text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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Couchtuner
how to download netflix shows on mac reddit – Change Your Netflix Account Information Or Subscription Billed Through Comcast Xfinity
Netflix / By admin
Since FAANG stocks are overweight in many portfolios, investors ought to keep a careful forward-looking watch on the landscape as well as on the price action in Apple, Amazon, Google and Facebook shares.
best netflix shows october 2019 – The 100 Best Movies To Watch On Netflix Right Now
On the Internet, there is nothing like Netflix. The third feature film by director Quentin Tarantino, Jackie Brown is an adaptation of author Elmore Leonard's 1992 novel Rum Punch, and stars blaxploitation legend Pam Grier in the titular role of a stewardess who runs afoul of a crime lord. In many ways a tribute to Grier's classics like Foxy Brown, Jackie Brown's star-studded cast also includes Samuel L. Jackson, Robert Forster, Bridget Fonda, Michael Keaton, and Robert De Niro. While not as highly regarded as predecessor Pulp Fiction, critics still loved Jackie Brown, and it's a prime Netflix pick.
And pressure has been building on the Los Gatos company to expand its library with new original content as media companies pull their popular shows like The Office" and Friends" from Netflix and place them on their own streaming services.
The movie: Ten years prior to his supreme satire Starship Troopers, Paul Verhoeven delivered this stonker, dripping with graphic violence. Detroit runs rampant with violent crime, leading the police department into privatisation. Enter shifty corporation, Omni Consumer Products, which brutally murders a beat cop Alex Murphy in order to use his barely-living body to test their new cyborg cop tech. That's all well and good, except Murphy retains much of his human memories, giving him an added edge and a score to settle with OCP.
Next, tap any video you want to watch from the interface. Wait for few minutes until the new page loads completely. The number of titles you can download is dependent upon the available storage space on your device. Not all devices are able to store 100 downloads.
They began creating their own content in 2013 with the widely successful House of Cards; in 2016 they released 126 titles (film and streaming shows) under their Netflix Original" banner. That's more than any other cable channel or network.
Free trials for VPNs are always for the full premium VPN service That means that you are getting all the same privacy and security features that you can expect to get if you pay. The free trial is specifically designed to let potential subscribers try the full service, with all its bells and whistles. This means that all of the free VPNs in this article are secure and provide watertight digital privacy.
Other big projects rumored for Apple TV+ include the animated series Central Park from Bob's Burgers creator Loren Bouchard, and On the Rocks, an original feature film that might reunite Oscar-winning Lost in Translation director Sofia Coppola with star Bill Murray.
For $7.99 per month (what Hulu charges for their limited-commercial plan), for example, someone could potentially gain access to the same content. The only tradeoff would be commercial breaks while streaming, something the service has never done before.
A Netflix user can tell if their account is being used illegitimately when random shows and movies appear in the Continue watching" field, and by receiving arbitrary recommendations of what to watch next. If your Netflix account is being used without your knowledge, you can check a list of content that has been recently watched via the Netflix website , then click on the downward-facing triangle beside your account found on the top right of the screen. Choose your account, and under your profile choose Viewing Activity". Click on the See recent account access" link at the top of the page to view which devices have also been checked in. You will see an option on your Netflix account page to Sign out of all devices". Lock outsiders out by changing your password by clicking on Change your password" under Membership & Billing".
Needless to say, this new mobile-only streaming plan isn't going to launch anytime soon in the US. In fact, Netflix might never make a similar plan available here, since it could potentially cannibalize a huge portion of Netflix subscribers who would have otherwise continued to pay higher prices.
Lost in Space has a long history of reimaginings. This new Netflix original series is a remake of both the 1998 film and the 1965 series that the film was based on, with all three properties being adaptations of the 1812 novel The Swiss Family Robinson. The story begins when a celestial object, the Christmas Star, crashes into Earth and threatens to wipe out civilization, leaving mankind forced to evacuate the planet and look for a new place to call home. The Robinson family is selected to be placed on the outgoing ship, christened the Resolute, but before they can reach their destination, an alien robot destroys the hull of the ship. The Robinsons, now crash-landed on a nearby planet, are forced to reckon with their new environment while also dealing with their own problems along the way. The show has been renewed for a second season.
The only downside is, it's difficult to access this library using a smart TV or an Apple TV device. The secret categories typically work the best using a web browser like Google Chrome. Just browse, find a movie and save it to your Netflix "My List." Then, fire up Netflix on whichever device you're using, and call up My List, and there you are.
Using these Netflix cookies, you can access Netflix premium accounts for free without any username and passwords. We will update all cookies daily so bookmark this page for future updates. Netflix offers three streaming video plans that start as low as $8.99 per month and top out at $15.99 per month.
Rooted or uncertified Android mobile devices are unable to download the Netflix app from the Play Store. Rooted or uncertified mobile devices are not blocked from accessing Netflix, but, depending on configuration, may not function properly.
No, you will get Netflix cookies absolutely free on this site. You can access Netflix account free using this cookie. You do not have to pay a rupee for this. If cookies do not work, then there is nothing to worry about. We update cookies every day.
This series hits all of the notes you need for some great historical drama: romance, intrigue, British accents, political drama, pretty people doing passionate things, and great writing against a backdrop of major historical events. If you want a less-soapy story of British aristocracy than "Downton Abbey" offered, here you go.
Since Netflix advertises that membership can be cancelled at any time, the one month trial may be a good test to ensure that you have the minimum bandwidth connection needed to stream movies and TV shows. According to experts, Netflix faces challenges with its streaming service in Latin America. High-speed broadband Internet reaches significantly fewer homes than in the U.S. Additionally, Netflix will have to compete with pirated movies that can be bought for a dollar on the street.
If you have multiple credit cards then you can create multiple Paypal accounts linked with credit cards to activate a free Netflix account. Forty-five analysts cover Netflix's stock, with a median price target of $410, down from $420 in late June. Its current price target is 60% above Netflix's current price of $255.
Greta Gerwig and director Noah Baumbach combine forces to create one of the best movies from the "mumblecore" genre. Gerwig plays a New York nomad who bounces around hoping one of her life aspirations will bear fruit.
From the producers of Riverdale comes a show that was originally destined for The CW as an official Riverdale spinoff: Chilling Adventures of Sabrina. At once a reboot of the Sabrina the Teenage Witch sitcom of the 1990s and an adaptation of the comic book series of the same name. Fans of the latter will find the adaptation accurate, but those who only know Sabrina from the 90s TGIF sitcom will find an all-new character here. Kiernan Shipka (Mad Men) plays Sabrina Spellman, a half-human, half-witch who finds herself balancing her normal high school life while also learning dark magic. Though the show retains plenty of the elements anyone familiar with Sabrina will know—Aunts Hilda and Zelda, Harvey Kinkle, Salem the cat—but with a higher influence places on dark magic, Satanism, and horror elements, this is a great adaptation of the comics series. The two-part first season is currently streaming. Season two, consisting of two eight-episode parts, should arrive next year.
Netflix's true story selection even includes some of the most notable Academy Award-nominated dramas of recent years, like Dallas Buyers Club , Lion, Lincoln, The Imitation Game, and Spotlight. If you've missed any of these throughout the years, now's the best time to go ahead and watch. Each of these movies is powerful in its own way, whether it's telling an unbelievable true story about familial bonds or portraying an important historical event. So next time you're not sure what to watch, check out these flicks and find yourself totally transported.
This comedy road movie swept the board at the 1988 Oscars, winning Best Picture, Best Director, Best Original Screenplay and Best Actor in a Leading Role for Dustin Hoffman. That's all for the Movies, and now let's check out the secret codes for TV Shows on Netflix.
It's one of the less essential Coen brothers movies, but there's still fun to be had from this cynical farce about a small group of very stupid people caught up in a blackmail and espionage yarn involving a retired CIA operative (John Malkovich), his mislaid memoirs, his wife (Tilda Swinton), her lover (George Clooney) and two dumb but ambitious gym employees (Frances McDormand, Brad Pitt).
If you decide you don't want to continue with Netflix after the 1-month trial, canceling is easy. To cancel, click on Your Account" on the Netflix website and follow the simple cancellation instructions.
However gradually, it has spread across the globe to places like Japan,Australia and also some parts of you would like to signup for premium version you can signup for netflix premium account from here.
Look below; you see date wise Netflix premium cookies list in the different browser. The first list is Netflix premium cookies for Chrome browser, and other is Netflix premium for Firefox. These cookies are updated daily, so I request you to find regular cookies Netflix then save our post to your bookmark.
Tamara Jenkins' heartbreakingly funny story about a couple trying to have a child and the young woman who agrees to donate her eggs to them possibly suffered from lack of recognition because of its late-fall debut on Netflix, but Jenkins is an incredible storyteller with a gift for humor and intimacy in the face of devastating personal trauma. Her original screenplay is worthy of a nomination by itself, but Kathryn Hahn has been doing great work for years and this somehow surpasses almost all of it as Lead Actress.
So I hope you have liked this post and managed to find at least one Free Netflix Account from here, If yes then don't forget to share this post with your friends and family maybe they want a Netflix Premium for free too. Thank you for visiting on Tricks Nation 🙂 don't forget to comment if you have any question or query related to Free Netflix Accounts with email and password.
There is no doubt as to the quality of Netflix and the variety of shows it has to offer. To top it off, the free trial adds to the allure of Netflix, making it the streaming service of choice for people all over the world. So, sign up for the Netflix free trial today and experience first-hand why everyone is so crazy about Netflix.
Like before, Navigate to My Download in netflix app. Next window the page says choose to multiple options. On below screenshot, I select one-month netflix account. Plans start at $4.17 per month or $49.99 per year, but you can try it out for seven days free of charge.
Netflix's Marvel Defenders shows are complicated too. Netflix has put out five original series based on Defenders characters in partnership with Disney. In 2018, Netflix canceled three of them: Daredevil, Luke Cage and Iron Fist. Then in 2019, Netflix canceled the last two : The Punisher and Jessica Jones. Kevin Mayer, the Disney executive in charge of Disney Plus, has said Disney Plus could possibly revive the canceled shows. But the terms of their original deal could restrict Disney Plus from any revivals until 2020 , according to a report.
As mentioned before, not every TV show and movie will be available to download. Many of them are only available to stream, but Netflix has made it easy to search for content that's available to download for offline viewing.
So, is there a download limit per title? Yes, there is. Sadly, Netflix doesn't reveal the limit set on each title. That means you've to resort to the tried and tested method of downloading the title and checking it.
Step 2: After Installing This Extension in your pcyou are able to Import and Export Cookies from your Browser. Now open the Netflix Official Website and click on the EditThisCookie Icon this will be appeared in your browser top right side. Now click on Import Button.
App users have the ability to download shows to watch when you're offline, as Netflix users do with certain content. Even due to any reason this cookie method doesn't work, use traditional email and password method to watch free Netflix.
According to research company Magid , around nine per cent of users share passwords for streaming services. Worryingly, that number gets larger the younger the users get with just 13 per cent of Baby Boomers, compared to 19 per cent of Generation X and 35 per cent of millennials.
Jake Gyllenhaal stars in Denis Villeneuve's (Arrival) creepy sci-fi thriller about a man who discovers he has a doppelganger. The double has been a literary trope for just about as long as people have been creating art (Enemy is based on Nobel laureate José Saramago's novel The Double), but Gyllenhaal's unnerving performance and Villeneuve's claustrophobic, monochromatic directing make Enemy a particularly sophisticated riff on a well-worn theme. It's a mind-bending exploration of identity, and the ending will leave you lying awake, puzzling over what it all means.
Terms of service – Regulations | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,610 |
<?php
namespace Hal\Config;
class Config {
public $setting = [];
private $db;
public function __construct($env) {
# Database Connection
$this->setting['db_host'] = $env->get_global_configuration('db_host');
$this->setting['db_name'] = $env->get_global_configuration('db_name');
$this->setting['db_user'] = $env->get_global_configuration('db_user');
$this->setting['db_pass'] = $env->get_global_configuration('db_pass');
$this->setting['default_controller'] = $env->get_global_configuration('default_controller');
# Define the site name
$this->setting['site_name'] = $env->get_global_configuration('site_name');
# Does your website/company have a tagline or slogan?
$this->setting['site_slogan'] = $env->get_global_configuration('site_slogan');
# Customer service or support email address
$this->setting['site_email'] = $env->get_global_configuration('site_email');
# Site admin name
$this->setting['site_admin'] = $env->get_global_configuration('site_admin');
$this->setting['admin_controller'] = $env->get_global_configuration('admin_controller');
# Address
$this->setting['street_address'] = $env->get_global_configuration('street_address');
$this->setting['city'] = $env->get_global_configuration('city');
$this->setting['state'] = $env->get_global_configuration('state');
$this->setting['zipcode'] = $env->get_global_configuration('zipcode');
# Phone
$this->setting['telephone'] = $env->get_global_configuration('telephone');
# Time Zone
$this->setting['time_zone'] = $env->get_global_configuration('time_zone');
$this->setting['error_reports'] = $env->get_global_configuration('error_reports');
$this->setting['debug_mode'] = $env->get_global_configuration('debug_mode');
$this->setting['log_errors'] = $env->get_global_configuration('log_errors');
# Controllers directory
$this->setting['controllers_path'] = BASE_PATH.$env->get_global_configuration('controllers_path');
# Models directory
$this->setting['models_path'] = BASE_PATH.$env->get_global_configuration('models_path');
$this->setting['log_file_max_size'] = $env->get_global_configuration('log_file_max_size');
# Name of the directory storing template files ( css/js/img, etc. )
$this->setting['template_name'] = $env->get_global_configuration('template_name');
# Name of the directory storing template files for administration area of website( css/js/img, etc. )
$this->setting['admin_template_name'] = $env->get_global_configuration('admin_template_name');
# Enable / disable breadcrumb links
$this->setting['breadcrumbs'] = $env->get_global_configuration('breadcrumbs');
# Put site in maintenance mode
$this->setting['maintenance_mode'] = $env->get_global_configuration('maintenance_mode');
# Check for common issues preventing system from running
$this->setting['system_startup_check'] = $env->get_global_configuration('system_startup_check');
$this->setting['signup_email_confirmation'] = $env->get_global_configuration('signup_email_confirmation');
$this->setting['compression'] = $env->get_global_configuration('compression');
$this->setting['login_math'] = $env->get_global_configuration('login_math');
# Image gallery settings
$this->setting['total_img_allowed'] = $env->get_global_configuration('total_img_allowed');
$this->setting['img_file_size'] = $env->get_global_configuration('img_file_size');
$this->setting['img_height'] = $env->get_global_configuration('img_height');
# Maximum image width in pixels. Set to zero for unlimited
$this->setting['img_width'] = $env->get_global_configuration('img_width');
$this->setting['img_type'] = $env->get_global_configuration('img_type');
/*----------------------------------------
* Global messenger inbox settings
*/
# Enable the messenger system by setting this to true
$this->setting['inbox_enabled'] = $env->get_global_configuration('inbox_enabled');
# Max number of saved messages in inbox
$this->setting['inbox_limit'] = $env->get_global_configuration('inbox_limit');
# Allow email addresses to be sent in messages?
$this->setting['inbox_allow_email'] = $env->get_global_configuration('inbox_allow_email');
# Allow URLs to be sent in messages?
$this->setting['inbox_allow_url'] = $env->get_global_configuration('inbox_allow_url');
# Allow links to be sent in messages?
$this->setting['inbox_allow_link'] = $env->get_global_configuration('inbox_allow_link');
# Allow images to be sent in messages?
$this->setting['inbox_allow_image'] = $env->get_global_configuration('inbox_allow_image');
# Allow HTML formatting ( <strong>, <em>, <h1>, etc. ) to be sent in messages?
$this->setting['inbox_allow_formatting'] = $env->get_global_configuration('inbox_allow_formatting');
$this->setting['site_url'] = $env->get_global_configuration('site_url');
#== Global system settings ==#
# Location of front controller
$this->setting['BASE_PATH'] = BASE_PATH;
# Location of the system directory
$this->setting['system_folder'] = $this->setting['BASE_PATH'].'app/code/core/system/';
# Location of the public directory
$this->setting['public_folder'] = $this->setting['BASE_PATH'].'public/';
# Location of template directory
$this->setting['template_folder'] = $this->setting['BASE_PATH'].'app/design/frontend/'.$this->setting['template_name'].'/';
# Location of admin template directory
$this->setting['admin_template_folder'] = $this->setting['BASE_PATH'].'app/design/admin/'.$this->setting['template_name'].'/';
# Template URL for fetching CSS / JS / IMG files
$this->setting['template_url'] = $this->setting['site_url'].'app/design/frontend/'.$this->setting['template_name'].'/';
# Admin Template URL for fetching CSS / JS / IMG files
$this->setting['admin_template_url'] = $this->setting['site_url'].'app/design/admin/'.$this->setting['admin_template_name'].'/';
# Convert image file size setting to kb
$this->setting['img_size'] = $this->setting['img_file_size']*1024;
$size = $this->setting['img_size'];
$unit = ['b', 'kb', 'mb', 'gb', 'tb', 'pb'];
$this->setting['notify_img_size'] = number_format(round($size/pow(1024, ($i = floor(log($size, 1024)))), 2)).' '.$unit[$i];
# Enable / disable Memcached helper
if (extension_loaded('memcached')) {
$this->setting['memcached'] = TRUE;
} else {
$this->setting['memcached'] = FALSE;
}
$this->setting['memcached_host'] = $env->get_global_configuration('memcached_host');
$this->setting['memcached_port'] = $env->get_global_configuration('memcached_port');
# Measure script execution time
$this->setting['execution_time'] = (microtime(true)-$_SERVER["REQUEST_TIME_FLOAT"]);
# Session settings
# Session cookie name
# Give this a unique name
$this->setting['session.name'] = $env->get_global_configuration('session.name');
# Recommended to leave this enabled for session security. 0 = disabled 1 = enabled
$this->setting['session.use_strict_mode'] = $env->get_global_configuration('session.use_strict_mode');
# Default setting is zero; i.e. until browser is closed
# Set this value in seconds if you wish to change the default behavior
$this->setting['session.cookie_lifetime'] = $env->get_global_configuration('session.cookie_lifetime');
# Leave blank for default settings; otherwise you can specify the host name of your server here
$this->setting['session.cookie_domain'] = $env->get_global_configuration('session.cookie_domain');
# Marks the cookie as accessible only through the HTTP protocol.
# This means that the cookie won't be accessible by scripting languages, such as JavaScript.
# This setting can effectively help to reduce identity theft through XSS attacks (although it is not supported by all browsers).
$this->setting['session.cookie_httponly'] = $env->get_global_configuration('session.cookie_httponly');
# Default is nocache. [nocache, private, private_no_expire, public]
# See http://php.net/manual/en/function.session-cache-limiter.php for more information about each setting.
$this->setting['session.cache_limiter'] = $env->get_global_configuration('session.cache_limiter');
# Release version
$this->setting['software_version'] = '1.0.0';
}
public final function setting($setting = null) {
return $this->setting["$setting"];
}
/**
* Private clone method to prevent cloning of the instance
*
* @return void
*/
private function __clone() {
}
/**
* Private unserialize method to prevent unserializing
*
* @return void
*/
private function __wakeup() {
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,607 |
<?php
declare(strict_types=1);
namespace Nelmio\Alice\Generator\Resolver\Parameter\Chainable;
use Nelmio\Alice\Generator\Resolver\ChainableParameterResolverInterface;
use Nelmio\Alice\Generator\Resolver\ParameterResolverAwareInterface;
use Nelmio\Alice\Generator\Resolver\ParameterResolverInterface;
use Nelmio\Alice\Generator\Resolver\ResolvingContext;
use Nelmio\Alice\IsAServiceTrait;
use Nelmio\Alice\Parameter;
use Nelmio\Alice\ParameterBag;
use Nelmio\Alice\Throwable\Exception\Generator\Resolver\ResolverNotFoundExceptionFactory;
use Nelmio\Alice\Throwable\Exception\ParameterNotFoundException;
use Nelmio\Alice\Throwable\Exception\ParameterNotFoundExceptionFactory;
final class StringParameterResolver implements ChainableParameterResolverInterface, ParameterResolverAwareInterface
{
use IsAServiceTrait;
const PATTERN = '/<{(?<parameter>[^<{]+?)}>/';
const SINGLE_PARAMETER_PATTERN = '/^<{(?<parameter>(?(?=\{)^[\>]|.)+)}>$/';
/**
* @var ParameterResolverInterface|null
*/
private $resolver;
public function __construct(ParameterResolverInterface $resolver = null)
{
$this->resolver = $resolver;
}
public function withResolver(ParameterResolverInterface $resolver)
{
return new self($resolver);
}
public function canResolve(Parameter $parameter): bool
{
return is_string($parameter->getValue());
}
/**
* @throws ParameterNotFoundException
*/
public function resolve(
Parameter $parameter,
ParameterBag $unresolvedParameters,
ParameterBag $resolvedParameters,
ResolvingContext $context = null
): ParameterBag {
$context = ResolvingContext::createFrom($context, $parameter->getKey());
$self = $this;
$value = preg_replace_callback(
self::PATTERN,
static function ($match) use ($self, $context, $unresolvedParameters, &$resolvedParameters, $parameter) {
$key = $match['parameter'];
$resolvedParameters = $self->resolveStringKey(
$self->resolver,
$parameter,
$key,
$unresolvedParameters,
$resolvedParameters,
$context
);
return $resolvedParameters->get($key);
},
$parameter->getValue()
);
return $resolvedParameters->with($parameter->withValue($value));
}
/**
* @param Parameter $parameter Parameter being resolved
* @param string $key Key of the parameter that need to be resolved to resolve $parameter
*/
private function resolveStringKey(
ParameterResolverInterface $resolver = null,
Parameter $parameter,
string $key,
ParameterBag $unresolvedParameters,
ParameterBag $resolvedParameters,
ResolvingContext $context
): ParameterBag {
if ($resolvedParameters->has($key)) {
return $resolvedParameters;
}
if (false === $unresolvedParameters->has($key)) {
throw ParameterNotFoundExceptionFactory::createForWhenResolvingParameter($key, $parameter);
}
$context->checkForCircularReference($key);
$context->add($key);
if (null === $resolver) {
throw ResolverNotFoundExceptionFactory::createUnexpectedCall(__METHOD__);
}
return $resolver->resolve(
new Parameter($key, $unresolvedParameters->get($key)),
$unresolvedParameters,
$resolvedParameters,
$context
);
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,455 |
Through a step-by-step process, the author breaks down the plotting of even complicated designs, allowing the freedom to create and plot your own designs. There are 224 designs; from original linework to identifying crossing points, and converting the knots to ribbonwork. Knots are grouped according to base element - including hearts, loops and lines - to give a grounding in the many permutations of the art. Corners and geometric designs are covered as well as zoomorphic motifs including fish, snakes, dogs, birds and plants. The final chapters give advice on embellishment and the use of colour and pattern in finishing a design. Techniques are illustrated throughout in full colour examples of the given designs. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,626 |
What To Expect From This is Us Season 5 Episode 7?
Anisha Dutta
In the latest episode of 'This is Us', we could not help but shed a tear or two. We finally learn about Randall's birth story, and it is a mix of tragedy and circumstances. It beautifully showcases how agony is a lifetime part of the journeys of some people, and it has the ability to change you. But more on that later. First, let's see what you can expect from 'This is Us' season 5 episode 7!
This is Us Season 5 Episode 7 Release Date
'This is Us' season 5 episode 7 is slated to premiere on January 19, 2021, at 9 pm ET/ 8 pm CT, on NBC. The fifth season has a total of 18 episodes that air on Tuesday each week.
Where to Watch This is Us Season 5 Episode 7 Online?
The easiest way to watch 'This is Us' season 5 episode 7 is by tuning in to NBC at the aforementioned time. You can also stream the show on NBC's official website and the NBC app. If you've ditched your cable subscription, don't worry, you can alternatively watch the show on Hulu, Direct TV, and Fubo TV. Viewers living in Canada can watch the previous seasons of 'This is Us' on Netflix (Canada). 'This is Us' season 5 is also available as a VOD service on YouTube TV. Additionally, you can also buy or rent the episodes on Amazon Prime.
This is Us Season 5 Episode 7 Spoilers
The upcoming episode of 'This is Us' is titled 'There,' and its official synopsis reads as follows – "Kevin embarks on a stressful road trip. Jack and young Kevin go to a football training camp." No, the outline does not give away much in terms of the storyline. However, it is not difficult to gauge what the episode will entail. We do know that Kevin's trip is packed with mishaps. Although he gets involved in a road accident, we do know that he is still alive. However, it is yet to be seen how this incident affects his homecoming journey and Madison's delivery. You can watch the new episode's promo below.
This is Us Season 5 Episode 6 Recap
Beth and Randall reach Hai's house, and the man informs them that the place had originally belonged to Laurel. Hai then proceeds to tell them the story: Laurel belonged to one of the most distinguished families in New Orleans. When her brother died, Lauren became close to her Aunt Mae, and this is when she met Hai – a Vietnamese refugee who had come to Louisiana with his parents after the War. The two eventually fell in love, but when her parents would not approve of the marriage, she left her home for Pittsburgh.
When Randall was born, Laurel was arrested for drug possession and sentenced to five years in prison. Hai explains: "She said there wasn't a single night that went by when she didn't dream about you. I think she was punishing herself. She felt she forfeited the right to be a mother." Laurel went back to New Orleans and met Hai again. Eventually, she succumbed to cancer with Hai by her side. The next day, we meet a lighter and fresher Randall, who seems relieved that he now knows his birth story.
Read More: Shows Like This Is Us | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,125 |
\section{Introduction}
Similar to the digital computer,
a rigid error-correcting system is required in the quantum computer.
Various quantum error-correcting codes (QECC) have been developed
such as the standard code
~\cite{Shor,Steane,Gottesman,Nielsen,Steane2,Knill1,Knill2,Reed,Moussa},
the subsystem codes~\cite{Kribs,Poulin,Bacon},
and the topological code~\cite{Kitaev1,Bombin,Bravyi,Raussendorf,Milman,Fowler}.
In QECC, it is necessary for many qubits to be coherently entangled for constructing logical qubits.
For instance, nine qubits are required for a logical qubit of the nine-qubit code~\cite{Shor},
seven qubits are required for the seven-qubit Steane code, which is the smallest
code of the general CSS code~\cite{Steane}, and so on~\cite{Gottesman,Nielsen}.
In any quantum codes, many operations and measurements are required for encoding,
decoding and error-correcting processes.
There are strict requirements concerning the maximum error rate
for the success of QECC~\cite{Gottesman,Nielsen,Steane2,Fowler}.
All manipulations of many qubits
should be done sufficiently within the coherence time.
In general, it is difficult to produce desired encoded states consisting of many qubits.
However, it is also difficult
to maintain each entangled state during the time required in a flow of quantum computation~\cite{Knill1,Knill2,Reed,Moussa}.
This problem arises when the encoded state is not the eigenstate of a system Hamiltonian.
The encoded state changes following the dynamics of the system Hamiltonian.
Assume that a computer system consists of many blocks.
Each block must correlate with every other block
to carry out a definite set of quantum computations.
As a simple structure of a computing system,
let us consider a system in which operations are synchronized to a system clock,
which is the case with the present widely-used digital computers.
Then, all operations are carried out step by step
as the system clock ticks the system time.
Entangled states produced by CNOT gates or other quantum gates
appear only periodically when the entangled states are not the
ground states of the system Hamiltonian.
In such case, if each block of a system includes an individual entangled state,
it will be difficult to control the synchronization of the total system
because the period of the desired entangled states
differs depending on the dynamics of each block.
Thus, it will be desirable for encoded states to be
the ground states of Hamiltonians of the blocks.
Moreover, because each block of a system changes its role as system time passes,
it is desirable that the Hamiltonian of each block
changes depending on each calculation step.
In this paper, we show how to efficiently implement standard QECC in
solid-state qubit systems with natural two-body interactions,
focusing on the stabilizer formalism.
Stabilizer operators $\{G_i| 1\le i\le l\}$ are mutually commuting operators
given by products of multiple Pauli matrices~\cite{Gottesman,Nielsen}.
Logical qubit states are encoded
into a mutual eigenspace $\mathcal{H}_S$ of dimension $2^l$ of these operators
through measurements. For $l$ different stabilizers and $n$ physical qubits, a
maximum number of $k = n - l$ logical qubits can be encoded into $\mathcal{H}_S$,
whereas $k < n - l$ in case of subsystem encoding~\cite{Kribs,Poulin,Bacon}.
Although preparation of some ``quantum memory" blocks to where logical qubit
states can be transferred or teleported is one solution to preserve logical qubit states,
we consider that it is better to change a system Hamiltonian
into a stabilizer Hamiltonian defined by $H_{\rm stab}\equiv -\sum_i G_i$,
because transformation or teleportation of encoded states requires more complexity.
We would also like to show how to generate encoded states without measurements.
The encoded states are generated by using operators that are modified from stabilizer operators.
Therefore, in this paper, we mainly describe the generation process of $H_{\rm stab}$.
In previous papers~\cite{stab,qecc-basel}, we showed that we can construct $G_i$
one by one based on the two-body Hamiltonian by using
the appropriate pulse sequence.
However, it is much more efficient to directly produce $H_{\rm stab}$.
In this paper, we show how to directly create $H_{\rm stab}$
starting from the two-body Hamiltonian.
$H_{\rm stab}$ has a complicated form of multiplied Pauli matrices.
We show that appropriate pulse sequences to generate
$H_{\rm stab}$ can be found by inversely tracing a transformation from $H_{\rm stab}$ into single-qubit Hamiltonian.
We show that the direct creation of $H_{\rm stab}$ greatly reduces
the number of operations compared with our previous method in Ref.~\cite{stab,qecc-basel}.
This reduction is remarkable in the case of qubits with $XY$ interaction.
For example, the number of single-qubit rotations $N_{\rm rot}$ and that of
qubit-qubit $XY$ interaction $N_{\rm int}$ are reduced from $N_{\rm rot}=44$ to $N_{\rm rot}=20$,
and from $N_{\rm int}=288$ to $N_{\rm int}=132$,
respectively, for the Steane code. Similar results are
obtained for the nine-qubit code and the five-qubit code.
Accordingly, operation time can also be reduced.
If we use a typical experimental parameter of superconducting qubits,
we can reduce the time required to generate $H_{\rm stab}$ by 48.4~\% ( 194~ns), 59.1~\% (127.5~ns) and 54.4~\% (257~ns)
for the nine-qubit code, the five-qubit code and the Steane code, respectively.
The present method has the advantage that, as pulse control technology progresses,
pulse error rate and speed are improved.
Pulse errors can be corrected by using NMR techniques such as the composite-pulse
method~\cite{Ernst,Haffner,Molmer,Hill,Torosov},
and the speed is increased by improving a control system operated by a digital computer.
We also investigate a possible architecture of standard codes for solid-state qubits on lattice sites.
In general, interactions between solid-state qubits are
restricted to their nearest-neighbor or next nearest-neighbor sites~\cite{Daniel,Yamamoto,You,Loss,Petta,Kane,tana}.
In order to prevent unexpected external noise,
it is preferable for physical qubits in a logical
qubit to be placed compactly in a small region.
Moreover, for logical qubits to interact effectively with one another,
it is desirable to place logical qubits side by side.
Therefore, it is natural to construct logical qubit by one-dimensional (1D) qubit arrays
and place them parallel as shown in Fig.~\ref{qubitlattice}.
In addition, frequent measurements in QECC require other qubit arrays for measurements.
We will discuss possible setups of a qubit system.
As a general case, we consider always-on Hamiltonian in this paper.
We think that we can separate logical qubits by effectively eliminating
qubit-qubit interaction through the use of appropriate pulse sequences.
This paper is organized as follows:
In Sec.~\ref{sec:form} we establish the general procedure of generating the stabilizer Hamiltonian.
In Sec.~\ref{sec:code}, we show examples of generating
the stabilizer Hamiltonian in the standard code,
and in Sec.~\ref{sec:initial} we show how to
generate the code state.
Finally, in Sec.~\ref{sec:architecture},
we consider possible qubit architecture realized by
solid-state qubits. We close with a summary and conclusions in
Sec.~\ref{sec:conclusion}.
\begin{figure}
\includegraphics[width=7cm,clip=true]{qeccfig1.eps}
\caption{Two-dimensional qubit array aiming at Shor's nine-qubit code. Boxes show qubits and bars
between the boxes show interactions between qubits. Horizontal qubits constitute logical qubits.
}
\label{qubitlattice}
\end{figure}
\section{Stabilizer Hamiltonian generation method}\label{sec:form}
\subsection{Stabilizer coding and stabilizer Hamiltonian}\label{sec:review}
In the stabilizer code~\cite{Gottesman,Nielsen},
encoding, decoding and error-correction are carried out based on the stabilizers,
which are mutually commutable and can be expressed by the Pauli matrices:
\begin{equation}
G_l=\otimes_{i=1}^{n} (X_i)^{x_i(G_l)}(Z_i)^{z_i(G_l)}
\label{ggg}
\end{equation}
($x_i(G_l), \ z_i(G_l) \in \{0,1\}$), where
Pauli matrices are given by
\begin{equation}
X=
\left(
\begin{array}{cc}
0 & 1 \\
1 & 0 \\
\end{array}
\right), \
Y=
\left(
\begin{array}{cc}
0 & -i \\
i & 0 \\
\end{array}
\right), \
Z=
\left(
\begin{array}{cc}
1 & 0 \\
0 & -1 \\
\end{array}
\right).
\end{equation}
The codeword $|\Psi_m\rangle$ obeys the eigenvalue equation
\begin{equation}
G_l|\Psi_m \rangle =|\Psi_m \rangle
\end{equation}
Conventionally,
in order to construct encoding states, starting from an initial state $\Pi_{i=1}^k|0\rangle_i$,
measurements over stabilizer operators of the selected code are repeated.
Depending on the measurement outcome,
the common eigenstate is fixed to be the desirable encoded state.
The correction procedure for the stabilizer code is
carried out by measuring all relevant stabilizer operators.
The stabilizer Hamiltonian $H_{\rm stab}$ is defined by
\begin{equation}
H_{\rm stab}=-\sum_{i=1}^{l}G_i,
\label{totalstab}
\end{equation}
where the summation is taken over the constituent stabilizers of each code.
Owing to the commutability of the stabilizers $G_i$,
the ground state of Eq.(\ref{totalstab}) is a common eigenstate of the
stabilizers, which is the encoded logical state.
For the sake of simplicity, we consider the standard codes without considering subsystem code
($k=n-l$).
\subsection{System Hamiltonian}
The solid-state Hamiltonian controlled by pulse signals
can be written as~\cite{You2,Regetti,tana2001}
\begin{equation}
H(t)=\sum_i \left[ \Omega_{0i} Z_i + 2\Omega_i \cos (\omega^{\rm rf}_i t+\delta_i) X_i\right] +\sum_{i<j}J_{ij}X_iX_j,
\label{app1}
\end{equation}
where $\Omega_i$ and $\omega^{\rm rf}_i$ are an amplitude and a frequency of a controlled signal
applied to a qubit $i$.
If we move to a frame rotating with the radio-frequency $\omega^{\rm rf}_i$ about the z-axis,
$H^r =R^{-1} H(t)R$,
with $R=\exp [-i \sum_i (\omega^{\rm rf}_i t/2) Z_i]$, then the transferred static Hamiltonian
$H'=H^r -\sum_i (\omega^{\rm rf}_i t/2) Z_i$ is approximately given by
\begin{eqnarray}
H' &=&\sum_i \left[ \left(\Omega_{0i} - \frac{\omega^{\rm rf}_i}{2}\right) Z_i
+\Omega_i (\cos \delta_i X_i -\sin \delta_i Y_i) \right]
\nonumber \\
&+&\sum_{i<j}\frac{J_{ij}}{2}
[X_iX_j +Y_iY_j].
\label{app2}
\end{eqnarray}
(high-frequency components $2\omega^{\rm rf}_i$ can be neglected).
If Eq.~(\ref{app1}) includes an interaction of $\sum_{i<j}J_{ij}Z_iZ_j$
instead of $\sum_{i<j}J_{ij}X_iX_j$,
the final Hamiltonian Eq.~(\ref{app2}) includes the Ising interaction.
The x-pulse and y-pulse for qubit $i$ are realized when
$\delta=0$ and $\delta=-\pi/2$ signals are respectively
applied to the qubit with $\omega^{\rm rf}_i=2\Omega_{0i}$.
We assume that each pulse is sufficiently strong for
interactions between qubits to be neglected during
the pulse sequences ($\Omega_i,\Omega_{0i} > J_{ij}$).
Then the qubit Hamiltonian
in the rotating frame of $\omega^{\rm rf}_i=2\Omega_{0i}$
is expressed by $H_q=H_0+H_{\rm int}$
where a single-qubit part $H_0$ is given by
\begin{equation}
H_0= \sum_i H_{0i}= \sum_i \Omega_i X_{i}
\label{H0}
\end{equation}
The interacting part
$H_{\rm int} =\sum_{ij}H_{\rm int}^{ij}$ is expressed by
\begin{equation}
H_{XY}=\sum_{i<j}H_{XY}^{ij}=\sum_{i<j}J_{ij}[X_i X_j+Y_iY_j],
\end{equation}
for $XY$ interaction, and
\begin{equation}
H_{\rm Ising}=\sum_{i<j}H_{\rm Ising}^{ij}=\sum_{i<j}J_{ij}Z_i Z_j,
\label{Hising}
\end{equation}
for Ising interaction.
\subsection{Dynamic generation of stabilizer Hamiltonian}
The generation of $H_{\rm stab}$ from $H_q$ consists of two steps.
The first step is to extract a
single-qubit part or a pure two-body interaction part from a qubit Hamiltonian $H_q$.
The second step is to construct $H_{\rm stab}$ dynamically with pulse sequences
by using a selected single qubit part $H_{\rm ini}$ and qubit-qubit interactions $H_{\rm int}^{ij}$.
Because the second step is the core framework of this paper,
we first describe the second step of dynamical transformation
to $H_{\rm stab}$. The extraction method is described in the next section~\ref{sec:extract}.
The transformation from the two-body Hamiltonian $H_q$ to the many-body
Hamiltonian $H_{\rm stab}$ is carried out
dynamically by using a time evolution of a system
starting from a simple initial Hamiltonian $H_{\rm ini}\propto X_i$, $Y_i$, or
$Z_i$ ~\cite{stab,qecc-basel}. The time evolution of the generation process is illustrated
with the schematic notation $\rho(0) \stackrel{t H}{\longrightarrow } \rho(t)$,
where $\rho(t) = \exp(-iHt) \rho(0) \exp(iHt)$ is the density matrix for a
time-independent Hamiltonian $H$, or for an effective $H$ in the sense of the
average Hamiltonian theory~\cite{Ernst}. After the application of mutually inverse, unitary
operations according to
\begin{equation}
\rho(0) \stackrel{\tau_{\rm op} H_{\rm op}
}{\longrightarrow } \ \ \stackrel{ \tau_{\rm ini} H_{\rm ini} }{\longrightarrow
} \ \ \stackrel{ -\tau_{\rm op} H_{\rm op} }{\longrightarrow } \rho(\tau_{\rm
ini}+2\tau_{\rm op}),
\label{rho}
\end{equation}
the system evolves as if propagated by the effective
Hamiltonian $\exp(-i \tau_{\rm op} H_{\rm op}) H_{\rm ini} \exp(i \tau_{\rm op}
H_{\rm op})$ for a time $\tau_{\rm ini}$~\cite{stab}.
To build $H_{\rm stab}$ from $H_{\rm ini}$,
we need two elementary transformations: one
that rotates arbitrary single-qubit terms through an angle of $\pi/2$ and
another that increases the order of Pauli-matrix terms by one.
Higher-order products of Pauli
matrices can be generated using the following transformations~\cite{stab}:
\begin{eqnarray}
\! e^{-i\theta [XY]_{12}} X_{1} e^{i\theta [XY]_{12}} \!\!\!&\!=\!&\!\! \cos (2\theta) X_{1} -\sin (2\theta) Z_{1} Y_{2}
\label{XYa}, \\
\! e^{-i\theta [XY]_{12}} Y_{1} e^{i\theta [XY]_{12}} \!\!\!&\!=\!&\!\! \cos (2\theta) Y_{1} +\sin (2\theta) Z_{1}X_{2}
\label{XYb}, \\
\! e^{-i\theta [XY]_{12}} Z_{1} e^{i\theta [XY]_{12}} \!\!\!&\!=\!&\!\! \cos^2 (2\theta) Z_{1} +\sin^2 (2\theta) Z_{2} \nonumber\\
\!\!\!&\!+\!&\!\!\frac{1}{2}\sin (4\theta)
[X_{1}Y_{2}-Y_{1}X_{2}]\:,
\label{XYc}
\end{eqnarray}
for $XY$ interaction.
When $\theta=\pi/4$ we can change the number of Pauli matrices given by
\begin{eqnarray}
X_{1} &\rightarrow & -Z_{1} Y_{2}, \label{XYa1} \\
Y_{1} &\rightarrow & Z_{1} X_{2}, \label{XYb1} \\
Z_{1} &\rightarrow & Z_{2}. \label{XYc1}
\end{eqnarray}
For Ising interaction, we use the relations given by
\begin{eqnarray}
& & e^{-i\theta Z_{1}Z_{2} } X_{1} e^{i\theta Z_{1}Z_{2}}
\!=\! \cos (2\theta) X_{1} +\sin (2\theta) Y_{1} Z_{2},
\label{ZZa}\\
& & e^{-i\theta Z_{1}Z_{2} } Y_{1} e^{i\theta Z_{1}Z_{2}}
\!=\! \cos (2\theta) Y_{1} -\sin (2\theta) X_{1} Z_{2}\:,
\label{ZZb}
\end{eqnarray}
Then, for $\theta=\pi/4$, we can
change the number of Pauli matrices given by
\begin{eqnarray}
& & X_{1} \rightarrow Y_{1} Z_{2}, \label{ZZa1} \\
& & Y_{1} \rightarrow -X_{1} Z_{2}, \label{ZZb1}\\
& & Z_{1} \rightarrow Z_{1}. \label{ZZc1}
\end{eqnarray}
By combining these equations with single-qubit rotations, we can change $H_q$ to
$H_{\rm stab}$.
\subsection{Extracting $H_{\rm ini}$ and $H_{\rm op}$ from a qubit Hamiltonian}\label{sec:extract}
In order to use the above-mentioned dynamic method, the important step is
to extract a
single-qubit part or a pure two-body interaction part from a qubit Hamiltonian $H_q$.
This process is carried out using the Baker-Campbell-Hausdorff (BCH) formula
\cite{Ernst}.
Here, we assume that qubits interact with their nearest-neighbor qubits.
Then, in order to define a logical qubit, we have to determine the locations of physical qubits in a logical qubit.
In this section, after we explain the BCH formula, we would like to
define a logical qubit arranged on lattice sites. Then, finally we will show
how to extract a single-qubit part $H_{\rm ini}$
and a pure two-body interaction $H_{\rm op}$ from
the Hamiltonian of a qubit lattice.
\subsubsection{Manipulation by using the Baker-Campbell-Hausdorff (BCH) formula}
A desirable part of the original Hamiltonian $H_q$
is extracted by using appropriate pulse sequences~\cite{stab}.
The basic idea can be illustrated by using the standard NMR Hamiltonian $H_{\rm nmr}=\sum_i
\varepsilon_i Z_i+ \sum_{i<j} J Z_i Z_j$.
In this case, because of the property $[H_0, H_{\rm int}]=0$, $H_0$ and
$H_{\rm int}$ can be separately obtained by using a simple pulse sequence.
The interaction part $H_{\rm Ising}$ can be extracted by using two
sandwiched $\pi$-pulses such as
$\exp(i\tau H_{\rm Ising})=e^{-i(\pi/2) \sum_j Y_j} e^{ i(\tau/2)
H_{\rm nmr}} e^{ i(\pi/2) \sum_j Y_j} e^{ i(\tau/2) H_{\rm nmr}}$.
For the general Hamiltonian
(Eqs.(\ref{H0})-(\ref{Hising})), because $[H_0, H_{\rm int}]\neq 0$,
we approximately obtain a desirable part by repeatedly applying the Baker-Campbell-Hausdorff (BCH) formula.
For $A=h_a+h_b$ (original
Hamiltonian) and $B=h_a-h_b$ (transferred by applying a $\pi$ pulse)
with $h_a=i\tau H_a$ and $h_b=i\tau H_b$,
we can extract $h_a=i\tau H_a$ by using the relation given by
\begin{equation}
(e^Ae^B)^n \approx \exp( i2 t_0 H_a + (t_0^2/n)[H_a,H_b])
\label{AB}
\end{equation}
($t_0\equiv n\tau$). Thus, as long as $(t_0/n)||H_b|| \ll 1$ where
$||A||=[\mathrm{Tr}(A^\dagger A)/d]^{1/2}$ is the standard operator norm in a
Hilbert space of dimension $d$, we can neglect the second term.
As the number $n$ of repetitions increases, this approximation improves.
In the following sections, we use an extended form of Eq.(\ref{AB}) described by
\begin{eqnarray}
\lefteqn{ (e^Ae^B e^{B'}e^{A'})^n \approx [\exp( 2h_a + [h_b,h_a]) \exp( 2h_a' - [h_b',h_a'])]^n } \nonumber \\
& \approx& \exp( 2n(h_a+h_a') + n[h_b,h_a]- n[h_b',h_a'] +4 n[h_a, h_a'] ),
\nonumber \\
\label{ABAB}
\end{eqnarray}
where $A'=h_a'+h_b'$ and $B'=h_a'-h_b'$. $2(h_a+h_a')$ is the target Hamiltonian.
In the following two subsections, we show how to extract a desirable interaction
term $H_{\rm int}^{ij}$ and a single-qubit part $H_0$ from $H_q$ by using
Eq.~(\ref{ABAB}).
\subsubsection{Qubit lattice and logical qubit}\label{sec:lattice}
We consider a qubit lattice in which physical qubits are arrayed on a lattice site
interacting with their neighboring qubits.
The simplest arrangement is a 1D array as shown in Fig.~\ref{qubitlattice}.
Then we can interact logical qubits with their nearest-neighbor logical
qubits by using interactions between physical qubits.
The number of qubits in each 1D array depends on how many physical qubits
are required to construct a single logical qubit.
In Fig.~\ref{qubitlattice}, nine qubits constitute a logical qubit.
\subsubsection{Selection of a single-qubit Hamiltonian}\label{sec:extractH_0}
Here we show how to extract $H_0$ from $H_q$ for 2D qubit lattice,
assuming always-on interactions between qubits.
As an example, we consider logical qubits consisting of five qubits.
In a 1D qubit array, $H_0$ is obtained by choosing
$B=-h_1+h_2-h_3+h_4-h_5-\sum_i h_{ij}$ and
$B'=h_1-h_2+h_3-h_4+h_5-\sum_i h_{ij}$ while
$A$ and $A'$ are $H_q$ in Eq.~(\ref{ABAB})
[$h_i=\tau H_{0i}$ and $h_{ij}=\tau H_{\rm int}^{ij}$].
This procedure can be extended to the 2D lattice case
by taking into account interactions between different logical qubits.
In this section, we treat Hamiltonians that include two types of Pauli matrices or fewer
such as Eq.~(\ref{app1}) or Eq.~(\ref{app2}) with $\omega_i^{\rm rf}=2\Omega_{0i}$.
For Eq.~(\ref{app1}), `$\pi$-pulse' corresponds to $\pi$-pulse around $y$-axis.
For Eq.~(\ref{app2}) with $\omega_i^{\rm rf}=2\Omega_{0i}$, `$\pi$-pulse'
corresponds to $\pi$-pulse around $z$-axis, which can also be produced by
$\pi$-pulse around $y$-axis after that around $x$-axis.
Extraction of $H_0$ and two-body interaction from the Hamiltonian Eq.~(\ref{app2})
with $\omega_i^{\rm rf}\neq 2\Omega_{0i}$
is described in Appendix~\ref{appendixA}.
The 2D lattice Hamiltonian is given by
\begin{equation}
H^{2D}=\sum_k H_q^{(k)},
\end{equation}
where
\begin{equation}
H_q^{(k)}=H_0^{(k)}+H_{\rm int}^{(k)}+H_{\rm int}^{(k,k+1)}.
\end{equation}
$H_{\rm int}^{(k,k+1)}$ shows an interaction term between $k$-th logical qubits and $k+1$-th qubits.
In order to separate different logical qubits, $H_{\rm int}^{(k,k+1)}$ should be erased.
