilde{c}c_{2}^{2}}{\alpha_{3}} \text{ and } L_{2} {=} \tfrac{\tilde{c}}{\tilde{b}^{2}}.
\end{align*}
\end{enumerate}
\end{theorem}
Prior to proving the above theorem, we show the following auxiliary lemmas:
\begin{lemma}
\label{lem:bcrc1}
Let $a,b$ in (\ref{eq:abd}) be coprime, let $Z = a/b$, and let $\tilde{a}, \tilde{b}, \tilde{c}$, $c_{0}$--$c_{2}$, $d_{0}$--$d_{2}$, $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as defined in Theorem \ref{thm:bcn1013}. If $Z$ is the impedance of $N_{11}$ in Fig.\ \ref{fig:bcr2} (resp., $N_{12}$, $N_{13}$, $N_{14}$), then there exists $x \geq 0$ such that the element values take the form indicated in item \ref{nl:bcev1} (resp., \ref{nl:bcev2}, \ref{nl:bcev3}, \ref{nl:bcev4}) of Theorem \ref{thm:bcn1013}.
\end{lemma}
\begin{IEEEproof}
Note initially from Definition \ref{def:bezd} that
\begin{align}
c(s)(s+x) &= a(s)b(-x)-b(s)a(-x), \text{ and}\label{eq:cbezr}\\
d(s)(s+x) &= b(-x)(b(s)c(-x)-b(-x)c(s)).\label{eq:dbezr}
\end{align}
We first consider the case where $Z$ is the impedance of $N_{11}$. With $x = R_{5}/L_{2}$, then $x \geq 0$, and we note that
\begin{align}
Z(s) &= R_{4} + \tfrac{1}{\frac{1}{Z_{2}(s)} + \frac{1}{L_{2}(s+x)}} \nonumber \\
&= R_{4} + \frac{Z_{2}(s)L_{1}(s+x)}{Z_{2}(s) + L_{2}(s+x)},\label{eq:zbcgf}
\end{align}
where $Z_{2}$ is the impedance of a network of the form of $N_{6}$ in Fig.\ \ref{fig:bq1c1l}. Since $Z$ is bicubic, then $Z_{2}$ must be biquadratic and the poles of $1/Z_{2}(s)$ and $1/L_{2}(s+x)$ must be distinct, which implies that $Z_{2}(-x) \neq 0$. Then, from (\ref{eq:zbcgf}), it follows that $R_{4} = Z(-x) = \tilde{a}/\tilde{b}$, and
\begin{align}
\frac{1}{Z_{2}(s)} &= \frac{1}{Z(s)-Z(-x)} - \frac{1}{L_{2}(s+x)} \\
&= \frac{b(s)b(-x)}{c(s)(s+x)} - \frac{1}{L_{2}(s+x)}.\label{eq:zbqrgf}
\end{align}
By multiplying both sides in the above equation by $s+x$ and taking the limit as $s \rightarrow -x$, we find that $L_{2} = \tilde{c}/\tilde{b}^{2}$. Thus, $R_{5} = \tilde{c}x/\tilde{b}^{2}$, and we have shown that $L_{2}, R_{4}$ and $R_{5}$ have the forms indicated in item \ref{nl:bcev1} of Theorem \ref{thm:bcn1013}. Moreover, (\ref{eq:zbqrgf}) implies that
\begin{equation}
\hspace*{-0.4cm} \frac{1}{Z_{2}(s)} = \frac{b(-x)(b(s)c(-x)-c(s)b(-x))}{(s+x)c(s)c(-x)} = \frac{d(s)}{c(s)c(-x)},
\end{equation}
so $Z_{2} = \tilde{c}c/d$. Direct calculation then gives $R_{0}(\tilde{c}c,d) = -\tilde{b}^{4}\tilde{c}^{3}R_{0}(a,b)$. Since, in addition, $Z_{2}$ is the impedance of a network of the form of $N_{6}$, then from Remark \ref{rem:z2lo2c}, we conclude that $R_{1}, R_{2}, G_{3}, C_{1}$ and $L_{1}$ have the forms indicated in item \ref{nl:bcev1} of Theorem \ref{thm:bcn1013}.
The case in which $Z$ is the impedance of $N_{12}$ (resp., $N_{13}$, $N_{14}$) is similar. In this case, $Z_{2}$ is the impedance of a network of the form of $N_{7}$ (resp., $N_{8}, N_{9}$).
\end{IEEEproof}
In the next lemma, we establish the following equivalent algebraic conditions to conditions \ref{nl:bcc1a}--\ref{nl:bcc1d} in Theorem \ref{thm:bcn1013}.
\begin{lemma}
\label{lem:bciec}
Let $a, b$ be as in (\ref{eq:abd}), let $\tilde{a}, \tilde{b}, \tilde{c}$, $c_{0}$--$c_{2}$, $d_{0}$--$d_{2}$, $\alpha_{1}$--$\alpha_{3}$, $\lambda_{1}$--$\lambda_{4}$, and $x \geq 0$ be as in Theorem \ref{thm:bcn1013}, let \ref{nl:bcc1a}--\ref{nl:bcc1d} be the sets of inequalities in that theorem, and consider the following additional sets of inequalities:
\begin{remunerate}
\labitem{(C\Alph{muni})}{nl:bcca1} $b_{i} > 0$ for at least one value of $i \in 0,\ldots,3$; $\tilde{a}$ and $\tilde{b}$ have the same sign; $\tilde{b} \neq 0$; $c_{0}d_{0}, d_{2}\lambda_{1}, \alpha_{2}d_{0}d_{2} \geq 0$; $d_{0}, d_{2} \neq 0$; $\tilde{c}, \alpha_{1} > 0$; and $R_{0}(a,b) > 0$.
\labitem{(C\Alph{muni})}{nl:bcca2} $b_{i} > 0$ for at least one value of $i \in 0,\ldots,3$; $\tilde{a}$ and $\tilde{b}$ have the same sign; $\tilde{b} \neq 0$; $c_{2}d_{2}, d_{0}\lambda_{3}, -\alpha_{2}d_{0}d_{2} \geq 0$; $d_{0}, d_{2} \neq 0$; $\tilde{c}, -\alpha_{3} < 0$; and $R_{0}(a,b) > 0$.
\labitem{(C\Alph{muni})}{nl:bcca3} $b_{i} > 0$ for at least one value of $i \in 0,\ldots,3$; $\tilde{a}$ and $\tilde{b}$ have the same sign; $\tilde{b} \neq 0$; $c_{0}d_{0}, c_{2}\lambda_{4}, -\alpha_{2}c_{0}c_{2} \geq 0$; $c_{0}, c_{2} \neq 0$; $\tilde{c}, -\alpha_{1} > 0$; and $R_{0}(a,b) > 0$.
\labitem{(C\Alph{muni})}{nl:bcca4} $b_{i} > 0$ for at least one value of $i \in 0,\ldots,3$; $\tilde{a}$ and $\tilde{b}$ have the same sign; $\tilde{b} \neq 0$; $c_{2}d_{2}, c_{0}\lambda_{2}, \alpha_{2}c_{0}c_{2} \geq 0$; $c_{0}, c_{2} \neq 0$; $\tilde{c}, \alpha_{3} > 0$; and $R_{0}(a,b) > 0$.
\end{remunerate}
Then \ref{nl:bcc1a} (resp., \ref{nl:bcc1b}, \ref{nl:bcc1c}, \ref{nl:bcc1d}) is satisfied if and only if \ref{nl:bcca1} (resp., \ref{nl:bcca2}, \ref{nl:bcca3}, \ref{nl:bcca4}) is satisfied.
\begin{comment}
The following hold:
\begin{enumerate}
\item \ref{nl:bcc1a} holds if and only if \ref{nl:bcca1} holds.\label{nl:bcpec1}
\item \ref{nl:bcc1b} holds if and only if \ref{nl:bcca2} holds.\label{nl:bcpec2}
\item \ref{nl:bcc1c} holds if and only if \ref{nl:bcca3} holds.\label{nl:bcpec3}
\item \ref{nl:bcc1d} holds if and only if \ref{nl:bcca4} holds.\label{nl:bcpec4}
\end{enumerate}
\end{comment}
\end{lemma}
\begin{IEEEproof}
We first let condition \ref{nl:bcc1a} hold and we prove that so too does condition \ref{nl:bcca1}. Since $b_{i} \geq 0$ for $i = 0,1,2,3$ and $R_{0}(a,b) \neq 0$, then $b_{i} > 0$ for at least one value of $i \in 0, \ldots , 3$. Next, we recall the relationships (\ref{eq:cbezr}) and (\ref{eq:dbezr}). From (\ref{eq:cbezr}), if $\tilde{b} = 0$, then $c(s)(s+x) = -b(s)a(-x)$. Since $c_{0}, c_{1}, c_{2}$ and $\tilde{a} = a(-x)$ have the same sign, $x \geq 0$, and $b_{i} \geq 0$ for $i = 0,1,2,3$, then we require $c_{0} = c_{1} = c_{2} = 0$, so $\begin{bmatrix}1& -x& x^{2}\end{bmatrix}\mathcal{B}(a,b) = 0$. This implies that $\abs{\mathcal{B}(a,b)} = R_{0}(a,b) = 0$. But $R_{0}(a,b) > 0$, so we conclude that $\tilde{b} \neq 0$. Next, from (\ref{eq:dbezr}), it follows that if $\tilde{c} = 0$ then $d(s)(s+x) = -b(-x)^{2}c(s)$. Since $c_{0}, c_{1}, c_{2}, d_{0}, d_{1}$ and $d_{2}$ have the same sign, $x \geq 0$, and $\tilde{b} \neq 0$, then this implies that $c_{0} = c_{1} = c_{2} = 0$, and similar to before we arrive at a contradiction. It follows that $\tilde{b} \neq 0$ and $\tilde{c} > 0$. Finally, direct calculation shows that $R_{0}(c,d) = -\tilde{b}^{4}\tilde{c}^{3}R_{0}(a,b) < 0$, and from Lemma \ref{lem:bqec} we conclude that $c_{0}d_{0}, d_{2}\lambda_{1}, \alpha_{2}d_{0}d_{2} \geq 0$; $d_{0}, d_{2} \neq 0$; and $\alpha_{1} > 0$. This proves that \ref{nl:bcc1a} $\Rightarrow$ \ref{nl:bcca1}.
We next
let condition \ref{nl:bcca1} hold, and we show that condition \ref{nl:bcc1a} is satisfied. Recall from before that $R_{0}(c,d) = -\tilde{b}^{4}\tilde{c}^{3}R_{0}(a,b)$. Thus, $R_{0}(c,d) < 0$, and from Lemma \ref{lem:bqec} we find that $c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}$ and $\lambda_{1}$ have the same sign and $\alpha_{2} \geq 0$. Next, we recall again the relationships (\ref{eq:cbezr}) and (\ref{eq:dbezr}). From (\ref{eq:dbezr}), $b(-x)c(-x)b(s) = b(-x)^{2}c(s) + d(s)(s+x)$. Since $c_{0}, c_{1}, c_{2}, d_{0}, d_{1}$ and $d_{2}$ have the same sign, $b(-x) \neq 0$, $c(-x) > 0$, and $b_{i} > 0$ for at least one value of $i \in 0, \ldots , 3$, then it follows that $\tilde{b} = b(-x)$ has the same sign as $c_{0}, c_{1}, c_{2}, d_{0}, d_{1}$ and $d_{2}$, and $b_{i} \geq 0$ for $i = 0, 1, 2, 3$. Since, in addition, $a(s)b(-x) = c(s)(s+x) + b(x)a(-x)$, and $a(-x)$ and $b(-x)$ have the same sign, then it follows that $a_{i} \geq 0$ for $i = 0, 1, 2, 3$. This proves that \ref{nl:bcca1} $\Rightarrow$ \ref{nl:bcc1a}.
A similar argument proves \ref{nl:bcc1b} $\iff$ \ref{nl:bcca2}, \ref{nl:bcc1c} $\iff$ \ref{nl:bcca3}, and \ref{nl:bcc1d} $\iff$ \ref{nl:bcca4}.
\end{IEEEproof}
\begin{IEEEproof}[Proof of Theorem \ref{thm:bcn1013}]
From Lemma \ref{lem:bcrc1}, if $Z$ is the impedance of a network from $\mathcal{N}_{11}$, then there exists $x \geq 0$ such that $R_{1}, R_{2}, G_{3}, R_{4}, R_{5}, C_{1}, L_{1}$ and $L_{2}$ have the form indicated in item \ref{nl:bcev1} of the present theorem. Since $R_{1}, R_{2}, G_{3}, R_{4}, R_{5} \geq 0$ and $C_{1}, L_{1}, L_{2} > 0$, then it is easily shown that condition \ref{nl:bcca1} of Lemma \ref{lem:bciec} holds. From that lemma, it follows that condition \ref{nl:bcc1a} of the present theorem also holds.
Conversely, by Lemma \ref{lem:bciec}, if condition \ref{nl:bcc1a} of the present theorem holds, then so too does condition \ref{nl:bcca1} in that lemma, and it follows that the element values in item \ref{nl:bcev1} of the present theorem satisfy $R_{1}, R_{2}, G_{3}, R_{4}, R_{5} \geq 0$ and $C_{1}, L_{1}, L_{2} > 0$. Also, direct calculation verifies that the impedance of $N_{11}$ is equal to $a/b$ when the elements take these values. We have thus shown that $Z$ is the impedance of a network from $\mathcal{N}_{11}$ if and only if there exists $x \geq 0$ such that condition \ref{nl:bcc1a} holds.
The proof of the remaining parts of the theorem can be shown in a similar manner.
\end{IEEEproof}
The proof of the following theorem is then similar to Theorem \ref{thm:bcn1013}
\begin{theorem}
\label{thm:bcn14}
Let $a,b$ in (\ref{eq:abd}) be coprime; let $b_{i} > 0$ for at least one value of $i \in 0, \ldots , 3$; let
\begin{comment}
\begin{align*}
&\begin{bmatrix}c_{0}& c_{1}& c_{2}\end{bmatrix} {=} -\begin{bmatrix}x^{2}& -x& 1\end{bmatrix}\mathcal{B}(a,b), \hspace{0.1cm} c(s) {=} c_{2}s^{3} + c_{1}s^{2}+ c_{0}s, \\
&\tilde{a} {=} -a_{0}x^{3}+a_{1}x^{2}-a_{2}x+a_{3}, \hspace{0.1cm} \tilde{b} {=} -b_{0}x^{3}+b_{1}x^{2}-b_{2}x+b_{3}, \hspace{0.1cm} \tilde{c} {=} c_{0}x^{2} -c_{1}x + c_{2}, \\
&\begin{bmatrix}d_{0}& d_{1}& d_{2}\end{bmatrix} {=} - \tilde{b}\begin{bmatrix}x^{2}& -x& 1\end{bmatrix}\mathcal{B}(b,c), \hspace{0.1cm} d(s) {=} d_{2}s^{2} + d_{1}s + d_{0};
\end{align*}
and let $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as in (\ref{eq:bqid1});
\end{comment}
\begin{comment}
\begin{enumerate}[label=(\roman*)]
\item $\begin{bmatrix}c_{0}& c_{1}& c_{2}\end{bmatrix} = -\begin{bmatrix}x^{2}& -x& 1\end{bmatrix}\mathcal{B}(a,b)$,
\item $c(s) = c_{2}s^{3} + c_{1}s^{2}+ c_{0}s$,
\item $\tilde{a} = -a_{0}x^{3}+a_{1}x^{2}-a_{2}x+a_{3}$,
\item $\tilde{b} = -b_{0}x^{3}+b_{1}x^{2}-b_{2}x+b_{3}$,
\item $\tilde{c} = c_{0}x^{2} -c_{1}x + c_{2}$,
\item $\begin{bmatrix}d_{0}& d_{1}& d_{2}\end{bmatrix} = - \tilde{b}\begin{bmatrix}x^{2}& -x& 1\end{bmatrix}\mathcal{B}(b,c)$,
\item $d(s) = d_{2}s^{2} + d_{1}s + d_{0}$, and
\item $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as in (\ref{eq:bqid1}),
\end{enumerate}
\end{comment}
\begin{itemize}[leftmargin=0cm]
\item[] $\begin{bmatrix}c_{0}& c_{1}& c_{2}\end{bmatrix} = -\begin{bmatrix}x^{2}& -x& 1\end{bmatrix}\mathcal{B}(a,b)$,
\item[] $c(s) = c_{2}s^{3} + c_{1}s^{2}+ c_{0}s$,
\item[] $\tilde{a} = -a_{0}x^{3}+a_{1}x^{2}-a_{2}x+a_{3}$,
\item[] $\tilde{b} = -b_{0}x^{3}+b_{1}x^{2}-b_{2}x+b_{3}$,
\item[] $\tilde{c} = c_{0}x^{2} -c_{1}x + c_{2}$,
\item[] $\begin{bmatrix}d_{0}& d_{1}& d_{2}\end{bmatrix} = - \tilde{b}\begin{bmatrix}x^{2}& -x& 1\end{bmatrix}\mathcal{B}(b,c)$,
\item[] $d(s) = d_{2}s^{2} + d_{1}s + d_{0}$, and
\item[] $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as in (\ref{eq:bqid1});
\end{itemize}
and consider the set of inequalities:
\begin{remunerate}
\setcounter{muni}{4}
\labitem{(C\arabic{muni})}{nl:bcc2} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2,3$; $\tilde{a}, \tilde{b}, c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}$ and $\lambda_{1}$ have the same sign; $\tilde{c}, \alpha_{2} \geq 0$; and $R_{0}(a,b) > 0$.
\end{remunerate}
Then $Z$ is the impedance of a network from $\mathcal{N}_{15}$ if and only if there exists $x \geq 0$ such that $Z = a/b$ and \ref{nl:bcc2} holds.
Moreover, if $x \geq 0$ is such that \ref{nl:bcc2} holds, then $\tilde{b}, d_{0}, \lambda_{1}, \tilde{c} \neq 0$, $\alpha_{1} > 0$, and $Z$ is the impedance of $N_{15}$ in Fig.\ \ref{fig:bcr6} with
\begin{align*}
&R_{1} {=} \tfrac{\tilde{c}c_{0}}{d_{0}}, \hspace{0.1cm} R_{2} {=} \tfrac{\tilde{c}\alpha_{1}^{2}}{d_{0}\lambda_{1}}, \hspace{0.1cm} G_{3} {=} \tfrac{d_{2}\lambda_{1}}{\tilde{c}^{2}\tilde{b}^{4}R_{0}(a,b)}, \hspace{0.1cm} R_{4} {=} \tfrac{\tilde{a}}{\tilde{b}},\\
&R_{5} {=} \tfrac{\tilde{c}x}{\tilde{b}^{2}}, \hspace{0.1cm} C_{1} {=} \tfrac{\tilde{b}^{2}}{\tilde{c}}, \hspace{0.1cm} L_{1} {=} \tfrac{\tilde{b}^{4}\tilde{c}^{2}\alpha_{1}R_{0}(a,b)}{\lambda_{1}^{2}} \text{ and } L_{2} {=} \tfrac{\tilde{c}\alpha_{1}}{d_{0}^{2}}.
\end{align*}
\end{theorem}
\begin{comment}
\begin{theorem}
\label{thm:bcn15}
Let $a,b \in \mathbb{R}[s]$ in (\ref{eq:abd}) be coprime, let $a_{i} > 0$ for at least one value of $i \in 0, \ldots , 3$, let $Z = a/b$, let
\begin{enumerate}[label=(\roman*)]
\item $\begin{bmatrix}d_{0}& d_{1}& d_{2}\end{bmatrix} = \begin{bmatrix}1& -x& x^{2}\end{bmatrix}\mathcal{B}(b,a)$,
\item $d(s) = d_{2}s^{2} + d_{1}s+ d_{0}$,
\item $\tilde{a} = a(-x)$,
\item $\tilde{b} = b(-x)$,
\item $\tilde{d} = d(-x)$,
\item $\begin{bmatrix}c_{0}& c_{1}& c_{2}\end{bmatrix} = \tilde{a}\begin{bmatrix}1& -x& x^{2}\end{bmatrix}\mathcal{B}(a(s),d(s))$,
\item $c(s) = c_{2}s^{2} + c_{1}s + c_{0}$, and
\item $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as in (\ref{eq:bqid1}),
\end{enumerate}
and consider the set of inequalities:
\begin{remunerate}
\setcounter{muni}{5}
\labitem{(C\arabic{muni})}{nl:bcc3} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2,3$; $\tilde{a}, \tilde{b}, c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}$ and $\lambda_{1}$ have the same sign; $\tilde{d}, \alpha_{2} \geq 0$; and $R_{0}(a,b) > 0$.
\end{remunerate}
Then $Z$ is the impedance of a network from $\mathcal{N}_{16}$ if and only if there exists $x \geq 0$ such that \ref{nl:bcc3} holds.
Moreover, if $x \geq 0$ is such that \ref{nl:bcc3} holds, then $\tilde{a}, d_{0}, \lambda_{1}, \tilde{d} \neq 0$, $\alpha_{1} > 0$, and $Z$ is the impedance of $N_{16}$ in Fig.\ \ref{fig:bcr7} with
\begin{align*}
&R_{1} {=} \tfrac{c_{0}}{\tilde{d}d_{0}}, \hspace{0.1cm} R_{2} {=} \tfrac{\alpha_{1}^{2}}{\tilde{d}d_{0}\lambda_{1}}, \hspace{0.1cm} G_{3} {=} \tfrac{d_{2}\lambda_{1}}{\tilde{a}^{4}R_{0}(a,b)}, \hspace{0.1cm} G_{4} {=} \tfrac{\tilde{b}}{\tilde{a}},\\
&G_{5} {=} \tfrac{\tilde{d}x}{\tilde{a}^{2}}, \hspace{0.1cm} C_{1} {=} \tfrac{\tilde{d}}{\tilde{a}^{2}}, \hspace{0.1cm} L_{1} {=} \tfrac{\tilde{a}^{4}\alpha_{1}R_{0}(a,b)}{\lambda_{1}^{2}} \text{ and } L_{2} {=} \tfrac{\alpha_{1}}{\tilde{d}d_{0}^{2}}.
\end{align*}
\end{theorem}
\begin{theorem}
\label{thm:bcn1619}
Let $a,b \in \mathbb{R}[s]$ in (\ref{eq:abd}) be coprime; let $a_{i} > 0$ for at least one value of $i \in 0, \ldots , 3$; let $Z = a/b$; let
\begin{enumerate}[label=(\roman*)]
\item $\begin{bmatrix}d_{0}& d_{1}& d_{2}\end{bmatrix} = -\begin{bmatrix}x^{2}& -x& 1\end{bmatrix}\mathcal{B}(b,a)$,
\item $d(s) = d_{2}s^{2} + d_{1}s+ d_{0}$,
\item $\tilde{a} = -a_{0}x^{3}+a_{1}x^{2}-a_{2}x+a_{3}$,
\item $\tilde{b} = -b_{0}x^{3}+b_{1}x^{2}-b_{2}x+b_{3}$,
\item $\tilde{d} = d_{0}x^{2} -d_{1}x + d_{2}$,
\item $\begin{bmatrix}c_{0}& c_{1}& c_{2}\end{bmatrix} = - \tilde{a}\begin{bmatrix}x^{2}& -x& 1\end{bmatrix}\mathcal{B}(a(s),s d(s))$,
\item $c(s) = c_{2}s^{2} + c_{1}s + c_{0}$, and
\item $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as in (\ref{eq:bqid1}),
\end{enumerate}
\begin{itemize}[leftmargin=0cm]
\item[] $\begin{bmatrix}d_{0}& d_{1}& d_{2}\end{bmatrix} = -\begin{bmatrix}x^{2}& -x& 1\end{bmatrix}\mathcal{B}(b,a)$,
\item[] $d(s) = d_{2}s^{2} + d_{1}s+ d_{0}$,
\item[] $\tilde{a} = -a_{0}x^{3}+a_{1}x^{2}-a_{2}x+a_{3}$,
\item[] $\tilde{b} = -b_{0}x^{3}+b_{1}x^{2}-b_{2}x+b_{3}$,
\item[] $\tilde{d} = d_{0}x^{2} -d_{1}x + d_{2}$,
\item[] $\begin{bmatrix}c_{0}& c_{1}& c_{2}\end{bmatrix} = - \tilde{a}\begin{bmatrix}x^{2}& -x& 1\end{bmatrix}\mathcal{B}(a(s),s d(s))$,
\item[] $c(s) = c_{2}s^{2} + c_{1}s + c_{0}$, and
\item[] $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as in (\ref{eq:bqid1});
\end{itemize}
and consider the following sets of inequalities:
\begin{remunerate}
\setcounter{muni}{6}
\labitem{(C\arabic{muni})}{nl:bcc4a} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2,3$; $\tilde{a}, \tilde{b}, c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}$ and $\lambda_{1}$ have the same sign; $\tilde{d}, \alpha_{2} \geq 0$; and $R_{0}(a,b) > 0$.
\labitem{(C\arabic{muni})}{nl:bcc4b} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2,3$; $\tilde{a}, \tilde{b}, c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}$ and $\lambda_{3}$ have the same sign; $\tilde{d}, -\alpha_{2} \geq 0$; and $R_{0}(a,b) > 0$.
\labitem{(C\arabic{muni})}{nl:bcc4c} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2,3$; $\tilde{a}, \tilde{b}, c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}$ and $\lambda_{4}$ have the same sign; $\tilde{d}, -\alpha_{2} \geq 0$; and $R_{0}(a,b) > 0$.
\labitem{(C\arabic{muni})}{nl:bcc4d} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2,3$; $\tilde{a}, \tilde{b}, c_{0}, c_{1}, c_{2}, d_{0}, d_{1}, d_{2}$ and $\lambda_{2}$ have the same sign; $\tilde{d}, \alpha_{2} \geq 0$; and $R_{0}(a,b) > 0$.
\end{remunerate}
Then $Z$ is the impedance of a network from $\mathcal{N}_{17}$ (resp., $\mathcal{N}_{18}$, $\mathcal{N}_{19}$, $\mathcal{N}_{20}$) if and only if there exists $x \geq 0$ such that \ref{nl:bcc4a} (resp., \ref{nl:bcc4b}, \ref{nl:bcc4c}, \ref{nl:bcc4d}) holds.
Moreover,
\begin{enumerate}
\item If $x \geq 0$ is such that \ref{nl:bcc4a} holds, then $\tilde{a}, d_{0}, d_{2}, \tilde{d} \neq 0$, $\alpha_{1} > 0$, and $Z$ is the impedance of $N_{17}$ in Fig.\ \ref{fig:bcr8} with\label{nl:bcev7}
\begin{align*}
&R_{1} {=} \frac{c_{0}}{\tilde{d}d_{0}}, \hspace{0.1cm} R_{2} {=} \frac{\alpha_{2}}{\tilde{d}d_{0}d_{2}}, \hspace{0.1cm} G_{3} {=} \frac{d_{2}\lambda_{1}}{\tilde{a}^{4}R_{0}(a,b)}, \hspace{0.1cm} G_{4} {=} \frac{\tilde{b}}{\tilde{a}},\\
&G_{5} {=} \frac{\tilde{d}x}{\tilde{a}^{2}}, \hspace{0.1cm} C_{1} {=} \frac{d_{2}^{2}\alpha_{1}}{\tilde{a}^{4}R_{0}(a,b)}, \hspace{0.1cm} L_{1} {=} \frac{\alpha_{1}}{\tilde{d}d_{0}^{2}} \text{ and } L_{2} {=} \frac{\tilde{a}^{2}}{\tilde{d}}.
\end{align*}
\item If $x \geq 0$ is such that \ref{nl:bcc4b} holds, then $\tilde{a}, d_{0}, d_{2}, \tilde{d} \neq 0$, $\alpha_{3} < 0$, and $Z$ is the impedance of $N_{18}$ in Fig.\ \ref{fig:bcr9} with\label{nl:bcev8}
\begin{align*}
&R_{1} {=} \frac{c_{2}}{\tilde{d}d_{2}}, \hspace{0.1cm} R_{2} {=} \frac{-\alpha_{2}}{\tilde{d}d_{0}d_{2}}, \hspace{0.1cm} G_{3} {=} \frac{d_{0}\lambda_{3}}{\tilde{a}^{4}R_{0}(a,b)}, \hspace{0.1cm} G_{4} {=} \frac{\tilde{b}}{\tilde{a}},\\
&G_{5} {=} \frac{\tilde{d}x}{\tilde{a}^{2}}, \hspace{0.1cm} C_{1} {=} \frac{-\tilde{d}d_{2}^{2}}{\alpha_{3}}, \hspace{0.1cm} L_{1} {=} \frac{-\tilde{a}^{4}R_{0}(a,b)}{d_{0}^{2}\alpha_{3}} \text{ and } L_{2} {=} \frac{\tilde{a}^{2}}{\tilde{d}}.
\end{align*}
\item If $x \geq 0$ is such that \ref{nl:bcc4c} holds, then $\tilde{a}, c_{0}, c_{2}, \tilde{d} \neq 0$, $\alpha_{1} < 0$, and $Z$ is the impedance of $N_{19}$ in Fig.\ \ref{fig:bcr10} with\label{nl:bcev9}
\begin{align*}
&G_{1} {=} \frac{\tilde{d}d_{0}}{c_{0}}, \hspace{0.1cm} G_{2} {=} \frac{-\tilde{d}\alpha_{2}}{c_{0}c_{2}}, \hspace{0.1cm} R_{3} {=} \frac{c_{2}\lambda_{4}}{\tilde{a}^{4}\tilde{d}^{2}R_{0}(a,b)}, \hspace{0.1cm} G_{4} {=} \frac{\tilde{b}}{\tilde{a}}, \\
&G_{5} {=} \frac{\tilde{d}x}{\tilde{a}^{2}}, \hspace{0.1cm} C_{1} {=} \frac{-\tilde{d}\alpha_{1}}{c_{0}^{2}}, \hspace{0.1cm} L_{1} {=} \frac{-c_{2}^{2}\alpha_{1}}{\tilde{a}^{4}\tilde{d}^{2}R_{0}(a,b)} \text{ and } L_{2} {=} \frac{\tilde{a}^{2}}{\tilde{d}}.
\end{align*}
\item If $x \geq 0$ is such that \ref{nl:bcc4d} holds, then $\tilde{a}, c_{0}, c_{2}, \tilde{d} \neq 0$, $\alpha_{3} > 0$, and $Z$ is the impedance of $N_{20}$ in Fig.\ \ref{fig:bcr11} with\label{nl:bcev10}
\begin{align*}
&G_{1} {=} \frac{\tilde{d}d_{2}}{c_{2}}, \hspace{0.1cm} G_{2} {=} \frac{\tilde{d}\alpha_{2}}{c_{0}c_{2}}, \hspace{0.1cm} R_{3} {=} \frac{c_{0}\lambda_{2}}{\tilde{a}^{4}\tilde{d}^{2}R_{0}(a,b)}, \hspace{0.1cm} G_{4} {=} \frac{\tilde{b}}{\tilde{a}}, \\
&G_{5} {=} \frac{\tilde{d}x}{\tilde{a}^{2}}, \hspace{0.1cm} C_{1} {=} \frac{\tilde{a}^{4}\tilde{d}^{2}R_{0}(a,b)}{c_{0}^{2}\alpha_{3}}, \hspace{0.1cm} L_{1} {=} \frac{c_{2}^{2}}{\tilde{d}\alpha_{3}} \text{ and } L_{2} {=} \frac{\tilde{a}^{2}}{\tilde{d}}.
\end{align*}
\end{enumerate}
\end{theorem}
\end{comment}
Further equivalent algebraic conditions to \ref{nl:bcc1a}--\ref{nl:bcc2} are provided in the following theorem, which allow us to obtain a minimal generating set for the bicubic impedances in $\mathcal{Z}_{1,2}$. Prior to stating this theorem, we recall from Section \ref{sec:ncm} that if the impedance of the series-parallel network $N$ takes the form
\begin{equation}
\hspace*{-0.3cm}\frac{a_{3}s^{3} + a_{2}s^{2} + a_{1}s + a_{0}}{b_{3}s^{3} + b_{2}s^{2} + b_{1}s + b_{0}}, \hspace{0.1cm} (b_{i}>0 \text{ for }i=3,2,1 \text{ or }0),\label{eq:zbcd1}
\end{equation}
then the impedance of $N^{p}$ takes the form
\begin{equation}
\hspace*{-0.3cm}\frac{b_{0}s^{3} + b_{1}s^{2} + b_{2}s + b_{3}}{a_{0}s^{3} + a_{1}s^{2} + a_{2}s + a_{3}}, \hspace{0.1cm} (a_{i}>0 \text{ for }i=3,2,1 \text{ or }0).\label{eq:zbcd2}
\end{equation}
\begin{theorem}
\label{thm:bcmrac}
Let $Z \in \mathbb{R}(s)$ be bicubic. Then $Z \in \mathcal{Z}_{1,2}$ if and only if there exists $x \geq 0$ such that $Z$ takes the form of (\ref{eq:zbcd1}) or (\ref{eq:zbcd2})
and at least
one of the following conditions holds:
\begin{enumerate}
\item Condition \ref{nl:bcc1a} of Theorem \ref{thm:bcn1013} holds and either (i) $x = 0$; (ii) $\lambda_{1} = 0$; or (iii) $c_{0} = 0$.\label{nl:bcmrc1}
\item Condition \ref{nl:bcc1b} of Theorem \ref{thm:bcn1013} holds and either (i) $x = 0$; (ii) $\lambda_{3} = 0$; or (iii) $c_{2} = 0$.\label{nl:bcmrc2}
\item Condition \ref{nl:bcc1c} of Theorem \ref{thm:bcn1013} holds and either (i) $x = 0$; (ii) $\lambda_{4} = 0$; or (iii) $d_{0} = 0$.\label{nl:bcmrc3}
\item Condition \ref{nl:bcc1d} of Theorem \ref{thm:bcn1013} holds and either (i) $x = 0$; (ii) $\lambda_{2} = 0$; or (iii) $d_{2} = 0$.\label{nl:bcmrc4}
\item Condition \ref{nl:bcc2} of Theorem \ref{thm:bcn14} holds and either (i) $x = 0$; (ii) $c_{0} = 0$; or (iii) $d_{2} = 0$.\label{nl:bcmrc5}
\begin{comment}
\item Condition \ref{nl:bcc3} of Theorem \ref{thm:bcn15} holds and either (i) $x = 0$; (ii) $c_{0} = 0$; or (iii) $d_{2} = 0$.\label{nl:bcmrc6}
\item Condition \ref{nl:bcc4a} of Theorem \ref{thm:bcn1619} holds and either (i) $x = 0$; (ii) $\lambda_{1} = 0$; or (iii) $c_{0} = 0$.\label{nl:bcmrc7}
\item Condition \ref{nl:bcc4b} of Theorem \ref{thm:bcn1619} holds and either (i) $x = 0$; (ii) $\lambda_{3} = 0$; or (iii) $c_{2} = 0$.\label{nl:bcmrc8}
\item Condition \ref{nl:bcc4c} of Theorem \ref{thm:bcn1619} holds and either (i) $x = 0$; (ii) $\lambda_{4} = 0$; or (iii) $d_{0} = 0$.\label{nl:bcmrc9}
\item Condition \ref{nl:bcc4d} of Theorem \ref{thm:bcn1619} holds and either (i) $x = 0$; (ii) $\lambda_{2} = 0$; or (iii) $d_{2} = 0$.\label{nl:bcmrc10}
\end{comment}
\end{enumerate}
\end{theorem}
\begin{IEEEproof}
Let $Z \in \mathbb{R}(s)$ be bicubic. Then, from Lemma \ref{lem:ai} and Theorem \ref{thm:z12nmgs}, $Z \in \mathcal{Z}_{1,2}$ if and only if $Z$ is the impedance of a network from one of the classes $\mathcal{N}_{11}$--$\mathcal{N}_{15}$. Also, since $Z$ is bicubic, then there exists $a,b$ as in (\ref{eq:abd}) such that $Z$ takes the form of (\ref{eq:zbcd1}) or (\ref{eq:zbcd2}) and $b_{i} > 0$ for at least one value of $i \in 0, \ldots , 3$. It then follows from Theorems \ref{thm:bcn1013} and \ref{thm:bcn14} that $Z \in \mathcal{Z}_{1,2}$ if and only if there exists $x \geq 0$ such that $Z$ takes the form of (\ref{eq:zbcd1}) or (\ref{eq:zbcd2}) and one of conditions \ref{nl:bcc1a}--\ref{nl:bcc2} holds.
We denote this $x$ by $x_{0}$, and we will show that there exists $0 \leq y \leq x_{0}$ such that at least one of conditions \ref{nl:bcc1a}--\ref{nl:bcc2} holds for all $y \leq x \leq x_{0}$ and at least one of the conditions of the present theorem holds when $x = y$.
Suppose initially that there exists an $x \geq 0$ such that one of conditions \ref{nl:bcc1a}--\ref{nl:bcc1d} holds, and denote this $x$ by $x_{0}$. We note that $\tilde{a}, \tilde{b}, \tilde{c}$, $c_{0}$--$c_{2}$, $d_{0}$--$d_{2}$, $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ as defined in Theorem \ref{thm:bcn1013} are all polynomial (hence continuous) functions of $x$. It follows that
either (a) one of conditions \ref{nl:bcc1a}--\ref{nl:bcc1d} holds for all $x \geq 0$ (in which case one of conditions \ref{nl:bcmrc1}(i), \ref{nl:bcmrc2}(i), \ref{nl:bcmrc3}(i) or \ref{nl:bcmrc4}(i) of the present theorem holds); or (b) there exists $0 < y \leq x_{0}$ and $\epsilon > 0$ such that one of conditions \ref{nl:bcc1a}--\ref{nl:bcc1d} holds for all $y \leq x \leq x_{0}$ and none of these conditions hold in the interval $y-\epsilon < x < y$. Accordingly, we suppose condition (b) holds, and we let $x_{0} \geq 0$, $\epsilon > 0$ and $0 < y \leq x_{0}$ be as defined in that condition. Also, we recall the equivalence of conditions \ref{nl:bcc1a}--\ref{nl:bcc1d} and conditions \ref{nl:bcca1}--\ref{nl:bcca4} of Lemma \ref{lem:bciec}. It follows that, for $x = y$,
then one of conditions \ref{nl:bcca1}--\ref{nl:bcca4} of Lemma \ref{lem:bciec} must hold, with one of the (non-strict) inequalities being satisfied with equality. In other words, for $x = y$, either one of conditions \ref{nl:bcmrc1}--\ref{nl:bcmrc4} of the present theorem holds, or $\alpha_{2} = 0$, or $\tilde{a} = 0$.
Next, we note that
\begin{equation*}
\frac{\tilde{c}}{\tilde{b}^{2}} = -\frac{d}{dx}\left(\frac{\tilde{a}}{\tilde{b}}\right).