We apply
$\pi$-pulses to
(i) qubits 1,3,5 of ...,$k-1$-th, $k+1$-th, ... arrays for $A$,
(ii) qubits 1,3,5 of ...,$k$-th, $k+2$-th, ... arrays for $B$,
(iii) qubits 2,4 of ...,$k-1$-th, $k+1$-th, ... arrays for $B'$,
and (iv) qubits 2,4 of qubits of ...,$k$-th, $k+2$-th, ... arrays for $A'$:
\begin{eqnarray}
\lefteqn{A = \cdots} \nonumber \\
&-&h_1^{(k-1)}+h_2^{(k-1)}-h_3^{(k-1)}+h_4^{(k-1)}-h_5^{(k-1)} -h_{\rm int}^{(k-1)}
\nonumber \\
&-& h_{11}^{(k-1,k)}+h_{22}^{(k-1,k)}-h_{33}^{(k-1,k)}+h_{44}^{(k-1,k)}
-h_{55}^{(k-1,k)} \nonumber \\
&+& h_q^{(k)}
\nonumber \\
&-& h_{11}^{(k,k+1)}+h_{22}^{(k,k+1)}-h_{33}^{(k,k+1)}+h_{44}^{(k,k+1)}
-h_{55}^{(k,k+1)} \nonumber \\
&-&h_1^{(k+1)}+h_2^{(k+1)}-h_3^{(k+1)}+h_4^{(k+1)}-h_5^{(k+1)} -h_{\rm int}^{(k+1)}
\nonumber \\
& & \cdots
\end{eqnarray}
\begin{eqnarray}
\lefteqn{B = \cdots +h_q^{(k-1)} } \nonumber \\
&-& h_{11}^{(k-1,k)}+h_{22}^{(k-1,k)}-h_{33}^{(k-1,k)}+h_{44}^{(k-1,k)}
-h_{55}^{(k-1,k)} \nonumber \\
&-&h_1^{(k)}+h_2^{(k)}-h_3^{(k)}+h_4^{(k)}-h_5^{(k)} -h_{\rm int}^{(k)}
\nonumber \\
&-& h_{11}^{(k,k+1)}+h_{22}^{(k,k+1)}-h_{33}^{(k,k+1)}+h_{44}^{(k,k+1)}
-h_{55}^{(k,k+1)} \nonumber \\
&+& h_q^{(k+1)} \cdots
\end{eqnarray}
\begin{eqnarray}
\lefteqn{B' = \cdots} \nonumber \\
&+&h_1^{(k-1)}-h_2^{(k-1)}+h_3^{(k-1)}-h_4^{(k-1)}+h_5^{(k-1)} -h_{\rm int}^{(k-1)}
\nonumber \\
&+& h_{11}^{(k-1,k)}-h_{22}^{(k-1,k)}+h_{33}^{(k-1,k)}-h_{44}^{(k-1,k)}
+h_{55}^{(k-1,k)} \nonumber \\
&+& h_q^{(k)}
\nonumber \\
&+& h_{11}^{(k,k+1)}-h_{22}^{(k,k+1)}+h_{33}^{(k,k+1)}-h_{44}^{(k,k+1)}
+h_{55}^{(k,k+1)} \nonumber \\
&+&h_1^{(k+1)}-h_2^{(k+1)}+h_3^{(k+1)}-h_4^{(k+1)}+h_5^{(k+1)} - h_{\rm int}^{(k+1)}
\nonumber \\
& & \cdots
\end{eqnarray}
\begin{eqnarray}
\lefteqn{A' = \cdots +h_q^{(k-1)} } \nonumber \\
&+& h_{11}^{(k-1,k)}-h_{22}^{(k-1,k)}+h_{33}^{(k-1,k)}-h_{44}^{(k-1,k)}
+h_{55}^{(k-1,k)} \nonumber \\
&+&h_1^{(k)}-h_2^{(k)}+h_3^{(k)}-h_4^{(k)}+h_5^{(k)} -h_{\rm int}^{(k)}
\nonumber \\
&+& h_{11}^{(k,k+1)}-h_{22}^{(k,k+1)}+h_{33}^{(k,k+1)}-h_{44}^{(k,k+1)}
+h_{55}^{(k,k+1)} \nonumber \\
&+& h_q^{(k-1)} \cdots
\end{eqnarray}
where $h_q^{(k)}=\tau (H_0^{(k)}+H_{\rm int}^{(k)})$.
By using Eq.(\ref{ABAB}), we obtain $H_{\rm eff}=2 \sum_k H_0^{(k)}$.
\subsubsection{Selection of two-body interaction}\label{sec:extractH_int}
Next, we show how to extract the interaction term
$H_{\rm int}^{ij}$ between two qubits
in order to use Eqs.(\ref{XYa})-(\ref{XYc}) or Eqs.(\ref{ZZa})-(\ref{ZZb})
for the 2D lattice qubits.
As an example, we consider a case of extracting $h_{23}=i\tau H_{\rm int}^{23}$ in
five-qubit array.
The required transformation is given by extending the results of Ref.~\cite{qecc-basel}.
$A$ in Eq.(\ref{ABAB}) is the original Hamiltonian such as $A=\tau (H_0 + H_{\rm int})$.
$B$ in Eq.(\ref{ABAB}) is given by applying $\pi$ pulse to qubits 2,3,5 of $(k+2n)$-th logical qubits
and qubits 1,4 of $(k+2n-1)$-th logical qubits ($n$ is an integer):
\begin{eqnarray}
\lefteqn{B= \cdots} \nonumber \\
&-& h_1^{(k-1)}+h_2^{(k-1)}+h_3^{(k-1)}-h_4^{(k-1)}+h_5^{(k-1)}
\nonumber \\
& & -h_{12}^{(k-1)}+h_{23}^{(k-1)}-h_{34}^{(k-1)}-h_{45}^{(k-1)}
-h_{\rm int}^{(k-1,k)} \nonumber \\
&+&h_1^{(k)}-h_2^{(k)}-h_3^{(k)}+h_4^{(k)}-h_5^{(k)}
\nonumber \\
& & -h_{12}^{(k)}+h_{23}^{(k)}-h_{34}^{(k)}-h_{45}^{(k)}
-h_{\rm int}^{(k,k+1)} \nonumber \\
&-&h_1^{(k+1)}+h_2^{(k+1)}+h_3^{(k+1)}-h_4^{(k+1)}+h_5^{(k+1)}
\nonumber \\
& & -h_{12}^{(k+1)}+h_{23}^{(k+1)}-h_{34}^{(k+1)}-h_{45}^{(k+1)}
-h_{\rm int}^{(k+1,k+2)} \nonumber \\
& & \cdots
\end{eqnarray}
where $h_i^{(k)} \equiv i\tau H_0^{(k)}$ and $h_{ij}^{(k)}=i\tau H_{\rm int}^{(k)}$.
$B'$ is given by applying $\pi$ pulse to qubits 2,3,5 of $(k+2n-1)$-th logical qubits
and qubits 1,4 of $(k+2n)$-th logical qubits ($n$ is an integer):
\begin{eqnarray}
\lefteqn{B'= \cdots} \nonumber \\
&+&h_1^{(k-1)}-h_2^{(k-1)}-h_3^{(k-1)}+h_4^{(k-1)}-h_5^{(k-1)}
\nonumber \\
& & -h_{12}^{(k-1)}+h_{23}^{(k-1)}-h_{34}^{(k-1)}-h_{45}^{(k-1)}
-h_{\rm int}^{(k-1,k)} \nonumber \\
&-& h_1^{(k)}+h_2^{(k)}+h_3^{(k)}-h_4^{(k)}+h_5^{(k)}
\nonumber \\
& & -h_{12}^{(k)}+h_{23}^{(k)}-h_{34}^{(k)}-h_{45}^{(k)}
-h_{\rm int}^{(k,k+1)} \nonumber \\
&+&h_1^{(k+1)}-h_2^{(k+1)}-h_3^{(k+1)}+h_4^{(k+1)}-h_5^{(k+1)}
\nonumber \\
& & -h_{12}^{(k+1)}+h_{23}^{(k+1)}-h_{34}^{(k+1)}-h_{45}^{(k+1)}
-h_{\rm int}^{(k+1,k+2)} \nonumber \\
& & \cdots
\end{eqnarray}
The $A'$ is obtained by applying $\pi$ pulse to all qubits given by
\begin{equation}
A'=\tau (-H_0 +H_{\rm int}).
\end{equation}
By using Eq.(\ref{ABAB}), we can obtain $\sum_k 4h_{23}^{(k)}$.
The perturbation terms in Eq.(\ref{ABAB})
are described in Appendix~\ref{appendixB}.
For the selection of $\sum_k 4h_{23}^{(k)}$,
the perturbation is estimated as
$|| H_{\rm pert} || \approx 10 \tau N_{\rm qubit} J\Omega$,
and for the case of $H_0$, we have
$|| H_{\rm pert} || \approx 20 \tau N_{\rm qubit} J\Omega$,
where $N_{\rm qubit}$ is the number of connected qubits.
As long as $N_{\rm qubit}$ is not large,
these perturbation terms can be neglected
by repeating Eq.(\ref{ABAB}) with $J_{ij}t_0/n \ll 1$.
Hereafter, we consider the case of $n=1$ for simplicity.
Note that the procedure described in this section can be easily extended to three-dimensionally (3D) arrayed qubits.
\subsection{Estimation of elapsed time}\label{sec:time}
In order to estimate an operation time of pulse manipulations,
we express the time for single-qubit rotation as $\tau_{\rm rot}$.
For preparing a single Hamiltonian $H_0$,
it takes an extra time of $5\tau_{\rm rot}$, because,
in Eq.~(\ref{ABAB}), four Hamiltonians $A$, $B$, $B'$, and $A'$
are transformed from $H_q$ by being sandwiched by $\pi$-pulses.
It also takes extra times of $4\tau_{\rm rot}$ and $5\tau_{\rm rot}$ to
obtain $\exp ( i \tau_{\rm op} H_{\rm op})$ and $\exp (-i \tau_{\rm op} H_{\rm op})$,
respectively, in Eq.(\ref{rho}).
In the latter case, $\tau_{\rm rot}$ is required to reverse the sign of $H_{\rm op}$.
Thus, for $N_{\rm op}$ qubit-qubit operations, it takes a time of
$N_{\rm op} [2\tau_{\rm op}+9\tau_{\rm rot}]$.
In the following, we would like to address the feasibility of our scheme in a
typical superconducting qubit system. Note that our qubit lattice model can
be applied not only to solid-state coupling qubits~\cite{Zagoskin,Grajcar,Niskanen,Ashhab},
but also to circuit-QED qubits~\cite{Schoelkopf,NTT,Houck,Paik}.
For two superconducting qubits in a circuit-QED setup
the effective inter-qubit interaction can be treated as $XY$
type~\cite{Blais++:04,toffoli}. For instance, for $g/\Delta=0.1$, $g/(2\pi)=200$
MHz, $\Delta/(2\pi)=2$ GHz, where $g$ is the Jaynes-Cummings coupling
constant and $\Delta$ the detuning between the resonator frequency and
the qubit splitting, we have $J/(2\pi)=20$ MHz. Thus, $\tau_{\rm op}\approx 6.25$~ns.
We also take $\tau_{\rm rot}\approx
1$ ns~\cite{qecc-basel}.
The criterion is whether all pulse sequences can be done during the dephasing time $T_2$.
We will show that all generation times are less than 300~ns.
Thus, if we assume $T_2 \sim$ 10 to 20~$\mu$s with well-controlled pulses,
which was realized by Paik {\it et al}~\cite{Paik},
we will be able to use the standard QECC process and correct qubit errors,
as long as the number of errors is small.
\section{Generation of stabilizer code from conventional Hamiltonian}\label{sec:code}
Here, we show concrete pulse sequences to produce the target stabilizer Hamiltonians
of the three major codes: the nine-qubit code, the five-qubit code, and the Steane code.
In general, it is difficult to find a pulse sequence of the transformation
from the conventional two-body solid-state Hamiltonian to
the target stabilizer Hamiltonian, because the target Hamiltonians have Pauli matrices whose form is complicated.
The best way to look for an appropriate pulse sequence is to change the target stabilizer Hamiltonian
into single-qubit Hamiltonian,
because it is easier to reduce
the number of multiplications of the Pauli matrices to single-qubit Hamiltonian.
In the following, we show the transformation process of $H_{\rm stab}$ of the three major codes
to the initial single-qubit Hamiltonian.
We also count the number of pulses and estimate generation time of the codes.
We show that the direct generation of $H_{\rm stab}$ is more effective
than the previous method~\cite{qecc-basel} in which $G_i$ is generated one by one.
The comparison of the present results with those of the previous results
is summarized in Tables I and II.
\subsection{Nine-qubit code}
We would like to start from Shor's nine-qubit code that was the first advanced QECC to be invented~\cite{Shor}.
This code can correct single-qubit error ($n=9$, $k=1$), and the number of
stabilizers is $l=8$. The stabilizers are given by $G_1=Z_1Z_2$, $G_2=Z_2Z_3$, $G_3=Z_4Z_5$, $G_4=Z_5Z_6$
$G_5=Z_7Z_8$, $G_6=Z_8Z_9$ $G_7=X_1X_2X_3X_4X_5X_6$, and $G_8=X_4X_5X_6X_7X_8X_9$~\cite{Gottesman,Nielsen}.
Then, the target stabilizer Hamiltonian is given by
$H^{\rm 9code}=\sum_{i=1}^{8}G_i$ in which $\Omega_i$ are omitted,
and we treat $H^{\rm 9code}=\sum_{i=1}^{8}G_i$ instead of $H^{\rm 9code}=-\sum_{i=1}^{8}G_i$
for clarity. We will treat the stabilizer Hamiltonians of the five-qubit code and the Steane code similarly.
We consider how this target Hamiltonian is transformed to
a single-qubit Hamiltonian by using Eqs.~(\ref{XYa1})-~(\ref{XYc1}) for the XY interaction
or Eqs.~(\ref{ZZa1})-~(\ref{ZZc1}) for the Ising interaction. Let us first consider a case of the
XY interaction. $H^{\rm 9code}$ is changed as follows:
\begin{widetext}
\begin{eqnarray}
H^{\rm 9code}&=& Z_1Z_2+Z_2Z_3+Z_4Z_5+Z_5Z_6 +Z_7Z_8+Z_8Z_9
+X_1X_2X_3X_4X_5X_6+X_4X_5X_6X_7X_8X_9, \ \
: (x \leftrightarrow z:2,4,6,8),
\nonumber \\
&\rightarrow & Z_1X_2+X_2Z_3+X_4Z_5+Z_5X_6 +Z_7X_8+X_8Z_9
-X_1Z_2X_3Z_4X_5Z_6-Z_4X_5Z_6X_7Z_8X_9, \ \
\nonumber \\
& & \hspace{10cm}: H_{XY}^{12}+H_{XY}^{34}+H_{XY}^{56}+H_{XY}^{78},
\nonumber \\
&\rightarrow &
-Y_1-Y_1Z_2Z_4-Y_3Z_4Z_6-Y_5 -Y_7-Y_7Z_8Z_9
+Y_2Y_4Y_6-Z_3Y_6Y_8X_9, \ \ : (y\leftrightarrow z:1,5,7)(x\leftrightarrow z:9)
\nonumber \\
&\rightarrow &
-Z_1-Z_1Z_2Z_4+Y_3Z_4Z_6-Z_5 -Z_7-Z_7Z_8X_9
+Y_2Y_4Y_6+Z_3Y_6Y_8Z_9, \ \ : H_{XY}^{34}+H_{XY}^{56}+H_{XY}^{89}
\nonumber \\
&\rightarrow &
-Z_1-Z_1Z_2Z_3+X_4Z_5-Z_6 -Z_7+Z_7Y_8
+Y_2X_3Z_4X_5Z_6+Z_4X_5Z_6X_9, \ \ : (x\leftrightarrow z:3,4,5,9)
\nonumber \\
&\rightarrow &
-Z_1-Z_1Z_2X_3-Z_4X_5-Z_6 -Z_7+Z_7Y_8
+Y_2Z_3X_4Z_5Z_6+X_4Z_5Z_6Z_9, \ \ : H_{XY}^{23}+H_{XY}^{45}+H_{XY}^{78}
\nonumber \\
&\rightarrow &
-Z_1+Z_1Y_2+Y_4-Z_6 -Z_8+X_7
-X_3Y_5Z_6-Y_5Z_6Z_9, \ \ : (y\leftrightarrow z:4)(x\leftrightarrow z:7), \ H_{XY}^{12}+H_{XY}^{56}+H_{XY}^{89}
\nonumber \\
&\rightarrow &
-Z_2+X_1+Z_4-Z_5 -Z_9-Z_7
-X_3X_6-X_6Z_8, \ \ : (x\leftrightarrow z:1,3)
\nonumber \\
&\rightarrow &
-Z_2-Z_1+Z_4-Z_5 -Z_9-Z_7
+Z_3X_6-X_6Z_8, \ \ : H_{XY}^{34}+H_{XY}^{78}
\nonumber \\
&\rightarrow &
-Z_2-Z_1+Z_3-Z_5 -Z_9-Z_8
+Z_4X_6-X_6Z_7, \ \ : H_{XY}^{45}+H_{XY}^{67}
\nonumber \\
&\rightarrow &
-Z_2-Z_1+Z_3-Z_4 -Z_9-Z_8
-Z_5Z_6Y_7+Y_7, \ \ : (x\leftrightarrow z:5)(y\leftrightarrow z:7)
\nonumber \\
&\rightarrow &
-Z_2-Z_1+Z_3-Z_4 -Z_9-Z_8
-X_5Z_6Z_7+Z_7, \ \ : H_{XY}^{56}
\nonumber \\
&\rightarrow &
-Z_2-Z_1+Z_3-Z_4 -Z_9-Z_8
+Y_6Z_7+Z_7, \ \ : H_{XY}^{67}
\nonumber \\
&\rightarrow &
-Z_2-Z_1+Z_3-Z_4 -Z_9-Z_8
+X_7+Z_6,
\label{9codeXY}
\end{eqnarray}
Applied pulses are shown after the colon in each line.
Regarding the notation, the $H_{XY}^{ij}$ on the right-hand side of each line
after the colon shows that we apply Eq.~(\ref{rho}) to the Hamiltonian of the left-hand side.
For example, the second line of the above equation means that
\begin{equation}
e^{ i\frac{\pi}{4}[H_{XY}^{12}+H_{XY}^{34}+H_{XY}^{56}+H_{XY}^{78}]} H^{\rm 9code}
e^{ -i\frac{\pi}{4}[H_{XY}^{12}+H_{XY}^{34}+H_{XY}^{56}+H_{XY}^{78}]}.
\end{equation}
The notation such as $(y\leftrightarrow z:1,5,7,9)$ shows that single-qubit $\pi$-rotation
is applied to qubits 1,5,7 and 9 around the $x$-axis.
Thus when we start an initial Hamiltonian given by
\begin{equation}
H_{\rm ini}^{\rm 9code}=\Omega_1 X_1+\Omega_2 X_2+\Omega_3 X_3+\Omega_4 X_4+\Omega_6 X_6
+\Omega_7 X_7+ \Omega_8 X_8+ \Omega_9 X_9,
\label{9codeini}
\end{equation}
we can produce the stabilizer Hamiltonian $H^{\rm 9code}$ by using the pulse sequence
described by the reverse operations of Eq.(\ref{9codeXY}).
The initial Hamiltonian Eq.(\ref{9codeini}) is obtained by
$e^{-i t H_0} e^{i \pi X_5/2 }e^{-i t H_0} e^{-i \pi X_5/2 }$ in which
$e^{-i t H_0}$ term is obtained from $H_q$ as shown in the previous section.
For the Ising interaction, we obtain
\begin{eqnarray}
H^{\rm 9code} &\rightarrow &
X_1Z_2+Z_2Z_3+X_4Z_5+Z_5X_6 +Z_7Z_8+Z_8X_9
-Z_1X_2X_3Z_4X_5Z_6-Z_4X_5Z_6X_7X_8Z_9, \ \
\nonumber \\
& & \hspace{10cm}: H_{\rm Ising}^{12}+H_{\rm Ising}^{56}+H_{\rm Ising}^{89}
\nonumber \\
&\rightarrow &
Y_1+Z_2Z_3+X_4Z_5+Y_6 +Z_7Z_8+Y_9
-Y_2X_3Z_4Y_5-Z_4Y_5X_7Y_8, \ \ : (y\leftrightarrow z:2,8)
\nonumber \\
&\rightarrow &
Y_1-Y_2Z_3+X_4Z_5+Y_6 -Z_7Y_8+Y_9
-Z_2X_3Z_4Y_5-Z_4Y_5X_7Z_8, \ \ : H_{\rm Ising}^{23}+H_{\rm Ising}^{45}+H_{\rm Ising}^{78}
\nonumber \\
&\rightarrow &
Y_1+X_2+Y_4+Y_6 +X_8+Y_9
+Y_3X_5+X_5Y_7, \ \ : H_{\rm Ising}^{34}+H_{\rm Ising}^{67}
\nonumber \\
&\rightarrow &
Y_1+X_2-Z_3X_4-X_6Z_7 +X_8+Y_9
-X_3Z_4X_5-X_5Z_6X_7, \ \ :(x\leftrightarrow z:3,4,5,6,7)
\nonumber \\
&\rightarrow &
Y_1+X_2+X_3Z_4+Z_6X_7 +X_8+Y_9
-Z_3X_4Z_5-Z_5X_6Z_7, \ \ : H_{\rm Ising}^{45}+H_{\rm Ising}^{67}
\nonumber \\
&\rightarrow &
Y_1+X_2+X_3Z_4+Y_7 +X_8+Y_9
-Z_3Y_4-Z_5Y_6, \ \ : H_{\rm Ising}^{34}+H_{\rm Ising}^{56}
\nonumber \\
&\rightarrow &
Y_1+X_2+Y_3+Y_7 +X_8+Y_9
+X_4+X_6, \ \
\label{9codeZZ}
\end{eqnarray}
\end{widetext}
After single-qubit rotations, we obtain an initial Hamiltonian:
\begin{eqnarray}
H_{\rm ini}^{\rm 9code}&=& \Omega_1 X_1+\Omega_2 X_2+\Omega_3 X_3+\Omega_4 X_4
\nonumber \\
&+&\Omega_6 X_6 +\Omega_7 X_7 +\Omega_8 X_8+\Omega_9 X_9.
\end{eqnarray}
This Hamiltonian is obtained by eliminating $X_5$ term in $H_0$ as in the case of the XY interaction.
Let us count the number of pulses necessary to obtain the nine-qubit code.
Because the present method mainly relies on the control of many pulses,
as the number of pulses increases, pulse errors become the principal origin of
decoherence. Thus, the number of pulses is an indicator
of decoherence in which it is desirable to have fewer pulses.
Eq.(\ref{9codeini}) shows that eight qubit-qubit interaction processes and five single-qubit
rotation processes are needed.
Note that because $\tau_{\rm op} > \tau_{\rm rot}$, the sixth operations of Eq.(\ref{9codeXY})
can be represented by $\tau_{\rm rot}$.
From the result of Sec.III, it takes a time of $N_{\rm op} [2\tau_{\rm op}+9\tau_{\rm rot}]$
for $N_{\rm op}$ uses of $H_{XY}^{ij}$. The initial state Eq.(\ref{9codeini}) is
obtained by twice using the generating process of $H_0$, and thus it takes a time of $10\tau_{\rm rot}$.
Thus, we need a time of
\begin{equation}
\tau^{\rm 9code(new)}_{XY}=8[2\tau_{\rm op}+9 \tau_{\rm rot}]+12\tau_{\rm rot}+10\tau_{\rm rot}=
16\tau_{\rm op}+94 \tau_{\rm rot}.
\end{equation}
In order to compare the present method with that of Ref.~\cite{qecc-basel},
let us consider constructing $H_{\rm stab}$ of the XY interaction by
summing up $G_i$ as in Ref.~\cite{qecc-basel}.
For $G_1 \sim G_6$, it takes a time of
$[2\tau_{\rm op}+9\tau_{\rm rot}] + (4+10)\tau_{\rm rot}
= 2\tau_{\rm op}+23 \tau_{\rm rot}$,
because $G_i \rightarrow Z_iX_{i+1} \rightarrow Y_i \rightarrow X_i$.
For $G_7$, we have
\begin{eqnarray}
G_7
&= & X_1X_2X_3X_4X_5X_6, \ \ : (x \leftrightarrow z:2,5)
\nonumber \\
&\rightarrow & X_1Z_2X_3X_4Z_5X_6, \ \ : H_{XY}^{12}+H_{XY}^{56}
\nonumber \\
&\rightarrow & Y_2X_3X_4Y_5, \ \ : (x\leftrightarrow z:3,4)
\nonumber \\
&\rightarrow & Y_2Z_3Z_4Y_5, \ \ : H_{XY}^{23}+H_{XY}^{45}
\nonumber \\
&\rightarrow & X_3X_4, \ \ : (x\leftrightarrow z:3)
\nonumber \\
&\rightarrow & -Z_3X_4, \ \ : H_{XY}^{34}
\nonumber \\
&\rightarrow & Y_3, \ \ : (y\leftrightarrow x:3)
\nonumber \\
&\rightarrow & X_3.
\end{eqnarray}
Thus, it takes a time of
$ 3[2\tau_{\rm op}+9\tau_{\rm rot}]+8\tau_{\rm rot}+10\tau_{\rm rot}=6\tau_{\rm op}+45\tau_{\rm rot}$
for obtaining $G_7$ and $G_8$. Thus, total generation time for the nine-qubit code
by the method of Ref.~\cite{qecc-basel} is given by $\tau^{\rm 9code(old)}_{XY}=24\tau_{\rm op}+228\tau_{\rm rot}$.
Thus, the number of the qubit-qubit interaction of the present method is reduced to two-thirds of that of the previous method and
the number of the single-qubit rotations is reduced to 41.2 \% of that of the previous method.
When we use the experimental values in Sec.\ref{sec:time},
$\tau^{\rm 9code(new)}_{XY}= 194$~ns and $\tau^{\rm 9code(old)}_{XY}=376$~ns, thus 48.7\% reduction
of the operation time is achieved.
For the Ising interaction, from Eq.~(\ref{9codeZZ}), we obtain a time of the operation given by
\begin{equation}
\tau^{\rm 9code(new)}_{\rm Ising}=10\tau_{\rm op}+63 \tau_{\rm rot}=125.5 {\rm ns}.
\end{equation}
Here, we used the experimental value of $\tau_{\rm op}\approx$ 6.25~ns and $\tau_{\rm rot}\approx$ 1~ns
(See Sec.~\ref{sec:time}).
$G_1 \sim G_6$ have the form of the two-body interaction, thus they are directly extracted from $H_q$
as shown in Sec.\ref{sec:extract}.
Thus it takes a time of $4\tau_{\rm rot}$ for each process.
$G_7$ and $G_8$ are reduced to $Z_3Z_4$ and $Z_6Z_7$ with a time of
$4\tau_{\rm op}+28\tau_{\rm rot}$, respectively. Therefore,
we obtain
$\tau^{\rm 9code(old)}_{\rm Ising}=8\tau_{\rm op}+80 \tau_{\rm rot}=130 {\rm ns}$.
For this case, 3.5~\% reduction of time is achieved.
\subsection{Five-qubit code}
Next, we consider $H_{\rm stab}$ of the five-qubit code ($n=5$ and $k=1$).
The stabilizers $G_i (i=1,..,4)$ of this code are given by
$G_1 = X_1 Z_2Z_3 X_4$,
$G_2 = X_2 Z_3 Z_4 X_5$,
$G_3 = X_3 Z_4 Z_5 X_1$, and
$G_4 = X_4 Z_5 Z_1 X_2$~\cite{Gottesman,Nielsen}.
The process of constructing $H^{\rm 5code}\equiv \sum_{i=1}^4 G_i$ is
obtained by changing $H^{\rm 5code}$
reversely into a single-qubit Hamiltonian.
For $XY$ model, this process is obtained by
\begin{widetext}
\begin{eqnarray}
H^{\rm 5code}&=&
X_1Z_2Z_3X_4 +X_2Z_3Z_4X_5+X_3Z_4Z_5X_1+X_4Z_5Z_1X_2, \ : H_{XY}^{12}+H_{XY}^{34}
\nonumber \\
&\rightarrow &
Y_2Y_3 -Y_1Z_2Z_3Z_4X_5+Y_4Z_5Z_1Y_2+Y_3Z_4Z_5Y_1, \ : (y\leftrightarrow z:2)
\nonumber \\
&\rightarrow &
Z_2Y_3 +Y_1Y_2Z_3Z_4X_5+Y_4Z_5Z_1Z_2+Y_3Z_4Z_5Y_1, \ : H_{XY}^{23}+H_{XY}^{45}
\nonumber \\
&\rightarrow &
X_2 -Y_1X_3Y_4+Z_1Z_3X_5+Y_1X_2Z_3Z_4Z_5, \ : (x\leftrightarrow z:2,3)
\nonumber \\
&\rightarrow &
-Z_2 +Y_1Z_3Y_4+Z_1X_3X_5-Y_1Z_2X_3Z_4Z_5, \ : H_{XY}^{12}+H_{XY}^{34}
\nonumber \\
&\rightarrow &
-Z_1 +Z_1X_2X_3-Z_2Z_3Y_4X_5+X_2Y_4Z_5, \ : (x\leftrightarrow z:2)(y\leftrightarrow z:4,5)
\nonumber \\
&\rightarrow &
-Z_1 -Z_1Z_2X_3-X_2Z_3Z_4X_5+Z_2Z_4Y_5, \ : H_{XY}^{23}+H_{XY}^{45}
\nonumber \\
&\rightarrow &
-Z_1 +Z_1Y_2-Y_3Y_4+Z_3X_4, \ : (y\leftrightarrow z:4)
\nonumber \\
&\rightarrow &
-Z_1 +Z_1Y_2-Y_3Z_4+Z_3X_4, \ :H_{XY}^{12}+H_{XY}^{34}
\nonumber \\
&\rightarrow &
-Z_2 +X_1-X_4-Y_3,
\label{5codeXY}
\end{eqnarray}
Thus, the initial Hamiltonian is given by
\begin{equation}
H_{\rm ini}^{\rm 5code}=\Omega_1 X_1+\Omega_2 X_2 +\Omega_3 X_3+\Omega_4 X_4.
\label{ini_five_xy}
\end{equation}
The time of constructing this code is given by
\begin{equation}
\tau^{\rm 5code(new)}_{XY}=5[2\tau_{\rm op}+9\tau_{\rm rot}]+10\tau_{\rm rot}+10\tau_{\rm rot}
=10\tau_{\rm op}+65\tau_{\rm rot}=127.5 {\rm ns}.
\label{time5code}
\end{equation}
If we use the previous method in Ref.~\cite{qecc-basel}, we have
$
\tau^{\rm 5code(old)}_{XY}=24 \tau_{\rm op}+162\tau_{\rm rot}=312 {\rm ns}.
$
This result is a little different from that in Ref.~\cite{qecc-basel}
in that here we start from 2D Hamiltonian.
Thus, 59.1 \% reduction of time is expected with the present method.
For the Ising interaction, we have
\begin{eqnarray}
H^{\rm 5code}&\rightarrow &
Z_1X_2X_3Z_4 +Z_2X_3X_4Z_5+Z_1Z_3X_4X_5+X_1Z_2Z_4X_5, \ : H_{\rm Ising}^{23}
\nonumber \\
&\rightarrow &
Z_1X_2X_3Z_4 +Y_3X_4Z_5+Z_1Z_3X_4X_5+X_1Z_2Z_4X_5, \ : (x\leftrightarrow z:1,2)(y\leftrightarrow z:3)
\nonumber \\
&\rightarrow &
-X_1Z_2X_3Z_4 +Z_3X_4Z_5-X_1Y_3X_4X_5-Z_1X_2Z_4X_5, \ : H_{\rm Ising}^{12}+H_{\rm Ising}^{34}
\nonumber \\
&\rightarrow &
-Y_1Y_3 +Y_4Z_5-Y_1Z_2Y_3X_4X_5-Y_2Z_4X_5, \ : (y\leftrightarrow z:1,2,4)(x\leftrightarrow z:5)
\nonumber \\
&\rightarrow &
-Z_1Y_3 +Z_4X_5-Z_1Y_2Y_3X_4Z_5-Z_2Y_4Z_5, \ : H_{\rm Ising}^{12}+H_{\rm Ising}^{45}
\nonumber \\
&\rightarrow &
-Z_1Y_3 +Y_5+X_2Y_3Y_4+Z_2X_4, \ : (x\leftrightarrow z:2)
\nonumber \\
&\rightarrow &
-Z_1Y_3 +Y_5-Z_2Y_3Y_4+X_2X_4, \ : H_{\rm Ising}^{23}
\nonumber \\
&\rightarrow &
Z_1Z_2X_3 +Y_5+X_3Y_4+Y_2Z_3X_4, \ : (y\leftrightarrow z:2) (x\leftrightarrow z:3,4)
\nonumber \\
&\rightarrow &
Z_1Y_2Z_3 +Y_5-Z_3Y_4-Z_2X_3Z_4, \ : H_{\rm Ising}^{23}
\nonumber \\
&\rightarrow &
-Z_1X_2 +Y_5-Z_3Y_4-Y_3Z_4, \ : H_{\rm Ising}^{12}+H_{\rm Ising}^{34}
\nonumber \\
&\rightarrow &
-Y_2 +Y_5+X_4+X_3,
\end{eqnarray}
\end{widetext}
Thus, the initial Hamiltonian from which $H_{\rm stab}$ is derived is given by
\begin{equation}
H_{\rm ini}^{\rm 5code}= \Omega_2 X_2+\Omega_3 X_3+\Omega_4 X_4 +\Omega_5 X_5,
\end{equation}
The time for the generation of this code is
$6[2\tau_{\rm op}+9\tau_{\rm rot}]+12\tau_{\rm rot}+10\tau_{\rm rot}
=151
$~ns.
Because
$G_1=X_1Z_2Z_3X_4 \rightarrow Z_1X_2X_3Z_4\rightarrow Y_2Y_3 \rightarrow Z_2Z_3$,
it takes a time of
$[2\tau_{\rm op}+9\tau_{\rm rot}]+4\tau_{\rm rot}+4\tau_{\rm rot}
=2\tau_{\rm op}+17\tau_{\rm rot}$ to obtain $G_1$ and $G_2$. $G_3$ is estimated
from
\begin{eqnarray}
G_3
&= & X_1X_3Z_4Z_5, \ \ : (x \leftrightarrow z:1,4)
\nonumber \\
&\rightarrow & -Z_1X_3X_4Z_5, \ \ : H_{XY}^{23}+H_{XY}^{45}
\nonumber \\
&\rightarrow & -Z_1Z_2Y_3Y_4, \ \ : (y\leftrightarrow z:2,4)
\nonumber \\
&\rightarrow & Z_1Y_2Y_3Z_4, \ \ : H_{XY}^{12}+H_{XY}^{34}
\nonumber \\
&\rightarrow & X_2X_3, \ \ : (x\leftrightarrow z:2,3)
\nonumber \\
&\rightarrow & Z_2Z_3.
\end{eqnarray}
Thus it takes $4\tau_{\rm op}+28\tau_{\rm rot}$. Similarly, it takes
$6\tau_{\rm op}+35\tau_{\rm rot}$ for $G_4$. Therefore, in total,
it takes $14\tau_{\rm op}+97\tau_{\rm rot}=184.5$~ns for summing up $G_1 \sim G_4$ in
the Ising interaction. In this case the present method reduces the generation
time by 18.2\%.
\subsection{Steane code}
The stabilizers of the Steane code are described by
$G_1 = X_1 X_2 X_3 X_4$,
$G_2 = X_1 X_2 X_5 X_6$,
$G_3 = X_1 X_3 X_5 X_7$,
$G_4 = Z_1 Z_2 Z_3 Z_4$,
$G_5 = Z_1 Z_2 Z_5 Z_6$, and
$G_6 = Z_1 Z_3 Z_5 Z_7$~\cite{Gottesman,Nielsen}.
Because $G_4$, $G_5$ and $G_7$ are obtained from $G_1$, $G_2$ and $G_3$ by applying $\pi$-pulses,
we first consider the generation process of $H_X^{Steane}\equiv G_1+G_2+G_3$.
The process of the construction of $H_X^{\rm Steane}$ is obtained by resolving it to a single-qubit Hamiltonian.
For the case of $XY$ Hamiltonian, this process is given by,
\begin{widetext}
\begin{eqnarray}
H_X^{\rm Steane}&=&
X_1X_2X_3X_4 +X_1X_2X_5X_6+X_1X_3X_5X_7, \ : (x\leftrightarrow z :2,3,5)\nonumber \\
&\rightarrow &
X_1Z_2Z_3X_4 +X_1Z_2Z_5X_6+X_1Z_3Z_5X_7, \ : H_{XY}^{12}+H_{XY}^{34}+H_{XY}^{56}\nonumber \\
&\rightarrow &
Y_2Y_3 +Y_2Y_5-Z_1Y_2Z_4Z_6X_7, \ : (y\leftrightarrow z:2,5) \nonumber \\
&\rightarrow &
Z_2Y_3 +Z_2Z_5-Z_1Z_2Z_4Z_6X_7, \ : H_{XY}^{23}+H_{XY}^{45}+H_{XY}^{67} \nonumber \\
&\rightarrow &
X_2 +Z_3Z_4+Z_1Z_3Z_5Y_6, \ : (x\leftrightarrow z:2,4) \nonumber \\
&\rightarrow &
-Z_2 +Z_3X_4+Z_1Z_3Z_5Y_6, \ : H_{XY}^{12}+H_{XY}^{34}+H_{XY}^{56} \nonumber \\
&\rightarrow &
-Z_1 -Y_3+Z_2Z_4X_5, \ : (y \leftrightarrow z:3) \nonumber \\
&\rightarrow &
-Z_1 -Z_3+Z_2Z_4X_5, \ : H_{XY}^{23}+H_{XY}^{45} \nonumber \\
&\rightarrow &
-Z_1 -Z_2-Z_3Y_4, \ : H_{XY}^{34} \nonumber \\
&\rightarrow &
-Z_1 -Z_2-X_3. \
\label{CSSXY}
\end{eqnarray}
\end{widetext}
Thus, we obtain the initial Hamiltonian:
\begin{equation}
H_{X:{\rm ini}}^{\rm Steane}= \Omega_1 X_1+\Omega_2 X_2+\Omega_3 X_3,
\end{equation}
The time of generation of $H_{\rm X}^{\rm Steane}$ is the same as Eq.(\ref{time5code}).
For the previous method, the time for obtaining $H_{\rm X}^{\rm Steane}$
is given by $22\tau_{\rm op}+143\tau_{\rm rot}$.
As mentioned above, because $H_Z^{\rm Steane}\equiv G_4+G_5+G_6$ is obtained by
\begin{equation}
H_Z^{\rm Steane}=e^{-i\pi (Y_1+Y_2+Y_3)/4}H_X^{\rm Steane}e^{i\pi (Y_1+Y_2+Y_3)/4},
\end{equation}
the generation time of $H_Z^{\rm Steane}$ is increased by $2\tau_{\rm rot}$
compared with that of $H_X^{\rm Steane}$. Then, the time of obtaining the
Steane code by the present method is given by
\begin{equation}
\tau^{\rm Steane(new)}_{XY}
= 20 \tau_{\rm op}+132\tau_{\rm rot} =257 {\rm ns}.
\label{timeSteane}
\end{equation}
When we use the previous method in Ref.~\cite{qecc-basel}, the time for the code
generation is given by $\tau^{\rm Steane(old)}_{XY}=44\tau_{\rm op}+288\tau_{\rm rot}=563$~ns.
Thus, 54.4\% reduction of time is expected.
\begin{widetext}
For the Ising Hamiltonian, we have
\begin{eqnarray}
H_X^{\rm Steane}&=&
X_1X_2X_3X_4 +X_1X_2X_5X_6+X_1X_3X_5X_7, \ : (z \leftrightarrow x:1,4,6) \nonumber \\
&\rightarrow &
Z_1X_2X_3Z_4 +Z_1X_2X_5Z_6-Z_1X_3X_5X_7, \ : H_{\rm Ising}^{12}+H_{\rm Ising}^{34}+H_{\rm Ising}^{56}
\nonumber \\
&\rightarrow &
Y_2Y_3 +Y_2Y_5-Z_1Y_3Z_4Y_5Z_6X_7, \ : (y\leftrightarrow z:3)(x\leftrightarrow z:6,7)
\nonumber \\
&\rightarrow &
Y_2Z_3 +Y_2Y_5+Z_1Z_3Z_4Y_5X_6Z_7, \ : H_{\rm Ising}^{23}+H_{\rm Ising}^{45}+H_{\rm Ising}^{67}
\nonumber \\
&\rightarrow &
-X_2 +X_2Z_3Z_4X_5-Z_1Z_3X_5Y_6, \ : (x\leftrightarrow z:2,3,4,5)
\nonumber \\
&\rightarrow &
Z_2 +Z_2X_3X_4Z_5+Z_1X_3Z_5Y_6, \ : H_{\rm Ising}^{23}+H_{\rm Ising}^{45}
\nonumber \\
&\rightarrow &
Z_2 +Y_3Y_4+Z_1Z_2Y_3Z_5Y_6, \ : (y\leftrightarrow z:2,4,5,6)
\nonumber \\
&\rightarrow &
-Y_2 +Y_3Z_4+Z_1Y_2Y_3Y_5Z_6, \ : H_{\rm Ising}^{12}+H_{\rm Ising}^{34}+H_{\rm Ising}^{56}
\nonumber \\
&\rightarrow &
Z_1X_2 -X_3-X_2X_3Z_4X_5, \ : (x\leftrightarrow z:2,4,5)
\nonumber \\
&\rightarrow &
-Z_1Z_2 -X_3-Z_2X_3X_4Z_5, \ : H_{\rm Ising}^{23}+H_{\rm Ising}^{45}
\nonumber \\
&\rightarrow &
-Z_1Z_2 -Z_2Y_3-Y_3Y_4,
\label{CSSIsing}
\end{eqnarray}
\end{widetext}
Thus, the initial Hamiltonian is given by
\begin{equation}
H_{X:{\rm ini}}^{\rm Steane}= J_{12}Z_1Z_2 +J_{23}Z_2Z_3+J_{34}Z_3Z_4,
\label{SteaneIsingIni}
\end{equation}
This Hamiltonian is obtained by erasing $H_{\rm Ising}^{45}$ from $H_{\rm Ising}$.
$H_{\rm Ising}$ is obtained by applying $\pi$-pulses to all qubits in $B$ of Eq.~(\ref{AB}).
Then, Eq.(\ref{SteaneIsingIni}) is obtained by applying a $\pi$-pulse
only to qubit 5 in $B$ of Eq.~(\ref{AB}) for $H_{\rm Ising}$. Then, the time of
the preparation of Eq.~(\ref{SteaneIsingIni}) is estimated by 4$\tau_{\rm rot}$.
Therefore, the total time of the generation of Eq.~(\ref{CSSIsing}) is given by
$5[2\tau_{\rm op}+9\tau_{\rm rot}]+12\tau_{\rm rot}+4\tau_{\rm rot}
=123.5$~ns, and
$\tau^{\rm Steane(new)}_{\rm Ising}
=20\tau_{\rm op}+124\tau_{\rm rot}=249$~ns.
On the other hand, when we use the previous method,
times for generating $G_1$, $G_2$ and $G_3$ are
$4\tau_{\rm op}+26\tau_{\rm rot}$,
$6\tau_{\rm op}+35\tau_{\rm rot}$, and
$8\tau_{\rm op}+48\tau_{\rm rot}$, respectively. Therefore, we obtain
$\tau^{\rm Steane(old)}_{\rm Ising}=36\tau_{\rm op}+220\tau_{\rm rot}=445$~ns,
resulting in 44\% reduction of time.
All the results of the above-mentioned three codes
are summarized in Tables I and II for the $XY$ interaction and Ising interaction, respectively.
From Tables I and II, we can see the large reduction of
the generation time is achieved in the $XY$ interaction.
\begin{table*}
\begin{tabular}{l|l|l|l|l|c}
\hline\hline
$XY$
& \multicolumn{2}{c|}{Previous generation time}
& \multicolumn{2}{c|}{New generation time
& Improvement \\
\hline
Nine-qubit code
& $24\tau_{\rm op}+228\tau_{\rm rot}$ & 378~ns
& $16\tau_{\rm op}+92\tau_{\rm rot}$ & 194~ns
& 48.7~\% \\
Five-qubit code
& $24\tau_{\rm op}+162\tau_{\rm rot}$ & 312~ns
& $10\tau_{\rm op}+65\tau_{\rm rot}$ & 127.5~ns
& 59.1~\% \\
Steane code
&$44\tau_{\rm op}+288\tau_{\rm rot}$ & 563~ns
&$20\tau_{\rm op}+132\tau_{\rm rot}$ & 257~ns
& 54.4~\% \\
\hline\hline
\end{tabular}
\begin{flushleft}
TABLE I. The generation time of the stabilizer Hamiltonian of the
$XY$ interaction. ``New generation time" is a generation time
of the stabilizer Hamiltonian by using the proposed method.
``Previous generation time" is a time,
estimated by using the previous method~\cite{qecc-basel}.
$\tau_{\rm op} = \pi/(4 J)$.
$\tau_{\rm rot}$ represents a time of a single qubit rotation.
We take $\tau_{\rm op}=6.25$~ns and $\tau_{\rm rot}=1$~ns
(Sec.~\ref{sec:time}).
``Improvement" is a ratio of reduction of time of the new generation,
calculated from the 3rd and 5th columns.
\end{flushleft}
\end{table*}
\begin{table*}
\begin{tabular}{l|l|l|l|l|c}
\hline\hline
Ising
& \multicolumn{2}{c|}{Previous generation time}
& \multicolumn{2}{c|}{New generation time
& Improvement \\
\hline
Nine-qubit code
& $8\tau_{\rm op}+80\tau_{\rm rot}$ & 130~ns
& $10\tau_{\rm op}+61\tau_{\rm rot}$ & 125.5~ns
& 3.5~\% \\
Five-qubit code
& $14\tau_{\rm op}+97\tau_{\rm rot}$ & 184.5~ns
& $12\tau_{\rm op}+76\tau_{\rm rot}$ & 151~ns
& 18.2~\% \\
Steane code
&$36\tau_{\rm op}+220\tau_{\rm rot}$ & 445~ns
&$20\tau_{\rm op}+124\tau_{\rm rot}$ & 249~ns
&44~\% \\
\hline\hline
\end{tabular}
\begin{flushleft}
TABLE II. The generation time of the stabilizer Hamiltonian from the
Ising model. Parameters are the same as those in Table I.
\end{flushleft}
\end{table*}
\section{Creation of the standard codes}\label{sec:initial}
As briefly reviewed in Sec.~\ref{sec:review}, encoded states are generated by
repeating measurements of the stabilizers
$G_i$ ($i=1,..,l)$ for an initial state $\Pi_{i=1}^k|0\rangle_n$~\cite{Gottesman,Nielsen}.
Considering that measurements induce extra decoherence, the effectiveness of this conventional method
is limited. In Ref.~\cite{qecc-basel}, we presented the more effective method
of directly generating logical states:
For any given code,
only those $G_j$ with $1 \le j \le m$ and $m \le n - k$ that contain
$X$ or $Y$ operators are needed for the preparation:
\begin{eqnarray}\label{eqn:CodeGen}
|\bar{c}_1...\bar{c}_k\rangle &=& (1+G_1)\cdots
(1+G_{m})\bar{X}^{c_1}_1
\cdots\bar{X}^{c_k}_k |0...0\rangle \nonumber \\
&= & \prod_{i=1}^{k} \bar{X}^{c_i}_i \prod_{j=1}^{m}
\exp\left( -i \frac{\pi}{4} \tilde{G}_j^{a_j} \right) |0...0\rangle\:,
\label{gen}
\end{eqnarray}
where $c_i=0,1$ and operators $\bar{X}_i$ act in the logical state space
$\{|\bar{0}\rangle_i$, $|\bar{1}\rangle_i\}$. Here, $\tilde{G}_j^{a_j}$ denotes
a modified stabilizer operator obtained from $G_j$
by replacing the $X$ operator acting on qubit $ a_j$ by a $Y$
operator, or vice versa. This is done in order to match the effect of an \emph{individual}
factor $\exp [i(\pi/4) \tilde{G}_j^{a_j} ]$ with
the action of the projector $(1+G_j)$ when qubit $ a_j$ is in state
$|0\rangle$. To fulfill Eq.~\eqref{eqn:CodeGen} for all $1 \le j \le m$ \emph{simultaneously}, all
the $a_j$ have to be different and the modified stabilizers have to be generated
in an order such that prior to $\tilde{G}_j^{a_j}$ none of the $\tilde{G}_k^{a_k}$
with $k < j$ have acted on qubit $a_j$ with an $X$ or $Y$.
The time for generating the encoded state is given by
$\tau_{\rm stab}+(\sum_i c_i)\tau_{\rm rot}$.