\end{equation*}
Since $\tilde{c}/\tilde{b}^{2} > 0$ for all $y \leq x \leq x_{0}$, then $\tilde{a}/\tilde{b}$ is a decreasing function of $x$ in this interval. Thus, for $x=y$, it is not possible for $\tilde{a}$ to be zero, and we conclude that either one of conditions \ref{nl:bcmrc1}--\ref{nl:bcmrc4} of the present theorem holds or $\alpha_{2} = 0$.
Now, suppose that condition \ref{nl:bcc1a} holds at $x = y$ but none of conditions \ref{nl:bcmrc1}--\ref{nl:bcmrc4} in the present theorem hold. Then $\alpha_{2} = 0$ when $x = y$. Since condition \ref{nl:bcc1a} holds at $x = y$ but $\lambda_{1}, c_{0} \neq 0$, then from Lemma \ref{lem:bciec} it follows that condition \ref{nl:bcca1} of that lemma holds and $c_{0}d_{0}, d_{2}\lambda_{1}, \alpha_{1} > 0$. Also, at $x = y$, since $\alpha_{2} = 0$, then $R_{0}(c,d) = \alpha_{1}\alpha_{3} < 0$, and it follows that $\alpha_{3} < 0$. Moreover, $\lambda_{1} = d_{1}\alpha_{1}$ and $\lambda_{3} = -d_{1}\alpha_{3}$, and hence $\lambda_{3}$ has the same sign as $d_{1}$. We conclude that condition \ref{nl:bcc1b} also holds at $x = y$. Since, in addition, condition \ref{nl:bcmrc2} of the present theorem does not hold at $x = y$, then from Lemma \ref{lem:bciec} it follows that condition \ref{nl:bcca2} of that lemma holds and $c_{2}d_{2}, d_{0}\lambda_{3}, -\alpha_{3} > 0$. Finally, since $\tilde{a} \neq 0$ at $x = y$, and either $\alpha_{2}d_{0}d_{2} \geq 0$ or $\alpha_{2}d_{0}d_{2} \leq 0$ for all $x \in \mathbb{R}$, then it follows that there exists $\gamma > 0$ such that either condition \ref{nl:bcca1} or condition \ref{nl:bcca2} of Lemma \ref{lem:bciec} holds for all $y - \gamma < x < y$. It follows from Lemma \ref{lem:bciec} that either condition \ref{nl:bcc1a} or condition \ref{nl:bcc1b} hold for all $y - \gamma < x < y$, which contradicts the assumption that none of conditions \ref{nl:bcc1a}--\ref{nl:bcc1d} hold in the interval $y - \epsilon < x < y$. We conclude that, if condition \ref{nl:bcc1a} holds at $x = y$, then one of conditions \ref{nl:bcmrc1}--\ref{nl:bcmrc4} in the present theorem hold. A similar argument then shows that, if condition \ref{nl:bcc1b} (resp., \ref{nl:bcc1c}, \ref{nl:bcc1d}) holds at $x = y$, then one of conditions \ref{nl:bcmrc1}--\ref{nl:bcmrc4} in the present theorem hold.
We have shown that, if there exists $x \geq 0$ such that one of conditions \ref{nl:bcc1a}--\ref{nl:bcc1d} holds, then there exists $x \geq 0$ such that one of conditions \ref{nl:bcmrc1}--\ref{nl:bcmrc4} of the present theorem hold. A similar argument shows that, if there exists $x \geq 0$ such that condition \ref{nl:bcc2} holds, then there exists $x \geq 0$ such that condition \ref{nl:bcmrc5} of the present theorem holds. This completes the proof of the present theorem.
\end{IEEEproof}
From Theorems \ref{thm:bcn1013} and \ref{thm:bcn14}--\ref{thm:bcmrac}, it is easily shown that if $Z \in \mathcal{Z}_{1,2}$ is bicubic, then $Z$ is realized by a network from one of the classes $\mathcal{N}_{16}$--$\mathcal{N}_{30}$ or $\mathcal{N}_{16}^{p}$--$\mathcal{N}_{30}^{p}$.
We are now in a position to provide an algebraic characterisation of the set of bicubic impedances in $\mathcal{Z}_{3}$. We first note the following characterisations of the sets of bicubic impedances in $\mathcal{Z}_{0,3}$ and $\mathcal{Z}_{3,0}$:
\begin{lemma}
\label{lem:bcrcars}
Let $Z \in \mathbb{R}(s)$ be bicubic, and consider the following set of inequalities:
\begin{remunerate}
\setcounter{muni}{5}
\labitem{(C\arabic{muni})}{nl:bcc1tre} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2,3$, and $b_{3}, b_{2}, b_{1}, b_{1}b_{2}-b_{0}b_{3}, R_{2}(a,b), R_{1}(a,b), R_{0}(a,b) > 0$.
\end{remunerate}
Then $Z \in \mathcal{Z}_{3,0}$ (resp., $Z \in \mathcal{Z}_{0,3}$) if and only if $Z$ (resp., $1/Z$) takes the form of (\ref{eq:zbcd1}) and condition \ref{nl:bcc1tre} holds.
\end{lemma}
\begin{IEEEproof}
First, suppose $Z \in \mathcal{Z}_{3,0}$. Then $Z$ can be written in the form of (\ref{eq:zbcd1}) where $a_{i}, b_{i} \geq 0$ for $i = 0,1,2,3$ by Kirchhoff's tree formula. Also, from \cite[Lemmas 5 and 7]{HugCF}, then (i) $Z$ is analytic in the open right half plane, and the only possible imaginary axis poles are simple and at the origin; and (ii) $R_{2}(a,b), R_{1}(a,b), R_{0}(a,b) > 0$. Moreover, in case (i), then either (ia) $b_{0} \neq 0$ and $b_{3}s^{3} + b_{2}s^{2} + b_{1}s + b_{0}$ has no zeros in the closed right half plane; or (ib) $b_{0} = 0$ and $b_{3}s^{3} + b_{2}s^{2} + b_{1}s$ has no zeros in the closed right half plane. In either case, we require $b_{3}, b_{2}, b_{1}, b_{1}b_{2}-b_{0}b_{3} > 0$ by the Routh-Hurwitz criteria, so $Z$ satisfies \ref{nl:bcc1tre}.
Conversely, suppose $Z$ takes the form of (\ref{eq:zbcd1}) and \ref{nl:bcc1tre} is satisfied. Then it is straightforward to show that (i) $Z$ is analytic in the open right half plane; (ii) $Z(s_{0}) \geq 0$ for all $s_{0} > 0$; and (iii) $R_{2}(a,b), R_{1}(a,b), R_{0}(a,b) > 0$. Thus, from \cite[Lemma 7 and Theorem 8]{HugCF}, we conclude that $Z$ is the impedance of a network from $\mathcal{N}_{10}$, and so $Z \in \mathcal{Z}_{3,0}$.
Finally, since $Z \in \mathcal{Z}_{3,0} \iff 1/Z \in \mathcal{Z}_{0,3}$, then $Z \in \mathcal{Z}_{0,3}$ if and only if $1/Z$ can be written in the form of (\ref{eq:zbcd1}) and condition \ref{nl:bcc1tre} holds.
\end{IEEEproof}
The set $\mathcal{Z}_{3}$ then has the following characterisation:
\begin{theorem}
\label{thm:z03arc}
Let $Z$ be bicubic. Then $Z \in \mathcal{Z}_{3}$ if and only if either (i) there exists $x \geq 0$ such that $Z$ or $1/Z$ takes one of the forms (\ref{eq:zbcd1})--(\ref{eq:zbcd2}) and one of conditions \ref{nl:bcmrc1}--\ref{nl:bcmrc5} of Theorem \ref{thm:bcmrac} holds; or (ii) $Z$ or $1/Z$ takes the form of (\ref{eq:zbcd1}) and condition \ref{nl:bcc1tre} of Lemma \ref{lem:bcrcars} is satisfied.
\end{theorem}
\begin{IEEEproof}
This follows from Theorem \ref{thm:bcmrac} and Lemma \ref{lem:bcrcars}, and the fact that $Z \in \mathcal{Z}_{m,n} \iff 1/Z \in \mathcal{Z}_{n,m}$.
\end{IEEEproof}
\begin{remark}
\label{rem:ercc}
In \cite{ZJ_ERBI}, the concept of an \emph{essential-regular} function was defined, and a minimal generating set was obtained for the class of bicubic essential-regular functions.
\begin{comment}
A bicubic impedance is defined as essential-regular if it is positive-real and, with the notation
\begin{align*}
H_{1}(s) &= \tfrac{1}{G_{1}(s)} {-} \tfrac{1}{s} \lim_{s \rightarrow 0}\left(\tfrac{s}{G_{1}(s)}\right) \text{ with } G_{1}(s) = Z(s) {-} Z(0), \\
H_{2}(s) &= \tfrac{1}{G_{2}(s)} {-} \tfrac{1}{s} \lim_{s \rightarrow \infty}\left(\tfrac{1}{sG_{2}(s)}\right) \text{ with } G_{2}(s) = Z(s) {-} Z(\infty), \\
H_{3}(s) &= \tfrac{1}{G_{3}(s)} {-} \tfrac{1}{s} \lim_{s \rightarrow 0}\left(\tfrac{s}{G_{3}(s)}\right) \text{ with } G_{3}(s) = \tfrac{1}{Z(s)} {-} \tfrac{1}{Z(0)}, \\
H_{4}(s) &= \tfrac{1}{G_{4}(s)} {-} \tfrac{1}{s} \lim_{s \rightarrow \infty}\left(\tfrac{1}{sG_{4}(s)}\right) \text{ with } G_{4}(s) = \tfrac{1}{Z(s)} {-} \tfrac{1}{Z(\infty)},
\end{align*}
then $G_{i}$ is positive-real and $H_{i}$ is regular (i.e., $H_{i} = c/d$ where one of conditions \ref{nl:bqcb1}--\ref{nl:bqcb4} in Lemma \ref{lem:bqrrc} hold) for at least one value of $i \in 1, \ldots , 4$.
\end{comment}
However, not every impedance $Z \in \mathcal{Z}_{3}$ is essential-regular. To see this, note initially from \cite{ZJ_ERBI} that, if $Z$ in (\ref{eq:zbcd1}) is essential regular and bicubic, then either (i) $a_{3}b_{2} - a_{2}b_{3}$, $a_{3}b_{1} - a_{1}b_{3}$ and $a_{3}b_{0} - a_{0}b_{3}$ have the same sign; or (ii) $a_{3}b_{0} - a_{0}b_{3}$, $a_{2}b_{0} - a_{0}b_{2}$ and $a_{1}b_{0} - a_{0}b_{1}$ have the same sign. Now, consider the impedance of network $N_{11}$ in Fig.\ \ref{fig:bcr2}, and let $R_{1} = \tfrac{4}{9}$, $R_{2} = \tfrac{128}{279}$, $G_{3} = \tfrac{31}{96}$, $R_{4} = 0$, $R_{5} = 1$, $C_{1} = \tfrac{405}{64}$, $L_{1} = \tfrac{576}{961}$, and $L_{2} = 1$. It can be verified that the impedance of this network takes the form of (\ref{eq:zbcd1}) with $a_{3} = 60$, $a_{2} = 124$, $a_{1} = 92$, $a_{0} = 28$, $b_{3} = 135$, $b_{2} = 291$, $b_{1} = 191$ and $b_{0} = 59$, and by direct calculation we find that neither (i) nor (ii) hold, so $Z$ is not essential regular.
\end{remark}
\begin{remark}
\label{rem:qe}
Theorem \ref{thm:z03arc} provides an implicit description of the sets $\mathcal{Z}_{1,2}$ and $\mathcal{Z}_{2,1}$ in terms of the quantifier $x$. An explicit description can be obtained using the techniques of quantifier elimination. The problem amounts to determining the existence of a root of a polynomial subject to a finite number of polynomial inequalities, which can be solved using the technique in \cite{CAN_QE}. However, the computation is very demanding in terms of both memory and computation time.
The method in \cite{CAN_QE} is formulated using so-called \emph{Sturm chains}, but can equivalently be stated in terms of so-called \emph{Cauchy indices}. Specifically, the Cauchy index of an $H \in \mathbb{R}(x)$, denoted $I_{-\infty}^{+\infty}(H)$, is the sum of the number of jumps in $H$ from $-\infty$ to $+\infty$ less the number of jumps from $+\infty$ to $-\infty$, as $x$ traverses the real axis from $-\infty$ to $+\infty$. It is then straightforward to verify that, given polynomials $f$ and $g$ in the indeterminate $x$, then $I_{-\infty}^{+\infty}(g\tfrac{df}{dx}/f)$ is equal to the number of roots of $f$ at which $g > 0$ minus the number of roots of $f$ at which $g < 0$. Judicious choices for the polynomial $g$ can then be used to determine the number of roots of $f$ which satisfy a certain set of inequalities (see \cite{CAN_QE}).
For example, consider the problem of determining the number of roots of $\lambda_{1} = 0$ for which $\alpha_{2} \geq 0$ (here, we consider $\lambda_{1}$ and $\alpha_{2}$ as polynomials in the indeterminate $x$). From the previous paragraph, it is easily verified that this is equal to $I_{-\infty}^{+\infty}(\tfrac{d \lambda_{1}}{dx}/\lambda_{1}) + \tfrac{1}{2}I_{-\infty}^{+\infty}(\alpha_{2}\tfrac{d \lambda_{1}}{dx}/\lambda_{1}) - \tfrac{1}{2}I_{-\infty}^{+\infty}(\alpha_{2}^{2}\tfrac{d \lambda_{1}}{dx}/\lambda_{1})$. Here, $\tfrac{d \lambda_{1}}{dx}/\lambda_{1}$ can be written as a ratio $u/v$ where $u$ (resp., $v$) is a polynomial of degree $8$ (resp., $9$) whose coefficients are polynomials in $a_{i}, b_{i}$ $(i = 0,1,2,3)$. Then, $I_{-\infty}^{+\infty}(\tfrac{d \lambda_{1}}{dx}/\lambda_{1})$ can be obtained by counting permutations and variations in a sequence of $2k \times 2k$ subdeterminants of a Sylvester matrix formed from the coefficients of $u$ and $v$ ($k = 1, \ldots , 9$). However, even the $6 \times 6$ subdeterminant is the product of $3$ polynomials in the coefficients $a_{i}, b_{i}$ ($i = 0,1,2,3$), of which one has multi-degree $14$ and $320$ terms.
The aforementioned discussion indicates that it is not likely to be informative, and may not be computationally feasible, to eliminate the quantifier $x$ from the description of the sets $\mathcal{Z}_{1,2}$ and $\mathcal{Z}_{2,1}$. Nevertheless, the algorithm in \cite{CAN_QE} can provide a definitive answer as to whether a given impedance $Z$ is in one of these sets, providing the coefficients in $Z$ are rational numbers. Also, to obtain a (approximate) realization of a given impedance $Z \in \mathcal{Z}_{1,2} \cup \mathcal{Z}_{2,1}$ is \emph{not} computationally demanding: it suffices to compute the roots of a handful of polynomials (to the desired accuracy) and test the signs of a handful of polynomials at each of these roots.
\end{remark}
\subsection{Biquadratic impedances in $\mathcal{Z}_{3}$}
In this final subsection, we consider the biquadratic impedances which are realized by the networks described in Theorem \ref{thm:z12nmgs}. In this case, we provide explicit expressions for the element values in these networks in terms of the polynomials
\begin{align}
a(s) &= a_{2}s^{2} + a_{1}s + a_{0}, \text{ and}\\
b(s) &= b_{2}s^{2} + b_{1}s + b_{0},\label{eq:abd2}
\end{align}
in a coprime factorization $Z = a/b$ of the impedance $Z$. These explicit expressions are provided in Theorems \ref{thm:bqn1013} and \ref{thm:bqn14}.
\begin{theorem}
\label{thm:bqn1013}
Let $a,b$ in (\ref{eq:abd2}) be coprime; let $b_{i} > 0$ for at least one value of $i \in 0, \ldots , 2$; let
\begin{comment}
\begin{enumerate}[label=(\roman*)]
\item $\tilde{a} = a(-x)$, $\tilde{b} = b(-x)$,
\item $\begin{bmatrix}f_{0}& f_{1}\end{bmatrix} = \begin{bmatrix}1& -x\end{bmatrix}\mathcal{B}(a,b)$,
\item $c_{2} = f_{1}, \hspace{0.1cm} c_{1} = f_{0}+xf_{1}, \hspace{0.1cm} c_{0} = xf_{0}$,
\item $h_{2} = b_{2}, \hspace{0.1cm} h_{1} = b_{1} - zf_{1}, \hspace{0.1cm} h_{0} = b_{0} - zf_{0}$,
\item $d_{2} = \tilde{b}h_{2}, \hspace{0.1cm} d_{1} = \tilde{b}h_{1}, \hspace{0.1cm} d_{0} = \tilde{b}h_{0}$,
\item $c(s) = c_{2}s^{2} + c_{1}s + c_{0}$,
\item $d(s) = d_{2}s^{2} + d_{1}s + d_{0}$,
\item $h(s) = h_{2}s^{2} + h_{1}s + h_{0}$,
\item $\tilde{h} = h(-x)$, and
\item $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as in (\ref{eq:bqid1}),
\end{enumerate}
\end{comment}
\begin{itemize}[leftmargin=0cm]
\item[] $\tilde{a} = a(-x)$, $\tilde{b} = b(-x)$, $\begin{bmatrix}f_{0}& f_{1}\end{bmatrix} = \begin{bmatrix}1& -x\end{bmatrix}\mathcal{B}(a,b)$,
\item[] $c_{2} = f_{1}, \hspace{0.1cm} c_{1} = f_{0}+xf_{1}, \hspace{0.1cm} c_{0} = xf_{0}$,
\item[] $h_{2} = b_{2}, \hspace{0.1cm} h_{1} = b_{1} - zf_{1}, \hspace{0.1cm} h_{0} = b_{0} - zf_{0}$,
\item[] $d_{2} = \tilde{b}h_{2}, \hspace{0.1cm} d_{1} = \tilde{b}h_{1}, \hspace{0.1cm} d_{0} = \tilde{b}h_{0}$,
\item[] $c(s) = c_{2}s^{2} + c_{1}s + c_{0}$, $d(s) = d_{2}s^{2} + d_{1}s + d_{0}$,
\item[] $h(s) = h_{2}s^{2} + h_{1}s + h_{0}$, $\tilde{h} = h(-x)$, and
\item[] $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as in (\ref{eq:bqid1});
\end{itemize}
and consider the following sets of inequalities:
\begin{remunerate}
\setcounter{muni}{6}
\labitem{(Q\arabic{muni})}{nl:bq3c3a} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2$; $\tilde{a}, \tilde{b}, z, d_{0}, d_{1}, d_{2}, f_{0}, f_{1}$ and $\lambda_{1}$ have the same sign; $\alpha_{2}, -\tilde{b}\tilde{h}R_{0}(a,b) {\geq} 0$; and $\tilde{h}, R_{0}(a,b) {\neq} 0$.
\labitem{(Q\arabic{muni})}{nl:bq3c3b} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2$; $\tilde{a}, \tilde{b}, z, d_{0}, d_{1}, d_{2}, f_{0}, f_{1}$ and $\lambda_{3}$ have the same sign; $\alpha_{2}, \tilde{b}\tilde{h}R_{0}(a,b) \leq 0$; and $\tilde{h}, R_{0}(a,b) \neq 0$.
\labitem{(Q\arabic{muni})}{nl:bq3c3c} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2$; $\tilde{a}, \tilde{b}, z, d_{0}, d_{1}, d_{2}, f_{0}, f_{1}$ and $\lambda_{4}$ have the same sign; $\alpha_{2}, \tilde{b}\tilde{h}R_{0}(a,b) {\leq} 0$; and $\tilde{h}, R_{0}(a,b) {\neq} 0$.
\labitem{(Q\arabic{muni})}{nl:bq3c3d}$a_{i}, b_{i} \geq 0$ for $i {=} 0,1,2$; $\tilde{a}, \tilde{b}, z, d_{0}, d_{1}, d_{2}, f_{0}, f_{1}$ and $\lambda_{2}$ have the same sign; $\alpha_{2}, -\tilde{b}\tilde{h}R_{0}(a,b) {\geq} 0$; and $\tilde{h}, R_{0}(a,b) {\neq} 0$.
\end{remunerate}
Then $Z$ is the impedance of a network from $\mathcal{N}_{11}$ (resp., $\mathcal{N}_{12}$, $\mathcal{N}_{13}$, $\mathcal{N}_{14}$) if and only if there exists $x \geq 0$ and $z \neq 0$ such that $Z = a/b$ and condition \ref{nl:bq3c3a} (resp., \ref{nl:bq3c3b}, \ref{nl:bq3c3c}, \ref{nl:bq3c3d}) is satisfied.
Moreover,
\begin{enumerate}[leftmargin=0.3cm]
\item If $x \geq 0$ and $z \neq 0$ are such that \ref{nl:bq3c3a} holds, then $\tilde{b}, d_{0}, d_{2} \neq 0$, $\alpha_{1} > 0$, and $Z$ is the impedance of $N_{11}$ with\label{nl:bqev1}
\begin{align*}
&R_{1} = \tfrac{c_{0}}{d_{0}}, \hspace{0.1cm} R_{2} = \tfrac{\alpha_{2}}{d_{0}d_{2}}, \hspace{0.1cm} G_{3} = \tfrac{-d_{2}\lambda_{1}}{\tilde{b}^{3}\tilde{h}R_{0}(a,b)}, \hspace{0.1cm} R_{4} = \tfrac{\tilde{a}}{\tilde{b}},\\
&R_{5} = \tfrac{x}{\tilde{b}z}, \hspace{0.1cm} C_{1} = \tfrac{-d_{2}^{2}\alpha_{1}}{\tilde{b}^{3}\tilde{h}R_{0}(a,b)}, \hspace{0.1cm} L_{1} = \tfrac{\alpha_{1}}{d_{0}^{2}} \text{ and } L_{2} = \tfrac{1}{\tilde{b}z}.
\end{align*}
\item If $x \geq 0$ and $z \neq 0$ are such that \ref{nl:bq3c3b} holds, then $\tilde{b}, d_{0}, d_{2} \neq 0$, $\alpha_{3} < 0$, and $Z$ is the impedance of $N_{12}$ with\label{nl:bqev2}
\begin{align*}
&R_{1} = \tfrac{c_{2}}{d_{2}}, \hspace{0.1cm} R_{2} = \tfrac{-\alpha_{2}}{d_{0}d_{2}}, \hspace{0.1cm} G_{3} = \tfrac{-d_{0}\lambda_{3}}{\tilde{b}^{3}\tilde{h}R_{0}(a,b)}, \hspace{0.1cm} R_{4} = \tfrac{\tilde{a}}{\tilde{b}},\\
&R_{5} = \tfrac{x}{\tilde{b}z}, \hspace{0.1cm} C_{1} = \tfrac{-d_{2}^{2}}{\alpha_{3}}, \hspace{0.1cm} L_{1} = \tfrac{\tilde{b}^{3}\tilde{h}R_{0}(a,b)}{d_{0}^{2}\alpha_{3}} \text{ and } L_{2} = \tfrac{1}{\tilde{b}z}.
\end{align*}
\item If $x \geq 0$ and $z \neq 0$ are such that \ref{nl:bq3c3c} holds, then $\tilde{b}, c_{0}, c_{2} \neq 0$, $\alpha_{1} < 0$, and $Z$ is the impedance of $N_{13}$ with\label{nl:bqev3}
\begin{align*}
&G_{1} = \tfrac{d_{0}}{c_{0}}, \hspace{0.1cm} G_{2} = \tfrac{-\alpha_{2}}{c_{0}c_{2}}, \hspace{0.1cm} R_{3} = \tfrac{-c_{2}\lambda_{4}}{\tilde{b}^{3}\tilde{h}R_{0}(a,b)}, \hspace{0.1cm} R_{4} = \tfrac{\tilde{a}}{\tilde{b}}, \\
&R_{5} = \tfrac{x}{\tilde{b}z}, \hspace{0.1cm} C_{1} = \tfrac{-\alpha_{1}}{c_{0}^{2}}, \hspace{0.1cm} L_{1} = \tfrac{c_{2}^{2}\alpha_{1}}{\tilde{b}^{3}\tilde{h}R_{0}(a,b)} \text{ and } L_{2} = \tfrac{1}{\tilde{b}z}.
\end{align*}
\item If $x \geq 0$ and $z \neq 0$ are such that \ref{nl:bq3c3d} holds, then $\tilde{b}, c_{0}, c_{2} \neq 0$, $\alpha_{3} > 0$, and $Z$ is the impedance of $N_{14}$ with\label{nl:bqev4}
\begin{align*}
&G_{1} = \tfrac{d_{2}}{c_{2}}, \hspace{0.1cm} G_{2} = \tfrac{\alpha_{2}}{c_{0}c_{2}}, \hspace{0.1cm} R_{3} = \tfrac{-c_{0}\lambda_{2}}{\tilde{b}^{3}\tilde{h}R_{0}(a,b)}, \hspace{0.1cm} R_{4} = \tfrac{\tilde{a}}{\tilde{b}}, \\
&R_{5} = \tfrac{x}{\tilde{b}z}, \hspace{0.1cm} C_{1} = \tfrac{-\tilde{b}^{3}\tilde{h}R_{0}(a,b)}{c_{0}^{2}\alpha_{3}}, \hspace{0.1cm} L_{1} = \tfrac{c_{2}^{2}}{\alpha_{3}} \text{ and } L_{2} = \tfrac{1}{\tilde{b}z}.
\end{align*}
\end{enumerate}
\end{theorem}
\begin{theorem}
\label{thm:bqn14}
Let $a,b \in \mathbb{R}[s]$ in (\ref{eq:abd2}) be coprime; let $b_{i} > 0$ for at least one value of $i \in 0, \ldots , 2$; let
\begin{comment}
\begin{enumerate}[label=(\roman*)]
\item $\tilde{a} = a_{0}x^{2} -a_{1}x + a_{2}$,
\item $\tilde{b} = b_{0}x^{2} - b_{1}x + b_{2}$,
\item $\begin{bmatrix}f_{0}& f_{1}\end{bmatrix} = \begin{bmatrix}-x& 1\end{bmatrix}\mathcal{B}(a,b)$,
\item $c_{2} = xf_{1}, \hspace{0.1cm} c_{1} = xf_{0}+f_{1}, \hspace{0.1cm} c_{0} = f_{0}$,
\item $h_{2} = b_{2}-zf_{1}, \hspace{0.1cm} h_{1} = b_{1} - zf_{0}, \hspace{0.1cm} h_{0} = b_{0}$,
\item $d_{2} = \tilde{b}h_{2}, \hspace{0.1cm} d_{1} = \tilde{b}h_{1}, \hspace{0.1cm} d_{0} = \tilde{b}h_{0}$,
\item $c(s) = c_{2}s^{2} + c_{1}s + c_{0}$,
\item $d(s) = d_{2}s^{2} + d_{1}s + d_{0}$,
\item $\tilde{h} = h_{0}x^{2} -h_{1}x + h_{2}$, and
\item $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as in (\ref{eq:bqid1}),
\end{enumerate}
\end{comment}
\begin{itemize}[leftmargin=0cm]
\item[] $\tilde{a} = a_{0}x^{2} -a_{1}x + a_{2}$, $\tilde{b} = b_{0}x^{2} - b_{1}x + b_{2}$,
\item[] $\begin{bmatrix}f_{0}& f_{1}\end{bmatrix} = \begin{bmatrix}-x& 1\end{bmatrix}\mathcal{B}(a,b)$,
\item[] $c_{2} = xf_{1}, \hspace{0.1cm} c_{1} = xf_{0}+f_{1}, \hspace{0.1cm} c_{0} = f_{0}$,
\item[] $h_{2} = b_{2}-zf_{1}, \hspace{0.1cm} h_{1} = b_{1} - zf_{0}, \hspace{0.1cm} h_{0} = b_{0}$,
\item[] $d_{2} = \tilde{b}h_{2}, \hspace{0.1cm} d_{1} = \tilde{b}h_{1}, \hspace{0.1cm} d_{0} = \tilde{b}h_{0}$,
\item[] $c(s) = c_{2}s^{2} + c_{1}s + c_{0}$, $d(s) = d_{2}s^{2} + d_{1}s + d_{0}$,
\item[] $\tilde{h} = h_{0}x^{2} -h_{1}x + h_{2}$, and $\alpha_{1}$--$\alpha_{3}$ and $\lambda_{1}$--$\lambda_{4}$ be as in (\ref{eq:bqid1});
\end{itemize}
and consider the set of inequalities:
\begin{remunerate}
\setcounter{muni}{10}
\labitem{(Q\arabic{muni})}{nl:bq3c4} $a_{i}, b_{i} \geq 0$ for $i = 0,1,2$; $\tilde{a}, \tilde{b}, z, d_{0}, d_{1}, d_{2}, f_{0}, f_{1}$ and $\lambda_{1}$ have the same sign; $\alpha_{2}, \tilde{b}\tilde{h}R_{0}(a,b) {\geq} 0$; and $\tilde{h}, R_{0}(a,b) {\neq} 0$.
\end{remunerate}
Then $Z$ is the impedance of a network from $\mathcal{N}_{15}$ if and only if there exists $x \geq 0$ and $z \neq 0$ such that $Z = a/b$ and condition \ref{nl:bq3c4} is satisfied.
Moreover, if $x \geq 0$ and $z \neq 0$ are such that \ref{nl:bq3c4} holds, then $\tilde{b}, d_{0}, \lambda_{1} \neq 0$, $\alpha_{1} > 0$, and $Z$ is the impedance of $N_{15}$ in Fig.\ \ref{fig:bcr6} with\label{nl:bqev5}
\begin{align*}
&R_{1} = \tfrac{c_{0}}{d_{0}}, \hspace{0.1cm} R_{2} = \tfrac{\alpha_{1}^{2}}{d_{0}\lambda_{1}}, \hspace{0.1cm} G_{3} = \tfrac{d_{2}\lambda_{1}}{\tilde{b}^{3}\tilde{h}R_{0}(a,b)}, \hspace{0.1cm} R_{4} = \tfrac{\tilde{a}}{\tilde{b}},\\
&R_{5} = \tfrac{x}{\tilde{b}z}, \hspace{0.1cm} C_{1} = \tilde{b}z, \hspace{0.1cm} L_{1} = \tfrac{\tilde{b}^{3}\tilde{h}\alpha_{1}R_{0}(a,b)}{\lambda_{1}^{2}} \text{ and } L_{2} = \tfrac{\alpha_{1}}{d_{0}^{2}}.
\end{align*}
\end{theorem}
The proofs of Theorems \ref{thm:bqn1013} and \ref{thm:bqn14} are similar to the proof of Theorem \ref{thm:bcn1013}, and are omitted for brevity.
Next, we recall from Section \ref{sec:ncm} that if the impedance of the series-parallel network $N$ takes the form
\begin{equation}
\frac{a_{2}s^{2} + a_{1}s+ a_{0}}{b_{2}s^{2} + b_{1}s + b_{0}},\label{eq:zbqd1}
\end{equation}
then the impedance of $N^{p}$ takes the form
\begin{equation}
\frac{b_{0}s^{2} + b_{1}s+ b_{2}}{a_{0}s^{2} + a_{1}s + a_{2}}.\label{eq:zbqd2}
\end{equation}
Then a similar proof to that of Theorem \ref{thm:bcmrac}, but this time varying the parameter $z$ as opposed to $x$ (in which case $c_{0}$--$c_{2}$ and $x$ are unchanged), leads to the following theorem:
\begin{theorem}
\label{thm:bq3remrc}
Let $Z \in \mathbb{R}(s)$ be biquadratic. Then $Z \in \mathcal{Z}_{1,2}$ if and only if either (a) $Z \in \mathcal{Z}_{1,1}$; (b) $Z \in \mathcal{Z}_{0,2}$; or (c) there exists $x \geq 0$ and $z \neq 0$
such that $Z(s)$ takes the form of (\ref{eq:zbqd1}) or (\ref{eq:zbqd2}) and at least one of the following conditions holds:
\begin{enumerate}
\item Condition \ref{nl:bq3c3a} of Theorem \ref{thm:bqn1013} holds and $\lambda_{1} = 0$.\label{nl:bq3remrc3}
\item Condition \ref{nl:bq3c3b} of Theorem \ref{thm:bqn1013} holds and $\lambda_{3} = 0$.\label{nl:bq3remrc4}
\item Condition \ref{nl:bq3c3c} of Theorem \ref{thm:bqn1013} holds and either (i) $\lambda_{4} = 0$, or (ii) $d_{0} = 0$.\label{nl:bq3remrc5}
\item Condition \ref{nl:bq3c3d} of Theorem \ref{thm:bqn1013} holds and either (i) $\lambda_{2} = 0$, or (ii) $d_{2} = 0$.\label{nl:bq3remrc6}
\item Condition \ref{nl:bq3c4} of Theorem \ref{thm:bqn14} holds and $d_{2} = 0$.\label{nl:bq3remrc7}
\begin{comment}
\item Condition \ref{nl:bq3c5} of Theorem \ref{thm:bqn15} holds and $c_{0} = 0$.\label{nl:bq3remrc8}
\item Condition \ref{nl:bq3c6a} of Theorem \ref{thm:bqn1619} holds and either (i) $\lambda_{1} = 0$, or (ii) $c_{0} = 0$.\label{nl:bq3remrc9}
\item Condition \ref{nl:bq3c6b} of Theorem \ref{thm:bqn1619} holds and either (i) $\lambda_{3} = 0$, or (ii) $c_{2} = 0$.\label{nl:bq3remrc10}
\item Condition \ref{nl:bq3c6c} of Theorem \ref{thm:bqn1619} holds and $\lambda_{4} = 0$.\label{nl:bq3remrc11}
\item Condition \ref{nl:bq3c6d} of Theorem \ref{thm:bqn1619} holds and $\lambda_{2} = 0$.\label{nl:bq3remrc12}
\end{comment}
\end{enumerate}
\end{theorem}
We then obtain the following algebraic characterisation of the biquadratic impedances in $\mathcal{Z}_{3}$:
\begin{theorem}
Let $Z \in \mathbb{R}(s)$ be biquadratic. Then $Z \in \mathcal{Z}_{3}$ if and only if either (a) $\mathcal{Z} \in \mathcal{Z}_{2}$; or (b) there exists $x \geq 0$ and $z \neq 0$ such that either $Z$ or $1/Z$ takes the form of (\ref{eq:zbqd1}) or (\ref{eq:zbqd2}) and one of conditions \ref{nl:bq3remrc3}--\ref{nl:bq3remrc7} in Theorem \ref{thm:bq3remrc} holds.
\end{theorem}
Finally, we prove Theorem \ref{thm:z12mgs}, and Theorem \ref{thm:z21mgs} follows since $Z \in \mathcal{Z}_{1,2} \iff 1/Z \in \mathcal{Z}_{2,1}$.
\begin{IEEEproof}[Proof of Theorem \ref{thm:z12mgs}]
Let $Z \in \mathcal{Z}_{1,2}$. From Lemmas \ref{lem:blgs} and \ref{lem:blec}, if $Z$ is constant or bilinear, then $Z$ is the impedance of a network from $\mathcal{N}_{1}, \mathcal{N}_{2}$ or $\mathcal{N}_{3}$. By Theorem \ref{thm:bq3remrc}, if $Z$ is biquadratic, either (a) $Z \in \mathcal{Z}_{1,1}$, (b) $Z \in \mathcal{Z}_{0,2}$, or (c) there exists $x \geq 0$ and $z \neq 0$ such that $Z(s)$ takes the form of (\ref{eq:zbqd1}) or (\ref{eq:zbqd2}) and one of conditions \ref{nl:bq3remrc3}--\ref{nl:bq3remrc7} of that theorem hold. In case (c), it is easily shown that $Z$ is the impedance of a network from $\mathcal{N}_{17}, \mathcal{N}_{20}, \mathcal{N}_{22}, \mathcal{N}_{23}, \mathcal{N}_{25}, \mathcal{N}_{26}, \mathcal{N}_{29}, \mathcal{N}_{17}^{p}, \mathcal{N}_{20}^{p}, \mathcal{N}_{22}^{p}, \mathcal{N}_{23}^{p}, \mathcal{N}_{25}^{p}, \mathcal{N}_{26}^{p}$ or $\mathcal{N}_{29}^{p}$. Finally, if $Z$ is bicubic, then there exists $x \geq 0$ such that $Z$ takes the form of (\ref{eq:zbcd1}) or (\ref{eq:zbcd2}) and one of conditions \ref{nl:bcmrc1}--\ref{nl:bcmrc5} of Theorem \ref{thm:bcmrac} hold. It then follows from Lemma \ref{lem:bciec} that $Z$ is the impedance of a network from one of the classes $\mathcal{N}_{16}$--$\mathcal{N}_{30}$ or $\mathcal{N}_{16}^{p}$--$\mathcal{N}_{30}^{p}$.
\end{IEEEproof}
\section{Conclusions}
In this paper, we developed a novel continuity-based argument to solve the minimal network realization problem for the class of impedance functions realized by series-parallel networks containing three energy storage elements and a finite number of resistors. As outlined in Section \ref{sec:pmc}, this represents an important contribution towards the design of passive mechanical controllers containing the inerter. Finally, we note that the mathematical structures encountered in electric circuits are also prevalent in other physical systems, such as multibody systems, hydraulic networks, chemical reaction networks and power systems; and similar network dynamics arise in other fields, such as consensus and clustering algorithms \cite{VDS_PHS}. The adaptation and extension of these results to the design of such systems is a topic for future research.