Here, we extend this idea further and consider
whether we can replace this equation by
\begin{eqnarray}
|\bar{0} \rangle &=& \exp \left( -i\frac{\pi}{4} \tilde{H}_{\rm stab} \right)|0\rangle, \\
\tilde{H}_{\rm stab} &\equiv & \sum_i \tilde{G}_i.
\label{gennew}
\end{eqnarray}
For the five-qubit code, we need
$\tilde{G}_1=Y_1Z_2Z_3X_4$, $\tilde{G}_2=X_2Z_3Z_4Y_5$, $\tilde{G}_3=X_1Y_3Z_4Z_5$,
$\tilde{G}_4=Z_1 Y_2 X_4 Z_5$, and the multiplication is
carried out in the following order:
$\exp[ i (\pi/4) \tilde{G}_2 ]$ $\exp[i (\pi/4) \tilde{G}_4 ]$
$\exp[ i (\pi/4) \tilde{G}_3 ]$ $\exp[i (\pi/4) \tilde{G}_1 ]$.
However, only $\tilde{G}_3$ and $\tilde{G}_4$ commute,
Thus, we cannot replace Eq.~(\ref{gen}) by Eq.~(\ref{gennew}).
For the Steane code, we need three generators:
\begin{eqnarray}
\tilde{G}_1&=&X_1X_2X_3Y_4, \\
\tilde{G}_2&=&X_1X_2X_5Y_6, \\
\tilde{G}_3&=&X_1X_3X_5Y_7.
\end{eqnarray}
Because these three generators mutually commute, such
as $[\tilde{G}_i,\tilde{G}_j]=0$. Therefore
we can apply Eq.~(\ref{gennew}) and reduce the generation time of the
encoded state.
Thus, it is observed that sparse distribution of the Pauli
operators in a logical qubit is preferable for the code generation,
because it results in simpler generation of encoded states.
Next, we consider an encoding of unknown state $a|0\rangle +b|1\rangle$ to
$a|\bar{0}\rangle +b|\bar{1} \rangle $ ($a$ and $b$ are arbitrary complex numbers).
Because, in Eq.(\ref{gen}), $\tilde{G}_j^{a_j}$ was
introduced to hold
$\exp[ -i (\pi/4) \tilde{G}_j^{a_j} ]|0\rangle=(1+G_j )|0\rangle$,
we need different operations for obtaining $|\bar{1} \rangle$.
For simplicity, we consider $|\bar{1}\rangle=\bar{X}|\bar{0} \rangle$.
Then, we can solve this problem if we can prepare a modified initial
state for $|1\rangle$ defined by
\begin{equation}
|\bar{1}\rangle' =\bar{M}^{-1} \bar{X} \bar{M} |0,...,0\rangle,
\end{equation}
with
$\bar{M}\equiv \prod_{j=1}^{m}
\exp [ -i (\pi/4) \tilde{G}_j^{a_j}] $.
This is because we can use the following relation:
\begin{eqnarray}
\bar{M} (a|0...0\rangle
+ b |\bar{1}\rangle')
=a|\bar{0}\rangle +b|\bar{1}\rangle.
\end{eqnarray}
For the five-qubit code, $\bar{X}$ is given by $\bar{X}=X_1X_2X_3X_4X_5$~\cite{Gottesman},
and the modified initial state $|\bar{1}\rangle'$ is expressed by
$|\bar{1}\rangle'=-\tilde{G}_3\tilde{G}_2 \bar{X} |00000\rangle=-|00010\rangle$.
This means that we can obtain an encoded unknown state $a|\bar{0}\rangle +b|\bar{1}\rangle$
when we encode an initial unknown state $a|0\rangle+b|1\rangle$ into
the fourth qubit described by
$|0\rangle_1|0\rangle_2|0\rangle_3(a|0\rangle_4-b|1\rangle_4)|0\rangle_5$
(the phase of $|1\rangle_4$ is changed).
For the Steane code, $\bar{X}$ is given by
$\bar{X}=X_5X_6X_7$~\cite{Gottesman}, and
the modified initial state $|\bar{1}\rangle'$ is expressed by
$|\bar{1}\rangle'=X_2X_3X_5 |00000\rangle=|0110100\rangle$.
Hence, we have to prepare
$a|0000000\rangle+b|0110100\rangle$ to which $\bar{M}$ is applied.
This state is transformed from $|0\rangle_1(a|0\rangle_2+b|1\rangle_2)|00000\rangle$
by applying CNOT gates in which qubits 3 and 5 are target qubits
while qubit 2 is the control qubit.
The nine-qubit codes can be generated in a different way, because the nine-qubit
code is expressed by the product of three parts given by~\cite{Shor}:
\begin{eqnarray}
|\bar{0} \rangle \equiv
(|000\rangle +|111\rangle)(|000\rangle +|111\rangle)(|000\rangle +|111\rangle),
\ \ \\
|\bar{1} \rangle \equiv
(|000\rangle -|111\rangle)(|000\rangle -|111\rangle)(|000\rangle -|111\rangle).
\ \ \label{9code}
\end{eqnarray}
Each three-qubit block is a Greenberger-Horne-Zeilinger(GHZ) state.
From
$
|000\rangle \pm |111\rangle =\exp[ \mp i (\pi/4) X_1Y_2X_3 ] |000\rangle
$, we have
\begin{eqnarray}
|\bar{0}\rangle \!\!&\!=\!&\!
\exp\left( -i \frac{\pi}{4} H_0^{\rm 9code} \right)
|0...0\rangle, \ \ \\
|\bar{1}\rangle \!\!&\!=\!&\!
\exp\left( i \frac{\pi}{4} H_0^{\rm 9code} \right)
|0...0\rangle. \ \ \ \ \
\end{eqnarray}
where the Hamiltonian $H_0^{\rm 9code}\equiv X_1Y_2X_3+X_4Y_5X_6+X_7Y_8X_9$ is obtained
starting from $X_1+X_4+X_7$ by applying
operations discussed in the previous sections.
The concrete pulse sequence is given by
(1)$H_{XY}^{12}+H_{XY}^{45}+H_{XY}^{67}$,
(2)$H_{XY}^{34}+H_{XY}^{56}+H_{XY}^{78}$,
and (3) single-qubit rotations.
Unknown state $a|0\rangle+b|1\rangle$ is encoded by
applying $\exp\left( -i \frac{\pi}{4} H_0^{\rm 9code} \right)$
to a changed state $a|0...0\rangle+b|1...1\rangle$ which can be
obtained by CNOT gate to $(a|0\rangle+b|1\rangle)|0...0\rangle$.
\begin{figure}
\includegraphics[width=7cm,clip=true]{qeccfig3.eps}
\caption{Measurement circuit for fault-tolerant quantum computation~\cite{Nielsen}.
In order to apply any kinds of QECC, measurement qubit is
required for every physical qubit in the logical qubit layer.
(a) Single-qubit measurement. (b)Multi-qubit measurement. $H$ shows a Hadamard gate. }
\label{FT}
\end{figure}
\section{qubit architecture}\label{sec:architecture}
Let us consider possible encoded qubit architectures for solid-state qubits
controlled by local gate electrodes.
In general, solid-state qubits are fabricated on some substrate and,
unlike optical qubits and ion trap qubits~\cite{Barreiro},
they cannot be moved,
being subject to the restriction that the interactions
between qubits are limited to the nearest qubits.
Thus, as discussed in Sec.~\ref{sec:lattice}, it is natural to set a logical qubit as a 1D array.
In order to construct various stabilizer codes,
every qubit should be accessed by an appropriate gate electrode.
This means that a gate electrode layer should be placed along logical qubits.
Because logical qubits interact with each other in a 2D plane, the gate electrode
layer will be constructed on or under the logical qubit layer.
Next, let us consider a structure of measurements.
For the fault-tolerant computation, additional measurement
circuits are required as described in Ref.~\cite{Steane2,Nielsen}.
Figure~\ref{FT} shows the measurement circuit for a single-qubit measurement
and the multi-qubit measurement.
The multi-qubit measurement is used for stabilizer formalism (Fig.~\ref{FT}(b)).
In Fig.~\ref{FT}(b), the number of qubits in the cat state $|0...0\rangle + |1....1\rangle$,
depends on the number of the Pauli matrices
of the stabilizer (Fig.~\ref{FT}(b) is the case of three-qubit stabilizer).
This means that the number of ancilla qubits for the whole measurement circuit is of
the same order as that of qubits in a logical qubit layer.
Therefore, so as to avoid direct measurements and achieve the fault-tolerant computation,
it is appropriate to set an independent qubit layer for measurements.
Because we already have a logical qubit layer, it is natural
that the additional measurement layer should be stacked as shown in Fig~\ref{stack}.
Note that physical qubits and electrodes in Fig.~\ref{stack} are described in a abstract form.
Real qubits and electrodes are more complicated than a box.
Thus, a stacked 3D qubit system will be straightforward architecture for an effective QECC system,
as long as we assume that the interaction between physical qubits is restricted to their neighboring qubits.
For generating the cat state of qubits in the measurement layer,
our method shown in the previous section regarding the nine-qubit code generation
is useful.
The stacked 3D qubit system can be applied to spin qubits and charge qubits.
However, not all qubits can be stacked in the 3D system.
Consider an example of standard superconducting flux qubits.
If we stack flux qubits, the same flux penetrates stacked two qubits, resulting in confusion of
signal between the stacked flux qubits.
In such case, we will be able to implement a single logical qubit
into a square form as shown in Fig.~\ref{2D}.
The 2D arrangement consists of four logical qubits placed at the peripheral
and ancilla qubits surrounded by the logical qubits.
The four logical qubits share their quantum information through
SWAP operation in the ancilla qubits and connect
to the four directions of the nearest logical qubits.
The ancilla qubits at the central region work for fault-tolerant measurements.
\begin{figure}
\includegraphics[width=8cm]{qeccfig2.eps}
\caption{Layered 3D QECC system.
There are two qubit layers; a logical qubit layer and
a measurement qubit layer.
Each qubit layer is connected
to a gate electrode layer by which physical qubits are controlled.
Boxes show qubits and electrodes. Dot lines show qubit-qubit interaction.
In the stabilizer coded, measurement is an indispensable process
for encoding and decoding.
Thus, the logical qubit layer is set close to the measurement qubit layer.}
\label{stack}
\end{figure}
\begin{figure}
\includegraphics[width=8.5cm]{qeccfig4.eps}
\caption{2D qubit layout. Small box shows physical qubits.
(a) Single logical qubit unit, which is composed of four peripheral logical qubits
and central ancilla qubits. The four peripheral qubits are processed to be equivalent.
They interact with logical qubits of other logical qubit units.
(b) 3$\times$4 logical qubit array where each square corresponds to the logical qubit of (a).}
\label{2D}
\end{figure}
\section{Robustness against pulse errors}\label{sec:robustness}
Since the codeword states are encoded in the twofold-degenerate ground-state
manifold $|\bar{0}\rangle$ and $|\bar{1}\rangle$ of $H_{\rm stab}$, the
robustness of this method is limited by the rate of leakage out of this
manifold. Thus, energy non-conserving single-qubit errors---often a prevalent
kind of errors created by a thermal bath--- are exponentially suppressed
for temperatures that are low compared to the Zeeman-splitting $\Omega$.
Hence, besides local imperfections and noise sources, unavoidable
pulse errors are likely to be the predominant cause of leakage, at
low temperatures.
In the present method, each logical qubit is constructed
by starting from a single-qubit Hamiltonian $\sum_i\Omega_i Z_i$, and
multiplying operators like $X_1 \rightarrow X_1X_2 \rightarrow X_1\cdots X_N$.
Hence, it is possible
that this process makes operation errors transmit through each logical qubit.
If we model the pulse errors by randomly distributed, unbiased, and
uncorrelated deviations
$\delta\theta$ with $\sigma_\theta = \sqrt{\langle \delta \theta^2 \rangle}$
from the ideal angle of $\pi / 2$. The leakage from the twofold-degenerate ground-state
manifold $|\bar{0}\rangle$ and $|\bar{1}\rangle$ can then be estimated
by looking at the average of the ground state fidelity
$\langle F (t) \rangle
\approx 1 - N_{\rm P} \sigma_\theta^2 t / (8 \mathcal{T})$, where $N_{\rm P}$
is the number of pulses in the sequence to generate $H_{\rm stab}$,
and $\mathcal{T}$ its duration~\cite{qecc-basel}.
Thus, the reduction of the number of pulses $N_{\rm P}$
for generating stabilizer codes (Tables I and II) is very important.
For the QECC scheme to succeed, the error rate of each qubit operation
should be less than 10$^{-7} \sim 10^{-5}$~\cite{Steane2,Nielsen}.
Thus the accuracy of operation pulses is crucial.
In this regard, we can also use one of many NMR techniques.
If we construct each single pulse by composite pulses,
the accuracy of the pulse increases dramatically~\cite{Ernst}.
The composite-pulse method generalizes the concept of spin echo, and
has already been applied in the field of quantum computation
to greatly improve both single-qubit rotations and CNOT operations~\cite{Haffner,Molmer,Hill,Torosov}.
As the number of pulses $N_{\rm P}$ decreases and the dephasing time $T_2$ increases,
more accurate composite pulses can be implemented, resulting in the success of QECC scheme.
\section{Summary and Conclusions}\label{sec:conclusion}
In summary, we showed how to produce stabilizer Hamiltonians starting from
natural two-body Hamiltonians by using appropriate pulse
sequences.
We demonstrated our method by using typical codes: the nine-qubit code, the five-qubit code and
the Steane code.
The key method of finding the pulse sequence is to inversely trace
the derivation process from the stabilizer Hamiltonian to the single-qubit Hamiltonian.
We also showed how to generate encoded states without using measurements.
Stabilizer Hamiltonians are important for preserving encoded states
as ground states of the system.
Effective preparation of stabilizers is considered to be critical to the succeed of QECC.
Many important experiments have been performed to enlarge
coherence time in solid-state qubits~\cite{Steffen}.
The criteria for the realization of quantum computing
is whether a sufficient number of quantum operations can be carried out
during a given coherence time.
Thus, manipulation speed of each quantum operation
is one of the most important factors for practical quantum computing.
Considering the fact that a quantum computer exceeds a digital computer
only in several fields such as search algorithm,
it will be natural to embed a quantum computer as a part of a digital computer system.
Moreover, as in the present experiments, a quantum circuit will be operated by
a digital computer.
Although the speed of a single processing unit of
a commercial digital computer seems to become saturated,
performance of digital computers
will continue to increase by parallel processing.
Accordingly, it is expected that the manipulation speed of a pulse
sequence will also increase.
Therefore, the approach presented in this paper enables
faster quantum operations by using the cutting-edge
technology of computer science.
How to achieve an appropriate and smooth connection between a quantum computer and a
digital computer will be a future problem.
\acknowledgements
We would like to thank C. Bruder, V. M. Stojanovi\'c, D. Becker,
A. Nishiyama, K. Muraoka, S. Fujita, H. Goto, Y.X. Liu and F. Nori
for discussions.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,601 |
\section{Introduction}
Recent years have witnessed an ever-increasing proliferation of techniques from
quantum-information theory into the field of condensed-matter physics~\cite{Laflorencie:16}.
The first major surge of interest in this direction entailed the use of bipartite entanglement and
the concept of entanglement entropy to characterize various quantum phase transitions~\cite{Amico:08},
primarily in the realm of strongly-correlated and quantum-spin systems. For a quantum system that can be
partitioned into two entangled subsystems, the entanglement entropy is defined as the von Neumann entropy
of the reduced density matrix pertaining to either one of the two subsystems, obtained by tracing out the
degrees of freedom of the other one. This entropy -- a single number -- represents a quantitative measure
of entanglement in any given state of a bipartite quantum system.
Over the past decade, the concept of the {\em entanglement spectrum} attracted considerable interest in the
context of the symmetry-protected topological states of matter~\cite{Li+Haldane:08}. It arises naturally -- simply
by noticing that each reduced density matrix can be written in the form $\rho=\exp(-H_{\textrm{E}})$, i.e., as the
canonical density matrix corresponding to a ``Hamiltonian'' $H_{\textrm{E}}$ at the inverse temperature
$\beta_{\textrm{E}}=1$~\cite{Chandran+:14}. In the same vein, the entanglement entropy can be thought of as the
thermodynamic entropy~\cite{Wehrl:78,HayashiBOOK} of a system described by $H_{\textrm{E}}$. This last
Hamiltonian, the negative logarithm of the reduced density matrix, became known as the modular (or entanglement)
Hamiltonian and the set of its eigenvalues the entanglement spectrum. Such spectra have
already proven their worth as they were shown to capture the edge physics of topologically ordered
phases~\cite{Li+Haldane:08,Pollmann+:10,Thomale+:10}, a research direction pioneered by Li and Haldane~\cite{Li+Haldane:08}.
They also led to nontrivial physical insights in other condensed-matter areas, e.g., interacting spin chains~\cite{Poilblanc:10,Laeuchli+Schliemann:12},
topological insulators and superconductors~\cite{Fidkowski:10}, integer quantum Hall effect~\cite{Schliemann:11}, interacting
bosons~\cite{Deng+Santos:11,Ejima+:14} and fermions~\cite{Toldin+Assad:18}, the Hofstadter problem~\cite{Huang+Arovas:12,Schliemann:13},
and many-body localization~\cite{Yang+:15,Geraedts+:17}. This can be attributed to the fact that entanglement spectrum
provides a more detailed characterization of the pattern of entanglement in a given system than the corresponding entropy~\cite{Li+Haldane:08}.
One area of condensed-matter physics that has not been explored yet from the entanglement-spectrum viewpoint is that of small
polarons~\cite{Emin:82,RanningerReview:06,AlexandrovDevreeseBook} -- quasiparticles emerging in lattice models based on the
molecular-crystal paradigm~\cite{Holstein:59}. Those models describe a short-ranged coupling of an itinerant excitation to
dispersionless phonons~\cite{Engelsberg+Schrieffer:63}, representing an abstraction for the physical situation in which an excess
charge carrier or an exciton in a crystal of a narrow-band semiconductor (or an insulator) interacts with optical phonons of
the host crystal. A strong excitation-phonon (e-ph) coupling leads to a heavily phonon-dressed excitation, that acquires at the same
time a large effective band mass. In particular, if the spatial extent of its wave function does not exceed one unit cell of
the host crystal the ensuing phonon-dressed quasiparticle is referred to as small polaron~\cite{Emin:82}.
Naturally arising from investigations of transport properties of narrow-band electronic
materials~\cite{Hannewald++:04,Slezak++:06,Fratini+Ciuchi:09,StojanovicGraphene}, in the
course of time studies of small-polaron models spawned a research area important in its own
right~\cite{Ranninger:92,Capone+:97,Wellein+Fehske,Jeckelmann+White:98,Bonca+:99,Zoli:03,Stojanovic+:04,
Edwards:06,Alvermann:07,Stojanovic+:12,Mei+:13,Chakraborty+:16,Jansen+:19}.
While the bulk of such studies have been devoted to the time-honored Holstein model~\cite{Holstein:59},
which captures the dependence of the excitation's on-site energy upon Einstein-phonon displacements
on the same site (local e-ph coupling), over the past two decades considerable attention was devoted to various models
with nonlocal-coupling mechanisms~\cite{Zoli:03,Stojanovic+:04,Edwards:06,Alvermann:07}. The most well known among them
is the Peierls-coupling mechanism (also known as Su-Schrieffer-Heeger- or off-diagonal coupling)~\cite{Stojanovic:08},
which accounts for the effective dependence of the hopping amplitude between adjacent lattice sites upon the difference of
local Einstein-phonon displacements on those sites.
An important point of distinction between various coupled e-ph models is provided by the Gerlach-L\"{o}wen
theorem~\cite{Gerlach+Lowen:87,GerlachLowenRMP:91}. This rigorous result rules out nonanalyticities in ground-state-related properties
for all models with e-ph vertex functions that are either completely momentum-independent (Holstein-type coupling~\cite{Holstein:59})
or depends on the phonon quasimomentum, but not on that of the excitation (e.g., Fr\"{o}hlich-type coupling~\cite{Froehlich:54}).
Thus, couplings that depend on both the excitation and phonon quasimomenta do not belong to the domain of applicability of
this theorem. Moreover, for some particular e-ph interactions of this type -- with the Peierls-type coupling being the prime
example -- level-crossing-type sharp transitions were shown to exist~\cite{Stojanovic:08,Sous+:17,Stojanovic+:14}. Namely,
at certain critical coupling strengths their ground states change their character from nondegenerate zero-quasimomentum ones to
twofold-degenerate ones corresponding to a symmetric pair of nonzero quasimomenta. To demonstrate one such transition, a
quantum simulator based on superconducting qubits and resonators was proposed~\cite{Stojanovic+:14,Stojanovic+Salom:19}.
In this paper, the sharp transition in a one-dimensional (1D) model with Peierls-type coupling is analyzed from
the point of view of the entanglement spectrum of the underlying (bipartite) e-ph system. In particular, the main aim of
this paper is to analyze the dependence of the entanglement-spectrum eigenvalues on the effective e-ph coupling strength.
Its principal finding is that the behavior of the entanglement entropy in the vicinity of the critical coupling strength is
to a large extent determined by the smallest eigenvalue. It is also demonstrated that -- as a consequence of the discrete translational
symmetry of the system -- the eigenvalues from the entanglement spectrum can be labeled by the bare-excitation quasimomentum
quantum numbers. This is complemented by the numerical finding that this quantum number in the model under consideration takes
values $0$ and $\pi$, including cases where a transition between the two occurs at a coupling strength far larger
than the critical one.
The remainder of this paper is organized as follows. In Sec.~\ref{ModelMethod} the relevant coupled e-ph Hamiltonian is introduced
(Sec.~\ref{ModelHamiltonian}), along with a short description of the computational methodology utilized here to compute its ground-state
properties (Sec.~\ref{Methodology}). In Sec.~\ref{entspectrum}, after recapitulating the most general properties of entanglement in
bipartite systems (Sec.~\ref{Bipartite}), basic aspects of entanglement spectra in such systems are briefly reviewed (Sec.~\ref{EntSpectrum}),
followed by some general symmetry-related considerations and their specific application to the coupled e-ph system at hand
(Sec.~\ref{symmconsid}). The main findings of this paper are presented and discussed in Sec.~\ref{ResultsDiscuss}. Finally, the paper
is summarized, with conclusions and some general remarks, in Sec.~\ref{SumConcl}. An involved mathematical derivation is relegated to
Appendix~\ref{KeExpr}.
\section{Model and method} \label{ModelMethod}
\subsection{Hamiltonian of the system} \label{ModelHamiltonian}
The system under consideration comprises a spinless-fermion excitation nonlocally coupled to dispersionless
phonons. It is described by a 1D e-ph model, whose Hamiltonian can succinctly be written as
\begin{equation} \label{Hamiltonian}
H = H_{\rm e} + H_{\rm ph} + H_{\textrm{e-ph}} \:.
\end{equation}
Here $H_{\rm e}$ is the excitation hopping (i.e., kinetic-energy) term in the tight-binding
representation, given by
\begin{equation}
H_{\rm e} = -t_{\rm e}\sum_n (c^\dagger_{n+1}c_n + \mathrm{H.c.}) \:,
\end{equation}
with $t_{\rm e}$ being the corresponding hopping amplitude; $c_{n}^{\dagger}$ ($c_{n}$) creates (destroys) an
excitation at site $n$ ($n=1,\ldots,N$). [For simplicity, the excitation on-site energy is set to zero in the
following.] At the same time $H_{\rm ph}$ stands for the free-phonon term ($\hbar=1$ in what follows)
\begin{equation}
H_{\rm ph} = \omega_{\textrm{ph}} \sum_n b^\dagger_n b_n \:,
\end{equation}
where $b_{n}^{\dagger}$ ($b_{n}$) creates (destroys) an Einstein phonon with frequency $\omega_{\textrm{ph}}$ at site $n$.
Finally, the e-ph coupling term describes the lowest-order (linear) dependence of the effective hopping amplitude
between two adjacent sites, say $n$ and $n+1$, on the difference of the local phonon displacements $u_{n+1}$ and
$u_n$ (where $u_n\propto b^\dagger_n + b_n$) at those sites (Peierls-type coupling). It is given by
\begin{equation}\label{Heph}
H_{\textrm{e-ph}} = g\omega_{\textrm{ph}} \sum_n (c^\dagger_{n+1}c_n + \mathrm{H.c.})
(b^\dagger_{n+1} + b_{n+1} - b^\dagger_n - b_n) \:,
\end{equation}
with $g$ being the dimensionless coupling strength.
The eigenstates of the Hamiltonian $H$ in Eq.~\eqref{Hamiltonian} ought to be good-quasimomentum states, i.e.,
joint eigenstates of $H$ and the total quasimomentum operator
\begin{equation}\label{totalcryst}
K_{\textrm{tot}}=\sum_{k} k\:c^{\dagger}_{k}c_{k}+\sum_{q}q\:b^{\dagger}_{q}b_{q} \:,
\end{equation}
since the latter commutes with $H$. In the following, the eigenvalues of $K_{\textrm{tot}}$ are labelled with $K$ and
quasimomenta are dimensionless, i.e., expressed in units of the inverse lattice period. In particular, use is made of
periodic boundary conditions, with $N$ permissible quasimomenta in the Brillouin zone given by $k_n=2\pi n/N$,
where $n=-N/2+1,\ldots, N/2$ ($N$ is assumed to be even).
By switching to momentum space, the e-ph coupling Hamiltonian of Eq.~\eqref{Heph} can be recast in the generic form
\begin{equation}\label{mscoupling}
H_{\mathrm{e-ph}}=N^{-1/2}\sum_{k,q}\gamma_{\textrm{e-ph}}(k,q)\:
c_{k+q}^{\dagger}c_{k}(b_{-q}^{\dagger}+b_{q}) \:,
\end{equation}
where its corresponding vertex function is given by
\begin{equation}\label{vertex_func}
\gamma_{\textrm{e-ph}}(k,q)=2ig\:\omega_{\textrm{ph}}\:[\:\sin k-\sin(k+q)] \:.
\end{equation}
Because the latter depends both on $k$ and $q$, the Peierls-coupling term in Eq.~\eqref{Heph} does not satisfy
the conditions for the applicability of the Gerlach-L\"{o}wen theorem~\cite{GerlachLowenRMP:91}.
Ground-state properties of small polarons are customarily discussed in terms of an effective coupling strength.
For the most general (momentum-dependent) vertex function $\gamma_{\textrm{e-ph}}(k,q)$, the effective coupling
strength is defined as $\lambda_{\textrm{eff}}=\langle|\gamma_{\textrm{e-ph}}(k,q)|^{2}\rangle_{\textrm{BZ}}/(2t_{\rm e}\:
\omega_{\textrm{ph}})$, where $\langle\ldots\rangle_{\textrm{BZ}}$ stands for the Brillouin-zone average. For
$\gamma_{\textrm{e-ph}}(k,q)$ given by Eq.~\eqref{vertex_func}, this reduces to $\lambda_{\textrm{eff}}\equiv
2g^{2}\:\omega_{\textrm{ph}}/t_{\rm e}$. In particular, the ground state of the Hamiltonian \eqref{Hamiltonian}
with Peierls-type coupling undergoes a sharp level-crossing-type transition (i.e., first-order nonanalyticity) at
a critical value $\lambda^{\textrm{c}}_{\textrm{eff}}\sim 1$ of $\lambda_{\textrm{eff}}$~\cite{Stojanovic:08,Sous+:17}.
For $\lambda_{\textrm{eff}}<\lambda^{\textrm{c}}_{\textrm{eff}}$ the ground state is the (nondegenerate) $K=0$ eigenvalue
of $K_{\mathrm{tot}}$, while for $\lambda_{\textrm{eff}}\ge\lambda^{\textrm{c}}_{\textrm{eff}}$ it is twofold-degenerate
and corresponds to a symmetric pair of nonzero quasimomenta $K=\pm K_{\textrm{gs}}$. Upon increasing $\lambda_{\textrm{eff}}$
beyond its critical value, $K_{\textrm{gs}}$ also changes -- which is reflected in the ground-state energy as a sequence
of further first-order nonanalyticities -- and saturates at $K_{\textrm{gs}}=\pi/2$ for a sufficiently large $\lambda_{\textrm{eff}}$.
Importantly, both $\lambda^{\textrm{c}}_{\textrm{eff}}$ and the values of $\lambda_{\textrm{eff}}$ that correspond
to the latter nonanalyticities are not universal, being dependent on the adiabaticity ratio $\omega_{\textrm{ph}}/t_{\rm e}$.
It is worthwhile to mention that a similar sharp transition was found~\cite{Stojanovic+:14,Stojanovic+Salom:19} in a model where
Peierls-type coupling is complemented by e-ph interaction of the breathing-mode type~\cite{Slezak++:06}. It is important to
stress that a dependence on both the excitation and phonon quasimomenta $(k,q)$ is not a sufficient condition for the existence
of a ground-state nonanalyticity; a counterexample is furnished, e.g., by the Edwards model~\cite{Edwards:06,Alvermann:07,Chakraborty+:16}.
\subsection{Computational methodology} \label{Methodology}
The ground-state properties of the e-ph system at hand are here computed using the conventional Lanczos diagonalization
method for sparse matrices~\cite{CullumWilloughbyBook,PrelovsekBoncaChapter:13}, combined with a controlled truncation of
the (otherwise infinite-dimensional) phonon Hilbert space.
The Hilbert space of the e-ph system is spanned by states of the form $|n\rangle_{\textrm{e}} \otimes |\mathbf{m}\rangle_\text{ph}$,
where $|n\rangle_{\textrm{e}}\equiv c_{n}^{\dagger}|0\rangle_{\textrm{e}}$ represents an excitation localized at site
$n$, $\mathbf{m}\equiv(m_1,\ldots,m_N)$ is the set of phonon occupation numbers, and $|\mathbf{m}\rangle_\text{ph} =
\prod_{i=1}^N(1/\sqrt{m_i!})(b_i^\dagger)^{m_i}|0\rangle_\text{ph}$ (here $|0\rangle_{\textrm{e}}$ and $|0\rangle_{\textrm{ph}}$
are the excitation and phonon vacuum states, respectively). With the restriction to a truncated phonon space comprising
states with at most $M$ phonons, all $m$-phonon states with $0\le m_i \le m$ are included, where $m=\sum_{i=1}^N m_i \le M$.
The dimension of the total Hilbert space is given by $D = D_\text{e} \times D_\text{ph}$, where $D_\text{e} = N$ and $D_\text{ph}=
(M+N)!/(M!N!)$. A generic state in this Hilbert space is given by
\begin{equation}\label{expandPsi}
|\psi\rangle=\sum_{n,\mathbf{m}}C_{n,\mathbf{m}}\:|n\rangle_{\textrm{e}}\otimes
|\mathbf{m}\rangle_\text{ph} \:,
\end{equation}
where the information about the phonon content of this state is contained in the coefficients $C_{n,\mathbf{m}}$.
The truncation of the phonon Hilbert space follows a well-established procedure in which the system size ($N$)
and maximum number of phonons retained ($M$) are gradually increased until the convergence for the ground-state energy
and phonon distribution is reached~\cite{Wellein+Fehske}. The convergence criterion adopted here is that the
relative error in these quantities upon further increase of $N$ and $M$ is not larger than $10^{-4}$. While
for Holstein-type coupling the system size is practically inconsequential~\cite{Ranninger:92}, this is not the
case for the nonlocal Peierls-type coupling investigated here. In particular, the stated criterion is here
satisfied for a system with $N=6$ sites and $M=8$ phonons, the values adopted in the following.
\section{Entanglement spectrum} \label{entspectrum}
To set the stage for further discussion, the concept of entanglement spectra for bipartite quantum systems
is briefly introduced here, complemented by its specific application to the coupled e-ph system under
consideration. To begin with, a reminder is presented about some basic aspects of entanglement in
bipartite systems, including the definition of von Neumann entanglement entropy (Sec.~\ref{Bipartite}).
The most general features of entanglement spectra, exemplified by their intimate connection to the Schmidt
decomposition~\cite{Schmidt:1907,Ekert+Knight:95}, are then briefly reviewed (Sec.~\ref{EntSpectrum}).
Finally, Sec.~\ref{symmconsid} is devoted to general considerations on labeling the entanglement-spectrum
eigenvalues with quantum numbers of certain symmetry-related observables, as well as their concrete
use in the coupled e-ph system at hand.
\subsection{Bipartite systems, entanglement entropy, and application to the coupled e-ph system} \label{Bipartite}
The Hilbert space of a quantum system that can be divided up into two subsystems $A$ and $B$ has the form of a tensor product
$\mathcal{H}=\mathcal{H}_{\textrm{A}}\otimes\mathcal{H}_{\textrm{B}}$ of the component spaces. In what follows the respective
dimensions of $\mathcal{H}_{\textrm{A}}$ and $\mathcal{H}_{\textrm{B}}$ will be denoted by $d_{\textrm{A}}$ and $d_{\textrm{B}}$.
In a pure state $|\Psi\rangle$ -- not necessarily normalized -- the density matrix of the full system is given by
\begin{equation}\label{rhogen}
\rho=\frac{|\Psi\rangle\langle\Psi|}{\langle\Psi|\Psi\rangle} \:.
\end{equation}
The reduced (marginal) density matrix $\rho_{\textrm{\tiny{A}}}$ of the subsystem
$A$ is obtained by tracing $\rho$ over the degrees of freedom of the subsystem $B$:
$\rho_{\textrm{A}}=\qopname\relax{no}{Tr}_{\textrm{B}}\rho$.
The von Neumann (entanglement) entropy, defined as
\begin{equation} \label{vonNeumannS}
S_{\textrm{E}}= -\textrm{Tr}_{\textrm{A}}(\rho_{\textrm{A}}\ln\rho_{\textrm{A}}) \:,
\end{equation}
describes the quantum correlations in the state $|\Psi\rangle$. Note that
$S_{\textrm{E}}=-\qopname\relax{no}{Tr}_{\textrm{A}}(\rho_{\textrm{A}}\ln\rho_{\textrm{\tiny{A}}})=
-\qopname\relax{no}{Tr}_{\textrm{B}}(\rho_{\textrm{\tiny{B}}}\ln\rho_{\textrm{B}})$, where the reduced density
matrix $\rho_{\textrm{B}}$ is obtained by tracing $\rho$ over the degrees of freedom of the
subsystem $A$.
In accordance with general relation in Eq.~\eqref{rhogen}, the density matrix corresponding to the ground
state $|\psi_{\textrm{gs}}\rangle$ of the coupled e-ph system ($A\rightarrow\textrm{e}$, $B\rightarrow \textrm{ph}$)
with the tensor-product Hilbert space ${\mathcal H}={\mathcal H}_{\textrm{e}}\otimes{\mathcal H}_{\textrm{ph}}$ is
given by
\begin{equation}\label{rho_eph}
\rho_{\textrm{e-ph}}=\frac{|\psi_{\textrm{gs}}\rangle
\langle\psi_{\textrm{gs}}|}{\langle\psi_{\textrm{gs}}|\psi_{\textrm{gs}}\rangle} \:.
\end{equation}
The reduced excitation density matrix is then given by
\begin{equation} \label{rho_e}
\rho_{\textrm{e}}= \textrm{Tr}_{\textrm{ph}}\big(\rho_{\textrm{e-ph}}\big) \:,
\end{equation}
and the ground-state entanglement entropy $S_{\textrm{gs}}$ of the system is defined as
\begin{equation} \label{S_gs}
S_{\textrm{gs}}= -\textrm{Tr}_{\textrm{e}}\big(\rho_{\textrm{e}}\ln
\rho_{\textrm{e}}\big) \:.
\end{equation}
\subsection{Entanglement spectrum: generalities}\label{EntSpectrum}
Let $\{|s_{\textrm{A}}\rangle, s_{\textrm{A}}=1,\ldots,d_{\textrm{A}}\}$ and
$\{|s_{\textrm{B}}\rangle, s_{\textrm{B}}=1,\ldots,d_{\textrm{B}}\}$ be orthonormal bases in the component
spaces $\mathcal{H}_{\textrm{A}}$ and $\mathcal{H}_{\textrm{B}}$ of the above bipartite system.
A generic pure quantum state $|\Psi\rangle$ of the bipartite system can be decomposed in the orthonormal
basis $\{|s_{\textrm{A}}\rangle \otimes |s_{\textrm{B}}\rangle\}$, i.e., the tensor product of
$\{|s_{\textrm{A}}\rangle\}$ and $\{|s_{\textrm{B}}\rangle\}$:
\begin{equation}\label{statePsi}
|\Psi\rangle = \sum_{s_{\textrm{A}}=1}^{d_{\textrm{A}}}\sum_{s_{\textrm{B}}=1}
^{d_{\textrm{B}}}\:c_{s_{\textrm{A}},s_{\textrm{B}}}\:
|s_{\textrm{A}}\rangle\otimes|s_{\textrm{B}}\rangle \:.
\end{equation}
The coefficients $c_{s_{\textrm{A}},s_{\textrm{B}}}$ in this last expansion can be thought of as the matrix
elements of a (generically rectangular) matrix $M$, which will henceforth be referred to as the entanglement
matrix. Through singular-value decomposition (SVD) this matrix can be recast as
\begin{equation}\label{SVDentMatrix}
M = UDV^{\dagger} \:,
\end{equation}
where $U$ is a matrix of dimension $d_{\textrm{A}}\times \textrm{min} (d_{\textrm{A}},d_{\textrm{B}})$ that satisfies
$U^{\dagger}U = \mathbbm{1}$ and $V$ a $d_{\textrm{B}}\times \textrm{min}(d_{\textrm{A}},d_{\textrm{B}})$ matrix which
satisfies $VV^{\dagger}=\mathbbm{1}$; $D$ is a diagonal square matrix of dimension $\textrm{min} (d_{\textrm{A}},d_{\textrm{B}})$
where all entries -- the singular values of the matrix $M$ -- are non-negative and can be written as $\{e^{-\xi_{\alpha}/2}|\:
\alpha=1,\ldots,\textrm{min} (d_{\textrm{A}},d_{\textrm{B}})\}$.
Using the above SVD of the entanglement matrix, one arrives at the Schmidt decomposition~\cite{Ekert+Knight:95}
\begin{equation}\label{SchmidtDecomp}
|\Psi\rangle = \sum_{\alpha=1}^{\alpha_{\textrm{max}}}\:e^{-\xi_{\alpha}/2}
|\psi^{\alpha}_\textrm{A}\rangle\otimes|\psi^{\alpha}_\textrm{B}\rangle \:,
\end{equation}
where $\alpha_{\textrm{max}}=\textrm{min}(d_{\textrm{A}},d_{\textrm{B}})$ and
\begin{equation} \label{psiAB}
|\psi^{\alpha}_\textrm{A}\rangle = \sum_{s_{\textrm{A}}=1}^{d_{\textrm{A}}}\:U^{\dagger}_{\alpha,s_{\textrm{A}}}
|s_{\textrm{A}}\rangle \:,\quad
|\psi^{\alpha}_\textrm{B}\rangle = \sum_{s_{\textrm{B}}=1}^{d_{\textrm{B}}}\:V^{\dagger}_{\alpha,s_{\textrm{B}}}
|s_{\textrm{B}}\rangle \:,
\end{equation}
are the singular vectors of the matrix $M$. The latter allow one to express the reduced density matrices as
\begin{eqnarray} \label{rhoArhoB}
\rho_{\textrm{A}} = \sum_{\alpha=1}^{\alpha_{\textrm{max}}}\:e^{-\xi_{\alpha}}|\psi^{\alpha}_{\textrm{A}}
\rangle\langle\psi^{\alpha}_{\textrm{A}}| \:, \nonumber \\
\rho_{\textrm{B}} = \sum_{\alpha=1}^{\alpha_{\textrm{max}}}\:e^{-\xi_{\alpha}}|\psi^{\alpha}_{\textrm{B}}
\rangle\langle\psi^{\alpha}_{\textrm{B}}| \:.
\end{eqnarray}
Thus the joint spectrum of $\rho_{\textrm{A}}$ and $\rho_{\textrm{B}}$ can be obtained from the Schmidt decomposition of
the state $|\Psi\rangle$ [cf. Eq.~\eqref{SchmidtDecomp}] (or, equivalently, from the SVD of the entanglement matrix)
and is given by the set $\{e^{-\xi_{\alpha}}\}$ (i.e., squares of the above singular values). In particular, the {\em
entanglement spectrum} corresponds to the set $\{\xi_{\alpha}\}$ of the negative logarithms of the joint eigenvalues of
$\rho_{\textrm{A}}$ and $\rho_{\textrm{B}}$.
\subsection{Symmetry-related considerations and application to the coupled e-ph system}\label{symmconsid}
In what follows, it is shown that the entanglement-spectrum eigenvalues of the e-ph system can be labeled
by the quantum number associated with the excitation quasimomentum operator, this being a special case of
more general symmetry-related considerations.
Consider a Hermitian operator (observable) $\mathcal{O}$ acting on the tensor-product Hilbert space $\mathcal{H}=\mathcal{H}_{\textrm{A}}
\otimes\mathcal{H}_{\textrm{B}}$ that can be decomposed as $\mathcal{O}=\mathcal{O}_{\textrm{A}}+\mathcal{O}_{\textrm{B}}$,
where $\mathcal{O}_{\textrm{A}}$ acts only on $\mathcal{H}_{\textrm{A}}$ and $\mathcal{O}_{\textrm{B}}$ only on
$\mathcal{H}_{\textrm{B}}$. Assuming that the state $|\Psi\rangle$ is an eigenstate of $\mathcal{O}$, it immediately
follows that its corresponding density matrix $\rho$ commutes with $\mathcal{O}$. Furthermore, $[\mathcal{O},\rho]=0$
implies that $\textrm{Tr}_{\textrm{B}}[\mathcal{O},\rho]=\textrm{Tr}_{\textrm{B}}[\mathcal{O}_{A},\rho]+
\textrm{Tr}_{\textrm{B}}[\mathcal{O}_{\textrm{B}},\rho]=0$. By virtue of the fact that $\textrm{Tr}_{\textrm{B}}
[\mathcal{O}_{\textrm{B}},\rho]=0$, which can be verified by evaluating this last trace in the
eigenbasis of the operator $\mathcal{O}_{\textrm{B}}$, and
\begin{equation}
\textrm{Tr}_{\textrm{B}}[\mathcal{O}_{\textrm{A}},\rho]=[\mathcal{O}_{\textrm{A}},\textrm{Tr}_{\textrm{B}}
\rho]\equiv [\mathcal{O}_{\textrm{A}},\rho_{\textrm{A}}]\:,
\end{equation}
one readily finds that $[\mathcal{O}_{\textrm{A}},\rho_{\textrm{A}}]=0$. Therefore, one can simultaneously
diagonalize $\rho_{\textrm{A}}$ and $\mathcal{O}_{\textrm{A}}$, and label the entanglement-spectrum eigenvalues
$\{\xi_{\alpha}\}$ according to the quantum number of $\mathcal{O}_{\textrm{A}}$.
It is pertinent to apply these general symmetry-related considerations to the coupled e-ph system at hand,
which possesses a discrete translational symmetry. Owing to this symmetry, mathematically expressed by
$[H,K_{\textrm{tot}}]=0$, the ground state $|\psi_{\textrm{gs}}\rangle$ of the system is an eigenstate of
the operator $K_{\textrm{tot}}$ [cf. Eq.~\eqref{totalcryst}]. This operator -- the generator of discrete
translations -- plays the role of the observable $\mathcal{O}$ in the above discussion. Namely, it can be
decomposed as $K_{\textrm{tot}}=K_{\textrm{e}}+K_{\textrm{ph}}$, where $K_{\textrm{e}}=\sum_{k} k\:c^{\dagger}_{k}c_{k}$
acts only on $\mathcal{H}_{\textrm{e}}$ and $K_{\textrm{ph}}=\sum_{q}q\:b^{\dagger}_{q}b_{q}$ on
$\mathcal{H}_{\textrm{ph}}$. Following the above general reasoning, one concludes that the operator
$K_{\textrm{e}}$ commutes with the reduced density matrix $\rho_{\textrm{e}}$
corresponding to $|\psi_{\textrm{gs}}\rangle$ [cf. Eq.~\eqref{rho_e}]. Thus, the operators $K_{\textrm{e}}$
and $\rho_{\textrm{e}}$ can be diagonalized simultaneously and the entanglement-spectrum eigenvalues $\{\xi_1,\ldots,\xi_N\}$
can be labeled by the quantum number of $K_{\textrm{e}}$, i.e., they correspond to different quasimomenta
in the Brillouin zone permissible by the periodic boundary conditions (cf. Sec.~\ref{ModelHamiltonian}). In
particular, the excitation-quasimomentum eigenvalue $K^{\alpha}_{\textrm{e}}\equiv\langle\xi_{\alpha}|\:K_{\textrm{e}}\:|\xi_{\alpha}\rangle$
corresponding to $\xi_{\alpha}$ ($\alpha=1,\ldots,N$) is given by Eq.~\eqref{FinalKe} in Appendix~\ref{KeExpr}.
\section{Results and Discussion} \label{ResultsDiscuss}
The strategy employed here to analyze the coupled e-ph system entails the following steps. After
the ground-state vector $|\psi_{\textrm{gs}}\rangle$ -- represented by the coefficients $C^{\textrm{gs}}_{n,\mathbf{m}}$
[cf. Eq.~\eqref{expandPsi}] -- is obtained through Lanczos diagonalization~\cite{CullumWilloughbyBook,PrelovsekBoncaChapter:13}
of the e-ph Hamiltonian \eqref{Hamiltonian} for each value of $\lambda_{\textrm{eff}}$ in the chosen range $[0,4]$, the reduced
density matrix is determined with the aid of Eqs.~\eqref{rho_eph} and \eqref{rho_e}. Its matrix elements
$(\rho_{\textrm{e}})_{nn'}$ ($n,n'=1,\ldots,N$) are given by
\begin{equation}\label{rho_e}
(\rho_{\textrm{e}})_{nn'} = \frac{\displaystyle\sum_{\mathbf{m}}\:C^{\textrm{gs}}_{n,\mathbf{m}}C^{\textrm{gs}\:*}
_{n',\mathbf{m}}}{\displaystyle\sum^{N}_{p=1}\sum_{\mathbf{m}}\:|C^{\textrm{gs}}_{p,\mathbf{m}}|^{2}} \:.
\end{equation}
The entanglement-spectrum eigenvalues and their associated eigenvectors are then obtained by simply solving the
($N\times N$)-dimensional eigenproblem of $\rho_{\textrm{e}}$. Alternatively, the same spectrum can be obtained
through a numerical SVD~\cite{NRcBook} of the corresponding entanglement matrix [cf. Eq.~\eqref{SVDentMatrix}].
\begin{figure}[b!]
\includegraphics[clip,width=8.45cm]{EntSpectFig1.eps}
\caption{\label{fig:EntropyPlot}Dependence of the ground-state e-ph entanglement entropy for a system of size
$N=6$ on the effective coupling strength, depicted for three different values of the adiabaticity ratio.}
\end{figure}
In what follows, the entire range of e-ph coupling strengths is analyzed -- from the weak-coupling regime characterized by a
weakly-dressed (quasi-free) excitation to the strong-coupling regime with a heavily-dressed one (small polaron). The analysis
was repeated for different values of the adiabaticity ratio, covering the adiabatic ($\omega_{\textrm{ph}}/t_{\rm e}<1$) and
antiadiabatic ($\omega_{\textrm{ph}}/t_{\rm e}>1$) regimes, as well as the intermediate case ($\omega_{\textrm{ph}}/t_{\rm e}=1$).
\begin{figure}[t!]
\includegraphics[clip,width=8.45cm]{EntSpectFig2.eps}
\caption{\label{fig:xi123Plot}Entanglement-spectrum eigenvalue $\xi_{\alpha}$ in the ground state of a system of size $N=6$
as a function of the effective coupling strength: (a) $\alpha=1$, (b) $\alpha=2$, and (c) $\alpha=3$.}
\end{figure}
Before embarking on the analysis of the ground-state entanglement spectrum of the system it is instructive to discuss its corresponding
entanglement entropy $S_{\textrm{gs}}$ [cf. Eq.~\eqref{S_gs}]. In Fig.~\ref{fig:EntropyPlot}, this quantity is depicted for three different
values of the adiabaticity ratio and clearly shows a first-order nonanalyticity at a critical value $\lambda^{\textrm{c}}_{\textrm{eff}}$
of $\lambda_{\textrm{eff}}$. This critical value decreases -- albeit rather slowly -- with $\omega_{\textrm{ph}}/t_{\rm e}$. Beyond this
critical value, the entanglement entropy grows monotonously and for a sufficiently large coupling strength saturates at the value $\ln N$
characteristic of maximally-entangled states~\cite{Zhao+:04} in this system; for $N=6$, this maximal value is $S^{\textrm{max}}_{\textrm{gs}}
\approx 1.79$ [cf. Fig.~\ref{fig:EntropyPlot}].