\bibliographystyle{IEEEtran}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,912 |
{"url":"https:\/\/terrytao.wordpress.com\/tag\/mathml\/","text":"You are currently browsing the tag archive for the \u2018MathML\u2019 tag.\n\nAs the previous discussion on displaying mathematics on the web has become quite lengthy, I am opening a fresh post to continue the topic. \u00a0I\u2019m leaving the previous thread open for those who wish to respond directly to some specific comments in that thread, but otherwise it would be preferable to start afresh on this thread to make it easier to follow the discussion.\n\nIt\u2019s not easy to summarise the discussion so far, but the comments have identified several existing formats for displaying (and marking up) mathematics on the web (mathMLjsMath, MathJaxOpenMath), as well as a surprisingly large number of tools for converting mathematics into web friendly formats (e.g. \u00a0LaTeX2HTMLLaTeXMathML, LaTeX2WPWindows 7 Math Inputitex2MMLRitexGellmumathTeXWP-LaTeXTeX4htblahtexplastexTtHWebEQtechexplorer, etc.). \u00a0Some of the formats are not widely supported by current software, and by current browsers in particular, but it seems that the situation will improve with the next generation of these browsers.\n\nIt seems that the tools that already exist are enough to improvise a passable way of displaying mathematics in various formats online, though there are still significant issues with accessibility, browser support, and ease of use. \u00a0Even if all these issues are resolved, though, I still feel that something is still missing. \u00a0 \u00a0Currently, if I want to transfer some mathematical content from one location to another (e.g. from a LaTeX file to a blog, or from a wiki to a PDF, or from email to an online document, or whatever), or to input some new piece of mathematics, I have to think about exactly what format I need for the task at hand, and what conversion tool may be needed. \u00a0In contrast, if one looks at non-mathematical content such as text, links, fonts, non-Latin alphabets, colours, tables, images, or even video, the formats here have been standardised, and one can manipulate this type of content in both online and offline formats more or less seamlessly (in principle, at least \u2013 there is still room for improvement), without the need for any particularly advanced technical expertise. \u00a0It doesn\u2019t look like we\u2019re anywhere near that level currently with regards to mathematical content, though presumably things will improve when a single mathematics presentation standard, such as mathML, becomes universally adopted and supported in browsers, in operating systems, and in other various pieces of auxiliary software.\n\nAnyway, it has been a very interesting and educational discussion for me, and hopefully for others also; I look forward to any further thoughts that readers have on these topics. \u00a0(Also, feel free to recapitulate some of the points from the previous thread; the discussion has been far too multifaceted for me to attempt a coherent summary by myself.)\n\nThe various languages and formats that make up modern web pages (HTML, XHTML, CSS, etc.) work wonderfully for most purposes, but there is one place where they are still somewhat clunky, namely in the presentation of mathematical equations and diagrams on web pages. While web formats do support very simple mathematical typesetting (such as the usage of basic symbols such as \u03c0, or superscripts such as x2), it is difficult to create more sophisticated (and non-ugly) mathematical displays, such as\n\n$\\displaystyle \\hbox{det} \\begin{pmatrix} 1 & x_1 & \\ldots & x_1^{n-1} \\\\ 1 & x_2 & \\ldots & x_2^{n-1} \\\\ \\vdots & \\vdots & \\ddots & \\vdots \\\\ 1 & x_n & \\ldots & x_n^{n-1} \\end{pmatrix} = \\prod_{1 \\leq i < j \\leq n} (x_j - x_i)$\n\nwithout some additional layer of software (in this case, WordPress\u2019s LaTeX renderer). These type of ad hoc fixes work, up to a point, but several difficulties still remain. For instance:\n\n1. There is no standardisation with regard to mathematics displays. For instance, WordPress uses $latex and$ to indicate a mathematics display, Wikipedia uses $and$, the current experimental Google Wave plugins use $$and$$, and so forth.\n2. Mathematical formulae need to be compiled from a plain text language (much as with LaTeX), rather than edited directly on a visual editor. This is in contrast to other HTML elements, such as links, boldface, colors, etc.\n3. One cannot easily cut and paste a portion of a web page containing maths displays into another page or file (although with WordPress\u2019s format, things are not so bad as the raw LaTeX code will be captured as plain text). Again, this is in contrast to other HTML elements, which can be cut and pasted quite easily.\n4. Currently, mathematical displays are usually rendered as static images and thus cannot be easily edited without recompiling the source code for that display. A related issue is that the images do not automatically resize when the browser scale changes; also, in some cases they do not blend well with the background colour scheme for the page.\n5. It is difficult to take an extended portion of LaTeX and convert it into a web page or vice versa, although tools such as Luca Trevisan\u2019s LaTeX to WordPress converter achieve a heroic (and very useful) level of partial success in this regard.\n\nThere are a number of extensions to the existing web languages that have been proposed to address some of these difficulties, the most well known of which is probably MathML, which is used for instance in the n-Category Caf\u00e9. So far, though, adoption of the MathML standard (and development of editors and other tools to take advantage of this standard) seems to not be too widespread at present.\n\nI\u2019d like to open a discussion, then, about what kinds of changes to the current web standards could help facilitate the easier use of mathematical displays on web pages. (I\u2019m indirectly in contact with some people involved in these standards, so if some interesting discussions arise here, I can try to pass them on.)\n\n Vivek Kumar Bagaria on 254A, Notes 2: The central lim\u2026 Ultrafilters, nonsta\u2026 on Soft analysis, hard analysis,\u2026 The Taxpayer on IMU Graduate Breakout Fellowsh\u2026 Terence Tao on IMU Graduate Breakout Fellowsh\u2026 valuevar on IMU Graduate Breakout Fellowsh\u2026 Anonymous on Open question: scarring for th\u2026 Various Items | Not\u2026 on IMU Graduate Breakout Fellowsh\u2026 Anonymous on Open question: scarring for th\u2026 louigiaddario on IMU Graduate Breakout Fellowsh\u2026 gninrepoli on P=NP, relativisation, and mult\u2026 Wolfgang Arendt on A quick application of the clo\u2026 Terence Tao on Ultrafilters, nonstandard anal\u2026 sagar on Ultrafilters, nonstandard anal\u2026 Anonymous on On time management Terence Tao on An elementary non-commutative\u2026","date":"2016-05-03 23:57:35","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 1, \"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\": 0.5393917560577393, \"perplexity\": 1773.4900591978865}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-18\/segments\/1461860121985.80\/warc\/CC-MAIN-20160428161521-00102-ip-10-239-7-51.ec2.internal.warc.gz\"}"} | null | null |
I surprised when heard tomato prices reach Rs.100 per kg 2 days before, I have doubt Pakistan is not producing tomato but imports that reason prices are so high.
Tomato prices must be under Rs.40 per kg see how the profit margin enjoys by traders. Why not Govt sincere to control the prices of all kitchen items like chicken – meat – vegetable – fruits – food stuff so every middle class group saved by over charged. Why not introduce UNIFORM rates of products area wise, if any traders overcharging must be fined.
If Govt fail to control the prices than what will be limit of inflation during holy month of Ramzan. I must say Pakistan is heaven for traders where they getting profits according to their wishes. | {
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Final Fantasy XIII, Final Fantasy XIII-2
Ark (Final Fantasy XIII)
Fifth Ark.
Arks (アーク, Āku?) are Gran Pulse armories in Final Fantasy XIII and Final Fantasy XIII-2. Two Arks are seen: the Fifth Ark hidden underneath Eden, and the Thirteenth Ark that floats above Academia four centuries later. A location called the Seventh Ark was considered to be a downloadable location if DLC was to be made for Final Fantasy XIII. There, Lightning and her companions would fight powerful enemies.[1] This never materialized, but hackers have unveiled the location as dummied content.
Lightning Returns: Final Fantasy XIII has a location known as the Ark, but it appears manmade rather than made by Pulse fal'Cie. The Academy may have named it in homage to the mythical Arks of the fal'Cie.
See also: Datalog (Final Fantasy XIII)/Analects
They say the fal'Cie made the Arks in preparation for battle against the menace that lurks beyond. Where is this "beyond" of which they speak? Do they mean Cocoon, and the demons that dwells within? If so, they are mistaken. The legends of the Arks date far before that sphere was even crafted; whispers even hint at Arks displaced around the time of Cocoon's creation, spirited away to be hidden in its shell.
What, then, is the "menace"? What distant threat confronts us, and to what purpose? The gods vanished from this place. Are they now residents of the "beyond"?
—On the Nature of Fal'Cie
According to legend, Arks are Gran Pulse's floating armories that hold living weapons in stasis in preparation for war with outside forces. Arks are hidden all around the world and most of their locations are unknown. Arks were built by the fal'Cie thousands of years ago, and even though their purpose is a mystery to humankind, they are used for training l'Cie to become strong enough to fulfill their Focus. The Arks levitate via the power of items known as Graviton Cores.
Gran Pulse citizens had searched for the legendary Arks, but never found them. On Cocoon, Lightning's party visits the Fifth Ark that they find hidden under Eden. As told in Final Fantasy XIII Gaiden Shōsetsu: Yumemiru Mayu, Akatsuki ni Otsu, a university student called René discovers another Ark hidden beneath Eden when being hunted by purger fal'Cie. The Arks being hidden underneath Eden may explain why Gran Pulse citizens never found them, if many were, in fact, brought to Cocoon.
Thirteenth Ark.
The purpose of Arks is not revealed in Final Fantasy XIII, but the official mythology on Fabula Nova Crystallis tells the god Bhunivelze created the world of Final Fantasy XIII in preparation to fight the goddess Mwynn. After tasking the god Pulse to find a way to the dimension where Mwynn resided, Bhunivelze turned himself into a crystal to sleep until time would be ripe. It can be thus gathered that the Arks were intended to be used in Bhunivelze's battle against Mwynn, as all that is known of their purpose is that they were constructed to fight a great menace in "the beyond".
The Seventh Ark.
[view · edit · purge]In the Bible, Noah carried seven pairs of each clean animal species and one pair of each unclean animal species, each pair a male and female, into a giant ship called the Ark. The Ark was built by God's command to survive the deluge He sent to purge the world from the sins committed by mankind.
There being thirteen Arks is an allusion to the number 13, a recurring theme in the Lightning Saga and all of the Fabula Nova Crystallis series.
Retrieved from "https://finalfantasy.fandom.com/wiki/Ark_(Final_Fantasy_XIII)?oldid=3212999" | {
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{"url":"http:\/\/mathoverflow.net\/questions\/93556\/finding-topological-obstructions-for-a-complex-manifold-to-be-kaehler","text":"# Finding topological obstructions for a complex manifold to be Kaehler\n\nWell, it is of the \"straightforward\" questions one may ask. I propose it here to see if someone could tell me more on the recent status of this quite long-standing problem.\n\nTo initiate, let me give a brief description of the \"classical\" invariants. Essentially they are provided by either the symplectic structure or the Hodge structure, most of which relating to vanishing theorems and integral theorems. As far as a compact manifold $M$ of complex dimension $n$ is concerned, we have the following:\n\n$b_2 \\ne 0$ for the symplectic structure and many more;\n\n$b_{2k+1}$ are even and $b_{k-2} \\leq b_k$ for $k \\leq n$ for the Hodge structure, which date back to Lefschetz;\n\nVarious integral results on Chern numbers by Hirzebruch-Riemann-Roch on Kaehler manifolds;\n\nand etc.\n\nTo make the question precise, I would like to ask:\n\nHas there been any \"higher\" or essentially new invariants discovered so far? Particularly, one may observe that the above invariants are all torsion-free(as Yau did in his problem list), so the torsion invariants would be rather interesting, suppose they do exist.\n\nAny comments are welcomed and thanks a lot!\n\n-\nThis may not address your question as stated but it is known, by a famous result of Deligne-Griffiths-Morgan-Sullivan, that compact Kahler manifolds are formal. What that entails, for instance, is that all triple Massey products in the cohomology ring vanish. Sometimes, this may be useful in detecting if a complex manifold admits a Kahler structure. \u2013\u00a0 Somnath Basu Apr 9 '12 at 17:27\nSomnath: My knowledge of rational homotopy theory is rather limited, certainly not adequate to understand all the mathematics evolved here. But after all many thanks\uff01 \u2013\u00a0 Zhang Xiao Apr 10 '12 at 1:10\n\nThe cohomology ring of $X$ is probably a good place to look for candidates for such invariants. For example, let $$B = \\lbrace \\alpha \\in H^{1,1}(X,\\mathbb R) \\mid \\text{Vol}(X,\\alpha) := \\int_X \\alpha^n\/n! > 0 \\rbrace$$ be the big cone of $X$. If $X$ is Kahler, then it contains the Kahler cone $K$ of $X$, but it is in general larger.\n\nThe big cone admits a (in general only pseudo-)Riemannian metric $g$, given by the Hessian of the smooth function $- \\log \\text{Vol}$. Conjecturally, the sectional curvature of this metric is seminegative if $X$ is Kahler (see Wilson's http:\/\/arxiv.org\/abs\/math\/0307260 for the first version of this question).\n\nSo, suppose we have a compact manifold $X$ and that we know its cohomology ring and its intersection product. Then we can calculate the big cone and the Riemannian metric $g$ and check if its sectional curvature is seminegative. If not, then $X$ should not be Kahler.\n\nOf course this may not be so easy to check in practice, but Wilson's article contains an interesting prototype of an example (Propositon 5.3) where one should obtain the non-existence of a certain type of Kahler structure on a differentiable manifold through these means, that one does not get by the more traditional invariants.\n\n-\nGunnar: I really appreciate your response. Since the criterion mentioned is rather differential-geometric than topological, I am wondering whether the seminegativeness of the sectional curvature of this \"moduli space\" has some implications on the topology of the original manifold:) \u2013\u00a0 Zhang Xiao Apr 10 '12 at 1:17\nWell, it has some implications, as in Wilson's example, but exactly what those are is very badly understood. One should also be careful to note that manifolds with wildly different topology can have the \"same\" Kahler cone and thus metric; f.ex. for a manifold with $h^{1,1} = 1$ the Kahler cone is a ray $\\mathbb R_+$ and the metric is basically the Poincar\u00e9 metric on the half-plane, restricted to the half-line $i\\mathbb R_+$. Now, the collection of comact manifolds with $h^{1,1} = 1$ is immense, it includes all Riemann surfaces, and all smooth hypersurfaces in $\\mathbb P^n$ ($n \\geq 4$). \u2013\u00a0 Gunnar Magnusson Apr 10 '12 at 6:13\nI am not familiar with the kaehler cone but all these stuffs sound rather interesting. By the way, personally I regard $h^{1,1}=1$ as a rather strong restriction since they are \"immense\" only in the cardinality sense but only consist of a few of \"types\":) \u2013\u00a0 Zhang Xiao Apr 10 '12 at 23:57\n\nYou might want to consult the really excellent book by Amoros,Burger, Corlete, Kotschick, Toledo called \"Fundamental groups of compact Kahler manifolds\".\n\n-\n...as well as Donu Arapura's survey \"Fundamental Groups of Smooth Projective Varieties\" (which almost entirely deals with Kahler groups). \u2013\u00a0 Misha Apr 9 '12 at 14:31\n@Misha: very true... \u2013\u00a0 Igor Rivin Apr 9 '12 at 15:58\nNice reference. My thanks to both Igor and Misha. Actually without knowing something has been done on the fundamental group, what I have in mind is the second homotopy group, which by Hurewicz a refinement of $H^2(M)$\uff0cthe latter being much more important in the Kaehler case than $H^1(M)$.Has any work been done on this direction? \u2013\u00a0 Zhang Xiao Apr 10 '12 at 1:23\nApart from the rational homotopy theory, I do not think there was anything done in this direction. A good introduction to RHT is \"Rational homotopy theory and differential forms\" by Griffiths and c Morgan. An interesting question would be to consider $\\pi_2(M)\\otimes {\\mathbb Q}$ as a $\\pi_1(M)$-module. The point is that much is known about $\\pi_1$ and $\\pi_2$, but, I do not think anybody looked (in the Kahler context) at the module. There are probably some interesting restrictions appearing here. Also, every fp group appears as $\\pi_1$ of a compact complex manifold (C.Taues), but ... \u2013\u00a0 Misha Apr 10 '12 at 20:07\n...his construction has nontrivial $\\pi_2$: The complex 3-fold constructed by Taubes is a twistor bundle over some ASD 4-manifold with huge $\\pi_2$. I do not know if there are any restrictions on the above module ($\\pi_2$ as a $\\pi_1$-module) in the context of compact complex manifolds. \u2013\u00a0 Misha Apr 10 '12 at 20:11","date":"2014-10-23 18:45:18","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.8746251463890076, \"perplexity\": 387.6607497551711}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2014-42\/segments\/1413558067214.90\/warc\/CC-MAIN-20141017150107-00059-ip-10-16-133-185.ec2.internal.warc.gz\"}"} | null | null |
using System;
using System.ComponentModel.DataAnnotations;
using System.ComponentModel.DataAnnotations.Schema;
using Abp.Domain.Entities;
using Abp.Domain.Entities.Auditing;
using Abp.Json;
namespace Abp.Notifications
{
/// <summary>
/// Used to store a notification subscription.
/// </summary>
[Table("AbpNotificationSubscriptions")]
public class NotificationSubscriptionInfo : CreationAuditedEntity<Guid>
{
/// <summary>
/// User Id.
/// </summary>
public virtual long UserId { get; set; }
/// <summary>
/// Notification unique name.
/// </summary>
[MaxLength(NotificationInfo.MaxNotificationNameLength)]
public virtual string NotificationName { get; set; }
/// <summary>
/// Gets/sets entity type name, if this is an entity level notification.
/// It's FullName of the entity type.
/// </summary>
[MaxLength(NotificationInfo.MaxEntityTypeNameLength)]
public virtual string EntityTypeName { get; set; }
/// <summary>
/// AssemblyQualifiedName of the entity type.
/// </summary>
[MaxLength(NotificationInfo.MaxEntityTypeAssemblyQualifiedNameLength)]
public string EntityTypeAssemblyQualifiedName { get; set; }
/// <summary>
/// Gets/sets primary key of the entity, if this is an entity level notification.
/// </summary>
[MaxLength(NotificationInfo.MaxEntityIdLength)]
public virtual string EntityId { get; set; }
/// <summary>
/// Initializes a new instance of the <see cref="NotificationSubscriptionInfo"/> class.
/// </summary>
public NotificationSubscriptionInfo()
{
}
/// <summary>
/// Initializes a new instance of the <see cref="NotificationSubscriptionInfo"/> class.
/// </summary>
public NotificationSubscriptionInfo(long userId, string notificationName, EntityIdentifier entityIdentifier = null)
{
NotificationName = notificationName;
UserId = userId;
EntityTypeName = entityIdentifier == null ? null : entityIdentifier.Type.FullName;
EntityTypeAssemblyQualifiedName = entityIdentifier == null ? null : entityIdentifier.Type.AssemblyQualifiedName;
EntityId = entityIdentifier == null ? null : entityIdentifier.Id.ToJsonString();
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
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Sit Resist is the second album by Laura Stevenson & The Cans. The album was released by Don Giovanni Records in 2011.
Track listing
References
Laura Stevenson albums
2011 albums
Don Giovanni Records albums | {
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REDUCED PRICE 23 Camden Road, Tunbridge Wells
Location Tunbridge Wells
Floor Area Approx. 1,394 sq ft [129.4 sq m]
Tenure Freehold / Leasehold
Description Freehold - Guide Price £199,000
Leasehold - Guide Rent £18,000 p.a.x.
Tunbridge Wells is an affluent and historic Spa town situated approximately 35 miles south east of Central London with a resident population of around 100,000 people.
The main arterial route to the north is via the A26 through Southborough linking with the A21 which serves the M25 at Junction 5. There is a main line station situated on Mount Pleasant Road with an average journey time of approx. 1 hour to the capital.
The property is situated on the northern side of Camden Road just a few metres from the entrance to NEXT.
Town centre retail unit with two self contained ground leasehold flats above.
Retail Sales: approx. 790ft² [73.3m²]
Stores / office approx. 604ft² [56.1m²]
Upper Floors
Flat 1 (First Floor front) – 1 Bed Flat
Flat 2 (First Floor rear) – 1 Bed Flat
Freehold subject to vacant possession of the retail premises and subject to the existing residential ground leases.
Flat 1 – ground lease for a term of 125 years from December 2012 at £100 per annum for first 33 years subject to phased increases.
Flat 2 – ground lease for a term of 99 years from March 2009 at £100 per annum for first 33 years subject to phased increases.
Guide Price £199,000.
The retail premises are available by way of a new effective full repairing and insuring lease for a term to be agreed. The provisions of Sections 24 - 28 inclusive of the Landlord and Tenant Act 1954 are to be excluded from the lease. The ingoing Tenant will be required to provide a rental deposit to be held throughout the Term.
Guide Rent £18,000 per annum exclusive.