The numerically-obtained entanglement spectrum has the same qualitative structure for all values of the adiabaticity ratios, which
appears to be consistent with the previously established general conclusion that the gross features of small polarons in the presence
of Peierls-type coupling are for the most part insensitive to the value of $\omega_{\textrm{ph}}/t_{\rm e}$~\cite{Capone+:97}. Its
corresponding eigenvalues, i.e., their dependence on $\lambda_{\textrm{eff}}$, are depicted in Figs.~\ref{fig:xi123Plot} ($\alpha=1,2,3$)
and \ref{fig:xi456Plot} ($\alpha=4,5,6$) for all three relevant regimes. While the nonanalytic behavior is manifested in all six eigenvalues,
what is noticeable from Figs.~\ref{fig:xi123Plot} and \ref{fig:xi456Plot} is that this nonanalyticity is much more pronounced in the three
eigenvalues shown in Fig.~\ref{fig:xi123Plot} than in those displayed in Fig.~\ref{fig:xi456Plot}.
\begin{figure}[b!]
\includegraphics[clip,width=8.45cm]{EntSpectFig3.eps}
\caption{\label{fig:xi456Plot}Entanglement-spectrum eigenvalue $\xi_{\alpha}$ in the ground state of a system of size $N=6$
as a function of the effective coupling strength: (a) $\alpha=4$, (b) $\alpha=5$, and (c) $\alpha=6$.}
\end{figure}
Importantly, from Fig.~\ref{fig:xi123Plot} it can be inferred that the behavior of the ground-state entanglement entropy
$S_{\textrm{gs}}=\sum^{6}_{\alpha=1}\xi_{\alpha}\:e^{-\xi_{\alpha}}$ [displayed in Fig.~\ref{fig:EntropyPlot}] -- especially
for $\lambda_{\textrm{eff}}\ge\lambda^{\textrm{c}}_{\textrm{eff}}$ -- is determined almost entirely by that of the smallest
entanglement-spectrum eigenvalue ($\alpha=1$)[cf. Fig.~\ref{fig:xi123Plot}(a)], i.e., the largest eigenvalue of the corresponding
reduced density matrix [cf. Eq.~\eqref{rho_e}]. Namely, the remaining five eigenvalues -- especially those corresponding
to $\alpha=2$ and $\alpha=4$, depicted in Figs.~\ref{fig:xi123Plot}(b) and ~\ref{fig:xi456Plot}(b), respectively -- have
a rather weak dependence on $\lambda_{\textrm{eff}}$ beyond the critical coupling strength, thus giving nearly constant
contributions to $S_{\textrm{gs}}$ for $\lambda_{\textrm{eff}}\ge\lambda^{\textrm{c}}_{\textrm{eff}}$. Another feature that
sets the $\alpha=1$ eigenvalue apart is that it is the only one which monotonously increases with $\lambda_{\textrm{eff}}$
below $\lambda^{\textrm{c}}_{\textrm{eff}}$, with all the other eigenvalues showing fairly similar decreasing behavior for
$\lambda_{\textrm{eff}}<\lambda^{\textrm{c}}_{\textrm{eff}}$. Interestingly, not only that the $\lambda_{\textrm{eff}}$-dependence of its
corresponding contribution $S_{\alpha=1}\equiv\xi_{\alpha=1}\:e^{-\xi_{\alpha=1}}$ (cf. Fig.~\ref{fig:Salpha1Plot}) mimics the behavior
of the total ground-state entanglement entropy $S_{\textrm{gs}}$, but this entanglement-spectrum eigenvalue itself also shows a very similar
dependence on $\lambda_{\textrm{eff}}$ as $S_{\alpha=1}$ and $S_{\textrm{gs}}$.
\begin{figure}[t!]
\includegraphics[clip,width=8.45cm]{EntSpectFig4.eps}
\caption{\label{fig:Salpha1Plot}Contribution $S_{\alpha=1}\equiv \xi_{\alpha=1}\:e^{-\xi_{\alpha=1}}$ of the $\alpha=1$
entanglement-spectrum eigenvalue to the ground-state entanglement entropy $S_{\textrm{gs}}$.}
\end{figure}
This last finding that the ground-state e-ph entanglement entropy $S_{\textrm{gs}}$ is to a large extent determined by
$\xi_{\alpha=1}$ -- i.e., by the smallest eigenvalue of the corresponding entanglement Hamiltonian -- squares with a conclusion
drawn in studies of other types of many-body systems. Namely, the interesting, universal part of the entanglement spectrum is typically
captured by the largest eigenvalues of the relevant reduced density matrix~\cite{Johri+:17}. Recalling that the entanglement
entropy corresponding to a certain reduced density matrix is equal to the thermodynamic entropy of the attendant entanglement
Hamiltonian $H_{\textrm{E}}$ at the inverse temperature $\beta_{\textrm{E}}=1$, this finding also becomes closely related to
the far more general issue as to when an entire Hamiltonian of a many-body system can be considered as being encoded in a single
eigenstate (typically its ground state). Such situations are not uncommon in many-body systems, but have so far been systematically
discussed only in the context of thermodynamic and entanglement entropies of single-component systems, such as interacting quantum
spin-$1/2$ chains or interacting hard-core bosons on a 1D lattice~\cite{Garrison+Grover:18}. The present study of the
entanglement spectrum in a (two-component) coupled e-ph system thus provides another, qualitatively different, example of a physical
system where this same issue becomes relevant.
As regards the relative importance of different entanglement-spectrum eigenvalues, a useful insight can be gleaned by evaluating the relative
contributions $S_{\alpha}/S_{\textrm{gs}}$ of those eigenvalues to the total entanglement entropy at different coupling strengths. The actual
calculation shows that the eigenvalues $\alpha=1, 4$, and $5$ give much larger contributions to $S_{\textrm{gs}}$ than the remaining ones. To
be more specific, they account for around $80\%$ of $S_{\textrm{gs}}$, with their maximal contributions being attained in the vicinity of the
critical coupling strength. Their individual relative contributions, depicted in Fig.~\ref{fig:Salpha1Sgs}, are completely
independent of the adiabaticity ratio (hence the value of $\omega_{\textrm{ph}}/t_{\rm e}$ is not indicated in the plot).
\begin{figure}[t!]
\includegraphics[clip,width=8.45cm]{EntSpectFig5.eps}
\caption{\label{fig:Salpha1Sgs}Relative contributions $S_{\alpha}/S_{\textrm{gs}}$ of the entanglement-spectrum
eigenvalues $\alpha=1,4,5$ to the total ground-state entanglement entropy (independent of the adiabaticity ratio).}
\end{figure}
As discussed in Sec.~\ref{symmconsid}, resulting from the presence of a discrete translational symmetry is the possibility to label
the entanglement-spectrum eigenvalues by the quantum number of the excitation-quasimomentum operator $K_{\textrm{e}}$; its values are
the quasimomenta $k_n$ in the Brillouin zone permitted by the periodic boundary conditions. Based on the expression given by Eq.~\eqref{FinalKe}
in Appendix~\ref{KeExpr}, it is straightforward to numerically determine the quasimomenta associated to different eigenvalues $\xi_{\alpha}$
for different coupling strengths and adiabaticity ratios.
The actual calculation shows that for $\omega_{\textrm{ph}}/t_{\rm e}\geq 1$ (i.e., in the antiadiabatic and intermediate cases) one eigenvalue,
more precisely $\alpha=3$, corresponds to the quasimomentum $\pi$ at all coupling strenghts, while the five remaining eigenvalues correspond
to $0$. This is illustrated in Fig.~\ref{fig:Kalpha}(a) for the special case $\omega_{\textrm{ph}}/t_{\rm e}=1$. The corresponding behavior
for $\omega_{\textrm{ph}}/t_{\rm e}<1$, i.e., in the adiabatic regime, has an additional interesting feature. Namely, while in this regime
there are eigenvalues corresponding to the bare-excitation quasimomenta $0$ and $\pi$ at all coupling strengths, one also finds cases where a
specific eigenvalue corresponds to $0$ in a certain interval of coupling strengths and to $\pi$ otherwise. For instance, Fig.~\ref{fig:Kalpha}(b)
illustrates one such example for $\omega_{\textrm{ph}}/t_{\rm e}=0.5$, where for a certain coupling strength slightly below
$\lambda_{\textrm{eff}}=3$ -- thus lying deeply in the strong e-ph coupling regime -- such a transition occurs between the quasimomenta $0$
and $\pi$ for the $\alpha=3$ and $\alpha=6$ entanglement-spectrum eigenvalues.
\begin{figure}[t!]
\includegraphics[clip,width=8.45cm]{EntSpectFig6.eps}
\caption{\label{fig:Kalpha}Quasimomentum $K^{\alpha}_{\textrm{e}}\equiv \langle\xi_{\alpha}|\:K_{\textrm{e}}\:|\xi_{\alpha}\rangle$
(expressed in units of $\pi$) associated to the $\alpha=3$ and $\alpha=6$ entanglement-spectrum eigenvalues
for (a) $\omega_{\textrm{ph}}/t_{\rm e}=1.0$, and (b) $\omega_{\textrm{ph}}/t_{\rm e}=0.5$.}
\end{figure}
The occurrence of this last generalized transition provides a differentiation between the e-ph entanglement pattern in the adiabatic regime
and the other relevant regimes (antiadiabatic, intermediate). This can be linked to the fact that this transition takes
place at a coupling strength for which maximally-entangled small-polaron states are still not reached in the adiabatic case, unlike in the
other two cases [cf. Fig.~\ref{fig:EntropyPlot}]. An immediate question is whether a concrete physical meaning can be attributed to it, this
being related to the much more general issue as to how universal is the entanglement spectrum~\cite{Chandran+:14}. In Ref.~\onlinecite{Chandran+:14},
based on several physical examples it was argued that the entanglement Hamiltonian of a physical system may undergo transitions in which
its ground state and low-energy spectrum exhibit singular changes, even when the
system actually remains in the same phase. In other words, the entanglement spectrum may exhibit spurious quantum phase transitions that do
not have any genuine physical counterpart, a property that it shares with the less general concept of entanglement entropy~\cite{Amico:08}.
While this issue was previously discussed in connection with broken-symmetry or topological phases of many-body systems, here it
comes up in the qualitatively different context of small-polaron states that do not spontaneously break the discrete translational symmetry
of the underlying excitation-phonon Hamiltonian.
\section{Summary and Conclusions} \label{SumConcl}
To summarize, in this paper the onset of nonanalytic behavior of ground-state-related properties in models with strongly momentum-dependent
excitation-phonon coupling was investigated from the point of view of the underlying entanglement spectrum. This was accomplished through
a case study of a lattice model with Peierls-type coupling whose entanglement spectrum was obtained in a numerically-exact fashion. The
accompanying analysis was carried out in the full range of the relevant effective excitation-phonon coupling strength -- from weak- (quasifree
excitation) to strong coupling (heavily-dressed excitation, i.e., small polaron) -- and in different regimes of the adiabaticity ratio.
The main finding of the present work is that the dependence of the ground-state entanglement entropy on the excitation-phonon coupling
strength -- and, in particular, the first-order nonanalyticity that it shows at the critical coupling strength -- chiefly originates
from the smallest entanglement-spectrum eigenvalue. Another nontrivial conclusion drawn is that this particular eigenvalue shows a very similar
dependence on the effective coupling strength as the entanglement entropy itself. In addition, as a special case of quite general symmetry-related
arguments it was demonstrated that the discrete translational symmetry of the system implies that the entanglement-spectrum eigenvalues can
be labeled by the bare-excitation quasimomentum quantum number. Finally, it was shown numerically that these eigenvalues are predominantly
associated to quasimomenta $0$ and $\pi$. Interestingly, it was also found that in particular in the adiabatic regime a generalized transition
between these two quasimomenta -- for specific entanglement-spectrum eigenvalues -- takes place deeply in the strong-coupling regime. This
feature sets apart the adiabatic regime from the other two relevant regimes.
The present work extends the range of applications of the concept of entanglement spectrum to polaronic systems. Generally speaking,
what makes the ground-state nonanalyticities in models of the kind investigated here particularly appealing is that they take place in a
system of finite size and are thus amenable to a rigorous numerical analysis. It would be interesting to test the generality of the conclusions
drawn here in a future work by studying other models with strongly momentum-dependent excitation-phonon coupling whose ground states show a
similar nonanalytic behavior. Furthermore, the local (single-qubit) addressability of the previously proposed analog quantum simulators of those
models~\cite{Stojanovic+:14,Stojanovic+Salom:19} may allow an experimental measurement of the corresponding entanglement spectra. Namely,
a completely general method for such measurements was recently suggested and applied to a specific class of locally-addressable systems (cold
atoms in optical lattices)~\cite{Pichler+:16}. This method -- based on an analogy to a many-body Ramsey interferometry~\cite{Ekert+:02} -- makes
use of the fact that the conditional evolution of a many-body system is determined by a copy of its density operator, which acts as the
Hamiltonian. It is conceivable that the ever-improving scalability and coherence properties of superconducting-qubit systems will allow
the realization of the aforementioned simulators in not-too-distant future, which will in turn make it possible to measure the relevant
entanglement spectra using the latter method.
\begin{acknowledgments}
The author acknowledges useful discussions on the numerical implementation with I. Salom
and thanks J. Sous for pointing out Ref.~[46]. This research was supported by the Deutsche
Forschungsgemeinschaft (DFG) as part of the project S4 within CRC 1119 CROSSING.
\end{acknowledgments}
| {
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Railway Recruitment Boards invite ONLINE applications for the post of Optometrist in Railway Recruitment Boards.
Minimum Educational Qualifications : B.Sc in Optometry or Diploma in Ophthalmic Technician (the course should be of 3 to 4 years duration). The candidate should have Registration with the concerned Council / Licensing body. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,505 |
{"url":"http:\/\/ncatlab.org\/nlab\/show\/K3+surface","text":"complex geometry\n\n# Contents\n\n## Definition\n\nA K3 surface is a Calabi-Yau variety of dimension $2$ (a Calabi-Yau algebraic surface). This means that the canonical bundle $\\omega_X=\\wedge^2\\Omega_X\\simeq \\mathcal{O}_X$ is trivial and $H^1(X, \\mathcal{O}_X)=0$.\n\n## Examples\n\n\u2022 A cyclic cover $\\mathbb{P}^2$ branched over a curve of degree $6$\n\n\u2022 A nonsingular degree $4$ hypersurface in $\\mathbb{P}^3$.\n\n## Properties\n\n### Moduli of higher line bundles and deformation theory\n\nIn positive characteristic $p$:\n\nThe N\u00e9ron-Severi group of a K3 is a free abelian group\n\nThe formal Brauer group is\n\n\u2022 either the formal additive group, in which case it has height $h = \\infty$, by definition;\n\n\u2022 or its height is $1 \\leq h \\leq 10$, and every value may occur\n\n(Artin 74), see also (Artin-Mazur 77, p. 5 (of 46))\n\nmoduli spaces of line n-bundles with connection on $n$-dimensional $X$\n\n$n$Calabi-Cau n-foldline n-bundlemoduli of line n-bundlesmoduli of flat\/degree-0 n-bundlesArtin-Mazur formal group of deformation moduli of line n-bundlescomplex oriented cohomology theorymodular functor\/self-dual higher gauge theory of higher dimensional Chern-Simons theory\n$n = 0$unit in structure sheafmultiplicative group\/group of unitsformal multiplicative groupcomplex K-theory\n$n = 1$elliptic curveline bundlePicard group\/Picard schemeJacobianformal Picard groupelliptic cohomology3d Chern-Simons theory\/WZW model\n$n = 2$K3 surfaceline 2-bundleBrauer groupintermediate Jacobianformal Brauer groupK3 cohomology\n$n = 3$Calabi-Yau 3-foldline 3-bundleintermediate JacobianCY3 cohomology7d Chern-Simons theory\/M5-brane\n$n$intermediate Jacobian\n\n## References\n\nOriginal sources include\n\nDiscussion of the deformation theory of K3-surfaces (of their Picard schemes) is (see also at Artin-Mazur formal group) in\n\nRevised on July 6, 2014 07:16:00 by Urs Schreiber (192.76.8.26)","date":"2014-09-19 09:49:57","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 22, \"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\": 0.9197381138801575, \"perplexity\": 11704.64348268141}, \"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-2014-41\/segments\/1410657131238.51\/warc\/CC-MAIN-20140914011211-00065-ip-10-196-40-205.us-west-1.compute.internal.warc.gz\"}"} | null | null |
Белого́рская — многозначный топоним:
Белогорская волость — волость в составе Берёзовского уезда, расположенная возле устья Иртыша, существовавшая с XVII века.
Белогорская улица — во многих городах России:
1-я Белогорская улица и 2-я Белогорская улица (Москва)
Белогорская улица (Екатеринбург)
Белогорская посадка — лесопосадка в Пензе.
См. также
Белогорский
Белогорское
Белогорск
Белогорье | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,077 |
#ifndef __GENERIC_IMAGING_IMAGEREADER__
#define __GENERIC_IMAGING_IMAGEREADER__
//==============================================================================
// EXTERNAL DECLARATIONS
//==============================================================================
#include <vector>
#include <string>
#include <map>
#include <ITL/Image/Image.h>
#ifdef WIN32
#ifdef GENERIC_EXPORTS
#define GENERIC_DLL_EXPORT __declspec(dllexport)
#else
#define GENERIC_DLL_EXPORT __declspec(dllimport)
#endif
#else // !defined(WIN32)
#define GENERIC_DLL_EXPORT
#endif // ] WIN32
//==============================================================================
// FORWARD DECLARATIONS
//==============================================================================
#ifdef GENERIC_NAMESPACE
namespace GENERIC_NAMESPACE {
#endif
class IImageFormatReader;
//==============================================================================
// INTERFACE ImageReader
//==============================================================================
class ImageReader
{
public:
GENERIC_DLL_EXPORT static ImageReader& instance();
~ImageReader();
public:
GENERIC_DLL_EXPORT IImageFormatReader* defaultReader(const std::string &format);
GENERIC_DLL_EXPORT void getReaders(const std::string &format, std::vector<IImageFormatReader*> *outImageLoaders);
GENERIC_DLL_EXPORT bool registerFormatReader(IImageFormatReader *imageReader);
GENERIC_DLL_EXPORT ITL::Image* read(const std::string &fileName);
GENERIC_DLL_EXPORT int setDefaultReader(const std::string &fileType, const std::string &imageReaderName);
GENERIC_DLL_EXPORT int setDefaultReader(const std::string &fileType, IImageFormatReader *imageReader);
private:
template<class TIMER>
void benchmark(const std::string &file, bool setDefault, IImageFormatReader **outBestImageLoader);
ImageReader();
std::map<std::string, IImageFormatReader*> mDefaultLoader;
std::multimap<std::string, IImageFormatReader*> mLoaders;
};
#ifdef GENERIC_NAMESPACE
}
#endif
#endif // __GENERIC_IMAGING_IMAGEREADER__
| {
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The Pennsylvania living will, Part III from the Commonwealth's Advance Directive, is a legal document that is designed as an avenue for a Declarant/Principal to state, in writing, prior to an end of life scenario, their own choices for how they would like their medical team to direct their end of life wishes without any dependency on relatives to have to make the difficult decision that often must be made when a relative is no longer able to make decisions on their own when the Declarant/Principal in no longer competent to do so.
Medical Power of Attorney – Part II of the Advance Directive that only allows for a proxy to handle health care decisions on the patient's best interest.
Step 1 – Declarant/Principal End of Life Decisions -This section addresses the decisions made by the Declarant while they are still of sound mind to decide for themselves how they would like their life to end. Begin by reading the selections provided.
If the Declarant/Principal, even though there will be no chance of recovery, would like to have tube fed nutrition, initial the line stating so. If no tune feeding is desired, initial the line preceding that option.
Step 2 – Healthcare Agent – If the Declarant/Principal has chosen a Healthcare Agent, they may select whether they wish for the agent to follow their initial instructions or they may select to have the healthcare agent only use the document as guidance as to how they should proceed with any further decision making with regard to the Declarant's health care.
Step 4 – Organ Donation – This area will address choices with regard to organ donation.
Step 6 – Notarization – Although notarization is optional in the state of Pennsylvania and not necessary, if the Declarant/Principal is in need of this document while in another state for any reason, the laws in other states may be more likely to honor the document with notarization. | {
"redpajama_set_name": "RedPajamaC4"
} | 35 |
Q: Other Script doesn't work on the same page I have a script when the button is clicked. An image will pop up..
And I want to have multiple buttons and multiple images but same div to have same animation on script.
But the problem is, the script only work on just 1 button.
And it doens't show the image.
HTML CODE
<button id="LearnMoreBtn">Learn More</button>
<div id="overlay"></div>
<div id="popup">
<div id="PopUpText">Popup contents here</div>
<button id="CloseBtn">Close</button>
</div> </div>
<div id="img2">
<button id="LearnMoreBtn">Learn More</button>
<div id="overlay"></div>
<div id="popup" img src="newfinal/images/portfolio6.jpg">
<div id="PopUpText">Popup contents here</div>
<button id="CloseBtn">Close</button>
</div> </div>
<div>
some other content that will be behind the popup
</div>
Javascript Code
window.onload = function() {
document.getElementById("LearnMoreBtn").onclick = function(){
var overlay = document.getElementById("overlay");
var popup = document.getElementById("popup");
overlay.style.display = "block";
popup.style.display = "block";
};
document.getElementById("CloseBtn").onclick = function(){
var overlay = document.getElementById("overlay");
var popup = document.getElementById("popup");
overlay.style.display = "none";
popup.style.display = "none";
}
};
Here's the demo..
http://jsfiddle.net/j4c7U/
A: You have the same id for both buttons. The id should be unique for each element.
Change LearnMoreBtn of the second button to LearnMoreBtn1 and check.
Here is the Fiddle
A: Because you use same id for all the buttons. getElementById will always return the first element with given id.
A: You can simplify by using query:
http://jsfiddle.net/j4c7U/129/
<button class="LearnMoreBtn">Learn More</button>
<button class="LearnMoreBtn">Learn More2</button>
$('.LearnMoreBtn').click(function(){
var overlay = document.getElementById("overlay");
var popup = document.getElementById("popup");
overlay.style.display = "block";
popup.style.display = "block";
});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,700 |
Q: Detox: How to pass -disableRNTestingOverride 1 parameter I'm using detox on the newest version and this commit (https://github.com/wix/Detox/commit/2507c1e4325936ed9f46c0f64571fa581c71ff5f) disabled the IS_TESTING field for our tests.
It mentions that we have to pass -disableRNTestingOverride 1 to disable this behavior, where do we have to set this?
Thank you in advance
A: You need to pass it as a launch argument to device.launchApp(), as shown here.
await device.launchApp({launchArgs: {disableRNTestingOverride: 1}});
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 483 |
{"url":"https:\/\/codeforces.com\/blog\/Frankenstein123","text":"Codeforces celebrates 10 years! We are pleased to announce the crowdfunding-campaign. Congratulate us by the link https:\/\/codeforces.com\/10years. \u00d7\n\n### Frankenstein123's blog\n\nBy\u00a0Frankenstein123, history, 6 months ago, ,\n\nSo this can be a little bit off-shore to problem solving in general, but I always wonder what to do the best contestants do before a contest. It would be very helpful if they share the pre-contest rituals which help them to calm down, hold their nerves and focus better during the contest. You can take the \"before\" to be from anything like a day before (does it matter? :p) to anything like 5 seconds before the round starts.\n\n\u2022 +22\n\nBy\u00a0Frankenstein123, history, 7 months ago, ,\n\nLet's suppose I need to calculate $a^{b^{c}}$ modulo $10^9 + 7$, with the constraints $1 \\leq a, b, c \\leq 10^{18}$. I can calculate $ans = b^c$ in $\\mathcal{O}(log(c))$, with modulo $10^9 + 6$, (probably everyone knows how) and then calculate $a^{ans}$ with modulo $10^9 + 7$.\n\nBut how does first taking $10^9 + 6$ and then $10^9 + 7$ work? Can anyone present a formal proof for this? Also are there any other methods to do this?\n\n\u2022 +25\n\nBy\u00a0Frankenstein123, 21 month(s) ago, ,\n\nSo after having searched at a number of places online, I have not yet completely understood how the different methods to fill a particular value at every place in a block of memory in C++. For the sake of clarity, I wish to know how the following methods differ.\n\n1)\n\nint arr[100] = {0};\n\n\n2)\n\nint arr[100];\nint main(){\nfill(arr, arr+100, 0);\n}\n\n\n3)\n\nint arr[100];\nint main(){\nmemset(arr, 0, sizeof(arr));\n}\n\n\nPlease describe the difference in the time complexity and performance of these operations and also what happens when they are used for a higher dimension array, like for declarations of the type int arr[100][100][100];.\n\n\u2022 0\n\nBy\u00a0Frankenstein123, history, 21 month(s) ago, ,\n\nI have a moderate experience of solving DP questions. I got informed about the hackerrank dp problem set. I found it quite challenging (I wasn't able to solve more than 10 problems all by myself). Can anyone give me remarks on the problem set, like its level of difficulty as compared to div2 problems and also general tips on solving non-standard dp problems and dp on graphs?","date":"2020-02-20 06:18:16","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\": 0.26253265142440796, \"perplexity\": 989.4078011484704}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875144637.88\/warc\/CC-MAIN-20200220035657-20200220065657-00078.warc.gz\"}"} | null | null |
{"url":"https:\/\/www.gamedev.net\/forums\/topic\/65862-speeding-up-a-sound\/","text":"\u2022 ### Popular Now\n\n\u2022 15\n\u2022 15\n\u2022 11\n\u2022 9\n\u2022 10\n\n#### Archived\n\nThis topic is now archived and is closed to further replies.\n\n# Speeding up a sound...\n\nThis topic is 5982 days old which is more than the 365 day threshold we allow for new replies. Please post a new topic.\n\n## Recommended Posts\n\nI''m trying to think of a way to speed up the rate that a sounds plays at, but NOT have it affect the pitch of the sound. Right now, I was thinking I could set up a system where I let the sound play at normal speed. Then, at set intervals of time, I calculate what position of the sound the player is supposed to be at, using the desired playrate for the calculations, and jump the player to that location. The end result is skipping, but the more times a second I jump the player, I smoother it would get, uness the sound it played so fast that it''s grossly noticable. Are they any other ways to do this, preferably ones that play the entire sound?\n\n##### Share on other sites\nPhysics.\n\nSound is a waveform; the \"speed\" of the sound is analogous to (actually the inverse of) its wavelength and its ptich is related to its amplitude (height). To modify the sound, modify the waveform. To obtain lower pitch, scale the waveform down (multiply y by a number < 1); to obtain higher speed, scale time down as well (multiply x by a number < 1).\n\nI wanna work for Microsoft!\n\n##### Share on other sites\nquote:\n\nSound is a waveform; the \"speed\" of the sound is analogous to (actually the inverse of) its wavelength and its ptich is related to its amplitude (height). To modify the sound, modify the waveform. To obtain lower pitch, scale the waveform down (multiply y by a number < 1); to obtain higher speed, scale time down as well (multiply x by a number < 1).\n\nI thought volume is amplitude and pitch is frequency, in which case you need to resample the sound but maintain the frequency. Can''t direct sound alter the pitch, so you can compensate for any increase\/decrease in playback speed?\n\n##### Share on other sites\nquote:\nOriginal post by invective\nI thought volume is amplitude and pitch is frequency, in which case you need to resample the sound but maintain the frequency.\n\nAck! I can''t believe I wrote that! You''re absolutely right\n\n\/me scuttles off in utter shame...\n\nI wanna work for Microsoft!\n\n##### Share on other sites\nquote:\nCan''t direct sound alter the pitch, so you can compensate for any increase\/decrease in playback speed?\n\nThat''s the catch. There isn''t any functions to change the playback speed, only the pitch! There are several effects available, but these are your standard distortion, echo, flanger, gargle, etc., no playback rate changing. There is a function to change the play offset, so my original plan would still work. Other than what I see in MSDN, I have no idea how to change playback rate if any function of the sort exists.\n\nYou mentioned resampling the sound but maintaining frequency... any way of doing it in practice? Don''t worry about the math, I can take anything\n\n##### Share on other sites\nCan be done, although you are going to struggle for real-time. Zipster, you were nearly there as well.\n\nEssentially, imagine a waveform a sampled at 100Hz - we have 100 discrete sample points per second, and playing this audio back will take exactly one second. Now, we change the playback rate to 200Hz. The sample now plays back at twice the pitch, and only takes half a second. This is obviously not what we want. Now, we will produce a resampled waveform b , by interpolating the amplitude at each sample point. Every OTHER sample in b will the same as every sample in a , whilst the remaining 'gaps' between these will be filled by using an interpolative scheme.\n\nImagine a simple linear interpolation scheme ( with interpolated values denoted by a * )\na[0] = 0a[1] = 10a[2] = 50a[3] = 20Resampled, we have :b[0] = 0b[1] = 5 *b[2] = 10b[3] = 30 *b[4] = 50b[5] = 35 *b[6] = 20\n\nLinear interpolation is fast, easy but produces generally poor results ( poor in the eyes, sorry, ears, of audio people - I doubt most casual users would notice. ) More exotic interpolation schemes can be used if you are willing to trade simplicity and speed for clarity - lagrange interpolation, cubic interpolation, beizer interpolation or cosine interpolation all produce superior results, whilst the best signal can be produced using bandlimited interpolation.\n\nAnd I apologise to any DSPers for this grossly simplified and lacklustre resampling 101 .\n\nEdited by - Colin Barry on November 5, 2001 4:23:30 AM\n\n##### Share on other sites\nYeah, real-time isn''t my friend in that situation\n\nNow I was glancing at GoldWave to see how it did its time warp, and it appears that there were two options: \"Similarity\" and \"FFT\". Now I don''t know if Similarity is a common term, but I''ve sure heard of FFT before. Any insight on these? What Colin told me is what I had in mind, only I wasn''t planning on interpolating the values inbetween, just jumping.\n\n##### Share on other sites\nNever heard of similarity before; I guess it is some technique developed by the Goldwave Author.\n\nFFT, on the other hand, stands for Fast Fourier Transform. The FFT is a way of decomposing a signal into an array of sinuosoids of a discreet amplitude, frequency and phase. From here, you can do all kinds of clever analysis stuff, or pitch-shift by transposing each individual sine component. Very nice.\n\nA good reference for FFT is Stephen Sprengers DSP Dimension, whilst if it is FFT implementation you want you should check out the Fastest Fourier Transform In The West. Yee Haw!\n\nEdited by - Colin Barry on November 6, 2001 4:07:55 AM\n\n##### Share on other sites\nDiffrent ideas:\n\nAnother way of doing it is splitting the sample up if it is drums, vocals, etc.\n\nAlso, you can loop the middle section of a note (like soundfonts, mods, etc), if this is approprite in your situation.\n\nANDREW RUSSELL STUDIOS\nLooking for my webpage? Funny that... Me too!\nResist nes8bit :: Bow Down to Linux Communisum","date":"2018-03-24 10:19:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.49849170446395874, \"perplexity\": 2417.9191263737903}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-13\/segments\/1521257650188.31\/warc\/CC-MAIN-20180324093251-20180324113251-00541.warc.gz\"}"} | null | null |
{"url":"http:\/\/www.kinberg.net\/wordpress\/stellan\/big-bang\/","text":"# Introduction\n\nThis post is an update of\u00a0a post published 17\/9 2017.\n\nI personally believe there is a God but not necessarily a God that was before and started Big Bang. Big Bang may have happened several times and several Universes may exist. To integrate the Big Bang and string theory in a belief of a God,\u00a0 I have to look at what holy books say about the Creation of the World.\n\n# Index\n\n## \u201cBig Bang\u201d science\n\nThis is a nice introduction to Big Bang made\u00a0 by www.\u00a0Minutephysics.com\n\nphysicians fully accepts the idea of a primeval atom calling it Singularity as Astronomer Penxlas and Wilson (2) found the radiation from the very first state of the Universe, called today\u00a0Big Bang CMB-radiation. read more inthis slide at\u00a0www.slideshare.net\n\nRadiation image built with data from a satellite.\nPhoto source:\u00a0http:\/\/faculty.washington.edu\n\n### The building blocks of the Universe\n\nDavid Tong, Professor of Theoretical Physics, call these \u201cThe real building blocks of the Universe\u201d\n\n\u201cMost of our Universe is dark. The secret of dark energy and matter is unknown.\u201d (Michio Kaku)\n\n### Look at muons with a cloud chamber\n\nThe positron(an antimatter electron)and the muon\u00a0 (the major component of cosmic rays) were discovered by\u00a0Carl Anderson with a cloud chamber 1932.\u00a0 read more in\u00a0www.orau.org\u00a0.\nYou can build your own cloud chamber with ice and alcohol. Look at this video:\u2019\n\nMichio Kaku talks about the particle zoo in his video minute 29:05\n\n### Big Bang is when two Universes split into two Universes\n\nOur Muliverse is 11 dimensional. Michio Kaku tells about this from minute 33:40 in this youtube:\n\n.\n\n## Quran verses:\n\n\u2022 The Quran says \u201cHave not those who disbelieve known that the\u00a0heavens and the earth were joined together as one united piece, then We parted them? And We have made from water every living thing. Will they not then believe?\u00a0\u201d (21:30)\n\u2013\u00a0\u00a0\u201cUnited piece\u201d (Singularity is what Scientists say today), \u201cparted them\u201d (=Universe expansion)\n\u2013 \u201cWater\u201d Science teaches that all were built with hydrogen, but who could understand what hydrogen atoms was year 500 ?\n\u2022 \u201cThen He rose over (Istawa) towards the heaven when it was smoke, and said to it and to the earth: \u201cCome both of you willingly or unwillingly.\u201d They both said: \u201cWe come willingly.\u201d (41:11)\nComment: \u201cSmoke\u201d The Earth would not have existed without a Supernova.\n\u2022 \u201cAnd He it is Who has created the night and the day, and the sun and the moon, each in an orbit floating.\u201d (21:33)\nComment: pretty accurate about our planetary system.\n\u2022 \u201cThe heavens, We have built them with power. And verily, We are expanding it\u201d (51:47).\nComment: Scientists confirm there was infinite energy and a unified Force at the beginning and then we had the expansion of the Universe.\n\n## The Bible\n\nI let this site speak for itself\n\nWith the sources given you have to find yourself the verses.\n\nuse e.g.\u00a0http:\/\/bible.catholic.net\/ to find them.\n\n## Zoroastrianism\n\nWikipedia user \u201cBeda67\u201d wrote in\n\nhttps:\/\/en.wikipedia.org\/wiki\/Zoroastrianism#Creation_of_the_universe\n\n\u201cAccording to the Zoroastrian\u00a0story of creationAhura Mazda\u00a0existed in light and goodness above, while\u00a0Angra Mainyu\u00a0existed in darkness and ignorance below. They have existed independently of each other for all time, and manifest contrary substances. Ahura Mazda first created seven abstract heavenly beings called\u00a0Amesha Spentas, who support him and represent beneficent aspects, along with numerous\u00a0yazads, lesser beings worthy of worship. He then created the universe itself in order to ensnare evil. Ahura Mazda created the floating, egg-shaped universe in two parts: first the spiritual (menog) and 3,000 years later, the physical (getig). Ahura Mazda then created\u00a0Gayomard, the archetypical perfect man, and the first bull.\n\nWhile Ahura Mazda created the universe and humankind, Angra Mainyu, whose instinct is to destroy, miscreated demons, evil\u00a0yazads, and noxious creatures (khrafstar) such as snakes, ants, and flies. Angra Mainyu created an opposite, evil being for each good being, except for humans, which he found he could not match. Angra Mainyu invaded the universe through the base of the sky, inflicting\u00a0Gayomard\u00a0and the bull with suffering and death. However, the evil forces were trapped in the universe and could not retreat. The dying primordial man and bull emitted seeds. From the bull\u2019s seed grew all beneficial plants and animals of the world, and from the man\u2019s seed grew a plant whose leaves became the first human couple. Humans thus struggle in a two-fold universe trapped with evil. The evils of this physical world are not products of an inherent weakness, but are the fault of Angra Mainyu\u2019s assault on creation. This assault turned the perfectly flat, peaceful, and ever day-lit world into a mountainous, violent place that is half night.\u201d\n\nAs a source he used\u00a0Cavendish, Richard; Ling, Trevor Oswald (1980),\u00a0Mythology: an Illustrated Encyclopedia, Rizzoli, pp.\u00a040\u201345,\u00a0ISBN\u00a00847802868\n\nCavendish, Richard; Ling, Trevor Oswald (1980),\u00a0Mythology: an Illustrated Encyclopedia, Rizzoli, pp.\u00a040\u201345,\u00a0ISBN\u00a00847802868\n\nInteresting verses in songs or Zarathushtra\n\n\u201cI pray to Thee, O Mazda, with uplifted hands, and to thy Holy Spirit, first of all and hope that through truths and righteousness I would enjoy the light of wisdom and a clean conscience, thus bringing solace to the Soul of (Mother Earth) Creation.\u201d (Yasna 28:1)\n\n## Debates\n\nI find the name of Creation in 11 more songs. The debate continues\u2026\n\nGet the Big Bang Theory in harmony with your belief in God, with a http:\/\/www.kinberg.net\/2017\/09\/18\/religious-pluralism\/pluralistic approach. Read more here.\n\n## A pluralist agnostic seeker\n\nInsert math as\n$${}$$","date":"2020-10-20 18:02:47","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.25338226556777954, \"perplexity\": 6661.962911271976}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-45\/segments\/1603107874026.22\/warc\/CC-MAIN-20201020162922-20201020192922-00102.warc.gz\"}"} | null | null |
Doljchim Craiova a fost o companie din România, specializată în producerea de îngrășăminte chimice minerale, de metanol și de intermediari.
Combinatul chimic a fost înființat în 1961, având ca obiect de activitate producerea îngrășămintelor chimice și a unor produse de sinteză organică prin chimizarea completă a gazelor naturale.
A fost cumpărată de către compania Petrom în anul 1998.
În anul 2010 a fost închisă și toate clădirile au fost demolate pentru decontaminarea zonei.
Adunarea Generală a Acționarilor OMV, a decis să închidă combinatul odată cu deschiderea centralei electrice de la Brazi, bazată pe gazul intern, găsind astfel soluția gazului românesc, combinatul a fost închis după modernizarea fabricii de metanol, modernizarea și înnoirea parcului de vagoane cisternă.
Totul a fost o afacere a conducerii Doljchim în parteneriat cu sindicatul liber Doljchim și Bordul OMV, pe fondul acceptării disponibilizărilor masive de personal, au fost vândute instalații și utilaje ca piese de schimb în regim de fier vechi fără plata de TVA, asta în timp ce clienții erau refuzați și contractele de vânzare reziliate. Ioan Niculae a declarat că OMV a vândut combinatul Doljchim la fier vechi și a tăiat tot, ca să nu mai poată folosi românii vreun utilaj.
Societatea Națională Petrom SA, cea mai mare companie românească a fost cumpărată în 1998, pe timpul guvernării Convenției Democratice, de către OMV, o companie de stat austriacă, cu drept de exploatare și explorare a zăcământului natural al României plătind cele mai mici redevențe din istoria Europei.
În anul 2006 a fost înregistra cel mai mare profit al combinatului de la înființare și în decembrie 2008 a fost comunicată decizia de închidere a Doljchimului.
În cadrul companiei OMV Petrom face parte și rafinăria Arpechim care este și ea în proces de dezafectare, ecologizare, dar Arpechimul producea materia prima a combinatului Oltchim Rm. Vâlcea și era legat prin rețele de conducte de acesta, astfel odată cu închiderea Combinatului Arpechim a fost hotărâtă soarta celor două combinate, considerate perla industriei petrochimice românești.
Cele trei combinate au fost realizate în timpul dictaturii comuniste cu mari sacrificii ale poporului român.
Număr de angajați: 1600 angajați
2012: 66
2010: 1.000
2009: 1.100
Note
Companii din industria chimică din România
Producători de îngrășăminte chimice din România
Companii din Craiova
Petrom | {
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{"url":"http:\/\/experiment-ufa.ru\/Prime-factorization-of-110","text":"# Prime factorization of 110\n\nIf it's not what You are looking for type in the field below your own integer, and You will get the solution.\n\nPrime factorization of 110:\n\nBy prime factorization of 110 we follow 5 simple steps:\n1. We write number 110 above a 2-column table\n2. We divide 110 by the smallest possible prime factor\n3. We write down on the left side of the table the prime factor and next number to factorize on the ride side\n4. We continue to factor in this fashion (we deal with odd numbers by trying small prime factors)\n5. We continue until we reach 1 on the ride side of the table\n\n 110 prime factors number to factorize 2 55 5 11 11 1\n\nPrime factorization of 110 = 1\u00d72\u00d75\u00d711= $1 \u00d7 2 \u00d7 5 \u00d7 11$\n\n## Related pages\n\nsin3x sinx sin2x50 000 pounds in dollars100-55sinxsinywhat does lnx equal50 000 pounds to dollarsroman numerals 19851965 in roman numeralssolve derivative calculatorroman numerals x1hcf of 72627.2prime factorization 252graph x 2y 10x 3-8 factoredpercent to a decimal calculatorwrite the prime factorization of 50simplify square root of 245factorise x squared xdecimals fractionsln3 ln2greatest common factor of a polynomial calculatorthe prime factorization of 114how do you write 20 as a decimalderivatives sin coslog3 3xsimplify 3x 2adding fractions calculator with mixed numberswhat is the prime factorization of 75lcm of 92020 in roman numerals8100-1what is the prime factorization of 212derivative of ln1xy equation solverx mc squaredsin3x formulawhat is the prime factorization for 54least to greatest decimals calculator4.3.2.1tan 4x-sec 4xprime factorisation of 47solve 2x-3y 12roman numerals 2003prime factorization of 759factors of49147 prime or compositex 2 3yequation solving calculator300-229700-270how to solve ln x 58y 7ycosx cos-xderivative of tan 3xroman numerals 1970ln45gcf of 32 and 72differentiate sin 2 3xsinx cosx cos2xwhat is the prime factorization of 108step by step simultaneous equations20 off of 19.99sin 2x cos 2x 16i mathtan 4x1956 in roman numerals250 in roman numeralsderivative of tan sinx3x 2 9xcos 2piwhat is sin45minimum common multiple calculator","date":"2017-09-22 04:33:41","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\": 0.3769795000553131, \"perplexity\": 9708.373197551842}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-39\/segments\/1505818688208.1\/warc\/CC-MAIN-20170922041015-20170922061015-00047.warc.gz\"}"} | null | null |
\section{Introduction}
The relationship between the coset construction
\cite{earlycoset,GKO,FSN=0} $G/H$ and the gauged Wess--Zumino--Witten
(WZW) model is well-known. The judicious use of the
Polyakov--Wiegmann identity and the careful consideration on the path
integral jacobians allows one to write the gauged WZW model in terms
of the original (ungauged) WZW model on $G$, an auxiliary WZW model on
$H$, and some anticommuting $H$-ghosts \cite{GWZW,FSN=0}. Further
analysis of this gauge theory reveals that in the BRST cohomology one
has a realisation of the coset Virasoro algebra, and when $G$ is
compact, restricting to a suitable subclass of representations of the
affine Lie algebra, one recovers
\cite{GK} precisely the unitary series of Virasoro representations
expected from the construction in \cite{GKO}.
The supersymmetric situation is somewhat different. The $N{=}1$ and
$N{=}2$ coset constructions were studied originally in
\cite{KazamaSuzuki,Schweigert}, where only reductive---that is,
semisimple $\times$ abelian---Lie algebras were considered. However a
satisfactory derivation of these coset constructions from a
supersymmetric gauged WZW model was lacking.\footnote{Superconformal
field theories were obtained in \cite{Schnitzer} (see also
\cite{Nakatsu}) from gauged supersymmetric WZW models, but these
theories do not coincide with the $N{=}1$ coset constructions---they
have different central charges.} Correct path integral constructions
were written down in \cite{Witten}, derived in \cite{Nojiri} (for a
particular model) and in \cite{Tseytlin} (in general). However a
conformal field theoretical derivation of the supersymmetric coset
constructions from a gauged WZW model, in the style of \cite{GWZW},
does not exist. The point of this letter is to remedy this situation.
This letter is organised as follows. In Section 2 we briefly set the
notation and review the conformal field theory describing the $N{=}1$
supersymmetric WZW model and the gauging of a diagonal subgroup. We
then describe the superconformal field theories described by these
models: the $N{=}1$ (affine) Sugawara construction in Section 3 and
the $N{=}1$ coset construction in Section 4. In Section 5 we prove
that the natural $N{=}1$ Virasoro algebra of the gauged $N{=}1$ WZW
model is both BRST invariant and BRST cohomologous to the one coming
from the coset construction. In Section 6 we discuss the $N{=}2$
cosets and we prove that when these cosets exist, the extra $N{=}2$
generators are also BRST invariant. In Section 7 we prove that the
BRST cohomology of the gauged supersymmetric WZW model reduces to that
of a gauged bosonic WZW model coupled to the coset fermions. In
particular, when discussing the topological gauging $G/G$, the two
theories---supersymmetric and bosonic---agree.
Finally a note on the notation. Because this work arose as part of
the programme initiated in \cite{FSN=0} and continued in \cite{FSN=1}
on WZW models based on Lie groups which are not necessarily compact,
we work under the assumption that Lie algebras are self-dual, but not
necessarily reductive. For nonreductive Lie algebras the notion of
{\em level\/} does not make sense and is replaced by that of an
invariant metric, which need not be proportional to any canonical
metric (as is the case in simple Lie algebras, for example).
Similarly, the dual Coxeter number is now replaced (roughly) by the
Killing form.
\section{The gauged supersymmetric WZW model}
The $N{=}1$ WZW model is defined classically by the action
\begin{eqnarray*}
(i\,\alpha)^{-1} I^S_\Omega[G] ={} &&
\int_{\Sigma_S}\pair<G^{-1}DG,G^{-1}{\bar D}G>\\
&& {} + \int_{B_S}\pair<G^{-1}\d_t G, \left[ G^{-1}DG,G^{-1}{\bar
D}G\right]>~,
\end{eqnarray*}
where $\Sigma_S$ is a super-Riemann surface, $B_S$ a supermanifold
with boundary $\Sigma_S$ and $G$ is a superfield whose
$\theta$-independent component takes values in a Lie group $\mathcal
G$ which we assume posses a bi-invariant metric
$\Omega$.\footnote{$\Omega$ is written as $\pair<-,->$ above to avoid
cluttering the notation.} Expanding into components and solving for
the auxiliary fields, the above model has a particularly simple
description in terms of a bosonic $\mathcal G$-valued field $g$ and
Majorana--Weyl fermions $\psi$ and $\bar\psi$ with values in the Lie
algebra $\gg$:
\begin{equation}
I^S_\Omega[G] = I_\Omega[g] + I_\Omega[\psi,\bar\psi;g]~,
\end{equation}
where $I_\Omega[g]$ is the bosonic WZW action defined in \cite{FSN=0},
and $I_\Omega[\psi,\bar\psi;g]$ is the action for Majorana--Weyl
fermions axially coupled to the bosonic currents.
The quantum theory is described by the path integral
\begin{equation}
Z = \int [dg][d\psi][d\bar\psi] e^{-I_\Omega[g] -
I_\Omega[\psi,\bar\psi;g]}~,
\end{equation}
or equivalently, by an $N{=}1$ affine Lie algebra
\cite{dVKPR,abdallas,Fuchs,KacTodorov} with data $(\gg, \Omega)$,
where we let $\Omega$ also denote the invariant metric on $\gg$. Such
Lie algebras are known as self-dual. (Of course, the theory has both a
holomorphic and an antiholomorphic sector, but as usual we concentrate
on the holomorphic one.) Fix once and for all a basis $\langle
X_a\rangle$ for $\gg$, relative to which $\Omega$ has components
$\Omega_{ab}$ and such that the structure constants are ${f_{ab}}^c$.
This $N{=}1$ affine Lie algebra is generated by currents $I_a(z)$ and
fermions $\psi_a(z)$, obeying the following OPEs:
\begin{eqnarray}
I_a(z) I_b(w) &=& \ope[2][\Omega_{ab}] + \ope[1][{f_{ab}}^c I_c(w)] +
\text{reg}\nonumber\\
I_a(z) \psi_b(w) &=& \ope[1][{f_{ab}}^c \psi_c(w)] + \text{reg}\nonumber\\
\psi_a(z) \psi_b(w) &=& \ope[1][\Omega_{ab}] +
\text{reg}~.\label{eq:KacTodorov}
\end{eqnarray}
Because $\Omega$ is nondegenerate, we can decouple the fermions from
the affine currents. Indeed, in terms of the modified currents:
\begin{equation}
J_a(z) \equiv I_a(z) - \half \Omega^{bd} {f_{ab}}^c
(\psi_c\psi_d)(z)\label{eq:modJ}
\end{equation}
the OPEs (\ref{eq:KacTodorov}) become
\begin{eqnarray}
J_a(z) J_b(w) &=& \ope[2][\Omega_{ab}-\half \kappa_{ab}] +
\ope[1][{f_{ab}}^c J_c(w)] + \text{reg}\nonumber\\
J_a(z) \psi_b(w) &=& \text{reg}\nonumber\\
\psi_a(z) \psi_b(w) &=& \ope[1][\Omega_{ab}] +
\text{reg}~,\label{eq:Affine+Fermions}
\end{eqnarray}
where $\kappa_{ab} = {f_{ac}}^d {f_{bd}}^c$ is the Killing form on
$\gg$. Since $\gg$ is not necessarily semisimple, $\kappa$ need not
be nondegenerate. Nevertheless, as shown in \cite{FSN=0,FSSD},
$\Omega - \half\kappa$ will generically be nondegenerate.
Now let ${\ensuremath{\mathfrak h}}\subset\gg$ be a Lie subalgebra such that the restriction
$\Omega|_{{\ensuremath{\mathfrak h}}}$ of $\Omega$ to ${\ensuremath{\mathfrak h}}$ remains nondegenerate. Assume
that we have chosen the basis for $\gg$ in such a way that a sub-basis
$\langle X_i\rangle$ is a basis for ${\ensuremath{\mathfrak h}}$. The condition on ${\ensuremath{\mathfrak h}}$
means that $\Omega_{ij}$ is an invariant metric on ${\ensuremath{\mathfrak h}}$. (In
particular, ${\ensuremath{\mathfrak h}}$ is also self-dual.) In \cite{FSN=1} it is proven
that this is a necessary and sufficient condition for the existence of
the $N{=}1$ coset construction $\gg/{\ensuremath{\mathfrak h}}$. In this letter we will see
that the diagonally gauged supersymmetric WZW model reproduces this
coset construction.
As shown in \cite{Nojiri,Tseytlin,FSN=1} gauging {\em both\/} the
fermionic and bosonic symmetries corresponding to the diagonal
subalgebra ${\ensuremath{\mathfrak h}}$ gives rise to the following path integral:
\begin{eqnarray*}
Z ={}&& \int [dg][d\psi][d\bar\psi][d\tilde
h][d\tilde\psi][d\bar{\tilde\psi}][db][dc][d\beta][d\gamma][d\bar
b][d\bar c][d\bar\beta][d\bar\gamma]\\
&&{} \times e^{-I_\Omega[g] - I_\Omega[\psi,\bar\psi;g]}\,
e^{I_{\Omega}[\tilde h] +
I_{\Omega}[\tilde\psi,\bar{\tilde\psi};\tilde h]}\,
e^{-I_{\mathrm{gh}}[b,c,\bar b, \bar c, \beta, \gamma, \bar\beta,
\bar\gamma]}~,
\end{eqnarray*}
where $(b,c)$ and $(\bar b, \bar c)$ are the ghosts familiar from the
nonsupersymmetric gauged WZW model, and $(\beta,\gamma)$ and
$(\bar\beta, \bar\gamma)$ are the (bosonic) ghosts corresponding to
the gauged fermionic symmetry. Notice that unlike in the
nonsupersymmetric case, the metric of the `auxiliary' ${\ensuremath{\mathfrak h}}$-sector
does not get shifted. This is because the jacobian responsible for
the shift receives now an equal but opposite contribution from the new
fermionic sector.
We may also describe this quantum theory as a superconformal field
theory consisting of three sectors coupled by a constraint, where
again we omit the antiholomorphic sector:
\begin{itemize}
\item[$\bullet$] the original $N{=}1$ affine Lie algebra with data
$(\gg,\Omega)$;
\item[$\bullet$] a second $N{=}1$ affine Lie algebra with data
$({\ensuremath{\mathfrak h}},-\Omega|_{\ensuremath{\mathfrak h}})$ generated by $\tilde I_i(z)$ and
$\tilde\psi_i(z)$ subject to the OPEs:
\begin{eqnarray}
\tilde I_i(z) \tilde I_j(w) &=& \ope[2][-\Omega_{ij}] +
\ope[1][{f_{ij}}^k \tilde I_k(w)] +
\text{reg}\nonumber\\
\tilde I_i(z) \tilde\psi_j(w) &=& \ope[1][{f_{ij}}^k \tilde\psi_k(w)]
+ \text{reg}\nonumber\\
\tilde\psi_i(z) \tilde\psi_j(w) &=& \ope[1][-\Omega_{ij}] +
\text{reg}~;\label{eq:gaugedHsector}
\end{eqnarray}
\item[$\bullet$] a supersymmetric ghost system with fermionic
$(b_i,c^i)$ and bosonic $(\beta_i,\gamma^i)$ generators subject to the
OPEs:
\begin{displaymath}
b_i(z) c^j(w) = \ope[1][\delta_i^j] + \text{reg}\qquad\hbox{and}\qquad
\beta_i(z) \gamma^j(w) = \ope[1][\delta_i^j] + \text{reg}~.
\end{displaymath}
\end{itemize}
The ghost sector also carries a realisation of the $N{=}1$ affine Lie
algebra with data $({\ensuremath{\mathfrak h}},0)$. Indeed, we can define
\begin{displaymath}
I_i^{\mathrm{gh}}(z) \equiv {f_{ij}}^k b_k c^j - {f_{ij}}^k \beta_k
\gamma^j\qquad\hbox{and}\qquad \psi_i^{\mathrm{gh}}(z) \equiv
{f_{ij}}^k \beta_k c^j~,
\end{displaymath}
which obey the OPEs
\begin{eqnarray}
I_i^{\mathrm{gh}}(z) I_j^{\mathrm{gh}}(w) &=& \ope[1][{f_{ij}}^k
I_k^{\mathrm{gh}}(w)] + \text{reg}\nonumber\\
I_i^{\mathrm{gh}}(z) \psi_j^{\mathrm{gh}}(w) &=& \ope[1][{f_{ij}}^k
\psi_k^{\mathrm{gh}}(w)] + \text{reg}\nonumber\\
\psi_i(z) \psi_j(w) &=& \text{reg}~.\label{eq:ghostsector}
\end{eqnarray}
Notice that since the metric is zero, the ``fermions''
$\psi_k^{\mathrm{gh}}$ cannot be decoupled; yet this will not
represent any problem.
Tensoring the three sectors we obtain a realisation of the $N{=}1$
affine Lie algebra with data $({\ensuremath{\mathfrak h}},0)$ generated by the {\em total\/}
fields:
\begin{displaymath}
I_i^{\mathrm{tot}}(z) \equiv I_i(z) + \tilde I_i(z) +
I_i^{\mathrm{gh}}(z)\quad\hbox{and}\quad\psi_i^{\mathrm{tot}}(z)
\equiv \psi_i(z) + \tilde \psi_i(z) + \psi_i^{\mathrm{gh}}(z)~.
\end{displaymath}
The absence of central terms in the algebra generated by
$I_i^{\mathrm{tot}}(z)$ and $\psi_i^{\mathrm{tot}}(z)$ just reiterates
the fact that we have gauged a non-anomalous symmetry and implies that
the BRST charge\footnote{We use the $[-,-]_n$ notation for operator
product expansions. The properties of these brackets are summarised
for example in \cite{GetzlerMT,Kris}.} $d=[j_{\hbox{\tiny BRST}},-]_1$,
defined by
\begin{equation}
j_{\hbox{\tiny BRST}} = (I_i+\tilde I_i) c^i - (\psi_i+\tilde\psi_i)
\gamma^i - {f_{ij}}^k \beta_k c^i \gamma^j - \half {f_{ij}}^k b_k c^i c^j
\end{equation}
squares to zero. In fact, the first order pole $[j_{\hbox{\tiny
BRST}},j_{\hbox{\tiny BRST}}]_1$ actually vanishes. It will be
convenient to introduce the linear combinations:~ $\psi_i^\pm \equiv
\psi_i \pm \tilde \psi_i$.
\section{The $N{=}1$ Virasoro algebras}
Associated with any $N{=}1$ affine Lie algebra based on a self-dual
Lie algebra, there is a supersymmetric Sugawara construction which
yields an $N{=}1$ Virasoro algebra. For the algebra defined by
(\ref{eq:KacTodorov}) we define:
\begin{displaymath}
G_\gg \equiv \Omega^{ab} J_a \psi_b - \fr{1}{6} f^{abc}
\psi_a\psi_b\psi_c\qquad\hbox{and}\qquad
T_{\gg} \equiv \half\Omega^{ab} J_aJ_b + \half\Omega^{ab}
\d\psi_a\psi_b~,
\end{displaymath}
where $f^{abc} \equiv \Omega^{ad} \Omega^{be} {f_{de}}^c$. Notice
that we have used the decoupled currents $J_a$ defined by equation
(\ref{eq:modJ}). $G_{\gg}$ and $T_{\gg}$ satisfy an $N{=}1$ Virasoro
algebra with central charge $c_{\gg} = \fr{3}{2}\dim\gg - \half
\Omega^{ab}\kappa^{\gg}_{ab}$, where we now let $\kappa^{\gg}$ denote
the Killing form of $\gg$.
Similarly for the $N{=}1$ affine Lie algebra defined by
(\ref{eq:gaugedHsector}) one defines
\begin{displaymath}
\tilde G_{\ensuremath{\mathfrak h}} \equiv -\Omega^{ij} \tilde J_i \tilde\psi_j - \fr{1}{6} f^{ijk}
\tilde\psi_i\tilde\psi_j\tilde\psi_k\quad\hbox{and}\quad
\tilde T_{{\ensuremath{\mathfrak h}}} \equiv -\half\Omega^{ij} \tilde J_i \tilde J_j -
\half\Omega^{ij} \d\tilde\psi_i\tilde\psi_j~,
\end{displaymath}
where $\tilde J_i$ are the decoupled currents in this sector which are
defined similarly to those in equation (\ref{eq:modJ}) but with metric
$-\Omega_{ij}$. The $N{=}1$ Virasoro algebra satisfied by $\tilde
G_{\ensuremath{\mathfrak h}}$ and $\tilde T_{{\ensuremath{\mathfrak h}}}$ has central charge $\tilde c_{{\ensuremath{\mathfrak h}}} =
\fr{3}{2}\dim{\ensuremath{\mathfrak h}} + \half \Omega^{ij}\kappa^{{\ensuremath{\mathfrak h}}}_{ij}$.
Finally we define the $N{=}1$ Virasoro algebra for the ghost system.
Since the fermionic ghosts have weights $(1,0)$ and the bosonic ghosts
have weights $(\half,\half)$, we write
\begin{displaymath}
G_{\mathrm{gh}} \equiv b_i\gamma^i + \beta_i\d c^i\qquad\hbox{and}\qquad
T_{\mathrm{gh}} \equiv -b_i \d c^i + \half \left( \beta_i\d\gamma^i -
\d\beta_i\gamma^i\right)~.
\end{displaymath}
The $N{=}1$ Virasoro algebra satisfied by $G_{\mathrm{gh}}$ and
$T_{\mathrm{gh}}$ has the expected central charge $c_{\mathrm{gh}} =
-3\dim{\ensuremath{\mathfrak h}}$.
Tensoring all three $N{=}1$ Virasoro algebras together we find a
realisation with total central charge
\begin{equation}
c_{\mathrm{tot}} \equiv c_{\gg} + \tilde c_{{\ensuremath{\mathfrak h}}} + c_{\mathrm{gh}} =
\fr{3}{2}\left(\dim\gg - \dim{\ensuremath{\mathfrak h}}\right) -
\half\left(\Omega^{ab}\kappa^{\gg}_{ab} -
\Omega^{ij}\kappa^{{\ensuremath{\mathfrak h}}}_{ij}\right)~,\label{eq:ctotal}
\end{equation}
which as we will see presently is the central charge of the $N{=}1$
coset construction.
\section{The $N{=}1$ coset construction}
Given a self-dual Lie algebra $(\gg,\Omega)$ and a Lie subalgebra
${\ensuremath{\mathfrak h}}\subset\gg$ such that the restriction $\Omega|_{{\ensuremath{\mathfrak h}}}$ is
nondegenerate, we can define an $N{=}1$ coset construction. This is
done as follows \cite{FSN=1}. Define currents $\hat J_i \equiv I_i -
\half \Omega^{j\ell} {f_{ij}}^k \psi_k\psi_{\ell}$. Notice that $\hat
J_i$ is not equal to the modified current defined in (\ref{eq:modJ})
(unless $\gg={\ensuremath{\mathfrak h}}$) but that nonetheless the new currents are decoupled
from the ${\ensuremath{\mathfrak h}}$-fermions:
\begin{displaymath}
\hat J_i(z) \psi_j(w) = \text{reg}~,
\end{displaymath}
and still define a realisation of an affine Lie algebra based on
${\ensuremath{\mathfrak h}}$:
\begin{displaymath}
\hat J_i(z) \hat J_j(w) = \ope[2][\Omega_{ij} - \half
\kappa^{{\ensuremath{\mathfrak h}}}_{ij}] + \ope[1][{f_{ij}}^k \hat J_k(w)] + \text{reg}~,
\end{displaymath}
where now $\kappa^{{\ensuremath{\mathfrak h}}}_{ij}$ is the Killing form on ${\ensuremath{\mathfrak h}}$, which need
not agree with the restriction to ${\ensuremath{\mathfrak h}}$ of the Killing form on $\gg$.
By assumption, the restriction of the metric $\Omega$ on $\gg$ to
${\ensuremath{\mathfrak h}}$ is nondegenerate, so we can decompose $\gg = {\ensuremath{\mathfrak h}} \oplus
{\ensuremath{\mathfrak h}}^\perp$, which, because of the invariance of the metric, is not
just a decomposition of vector space but also one of ${\ensuremath{\mathfrak h}}$-modules.
If we let $\langle X_\alpha \rangle$ denote a basis for ${\ensuremath{\mathfrak h}}^\perp$,
we can summarise this discussion by saying that $\Omega_{i\alpha} = 0$
and that ${f_{i\alpha}}^j = 0$. (Notice, however, that there is no
restriction on ${f_{\alpha\beta}}^\gamma$, hence the above
decomposition, although reductive, need not be symmetric.)
Define now the following $N{=}1$ Virasoro generators:
\begin{displaymath}
G_{\ensuremath{\mathfrak h}} \equiv \Omega^{ij} \hat J_i \psi_j - \fr{1}{6} f^{ijk}
\psi_i\psi_j\psi_k\qquad\hbox{and}\qquad
T_{{\ensuremath{\mathfrak h}}} \equiv \half\Omega^{ij} \hat J_i \hat J_j + \half\Omega^{ij}
\d\psi_i\psi_j~,
\end{displaymath}
whose central charge is given by $c_{{\ensuremath{\mathfrak h}}} = \fr{3}{2} \dim{\ensuremath{\mathfrak h}} - \half
\Omega^{ij} \kappa^{{\ensuremath{\mathfrak h}}}_{ij}$.
The $N{=}1$ coset theory is generated by
\begin{displaymath}
G_{\gg/{\ensuremath{\mathfrak h}}} \equiv G_{\gg} - G_{{\ensuremath{\mathfrak h}}}\qquad\hbox{and}\qquad
T_{\gg/{\ensuremath{\mathfrak h}}} \equiv T_{\gg} - T_{{\ensuremath{\mathfrak h}}}~.
\end{displaymath}
These fields obey an $N{=}1$ Virasoro algebra with central charge
$c_{\gg/{\ensuremath{\mathfrak h}}} \equiv c_\gg - c_{\ensuremath{\mathfrak h}}$ which agrees with
(\ref{eq:ctotal}), and, more importantly, they have regular OPEs with
$G_{\ensuremath{\mathfrak h}}$ and $T_{\ensuremath{\mathfrak h}}$.
\section{$N{=}1$ coset theory in BRST cohomology}
The equality between the central charge $c_{\gg/{\ensuremath{\mathfrak h}}}$ of the coset
construction and the central charge $c_{\mathrm{tot}}$ of the gauged
supersymmetric WZW model suggests that the two theories are actually
equivalent. In fact, we now show that the BRST cohomology of the
gauged supersymmetric WZW model admits a realisation of the coset
theory. Just like in the nonsupersymmetric case \cite{GWZW,FSN=0},
all we need to show is that the generators of the coset SCFT
$(G_{\gg/{\ensuremath{\mathfrak h}}}, T_{\gg/{\ensuremath{\mathfrak h}}})$ and of the gauged supersymmetric WZW SCFT
$(G_{\mathrm{tot}}, T_{\mathrm{tot}})$ are not just BRST-invariant but
also BRST-cohomologous, so that their differences
\begin{displaymath}
G' \equiv G_{\mathrm{tot}} - G_{\gg/{\ensuremath{\mathfrak h}}}\qquad\hbox{and}\qquad
T'\equiv T_{\mathrm{tot}} - T_{\gg/{\ensuremath{\mathfrak h}}}
\end{displaymath}
are BRST-exact. Since the BRST operator $d$ is a derivation over the
operator product and since, under the operator product, it is
$G_{\mathrm{tot}}$ and $G_{\gg/{\ensuremath{\mathfrak h}}}$ which generate the algebra, it is
enough to show our claim on these generators. Indeed, a short
calculation shows that both $G_{\mathrm{tot}}$ and $G_{\gg/{\ensuremath{\mathfrak h}}}$ are
BRST-invariant, and moreover that $G_{\mathrm{tot}} - G_{\gg/{\ensuremath{\mathfrak h}}} =
d\Theta$, with
\begin{displaymath}
\Theta = \half \Omega^{ij} b_i\psi^-_j + \half \Omega^{ij} \beta_i
(I_j - \tilde I_j) + \fr{1}{3} f^{ijk}\beta_i \left(\psi_j\psi_k +
\tilde\psi_j\tilde\psi_k - \psi_j\tilde\psi_k\right)~.
\end{displaymath}
\section{$N{=}2$ cosets}
Under certain circumstances the $N{=}1$ coset theory admits an extra
supersymmetry giving rise to an $N{=}2$ coset. For $\gg$ a reductive
Lie algebra---that is, semisimple $\times$ abelian---this is the
celebrated Kazama--Suzuki construction
\cite{KazamaSuzuki,Schweigert,GetzlerMT}. For a general self-dual Lie
algebra $(\gg,\Omega)$ the conditions for the existence of an $N{=}2$
theory extending the $N{=}1$ coset $\gg/{\ensuremath{\mathfrak h}}$ are the following
\cite{FSN=1}. Let ${\ensuremath{\mathfrak k}}\equiv {\ensuremath{\mathfrak h}}^\perp \subset \gg$ be the orthogonal
complement of ${\ensuremath{\mathfrak h}}\subset\gg$. Then ${\ensuremath{\mathfrak k}}$ must possess {\em an
${\ensuremath{\mathfrak h}}$-invariant, integrable complex structure compatible with the
restriction to ${\ensuremath{\mathfrak k}}$ of the metric $\Omega$}. Comparing with
\cite{HullWitten}, this condition supports our intuition that the CFT
defined by the $N{=}2$ coset is indeed described (at least
classically) by a $\sigma$-model on the coset space $G/H$.
Let $A: {\ensuremath{\mathfrak k}} \to {\ensuremath{\mathfrak k}}$ denote the complex structure. Relative to the
basis $\langle X_\alpha \rangle$ for ${\ensuremath{\mathfrak k}}$, $A$ has components
${A^\alpha}_\beta$. Because $A$ is compatible with the metric
$A^{\alpha\beta} = - A^{\beta\alpha}$, where $A^{\alpha\beta} =
{A^\alpha}_\gamma \Omega^{\beta\gamma}$. Define ${\ensuremath{\mathsf J}}$ and ${\ensuremath{\mathsf G}}^2$ by
\begin{eqnarray*}
2i\,{\ensuremath{\mathsf J}} &\equiv& A^{\alpha\beta} \psi_\alpha\psi_\beta -
A^{\alpha\beta} {f_{\alpha\beta}}^c I_c\\
{\ensuremath{\mathsf G}}^2 &\equiv& A^{\alpha\beta} J_\alpha\psi_\beta + \fr{1}{6}
A^{\alpha\alpha'} A^{\beta\beta'} A^{\gamma\gamma'}
f_{\alpha\beta\gamma} \psi_{\alpha'}\psi_{\beta'}\psi_{\gamma'}~.
\end{eqnarray*}
Then together with ${\ensuremath{\mathsf G}}^1\equiv G_{\gg/{\ensuremath{\mathfrak h}}}$ and ${\ensuremath{\mathsf T}} \equiv
T_{\gg/{\ensuremath{\mathfrak h}}}$, they obey an $N{=}2$ Virasoro algebra.
If the gauged supersymmetric WZW model is to describe the $N{=}1$
coset theory, any extended symmetry of the $N{=}1$ Virasoro algebra
which the coset theory admits, must be already present (maybe up
to BRST-exact terms) among the BRST-invariant fields in the WZW model.
Therefore we expect that the $N{=}2$ extension, whenever it exists,
must be BRST-invariant or, in this case, since they don't involve the
ghosts, actually gauge invariant. Since ${\ensuremath{\mathsf J}}$ and ${\ensuremath{\mathsf G}}^1$ generate
the rest of the $N{=}2$ Virasoro algebra, and ${\ensuremath{\mathsf G}}^1$ is already
BRST-invariant, all we need to show is that ${\ensuremath{\mathsf J}}$ is BRST-invariant.
But this follows trivially from the ${\ensuremath{\mathfrak h}}$-invariance of the complex
structure.
This proves that the gauged supersymmetric WZW model does provide a
lagrangian realisation of the $N{=}2$ coset construction.
\section{Decoupling the ${\ensuremath{\mathfrak h}}$-fermions}
The structure of the BRST current $j_{\hbox{\tiny BRST}}$ is very
suggestive. Notice that it can be written as the sum of two terms
whose charges separately square to zero. Indeed, let us write
$j_{\hbox{\tiny BRST}} = j_0 + j_1$, where
\begin{displaymath}
j_0 = -\psi^+_i\gamma^i\quad\hbox{and}\quad
j_1 = (I_i+\tilde I_i)c^i - {f_{ij}}^k \beta_k c^i \gamma^j - \half
{f_{ij}}^k b_k c^i c^j~.
\end{displaymath}
The subscripts refer to the $(b,c)$ ghost number. Given that both
$(b,c)$ and $(\beta,\gamma)$ ghosts numbers are separately conserved,
we have that the respective differentials $d_0 \equiv [j_0,-]_1$ and
$d_1 \equiv [j_1,-]_1$ form a double complex:
\begin{displaymath}
d_0^2 = d_1^2 = d_0 d_1 + d_1 d_0 = 0~.
\end{displaymath}
Moreover the form of $j_0$ is reminiscent of a Koszul complex.
Indeed, $(\psi^\pm_i, \beta_i,\gamma^i)$ forms a Kugo--Ojima quartet
(see, for instance, \cite{FKKO} for the relevant notions) and decouple
from the theory. We can see this in either of two ways.
First of all, as in all double complexes, there is a spectral sequence
converging to the BRST cohomology whose $E_1$ and $E_2$ terms are the
cohomology $H_{d_0}$ of $d_0$ and $H_{d_1}(H_{d_0})$ respectively.
Since the cohomology of $d_0$ is isomorphic to the CFT obtained by
decoupling the above Kugo--Ojima quartet, the spectral sequence
degenerates at the $E_2$ term. Thus the BRST cohomology is isomorphic
to the cohomology of $d_1$ acting on the remaining fields.
Alternatively, we can just change variables. Let's introduce the
field
\begin{displaymath}
r(z) \equiv -\half \Omega^{j\ell} {f_{ij}}^k \beta_\ell \psi^-_k c^i~.
\end{displaymath}
Let $R = \oint r(z)$ denote its charge. If $\phi(z)$ is any field, we
define the conjugation by $R$ as follows:
\begin{displaymath}
e^R\,\phi(z)\,e^{-R} \equiv \sum_{k\geq 0} {1\over k!}
\underbrace{[r,\cdots [r,[r}_k,\phi(z)]_1]_1\cdots ]_1~.
\end{displaymath}
It is easy to see that with the $r(z)$ defined above, this sum is
actually finite. Applying this conjugation to the BRST current, we
find
\begin{displaymath}
e^R\,j_{\hbox{\tiny BRST}}\,e^{-R} = j_0 +
j'~,\quad\hbox{where}\quad j' = (\hat J_i + \tilde J_i) c^i -
\half {f_{ij}}^k b_k c^ic^j~.
\end{displaymath}
Notice that now $j_0$ and $j'$ are completely decoupled, since $j'$
involves the currents $\hat J_i$ and $\tilde J_i$ which do not
interact with the ${\ensuremath{\mathfrak h}}$-fermions. At the level of the fields,
conjugation by $R$ induces a change of variables which, as will be
shown in \cite{FSN=1}, factorises the path integral. In the factor
involving the Kugo--Ojima quartet, the contribution from the bosonic
ghosts cancels precisely the contribution coming from the
${\ensuremath{\mathfrak h}}$-fermions, {\em including the zero modes\/}: since all fields
$(\beta_i,\gamma^i)$ and $\psi^\pm_i$ have weight $\half$. At the end
of the day, one is left with a theory of coset fermions $\psi_\alpha$
and a WZW model in which we have gauged the diagonal ${\ensuremath{\mathfrak h}}$ symmetry.
Notice that the coset fermions {\em are\/} gauged, since it is $\hat
J_i$ and not $J_i$ which appears in $j'$. This theory is precisely
the starting point of \cite{Witten}.
As a consequence we have two equivalent descriptions of the physical
sector of the supersymmetric gauged WZW model (or equivalently of the
$N{=}1$ coset theory):
\begin{itemize}
\item[$\bullet$] the cohomology of $d$ acting on the superconformal
field theory generated by $(I_a,\tilde I_i, \psi_a, \tilde\psi_i,
b_i,c^i,\beta_i,\gamma^i)$; and
\item[$\bullet$] the cohomology of $d' \equiv [j',-]_1$ acting on the
superconformal field theory generated by $(\hat J_a, \tilde J_i,
\psi_\alpha, b_i,c^i)$.
\end{itemize}
When ${\ensuremath{\mathfrak h}}=\gg$, there are no coset fermions, and the gauged
supersymmetric WZW model reduces to an ordinary bosonic gauged WZW
model.
Finally let us remark that from the results in \cite{FMech}---where
the above operator $R$ was introduced---it follows that any given
bosonic gauged WZW model embeds into a supersymmetric gauged WZW model
in such a way that their BRST cohomologies are the same. What we have
proven above is that in the case of the topological gauging ${\ensuremath{\mathfrak h}}=\gg$,
the converse also holds. In other words, the bosonic and
supersymmetric $G/G$ gauged WZW models are equivalent.
\begin{ack}
Some of this work was presented by one of us (SS) at the collaboration
meeting {\em Superstrings and the Physics of Fundamental
Interactions\/} held in London on October 30-31, 1995. SS would like
to thank the organisers of the meeting for the invitation to speak and
the members of the String Theory group of Queen Mary and Westfield
College and Chris Hull in particular for the hospitality. We
benefited once more from the excellent {\em Mathematica\/} package
{\em OPEdefs}, written by Kris Thielemans \cite{OPEdefs}. We are
grateful to Jaume Roca for a careful reading of the manuscript.
After most of this work was finished we learnt from Henric Rhedin
\cite{Rhedin} that he has independently obtained some of the results
pertaining to the (reductive) $N{=}1$ coset theory. It is a pleasure
to thank him for conversations on this and other related topics.
\end{ack}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,566 |
Q: How would I create this layout I understand this isn't possibly the best title so please edit if you have a better title.
Ok, so I have a mock up of part of the design I want to create in android, which I will post below.
I am not the best in working with custom shapes so I thought I could possibly go down the route of Images with clickable areas. This would mean I just import the image and just monitor if the user clicks a section of the screen.
What would be the best approach?
If creating it with XML is better do you have a good tutorial you could point me to.
Thanks
A: This May Help
<?xml version="1.0" encoding="utf-8"?><RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:tools="http://schemas.android.com/tools"
android:id="@+id/activity_main"
android:layout_width="match_parent"
android:layout_height="match_parent"
android:paddingBottom="16dp"
android:paddingLeft="16dp"
android:paddingRight="16dp"
android:paddingTop="16dp">
<LinearLayout
android:layout_width="match_parent"
android:layout_height="match_parent"
android:orientation="vertical"
android:weightSum="2">
<LinearLayout
android:layout_width="match_parent"
android:layout_height="match_parent"
android:layout_weight="1"
android:weightSum="2"
android:orientation="horizontal">
<Button
android:layout_width="match_parent"
android:layout_height="match_parent"
android:layout_weight="1"
android:text="Button 1"/>
<Button
android:layout_width="match_parent"
android:layout_height="match_parent"
android:layout_weight="1"
android:text="Button 2"/>
</LinearLayout>
<LinearLayout
android:layout_width="match_parent"
android:layout_height="match_parent"
android:layout_weight="1"
android:weightSum="2"
android:orientation="horizontal">
<Button
android:layout_width="match_parent"
android:layout_height="match_parent"
android:layout_weight="1"
android:text="Button 3"/>
<Button
android:layout_width="match_parent"
android:layout_height="match_parent"
android:layout_weight="1"
android:text="Button 4"/>
</LinearLayout>
</LinearLayout>
<Button
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:layout_centerInParent="true"
android:background="@drawable/circle"/></RelativeLayout>
Keep this file in drawable folder for circle button background named as circle
<?xml version="1.0" encoding="utf-8"?><shape xmlns:android="http://schemas.android.com/apk/res/android" android:shape="oval">
<size android:width="200dp" android:height="200dp" />
<solid android:color="@android:color/holo_red_light" /></shape>
Looks Like This
A: Here is a fully tested layout
<?xml version="1.0" encoding="utf-8"?>
<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
android:layout_width="wrap_content"
android:layout_height="wrap_content">
<Button
android:id="@+id/btn_1"
android:layout_width="100dp"
android:layout_height="wrap_content"
android:text="Button one"/>
<Button
android:id="@+id/btn_2"
android:layout_width="100dp"
android:layout_height="wrap_content"
android:layout_toRightOf="@id/btn_1"
android:text="Button two"/>
<Button
android:id="@+id/btn_3"
android:layout_width="100dp"
android:layout_height="wrap_content"
android:text="Button three"
android:layout_below="@id/btn_1"/>
<Button
android:id="@+id/btn_4"
android:layout_width="100dp"
android:layout_height="wrap_content"
android:layout_below="@id/btn_2"
android:layout_toRightOf="@id/btn_3"
android:text="Button four"/>
<ImageView
android:id="@+id/logo"
android:layout_width="wrap_content"
android:layout_height="wrap_content"
android:layout_centerInParent="true"
android:translationZ="10dp"
android:src="@drawable/ti_logo"/>
</RelativeLayout>
A: @ SammyG you can edit in this layout according to your requirements..
<RelativeLayout xmlns:android="http://schemas.android.com/apk/res/android"
xmlns:tools="http://schemas.android.com/tools"
android:id="@+id/activity_main"
android:layout_width="match_parent"
android:layout_height="match_parent"
tools:context="com.example.android.stackoverflow_test.MainActivity">
<LinearLayout
android:layout_width="match_parent"
android:layout_height="match_parent"
android:orientation="vertical"
android:weightSum="2">
<LinearLayout
android:layout_width="match_parent"
android:layout_height="0dp"
android:layout_weight="1"
android:orientation="horizontal">
<TextView
android:layout_width="0dp"
android:layout_height="match_parent"
android:layout_weight="1"
android:background="@android:color/holo_green_dark"
android:padding="20dp" />
<TextView
android:layout_width="0dp"
android:layout_height="match_parent"
android:layout_weight="1"
android:background="@android:color/black"
android:padding="20dp" />
</LinearLayout>
<LinearLayout
android:layout_width="match_parent"
android:layout_height="0dp"
android:layout_weight="1"
android:orientation="horizontal">
<TextView
android:layout_width="0dp"
android:layout_height="match_parent"
android:layout_weight="1"
android:background="@android:color/holo_red_dark"
android:padding="20dp" />
<TextView
android:layout_width="0dp"
android:layout_height="match_parent"
android:layout_weight="1"
android:background="@android:color/holo_blue_bright"
android:padding="20dp" />
</LinearLayout>
</LinearLayout>
<ImageView
android:layout_width="150dp"
android:layout_height="150dp"
android:layout_centerInParent="true"
android:background="@mipmap/ic_launcher" /></RelativeLayout>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,344 |
About 72 per cent of Thai graduates at risk of unemployment by 2030 because of new technologies
Digital transformation is redefining the ways in which people work, live and learn. For Thailand's Deputy Education Minister, "Some jobs will disappear, others will grow, and jobs that don't even exist today could become commonplace." The skills most requested by employers will include cognitive skills like problem solving, critical thinking, analysis and creative work.
Bangkok (AsiaNews/Agencies) – By 2030, 72 per cent of university graduates could be either unemployed or working at jobs that do not require a bachelor's degree. Artificial Intelligence (AI) and robots are the main threats to employment for Thailand's young people, this according to the country's Deputy Minister for Education Udom Kachinthorn.
Mr Udom spoke two days ago at a seminar organised by the Education Council in Bangkok. Citing a report published by the World Bank, he said, "if Thai universities do not adapt and cannot build a workforce with future-proof skills, the country may have to cope with the largest ever rate of unemployment."
As things stand, digital transformation will totally redefine the way people work, live and learn. "Some jobs will disappear, others will grow, and jobs that don't even exist today could become commonplace."
The deputy minister warned that administrative and office workers who lack all but routine skills will be the most vulnerable of being made "redundant" and being replaced by machines.
High-in-demand skills will include cognitive abilities like problem solving, critical thinking, analysis or creative work, Udom said.
He warned that universities and teachers must embrace change by using digital technology to make their classes and content more lively, relevant and responsive to the demands and lifestyles of a new generation.
"The universities of the future must teach students how to become learners. Schools and universities must change from being just classrooms to becoming learning spaces. Pedagogues must assume new roles as coaches who provide guidance, not only giving lectures," he explained.
"To survive, Thai universities need to adjust their strategies, modernise, shut down outdated majors that are not in high demand, and collaborate more," he added.
Church leads the way in helping Vietnam cope with its educational emergency
Growing unemployment in the Philippines, also due to corruption and waste
Buddhist monk robot chants and taps drum at funerals
Argentine Rabbi: intelligence comes from God, not from machines
Dr Carvalho calls for robots to be used for humanity, not wars | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,783 |
\section{Introduction}
Task-oriented dialogue systems aim to help user accomplish specific tasks via natural language interfaces such as restaurant reservation, hotel booking and weather forecast. There are many commercial applications of this kind (e.g. Amazon Alexa, Google Home, and Apple Siri) which make our life more convenient. Figure \ref{dialogue_example} illustrates such an example where a customer is asking for the information about restaurants. By querying the knowledge base (KB), the agent aims to provide the correct restaurant entities from the KB to satisfy the customer in a natural language form. Hence, the ability to understand the dialogue history, and to retrieve relevant information from the KB is essential in task-oriented dialogue systems.
\begin{figure}[t]
\setlength{\belowcaptionskip}{-3mm}
\centering
\includegraphics[width=3.0in]{TaskExample_0929_0.pdf}
\caption{An example dialogue in the restaurant booking domain. The top part is knowledge base (KB) information that represented by a graph and the bottom part is the conversation between a customer and the agent. Our aim is to predict the agent responses given KB information and the customer utterances.}
\label{dialogue_example}
\end{figure}
One approach for designing task-oriented dialogue systems is the pipeline approach \citep{williams2007partially,lee2009example,young2013pomdp}, but it suffers from the difficulty in credit assignment and adaption to new domains. Another popular approach is the end-to-end models \citep{serban2016building,wen2016network,williams2017hybrid,zhao2017generative,serban2017hierarchical}, which directly map the dialogue history to the output responses. This approach has attracted more attention in the research community recently as it alleviates the drawbacks of the pipeline approach.
However, end-to-end dialogue models usually suffer from ineffective use of knowledge bases due to the lack of appropriate framework to handle KB data.
To mitigate this issue, recent end-to-end dialogue studies \citep{eric2017key,mem2seq} employ memory networks \citep{weston2014memory,sukhbaatar2015end} to support the learning over KB, and have achieved promising results via integrating memory with copy mechanisms \citep{gulcehre2016pointingunknownwords,eric2017copyaugmented}. By using memory, they assume that the underlying structure of KB is linear since memory can be viewed as a list structure. As a result, the relationships between entities are not captured. However, since KB is naturally a graph structure (nodes are entities and edges are relations between entities). By overlooking such relationships, the model fails to capture substantial information embedded in the KB including the semantics of the entities which may significantly impact the accuracy of results. Moreover, structural knowledge such as dependency relationships has recently been investigated on some tasks (e.g., relation extraction) \citep{peng2017cross,song2018n} and shown to be effective in the model's generalizability. However, such dependency relationships (essentially also graph structure) have not been explored in dialogue systems, again missing great potential for improvements.
With the above insight, we propose a novel \textbf{graph}-based end-to-end task-oriented \textbf{dialog}ue model (\textbf{GraphDialog}) aimed to exploit the graph knowledge both in dialogue history and KBs. Unlike traditional RNNs such as LSTM \citep{hochreiter1997long} and GRU \citep{cho2014learning}, we design a novel recurrent unit (Section \ref{graphencodercell}) that allows multiple hidden states as inputs at each timestep such that the dialogue history can be encoded with graph structural information. The recurrent unit employs a masked attention mechanism to enable variable input hidden states at each timestep. Moreover, We incorporate a graph structure (Section \ref{graphknowledge}) to handle the external KB information and perform multi-hop reasoning on the graph to retrieve KB entities.
Overall, the contributions of this paper are summarized as follows:
\begin{itemize}
\item We propose a novel graph-based end-to-end dialogue model for effectively incorporating the external knowledge bases into task-oriented dialogue systems.
\item We further propose a novel recurrent cell architecture to exploit the graph structural information in the dialogue history. We also combine the multi-hop reasoning ability with graph to exploit the relationships between entities in the KB.
\item We evaluate the proposed model on two real-world task-oriented dialogue datasets (i.e., SMD and MultiWOZ 2.1). The results show that our model outperforms the state-of-the-art models consistently.
\end{itemize}
\begin{figure*}
\setlength{\belowcaptionskip}{-1mm}
\centering
\includegraphics[width=6.3in]{Framework_0528.png}
\caption{Overview of the proposed architecture. (a) Graph Encoder, top is forward graph and bottom is backward graph. (b) Decoder and Knowledge Graph with multi-hop reasoning mechanism. (c) Self-Attention Mechanism.}
\label{overallframework}
\end{figure*}
\section{Related Work}
Task-oriented dialogue system has been a long-standing studied topic \citep{williams2007partially,lee2009example,huang2020semi} and can be integrated into many practical applications such as virtual assistant \citep{sun2016contextual,sun2017collaborative}. Traditionally, task-oriented dialogue systems are built in the pipeline approach, which consists of four essential components: natural language understanding \citep{chen2016end}, dialogue state tracking \citep{lee2016task,zhong2018global,wu2019transferable}, policy learning \citep{su2016line,peng2018deep,su2018discriminative} and natural language generation \citep{sharma2016natural,chen2019semantically,huang2019mala}. Another recent approach is the end-to-end models \citep{wu2018end,lei2018sequicity}, which directly map the user utterances to responses without heavy annotations. \citet{bordes2016endtoendtaskorientedlearning} apply end-to-end memory networks \citep{sukhbaatar2015end} for task-oriented dialogues and shown that end-to-end models are promising on the tasks. To produce more flexible responses, several generative models are proposed \citep{zhao2017generative,serban2016building}. They formulate the response generation problem as a translation task and apply sequence-to-sequence (Seq2Seq) models to generate responses. Seq2Seq models have shown to be effective in language modeling but they struggle to incorporate external KB into responses. To mitigate this issue, \citet{eric2017copyaugmented} has enhanced the Seq2Seq model by adding copy mechanism. \citet{mem2seq} combines the idea of pointer with memory networks and obtained improved performance. \citet{wu2019global} incorporates global pointer mechanism and achieved improved performance. Our study differs from those works in that we exploit the powerful graph information both contained in the dialogue history and in the KBs to effectively incorporate KBs into dialogue systems.
\section{Proposed Model}
Our proposed model consists of three components: an encoder (Section \ref{graphencoder}), a decoder (Section \ref{graphdecoder}) and a knowledge graph with multi-hop reasoning ability (Section \ref{graphknowledge}). Formally, let \textit{X} = \{$x_1$,...,$x_n$\} be a sequence of tokens, where each token $x_i\in$ \textit{X} corresponds to a word in the dialogue history. We first obtain a \textit{dialogue graph} \textit{$\widehat{G}$} (Section \ref{dialoguegraph}), which is the dependency parsing graph of the sentences in the dialogue history \textit{X}, as the input of the encoder. The encoder then learns a fixed-length vector as the encoding of the dialogue history based on \textit{$\widehat{G}$}, which is then fed to the decoder for hidden state initialization. The knowledge graph adopts another graph \textit{G} = \{\textit{V},\textit{E}\} to store and retrieve the external knowledge data (Section \ref{graphcontent}), where \textit{V} denotes the entities and \textit{E} denotes the edges. The decoder generates the system response \textit{Y} = \{$y_1$,...,$y_m$\} token-by-token either by copying entities from graph \textit{G} via querying the knowledge graph or by generating tokens from vocabularies. Figure \ref{overallframework} illustrates the overall architecture of the proposed model. In the following sections, we describe each component in detail.
\subsection{Graph Encoder}\label{graphencoder}
\subsubsection{Dialogue Graph}\label{dialoguegraph}
To enable learning semantic rich representations of words with various relationships, such as adjacency and dependency relations, we first use the off-the-shelf tool \textit{spacy}\footnote{\url{https://spacy.io/}} to extract the dependency relations among the words in the dialogue history \textit{X}. Figure \ref{dependencyparsingresult} gives an example of the dependency parsing result. The bi-directional edges among words allow information flow both from dependents to heads and from heads to dependents. The intuition is that the representation learning of the head words should be allowed being influenced by the dependent words and vice versa, thus allowing the learning process to capture the mutual relationships between the head words and the dependent words to provide richer representation.
We compose the dialogue graph by combining the obtained dependency relations with the sequential relations (i.e., \textit{Next} and \textit{Pre}) among words, which serves as the input to the graph encoder. To further support bi-directional representation learning, we split the obtained dialogue graph into two parts: the \textit{forward graph} (from left to right) and the \textit{backward graph} (from right to left).
\begin{figure}[t]
\centering
\includegraphics[width=3.0in]{DialogueGraph_0516_V1.png}
\caption{An example of dialogue graph.}
\label{dependencyparsingresult}
\end{figure}
\subsubsection{Recurrent Cell Architecture}\label{graphencodercell}
The recurrent cell architecture (Figure \ref{recurrentcell}) is the core computing unit of the graph encoder, and is used to compute the hidden state of each word in the obtained dialogue graph. The cell traverse the words in the dialogue graph sequentially according to the word order in the dialogue history. Next, we show how to compute the cell hidden state \textit{$h_t$} at timestep \textit{t}.
\begin{figure}[t]
\setlength{\belowcaptionskip}{-1mm}
\centering
\includegraphics[width=2.5in]{GraphCell_0529.png}
\caption{Overview of the proposed recurrent unit.}
\label{recurrentcell}
\end{figure}
Let us define \textit{$x_t$} as the input word representation at timestep \textit{t}. \textit{P(t)} = \{$p_1$,$p_2$,\dots,$p_k$\} is the set of precedent words for \textit{$x_t$} where each $p_i\in$ \textit{P(t)} denotes a word in the dialogue graph that connects to \textit{$x_t$}, and \textit{k} is the total number of the precedents of \textit{$x_t$}. \textit{H} = \{$h_1$,$h_2$,\dots,$h_k$\} is a set of hidden states where each element $h_j\in$ \textit{H} denotes the hidden state of the \textit{j}-th predecessor $p_j\in$ \textit{P(t)}.
The input of the cell consists of two parts: the input word vector \textit{$x_t$}, and the predecessor hidden states \textit{H}. First, we loop over the \textit{k} hidden states in \textit{H} and compute a \textit{reset gate} for each of them. Specifically, we compute \textit{$r_j$} for the \textit{j}-th hidden state using:
\vspace{-2mm}
\begin{equation}
\textit{$r_j$} = \sigma\left(W_r x_t + U_r h_j \right) \label{resetgate}
\end{equation}
\noindent where $\sigma$ is the logistic sigmoid function, \textit{$x_t$} and \textit{$h_j$} are the current input and the hidden state of the \textit{j}-th predecessor at timestep \textit{t} respectively. \textit{$W_r$} and \textit{$U_r$} are parameters which will be learned. We then compute a candidate hidden state $\widetilde{h_t}$ using:
\vspace{-3mm}
\begin{equation}
\widetilde{h_t} = \phi\left(W_n x_t + \frac{1}{k} \sum_{j=1}^{k} r_j * \left( U_n h_j \right) \right)
\end{equation}
\noindent where $\phi$ is the hyperbolic tangent function, \textit{k} is the number of predecessors of word \textit{$x_t$}, \textit{$W_n$} and \textit{$U_n$} are the learnable weight matrices. Intuitively, $\widetilde{h_t}$ is the contextualized representation of current input \textit{$x_t$}.
Next, we combine the obtained candidate hidden state $\widetilde{h_t}$ with the predecessor hidden states \textit{H}, and use an \textit{masked attention mechanism} (Equation \ref{maskedattentionequation}) to aggregate them together to yield the output hidden state \textit{$h_t$} at timestep \textit{t}. To obtain sufficient expressive power, we first apply linear transformations to the input \textit{$x_t$} and the hidden states \textit{$h_j\in$} \textit{H} using:
\vspace{-3mm}
\begin{equation}
\textit{$x_t^{'}$} = W_z x_t
\end{equation}
\vspace{-3mm}
\begin{equation}
\textit{$h_j^{'}$} = U_z h_j
\end{equation}
\noindent where \textit{$W_z$}, \textit{$U_z$} are parameters which are learned, \textit{t} is the current timestep. We denote \textit{$H^{'}$}=\{\textit{$h_1^{'}$},\textit{$h_2^{'}$},\dots,\textit{$h_k^{'}$}\} as the transformed set of hidden states. Then we add the previously obtained candidate hidden state $\widetilde{h_t}$ into the transformed set of hidden states \textit{$H^{'}$} and obtain \textit{$H^{''}$}=\{$h_1^{'}$,$h_2^{'}$,\dots,$h_k^{'}$,$\widetilde{h_t}$\}. The intuition is that the output hidden state depends on both the history information ($h_1^{'}$ to $h_k^{'}$) and the current input ($\widetilde{h_t}$).
Then we perform attention mechanism by using the hidden states \textit{$H^{''}$} as keys and the current input \textit{$x_t$} as query. Intuitively, different inputs (e.g. different predecessors in \textit{$H^{''}$}) should have different impacts on the output hidden state \textit{$h_t$}, and we expect our model to capture that. However, the inputs may have different number of predecessors at different timesteps. To handle this, inspired by \citep{vaswani2017attention}, we employ an \textit{masked attention mechanism} to learn the importance of each predecessor at every timestep, thus avoiding the pad information affecting the learning process. We compute the attention using:
\vspace{-1mm}
\begin{equation}
\textit{$e_j$} = \textit{$v^{T}$} \phi\left(x_t^{'} + h_j^{'} \right)
\end{equation}
\vspace{-1mm}
\begin{equation}
\alpha_{j} = \textit{softmax} \left( [e_j]_{m} \right) \label{maskedattentionequation}
\end{equation}
\noindent where \textit{$v$} is a learnable parameter, \textit{$h_j^{'}$} is the \textit{j}-th vector in \textit{$H^{''}$}, \textit{$softmax$}(\textit{$z_i$})={\textit{$e^{z_i}$}}/{$\sum_{j}e^{z_j}$}, $\alpha_{j}$ denotes the attention weight on the \textit{j}-th vector in \textit{$H^{''}$}, $\left[\cdot\right]_{m}$ denotes the mask operation. In our implementation, we simply set the number to negative infinity if the \textit{j}-th hidden state corresponds to a pad token. Finally, we compute the weighted sum to obtain the cell output hidden state \textit{$h_t$} at timestep \textit{t} using:
\vspace{-1mm}
\begin{equation}
\textit{$h_t$} = \sum_{j=1}^{k+1} \alpha_j h_j^{'}
\end{equation}
Intuitively, the \textit{reset gate} controls the information flow from the multiple predecessors to the hidden state of current timestep. If a precedent word is more correlated to the current input word, then it is expected to let the information of the precedent word flow through the gate to affect the representation of current timestep.