We are advised that VAT is not applicable.
Enquiries of the VOA Website indicate that the shop has a Rateable Value of £21,250.
The small business rates multiplier for 2020 / 2021 is 49.9 pence in the £.
Any interested parties are strongly advised to verify this information with the Local Rating Authority.
Each party to be responsible for their own legal costs save that the Proposed Purchaser / Tenant will be required to provide an undertaking for any abortive legal costs.
Google Map (coming soon)
Street View (coming soon)
For more information on this property, please contact:
Darrell Barber
Bracketts Tunbridge Wells office
Phone (01892) 559507
darrell@bracketts.co.uk | {
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<!DOCTYPE html>
<html xmlns="http://www.w3.org/1999/xhtml">
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Lîle Trébéron est située dans la baie de Roscanvel, au sud-ouest de la rade de Brest, entre la presqu'île de Quélern et l'île Longue. À à l'ouest se trouve l'île des Morts, à au sud-est se trouve l'île Perdue. Une impasse de la ville de Brest porte son nom.
Historique
En 1720, l'île de Trébéron devient un lieu de quarantaine. Le lazaret qui y est construit accueille les marins de retour d'expédition en zone tropicale où règnent des maladies épidémiques (fièvre jaune, typhus, choléra…), les équipages « trop » malades (scorbut, gale…), ou les bagnards arrivant au bagne trop épuisés. Les morts sont enterrés sur l'Île des Morts toute proche, les survivants, au bout de 40 jours, ont le droit de regagner la Penfeld.
Le lazaret est prévu pour accueillir 150 personnes, même si, à l'occasion, un afflux de malades nécessite la construction de structures provisoires. Les Sœurs de la Sagesse constituèrent longtemps une part importante du personnel infirmier.
Au début du , l'île des Morts accueille une poudrière et perd sa vocation de cimetière du lazaret. Ce dernier est reporté sur la pointe nord de Trébéron.
Par ailleurs, à partir de 1826, l'ensemble du lazaret est réaménagé. De nouveaux bâtiments sont construits, dont le pavillon réservé aux officiers, dont les vestiges sont encore visibles aujourd'hui. En 1828, l'île est partagée en deux par un mur, séparant la zone de convalescence de la zone contagieuse.
Cette description de l'Île Trébéron date de 1895 :
Au début du , l'île perd une partie de sa vocation de lazaret pour se transformer en sanatorium, où les marins ou ouvriers de l'Arsenal sont mis au plein air, à l'écart de l'alcool et du tabac.
Pendant la Première Guerre mondiale, l'île est utilisée comme centre de convalescence pour les soldats et marins alliés ou ennemis soignés à l'hôpital maritime de Brest.
À la fin des années 1960, l'île entre dans le périmètre de protection érigé autour de l'Île Longue, et est interdite d'accès.
Aujourd'hui
Interdite d'accès (zone protégée), l'île est entretenue par la marine, en partenariat avec le Conservatoire du littoral, dans l'optique d'une éventuelle mise en valeur des vestiges. Par ailleurs, la faune y a pris ses aises, et régulièrement, des campagnes de dératisation doivent être menées.
Notes et références
Source
Bretagne Magazine 31 (novembre--janvier 2006)
Île de la Rade de Brest | {
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Conacul Zsombory este o construcție de tip conac, reprezentativă stilului baroc, din Zimbor, județul Sălaj. Acesta este înscris pe lista monumentelor istorice din județul Sălaj, elaborată de Ministerul Culturii și Patrimoniului Național din România în anul 2010.
Istoric
Clădirea a fost construită de Zsombory Lajos și soția sa, Kozma Mária, în locul unui vechi conac. Pe ornamentul unuia dintre turnuri se vede anul construirii, 1892. La începutul anilor 1900, groful a pierdut castelul la cărți, în urma unui joc de kanaszta (un joc de cărți specific nobilimii maghiare, destul de asemănător cu remi-ul și cu o formă simplificată a bridge-ului), ajungând astfel în posesia ministrului de finanțe Săulescu. El nu a stat multă vreme la castel, așadar soția Maria, cunoscută drept "cucoana", a îngrijit castelul și grădina ce-l înconjoară.
Castelul a rămas în proprietatea familiei Săulescu până după Al Doilea Război Mondial, iar odată cu începerea "marii naționalizări", castelul a fost trecut în proprietatea statului și dat în folosință Ministerului Sănătății și Ocrotirii Sociale. De atunci, în castel a funcționat, întâi, un spital-sanatoriu, după care în imobil a fost instalat dispensarul uman medical al comunei. Chiar și Primăria și-a avut sediul în conac pentru câțiva ani.
Până în prezent nimeni nu a revendicat castelul. Imobilul a fost vândut pe 10 august 2005, în urma unei licitații organizate de către Consiliul Local Zimbor. Proprietarii actuali, români din Australia, se pare că vor să scape de clădire. Cei doi au plătit o sumă de 9,6 miliarde de lei pentru achiziționarea castelului. Deși construită cu atenție chiar și la cele mai mici detalii, ferestrele și ușile fiind aduse din Austria, soarta i-a fost potrivnică acestui castel.
Construcția are 14 camere, holuri, beci, anexe și un parc de 18 ari. O parte din parcul castelului a fost păstrată în forma sa originală și a devenit parcul satului.
Vezi și
Lista monumentelor istorice din județul Sălaj
Referințe
Conace din județul Sălaj
Monumente istorice din județul Sălaj | {
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Ronald Tétrault, né le à Val-d'Or, est un homme politique québécois. Il est député à l'Assemblée nationale du Québec pour la circonscription d'Abitibi-Est de 1970 à 1973 sous la bannière du Ralliement créditiste. Il est par la suite candidat libéral lors des élections fédérales de 1979 mais ne parvint pas à être élu.
Biographie
Résultats électoraux
Notes et références
Personnalité politique québécoise
Naissance en juin 1936
Naissance en Abitibi-Témiscamingue | {
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Using studio6/avr-gcc the preprocessors are not evaluated correctly.
Line 5 below is always evaluated TRUE and thus USE_SPI is always set.
Is there a flag needed to be set to have avr-gcc evaluate preprocessors? Any help appreciated.
"The fool wonders, the wise man asks"
My code was just a simplified example stripped down to least non working code. My code is written to be used on several diferent HW, and each HW may also come in diferent flavours therefor it is nice to have a set of definitions wich represents each possible variants in an easy to understand textual description and which BUILD_VARIANT is set to.
You can probably make all your variants depend on one #define.
After all, a 32-bit number gives you 4000M variants.
In practice, you probably use less than 32 on-off options.
Just sit down with your current conditionals and tidy them up.
And decide which might be 'runtime' options.
CPP runs before the compiler proper. CPP knows nothing about the C language. CPP is a text processing thing. It identifies things starting with a hash-sign (#) and acts on those.
A #define creates an entry in the CPP symbol table - each entry being a mapping from one text (symbol) to another (definition). Whenever the symbol then is encountered in the source code it is replaced by the definition. It is pure textual substitution.
A #include is replaced by the contents of the file #include'd.
For #ifdef the text between #ifdef and #endif is kept intact (and also preprocessed!) if the symbol in the #ifdef is in the CPP symbol table. Else the whole text block is simply removed, and the compiler will never see it.
Similar for #if, but the symbol is tested for a specific value.
All this is done before the compiler proper has started. All this is done without any knowledge at all about the C language. There are no known types. No known language constructs (in this case enum's specifically). The only "language" known is the CPP language itself. Everything else is just text.
You could use the CPP to mangle any text you wish. If you had the complete text of Don Quixote in text file format you could use CPP and two #defines to get a story about Laurel and Hardy.
The CPP is not a part of the C language. Understanding this is key to using the CPP. | {
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Q: Different ways to save form values to the database I have started learning Django recently using a Udemy course. While going through the course instructor asked to save values from a Form to database.
After searching on the internet I figured out how to put form values into database and everything is working fine. Below is my views.py and forms.py files.
forms.py
class FormName(forms.Form):
fname = forms.CharField( label="First Name")
lname = forms.CharField(label="Last name:")
email = forms.EmailField()
verify_email = forms.EmailField(label='Verify Email:')
def clean(self):
all_clean_data = super().clean()
email = all_clean_data['email']
vmail = all_clean_data['verify_email']
if email != vmail:
raise forms.ValidationError("Check the emails")
views.py
def signup(request):
form = forms.FormName()
if request.method == 'POST':
form = forms.FormName(request.POST)
if form.is_valid():
post = User()
post.fname=request.POST.get('fname')
post.lname=request.POST.get('lname')
post.email=request.POST.get('email')
post.save()
return render(request,'third_app/greet.html')
else:
return render(request,'third_app/oops.html',{'form':form})
return render(request, 'third_app/signup.html',{'form':form})
Now coming to question, the instructor is using Meta class to store the form values to the database. Below are his forms.py and views.py files. I am curious about what the difference is between my method and the instructor's.
forms.py
class FormName(forms.ModelForm):
class Meta():
model = User
fields = 'all'
views.py
def signup(request):
form = forms.FormName()
if request.method == 'POST':
form = forms.FormName(request.POST)
if form.is_valid():
form.save(commit=True)
return render(request,'third_app/greet.html')
else:
return render(request,'third_app/oops.html',{'form':form})
return render(request, 'third_app/signup.html',{'form':form})
Thanks.
A: The Django docs explain this very well. It's what is known as a ModelForm:
If you're building a database-driven app, chances are you'll have forms that map closely to Django models. For instance, you might have a BlogComment model, and you want to create a form that lets people submit comments. In this case, it would be redundant to define the field types in your form, because you've already defined the fields in your model.
For this reason, Django provides a helper class that lets you create a Form class from a Django model.
So, to answer your question, your method uses a regular form (forms.Form) where you define the form fields, perform validation and then save each field individually in your view. When using form.ModelForm, field validation and saving is taken care of for you. Seeing as you have already defined what your fields are, the ModelForm uses this to perform the validation. The save() method conveniently saves each field to the database.
| {
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echo -e -n "
---------------------------------------------------------------------------------
\033[40;37mYou are using Bananian $READABLE_VERSION\033[0m
\033[32;40mThis script will upgrade your installation to Bananian 14.09 r01\033[0m
---------------------------------------------------------------------------------
The following files will be modified:
-------------------------------------
boot partition:
uImage
script.bin
uEnv.txt
root filesystem:
/lib/modules/*
/lib/firmware/*
/etc/inittab
/etc/asound.conf
/etc/rc.local
/usr/local/bin/bananian-config
/usr/local/bin/raspi-config
Packages to be installed:
-------------------------
console-data console-setup keyboard-configuration curl cpufrequtils libnss-myhostname (+ dependencies)
Incompatible changes:
--------------------
- USB OTG port enabled. Make sure you use the 'DC in' port for power supply!
- Video acceleration engine disabled. Enable it in 'bananian-config' if needed
- ALSA default output is now HDMI (see /etc/asound.conf)
For a list of all changes see our changelog:
https://dev.bananian.org/changelog_page.php?version_id=2
\033[0;31mDo you want to continue (yes/no)?\033[0m "
read start_upgrade
echo -e ""
if [ "$start_upgrade" != "yes" ]; then {
echo -e "---------------------------------------------------------------------------------"
echo -e "\033[40;31mupgrade canceled by user...\033[0m \n"
rm -rf $TMPDIR
exit
} fi
echo -e "---------------------------------------------------------------------------------"
echo -e "replacing files on mmcblk0p1... \n"
mkdir $TMPDIR/mnt
mount /dev/mmcblk0p1 $TMPDIR/mnt
tar -xzf mmcblk0p1.tar.gz -C $TMPDIR/mnt
umount $TMPDIR/mnt
echo -e "---------------------------------------------------------------------------------"
echo -e "replacing files in /lib/modules... \n"
tar -xzf kernel-modules.tar.gz -C /lib/
echo -e "---------------------------------------------------------------------------------"
echo -e "upgrading software... \n"
aptitude update
aptitude upgrade
echo -e ""
echo -e "---------------------------------------------------------------------------------"
echo -e "installing necessary software... \n"
aptitude install -y console-data console-setup keyboard-configuration curl cpufrequtils libnss-myhostname firmware-atheros firmware-brcm80211 firmware-libertas firmware-ralink firmware-realtek
echo -e "---------------------------------------------------------------------------------"
echo -e "upgrading bananian-config... \n"
mv bananian-config /usr/local/bin/bananian-config && chmod 700 /usr/local/bin/bananian-config
echo -e "---------------------------------------------------------------------------------"
echo -e "upgrading raspi-config... \n"
mv raspi-config /usr/local/bin/raspi-config && chmod 700 /usr/local/bin/raspi-config
echo -e "---------------------------------------------------------------------------------"
echo -e "enable serial console... \n"
sed -i '/enable serial console/d' /etc/inittab
sed -i '/T0:23:respawn:\/sbin\/getty -L ttyS0 115200 vt100/d' /etc/inittab
cat <<EOF >> /etc/inittab
# enable serial console
T0:23:respawn:/sbin/getty -L ttyS0 115200 vt100
EOF
echo -e "---------------------------------------------------------------------------------"
echo -e "creating alsa config (/etc/asound.conf) for HDMI sound... \n"
if [ -f /etc/asound.conf ]; then {
echo -e "\033[40;33m/etc/asound.conf exists, skipping...\033[0m \n"
} else {
cat <<EOF > /etc/asound.conf
pcm.!default {
type hw
card 1 # for headphone, turn 1 to 0
device 0
}
ctl.!default {
type hw
card 1 # for headphone, turn 1 to 0
}
EOF
} fi
echo -e "---------------------------------------------------------------------------------"
echo -e "configuring cpufreq... \n"
sed -i '/cpu frequency/d' /etc/rc.local
sed -i '/\/sys\/devices\/system\/cpu/d' /etc/rc.local
sed -i '/^exit 0$/i \
# cpu frequency \
echo ondemand > /sys/devices/system/cpu/cpu0/cpufreq/scaling_governor \
echo 600000 > /sys/devices/system/cpu/cpu0/cpufreq/scaling_min_freq \
echo 1008000 > /sys/devices/system/cpu/cpu0/cpufreq/scaling_max_freq \
echo 25 > /sys/devices/system/cpu/cpufreq/ondemand/up_threshold \
echo 10 > /sys/devices/system/cpu/cpufreq/ondemand/sampling_down_factor \
echo 1 > /sys/devices/system/cpu/cpufreq/ondemand/io_is_busy \
' /etc/rc.local
echo -e "---------------------------------------------------------------------------------"
echo -e "setting new version number... \n"
echo 140901 > /etc/bananian_version
echo -e "---------------------------------------------------------------------------------"
echo -e "\033[32;40mdone! please reboot your system now! (shutdown -r now)\033[0m \n"
| {
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{"url":"https:\/\/gilkalai.wordpress.com\/2019\/02\/28\/dan-romik-studies-the-riemanns-zeta-function-and-other-zeta-news\/","text":"## Dan Romik Studies the Riemann\u2019s Zeta Function, and Other Zeta\u00a0News.\n\nUpdates to previous posts: Karim Adiprasito expanded in a comment to his post on the g-conjecture on how to move from vertex-decomposable spheres to general spheres. Some photos were added to the post: Three pictures.\n\n## Dan Romik on the Zeta function and its hypergeometric expansions\n\nMy friend Dan Romik wrote an impressive paper Orthogonal polynomial expansions for the Riemann xi function about expansion of the Riemann Zeta function. (I thank Dan for telling me about it.) Dan kindly agreed to write a blog post about his work in May 2019.\n\nDan Romik (click once to enlarge, twice to enlarge further)\n\n## Michel Griffin, Ken Ono, Larry Rolen, and Don Zagier on Zeta and the hyperbolicity of Jensen polynomials in low degrees\n\nLet me mention also another impressive paper by Michael Griffin, Ken Ono, Larry Rolen and Dan Zagier: Jensen polynomials for the Riemann zeta function and other sequences. (I learned about it from Ken Ono over Facebook.\u00a0Ken writes: \u201cAfter almost 2 years and 83 drafts, I am happy to share this paper on the Riemann Hypothesis. Amazingly, this paper has been recommended for publication 6 days after submission!\u201d ). Here is a MO question regarding this development. (Update: There was a lot of media attention to the result. Here is a GLL post by R. J.\u00a0 Lipton.)\n\n## (Belated news) Polymath15 over Terry Tao\u2019s blog on Noising and Denoising of the Riemann Zeta Function.\n\nIt is now the 10 year anniversary of polymath projects and let me belatedly mention a successful project polymath15 that took place over Terry Tao blog. The problem is related to the Hermite expansion of the Zeta function. Roughly speaking you can apply a noise operator on Zeta that causes higher degrees to decay exponentially. (The noise can be \u201cnegative\u201d and then the higher degrees explode exponentially.) The higher the amount\u00a0 t of noise the easier the assertion of the Riemann Hypothesis (RH) becomes. Let $\\Lambda$, the de Bruijn-Newman constant, be the smallest amount of noise under which the assertion of RH is correct.\u00a0 It was conjectured that $\\Lambda \\ge 0$ namely the assertion of the RH fails when you apply negative noise no matter how small. This follows from known conjectures on the spacing between zeroes of the Zeta function since when the assertion of RH is known for some level (positive or negative) of noise, then with a higher level of noise spacing between zeroes become more boring. The conjecture was settled by Brad Rodgers and Terry Tao. (See this blog post by Terry Tao.)\n\nThe task of polymath15 (proposed here, launched here, last (11th) post here) was to use current analytic number theory technology that already has yielded information on the zeroes of the zeta function (in the direction of RH and related conjectures) to deduce sharper upper bounds for $\\Lambda$ than the previously known 0.5 value. The collaborative efforts led to\n\nTheorem (Polymath 15): $\\Lambda \\le 0.22$\n\nA remarkable success! (Of course, proving RH was not an objective of polymath15.) There were also\u00a0 interesting comments of general nature regarding the RH and other big conjectures like this nice comment by anonymous.\n\nIt is even possible that\u00a0$\\Lambda =0$\u00a0(i.e. RH is true) and for each\u00a0$t >0$\u00a0there is a simple Hermitian operator (possibly related to a probabilistic interpretation of\u00a0$H_t$) having\u00a0$H_t$\u00a0zeros as its spectrum (i.e. realizing Hilbert-Polya approach for\u00a0$H_t$) but no such operator for\u00a0$t=0$\u00a0(which may explain the failed attempts so far to find it). Since the zeros of\u00a0$H_t$\u00a0converge to that of\u00a0$H_0$, it is possible that there is a proof of RH via the functions\u00a0$H_t$\u00a0(whose zeros are more regularly distributed) but not via direct attack on\u00a0$H_0$\u00a0itself!\nSuch \u201chomotopic approach\u201d to study\u00a0$H_0$\u00a0via\u00a0$H_t$\u00a0reminds similar methods used in the past to solve big problems (e.g. de Branges solution \u2013 via Loewner\u2019s PDE with flow parameter \u2013 of the Bieberbach conjecture, and Perelman\u2019s solution \u2013 via Hamilton\u2019s Ricci flow PDE \u2013 of the Poincare conjecture).\n\nThis entry was posted in Number theory, Updates and tagged , , , , , , , , . Bookmark the permalink.\n\n### 4 Responses to Dan Romik Studies the Riemann\u2019s Zeta Function, and Other Zeta\u00a0News.\n\n1. Jake says:\n\nThe link to the Dan Romik paper is broken \u2026.\n\nGK: corrected, thanks!\n\n2. Gil Kalai says:\n\nIt is quite remarkable that wordpress\u2019s (deep?) learning algorithm found a connection with the post \u201cA Breakthrough by Maryna Viazovska Leading to the Long Awaited Solutions for the Densest Packing Problem in Dimensions 8 and 24\u201d. Also Dan\u2019s paper mentions works of Henry Cohn and of Maryna Viazovska.\n\n3. Gil Kalai says:\n\nI saw a very interesting paper by Keith Ball on rational approximations of the zeta function. https:\/\/warwick.ac.uk\/fac\/sci\/maths\/people\/staff\/keith_ball\/rappcf.pdf","date":"2021-09-28 16:09:57","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\": 16, \"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\": 0.861798107624054, \"perplexity\": 1242.116247476106}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-39\/segments\/1631780060877.21\/warc\/CC-MAIN-20210928153533-20210928183533-00581.warc.gz\"}"} | null | null |
Бэйлешти () — город на юге Румынии, в жудеце Долж.