\subsubsection{Bi-directional Representation}
To obtain a bi-directional representation for the dialogue history, we use the same cell architecture (Section \ref{graphencodercell}) to loop over the \textit{forward graph} and \textit{backward graph} separately, and compute a forward representation $\stackrel{\rightarrow}{h_n}$ and a backward representation $\stackrel{\leftarrow}{h_n}$, respectively. Then we concatenate them together to serve as the final representation of dialogue history \textit{$h_n^{e}$}=[$\stackrel{\rightarrow}{h_n}$;$\stackrel{\leftarrow}{h_n}$], which will become a part of the inputs to the decoder.
\subsection{Multi-hop Reasoning Mechanism over Knowledge Graph}\label{graphknowledge}
A straightforward way to explore the graph information in KB is to represent the KB as a graph structure, and then query the graph using attention mechanism with the decoder hidden states. However, our preliminary experiments didn't show a good performance using this approach. We conjecture that it may be due to the poor reasoning ability of this method. To address this issue, we extend the graph with multi-hop reasoning mechanism, which aimed to strengthen the reasoning ability over graph as well as to capture the graph structural information between entities via self-attention. We call it knowledge graph module in the following sections.
Formally, the knowledge graph module contains two sets of trainable parameters \textit{C} = \{\textit{$C^{1}$},\textit{$C^{2}$},\dots,\textit{$C^{K+1}$}\}, where each \textit{$C^{k}$} is an embedding matrix that maps tokens to vector representations, and \textit{V} = \{\textit{$V^{1}$},\textit{$V^{2}$},\dots,\textit{$V^{K+1}$}\}, where each \textit{$V^{k}$} is a weight vector for computing self-attention coefficients, and \textit{K} is the maximum number of hops.
Now we describe how to compute the output vector of the knowledge graph. The model loops over \textit{K} hops on an input graph. At each hop \textit{k}, a query vector \textit{$q^{k}$} is employed as the reading head. First, the model uses an embedding layer \textit{$C^{k}$} to obtain the continuous vector representations of each node \textit{i} in the graph as \textit{$C^{k}_{i}$}, where \textit{$C^{k}_{i}$}=\textit{$C^{k}$}(\textit{$n_i$}) and $n_i$ is the \textit{i}-th node in the graph. Then we perform \textit{self-attention mechanism} on the nodes and compute the attention coefficients using:
\vspace{-1mm}
\begin{equation}
\textit{$e_{ij}$} = \varphi\left(\left(\textit{{$V^{k}$}}\right)^{T}[\textit{$C^{k}_{i}$}||\textit{$C^{k}_{j}$}]\right) \label{gatlogits}
\end{equation}
\noindent where $\varphi$ is the LeakyReLU activation function (with negative input slope $\alpha$ = 0.2), \textit{$V^{k}$} is the parametrized weight vector of the attention mechanism at hop \textit{k}, \textit{$C^{k}_{i}$} and \textit{$C^{k}_{j}$} are the node vectors for the \textit{i}-th and \textit{j}-th node in the graph at hop \textit{k}, and $\|$ is the concatenation operation. We then normalize the coefficients of each node \textit{i} with respect to all its first-order neighbors using the softmax function:
\vspace{-1mm}
\begin{equation}
\alpha_{ij} = \frac{exp(e_{ij})}{\sum_{k\in\textit{$N_i$}} exp(e_{ik})} \label{gatattention}
\end{equation}
\noindent where \textit{$N_i$} is the first-order neighbors of node \textit{i} (including \textit{i}), \textit{exp} is the exponential function.
Then we update the representation of each node \textit{i} by a weighted sum of its neighbors in \textit{$N_i$} using:
\vspace{-1mm}
\begin{equation}
\left(\textit{$C^{k}_{i}$}\right)^{'} = \sum_{j\in\textit{$N_i$}} \alpha_{ij} C^{k}_{j} \label{gatupdate}
\end{equation}
\noindent Next, the query vector \textit{$q^k$} is used to attend to the updated nodes in the graph and compute the attention weights for each node \textit{i} at hop \textit{k} using:
\vspace{-1mm}
\begin{equation}
\textit{$p^{k}_{i}$} = softmax \left(\left(q^{k}\right)^{T} \left(\textit{$C^{k}_{i}$}\right)^{'} \right)
\end{equation}
To obtain the output of the knowledge graph, we apply the same self-attention mechanism (Equations \ref{gatlogits} and \ref{gatattention}) and update strategy (Equation \ref{gatupdate}) to the node representation \textit{$C^{k+1}_{i}$}. We use \textit{$C^{k+1}$} here since the adjacent weighted tying strategy is adopted. The updated node representation for output is denoted as $\textit{$\left(\textit{$C^{k+1}_{i}$}\right)$}^{'}$. Once obtained, the model reads out the graph \textit{$o^k$} by the weighted sum over it using:
\vspace{-1mm}
\begin{equation}
\textit{$o^{k}$} = \sum_{i} p^{k}_{i} \left(C^{k+1}_{i}\right)^{'}
\end{equation}
Then the query vector \textit{$q^k$} is updated for the next hop using \textit{$q^{k+1}$} = \textit{$q^k$} + \textit{$o^k$}. The final output of the knowledge graph is \textit{$o^{K}$}, which will become a part of the inputs to the decoder.
\subsubsection{Graph Construction}\label{graphcontent}
In practice, dialogue systems usually use KBs (mostly in a relational database format) to provide external knowledge. We have converted the original relational database into a graph structure to exploit the relation information between KB entities. First, we find all the entities in the relational database as the nodes of the graph. Then we assign an edge to a pair of entities if there exists relationship between them according to the records in the relational database. Thus we can obtain the graph structured external knowledge.
\subsection{Decoder}\label{graphdecoder}
We use a standard Gated Recurrent Unit (GRU) \citep{cho2014learning} as the decoder to generate the system response word-by-word. The initial hidden state \textit{$h_0$} consists of two parts: the graph encoder output and the knowledge graph output. We take the output hidden state of the graph encoder \textit{$h_{n}^{e}$} as the initial query vector \textit{$q^{0}$} to attend to the knowledge graph and obtain the output \textit{$o^{K}$}. The initial hidden state \textit{$h_0$} is then computed using:
\vspace{-1mm}
\begin{equation}
\textit{$h_0$} = [h_{n}^{e} || o^{K}]
\end{equation}
At each decoder timestep \textit{t}, the GRU takes the previously generated word \textit{$\widehat{y}_{t-1}$} and the previous hidden state \textit{$h_{t-1}$} as the input and generates a new hidden state \textit{$h_{t}$} using:
\vspace{-1mm}
\begin{equation}
\textit{$h_t$} = GRU\left(\widehat{y}_{t-1},h_{t-1} \right)
\end{equation}
Next, we follow \citep{wu2019global} that the decoder learns to generate a sketch response that the entities in the response are replaced with certain tags. The tags are obtained from the provided ontologies in the training data. The hidden state \textit{$h_t$} are used for two purposes. The first one is to generate a vocabulary distribution \textit{$P_{vocab}$} over all the words in the vocabulary using:
\vspace{-1mm}
\begin{equation}
\textit{$P_{vocab}$} = softmax\left(W_o h_t \right)
\end{equation}
\noindent where $W_o$ is the learnable parameter. The second one is to query the knowledge graph to generate a graph distribution \textit{$P_{graph}$} over all the nodes in the graph. We use the attention weights at the last hop of the knowledge graph \textit{$p^{K}_{t}$} as \textit{$P_{graph}$}.
At each timestep \textit{t}, if the generated word from \textit{$P_{vocab}$} (the word has the maximum posterior probability) is a tag, then the decoder choose to copy from the graph entities that has the largest attention value according to \textit{$P_{graph}$}. Otherwise, the decoder will generate the target word from \textit{$P_{vocab}$}. During training, all the parameters are jointly learned via minimizing the sum of two cross-entropy losses: one is between \textit{$P_{vocab}$} and \textit{$y_t\in$} \textit{Y}, and the other is between \textit{$P_{graph}$} and \textit{$G_{t}^{Label}$}, where \textit{$G_{t}^{Label}$} is the node id that corresponds to the current output \textit{$y_t$}.
\section{Experiments}
\subsection{Dataset}
To validate the efficacy of our proposed model, we evaluate it on two public multi-turn task-oriented diaglogue datasets: Stanford multi-domain dialogue (SMD) \citep{eric2017key} and MultiWOZ 2.1 \citep{eric2019multiwoz}. The SMD is a human–human dataset for in-car navigation task. It includes three distinct task domains: \textit{point-of-interest navigation}, \textit{calendar scheduling} and \textit{weather information retrieval}. The MultiWOZ 2.1 dataset is a recently released human–human dialogue corpus with much larger data size and richer linguistic expressions that make it a more challenging benchmark for end-to-end task-oriented dialogue modeling. It consists of seven distinct task domains: \textit{restaurant}, \textit{hotel}, \textit{attraction}, \textit{train}, \textit{hospital}, \textit{taxi} and \textit{police}. We select four domains (\textit{restaurant}, \textit{hotel}, \textit{attraction}, \textit{train}) to test our model since the other three domains (\textit{police}, \textit{taxi}, \textit{hospital}) lack KB information which is essential to our task. We will make our code and data publicly available for further study. To the best of our knowledge, we are the first to evaluate end-to-end task-oriented dialogue models on MultiWOZ 2.1. The train/validation/test sets of these two datasets are split in advance by the providers.
\begin{table}[t]
\centering
\small
\begin{tabular}{l|c|c}
\toprule[1pt]
Metrics& SMD& MultiWOZ 2.1 \\
\midrule[0.5pt]
\textit{Avg. Turns per dialog}& 5.25& 13.46 \\
\textit{Avg. Tokens per turn}& 8.02& 13.13 \\
\textit{Total number of turns}& 12732& 113556 \\
\midrule[0.5pt]
\textit{Vocabulary}& 1601& 23689 \\
\textit{Train dialogs}& 2425& 8438 \\
\textit{Val dialogs}& 302& 1000 \\
\textit{Test dialogs}& 304& 1000 \\
\bottomrule[1pt]
\end{tabular}
\caption{Dataset statistics for SMD and MultiWOZ 2.1.}
\label{datasetstatistics}
\end{table}
\begin{table*}[!h]
\centering
\small
\begin{tabular}{r|ccccccc|ccc}
\toprule[1pt]
\multirow{2}{*}{Model}& \multirow{2}{*}{S2S} & \multirow{2}{*}{S2S + Attn} & \multirow{2}{*}{Ptr-Unk} & \multirow{2}{*}{GraphLSTM} & \multirow{2}{*}{BERT} & \multirow{2}{*}{Mem2Seq} & \multirow{2}{*}{GLMP} & \multicolumn{3}{c}{GraphDialog} \\
\cline{9-11}
& & & & & & & & K=1 & K=3 & K=6 \\
\midrule[0.5pt]
BLEU& 8.4& 9.3& 8.3& 10.3& 9.13& 12.6& 12.2& 12.96& \textbf{13.66}& 12.74 \\
\midrule[0.5pt]
Entity F1& 10.3& 19.9& 22.7& 50.8& 49.6& 33.4& 55.1& 56.14& \textbf{57.42}& 55.90 \\
\midrule[0.5pt]
Schedule F1& 9.7& 23.4 & 26.9& 69.9& 57.4& 49.3& 67.3& 70.96& \textbf{71.90}& 71.84 \\
Weather F1& 14.1& 25.6& 26.7& 46.6& 47.5& 32.8& 54.1& 56.89& \textbf{59.68}& 54.36 \\
Navigation F1& 7.0& 10.8& 14.9& 43.2& 46.8& 20.0& 48.4& 48.37& \textbf{48.58}& 47.55 \\
\bottomrule[1pt]
\end{tabular}
\caption{Evaluation on SMD dataset. Human, rule-based and KV Retrieval Net results are reported from \citep{eric2017key}, which are not directly comparable since the problem is simplified to canonicalized forms. K denotes the maximum number of hops for knowledge graph. Ours achieves highest BLEU and entity F1 score over baselines.}
\label{smdresults}
\end{table*}
\begin{table*}[!h]
\setlength{\belowcaptionskip}{-1mm}
\centering
\small
\begin{tabular}{r|ccccccc|ccc}
\toprule[1pt]
\multirow{2}{*}{Model}& \multirow{2}{*}{S2S} & \multirow{2}{*}{S2S + Attn} & \multirow{2}{*}{Ptr-Unk} & \multirow{2}{*}{GraphLSTM} & \multirow{2}{*}{BERT} & \multirow{2}{*}{Mem2Seq} & \multirow{2}{*}{GLMP} & \multicolumn{3}{c}{GraphDialog} \\
\cline{9-11}
& & & & & & & & K=1 & K=3 & K=6 \\
\midrule[0.5pt]
BLEU& 2.5& 3.0& 2.3& 3.4& 3.9& 4.1& 4.3& 5.47& \textbf{6.17}& 5.14 \\
\midrule[0.5pt]
Entity F1& 1.3& 2.1& 2.5& 4.7& 4.1& 3.2& 6.7& 9.56& \textbf{11.28}& 8.74 \\
\midrule[0.5pt]
Restaurant F1& 1.6& 2.2& 2.3& 9.8& 7.3& 2.9& 11.4& 15.27& \textbf{15.95}& 13.25 \\
Hotel F1& 1.5& 3.4& 3.8& 2.1& 1.6& 4.5& 3.9& 7.54& \textbf{10.79}& 7.05 \\
Attraction F1& 0.8& 1.4& 1.7& 7.2& 8.4& 2.1& 9.4& 5.78& \textbf{14.12}& 7.89 \\
Travel F1& 0.2& 0.7& 0.9& 1.8& 2.1& 1.5& 3.5& 3.41& \textbf{4.39}& 3.53 \\
\bottomrule[1pt]
\end{tabular}
\caption{Evaluation on MultiWOZ 2.1 dataset. Ours achieves highest BLEU and entity F1 score over baselines.}
\label{multiwozresults}
\end{table*}
\subsection{Training Details}
We implement our model\footnote{Code and data are available at: \url{https://github.com/shiquanyang/GraphDialog}} in Tensorflow and is trained on NVIDIA GeForce RTX 2080 Ti. We use grid search to find the best hyper-parameters for our model over the validation set (use BLEU as criterion for both datasets). We randomly initialize all the embeddings in our implementation. The embedding size is selected between [16,512], which is also equivalent to the RNN hidden state (including the encoder and the decoder). We also use dropout for regularization on both the encoder and the decoder to avoid over-fitting and the dropout rate is set between [0.1,0.5]. We use Adam optimizer \citep{kingma2014adam} to accelerate the convergence with a learning rate chosen between [\textit{$1e^{-3}$},\textit{$1e^{-4}$}]. We simply use a greedy strategy to search for the target word in the decoder without advanced techniques like beam-search.
\subsection{Evaluation Metrics}
We use two common evaluation metrics in dialogue studies including BLEU \citep{papineni2002bleu} (using Moses \verb|multi-bleu.perl| script) and Entity F1 \citep{eric2017key,mem2seq} for evaluations.
\subsection{Effect of Models}
We compare our model with several existing models: standard sequence-to-sequence (Seq2Seq) models with and without attention \citep{luong2015effective}, pointer to unknown (Ptr-Unk, \citep{gulcehre2016pointingunknownwords}), GraphLSTM \citep{peng2017cross}, BERT \citep{devlin2019bert}, Mem2Seq \citep{mem2seq} and GLMP \citep{wu2019global}. Note that the results we listed in Table \ref{smdresults} for GLMP is different from the original paper, since we re-implement their model in Tensorflow according to their released Pytorch code for fair comparison.
\textbf{Stanford Multi-domain Dialogue}. Table \ref{smdresults} has shown the results on SMD dataset. Our proposed model achieves a consistent improvement over all the baselines with the highest BLEU score 13.6 and 57.4\% entity F1 score. The performance gain in BLEU score suggests that the generation error in the decoder has been reduced. The improvement on entity F1 indicates that our model can retrieve entities from the external knowledge data more accurately than those baselines. We also conduct comparisons with BERT to validate the effectiveness of our proposed model. Specifically, we use the bert-base-uncased model (due to GPU memory limit) from huggingface library\footnote{\url{https://github.com/huggingface}} as our encoder to encode the dialogue history and the remaining parts are the same as our model. We then fine-tune BERT on our dialogue dataset. We can find that our mode significantly outperforms the fine-tuned BERT by a large margin which further demonstrates the effectiveness of our proposed model. We conjecture that the reasons may lie in two aspects. First, the context of the corpus used for pretraining BERT differs from our dialogue dataset. Secondly, the model complexity of BERT may cause overfitting issue on small-scale datasets like SMD etc.
\textbf{MultiWOZ 2.1}. Table \ref{multiwozresults} shows the results on a more complex dataset MultiWOZ 2.1. Our model outperforms all the other baselines by a large margin both in entity F1 and BLEU score, which confirms our model has a better generalization ability than those baselines. One may find that the entity F1 and BLEU score has a huge gap between MultiWOZ 2.1 and SMD. This performance degradation phenomenon has also been observed by other dialogue works \citep{budzianowski2018multiwoz} which implies that the MultiWOZ corpus is much more challenging than the SMD dataset for dialogue tasks.
\begin{figure*}[!h]
\centering
\subfigure[Generation \textit{timestep 0}]{\includegraphics[width=1.5in]{T0.pdf}}
\subfigure[Generation \textit{timestep 1} ]{\includegraphics[width=1.5in]{T1.pdf}}
\subfigure[Generation \textit{timestep 2}]{\includegraphics[width=1.5in]{T2.pdf}}
\subfigure[Generation \textit{timestep 3}]{\includegraphics[width=1.5in]{T3.pdf}}
\caption{Knowledge graph attention visualization when generating responses in the SMD navigation domain. Based on the question ``Where is a nearby parking\_garage?", the generated response of our model is ``palo\_alto\_garage is 1\_miles away". Specifically, the attention results at each generation timestep for the knowledge graph information of this example are shown in (a), (b), (c) and (d) respectively. The color and size of the nodes represent their attention weights. The darker and bigger the nodes are, the larger their attention weights are. Our model successfully learns to attend to the correct KB entities (i.e., \textit{palo\_alto\_garage} and \textit{1\_miles} at generation timesteps 0 and 2) which have the highest attention, and the model copies them to serve as the output words. During timesteps 1 and 3, the model generates output words (i.e., \textit{is} and \textit{away}) from the vocabulary.}
\label{kgattention}
\end{figure*}
\begin{table*}[h]
\centering
\small
\begin{tabular}{l|cccc}
\toprule[1pt]
& \multicolumn{2}{c|}{SMD}& \multicolumn{2}{c}{MultiWOZ 2.1} \\
\midrule[0.5pt]
Model& BLEU& \multicolumn{1}{c|}{Entity F1(All)}& BLEU& Entity F1(All) \\
\midrule[0.5pt]
GraphDialog & 13.66(-)& \multicolumn{1}{c|}{57.42(-)} & 6.17(-)& 11.28(-) \\
GraphDialog w/o Graph Encoder & 12.35(-1.31)& \multicolumn{1}{c|}{56.61(-0.81)}& 4.57(-1.60)& 10.13(-1.15) \\
GraphDialog w/o Knowledge Graph & 13.13(-0.53)& \multicolumn{1}{c|}{55.28(-2.14)}& 5.35(-0.82)& 7.41(-3.87) \\
\bottomrule[1pt]
\end{tabular}
\caption{Model ablation study: Effects of Graph Encoder and Knowledge Graph. Number in the parentheses means the absolute value gap between the full version and the ablation one on corresponding metrics.}
\label{ablationstudy}
\end{table*}
\textbf{Ablation Study.} Table \ref{ablationstudy} shows the contributions of each components in our model. Ours without graph encoder means that we do not use the dependency relations information and the proposed recurrent cell architecture. We simply use a bi-directional GRU to serve as the encoder and the other parts of the model remain unchanged. We can observe that our model without the graph encoder has a 1.6\% absolute value loss (over 25\% in ratio) in BLEU score and a 1.1\% absolute value loss (9.8\% in ratio) in entity F1 on MultiWOZ 2.1, which suggests that the overall quality of the generated sentences are better improved by our graph encoder. On the other hand, ours without knowledge graph means that we do not use the graph structure to store and retrieve the external knowledge data. Instead we use memory networks \citep{sukhbaatar2015end} that has been shown useful to handle the knowledge base similar to \citep{wu2019global}. We can find a significant entity F1 drop (3.8\% in absolute value and 33.9\% in ratio) on MultiWOZ 2.1, which verifies the superiority of the proposed graph-based module with multi-hop reasoning ability in retrieving the correct entities, even compared to the strong memory-based baselines.
\textbf{Model Training Time.} We also compare the training time of GraphDialog with those baselines. GraphDialog is about 3 times faster than BERT since its model complexity is smaller. The number of parameters for GraphDialog is almost 90\% less than BERT, which also saves space for model storage. GraphDialog is slower than GLMP, which is expected as it needs to encode more information. However, the gap of the training time is up to 69\%, and we can complete the whole training process within one day which seems reasonable.
\begin{table}[!t]
\setlength{\belowcaptionskip}{-4mm}
\centering
\small
\begin{tabular}{c|c|c|c|c}
\toprule[1pt]
& \multicolumn{4}{c}{Edge Path Distance} \\
\midrule[0.5pt]
Dataset& 1& $\geq$ 2& $\geq$ 10& $\geq$ 15 \\
\midrule[0.5pt]
SMD& 52.82\%& 33.68\%& 10.61\%&2.89\% \\
\midrule[0.5pt]
MultiWOZ 2.1& 50.29\%& 35.41\%& 11.26\%&3.04\% \\
\bottomrule[1pt]
\end{tabular}
\caption{Edge path distance distribution on different datasets.}
\label{edgepath}
\end{table}
\subsection{Analysis and Discussion}\label{analysis}
\textbf{Why does dependency relations help?} We have conducted in-depth analyses from the edge path distance perspective. Table \ref{edgepath} shows the edge path distance distribution in the dialogue graph (Section \ref{dialoguegraph}) on both SMD and MultiWOZ 2.1. The edge path distance is defined as the the number of words between the head word and the tail word along the linear word sequence plus one. For example, for the sentence ``There is a supermarket", the edge distance of the ``Next" edge between ``There" and ``is" is 1, the edge path distance of the ``nsubj" edge between ``is" and ``supermarket" is 2. We can find that although many edges have small edge path distances, there are still a considerable number of edges with relatively large distances, which could encourage more direct information flow between distant words in the input. This may partly explain the benefits of using information such as dependency relations in encoding the dialogue history.
\textbf{Attention Visualization.} To further understand the model dynamics, we analyze the attention weights of the knowledge graph module to show its reasoning process. Figure \ref{kgattention} has shown an example of the attention distribution over all the nodes at the last hop of the knowledge graph. Based on the question ``Where is a nearby parking\_garage?" asked by the user, the generated response of our model is ``palo\_alto\_garage is 1\_miles away", and the gold answer is ``The nearest one is palo\_alto\_garage, it's just 1\_miles away". We can find that our model has successfully learned to copy the correct entities (i.e., \textit{palo\_alto\_garage} at timestep 0 and \textit{1\_miles} at timestep 2) from the knowledge graph.
\textbf{Error Analysis.} To inspire future improvements, we also inspect the generated responses manually. We find that the model tends to omit entities when the responses contain multiple KB entities. Besides, about 10\% of the generated responses contain duplicate KB entities. For example, ``\textit{The temperature in New York on Monday is 100F, 100F}". This may be attributed to the training of GRU in the decoder, and we aim to solve the problem in future work.
\section{Conclusion}
In this work, we present a novel graph-based end-to-end model for task-oriented dialogue systems. The model leverages the graph structural information in dialogue history via the proposed recurrent cell architecture to capture the semantics of dialogue history. The model further exploits the relationships between entities in the KB to achieve better reasoning ability by combining the multi-hop reasoning ability with graph.
We empirically show that our model outperforms the state-of-the-art models on two real-world task-oriented dialogue datasets. Our model may also be applied to end-to-end open-domain chatbots since the goal is to generate responses given inputs and external knowledge, which is what our model can do. We will explore this direction in future work.
\section*{Acknowledgements}
We would like to thank Bayu Distiawan Trisedya for his insightful discussions and the valuable feedbacks from all anonymous reviewers. This work is supported by Australian Research Council (ARC) Discovery
Project DP180102050.
\bibliographystyle{acl_natbib}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,194 |
\section{Introduction} \label{intro}
It is notoriously hard to unravel the nature of a finitely presented group $\Gamma$. In order to do so, one must explore how the group can act on different kinds of objects. The most primitive objects to consider are finite sets. Actions on finite sets capture only the finite images of groups, so the power of such actions to explain the nature of $\Gamma$ is limited by the answer to the fundamental question: to what extent is $\Gamma$ determined by its set of finite quotients? This compelling question has re-emerged with different emphases throughout the history of group theory, and in recent years it has been animated by a rich interplay with geometry and low-dimensional topology, e.g. \cite{BF}, \cite{BCR}, \cite{BridR}, \cite{BRW}, \cite{Fu}, \cite{He}, \cite{Wilk} and \cite{WZ}.
The finite images of $\Gamma$ are encoded in its profinite completion $\widehat{\Gamma}$ (see \S \ref{prelims_profinite}). If $\Gamma$ has elements that do not survive in any finite quotient, then one cannot hope to recover $\Gamma$ by studying $\widehat{\Gamma}$, so it is natural to restrict attention to residually finite groups, i.e.~groups where every finite subset injects into some finite quotient. In its rawest form, the fundamental question then becomes: which finitely generated, residually finite groups $\Gamma$ are {\em profinitely rigid in the absolute sense}, i.e.~have the property that if $\Lambda$ is finitely generated and residually finite, then $\widehat{\Lambda}\cong \widehat{\Gamma}$ implies ${\Lambda}\cong {\Gamma}$.
Variations on this problem have driven many advances in group theory in recent years but there has been very little progress towards identifying infinite groups that are profinitely rigid in the absolute sense. It is easy to see that finitely generated abelian groups are rigid, but beyond that one immediately struggles; virtually cyclic groups need not be rigid for example \cite{Baum74}. If $\Gamma$ satisfies a group law -- for example if it is nilpotent or solvable -- then $\widehat{\Lambda}\cong \widehat{\Gamma}$ will imply that $\Lambda$ satisfies the same law, provided it is residually finite. In such cases, the question of absolute profinite rigidity reduces to a question of {\em relative} profinite rigidity, where one asks if $\widehat{\Gamma}$ distinguishes $\Gamma$ from all other groups in a restricted class. If the relative context is sufficiently tame, this observation allows one to identify examples of profinitely rigid groups: for example, the free nilpotent group of fixed class on a fixed n
umber of generators is profinitely rigid (although many other nilpotent groups are not, \cite{GS}).
The pursuit of relative profinite rigidity theorems has provided a focal point for a rich body of research in recent years, particularly in geometric contexts. This includes many settings in which the groups are {\em{full-sized}} in the sense that they contain non-abelian free subgroups. (Note that a full-sized group cannot satisfy a law.) For example, every Fuchsian group can be distinguished from any other lattice in a connected Lie group by its finite quotients \cite{BCR}. Such relative theorems do not lead to absolute profinite rigidity, however, because in the absence of a group law it is extremely difficult to rule out the possible existence of an utterly exotic $\Lambda$, finitely generated and residually finite, with $\widehat{\Lambda}\cong \widehat{\Gamma}$. Thus, despite our understanding of Fuchsian groups, it remains unknown whether finitely generated free groups are profinitely rigid in the absolute sense (which is an old question of Remeslennikov \cite[Question 5.48]{Rem}).
In this article we provide the first examples of full-sized groups that are profinitely rigid in the absolute sense. Our examples are fundamental groups of 3--dimensional hyperbolic orbifolds, i.e.~lattices in $\PSL(2,\C)$. Our main result is the following.
\begin{theorem}\label{t:main}
There exist arithmetic lattices in $\PSL(2,\C)$ that are profinitely rigid in the absolute sense.
\end{theorem}
These lattices are described in \S \ref{organisation} below, and in more detail in \S \ref{mainplayers}. We provide both uniform and non-uniform examples. For the moment we can construct only finitely many of each, but from these one can manufacture infinitely many commensurability classes of full-sized groups that are profinitely rigid. For instance, in \S \ref{full_sized} we prove that for one of our examples $\Gamma_W$, the group $\Gamma_W\times\mathbb{Z}^r$ is profinitely rigid for every $r\in\mathbb{N}$. Ultimately, one hopes to prove that all Kleinian groups are profinitely rigid, but such a result seems beyond the reach of current technology.
The lattices that we exhibit to prove Theorem \ref{t:main} have several important features in common, some arithmetic, some geometric, and some algebraic. A crucial algebraic characteristic is {\em representation rigidity}. The arguments that we develop around this idea are not specific to dimension $3$, are of a quite general nature, and apply especially in the setting of higher rank lattices. On the other hand, our geometric arguments make very strong use of the subgroup structure of Kleinian groups and $3$--manifold topology, and the corresponding parts of our proof do not extend to lattices in other settings. Indeed, any progress in this direction will require a far better understanding of the infinite index subgroups of general lattices than exists at present.
We now outline how the strategy of our proof proceeds and describe how the key features of the examples enter the argument. First, representation rigidity: each of the lattices $\Gamma$ that we study has very few irreducible representations into $\SL(2,\C)$ up to conjugacy, and all of the representations arise from the arithmetic structure of the lattice. The explanation for this is different in each case, as is the way in which it is exploited, but it is an indispensable feature of our arguments. In particular it allows us to resolve the basic concern that an arbitrary finitely generated, residually finite group $\Lambda$ with $\widehat{\Lambda}\cong\widehat{\Gamma}$ may have no interesting representations to $\SL(2,\C)$. It turns out that one can always construct a useful representation by working one prime at a time. In this way, we relate representations $\Gamma\to{\rm{SL}}(2,\C)$ to bounded representations $\widehat{\Gamma}\to{\rm{SL}}(2,\overline{\Q}_p)$. The restrictions of th
e latter to $\Lambda < \widehat{\Lambda}=\widehat{\Gamma}$ fit together to provide a Zariski-dense representation $\rho\colon\Lambda\to \SL(2,\C)$, and for the lattices that we consider, one can use constraints coming from the arithmetic of $\Gamma$ to force the image of $\rho$ to lie in $\Gamma$, or perhaps a finite extension of $\Gamma$. For more general lattices, it is difficult to control the images of representations constructed by such local techniques. (See the comments below concerning the $(2,3,7)$--triangle group.)
The arguments up to this point (\S \ref{s:rep}) are those that can be modified so as to apply quite generally; this theme is taken up in \cite{McSpit} and \cite{Spitler}. Thus, for example, one can prove that if $\Lambda$ is a finitely generated group with the same finite quotients as ${\SL}(n,\mathbb{Z})$, and $n\ge 3$, then there is a Zariski-dense representation $\Lambda\to {\SL}(n,\mathbb{Z}) < \SL(n,\mathbb{R})$ that induces an isomorphism of profinite completions.
Returning to our setting, it is the special way in which arithmetic and geometry dovetail in dimension $3$ that allows us to complete the proof. The arithmetic structure of a lattice $\Gamma<{\rm{PSL}}(2,\C)$ is encoded in its invariant trace-field $K\Gamma = \Q({\rm{tr}}(\gamma^2) \colon \gamma\in\Gamma)$ and in the quaternion algebra $A\Gamma$ over $K\Gamma$. In all of our examples, $K\Gamma$ is a number field of low degree ($2$ or $3$), $\Gamma$ is closely related to the unit group of a maximal order in $A\Gamma$, and the ramification of $A\Gamma$ at finite places of $K\Gamma$ is highly constrained. On the geometric side, in each case, we also exploit special features of the low index subgroups of $\Gamma$, including the fact that the orbifold $\mathbb{H}^3/\Gamma$ has a finite-sheeted covering of small index that is a surface bundle over the circle, explicit features of which are exploited heavily. Mostow Rigidity and volume calculations also play an important role.
Once we have a Zariski-dense representation $\rho\colon\Lambda\to\Gamma$ in hand, we argue, roughly speaking, that if $\rho$ were not surjective then $\Lambda$ would have a finite quotient that $\Gamma$ does not have, contrary to hypothesis; and since $\widehat{\Lambda}\cong\widehat{\Gamma}$ is Hopfian, the surjectivity of $\hat\rho\colon\widehat{\Lambda} \to \widehat{\Gamma}$ implies injectivity, so $\rho$ is an isomorphism. In each case, arguments from classical 3--manifold topology reduce us quickly to the case where the image of $\rho$ has finite index in $\Gamma$, and at that stage bespoke arguments about the low index subgroups of $\Gamma$ and the finite linear quotients of $\Gamma$ are invoked.
We wish to emphasize that our strategy for proving Theorem \ref{t:main} relies on the remarkable fact that the diverse array of arithmetic, geometric and algebraic features needed in the proof can be found in specific examples. It does not extend in an obvious way to any general classes of Kleinian groups. Looking further afield to the case of $\SL(n,\mathbb{Z})$ with $n\geq 3$, this strategy reduces the issue of profinite rigidity to a question about finite quotients of finitely generated, infinite index, Zariski-dense subgroups of $\SL(n,\mathbb{Z})$; but at this point we are blocked by how little is known about the infinite index subgroups of $\SL(n,\mathbb{Z})$ and the argument goes no further.
It is also interesting to consider how our argument breaks down if one replaces $\Gamma$ by the triangle group $T = \Delta(2,3,7)$, which one also suspects might be profinitely rigid. One again has sufficient control to construct interesting representations of any group $\Lambda$ with $\widehat{\Lambda}\cong\widehat{T}$, but the arithmetic of the invariant trace-field of $T$ imposes less constraint than in the case of the 3--dimensional lattices we consider. Instead of concluding from $\widehat{\Lambda}\cong\widehat{T}$ that there exists a useful homomorphism $\Lambda\to T$, one has to contend with another possibility arising from the three real places of the invariant trace-field $K$. In this case, $\Lambda$ instead has a Zariski-dense representation to the Hilbert modular group $\PSL(2,R_K)$, about whose infinite index subgroups we again know embarrassingly little. There are, however, hyperbolic triangle groups where one has stricter control over their arithmetic, and in \cite{BMRS2} we
shall explain how the methods of the present paper can be extended to prove that certain of these co-compact Fuchsian groups are profinitely rigid in the absolute sense.
\subsection{Organisation of the paper}\label{organisation}
After gathering some basic facts about profinite completions in \S \ref{prelims_profinite} and trace-fields in \S \ref{s:algtraces}, in \S \ref{s:rep}, we prove the general results that lead to the construction of $\rho\colon\Lambda\to{\rm{PSL}}(2,\C)$ as described above. Theorem \ref{T1} is the key result of a general nature in this section, and Corollary \ref{C1} applies it to the lattices $\Gamma< {\rm{PSL}}(2,\C)$ that are the focus of subsequent sections. These lattices are presented in \S \ref{mainplayers}. Prominent among them is $\Gamma=\PSL(2,\mathbb{Z}[\omega])$, where $\omega$ denotes a primitive cube root of unity, i.e.~$\omega^2 + \omega + 1 = 0$. The orbifold $\mathbb{H}^3/\Gamma$ associated to this Bianchi group is a non-compact orientable arithmetic 3--orbifold, which is a 4--sheeted cover of the non-orientable orbifold of minimal volume. We also describe all of the lattices that contain $\Gamma$, as well as the Weeks manifold, which is the unique closed hyperbolic 3--manifold of minimal volume. In \S \ref{fewchars} we prove that, up to conjugacy, the only irreducible representations $\PSL(2,\mathbb{Z}[\omega]) \to{\rm{PSL}}(2,\C)$ are the inclusion and its complex conjugate; this is an instance of {\em Galois rigidity}, a concept that plays an important role in \S \ref{s:rep}. Particular features of the arithmetic of $\Q(\omega)$, the invariant trace-field of $\PSL(2,\mathbb{Z}[\omega])$, also play a key role in \S \ref{fewchars}, as does an extension of Paoluzzi and Zimmermann's detailed analysis \cite{PZ} of epimorphisms $\Gamma\to\PSL(2, \F)$, where $\F$ is a finite field. In \S \ref{s:rigid_bianchi} we prove that $\PSL(2,\mathbb{Z}[\omega])$ is profinitely rigid and in \S \ref{s:others} we prove that all of the lattices that contain it are also rigid. In \S \ref{s:weeks_rigidity} we turn our attention to uniform, torsion-free lattices and prove that the fundamental group of the Weeks manifold is profinitely rigid.
Several of our proofs rely on computer calculations, implemented via Magma \cite{Bos}. The full details of each of these calculations can be found in the supplementary document \cite{magma_calcs}.
\bigskip
\noindent{\bf{Acknowledgements:}}~We want to acknowledge the great intellectual debt that we owe to Alexander Lubotzky. His seminal results and unparalleled vision underpin many aspects of the modern understanding of profinite completions of discrete groups; we thank him for these insights, as well as many helpful conversations. We also thank Matt Stover for helpful conversations. During the period that this work was carried out, we benefited from the hospitality of many institutions, amongst which we mention The Hausdorff Institute for Mathematics (Bonn) which saw the final version of this work put in place. We thank all of them. We acknowledge with gratitude the financial support of the Royal Society (MRB), the National Science Foundation (DBM and AWR), and the Purdue Research Foundation (RS).
\section{Preliminaries concerning profinite groups}\label{prelims_profinite}
In this section we gather some results and remarks about profinite groups that we shall need. Let $\Gamma$ be a finitely generated group. The profinite completion of $\Gamma$ is defined as $\widehat{\Gamma} = \varprojlim \Gamma/N $ where the inverse limit is taken over the normal subgroups of finite index $N\ns\Gamma$ ordered by reverse inclusion. $\widehat{\Gamma}$ is a compact, totally disconnected topological group.
The natural homomorphism $i\colon\Gamma\to\widehat\Gamma$ is injective if and only if $\Gamma$ is residually finite, and the image is always dense. Hence, the restriction to $\Gamma$ of any continuous epimomorphism from $\widehat{\Gamma}$ to a finite group is onto. A deep theorem of Nikolov and Segal \cite{NS} implies that if $\Gamma$ is finitely generated then {\em every} homomorphism from $\widehat{\Gamma}$ to a finite group is continuous, and so every finite index subgroup of $\widehat{\Gamma}$ is open. The following proposition records the correspondence between subgroups of finite index in a discrete group and its profinite completion (see \cite[Prop 3.2.2]{RZ}). Note that we have used \cite{NS} to replace ``open subgroup'' in the profinite setting by ``finite index subgroup''. Given a subset $X$ of a profinite group $G$, we write $\overline X$ to denote the closure of $X$ in $G$.
\begin{proposition}\label{correspondence}
For every finitely generated, residually finite group $\Gamma$, there is a bijection from the set $\mathcal{X}$ of all finite index subgroups of $\Gamma$ to the set $\mathcal{Y}$ of all finite index subgroups of $\widehat{\Gamma}$. If $\Gamma$ is identified with its image in $\widehat{\Gamma}$, then this bijection takes $\Omega\in \mathcal{X}$ to $\overline{\Omega}$, while its inverse takes $\Lambda\in \mathcal{Y}$ to $\Lambda \cap \Gamma$. Also $[\Gamma : \Omega] = [\widehat{\Gamma}: \overline{\Omega}]$. Moreover, $\overline \Omega$ is normal in $\widehat\Gamma$ if and only if $\Omega$ is normal in $\Gamma$, in which case $\Gamma/\Omega\cong\widehat{\Gamma}/\overline \Omega$.
\end{proposition}
\begin{corollary}\label{correspond_corollary}
If $\Gamma_1$ and $\Gamma_2$ are finitely generated groups with $\widehat{\Gamma_1}\cong\widehat{\Gamma_2}$, then there is a one-to-one correspondence between the subgroups of finite index in $\Gamma_1$ and the subgroups of finite index in $\Gamma_2$; this correspondence preserves index and takes normal subgroups to normal subgroups.
\end{corollary}
\begin{proof}
Fixing an identification $\widehat{\Gamma_1}=\widehat{\Gamma_2}$, the correspondence is $\Omega \leftrightarrow \overline{\Omega}\cap\Gamma_2$ for $\Omega<\Gamma_1$.
\end{proof}
We will also find it useful to count conjugacy classes of finite index subgroups.
\begin{lemma}\label{l:conj-classes}
Let $\Gamma_1$ and $\Gamma_2$ be finitely generated groups. If $\widehat{\Gamma_1}\cong\widehat{\Gamma_2}$ then, for every integer $d$, the number of conjugacy classes of subgroups of index $d$ in $\Gamma_1$ is equal to the number in $\Gamma_2$. Moreover, the sizes of the conjugacy classes are the same in $\Gamma_1$ and $\Gamma_2$.
\end{lemma}
\begin{proof}
Each of $\Gamma$ and $\widehat{\Gamma}$ acts on its set of index $d$ subgroups by conjugation and the bijection between these sets $H\mapsto \overline{H}$ is equivariant with respect to $\Gamma\to\widehat{\Gamma}$. Thus the orbit structure for the action of $\Gamma$ is an invariant of $\widehat{\Gamma}$.
\end{proof}
A standard argument shows that finitely generated groups $\Gamma_1,\Gamma_2$ have the same set of finite quotients if and only if $\widehat{\Gamma}_1\cong \widehat{\Gamma}_2$ (see \cite{DFPR}). We also require two other basic facts. First, if ${\rm{Epi}}(\Gamma,Q)$ denotes the set of epimorphisms from the group $\Gamma$ to the finite group $Q$, there is a bijection ${\rm{Epi}}(\widehat{\Gamma},Q)\to {\rm{Epi}}(\Gamma,Q)$ (so if $\widehat{\Gamma}_1 \cong \widehat{\Gamma}_2$ then $\abs{{\rm{Epi}}(\Gamma_1,Q)}=\abs{{\rm{Epi}}(\Gamma_2,Q)}$). Second, we require the following elementary but useful lemma. In the statement of the lemma, $H_1(\Gamma,\mathbb{Z})$ denotes the first integral homology of $\Gamma$ and $b_1(\Gamma)$ denotes the first Betti number of $\Gamma$.
\begin{lemma}\label{l:abel}
Let $\Gamma_1$ and $\Gamma_2$ be finitely generated groups. If $\Gamma_1$ surjects a dense subgroup of $\widehat\Gamma_2$ then $b_1(\Gamma_1)\geq b_1(\Gamma_2)$. If $\widehat\Gamma_1\cong\widehat\Gamma_2$, then $H_1(\Gamma_1,\mathbb{Z})\cong H_1(\Gamma_2,\mathbb{Z})$.
\end{lemma}
Finally, we need to consider the relationship between the profinite completion of a group and the profinite completions of its subgroups. Let $\Gamma$ be a finitely generated, residually finite group with subgroup $\Delta < \Gamma$. The inclusion $\Delta\hookrightarrow\Gamma$ induces a continuous homomorphism $\widehat{\Delta}\to\widehat{\Gamma}$ whose image is $\-{\Delta}$. The map $\widehat{\Delta}\to\widehat{\Gamma}$ is injective if and only if $\-{\Delta}\cong\widehat{\Delta}$; and we say that {\em $\Gamma$ induces the full profinite topology on $\Delta$} when that holds. As $\Gamma$ is finitely generated, this is equivalent to the statement that for every finite index subgroup $I<\Delta$ there is a finite index subgroup $S<\Gamma$ such that $S\cap\Delta \subset I$. Note that if $\Delta<\Gamma$ is of finite index, then $\Gamma$ induces the full profinite topology on $\Delta$.
\section{Finitely generated subgroups of $\PSL(2,\C)$ with algebraic traces}\label{s:algtraces}
To fix notation, it will be convenient to record some basic facts about trace-fields of finitely generated subgroups of $\PSL(2,\C)$. Kleinian groups are subgroups of $\PSL(2,\C)$, but throughout the paper it is often convenient to pass to $\SL(2,\C)$. Indeed many of the arguments are given in $\SL(2,\C)$ and are applied to the pre-image of a subgroup of $\PSL(2,\C)$. For concreteness, let $\phi\colon \mathrm{SL}(2,\mathbb{C}) \to \mathrm{PSL}(2,\mathbb{C})$ be the quotient homomorphism. For $\Delta$ a finitely generated subgroup of $\PSL(2,\mathbb{C})$, we define $\Delta_1 = \phi^{-1}(\Delta)$. It will be convenient to refer to $\Delta$ as being Zariski-dense in $\PSL(2,\C)$, by which we mean $\Delta_1$ is a Zariski-dense subgroup of $\SL(2,\C)$.
\subsection{Trace-fields}\label{ss:traces}
The \textit{trace-field} of $\Delta$ is defined to be $K_\Delta=\mathbb{Q}(\mathrm{tr}(\delta)~\colon~ \delta \in \Delta_1)$. We say that $\Delta$ has {\em integral traces} if $\mbox{\rm{tr}}\, (\delta) \in R_{K_\Delta}$ for all $\delta \in \Delta_1$ where $R_{K_\Delta}$ is the ring of algebraic integers in $K_\Delta$. The field $K_\Delta$ need not be a number field. However, one well-known situation when $K_\Delta$ is a number field is when $\Delta$ is \emph{rigid} (i.e. when $\Delta$ has only finitely many Zariski-dense representations to $(\mathrm{P})\SL(2,\C)$ up to conjugation). Let $\mathrm{X}_{\mathrm{zar}}(\Delta,\C)$ denote the set of Zariski-dense representations of $\Delta$ up to conjugacy in $(\mbox{\rm{P}})\SL(2,\C)$. The following lemma is a consequence of \cite[Prop 6.6]{Rag}.
\begin{lemma} \label{trace_field_number_field}
If $\Delta<(\mathrm{P}) \SL(2,\C)$ is finitely generated and $\abs{\mathrm{X}_{\mathrm{zar}}(\Delta,\C)} < \infty$, then $[K_\Delta:\Q]<\infty$.
\end{lemma}
\subsection{Invariant trace-field and quaternion algebra}\label{invariant}
The \textit{invariant trace-field} of $\Delta$ is defined to be $K\Delta = \mathbb{Q}(\mathrm{tr}(\delta^2) ~\colon~ \delta \in \Delta_1)$. Alternatively, $K\Delta = K_{\Delta^{(2)}}$ where $\Delta^{(2)}$ is the subgroup of $\Delta$ generated by $\{\delta^2 ~\colon~ \delta \in \Delta\}$. As $K_\Delta/K\Delta$ is a multi-quadratic extension of degree $2^s$ for some $s \geq 0$, if $\Delta$ has no $\mathbb{Z}/2\mathbb{Z}$ quotient, then $s=0$ and $K_\Delta = K\Delta$. The group $\Delta_1$ generates a $K_\Delta$--quaternion algebra $A_0\Delta$ and $\Delta_1^{(2)} \leq \Delta_1$ generates a $K\Delta$--quaternion algebra $A\Delta$ called the \textit{invariant quaternion algebra}. When $\Delta_1$ has integral traces, $\Delta_1$ generates an $R_{K_\Delta}$--order $\mathcal{O}\Delta$ in $A_0\Delta$ (see \cite[Ch 3]{MR} for more details on this material). Conversely, if $\Delta_1$ is contained in an order of $A_0\Delta$, then $\Delta_1$ has integral traces. Similar statements hold fo
r the invariant trace-field and quaternion algebra.
\begin{remark} \label{Vinberg}
An alternative description of the invariant trace-field was given by Vinberg \cite{Vin} who showed that $K\Delta = \mathbb{Q}(\mathrm{tr}(\mathrm{Ad}(\delta))~:~ \delta \in \Delta_1)$, where $\mathrm{Ad}\colon \mathrm{SL}(2,\mathbb{C}) \to \mathrm{Aut}(\mathfrak{sl}(2,\mathbb{C}))$ is the adjoint representation.
\end{remark}
\section{Linear Representations and Profinite Completions}\label{s:rep}
In this section we establish the representation theoretic results that will serve as the starting point for the proofs of our main result. The main result of this section is Theorem \ref{T1}, from which we isolate the special cases that will apply to the Kleinian groups described in the introduction; see Corollary \ref{C1} and Examples \ref{ex1}, \ref{ex2}. The material in this section can be adapted to other reductive algebraic groups (see \cite{McSpit}) but we will focus exclusively on ($\mathrm{P}$)${\rm{SL}}_2$, which is the setting for the rigidity results here and in \cite{BMRS2}.
Each of our proofs of profinite rigidity begins with an arithmetic Fuchsian or Kleinian group $i\colon\Gamma\hookrightarrow$ ($\mbox{\rm{P}}$)$\SL(2,\C)$ whose only Zariski-dense representations, up to conjugacy, are the Galois conjugates of the inclusion (see \S 4.3). By way of arithmeticity, this {\em{``Galois rigidity"}} enables us to control the continuous representation theory of $\widehat{\Gamma}$ into ($\mbox{\rm{P}}$)$\SL(2, \overline{\Q}_p)$, for any finite rational prime $p$, and hence gain control on the representation theory of any finitely generated group $\Delta$ with $\widehat{\Gamma}\cong\widehat{\Delta}$. In particular, we obtain a bijection between the conjugacy classes of the Zariski-dense representations of $\Delta$ to $(\mbox{\rm{P}})\SL(2,\C)$ and those of $\Gamma$. Moreover, the arithmetic structure associated to these representations of $\Delta$ and $\Gamma$ are closely related. In the cases that are of particular interest to us, the specific control on the arithmetic data of $\Gamma$ is sufficient to force $\Delta$ to have a Zariski-dense representation whose image lies in $\Gamma$ (or a small index extension of it) and this representation is the key output from this section with regard to profinite rigidity.