История
Впервые упомянут в письменных источниках в 1536 году.
В ходе русско-турецкой войны 1828—1829 годов 14 (26) сентября 1828 года в сражении при «Боэлешти» передовой отряд русских войск генерал-майора барона Гейсмара разгромил превосходящие силы сераскира Видинского.
География
Расположен в 57 км к юго-западу от города Крайова, в 32 км к северо-востоку от города Калафат и в 18 км к северу от реки Дунай и границы с Болгарией.
Население
Согласно данным переписи 2011 года население города составляет 17 437 человек. По данным прошлой переписи 2002 года оно насчитывало 20 083 человека. Большую часть населения составляли румыны — 84,98 %; второй крупнейшей этнической группой были цыгане — 9,34 %. По данным переписи населения 2016 года, в городе проживало 19 993 человека. Таким образом, это второй по численности населения город жудеца, после Крайовы.
Динамика численности населения:
Известные уроженцы
Ион Митилинеу — румынский государственный деятель, дипломат, министр иностранных дел Румынии (1926-1927).
Амза Пелля — румынский актёр
Марчел Юреш — румынский актёр театра и кино
Адриана Некита — румынская гандболистка
Джорджана Чучулете — румынская гандболистка
Валерикэ Гэман — румынский футболист
Примечания
Ссылки
Официальный сайт .
Города Румынии | {
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\section{Introduction}
The sensitivity of LEP experiments to the direct production of new
particles has not increased significantly with respect to that
achieved after the first-year runs. However, the accumulation of large
statistics and the improvements on the systematics now allows not only
to test with much more detail the predictions and consistency of the
standard model (SM),
including the virtual effects of the top quark
and Higgs boson, but also to improve considerably the ability to
search for (or constrain) some subtle indirect manifestations of new
physics beyond the SM.
Among these last ones, the implications of the combined LEP
measurements up to 1993 are of particular importance for new neutral
gauge bosons that could mix with the standard $Z_0$ (so that the $Z$ boson
mass eigenstate has a small component with non-standard couplings) and
also for heavy fermions mixed with the known ones. In fact if the new
fermions have non-canonical $SU(2)\times U(1)$ quantum numbers (e.g.
left-handed singlets or right-handed doublets) they modify the
couplings of the electroweak gauge bosons with the light particles.
These new kinds of physics are a common feature of many GUT theories,
such as $SO(10)$ and E$_6$. The search for the tiny effects mentioned
above then allows us to look indirectly for the new states
predicted by these models, even if their direct production
is unaccessible at the energies
achievable with present colliders. Global
constraints on these effects have been regularly
performed in the past using the available electroweak data,
\cite{ll}--\cite{zp-new}. In
this paper we show that the inclusion of LEP and SLC data up to 1993
allows a significant improvement of the constraints on the deviations of the
fermion couplings with respect to their SM values and hence strengthen
the bounds on the above-mentioned mixings, in some cases even by an
order of magnitude. The value of the top quark mass recently
announced by the CDF collaboration \cite{cdf-mtop}, $m_t=174\pm
10^{+13}_{-12}$,
is also relevant for this analysis, since some bounds (such
as those on $Z_0$ mixing with an additional gauge boson or those on the
mixing of the $b$ quark) are correlated with it.
Finally we briefly discuss whether it is possible
that new physics effects of the kind discussed here could
account for the deviation from the SM expectations of
some measurements, such as
$\Gamma_b^{LEP}$, $A_{LR}^{SLC}$ and $A^{FB}_\tau$.
We can anticipate that we find essentially negative results.
\section{$Z_0$--$Z_1$ mixing}
The formalism describing the mixing of the standard neutral $Z_0$
boson of the electroweak gauge group
${\cal G}_{SM}=SU(2)\times U(1)$ with a new $Z_1$ associated
with an extra $U'(1)$ factor has been discussed at length in the past
\cite{fit6,zp-new}. Here we just recall a few relevant points.
In order to span a wide range of $Z'$ models, we will as usual take
the $U'(1)$ as a combination of the two additional Abelian factors in
the decomposition $E_6\to SO(10)\times U(1)_\psi\to SU(5)\times
U(1)_\chi\times U(1)_\psi$, where ${\cal G}_{SM}$ is assumed to
be embedded
in the $SU(5)$ factor. We hence parametrize the new gauge boson
as
\begin{equation}
Z_1=s_\beta Z_\psi+c_\beta Z_\chi,
\end{equation}
where $s_\beta \equiv \sin
\beta$, $c_\beta \equiv \cos\beta$. We will present results for the most
commonly considered $\chi$, $\psi$ and $\eta$ models, corresponding
respectively to $s_\beta=0$, 1 and $-\sqrt{5/8}$.
A mixing between $Z_0$ and $Z_1$ leads us to the two mass eigenstates
\begin{equation}
\left(\begin{array}{c}Z\\Z'\end{array}\right)=
\left(\begin{array}{cc}c_\phi&s_\phi\\-s_\phi&c_\phi\end{array}\right)
\left(\begin{array}{c}Z_0\\Z_1\end{array}\right).
\end{equation}
Although one may consider
$\phi$ as being a free parameter, one should remember that in
any given model one generally has $\phi\simeq CM_Z^2/M_{Z'}^2$, where
$C\sim O(1)$ is fixed once the vacuum expectation values
(VEVs) of the Higgs fields giving
masses to the gauge bosons are specified. This theoretical relation
between $M_{Z'}$ and $\phi$ has the important implication
that the very stringent constraints on the mixing angle $\phi$
obtained by LEP at the $Z$-pole (see below) induce, once a model
fixing $C$ is assumed, an indirect bound on $M_{Z'}$ typically much
stronger ($M_{Z'}\ \rlap{\raise 2pt \hbox{$>$}}{\lower 2pt \hbox{$\sim$}}\ $ 1 TeV) than those arising from direct $Z'$
searches at the Tevatron ($M_{Z'}\ \rlap{\raise 2pt \hbox{$>$}}{\lower 2pt \hbox{$\sim$}}\ 450$ GeV for 25 $pb^{-1}$
of integrated luminosity
\cite{zp-paco}) or those resulting
from the effects of $Z'$ exchange on low-energy neutral current
experiments ($M_{Z'}\geq 200-300$ GeV)
\cite{fit6,zp-new,zp-paco}.
In view of these bounds we will neglect
in the following $Z'$ exchange and $Z$--$Z'$ interference
effects in the neutral current (NC) processes,
and we will only consider the modifications of the
$Z$ couplings to fermions induced by the small
admixture with the $Z_1$.
Due to the $Z_0$--$Z_1$ mixing,
the vector and axial-vector fermion couplings
appearing in the NC
$J_Z^\mu=\bar\Psi^f(v^f+a^f\gamma_5)\gamma_\mu\Psi^f$,
which couples to the physical $Z$ boson, read\footnote{The sine of the weak
mixing angle $s_W$ appears due to the normalization of the $U'(1)$ coupling
\cite{fit6}.}
\begin{eqnarray}
v^f&=&c_\phi v^f_0+s_\phi s_Wv^f_1,\\
a^f&=&-s_\phi a^f_0+c_\phi s_Wa^f_1.
\end{eqnarray}
Within the SM, and including radiative corrections, one has
\begin{equation}
v^f_0=\sqrt{\rho_f}\, [t_3(f_L)-2\,Q^f \sin^2\theta_{eff}^f]~~~~,~~~~
a^f_0=\sqrt{\rho_f}\, t_3(f_L),
\end{equation}
where $\sin^2\theta_{eff}^f$ and the $\rho_f$ factors
have been evaluated by means
of the ZFITTER code\footnote{We thank D. Bardin for
providing us with the 1994 updated version of the program.} \cite{zfitter},
as functions of the input parameters $m_t$, $\alpha_s(M_Z)$ and $m_H$.
The $Z_1$
couplings $v_1$ and $a_1$ depend on the assumed $U'(1)$ model (i.e. on
$s_\beta$) and can be found in refs. \cite{fit6,zp-new}.
The effects of the SM radiative correction
induced by the mixings with the new particles,
as well as the
radiative effects of new physics, are expected to be small and
have been neglected. A more detailed justification of
this assumption can be found in \cite{fit6}.
Since we are neglecting $Z^\prime$ propagator effects, the
only quantity in which the $Z'$ mass appears explicitly is
$\rho_{mix}=1+(M_{Z'}^2/M_Z^2-1)s_\phi^2$.
This term affects the $SU(2)$ gauge coupling
deduced using as numerical inputs $G_F$,
$\alpha$ and the value of $M_Z$ measured at LEP, thus
modifying both the overall strengths $\rho_f$ and the
$\sin^2\theta_{eff}^f$ factors. Since the effects of
$\rho_{mix}$ in the LEP observables are crucial to constrain the mixing
$\phi$,
the limits on the $Z_0$--$Z_1$ mixture will depend on the
$Z'$ mass, generally improving with larger $M_{Z'}$ values.
A second remark is that $\rho_{mix}$
enters as a multiplicative factor in the effective $\rho$ parameter.
Then the combined appearance of $\rho_{mix} \cdot \rho_{top}$,
with $\rho_{top}\simeq 1+{3G_Fm_t^2\over 8\sqrt{2}\pi^2}$, induces
a strong
correlation between the gauge boson mixing and the top mass.
Hence the top mass measurement by CDF \cite{cdf-mtop}
turns out to be relevant to
establish precise bounds on the mixing angle $\phi$.
\section{Fermion mixing}
A mixture of the known fermions with new heavy states can in general
induce both flavour changing (FC) and non-universal flavour diagonal
vertices among the light states. The first ones are severely
constrained (for most of the charged fermions) by the limits on rare
processes \cite{fc-limits}. Here we aim to constrain the second ones
by means of the large set of precise electroweak data.
Due to the extremely tight constraints on the FC
mixings \cite{fc-limits},
neglecting them will not affect our numerical analysis
on the flavour diagonal ones,
since in general the limits on the latter ones turn out to be larger
by some orders of magnitude.
From a theoretical point of view,
the absence of FC parameters in the formalism that we will
outline here is equivalent to the assumption that
different light mass eigenstates have no
mixtures with the same new state \cite{ll}.
The couplings
of the light charged fermions can then be described with just two
parameters for each flavour: $(s^f_\alpha)^2\equiv
\sin^2\theta^f_\alpha$, $\alpha=L,R$, which account for the mixing
with exotic states (i.e. having non-canonical $SU(2)\times
U(1)$ quantum numbers) of each of the two fermion chiralities.
Since the mixing always involves states of equal electric charges,
only the piece proportional to the weak isospin $t_3(f)$ in (5) is
affected by the fermionic mixing. In particular, the chiral couplings
$\epsilon_{L,R}^f=(v^f\pm a^f)/2$ are modified according to (see eq.
2.15 of ref. \cite{fit6})
\begin{equation}
\epsilon_\alpha^f=t_3(f_\alpha)-Q^f\sin^2\theta^f_{eff}+\left[
t_3(f^{\cal N}_\alpha)-t_3(f_\alpha)\right] (s^f_\alpha)^2~~,
{}~\alpha=L,R~~,
\end{equation}
where $t_3(f_\alpha^{\cal N})$ is the isospin of the new state
$f^{\cal N}$ that mixes with the known state $f$.
(For notational simplicity we omit hereafter the $\sqrt{\rho_f}$ factors in
the expressions for the couplings.)
Eq. (6) shows that when a doublet state is mixed with a singlet,
the isospin-dependent part of the coupling
is reduced by a factor $(c^f_L)^2$, while the mixing of a
singlet ($t_3(f_R)=0$) with a new exotic doublet ($t_3(f^{\cal
N}_R)=\pm 1/2$) induces a coupling
proportional to $t_3(f_R^{\cal N})(s^f_R)^2$. Clearly, a mixing
between
states of the same isospin does not affect the overall electroweak
couplings.
Here we will only consider
mixings with new states that are either exotic singlets or
exotic doublets, i.e. $t_3(f_L^{\cal N})=t_3(f_R)=0$ and
$t_3(f_R^{\cal N})=t_3(f_L)=\pm 1/2$. Then,
in the absence of extra new gauge bosons, we have
\begin{eqnarray}
v^f&=&t_3(f_L)[1-(s^f_L)^2+(s^f_R)^2] -2Q^f\sin^2\theta_{eff}^f\\
a^f&=&t_3(f_L)[1-(s^f_L)^2-(s^f_R)^2].
\end{eqnarray}
\noindent
The mixing among the neutral fermionic states is not so simple, both
because of the lack of strong evidence against FCNC among neutrinos and
because of the possible existence of more than one type of exotic
states (singlets, exotic doublets with $t_3(N_L)=-1/2$, etc.
\cite{ll,fitnu}).
However, after summing over the undetected final neutrinos
and neglecting $O(s^4)$ terms, the
different NC observables can be obtained
by replacing the neutrino
couplings in the SM expressions by effective couplings,
which depend on just one mixing angle for each flavour:
\begin{equation}
v_{\nu_i}=a_{\nu_i}={1\over 2}-{\Lambda_i\over 4}(s^{\nu_i}_L)^2.
\end{equation}
The additional parameter $\Lambda$ describes the type of state
involved in the mixing and, for instance,
for a mixing with new ordinary, singlet or exotic doublet neutrinos
we have $\Lambda=0$, 2 or 4 respectively.
An important indirect effect of the presence of new fermions is to
alter the prediction for $\mu$ decay, in such a way that the effective
$\mu$-decay constant $G_\mu = 1.16637(2) \times 10^{-5}$ GeV$^{-2}$ is
related to the fundamental coupling $G_F$ through the fermion mixing
angles \cite{ll,fit},
\begin{equation}
G_\mu = G_F c_L^{e}c_L^{\mu}c_L^{\nu_e}c_L^{\nu_\mu}.
\label{eq:fitnu9}
\end{equation}
As a consequence, all the observables that depend
on the strength of the weak interactions $G_F$ are affected by
the mixing angles $\theta_L^{e}$, $\theta_L^{\mu}$,
$\theta_L^{\nu_e}$ and $\theta_L^{\nu_\mu}$.
This is the case, for instance, for the $W$ boson mass, for the effective
couplings of the fermions with the $Z$ boson, and
for the Cabibbo--Kobayashi--Maskawa (CKM) matrix elements
\cite{ll,fit}.
The complete formalism describing fermion mixings and
also the simultaneous
presence of $Z_0$--$Z_1$ mixing is given in ref. \cite{fit6}.
\section{Theoretical expectations for the fermion mixings}
As regards the theoretical expectations for the
mixing of the known fermions with new heavy states,
there is no exact model-independent relation
between the masses of the heavy partners and the
corresponding mixings.
However, in the framework of some classes of models,
it is still possible to make some general statements
and/or work out some order-of-magnitude estimates for the
mixings.
For the charged states, the L (or R) mixing angles result
from the diagonalization of the $N\times N$ symmetric
squared mass matrix for the known and the new states
${\cal M}{\cal M}^\dagger$ ( or ${\cal M}^\dagger{\cal M}$ ).
We know that the relevant eigenvalues must satisfy the hierarchy
$m^2_{\rm light} \ll m^2_{\rm heavy}$ (with $m_{\rm heavy}\ \rlap{\raise 2pt \hbox{$>$}}{\lower 2pt \hbox{$\sim$}}\ 100$
GeV),
and we can outline two main mechanisms that would
naturally produce
such a pattern for the light and heavy masses.
\noindent
{\it a)}\quad {\it See-saw models} \hfill\break
In these models the general form of the squared mass matrix is
\begin{equation}
{\cal M}{\cal M}^\dagger \sim
\left(\begin{array}{cc}\delta^2 &d^2 \\
d^2 &\sigma^2 \end{array}\right),
\label{mass-matrix}
\end{equation}
with $\delta$, $d \ll \sigma$.
If $\delta \sim d$,
as is the case if both these entries are
generated by VEVs of standard Higgs doublets,
we expect for the mass eigenvalues
$m_{\rm light} \sim \delta$, $m_{\rm heavy}\sim \sigma$, and
$s_{L,R} \sim d^2/\sigma^2 \sim
m^2_{\rm light} / m^2_{\rm heavy}$.
A different scenario appears when $\delta \ \rlap{\raise 2pt \hbox{$<$}}{\lower 2pt \hbox{$\sim$}}\ d^2/\sigma$, for
which $m_{\rm light} \sim d^ 2 /\sigma$, $m_{\rm heavy}\sim \sigma$,
and $s_{L,R} \sim d^2/\sigma^2 \sim m_{\rm light} / m_{\rm heavy}$.
Assuming $m_{\rm heavy}\ \rlap{\raise 2pt \hbox{$>$}}{\lower 2pt \hbox{$\sim$}}\ 100\,$GeV,
we see that in the Dirac see-saw case
the expectations for the
mixings are quite small. In the most favourable case of the bottom
quark mixing, it can be as large as $(s^b_{L,R})^2 \sim
2\times 10^{-3}$, which
turns out to be at the limit of the present experimental sensitivity.
\noindent
{\it b)}\quad {\it Quasi-degenerate mass matrices} \hfill\break
It can happen that, as a consequence of some symmetries,
in first approximation
the light--heavy fermion mass matrices are degenerate.
This implies that even if all the entries in
the mass matrices are large, some states (corresponding to the
light fermions) are massless, and would
acquire tiny masses due to
small flavour-dependent perturbations.
To give a simple example of this mechanism,
let us introduce a vector-like singlet of new fermions $F_L$ and $F_R$,
of the same charge and colour quantum numbers as those of the
$f_L$ component of a standard electroweak doublet, and of the
corresponding electroweak singlet $f_R$.
The general mass term reads
\begin{equation}
{\cal L}_{mass} =
\lambda_0 \overline {F_L} F_R S +
\lambda_1 \overline {F_L} f_R S +
\gamma_0 \overline {f_L} F_R D +
\gamma_1 \overline {f_L} f_R D ,
\label{l-mass}
\end{equation}
where $S$ and $D$ are respectively a singlet and a doublet VEV.
Let us also assume that because of some symmetries,
in first approximation
$\lambda_0 \simeq \lambda_1$ and $\gamma_0 \simeq \gamma_1$,
and let us absorb these Yukawas in the $D$ and $S$ VEVs.
Then, the light--heavy mass matrix squared that determines
the ordinary--exotic L mixing angle reads
\begin{equation}
{\cal M}{\cal M}^\dagger \sim
2\, \left(\begin{array}{cc}D^2 &D S \\
D S &S^2 \end{array}\right),
\label{deg-mass-matrix}
\end{equation}
and is clearly degenerate,
implying
$m_{\rm light} \simeq 0$ up to perturbations. At the same time,
the ordinary--exotic L mixings are expected to be large, and
could even be close to maximal.
The expectations for the neutral sector were
described in \cite{fitnu}, where it was shown that a
similar mechanism can also generate large light--heavy mixings even for
massless neutrinos.