\subsection{Preliminaries}\label{prelims}
We recall some standard terminology and fix notation. The set of primes $p \in \mathbb{N}$ together with $\infty$ will be denoted by $\mathrm{P}$. For each $p \in \mathrm{P}$, the metric completion of $\mathbb{Q}$ with respect to the $p$--adic metric will be denoted by $\mathbb{Q}_p$; note that $\mathbb{Q}_\infty=\mathbb{R}$. We will refer to the metric topology on $\mathbb{Q}_p$ and associated spaces (subsets, varieties over $\mathbb{Q}_p$, quotients, etc.) as the \emph{$p$--adic analytic topology}. For each $p \in \mathrm{P} \smallsetminus \set{\infty}$, the $p$-adic norm extends uniquely to the algebraic closure
$\overline{\mathbb{Q}_p}$ of $\mathbb{Q}_p$ which is isomorphic to $\mathbb{C}$ as a field but not as a metric field: any isomorphism will induce maps on associated varieties which are continuous with respect to the Zariski topology but not the $p$--adic analytic topologies.
Given a number field $K/\mathbb{Q}$, we denote the degree of $K$ over $\mathbb{Q}$ by $n_K$ and the ring of algebraic integers of $K$ by $R_K$. For each $p \in \mathrm{P}$, the set of \emph{embeddings} (i.e.,~injective field homomorphisms) $\sigma\colon K \to \overline{\mathbb{Q}_p}$ is denoted by $E_K^p$. Note that $\abs{E_K^p} = n_K$. The set of embeddings of $K$ over all $p \in \mathrm{P}$ is denoted by $E_K$. The set of embeddings over all finite $p$ is denoted by $E_K^f$. For $\sigma \in E_K^\infty$, we say that $\sigma$ is \emph{real} if $\sigma(K) \subset \mathbb{R}$ and that it is \emph{complex} otherwise. The group of continuous (analytic) automorphisms $\Aut_c(\overline{\mathbb{Q}_p})$ acts on $E_K^p$ by post-composition. The orbits are called the \emph{places of $K$ over $p$}. The set of places of $K$ over $p$ is denoted by $V_K^p$. The union of these sets over all $p \in \mathrm{P}\smallsetminus\{\infty\}$ is the set of \emph{finite} places $V_K^f$, and when the
set $V_K^\infty$ of infinite places is added we get the set $V_K$ of all places. Specific infinite places $v \in V_K^\infty$ can be termed \emph{real} or \emph{complex} as the action of $\Aut_c(\mathbb{C})\cong\mathbb{Z}/2$ preserves $\mathbb{R}$. Given $v \in V_K$ associated to $\sigma\colon K \to \overline{\mathbb{Q}_p}$, we denote the metric closure of $\sigma(K)$ in $\overline{\mathbb{Q}_p}$ by $K_v$, noting that the isomorphism class of this locally compact field depends only on $v$.
Given a quaternion algebra $B/K$ and $\sigma \in E_K^p$ with associated $v \in V_K^p$, we set $B_v = B \otimes_{\sigma(K)} K_v$. Note that $B_v$ is isomorphic to the topological closure of $B$ in $B \otimes_{\sigma(K)} \overline{\mathbb{Q}_p} \cong \mathrm{M}(2,\overline{\mathbb{Q}_p})$ in the $p$--adic analytic topology. We say that $B$ is \emph{ramified} at $v \in V_K$ if $B_v$ is a division algebra and we denote the set of ramified places by $\mathrm{Ram}(B)$. We denote the subsets of finite and infinite ramified places of $B$ by $\mathrm{Ram}_f(B)$ and $\mathrm{Ram}_\infty(B)$. By class field theory, a quaternion algebra $B/K$ is determined up to $K$--isomorphism by the set of $K_v$--isomorphism classes $\set{B_v}_{v \in V_K}$; in particular, $B$ is determined by $\mathrm{Ram}(B)$, which is finite and of even cardinality (see for example \cite{MR}). We denote the group of units of $B$ by $B^\times$ and the group of reduced norm one elements by $B^1$. Given an $R_K$--orde
r $\mathcal{O}$ of $B$ set $\mathcal{O}^\times = \mathcal{O} \cap B^\times$ and $\mathcal{O}^1 = \mathcal{O} \cap B^1$. Given $v \in V_K$ associated to $\sigma \in E_K$, we denote the closure of $\sigma(\mathcal{O})$ in $B_v$ by $\mathcal{O}_{v}$ and the closure of $\sigma(R_K)$ in $K_v$ by $R_{v}$. We note that $\mathcal{O}_{v}$ is an $R_{v}$--order in $B_v$.
\subsection{Bounded Representations and Profinite Completions}
Given a finitely generated group $\Lambda$, a homomorphism $ \Lambda \to \mathrm{SL}(2,\overline{\mathbb{Q}_p})$ is said to be \emph{bounded} if its image is bounded (i.e.~pre-compact) in the $p$--adic analytic topology. For all finite primes $p$, the universal property of the profinite completion $\widehat{\Lambda}$ provides a correspondence between representations $\widehat{\Lambda}\to \mathrm{SL}(2,\overline{\mathbb{Q}_p})$ that are continuous in the $p$--adic analytic topology and bounded representations $\Lambda \to \mathrm{SL}(2,\overline{\mathbb{Q}_p})$; see Lemma \ref{L2}. Conjugacy classes are preserved by this correspondence, as is Zariski-denseness. Thus we obtain five related sets of representations, for which it is convenient to have names:
\begin{align*}
\mathrm{X}_b(\Lambda,\overline{\mathbb{Q}_p}) &= \set{\textrm{the conjugacy classes of bounded representations}~~\Lambda \to \mathrm{SL}(2,\overline{\mathbb{Q}_p})} \\
\mathrm{X}_c(\widehat{\Lambda},\overline{\mathbb{Q}_p}) &= \set{\textrm{the conjugacy classes of continuous representations}~~\widehat{\Lambda} \to \mathrm{SL}(2,\overline{\mathbb{Q}_p})}\\
\mathrm{X}_{\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_p}) &= \set{\textrm{the conjugacy classes of Zariski-dense representations}~~\Lambda \to \mathrm{SL}(2,\overline{\mathbb{Q}_p})} \\
\mathrm{X}_{c,\mathrm{zar}}(\widehat{\Lambda},\overline{\mathbb{Q}_p}) &= \set{\textrm{the conjugacy classes of Zariski-dense continuous representations}~~\widehat{\Lambda} \to \mathrm{SL}(2,\overline{\mathbb{Q}_p})}.
\end{align*}
The final set is $\mathrm{X}_{b,\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_p})= \mathrm{X}_b(\Lambda,\overline{\mathbb{Q}_p}) \cap \mathrm{X}_{\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_p})$. Given $p_1,p_2 \in \mathrm{P}$, any field isomorphism $\theta\colon \overline{\mathbb{Q}_{p_1}} \to \overline{\mathbb{Q}_{p_2}}$ induces an isomorphism of abstract groups $\mathrm{SL}(2,\overline{\mathbb{Q}_{p_1}}) \to \mathrm{SL}(2,\overline{\mathbb{Q}_{p_2}})$ and a bijection $\theta_*\colon \mathrm{X}_{\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_{p_1}}) \longrightarrow \mathrm{X}_{\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_{p_2}})$, so
\begin{equation}\label{Eq:SillyEq}
\abs{\mathrm{X}_{\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_{p_1}})} = \abs{\mathrm{X}_{\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_{p_2}})}.
\end{equation}
\begin{proposition}\label{c:new}
If $\Lambda$, $\Delta$ are finitely generated groups with $\widehat{\Lambda}\cong\widehat{\Delta}$ and $\abs{\mathrm{X}_{\mathrm{zar}}(\Lambda,\mathbb{C})}<\infty$, then $\abs{\mathrm{X}_{\mathrm{zar}}(\Delta,\mathbb{C})}=\abs{\mathrm{X}_{\mathrm{zar}}(\Lambda,\mathbb{C})}$.
\end{proposition}
To prove this proposition, we need two lemmas.
\begin{lemma}\label{L2a}
Let $\Lambda$ be a finitely generated group. If $\phi_1,\dots,\phi_n\colon\Lambda\to \SL(2,\C)$ are Zariski-dense representations then for all but finitely many $p\in P\smallsetminus\{\infty\}$ there is a field isomorphism $\theta\colon\C\to \overline{\Q_p}$ such that each of the representations $\theta_*(\phi_i)$ is bounded. If $\mathrm{X}_{\mathrm{zar}}(\Lambda,\mathbb{C})$ is finite then $\abs{\mathrm{X}_{\mathrm{zar}}(\Lambda,\mathbb{C})} = \abs{\mathrm{X}_{b,\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_p})}$ and $\mathrm{X}_{\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_p}) = \mathrm{X}_{b,\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_p})$ for all but finitely many $p\in P\smallsetminus\{\infty\}$.
\end{lemma}
\begin{proof}
Let $R$ be the finitely generated subring of $\mathbb{C}$ generated by the matrix coefficients $\phi_i(\Lambda)$ for $i=1,\dots,n$. By Noether normalization, there exists $\alpha \in \mathbb{Z}$ such that $R[\alpha^{-1}]$ is isomorphic to a finite extension of $\mathbb{Z}[\alpha^{-1}][x_1,...,x_m]$. For any finite $p$ which does not divide $\alpha$, we obtain a ring embedding $\theta\colon R \to \overline{\mathbb{Q}_p}$ such that $\theta(R)$ is bounded by sending the transcendentals $x_1,...,x_m$ to $m$ independent transcendentals in $\mathbb{Z}_p$. This embedding has a unique extension to the field of fractions of $R$, which we identify with a subfield of $\mathbb{C}$. Using the axiom of choice, this in turn can be extended to the desired field isomorphism $\theta\colon \mathbb{C} \to \overline{\mathbb{Q}_p}$. If $\mathrm{X}_{\mathrm{zar}}(\Lambda,\mathbb{C})=\{\phi_1,\dots,\phi_n\}$, then $\theta_*$ defines an injection
$\mathrm{X}_{\mathrm{zar}}(\Lambda,\mathbb{C}) \hookrightarrow \mathrm{X}_{b,\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_p})$ for all $p$ not dividing $\alpha$, and \eqref{Eq:SillyEq} completes the proof.
\end{proof}
When $\widehat{\Gamma}\cong\widehat{\Lambda}$, one can indirectly relate the complex representation theory of $\Gamma$ to that of $\Lambda$ via $\mathrm{X}_c(\widehat{\Lambda},\overline{\mathbb{Q}_p})$.
\begin{lemma}\label{L2}
If $\Lambda$ is finitely generated group, then for each finite $p$, the map $ \mathrm{X}_c(\widehat{\Lambda},\overline{\mathbb{Q}_p}) \to \mathrm{X}_b(\Lambda,\overline{\mathbb{Q}_p})$ given by composing representations with the canonical map $\Lambda\to\widehat{\Lambda}$ is a bijection, and it restricts to a bijection from $\mathrm{X}_{c,\mathrm{zar}}(\widehat{\Lambda},\overline{\mathbb{Q}_p})$ to $\mathrm{X}_{b,\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_p})$.
\end{lemma}
\begin{proof}
The image of any continuous representation $\widehat{\phi}\colon \widehat{\Lambda} \to \mathrm{SL}(2,\overline{\mathbb{Q}_p})$ is compact, so its restriction to the image of $\Lambda$ is bounded. Conversely, given $\phi\colon \Lambda \to \mathrm{SL}(2,\overline{\mathbb{Q}_p})$, since $\phi(\Lambda)$ is finitely generated, there exists a finite extension $F/\mathbb{Q}_p$ such that $\phi(\Lambda) < \mathrm{SL}(2,F)$. The closed subgroup $\mathrm{SL}(2,F) < \mathrm{SL}(2,\overline{\mathbb{Q}_p})$ is locally profinite (i.e.~Hausdorff, locally compact, and totally disconnected), so if $\phi$ is bounded then the topological closure $\overline{\phi(\Lambda)}$ of $\phi(\Lambda)$ is a profinite group, and hence $\phi$ extends to a continuous representation of $\widehat{\Lambda}$. As conjugation in $\mathrm{SL}(2,\overline{\mathbb{Q}_p})$ is continuous and the image of $\Lambda$ is dense in $\widehat{\Lambda}$, the correspondence between continuous representations of $\widehat{\Lambda}$ and bounded r
epresentations of $\Lambda$ induces a bijection $\mathrm{X}_c(\widehat{\Lambda},\overline{\mathbb{Q}_p}) \to \mathrm{X}_b(\Lambda,\overline{\mathbb{Q}_p})$. The correspondence preserves Zariski-denseness because the $p$--adic analytic topology is finer than the Zariski topology.
\end{proof}
\begin{proof}[Proof of Proposition \ref{c:new}]
If $\abs{\mathrm{X}_{\mathrm{zar}}(\Delta,\mathbb{C})}$ were greater than $N(\Lambda):=\abs{\mathrm{X}_{\mathrm{zar}}(\Lambda,\mathbb{C})}$, then Lemma \ref{L2a} would imply for almost all finite primes $\abs{\mathrm{X}_{b,\mathrm{zar}}(\Delta,\overline{\mathbb{Q}_p})}>N(\Lambda)$. By Lemma \ref{L2},
\[ \abs{\mathrm{X}_{b,\mathrm{zar}}(\Delta,\overline{\mathbb{Q}_p})} = \abs{\mathrm{X}_{c,\mathrm{zar}}(\widehat{\Delta},\overline{\mathbb{Q}_p})} = \abs{\mathrm{X}_{c,\mathrm{zar}}(\widehat{\Lambda},\overline{\mathbb{Q}_p})} = \abs{\mathrm{X}_{b,\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_p})}. \]
However,
$N(\Lambda)=\abs{\mathrm{X}_{b,\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_p})}$ for almost all $p$ by Lemma \ref{L2a}, which is a contradiction.
\end{proof}
\subsection{Galois Rigidity}\label{s:galois-rigid}
In order to produce the necessary representations needed for profinite rigidity, we require lattices with the fewest possible Zariski-dense representations. Consider the situation where $\Gamma$ is a finitely generated group with $\rho\colon \Gamma\rightarrow (\mbox{\rm{P}})\SL(2,\C)$ a Zariski-dense representation such that $K=K_{\rho(\Gamma)}$ is a number field of degree $n_K$. As $K=\Q(\theta)$ for some algebraic number $\theta$, the Galois conjugates $\theta=\theta_1,\dots,\theta_{n_K}$ of $\theta$ provide embeddings $\sigma_i\colon K\to\C$ defined by $\theta\mapsto\theta_i$. These in turn can be used to build $n_K$ Zariski-dense non-conjugate representations $\rho_{\sigma_i}\colon \Gamma \to (\mbox{\rm{P}})\SL(2,\C)$ with the property that $\mbox{\rm{tr}}\, (\rho_{\sigma_i}(\gamma))=\sigma_i(\mbox{\rm{tr}}\, \rho(\gamma))$ for all $\gamma\in \Gamma$. We will refer to these as {\em Galois conjugate representations}. We see from this construction that $|\mathrm{X}_{\mathrm{zar}}(\Gamma,\mathbb{C})|\geq n_K$ which motivates the following defini
tion.
\begin{definition}[Galois Rigid]\label{def:galois-rigid}
Let $\Gamma$ be a finitely generated group and let $\rho\colon\Gamma\to (\mbox{\rm{P}})\SL(2,\C)$ be a Zariski-dense representation whose trace-field $K_{\rho(\Gamma)}$ is a number field. If $|\mathrm{X}_{\mathrm{zar}}(\Gamma,\mathbb{C})|= n_{K_{\rho(\Gamma)}}$, we say that $\Gamma$ is {\em Galois rigid} (with associated field $K_{\rho(\Gamma)}$).
\end{definition}
When we say that a subgroup of $(\mbox{\rm{P}})\SL(2,\C)$ is Galois rigid, we implicitly take $\rho$ to be the inclusion map. Note that if $\Gamma$ is Galois rigid, then any irreducible representation with infinite image can serve as $\rho$ as all such representations are Galois conjugate. In particular, $K_{\rho(\Gamma)}$ is an intrinsic invariant of $\Gamma$, as is the quaternion algebra $A_0\Gamma:=A_0\rho(\Gamma)$ and the group homomorphism
$\Gamma \to \rho(\Gamma) \hookrightarrow A_0\Gamma^1$. Integrality of traces plays a key role in what follows and is a necessary
property beyond Galois rigidity. From the discussion in \S \ref{invariant} integrality is assured if $\Gamma$ is contained in an order $\mathcal{O}<A_0\Gamma$.
\begin{lemma}\label{l:integral-traces}
If $\Gamma$ is a finitely generated, residually finite group with the property that for each Zariski-dense representation $\rho\colon\Gamma\to(\mbox{\rm{P}})\SL(2,\C)$ we have $\mathrm{tr}(\rho(\gamma)) \in R_{K_{\rho(\Gamma)}}$ for all $\gamma \in \Gamma$ and for some number field $K_{\rho(\Gamma)}$, then $\mathrm{X}_{\mathrm{zar}}(\Gamma,\overline{\mathbb{Q}_p}) = \mathrm{X}_{b,\mathrm{zar}}(\Gamma,\overline{\mathbb{Q}_p})$ for all primes $p$.
\end{lemma}
\begin{proof}
For convenience we work with $\rho\colon\Gamma\to \SL(2,\C)$, a Zariski-dense representation as in the statement of lemma. We replace $\mathbb{C}$ with $\overline{\Q_p}$ by fixing a field isomorphism $\C\to\overline{\Q_p}$; note that this preserves algebraic integers. Then $K=K_{\rho(\Gamma)}$ is a number field, and the traces of elements $\rho(\gamma)$ are algebraic integers in $K\subset \overline{\Q_p}$. In particular, for any place $v\in V_K^p$, $K_v$ is a finite extension of $\Q_p$, and integrality implies that $\mbox{\rm{tr}}\, (\rho(\gamma))\in R_v$ for all $\gamma \in \Gamma$. In particular, since $R_v$ is compact, $\mbox{\rm{tr}}\, (\rho(\gamma))$ is bounded. Thus the representation $\rho\colon \Gamma\to{\rm{SL}}(2,\overline{\Q_p})$ has bounded traces. But it is a standard argument that a representation with bounded traces is bounded. Briefly, following the construction of \S \ref{invariant}, taking the $R_v$ span of $\rho(\Gamma)$ in $\mathrm{M}(2,\overline{\Q_p})$ determines an $R_v$--order of a quaternion alg
ebra $B_v$. The elements of norm $1$ in this order form a compact subgroup of $\rm{SL}(2,\overline{\Q_p})$ (see \cite[Ch 7]{MR} for example), and so the representation is bounded.
\end{proof}
Combining \eqref{Eq:SillyEq} with Lemmas \ref{L2}, \ref{l:integral-traces} and Proposition \ref{c:new} we have:
\begin{lemma}\label{L:SimRig}
Let $\Lambda$ be a Galois rigid group with associated field $K$, each of whose Zariski-dense representations to $(\mbox{\rm{P}})\SL(2,\C)$ has integral traces. Suppose that $\Delta$ is a finitely generated group with $\widehat{\Delta} \cong \widehat{\Lambda}$. Then,
\begin{itemize}
\item[(i)]
$\abs{\mathrm{X}_{b,\mathrm{zar}}(\Lambda,\overline{\mathbb{Q}_p})} = n_K$ for all finite $p \in \mathrm{P}$,
\item[(ii)]
$\abs{\mathrm{X}_{\mathrm{zar}}(\Delta,\mathbb{C})} = n_K$,
\item[(iii)]
$\mathrm{X}_{b,\mathrm{zar}}(\Delta,\overline{\mathbb{Q}_p}) = \mathrm{X}_{\mathrm{zar}}(\Delta,\overline{\mathbb{Q}_p})$ for all finite $p$.
\end{itemize}
\end{lemma}
\begin{remark} \label{galois_FA}
If $\Gamma$ is a finitely generated group with \emph{Serre's Property FA} (i.e.~$\Gamma$ cannot act on a tree without a global fixed point), then the $(\mbox{\rm{P}})\SL(2,\C)$--character variety $\mathrm{X}(\Gamma,\C)$ is finite (see \cite{BZ}). In particular, any group with Property FA has the property that the set $\mathrm{X}_{\mathrm{zar}}(\Gamma,\C) \subset \mathrm{X}(\Gamma,\C)$ is finite. However, Property FA and Galois rigidity are distinct forms of rigidity. For instance, the triangle group $\Delta(6,6,6)$ has Property FA but is not Galois rigid. On the other hand, there are rational homology 3--manifolds $\Sigma$ such that $\Sigma$ is hyperbolic and $\pi_1\Sigma$ is Galois rigid, but $\pi_1\Sigma$ does not have property FA because $\Sigma$ is Haken; the result of $(10,1)$--Dehn surgery on the knot $5_2$ is such a manifold. There are also hyperbolic 3--manifolds $M$ with $b_1(M) >0$ whose fundamental group is Galois rigid, for example the manifold $M_6$ described in \S \ref{s:weeks_rigidity} below.
\end{remark}
\subsection{Profinite rigidity via Galois rigidity}\label{profinite_from_galois}
We now fix a number field $K$, a quaternion algebra $B/K$, and a maximal order $\mathcal{O} < B$. We will assume that $\Gamma < \mathcal{O}^1$ is a finitely generated subgroup such that $K_\Gamma= K$ and we identify $A_0\Gamma$ with $B$. We denote the inclusion $\Gamma \to B^1$ by $\phi$. We now state the main technical result of this section.
\begin{theorem}\label{T1}
Let $K,B,\mathcal{O}$, and $\Gamma$ be as above, assume that $\Gamma$ is Galois rigid and assume that $\Delta$ is a finitely generated residually finite group with $\widehat{\Delta} \cong \widehat{\Gamma}$. Then there is a number field $K'$, a quaternion algebra $B'/K'$, a maximal order $\mathcal{O}' < B'$, and a Zariski-dense homomorphism $\phi'\colon \Delta \to (\mathcal{O}')^1 < (B')^1 \subset \SL(2,\C)$ such that the following conditions hold:
\begin{itemize}
\item[(i)]
$\Delta$ is Galois rigid with associated field $K'$.
\item[(ii)]
There are bijective functions $\tilde{\tau}\colon E_{K'} \to E_K$ and $\tau\colon V_{K'} \to V_{K}$ with $K'_w \cong K_{\tau(w)}$ for all $w \in V_{K'}$. Hence, $K$ and $K'$ are arithmetically equivalent and have isomorphic adele rings.
\item[(iii)]
$B'_w \cong B_{\tau(w)}$ for all $w \in V_{K'}^f$.
\item[(iv)]
Up to isomorphism, there are only finitely many possibilities for $K'$, $B'$, and $\mathcal{O}'$.
\end{itemize}
\end{theorem}
\noindent An extension of Theorem \ref{T1} covering more general algebraic groups is developed in the Ph.D thesis of the fourth author \cite{Spitler}.
\begin{proof}[Proof of Theorem \ref{T1}]
By Lemma \ref{L:SimRig} (ii), $\mathrm{X}_{\mathrm{zar}}(\Delta,\mathbb{C})$ is nonempty. Let $\psi \in \mathrm{X}_{\mathrm{zar}}(\Delta,\mathbb{C})$, set $K' =K_{\psi(\Delta)}$, let $B'/K'$ be the quaternion algebra $A_0\psi(\Delta)$ and let $\phi'\colon \Delta \to (B')^1$ denote the homomorphism given by viewing $B'$ as an abstract quaternion algebra. By Lemma \ref{L:SimRig} (ii), we have $\abs{\mathrm{X}_{\mathrm{zar}}(\Delta,\mathbb{C})} = n_K$, so $n_{K'}\leq n_K$, and we conclude that $K'$ is a number field. To show that $\Delta$ is Galois rigid we need to prove that $n_{K'} = n_K$. This will be done by establishing that $K$ and $K'$ are arithmetically equivalent, from which $n_K = n_{K'}$ follows (see \cite{Per}). Indeed we will prove that $\mathbb{A}_K$ and $\mathbb{A}_{K'}$, the adele rings of $K$ and $K'$, are isomorphic.
The first part of this is to construct bijective functions $\tau_p\colon V_{K'}^p \to V_K^p$ for each $p \in \mathrm{P}\setminus \{\infty\}$ such that $K_{v'}' \cong K_{\tau_p(v')}$ for each $v' \in V_{K'}^p$. To that end, using the homomorphism $\phi'\colon \Delta \to (B')^1$ and $p \in \mathrm{P}\setminus \{\infty\}$, we obtain $n_{K'}$ non-conjugate, Zariski-dense representations $\{\phi'_{\sigma'_1},\dots,\phi'_{\sigma'_{n_{K'}}}\}$ into $\mathrm{SL}(2,\overline{\mathbb{Q}_p})$ as follows. For $\sigma' \in E_{K'}^p$ associated to $w \in V_{K'}^p$, we have an algebra injection $B' \to B'\otimes_{\sigma'(K')} K'_w \subset \mathrm{M}(2,\overline{\mathbb{Q}_p})$ which induces an injective homomorphism when restricted to $\phi'(\Delta)$ and hence a homomorphism $\phi'_{\sigma'}\colon \Delta \to \mathrm{SL}(2,\overline{\mathbb{Q}_p})$ with Zariski dense image. Hence $\phi'_{\sigma'} \in \mathrm{X}_{\mathrm{zar}}(\Delta,\overline{\mathbb{Q}_p})$.
For distinct $\sigma'_1,\sigma'_2 \in E_{K'}^p$, the representations $\phi'_{\sigma'_1},\phi'_{\sigma'_2}$ are not conjugate as they have distinct characters.
Since $\widehat{\Gamma}\cong\widehat{\Delta}$, from Lemma \ref{L:SimRig} (i) we have $\abs{\mathrm{X}_{b,\mathrm{zar}}(\Gamma,\overline{\mathbb{Q}_p})} = n_K$, and by Lemma \ref{L:SimRig} (iii), $\mathrm{X}_{b,\mathrm{zar}}(\Delta,\overline{\mathbb{Q}_p}) = \mathrm{X}_{\mathrm{zar}}(\Delta,\overline{\mathbb{Q}_p})$ for all finite $p$.
Given this set up, for $\sigma' \in E_{K'}^p$, we can define functions $\tilde{\tau}_p : E_{K'}^p \to E_K^p$ as follows: as noted above, the representation $\phi'_{\sigma'}\colon \Delta \to \mathrm{SL}(2,\overline{\mathbb{Q}_p})$ is Zariski-dense with bounded image; using $\widehat{\Gamma}\cong \widehat{\Delta}$, we deduce from Lemma \ref{L2} that there is a unique $\sigma \in E_K^p$ so that $\widehat{\phi'_{\sigma'}}$ and $\widehat{\phi_\sigma}$ are conjugate representations, and so we set $\tilde{\tau}_p(\sigma') = \sigma$. Note that $\tilde{\tau}_p\colon E_{K'}^p \to E_K^p$ is injective since the representations induced by distinct embeddings of $K'$ are not conjugate.
To show that the functions $\tilde{\tau}_p$ give rise to injective functions $\tau_p\colon V_{K'}^p \to V_K^p$, we must show that if $\sigma'_1,\sigma'_2 \in E_{K'}^p$ are $\mathrm{Aut}_c(\overline{\mathbb{Q}_p})$--equivalent, then $\tilde{\tau}_p(\sigma'_1),\tilde{\tau}_p(\sigma'_2) \in E_K^p$ are $\mathrm{Aut}_c(\overline{\mathbb{Q}_p})$--equivalent. Let $\sigma'_1,\sigma'_2 \in E_{K'}^p$ with $\sigma'_2 = \varphi \circ \sigma'_1$ for some $\varphi \in \mathrm{Aut}_c(\overline{\mathbb{Q}_p})$ and set $\sigma_i = \tilde{\tau}_p(\sigma'_i)$ for $i=1,2$. By definition of $\tilde{\tau}_p$, we have continuous Zariski-dense representations
\[ \widehat{\phi'_{\sigma'_1}},~\widehat{\phi'_{\sigma'_2}},~\widehat{\phi_{\sigma_1}},~\widehat{\phi_{\sigma_2}}\colon \widehat{\Gamma}\longrightarrow \mathrm{SL}(2,\overline{\mathbb{Q}_p}) \]
such that $\widehat{\phi'_{\sigma'_i}}$ is conjugate to $\widehat{\phi_{\sigma_i}}$ for $i=1,2$ and $\widehat{\phi'_{\sigma'_2}} = \varphi_* \circ \widehat{\phi'_{\sigma'_1}}$ where $\varphi_*\colon \mathrm{SL}(2,\overline{\mathbb{Q}_p}) \to \mathrm{SL}(2,\overline{\mathbb{Q}_p})$ is the automorphism induced by $\varphi$. So up to conjugation, $\widehat{\phi_{\sigma_2}} = \varphi_* \circ \widehat{\phi_{\sigma_1}}$, and restricting to $\Gamma$ gives $\phi_{\sigma_2} = \varphi_* \circ \phi_{\sigma_1}$ up to conjugation, so $\sigma_2 = \varphi \circ \sigma_1$. Thus, $\tilde{\tau}_p$ induces an injective function $\tau_p\colon V_{K'}^p \longrightarrow V_K^p$. Varying $p \in \mathrm{P}\setminus \set{\infty}$ induces injective functions $\tilde{\tau}_f\colon E_{K'}^f \to E_K^f$ and $\tau_f\colon V_{K'}^f \to V_K^f$.
Since $\Delta$ and $\Gamma$ are dense in $\widehat{\Delta}\cong\widehat{\Gamma}$, for each finite embedding $\sigma' \in E_{K'}^f$, the sets $\set{\mathrm{tr}~\phi'_{\sigma'}(\delta)}_{\delta \in \Delta}$ and $\set{\mathrm{tr}~\phi_{\tilde{\tau}(\sigma')}(\gamma)}_{\gamma \in \Gamma}$ are dense in
\[ \set{\mathrm{tr}~\widehat{\phi'_{\sigma'}}(x)}_{x \in \widehat{\Delta}} = \set{\mathrm{tr}~\widehat{\phi_{\tilde{\tau}(\sigma')}}(x)}_{x \in \widehat{\Gamma}}. \]
Hence, the fields generated by $\set{\mathrm{tr}~\phi'_{\sigma'}(\delta)}_{\delta \in \Delta}$ and $\set{\mathrm{tr}~\phi_{\tilde{\tau}(\sigma')}(\gamma)}_{\gamma \in \Gamma}$ have the same closures and so $K'_w \cong K_{\tau_f(w)}$ where $w \in V_{K'}^f$ is associated to $\sigma'$. To establish that $\tau_f$ is a bijection we need the following lemma. This appears to be well-known, but we could not find a proof in the literature, so we give it in the Appendix.
\begin{lemma}\label{L4}
If $K$, $K'$ are number fields and $\tau_f\colon V_{K'}^f \to V_K^f$ is an injective map with $K'_w \cong K_{\tau_f(w)}$ for all $w \in V_{K'}^f$, then $\tau_f$ is a bijection.
\end{lemma}
Given this, $\tau_f$ is a bijection, and hence $\tilde{\tau}_p,\tilde{\tau}_f,\tau_p$, and $\tau_f$ are all bijections. That $\tilde{\tau}_f, \tau_f$ can be extended to bijective functions $\tilde{\tau}\colon E_{K'} \to E_K$ and $\tau\colon V_{K'} \to V_K$ follows from the fact that $\tau_f$ being a bijection implies that $K$ and $K'$ are arithmetically equivalent (see \cite{Per}), and hence $K$ and $K'$ have the same number of real and complex embeddings/places. Hence, $n_K = n_{K'}$. This establishes (i).
Now Iwasawa \cite{Iwasawa} proved that the existence of $\tau$ is equivalent to the fields $K$ and $K'$ having isomorphic adele rings. This establishes (ii).
To prove (iii), we argue as follows. The algebras $B$ and $B'$ are generated over $K$ and $K'$ by $\phi(\Gamma)$ and $\phi'(\Delta)$ respectively. Using the bijection $\tau$ established in (ii), these quaternion algebras are also generated over $K$ and $K'$ by $\phi_{\tilde{\tau}(\sigma')}(\Gamma)$ and $\phi'_{\sigma'}(\Delta)$ respectively (up to isomorphism). Moreover, if we now take the algebras generated by $\phi_{\tilde{\tau}(\sigma')}(\Gamma)$ and $\phi'_{\sigma'}(\Delta)$ over $ K_{\tau(w)} = K'_w$ we obtain quaternion algebras $B_{0,\tau(w)}$ and $B'_{0,w}$ that are isomorphic to $B_{\tau(w)}$ and $B'_w$ respectively. Now these quaternion algebras are also generated over $K_{\tau(w)} = K'_w$ by $\widehat{\phi_{\tilde{\tau}(\sigma')}}(\widehat{\Gamma})$ and $\widehat{\phi'_{\sigma'}}(\widehat{\Delta})$. By the construction in the proof of (i), these groups are conjugate, and so the quaternion algebras $B_{0,\tau(w)}$ and $B'_{0,w}$ are isomorphic. Hence $B_{\tau(w)}
$ and $B'_w$ are isomorphic as required.
It remains to show that $\phi'$ has image contained in some maximal order $\mathcal{O}'$ of $B'$. As noted in \S \ref{invariant}, it suffices to show $\set{\mathrm{tr}~\phi'(\delta)}_{\delta \in \Delta} \subset R_{K'}$ To that end, let $R_{w'}$ be the associated local ring for $w' \in V_{K'}^f$. As the representations $\phi'_{\sigma'}$ are bounded for each $\sigma' \in E_{K'}^p$ and each finite $p$, we have
\[ \set{\mathrm{tr}~\phi'_{\sigma'}(\delta)}_{\delta \in \Delta} \subset R_{w'} \]
for each $w \in V_{K'}^f$. Since $\set{\mathrm{tr}~\phi'_{\sigma'}(\delta)}_{\delta \in \Delta} \subset K'$ for all $\sigma' \in E_{K'}^f$ and
\[ R_{K'} = \bigcap_{w' \in V_{K'}^f} (K' \cap R_{w'}), \]
we have $\set{\mathrm{tr}~\phi'(\delta)}_{\delta \in \Delta} \subset R_{K'}$ as needed. Thus, $\phi'(\Delta)$ generates an order over $R_{K'}$ which is contained in some maximal order $\mathcal{O}'$.
For (iv), we must analyze how $K'$, $B'$, and $\mathcal{O}'_{B'}$ can fail to be uniquely determined. Since $K$ and $K'$ are arithmetically equivalent, they have the same Galois closure, and so there are a finite number of possibilities for $K'$. For each possible $K'$, the quaternion algebra $B'/K'$ is determined by the following information:
\begin{itemize}
\item[(1)]
For each $v \in \mathrm{Ram}_f(B)$, a choice of $w \in \mathrm{Ram}_f(B')$ with $K_v \cong K'_w$.
\item[(2)]
With (1), we get a bijective function $\mathrm{Ram}_f(B') \to \mathrm{Ram}_f(B)$ and this determines $\mathrm{Ram}_f(B')$. It only remains to determine the possible choices for $\mathrm{Ram}_\infty(B')$. If $\abs{\mathrm{Ram}_f(B)}$ is even, then we can take $S=\mathrm{Ram}_\infty(B')$ for any set $S$ of real places with $\abs{S}$ even. If $\abs{\mathrm{Ram}_f(B)}$ is odd, then we can take $S=\mathrm{Ram}_\infty(B')$ for any set $S$ of real places with $\abs{S}$ odd.
\end{itemize}
As $\mathrm{Ram}(B')$ is finite, there are only finitely many possibilities for $B'$. Finally, fixing $B'$, there are only finitely many maximal orders $\mathcal{O}'_{B'}$ up to isomorphism. Hence, we can then take our list of possible codomain groups for $\phi'$ to be $(\mathcal{O}'_{B'})^1$ where $K'$, $B'$, and $\mathcal{O}'_{B'}$ each range over all of the above choices.
\end{proof}
We record a specific corollary that will be utilized in our proofs (a variation of which is also used in \cite{BMRS2}).
\begin{definition}[Locally uniform]
We say that a quaternion algebra $B/K$ is \emph{locally uniform} if for each $v,v' \in V_K^f$ with $K_v \cong K_{v'}$ we have $B_v \cong B_{v'}$.
\end{definition}
\begin{corollary}\label{C1}
Let $\Gamma,\Delta$, $K,K'$, $B$, $B'$ be as in Theorem \ref{T1}.
\begin{itemize}
\item[(i)]
If $K$ is Galois or has exactly one complex place, then $K' \cong K$.
\item[(ii)]
If $K' \cong K$, $K$ has at most one real place, and $B$ is locally uniform, then $B' \cong B$.
\item[(iii)]
\begin{itemize}
\item[(a)]
If $K$ is imaginary quadratic and $\mathrm{Ram}(B)=\emptyset$, then $K \cong K'$ and $B \cong B'$.
\item[(b)]
If $n_K=3$, $K$ has one real place, and $\mathrm{Ram}(B) = \set{v_1,v_2}$ where $v_1$ is a real place and $v_2 \in V_K^p$ for a finite $p$ with $V_K^p = \set{v_2}$, then $K \cong K'$ and $B \cong B'$.
\end{itemize}
\item[(iv)]
If $K,B$ satisfy (ii), (iii) (a) or (iii) (b), then there exists a Zariski-dense representation $\phi'\colon \Delta \to \mathcal{O}_B^1$ for some maximal order $\mathcal{O}_B < B$.
\end{itemize}
\end{corollary}
\begin{proof}
For (i), by Theorem \ref{T1} (ii), $K,K'$ are arithmetically equivalent, and so have isomorphic Galois closures by \cite[Thm 1]{Per}. In particular, when $K$ is Galois, then $K \cong K'$. If $K$ has exactly one complex place, then $K \cong K'$ by \cite[Cor 1.2]{CHLR}.
For (ii), we now assume that $K \cong K'$ and view $B'$ as a quaternion algebra over $K$. We also view $\tau_f$ as a bijective function $\tau_f\colon V_K^f \to V_K^f$ with $K_v \cong K_{\tau(v)}$ and $B'_v \cong B_{\tau(v)}$ for each $v \in V_K^f$. To prove that $B,B'$ are isomorphic, it suffices to prove that $B_v \cong B_v'$ for all $v \in V_K$. For $v \in V_K^f$, set $w = \tau_f^{-1}(v)$. As $K_v \cong K_w$ and $B$ is locally uniform, $B_v \cong B_w$. By Theorem \ref{T1}, we have $B_w \cong B_v'$ and so $B_v \cong B_v'$. It remains to prove $B_v \cong B_v'$ when $v \in V_K^\infty$. At each complex place $v$, we have $B_v \cong B_v' \cong \mathrm{M}(2,\mathbb{C})$. As $K$ has at most one real place, whether or not $\mathrm{Ram}(B)$, $\mathrm{Ram}(B')$ contains the real place (if it exists) depends only on the even/odd parity of $\abs{\mathrm{Ram}_f(B)}$, $\abs{\mathrm{Ram}_f(B')}$. As $\mathrm{Ram}_f(B) = \mathrm{Ram}_f(B')$ by the above, we see that either both $B,B'$ ram
ify at the real place or both $B,B'$ split over the real place. In particular, $B_v \cong B_v'$ when $v$ is real. This proves that $B_v \cong B_v'$ for all $v \in V_K$ as needed.
Parts (a) and (b) of (iii) follow from (i) and (ii). Part (iv) follows from (a) and (b) of (iii) and Theorem \ref{T1}.
\end{proof}
\noindent We single out two particular cases of Corollary \ref{C1} that will be important in what follows.
\begin{example}\label{ex1}
In the setting of Corollary \ref{C1}, suppose that $\Gamma$ is a non-uniform lattice in $\PSL(2,\C)$ with $K_\Gamma=\Q(\sqrt{-3})$. In this case $B\cong \mathrm{M}(2,\Q(\sqrt{-3}))$, and selecting $\mathcal{O} = \mathrm{M}(2,\mathbb{Z}[\omega])$ where $\omega^2+\omega+1=0$, Corollary \ref{C1}(iii)(a) applies. By Corollary \ref{C1}(iv), if $\Delta$ is a finitely generated, residually finite group with $\widehat{\Delta}\cong \widehat{\Gamma}$, then there is a maximal order $\mathcal{L} < \mathrm{M}(2,\Q(\sqrt{-3}))$ and we have $\phi':\Delta \to \mathcal{L}^1$. In this case, since the class number of $\Q(\sqrt{-3})$ is $1$, $\mathrm{M}(2,\Q(\sqrt{-3}))$ has type number $1$: there is a unique conjugacy class of maximal order in $\mathrm{M}(2,\Q(\sqrt{-3}))$ (see \cite[Ch 7.6]{MR}). Thus we may conjugate so that both $\Gamma$ and $\phi'(\Delta)$ are contained as subgroups of $(\mbox{\rm{P}})\SL(2,\mathbb{Z}[\omega])$.
\end{example}
\begin{example}\label{ex2}
Assume $\Gamma$ is as in Corollary \ref{C1}, that $K_\Gamma=\Q(\theta)$, that $\theta^3-\theta^2+1=0$, and that $B$ is ramified at both the real place of $K_\Gamma$ and the unique place $\nu$ of norm $5$. Then Corollary \ref{C1}(iii)(b) applies. As in Example \ref{ex1}, we can deduce from Corollary \ref{C1}(iv) that for any finitely generated, residually finite group with $\widehat{\Delta}\cong \widehat{\Gamma}$ there is a maximal order $\mathcal{O}_B < B$ with $\Delta \to \mathcal{O}_B^1$. As in Example \ref{ex1}, the class number of $K_\Gamma$ is $1$, so $B$ has type number $1$, and hence we can conjugate so that $\Gamma$ and the image of $\Delta$ are both contained as subgroups of $\mathcal{O}_B^1$.
\end{example}
\subsection{Congruence quotients}\label{congruence_quotients}
The results and proofs contained in this section allow us to make some additional conclusions about congruence quotients that will be useful later. Suppose that $\Gamma$ is as in the statement of Theorem \ref{T1}, and suppose also that $\Gamma$ has the following property: {\em Every finite quotient of $\Gamma$ the form $(\mbox{\rm{P}})\SL(2,\mathbb{F}_{p^n})$ arises as a quotient of $\widehat{\phi_\sigma}(\widehat{\Gamma})$ for some $\sigma$; i.e.~as a congruence quotient.} Then this property passes to $\Delta$, in the following sense.
Theorem \ref{T1} gives a Zariski-dense representation $\phi'$ of $\Delta$ and the proof of that theorem exhibits a family of finite quotients of the form $(\mbox{\rm{P}})\SL(2,\mathbb{F}_{p^n})$
by restriction of $\widehat{\phi'_{\tilde{\tau}^{-1}(\sigma)}}$ to $\Delta \subset \widehat{\Delta}$. Moreover, since $\widehat{\Delta}\cong \widehat{\Gamma}$, all such quotients of $\Delta$ (and $\phi'(\Delta)$) will arise as a quotient of $\widehat{\phi'_{\sigma'}}(\widehat{\Delta})$ for some $\sigma' \in E_{K'}^p$, i.e. as a congruence quotient.
This construction can be made more explicit in the setting of Corollary \ref{C1}, where we have a Zariski-dense representation $\phi'\colon \Delta \to \mathcal{O}^1$ for some maximal $R_K$--order $\mathcal{O} \subset B$ (with $K=K_\Gamma=K_{\phi'(\Delta)}$). For convenience
assume that $\phi'(\Delta)=L < \Gamma$. Let $v\in V_K^f$ with local ring $R_v$ having a local uniformizer $\pi_v$. Assume that $B$ is unramified at $v$. In such a case, the finite quotients of the form $(\mbox{\rm{P}})\SL(2,\mathbb{F}_{p^n})$ come from restricting the following sequence of homomorphisms to $\Gamma$ and $L$:
\[ \mathcal{O}^1\hookrightarrow \SL(2,R_v)\rightarrow \SL(2,R_v/\pi_vR_v) \cong \SL(2,\mathbb{F}_{p^n}). \]
If we denote the composition of these homomorphisms by $\eta_v$, and restrict to $\Gamma$ and to $L$, then $\ker\eta_{v}$ determines subgroups $\Gamma(v)=\Gamma\cap \ker\eta_v$ and $L(v)=L\cap \ker\eta_v$. Our arguments show that $\Gamma/\Gamma(v) = L/L(v)$.
\section{The main players}\label{mainplayers}
In this section we describe some of the important features of the groups that we shall prove are profinitely rigid.
\subsection{The group $\Gamma=\PSL(2,\mathbb{Z}[\omega])$ and other non-uniform lattices of small covolume in ${\rm{Isom}}(\mathbb{H}^3)$}\label{s:small}
Consider a regular ideal tetrahedron $\tau$ in $\mathbb{H}^3$; this is unique up to isometry and we write $v_0$ to denote its volume. Let $c_0$ be the centre of the inscribed sphere in $\tau$. This sphere touches each 2--dimensional face $f$ at the centre $c(f)$ of the inscribed circle in $f$ and this circle touches each edge $e$ in the boundary of $f$ at a point we denote $c(e)$. Let $\xi$ be an ideal vertex of $e$ and consider the ideal tetrahedron $T_0$ with vertices $(\xi,c(e),c(f),c_0)$. Note that $\tau$ decomposes into $24$ copies of $T_0$ corresponding to the different choices of $(\xi,e,f)$; thus $T_0$ has volume $v_0/24$. The dihedral angles of $T_0$ can be calculated by observing the number of translates of $T_0$ around each edge; the angles at the ideal vertices are $\pi/6, \pi/2, \pi/3$ and at the opposite edges they are $\pi/3, \pi/2, \pi/2$. Thus, in the notation of \cite{BLW}, $T_0=T[3,2,2; 6,2,3]$. We write $\Lambda_0$ to denote the subgroup of ${\rm{Isom}}(\mathbb{H}^3)$ gen
erated by reflections in the hyperplanes containing the faces of $T_0$. This is (up to conjugacy) the unique non-uniform lattice of smallest co-volume in ${\rm{Isom}}(\mathbb{H}^3)$ -- see \cite{Mey}; it has $T_0$ as fundamental domain, so its co-volume is $v_0/24$. Below is a presentation for $\Lambda_0$ that we will use later:
\[ \Lambda_0 = \innp{ x,y,z,w~\mid~ x^2 = y^2 = z^3 = w^2 = (xy)^3 = (xz)^2 = (xw)^2 = (yz)^3 = (yw)^2 = (zw)^6 = 1}. \]
The index $2$ subgroup $\Gamma_0<\Lambda_0$ consisting of orientation-preserving isometries is $\PGL(2,\mathbb{Z}[\omega])$, and from \cite{BLW} we have the presentation
\[ \Gamma_0 = \innp{ x,y,z ~\mid~ x^3= y^2= z^2 = (yx^{-1})^3 = (zx^{-1})^2 = (yz)^6 =1}. \]
$T_0$ has one face in the boundary of $\tau$ and the union of $T_0$ with its reflection in this face is the tetrahedron $T_1=T[3,2,2; 3,3,3]$. We write $\Lambda_1$ to denote the subgroup of ${\rm{Isom}}(\mathbb{H}^3)$ generated by reflections in the hyperplanes containing the faces of $T_1$; it has covolume $v_0/12$ and is an index 2 subgroup of $\Lambda_0$. The following is the natural (Coxeter) presentation:
\[ \Lambda_1 = \innp{ x,y,z,w ~\mid~ x^2 = y^2 = z^2 = w^2 = (xy)^2 = (xz)^2 = (xw)^3 = (yz)^3 = (yw)^3 = (zw)^3 =1}. \]
The index 2 subgroup of orientation preserving isometries in $\Lambda_1$ is $\Gamma=\PSL(2,\mathbb{Z}[\omega])$; thus $\Gamma$ has covolume $v_0/6$ and as in \cite{BLW} we have the presentation
\begin{equation}\label{present}
\Gamma = \innp{ x,y,z ~\mid~ x^3 = y^2 = z^2 = (yx^{-1})^3 = (zx^{-1})^3 = (yz)^3 = 1}.
\end{equation}
\begin{lemma}\label{l:abelG}
The abelianizations of $\Gamma, \Gamma_0, \Lambda_0, \Lambda_1$ are, respectively, $\mathbb{Z}/3\mathbb{Z}, \mathbb{Z}/2\mathbb{Z}, (\mathbb{Z}/2)^2$ and $\mathbb{Z}/2\mathbb{Z}$.