Clearly, in contrast to the see-saw case,
models of this kind can be effectively constrained by
analysing the most precise electroweak data, and in fact the tight
bounds that we will derive for some mixings tend
to disfavour this mechanism
for the generation of the known fermion masses.
\section{Experimental constraints}
Within the SM, the precise electroweak experiments allow to constrain
the values of the input parameters $m_t$, $\alpha_s(M_Z)$ and $m_H$,
and an overall satisfactory agreement is found whith the predictions
for a heavy top mass \cite{pietrzyk}, compatible with the range
obtained by CDF. For instance, for $m_t=170$ ~GeV, $\alpha_s=0.12$ and
keeping hereafter the Higgs mass fixed at $m_H=250$~GeV, for most
observables the measured value is actually very close to the
theoretical predictions, making the total $\chi^2$ per degree of
freedom reasonably low ($<2$). However, there are a few exceptions for
which recent data show some noticeable disagreement with respect to
the SM expectations. A well-known case is the SLC measurement of the
left--right polarized asymmetry $A_{LR}$ \cite{alr-slc} ($\chi^2\sim
10$ for the above mentioned choice of input parameters). Some LEP
results also show sizeable deviations. This is the case for the ratio
of the $Z$ width into $b$ quarks to the total hadronic width,
$R_b\equiv \Gamma_b/\Gamma_h$ ($\chi^2\sim4.5$), and for the $\tau$
forward--backward asymmetry $A^{FB}_\tau$ ($\chi^2\sim7$)
\cite{lep94}. (Clearly the actual value of the $\chi^2$ function
depends on the values adopted for the input parameters.)
For our analysis we have used the CC constraints on
lepton universality and on CKM unitarity, the $W$ mass measurement,
as well as the NC constraints from the LEP and SLC measurements
at the $Z$ peak.
The best test of $e$--$\mu$ universality comes
from $\pi\to e\nu$ compared to $\pi\to \mu\nu$.
The ratio of the electron to the muon couplings to the $W$ boson,
extracted from the TRIUMF \cite{triumf92} and PSI \cite{psi92}
measurements, is
$
\left(g_e/ g_\mu\right)^2=0.9966\pm0.0030
$ \cite{fitnu}.
Universality among the $\mu$ and $\tau$ leptons is tested by
the $\tau$ leptonic decays compared to $\mu$ decay, giving
$(g_\tau/g_\mu)^2=0.989\pm0.016$ \cite{roney}.
A second test comes from $\tau\to\pi(K)\nu_\tau$,
which gives $(g_\tau/g_\mu)^2=1.051\pm0.029$ \cite{roney};
this is almost $2\sigma$ off the SM,
and hardly compatible with the above determination from $\tau$ decays.
The use of this determination affects mainly our bounds for the mixing of
the $\tau$ neutrino with new ordinary states, as discussed in Ref.
\cite{fitnu}.
For the test of the unitarity of the first row of the CKM matrix,
we use the determination
$\sum_{i=1}^3|V_{ui}|^2 = 0.9992\pm 0.0014 $
of Ref. \cite{sirlin93}, and for the $W$ mass we take the average
$M_W=80.23\pm0.18$ \cite{pietrzyk} of the CDF and UA2 experimental
values.
For the $Z$-peak data, we have included
the measurements of the total $Z$ width
$\Gamma_Z$, the hadronic peak cross section $\sigma_h^0$, the ratios
$R_e$, $R_\mu$, $R_\tau$ of the total hadronic width to the
flavour-dependent leptonic ones,
the bottom and charm ratios $R_b$ and $R_c$
and forward--backward asymmetries $A^{FB}_b$ and $A^{FB}_c$,
and the leptonic unpolarized asymmetries
$A^{FB}_e$, $A^{FB}_\mu$ and $A^{FB}_\tau$. All the data up to 1993 as
well as all the relevant experimental correlations given in Ref.
\cite{lep94} have been taken into account in our analysis.
We also include in our set of constraints the measurements of the
left--right polarization asymmetry at SLC, $A_{LR}=0.1637\pm0.0075$
\cite{alr-slc}, and the measurement of the ``theoretically equivalent"
quantity
$A_e^0={2 a_e v_e\over a_e^2 + v_e^2}=
0.120\pm0.012$ which has been inferred by the LEP collaborations from
the angular distribution of the $\tau$ decay products \cite{lep94}.
These two different determinations of the same theoretical quantity
are both more than $2 \sigma$ off the
SM value ($A^0_e=0.1419$ for our set of input parameters) and are in even
more serious conflict between them, possibly
indicating some problem in the analysis of the experimental data or an
unlucky fluctuation.
We always use
values for the observables that are extracted from the data without
assuming universality, which is expected to be violated by the fermion
mixings in the models we are considering. It is interesting to notice that,
while the experimental leptonic partial width of the $Z$ boson are in good
agreement with the hypothesis of universality, some hint of a
discrepancy may be present
in the fitted flavour-dependent forward--backward asymmetries, which
are $A^{FB}_e=0.0158\pm0.0035$, $A^{FB}_\mu=0.0144\pm 0.0021$ and
$A^{FB}_\tau=0.0221\pm 0.0027$ \cite{lep94,pietrzyk}.
Finally, we have also included in our data set the (updated)
low-energy NC constraints (deep inelastic $\nu$ scattering and atomic
parity violation). Although less effective than the
$Z$ peak data for constraining the kind of physics we are considering,
they turn out to be relevant for our analysis in
the case of the `joint' fits to be discussed below.
\section{Results}
After constructing a $\chi^2$ function with all
the experimental measurements discussed in the previous section,
we have derived bounds on the mixing parameters
by means of the MINUIT package.
\begin{table}[p]
\begin{center}
\caption{90\% c.l. lower ($\phi_-$) and upper ($\phi_+$) bounds on the
the $Z-Z'$ mixing angle
$\phi$, in units of $10^{-2}$, for the $\psi$, $\eta$ and $\chi$ models.
The limits correspond to different values
of the top mass $m_t$ and the strong coupling constant $\alpha_s$, with the
Higgs mass fixed to $m_H=250$ GeV. They have been obtained by chosing the
most conservative values as
$M_{Z'}$ is allowed to vary from $\sim500$ GeV to infinity.}
\vskip 1cm
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$m_t$ [GeV] & $\alpha_s$ & $E_6$ model
& $\phi_- \, [10^{-2}]$ & $\phi_+\, [10^{-2}]$\\
\hline
\hline
150 & 0.11 & $\psi$ & 0 & 1.1 \\
& & $\eta$ & 0 & 1.3 \\
& & $\chi$ & 0 & 0.75 \\
\hline
& 0.12 & $\psi$ & 0 & 0.81\\
& & $\eta$ & 0 & 1.0 \\
& & $\chi$ & -0.31& 0.43\\
\hline
& 0.13 & $\psi$ & -1.0 & 0\\
& & $\eta$ & -1.0 & 0 \\
& & $\chi$ & -0.70& 0\\
\hline
200 & 0.11 & $\psi$ & -0.03& 0.57 \\
& & $\eta$ & -0.19& 0.60 \\
& & $\chi$ & 0 & 0.48 \\
\hline
& 0.12 & $\psi$ & -0.28& 0.32\\
& & $\eta$ & -0.33& 0.39\\
& & $\chi$ & -0.18& 0.25\\
\hline
& 0.13 & $\psi$ & -0.43 & 0.11\\
& & $\eta$ & -0.38 & 0.23\\
& & $\chi$ & -0.38& 0.06\\
\hline
\end{tabular}
\end{center}
\end{table}
Regarding the gauge boson mixing $\phi$, we give for the
{\it unconstrained models} (e.g. with $M_{Z'}$ independent of
$\phi$) conservative bounds
obtained letting the $Z'$ mass to take values in the range
$M_{Z'}>500$~GeV and taking the extreme values $\phi_\pm$ that
remain allowed at 90\% c.l.. In this way we obtain
$$-0.0056<\phi<0.0055 \ \ \ \ \ (\psi\ \mbox{model})$$
\begin{equation}
-0.0087<\phi<0.0075 \ \ \ \ \ (\eta\ \mbox{model})\end{equation}
$$-0.0032<\phi<0.0031 \ \ \ \ \ (\chi\ \mbox{model})$$
These results have been obtained chosing for the input parameters the
values
$m_t=170$ GeV, $m_H=250$ GeV and $\alpha_s=0.12$, which provide
a good agreement between the experimental observables and
the SM predictions (corresponding to vanishing $Z$-$Z'$
and fermion mixings).
Since the bounds on $\phi$ depend on the choice of input parameters,
we show in Table 1 how the constraints are modified for $m_t=150$ and
200 GeV and for $\alpha_s=0.11$, 0.12 and 0.13\footnote{For
a detailed discussion of the $m_t$ (and $m_H$)
dependence, see the last two references in \cite{zp-new}.}.
It is apparent that
the bounds become tighter for increasing $m_t$.
This can be easily traced back to the fact that
larger (absolute) values of $\phi$ and of $m_t$ both tend to increase the
value of the effective $\rho$ parameter
$\sim \rho_{mix} \cdot \rho_{top}$.
For this reason the CDF lower limit on
$m_t$ is relevant for constraining $\phi$.
On the other hand, in the models considered here,
increasing values of $\alpha_s$ lead to a shift towards negative
$\phi$ values of the allowed region.
The previous bounds get also somewhat relaxed if one
allows for the simultaneous presence of the fermion mixings,
which can produce
compensating effects. In this case, keeping from now on the same choice
($m_t=170$ GeV, $m_H=250$ GeV, $\alpha_s=0.12$) for the input parameters,
we get the 90\% c.l. constraints
$$-0.0066<\phi<0.0071\ \ \ \ (\psi\ \mbox{model}),$$
\begin{equation}
-0.0087<\phi<0.010\ \ \ \ (\eta\ \mbox{model}),
\end{equation}
$$-0.0032<\phi<0.0079\ \ \ \ (\chi\
\mbox{model}).$$
In contrast, tighter bounds result if one considers {\it constrained
models}, that is assuming a relation between the gauge boson mixing and
$M_{Z'}$ of the form $\phi\simeq C M_Z^2/M_{Z'}^2$, where $C$ can be
evaluated once the Higgs sector is specified.
In this case the bounds on $\phi$ translate also into indirect
constraints on $M_{Z'}$.
The following results have been derived
by assuming for each model a {\it minimal} Higgs content and the absence
of singlet VEVs.
For the $\psi$ model, denoting by $\sigma\equiv (v_u/v_d)^2$
the square of the ratio of the scalar VEVs giving masses respectively
to the $u$ and $d$-type quarks, we have
$C=-{\sqrt{10}\over 3}s_W
{\sigma-1\over \sigma+1}$.
For $\sigma\to\infty$ we obtain $0\geq\phi>-0.0042$, which
implies the indirect constraint $M_{Z'}>1.0$
TeV, while, for instance, for $\sigma=2$ we obtain $0\geq\phi>-0.0052$,
corresponding to $M_{Z'}>0.52$ TeV.
For the $\eta$ model ($C={4\over 3}s_W
{\sigma-1/4\over \sigma+1}$) the bound
for $\sigma\to\infty$ is $0\leq\phi<0.0035$, implying $M_{Z'}>1.2$
TeV, while
for $\sigma=2$ we obtain $0\leq\phi<0.0054$, implying $M_{Z'}>0.76$
TeV.
We recall that the $Z_\chi$ of
the $\chi$ model is equivalent to the $Z'$ present in
$SO(10)$, being the two models different only with respect
to the fermion and scalar representations.
For the minimal Higgs content of $SO(10)$
($C=s_W\sqrt{2/3}$ \cite{lang-luo})
we obtain the constraint
$0\leq\phi<0.0028$, which implies
$M_{Z'}>1.2$ TeV for a $Z'$ from $SO(10)$.
\begin{table}[p]
\begin{center}
\caption{The 90\% c.l. upper bound on the
ordinary--exotic fermion mixing
parameters. The `single' limits in the first column are obtained when
the remaining mixing parameters are set to zero. For the `joint' bounds
in the second column, cancellations among the effects of all the different
possible fermion mixings are allowed.
The third column gives the `joint' bound in
the $\chi$ model, taking into account the possible cancellations among
the effects of
all the ordinary--exotic mixing parameters present in $E_6$
as well as
of a $Z_0-Z_\chi$ mixing.
All the results presented correspond to the value $\Lambda
=2$ of the parameter describing the type
of new neutrinos involved in the mixing, with the fixed values
$m_t=170$ GeV, $m_H=250$ GeV and $\alpha_s=0.12$. }
\vskip 1cm
\begin{tabular}{|c||c|c|c|c|c|}
\hline
& Single limit & Joint limit & $\chi$ model \\
\hline
\hline
$(s_L^e)^2$ & 0.0018 & 0.0065 & \\
\hline
$(s_R^e)^2$ & 0.0020 & 0.0020 & 0.0024 \\
\hline
$(s_L^\mu)^2$ & 0.0017 & 0.0076 & \\
\hline
$(s_R^\mu)^2$ & 0.0034 & 0.0059 & 0.0045 \\
\hline
$(s_L^\tau)^2$ & 0.0016 & 0.0058 & \\
\hline
$(s_R^\tau)^2$ & 0.0030 & 0.0055 & 0.0037 \\
\hline
$(s_L^u)^2$ & 0.0024 & 0.012 & \\
\hline
$(s_R^u)^2$ & 0.0090 & 0.015 & \\
\hline
$(s_L^d)^2$ & 0.0023 & 0.013 & 0.0064 \\
\hline
$(s_R^d)^2$ & 0.019 & 0.029 & \\
\hline
$(s_L^s)^2$ & 0.0036 & 0.0087 & 0.019 \\
\hline
$(s_R^s)^2$ & 0.021 & 0.060 & \\
\hline
$(s_L^c)^2$ & 0.0042 & 0.019 & \\
\hline
$(s_R^c)^2$ & 0.010 & 0.17 & \\
\hline
$(s_L^b)^2$ & 0.0020 & 0.0025 & 0.0045 \\
\hline
$(s_R^b)^2$ & 0.010 & 0.015 & \\
\hline
$(s_L^{\nu_e})^2$ & 0.0050 & 0.0066 & 0.0064 \\
\hline
$(s_L^{\nu_\mu})^2$ & 0.0018 & 0.0060 & 0.0046 \\
\hline
$(s_L^{\nu_\tau})^2$ & 0.0096 & 0.018 & 0.017 \\
\hline
\end{tabular}
\end{center}
\end{table}
Turning now to the fermion mixings, we have listed in table 2 the
updated 90\% c.l. bounds
obtained by allowing just one mixing to be present
(single bounds) or allowing for the simultaneous presence of all types
of fermion mixings (joint bounds). In the last case the constraints
are generally relaxed due to possible accidental cancellations among
different mixings. The bounds on the fermion mixings that can appear
in $E_6$ models are given in the third column. In this case we have
also allowed for the presence of mixing among the gauge bosons, which
somewhat relaxes the limits. We present the results obtained in the
$\chi$ model with the $Z_0$--$Z_\chi$ mixing as an additional free
parameter.
The constraints we have listed in table 2 correspond to the
particular value $\Lambda=2$. However we stress that only the bound on
$s^{\nu_\tau}_L$ depends significantly on the adopted value of
$\Lambda$, since the $\nu_e$ and $\nu_\mu$ mixings are mainly
constrained by CC observables, which do not depend on this parameter.
The LEP data alone already imply
$(s^{\nu_\tau}_L)^2<0.002/\Lambda_\tau$, which, due to the improvement
in the determination of the invisible width, is significantly better
than what obtained in previous analyses. For $\Lambda_\tau\simeq 0$
the constraint on $s_L^{\nu_\tau}$ arises from CC observables and can
be found in ref. \cite{fitnu}.
The results in table 2 were obtained for the reference values
$m_t=170$ GeV and $\alpha_s=0.12$.
Allowing $m_t$ to vary in the range 150
to 200 GeV does not affect significantly the constraints
on the fermion mixings. In contrast,
increasing $\alpha_s$ up to $\alpha_s=0.13$
worsens the limits on
some of the hadronic mixings
up to a factor $\sim 2$.
Besides strengthening the bounds on the new physics, one may also
wonder whether it could be possible to account for some of the
deviations with respect to the SM predictions that we have
mentioned previously, by means of the new physics effects that we
have been discussing here.
Regarding the $\sim 2 \sigma$ excess reported in the measurement of
$R_b$, the observed deviation ($\Gamma_b^{exp} > \Gamma_b^{SM}$)
has the opposite sign than the one
resulting from a mixing of the bottom quark with exotic states. In fact,
since
$\Gamma_b\propto v_b^2+a_b^2$, at $O(s_{L,R}^2)$ we have
\begin{equation}
{\Gamma_b\over\Gamma_b^{SM}}\simeq 1 +
(s_L^b)^2{v_0^b+a_0^b\over (v_0^b)^2 + (a_0^b)^2} +
(s_R^b)^2{a_0^b-v_0^b\over (v_0^b)^2+(a_0^b)^2}
\simeq 1-2.2(s_L^b)^2-0.2(s_R^b)^2 .
\end{equation}
Hence, non-vanishing values for both
$s^b_R$ and $s^b_L$ have the effects of reducing
$\Gamma_b$, thus increasing the disagreement with the measurements.
Of course this behaviour is in part responsible for the drastic
improvement in the constraints on the $b$ mixing angles.
In addition, due to the effect of the top mass
on the $Zbb$ vertex corrections, the constraint arising from $R_b$
slightly improves with larger $m_t$
(the measured $R_b$ favours a lower $m_t$ value).
In the case of the different leptonic asymmetries, the LEP experimental
values are not in complete agreement with the
assumption of universality, since $A^{FB}_\tau$ is somewhat larger
than $A^{FB}_{e,\mu}$. The very small SM value of the charged
lepton vector coupling
$v_0^l\simeq -0.036 $
implies that $A^0_l\simeq v^l/a^l$ is very sensitive to tiny
effects of new physics affecting $v^l$, as for example the shift
$\delta v^l=[(s_L^l)^2-(s_R^l)^2]/2$
induced by a mixing of the leptons. An
increase in $A^{FB}_\tau$ could then result from a non-zero
$s_R^\tau$. However, since this fermion mixing would modify
simultaneously the axial coupling $a^\tau$ by a similar amount, it is
easy to check that the constraints from $\Gamma_\tau$ do not allow the
50\% increase required to explain the measured $A_\tau^{FB}$
(in the presence of $s_R^\tau$,
$\delta A_\tau/A_\tau\simeq -2\delta \Gamma_\tau/\Gamma_\tau$).
New physics effects could be able to account
for these deviations only if they affect mainly
the $\tau$ vector coupling, while leaving the axial-vector
coupling close to its SM value.
Regarding the measurement of
$A_{LR}^{SLC}$, even if
one were to ignore the discrepancy with the
LEP measurement of $A^0_e$, the same type of argument would prevent
the possibility of explaining the measured value
by means of a mixing of the electron.
Clearly the deviations in $\Gamma_b$ and $A^{FB}_\tau$
cannot be explained either by introducing a
$Z'$ boson of the type we have considered here, since these
new gauge interactions are universal and
would affect all generations. However, some
models involving a new gauge boson coupling mainly to the third
generation have been discussed in this context \cite{holdom94}.
In conclusion, LEP provides a powerful tool for the indirect search
of several types of physics beyond the SM.
Present observations do not hint to any of the new physics effects
that have been discussed here,
thus allowing for a
significant improvement of the limits on
the indirect effects induced by some of the new particles
that appear in many extensions of the SM.
\vskip 1.5 truecm
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,036 |
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