\end{lemma}
\begin{proof}
The abelianizations of $\Gamma,\, \Gamma_0$ and $\Lambda_1$ are readily calculated from the above presentations. As $\Gamma_0$ has index 2 in $\Lambda_0$, the abelianization of $\Lambda_0$ has order at most 4. And since $\Lambda_0$ has a further subgroup of index 2, namely $\Lambda_1$, we have $H_1(\Lambda_0,\mathbb{Z}) = (\mathbb{Z}/2\mathbb{Z})^2$.
\end{proof}
Since $H_1(\Lambda_0,\mathbb{Z}) = (\mathbb{Z}/2\mathbb{Z})^2$, there is a third subgroup of index $2$ in $\Lambda_0$ besides $\Gamma_0$ and $\Lambda_1$. We denote this subgroup by $\Lambda_2$, and note that since $\Gamma$ is the commutator subgroup of $\Lambda_0$, we have $\Gamma<\Lambda_2$. We will make use of the following presentation for $\Lambda_2$ in the proofs of the main results:
\[ \Lambda_2 = \innp{x,y,z ~\mid~ x^2 = y^2= z^3 = (zx)^3 = (xy)^6 = zyz^{-1}y = 1}. \]
Note that the abelianization of $\Lambda_2$ is isomorphic to $\mathbb{Z}/6\mathbb{Z}$.
We also include a diagram of the above subgroups for the reader's convenience
\begin{equation}\label{Eq:Natural}
\begin{tikzcd}
& & \Lambda_0 \arrow[rrdd,"2",dash] \arrow[lldd,"2"',dash] \arrow[dd,"2"',dash] & & \\ & & & & \\ \Lambda_1 \arrow[rrdd,"2"', dash] & & \Gamma_0 = \PGL(2,\mathbb{Z}[\omega]) \arrow[dd,"2"', dash] & &\Lambda_2 \arrow[lldd,"2", dash] \\ & & & & \\ & & \Gamma = \PSL(2,\mathbb{Z}[\omega]) & &
\end{tikzcd}
\end{equation}
\subsection{The five lattices $\Gamma, \Gamma_0, \Lambda_0$, and $\Lambda_1$, $\Lambda_2$}
From consideration of the smallest $3$--dimensional non-compact hyperbolic orbifold, we have identified four lattices containing $\Gamma$, namely $\Gamma_0$, $\Lambda_0$, $\Lambda_1$ and $\Lambda_2$.
\begin{lemma}\label{l:upLattice}
The only lattices in $\rm{Isom}(\mathbb{H}^3)$ that contain $\Gamma$ are $\Gamma$, $\Gamma_0$, $\Lambda_0$, $\Lambda_1$, and $\Lambda_2$.
\end{lemma}
\begin{proof}
$\Lambda_0$ is the unique non-uniform lattice of minimal co-volume $v_0/24$ and all other lattices commensurable with $\Lambda_0$ have co-volume at least $v_0/12$ (see \cite{Ad} and \cite{Mey}). It follows that $\Gamma$, which has co-volume $v_0/6$, must have index $2$ in any lattice $\Delta\neq \Lambda_0$ that properly contains it. This forces $\Gamma$ to be normal in $\Delta$. By Mostow Rigidity, the normaliser of $\Gamma$ is itself a lattice; and since it contains $\Lambda_0$ it must be equal to $\Lambda_0$. Thus $\Delta$ is a subgroup
of index 2 in $\Lambda_0$.
\end{proof}
\subsection{The orbifolds $\mathbb{H}^3/\Gamma$ and $\mathbb{H}^3/\Gamma_0$}
The quotient orbifolds $\mathbb{H}^3/\Gamma_0$ and $\mathbb{H}^3/\Gamma$ are shown in Figure 1. Note that each of these orbifolds has a single cusp, with cusp cross-section homeomorphic to a Euclidean $2$--orbifold which is a $2$--sphere with three cone points. In the case of $\mathbb{H}^3/\Gamma_0$, this orbifold is $S^2(2,3,6)$, which is $S^2$ with three cone points with cone angles $\pi$, $2\pi/3$ and $\pi/3$. In the case of $\mathbb{H}^3/\Gamma$ it is $S^2(3,3,3)$; i.e.~the cone angles are all $2\pi/3$.
\begin{figure}[h]
\centering
\begin{tikzpicture} [scale=.5]
\coordinate (O) at (0,0);
\coordinate (A) at (0,3.4);
\coordinate (B) at (0,1.2);
\coordinate (C) at (-1.6,-1.4);
\coordinate (D) at (1.6,-1.4);
\coordinate (E) at (-3.2,-2.4);
\coordinate (F) at (3.2,-2.4);
\filldraw (A) circle (1pt);
\filldraw (B) circle (1pt);
\filldraw (C) circle (1pt);
\filldraw (D) circle (1pt);
\filldraw (E) circle (1pt);
\filldraw (F) circle (1pt);
\draw (A) to [edge label = $3$] (B);
\draw (B) to [edge label' = $2$] (C);
\draw (B) to [edge label = $3$] (D);
\draw (C) to [edge label' = $2$] (D);
\draw (C) to [edge label' = $3$] (E);
\draw (D) to [edge label = $3$] (F);
\draw [densely dashed] (-4,0) arc [start angle = 180, end angle = 0,
x radius = 40mm, y radius = 8mm];
\draw [white, line width = 5pt](-4,0) arc [start angle=-180, end angle = 0,
x radius = 40mm, y radius = 8mm];
\draw (-4,0) arc [start angle=-180, end angle = 0,
x radius = 40mm, y radius = 8mm];
\draw (O) circle [radius=4cm];
\draw node at (0,-5) {$\mathbb{H}^3/\Gamma$};
\coordinate (O) at (10,0);
\coordinate (A) at (10,3.4);
\coordinate (B) at (10,1.2);
\coordinate (C) at (8.4,-1.4);
\coordinate (D) at (11.6,-1.4);
\coordinate (E) at (6.8,-2.4);
\coordinate (F) at (13.2,-2.4);
\filldraw (A) circle (1pt);
\filldraw (B) circle (1pt);
\filldraw (C) circle (1pt);
\filldraw (D) circle (1pt);
\filldraw (E) circle (1pt);
\filldraw (F) circle (1pt);
\draw (A) to [edge label = $6$] (B);
\draw (B) to [edge label' = $2$] (C);
\draw (B) to [edge label = $2$] (D);
\draw (C) to [edge label' = $3$] (D);
\draw (C) to [edge label' = $3$] (E);
\draw (D) to [edge label = $2$] (F);
\draw [densely dashed] (6,0) arc [start angle = 180, end angle = 0,
x radius = 40mm, y radius = 8mm];
\draw [white, line width = 5pt](6,0) arc [start angle=-180, end angle = 0,
x radius = 40mm, y radius = 8mm];
\draw (6,0) arc [start angle=-180, end angle = 0,
x radius = 40mm, y radius = 8mm];
\draw (O) circle [radius=4cm];
\draw node at (10,-5) {$\mathbb{H}^3/\Gamma_0$};
\end{tikzpicture}
\caption{}
\end{figure}
These cusps are {\em{rigid}} in the sense that no non-trivial Dehn surgery can be performed on them (see \cite{DuMe}). This cusp-rigidity is a crucial feature of $\Gamma$ and $\Gamma_0$. Both groups have Serre's property FA and $\Gamma$ is the unique Bianchi group with this property; see \cite{Fi}. As noted earlier (see Remark \ref{galois_FA}), it follows that the $\PSL(2,\C)$--character varieties of these groups, $\mathrm{X}(\Gamma,\C)$ and $\mathrm{X}(\Gamma_0,\C)$ are both finite. Unfortunately, our proof requires that $\Gamma$ be Galois rigid. We shall deduce this from the tight control that one can get on the representations of $\Gamma$ and $\Gamma_0$ over finite fields; this is the subject of \S \ref{fewchars}.
\subsection{The Weeks manifold}\label{weeks}
Let $M_W={\Bbb H}^3/\Gamma_W$ denote the Weeks manifold; this is the unique closed orientable hyperbolic 3--manifold of minimal volume. It can be obtained by performing $(-5,1)$ surgery on one component of the Whitehead link and $(5,2)$ surgery on the other. Using SnapPy \cite{CDW}, a presentation for $\Gamma_W$ can be computed and we record it for future use:
\[ \innp{a,b~|~ababa^{-1}b^2a^{-1}b,abab^{-1}a^2b^{-1}ab}. \]
From this, we see that $H_1(M_W,{\Bbb Z})\cong {\Bbb Z}/5{\Bbb Z}\times {\Bbb Z}/5{\Bbb Z}$.
We shall need some facts about the arithmetic structure of the Weeks manifold $M_W$ (see \cite[Ch 4.8.3, Ch 9.8.2]{MR} for more details). The invariant trace-field of $\Gamma_W$ is $K=K\Gamma_W={\Bbb Q}(\theta)$ where $\theta^3-\theta^2+1=0$ is as in Example \ref{ex2} (a field of discriminant $-23$). In addition, the invariant quaternion algebra $B_W$ coincides with the algebra $B$ of Example \ref{ex2}. As noted in Example \ref{ex2}, there is a unique conjugacy class of maximal orders in $B_W$ and for a choice of maximal order $\mathcal{O}\subset B_W$, the group $\Gamma_W$ is a normal subgroup of index $3$ in the group of units $\Gamma_\mathcal{O}^1$. As shown in \cite{MedV}, the orbifold $\mathbb{H}^3/\Gamma_\mathcal{O}^1$ can be described as $(3,0)$ Dehn surgery on $5_2$ (i.e.~$M_W$ is the $3$--fold cyclic branched cover of the knot $5_2$). Furthermore, the minimal orbifold in the commensurability class of $\Gamma_W$ arises from a maximal arithmetic group $\Gamma_{\mathcal{O}}$ which is the normalizer of $\Gamma_\mathcal{O}^1$ in $\PSL(2,\C)$, and $\Gamma_{\mathcal{O}}/\Gamma_\mathcal{O}^1\cong \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$. Since $M_W$ is non-Haken, $\mathrm{X}(\Gamma_W,\C)$ consists of a finite number of points \cite{BZ}. By \cite[Cor 6.3]{ReWan}, $\Gamma_W$ has very few irreducible representations into $\PSL(2,\C)$.
\begin{proposition}\label{reidwang}
In $\mathrm{X}(\Gamma_W,\C)$, there only three characters of irreducible representations, namely the characters of the faithful discrete representation, its complex conjugate, and a $\PSU(2)$--representation arising from the real ramified place of $B_W$.
\end{proposition}
\begin{corollary}\label{weeks_galois_rigid}
$\Gamma_W$ is Galois rigid.
\end{corollary}
\section{$\PSL(2,\mathbb{Z}[\omega])$ has very few characters}\label{fewchars}
In this section we prove the following result, which establishes that $\PSL(2,\mathbb{Z}[\omega])$ is Galois rigid.
\begin{theorem}\label{2chars}
$\Gamma=\PSL(2,\mathbb{Z}[\omega])$ has two characters of Zariski-dense representations in $\PSL(2,\C)$, namely the characters associated to the inclusion homomorphism and its complex conjugate. In particular, $\Gamma$ is Galois rigid.
\end{theorem}
For brevity, throughout this section $\Gamma$ will always denote $\PSL(2,\mathbb{Z}[\omega])$.
\subsection{$\PSL(2,\F)$--quotients}\label{Paoluzzi_Zimmermann}
We begin by describing $\PSL(2,{\F})$ quotients where $\F$ is a finite field. We shall see that homomorphisms from $\Gamma$ onto $\PSL(2,\F)$, with $\F$ a finite field, arise exclusively from the arithmetic of $\mathbb{Z}[\omega]$. This result is due in large part to L. Paoluzzi and B. Zimmermann \cite[Thm 6.3]{PZ}. First we need some notation. Given a prime $p\in \mathbb{Z}$, the ideal $p\mathbb{Z}[\omega]$ is a prime ideal of $\mathbb{Z}[\omega]$ if $p= -1~\rm{mod}~6$ or $p=2$, and splits as a product of two distinct prime ideals $\mathcal{P}_1\mathcal{P}_2$ if
$p = 1~\rm{mod}~6$. If $p=3$ then $3\mathbb{Z}[\omega] = \<\sqrt{-3}\>^2$, the square of a prime ideal. In the first of these three cases, the residue class field $\mathbb{Z}[\omega]/\mathcal{P}$ is a field with $p^2$ elements. In the remaining cases, for each prime $\mathcal{P}$ that arises, the field $\mathbb{Z}[\omega]/\mathcal{P}$ has $p$ elements. Each of the ring homomorphisms $\mathbb{Z}[\omega]\rightarrow \mathbb{Z}[\omega]/\mathcal{P}$ induces a {\em reduction homomorphism} $\pi_{\mathcal{P}}\colon\PSL(2,\mathbb{Z}[\omega])\rightarrow \PSL(2,\mathbb{Z}[\omega]/\mathcal{P})$. These homomorphisms are onto, and therefore pick out the following collection of finite quotients of $\Gamma$:
\begin{enumerate}
\item $\Gamma\twoheadrightarrow\PSL(2,{\Bbb F}_{p^2})$ when $p=2$ or $p=-1~\rm{mod}~6$;
\item a pair of quotient maps $\Gamma\twoheadrightarrow\PSL(2,{\F}_p)$ when $p=1~\rm{mod}~6$;
\item a single quotient map $\Gamma\twoheadrightarrow\PSL(2,{\F}_p)$ when $p=3$.
\end{enumerate}
\noindent We will denote this collection of finite quotients of $\Gamma$ by ${\bf P}$. When the meaning is clear, we shall say $\PSL(2,{\F}_p)\in {\bf P}$, but more formally the elements of ${\bf P}$ are the normal subgroups $\Gamma(\mathcal{P}) < \Gamma$ that arise as the kernels of the reduction homomorphisms $\pi_{\mathcal{P}}$. (Thus we identify two surjections if they differ by composition with an automorphism of the target.) The kernels $\Gamma(\mathcal{P})$ are torsion-free apart from when $\mathcal{P}=\<\sqrt{-3}\>$, in which case $\Gamma(\mathcal{P})$ has elements of order $3$, for example the image in $\PSL(2,\C)$ of the element $\begin{pmatrix} \omega & 0\cr 0& \omega^2\cr\end{pmatrix}$. It is shown in \cite{Alp} that there is a unique normal subgroup of index $12$ in $\Gamma$, and this coincides with $\Gamma(\<\sqrt{-3}\>)$. (Although \cite{Alp} mistakenly labels $\Gamma$ as $\PSL(2,\mathbb{Z}[\sqrt{-3}])$, the results in the paper actually refer to $\Gamma$.)
The results of \cite{PZ} (in particular Theorem 6.3) classify {\em admissible} epimorphisms of $\Gamma$ onto groups of the form $\PSL(2,\F)$ with $\F$ a finite field, where admissible means that the kernel is assumed to be torsion-free. In the applications that we require, torsion cannot be avoided, and we therefore require the following lemma to augment \cite{PZ}.
\begin{lemma}\label{torsion_normal}
Let $\Omega$ be a proper normal subgroup of $\Gamma$ and suppose that $\Omega$ contains a non-trivial element of finite order. Then $\Omega$ has index $3$ or $12$ in $\Gamma$: in the first case, $\Omega$ is the commutator subgroup of $\Gamma$, and in the second case $\Omega=\Gamma(\<\sqrt{-3}\>)$.
\end{lemma}
\begin{proof}
By the Cartan fixed-point theorem, every finite subgroup of $\Gamma$ fixes a point in $\mathbb{H}^3$ and is therefore conjugate into one of the vertex-stabilizer groups for the orbifold description of $\mathbb{H}^3/\Gamma$ given in \S \ref{s:small}. In terms of the presentation (\ref{present}) of $\Gamma$, these stabilizers are
\[ \Omega_1= \innp{y,z~\mid~ y^2=z^2=(yz)^3=1} \]
which is isomorphic to $S_3$, and two copies of the alternating group $A_4$, namely
\begin{align*}
\Omega_2 &=\innp{x,z\mid z^2=x^3=(zx^{-1})^3=1} \\
\Omega_3 &= \innp{x,y\mid y^2=x^3=(yx^{-1})^3=1}.
\end{align*}
Thus if a normal subgroup $\Omega$ contains a non-trivial element of finite order, then it intersects at least one of $\Omega_1$, $\Omega_2$ or $\Omega_3$ non-trivially.
Setting $x=1$ trivializes $\Gamma$, so $N$ cannot contain $\Omega_2$ or $\Omega_3$. The only proper normal subgroup of $A_4$ is the commutator subgroup, which has order $4$. The commutator subgroup $\Omega_2'$ contains $z$ while the commutator subgroup $\Omega_3'$ contains $y$, and setting either $y=1$ or $z=1$ in $\Gamma$ reduces $\Gamma$ to its abelian quotient $\mathbb{Z}/3\mathbb{Z}$. So either $\Omega\cap \Omega_i=\Omega_i'$ for $i=2,3$, in which case $\Omega=[\Gamma,\Gamma]$, or else $\Omega$ intersects both $\Omega_2$ and $\Omega_3$ trivially. In the latter case, $\Omega$ contains neither $y$ nor $z$.
If $\Omega$ intersects $\Omega_2$ and $\Omega_3$ trivially, then it must intersect $\Omega_1\cong S_3$ in a proper normal subgroup, and the only such is $\<yz\>$. A direct calculation shows that setting $yz=1$ in $(\ref{present})$ yields $A_4$ as a quotient. Thus, in this case, either $\Omega$ has index $12$ in $\Gamma$ (and then by \cite{Alp} $\Omega=\Gamma(\<\sqrt{-3}\>)$), or else $\Gamma/\Omega$ is the unique quotient $\mathbb{Z}/3\mathbb{Z}$ of $A_4$. But this last possibility would contradict the assumption $\Omega_2\cap \Omega=1$, because $y$ has order $2$.
\end{proof}
\begin{theorem}\label{controllingmodp}
Let $\F$ be a finite field and let $\phi\colon\Gamma \rightarrow \PSL(2,\F)$ an epimorphism. Then $\PSL(2,\F)\in{\bf P}$ and $\phi$ is a reduction homomorphism $\pi_{\mathcal{P}}$.
\end{theorem}
\begin{proof}
Paoluzzi and Zimmermann \cite[Thm 6.3]{PZ} proved this theorem for epimorphisms with torsion-free kernel, and Lemma \ref{torsion_normal} removes the need to assume the kernel is torsion-free.
\end{proof}
\begin{remark}
A description of admissible homomorphisms for $\PGL(2,\mathbb{Z}[\omega])$ is also given in \cite{PZ}, however we will not need to appeal to that here.
\end{remark}
\subsection{Some comments on Strong Approximation}\label{strongapprox}
The proof of Theorem \ref{2chars} requires the Strong Approximation Theorem \cite{Wei}. Suppose that $\rho\colon \Gamma \to \mathrm{PSL}(2,\mathbb{C})$ is a representation with Zariski-dense image $\Delta$. Since $\Gamma^{\rm{ab}}\cong \mathbb{Z}/3\mathbb{Z}$, $\Delta$ cannot have any $\mathbb{Z}/2\mathbb{Z}$ quotients and so we deduce that $K_\Delta=K\Delta$, and $A_0\Delta=A\Delta$ (recall \S \ref{s:algtraces}). As $\Gamma$ is rigid (see Remark \ref{galois_FA}), we have $[K_\Delta:\Q]<\infty$ by Lemma \ref{trace_field_number_field}. Hence, $A\Delta^1$ is an absolutely almost simple, simply connected algebraic group defined over the number field $K\Delta$. The following is a consequence of \cite[Thm 8.1]{Wei} stated in a form that is useful for us (Remark \ref{Vinberg} is pertinent here).
\begin{theorem} [Weisfeiler]\label{RepSec:T1}
In the notation above, for all but a finite number of $K\Delta$--primes $\mathcal{P}$ with residue class field $\F_{\mathcal{P}}$, there is a reduction homomorphism $\Delta\to \PSL(2,\F_{\mathcal{P}})$ which is onto.\end{theorem}
\subsection{Proof of Theorem \ref{2chars}}
Suppose that $\rho\colon \Gamma\to \PSL(2,\C)$ is a Zariski-dense representation. Since $\Gamma$ has Property FA, there are only a finite number of Zariski-dense representations up to conjugacy, and they all have integral traces \cite[Ch 1, \S 6]{Serre}. In particular, if $\Delta=\rho(\Gamma)$ then $\mbox{\rm{tr}}\, (\Delta) \subset R_{K\Delta}$. By Theorem \ref{RepSec:T1} we get for all but a finite number of prime ideals $\mathcal{P}$ of $R_{K\Delta}$, an epimorphism $\Gamma\colon \Delta\to \PSL(2,{\F}_q)$ where $q$ is the cardinality of the residue class field $\F_q=R_{K\Delta}/\mathcal{P}$. By Theorem \ref{controllingmodp}, for all but a finite number of primes $\mathcal{P}$ the finite groups $\PSL(2,{\F}_q)$ correspond to those in $\bf P$. Hence for all but a finite number of rational primes $p$, the field $K\Delta$ and $\Q(\omega)$ have the same splitting type for prime ideals. Since $[\Q(\omega):\Q]=2$, it follows from \cite{Per} that $K\Delta=\Q(\omega)$.
The algebra $A\Delta$ is defined over $\Q(\omega)$, and $\mathcal{O}=\mathcal{O}\Delta \subset A\Delta$ is an order since $\Delta$ has integral traces. We shall prove that $A\Delta=\mathrm{M}(2,\Q(\omega))$. Once this is established, we complete the argument as follows. As noted in Example \ref{ex1}, there is only one type of maximal order in $\mathrm{M}(2,\Q(\omega))$, and so we can conjugate so that $\Delta<\Gamma$. Now $\Gamma$ is residually $\PSL(2,{\F})$ (using the proof of residual finiteness), and so if $g\in\ker(\Gamma\to \Delta)$ then we can find a prime ideal $\mathcal{P}$ of $\mathbb{Z}[\omega]$ so that the image of $g$ in the composition $\Gamma\to \Delta\to \PSL(2,\mathbb{Z}[\omega]/\mathcal{P})$ is not trivial, so cannot lie in the kernel. Hence $\Gamma\to \Delta$ is injective, and so by Mostow Rigidity $\Delta=\Gamma$ and $\rho:\Gamma\rightarrow \Delta$ is the inclusion map up to conjugation in $\rm{Isom}(\mathbb{H}^3)$. Thus the character of $\rho$ is
that of the inclusion map or its complex conjugate.
We now prove that $A\Delta\cong \mathrm{M}(2,\Q(\omega))$. Suppose that this is not the case, then by the classification theorem for quaternion algebras, we can find at least two distinct primes $\mathcal{P}_1$ and $\mathcal{P}_2$ with $A\Delta$ ramified at these primes. Assume that the residue class fields $\mathbb{Z}[\omega]/\mathcal{P}_i$ have characteristic $p_i$ for $i=1,2$. Let $A_i=A\Delta\otimes_{\Q(\omega)} \Q(\omega)_{\mathcal{P}_i}$ for $i=1,2$ and $\mathcal{O}_i$ the unique maximal order in these division algebras (see \cite[Ch 6.4]{MR} for these and the following details). Now $\mathcal{O}_i$ admits a filtration that restricts to $\mathcal{O}_i^1$ to provide a filtration of the following form $\mathcal{O}_i^1 > G^i_1 > G^i_2 \ldots $, where each of these subgroups is normal in $\mathcal{O}_i^1$, $\cap G_j^i=1$, $\mathcal{O}_i^1/G_1^i$ is cyclic of order dividing $p_i^4-1$ and the $G_j^i/G_{j+1}^i$ are abelian $p_i$--groups. Note that for $i=1,2$, and all $j\geq 1$, the
groups $G_j^i$ are are pro--$p_i$ groups whose intersection is the identity subgroup. Hence any element of finite order in $G_j^i$ has order a power of $p_i$ for $i=1,2$. Identifying $\Delta$ with its image in $\mathcal{O}_i^1$ under the inclusion map, first note that for $i=1,2$, $\Delta$ cannot be a subgroup of $G_1^i$. The reason is as follows. From \S \ref{s:small}, we see that $\Gamma$ is generated by elements of orders $2$ and $3$, hence $\Delta$ is generated by elements of orders $2$ and $3$. Indeed from the proof of Lemma \ref{torsion_normal}, $\rho(x)$, $\rho(y)$ and $\rho(z)$ are all non-trivial elements of order $2$ (in the case of $\rho(y)$ and $\rho(z)$) or $3$ (in the case of $\rho(x)$). However, from the previous paragraph, the groups $G_1^i$ cannot contain elements of both orders $2$ and $3$.
Thus we may now assume that there are epimorphisms $\Delta\rightarrow C_i$ with the order of $C_i$ dividing $p_i^4-1$. From the previous paragraph both of $C_i\neq 1$. Since $\Gamma^{\rm{ab}} \cong \mathbb{Z}/3\mathbb{Z}$, we can assume $C_1$ say is cyclic of order $3$. Hence $\Delta\cap G_1^1$ is a normal subgroup of index $3$ which must coincide with $\rho([\Gamma,\Gamma])$. Since $[\Gamma,\Gamma]$ has abelianization $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$ (see \cite[\S 2]{magma_calcs}), and using the nature of the filtration described above, the only possibility for the prime $p_1$ is $p_1=2$. We now deal with $p_2$. We have $\Delta\to C_2$ a cyclic group of order dividing $p_2^4-1$, with kernel $\Omega$ that surjects a cyclic group of $p_2$--power order. Again, using $\Gamma^{\rm{ab}} \cong \mathbb{Z}/3\mathbb{Z}$, $C_2$ can only be a cyclic group of order $3$. Since $\Gamma$ admits a unique epimorphism to $\mathbb{Z}/3\mathbb{Z}$, it follows that is $\Omega=\rho([\Gamma,\Gamma])$ with $\Omega$ having a cyclic group of $p_2$--power order as a quotient. Once again we
use the fact that $[\Gamma,\Gamma]$ has abelianization $\mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$, and so the only possibility for $p_2$ is $p_2=2$. However, $2$ is inert to $\Q(\omega)$, that is to say there is only one prime of residue class degree a power of $2$, a contradiction.
$\sqcup\!\!\!\!\sqcap$
\section{Profinite rigidity for $\Gamma=\PSL(2,\mathbb{Z}[\omega])$}\label{s:rigid_bianchi}
In this section we exhibit the first example satisfying Theorem \ref{t:main}.
\begin{theorem}\label{main_bianchi}
$\Gamma=\PSL(2,\mathbb{Z}[\omega])$ is profinitely rigid.
\end{theorem}
Before embarking on the proof, we will require some additional information about subgroups of low index and their abelianizations.
\subsection{Small index subgroups of $\Gamma$ and their abelianizations}
We saw in Lemma \ref{l:abelG} that $\Gamma$ has finite abelianization, in other words its first Betti number $b_1(\Gamma)=0$. But since $\mathbb{H}^3/\Gamma$ is a non-compact finite volume hyperbolic 3--orbifold, any finite sheeted {\em{manifold}} cover of $\mathbb{H}^3/\Gamma$ has positive $b_1$; in other words, every torsion-free subgroup of $\Gamma$ has positive $b_1$. We will exploit the following results about the abelianizations of subgroups of index at most $12$ in $\Gamma$.
\begin{lemma}\label{indexatmost12}
If $\Delta < \Gamma$ is a subgroup with $[\Gamma:\Delta] \leq12$, then $b_1(\Delta)\leq 1$.
\end{lemma}
\begin{proof}
We make some preliminary comments and then refer the reader to \cite[\S 2]{magma_calcs} for the Magma routine that completes the proof of the lemma. By Lemma \ref{torsion_normal} $\Gamma^{\rm{ab}}\cong \mathbb{Z}/3\mathbb{Z}$, and so $\Gamma$ has no subgroups of index $2$. Magma also shows that there are no subgroups of index $11$. In addition, Magma shows that there is a unique conjugacy class of subgroups of index $3$, $4$, $5$, $9$ and $10$, two conjugacy classes of subgroups of index $6$ and $8$, and four of index $7$. There are $7$ of index $12$. The Betti number for each of these subgroups is provided.
\end{proof}
The enumeration of small index subgroups shows that $\Gamma$ has a unique conjugacy class of index $6$ subgroups with infinite abelianization, one class of index $10$, and three of index $12$. Only one of these conjugacy classes also contains $5$--torsion in its first homology group -- this class contains the index $12$ subgroup that we shall denote $\Gamma_s$. This is the fundamental group of the once-punctured torus bundle $X$ that is known as the {\em{sister of the figure-eight knot complement}}. (The figure-eight knot complement itself represents the class of index $12$ subgroups with abelianization $\mathbb{Z}$, but we shall not pursue this.) For future reference we record that the monodromy of this once-punctured torus bundle for $\Gamma_s$ is given by the matrix $\phi_s = \begin{pmatrix} -3& 1\cr -1& 0\cr\end{pmatrix}$, from which one easily computes $H_1(\Gamma_s,\mathbb{Z}) = \mathbb{Z} \times \mathbb{Z}/5\mathbb{Z}$.
It is known that $\Gamma_s$ is a congruence subgroup of $\Gamma$ containing the principal congruence subgroup $\Gamma(2)$ (see for example the proof of Lemma 3.1 of \cite{BaR}). From \S 3.2, we have that $\Gamma/\Gamma(2) \cong \PSL(2,\F_4)$, and so $[\Gamma_s:\Gamma(2)]=5$. It is well-known that $b_1(\Gamma(2))=5$, and this can be confirmed by direct calculation of the abelianizations of the kernels of the maps from $\Gamma_s$ onto the torsion part of $H_1(\Gamma_s,\mathbb{Z})$. Since none of the other subgroups $\Delta<\Gamma$ of index at most $12$ with $b_1(\Delta)=1$ contains $5$--torsion in its abelianization, in these cases the only map from $\Delta$ to $\mathbb{Z}/5\mathbb{Z}$ is the one that factors through the unique epimorphism $\Delta\to\mathbb{Z}$. It is easy to compute the abelianization of the kernel of $\Delta\to\mathbb{Z}/5\mathbb{Z}$ using Magma (see \cite[\S 3]{magma_calcs}), and in each case one finds that the first Betti number is $1$. Summarizing this discussion, we have proved:
\begin{proposition}\label{p:Sis-is-it}
Up to conjugacy, the only subgroup $\Delta<\Gamma$ that fits into a chain $\Omega \triangleleft \Delta < \Gamma$ with $[\Gamma:\Delta]\le 12,\ [\Delta:\Omega]= 5,\ b_1(\Delta)\ge 1$ and $b_1(\Omega)\ge 5$ is $\Delta=\Gamma_s$.
\end{proposition}
We shall also need the following observation about the homology of cyclic covers of the once-punctured torus bundle with fundamental group $\Gamma_s$.
\begin{lemma}\label{l:ab_cyclic_covers}
If $d\ge 3$ then the abelianization of $F_2\rtimes_{\phi_s^d}\mathbb{Z}$ is $T\times\mathbb{Z}$, where $T$ is torsion and $|T|>5$.
\end{lemma}
\begin{proof}
For any hyperbolic matrix $\psi$ of determinant $1$, a direct calculation of the abelianization of $G_\psi=F_2\rtimes_\psi\mathbb{Z}$ yields $\mathbb{Z}\times T_\psi$ where $\abs{T_\psi} = \abs{{\rm{tr}}(\psi)-2}$ (see \cite[Lemma 3.5]{BridR}, for example). Note that $\phi_s^2$ has trace $7$, and so we have $\abs{T}=5$, as it is for $\Gamma_s$.
Since $\phi_s^d$ is hyperbolic, the absolute value of its largest eigenvalue is strictly greater than $1$. It follows from elementary considerations that $\abs{\mathrm{tr}(\phi_s^d)}$ is strictly monotonically increasing in $d$. Hence $\abs{\mathrm{tr}(\phi_s^d) - 2} \geq \abs{\mathrm{tr}(\phi_s^d)} - 2$ is greater than $5$ for all $d\geq 3$, as needed.
\end{proof}
\subsection{Proof of Theorem \ref{main_bianchi}}\label{s:bianchi_mainproof}
We turn to the proof of our main result. We are assuming that $\Delta$ is a finitely generated, residually finite group with $\widehat{\Delta}\cong\widehat{\Gamma}$. From \S \ref{s:rep} (see in particular Example \ref{ex1}), we obtain a homomorphism $\rho\colon\Delta\to\Gamma$ whose image is Zariski-dense; we write $L$ to denote the image of $\rho$. Our purpose now is to show that $L=\Gamma$ and that $\rho$ is injective.
The construction of $L$ from \S \ref{s:rep} provides considerable additional information about $L$. This guides our proof and can be used to shorten it (see Remark \ref{r:use_more}), but we attack the problem of showing $L=\Gamma$ more directly and give an argument which shows that any non-elementary finitely generated subgroup of $\Gamma$ has a finite quotient that $\Gamma$ does not have (although the arguments are not phrased explicitly in this way). Our strategy is as follows. First we use arguments from 3--manifold topology to argue that if $L$ had infinite index in $\Gamma$, then $L$ would have a subgroup of index at most $12$ with first Betti number greater than $1$. But $\Gamma$ does not have such a subgroup (Lemma \ref{indexatmost12}), so $\widehat{\Gamma}$ cannot map onto $\widehat{L}$. This brings us to the heart of the argument: the case where $L$ has finite index in $\Gamma$. In this case, we focus attention on the subgroup $L_s = L\cap\Gamma_s$ and use calculations of virtual Betti numbers to argue that if $\widehat{\Gamma}$ maps onto $\widehat{L}$ then $L_s = \Gamma_s$. Considerations of co-volume and the Hopf property for finitely generated profinite groups then complete the proof.\\[\baselineskip]
\noindent{{\bf Notation:}} We focus on $\rho\colon\Delta\twoheadrightarrow L$ and $L_s = L\cap\Gamma_s$. Define $\Delta_s = \rho^{-1}(\Gamma_s)$ and $\Delta(2)=\rho^{-1}(\Gamma(2))$.
\subsection{Ruling out infinite index image}\label{s:infindex}
\begin{lemma}\label{l:Lsis.b1}
$b_1(L_s)\le 1$.
\end{lemma}
\begin{proof}
By definition, $L_s$ is a quotient of $\Delta_s=\rho^{-1}(\Gamma_s)$, so $b_1(L_s)\le b_1(\Delta_s)$. But $\Delta_s$ has index at most $12$ in $\Delta$, so by the Correspondence Theorem (Proposition \ref{correspondence}), it has the same abelianization as some subgroup of index at most 12 in $\Gamma$. Lemma \ref{indexatmost12} tells us that these subgroups of $\Gamma$ all have first Betti number at most one. Thus $b_1(\Delta_s)\le 1$.
\end{proof}
\begin{proposition}\label{p:not-inf}
$L$ has finite index in $\Gamma$.
\end{proposition}
\begin{proof}
By construction, $L$ is Zariski-dense. In particular neither $L$ nor $L_s=L\cap\Gamma_s$, which has index at most $12$ in $L$, is abelian. Since $L$ is finitely generated, so is $L_s$. And since $\Gamma_s$ is torsion-free, so is $L_s$. Thus $M=\mathbb{H}^3/L_s$ is a non-elementary orientable hyperbolic manifold with finitely generated fundamental group. Classical $3$--manifold topology provides a compact core for $M$ (\cite{Scott}), i.e.~a compact 3--manifold with boundary $N\subset M$ that is homotopy equivalent to $M$. If $L_s$ were of infinite index in $\Gamma_s$, then $M$ would have infinite volume and therefore a big end, i.e.~at least one of the connected components of $\partial N$ would have genus at least $2$. A standard duality argument establishes the following well-known ``half lives, half dies" principle:
\medskip
\noindent{\em If $N$ is a compact orientable 3--manifold with non-empty boundary $\partial N$, then the kernel and image of the natural map $H_1(\partial N, \mathbb R)\to H_1(N,\mathbb R)$ are of equal dimension. In particular, if $\partial N$ has a component of genus at least $2$, then $b_1(N)\ge 2$.}
\medskip
Thus if $L$ were of infinite index in $\Gamma$ then we would have $b_1(L_s) = b_1(N)\ge 2$, contradicting Lemma \ref{l:Lsis.b1}.
\end{proof}
\subsection{$\Delta_s=\Gamma_s$}
At this stage we have a finitely generated, residually finite group $\Delta$ with $\widehat{\Delta}\cong\widehat{\Gamma}$ mapping onto a subgroup of finite index $L<\Gamma$. We focus on $\Delta_s$, the preimage of $L\cap\Gamma_s$.
\begin{lemma}\label{l:sissy}
$\widehat{\Delta_s} \cong \widehat{\Gamma_s}$. Moreover, the subgroups of index $12$ in $\Delta$ corresponding to the conjugates of $\Gamma_s$ are the conjugates of $\Delta_s$.
\end{lemma}
\begin{proof}
As $L_s$ has finite index in $\Gamma_s$, we have $b_1(L_s)\ge b_1(\Gamma_s) =1$. From Lemma \ref{l:Lsis.b1} we deduce $b_1(L_s)=1$. And as $L_s\cap \Gamma(2)$ has finite index in $\Gamma(2)$, we have $b_1(L_s\cap\Gamma(2))\ge b_1(\Gamma(2))=5$. Now, $L_s\cap \Gamma(2)$ is normal in $L_s$ with index $1$ or $5$, and in fact the index must be $5$ since the Betti number of the subgroup is greater than that of $L_s$. Thus $\Delta_s = \rho^{-1}(\Gamma_s)$ is a subgroup of index at most $12$ in $\Delta$ with $b_1(\Delta_s)$ positive; moreover it contains $\Delta(2)=\rho^{-1}(\Gamma(2))$ as a subgroup of index $5$ and $b_1(\Delta(2))\ge 5$. So in $\Delta$ we have a chain of subgroups $\Delta(2) \ns \Delta_s < \Delta$ with $[\Delta:\Delta_s]\le 12,\ [\Delta_s:\Delta(2)]= 5,\ b_1(\Delta_s)\ge 1$ and $b_1(\Delta(2))\ge 5$.
We pass this chain across to a chain of subgroups in $\Gamma$ using Proposition \ref{correspondence}. Explicitly, replacing each of the groups $\Delta$, $\Delta_s$, and $\Delta(2)$ in this sequence with $\overline{\Delta}\cap\Gamma$, $\overline{\Delta_s} \cap \Gamma$, and $\overline{\Delta(2)} \cap \Gamma$, we obtain a chain of subgroups in $\Gamma$ with the same properties. From Proposition \ref{p:Sis-is-it} we deduce that, up to conjugacy in $\Gamma$, we have $\overline\Delta_s\cap \Gamma = \Gamma_s$. In particular, $\widehat{\Delta_s}\cong \widehat{\Gamma_s}$.
\end{proof}
\subsection{The Final Argument}
We now have $\widehat{\Gamma_s}\cong \widehat{\Delta_s}$ with $\Delta_s$ mapping onto the finite index subgroup $L_s < \Gamma_s$. The description of $\Gamma_s$ as a punctured-torus bundle gives a short exact sequence $1\to F \to \Gamma_s \to \mathbb{Z}\to 1$, where $F$, a free group of rank $2$, is the fundamental group of the fibre. This restricts to a short exact sequence $1\to F_L \to L_s \to \mathbb{Z}\to 1$ where $F_L=F\cap L$. At this point it is important to note that $\Gamma_s$ induces the full profinite topology on $F$ and that $L_s$ induces the full profinite topology on $F_L$ (see for example \cite[Lemma 2.2]{BridR}).
Since $b_1(L_s)$ is one, $F_L$ is the kernel of the unique map $L_s\to \mathbb{Z}$. Similarly, $\widehat{F}<\widehat{\Gamma_s}$ is the kernel of the unique epimorphism $\widehat{\Gamma_s}\to \widehat{\mathbb{Z}}$ and $\widehat{F_L}<\widehat{L_s}$ is the kernel of the unique epimorphism $\widehat{L_s}\to \widehat{\mathbb{Z}}$, so the epimorphism $\widehat{\Gamma_s}\to \widehat{L_s}$ must send $\widehat{F}$ onto $\widehat{F_L}$. Therefore the free group $F_L$ has rank $2$, and since it has finite index in $F$, we conclude that $F_L=F$; in other words $F<L$. As $L_s$ contains the fibre group $F<\Gamma_s$, it is the fundamental group of a cyclic covering of $\mathbb{H}^3/\Gamma_s$. Algebraically, $\Gamma_s = F\rtimes_{\phi_s}\mathbb{Z}$ and $L_s = F\rtimes_{\phi_s^d}\mathbb{Z}$. If $d>2$, then it follows from Lemma \ref{l:ab_cyclic_covers}, that $L_s$ would have a finite abelian quotient that $\Gamma_s$ does not have, hence in this case we can conclude that $L_s=\Gamma_s$.
We now deal with the case $d=2$. Assume that $L_s = F\rtimes_{\phi_s^2}\mathbb{Z}$. In this case $L_s$ has a unique $2$--fold cover $N$, for which the order of $T$ (the torsion subgroup of $H_1(N,\mathbb{Z})$) is $45$. But the index $2$ subgroup of $\Gamma_s$ must surject $\pi_1(N)$ and this cannot happen as this index $2$ subgroup only has torsion subgroup of order $5$. At this stage we know that $\Gamma_s = L_s \le L \le \Gamma$. Moreover, $[L:L_s]=12$ because $\Delta_s = \rho^{-1}(L_s)$ has index $12$ in $\Delta$ (as it corresponds to $\Gamma_s$). As $[\Gamma:\Gamma_s]=12$, we conclude that $L=\Gamma$.
Finally, we have $\widehat{\Gamma}\cong \widehat{\Delta}\overset{\hat{\rho}}\to\widehat{L}=\widehat{\Gamma}$, and as finitely generated profinite groups are Hopfian (see \cite[Prop 2.5.2]{RZ}), we conclude that $\widehat{\rho}$ and $\rho$ are injective. Thus $\rho\colon \Delta\to L = \Gamma$ is an isomorphism.
$\square$
\begin{remark} \label{r:use_more}
We close this section by explaining our earlier comment that \S \ref{s:rep} can be used to avoid parts of the above analysis of subgroups of $\Gamma$. The point is to exploit more explicitly the fact that $\Gamma_s$ contains the principal congruence subgroup $\Gamma(2)$. From \S \ref{Paoluzzi_Zimmermann} we have that $\Gamma/\Gamma(2) \cong \PSL(2,\F_4)$, and so $[\Gamma_s:\Gamma(2)]=5$. Moreover, by Theorem \ref{controllingmodp}, there is a unique $ \PSL(2,\F_4)$ quotient of $\Gamma$. From the discussion in \S \ref{congruence_quotients} we have a subgroup $L(2)<L$ with $L/L(2)\cong \Gamma/\Gamma(2) \cong \PSL(2,\F_4)$, and $L(2)=L\cap \Gamma(2)$. In addition, \S \ref{congruence_quotients} also provides us with a subgroup $L_s=L\cap \Gamma_s$ which is a subgroup of index $12$ in $L$ containing $L(2)$ of index $5$.
Using the epimorphism $\Delta \rightarrow L$ and the isomorphism $\widehat\Delta\cong\widehat\Gamma$, we have an epimorphism $\phi:\widehat{\Gamma}\rightarrow \widehat{L}$, and hence there is a subgroup $\Pi_\Gamma<\Gamma$ of index $12$ with $\phi(\widehat{\Pi_\Gamma})=\widehat{L}_s$. As above, it can be shown that the only possibility for $\Pi_\Gamma$ is $\Pi_\Gamma=\Gamma_s$.
\end{remark}
\begin{remark}[Exhibiting Additional Quotients]
The techniques of this section enable one to show that if $\Delta<\Gamma=\PSL(2,\mathbb{Z}[\omega])$ is a proper finitely generated infinite subgroup, then $\Delta$ has a finite quotient that $\Gamma$ does not have: there does not exist a continuous epimorphism $\widehat{\Gamma}\twoheadrightarrow\widehat{\Delta}$. When $\Delta$ has infinite index, alternative techniques allow one to say something more: $\Delta$ will have a congruence quotient ${\rm{PSL}}(2,\mathbb{F})$ that $\Gamma$ does not have. This will follow from a more general result that we prove in \cite{BMRS2} using techniques from Teichm\"uller theory and the study of character varieties. This more general result is a component in our proof of absolute profinite rigidity for certain cocompact Fuchsian triangle groups. \end{remark}
\section{Profinite rigidity for the lattices containing $\PSL(2,\mathbb{Z}[\omega])$}\label{s:others}
In this section we explain how Mostow Rigidity can be used to promote the profinite rigidity of $\PSL(2,\mathbb{Z}[\omega])$ to profinite rigidity for each of the lattices in ${\rm{Isom}}(\mathbb{H}^3)$ that contain it; these lattices were described in \S \ref{s:small}. To maintain brevity, throughout this section $\Gamma$ will always denote $\PSL(2,\mathbb{Z}[\omega])$.
\begin{lemma}\label{l:up2}
If $\Delta$ is a group with a subgroup of index 2 that is isomorphic to $\Gamma$, then $\Delta$ is isomorphic to one of $\Gamma_0, \Lambda_1$, $\Lambda_2$ or $\Gamma\times\mathbb{Z}/2\mathbb{Z}$.
\end{lemma}
\begin{proof}
By Mostow Rigidity, $\Lambda:={\rm{Aut}}(\Gamma)$ is a lattice in $\PSL(2,\C)$ containing $\Gamma$ (which has trivial centre) as the group of inner automorphisms -- a subgroup of finite index. The action of $\Delta$ by conjugation on $\Gamma$ defines a homomorphism $\Delta \to \Lambda $ whose image contains $\Gamma$ and whose kernel $\Omega$ commutes with $\Gamma$. If $\Omega=1$ then $\Delta< \Lambda$ is a lattice and Lemma \ref{l:upLattice} completes the proof. Otherwise $\Omega$ has order $2$, the image of $\Delta$ in $\Lambda$ is $\Gamma$, and the sequence splits.
\end{proof}
\begin{theorem}\label{t:rigid2}
Each of the groups $\Gamma_0$, $\Lambda_1$, $\Lambda_2$ and $\Gamma\times\mathbb{Z}/2\mathbb{Z}$ is profinitely rigid.
\end{theorem}
\begin{proof}
Let $\Omega_0$ be a finitely generated, residually finite group that has the same profinite completion as one of the groups $\Gamma_0, \ \Lambda_1$, $\Lambda_2$ or $\Gamma\times\mathbb{Z}/2\mathbb{Z}$. Then, by the Proposition \ref{correspondence}, $\Omega_0$ has a subgroup $\Omega$ of index $2$ with the same profinite completion as $\Gamma$. Since $\Gamma$ is profinitely rigid, $\Omega \cong \Gamma$, so Lemma \ref{l:up2} tells us that $\Omega_0$ is isomorphic to one of $\Gamma_0, \ \Lambda_1$, $\Lambda_2$ or $\Gamma\times\mathbb{Z}/2$. From Lemma \ref{l:abelG}, the first two of these groups have abelianization $\mathbb{Z}/2\mathbb{Z}$, and $\Gamma\times\mathbb{Z}/2\mathbb{Z}$ has abelianization $\mathbb{Z}/6\mathbb{Z}$. From \S \ref{s:small},
$\Lambda_2$ also abelianization $\mathbb{Z}/6\mathbb{Z}$. Hence we can distinguish $\Gamma_0$ and $\Lambda_1$ from $\Lambda_2$ and $\Gamma\times \mathbb{Z}/2\mathbb{Z}$ by abelianization. It remains to distinguish $\Gamma_0$ from $\Lambda_1$, and $\Lambda_2$ and $\Gamma\times \mathbb{Z}/2\mathbb{Z}$. We can distinguish $\widehat\Gamma_0$ from $\widehat{\Lambda}_1$ using Magma (see \cite[\S 4]{magma_calcs}) to count the number of conjugacy classes of index $8$ subgroups: in $\Lambda_1$ there is only one conjugacy class, whereas in $\Gamma_0=\PGL(2,\mathbb{Z}[\omega])$ there are three. It follows from Lemma \ref{l:conj-classes} that $\widehat\Gamma_0\not\cong\widehat{\Lambda}_1$.
Finally, referring to \cite[\S4] {magma_calcs}, we see that $\Lambda_2$ has no subgroups of index $7$, whereas $\Gamma\times \mathbb{Z}/2\mathbb{Z}$ has $4$ conjugacy classes of such subgroups.
\end{proof}
\begin{theorem}\label{t:rigid4}
$\Lambda_0$ is profinitely rigid.
\end{theorem}
\begin{proof}
As $\Gamma_0$, which has index 2 in $\Lambda_0$, is profinitely rigid, by repeating the argument of Lemma \ref{l:up2} we deduce that a finitely generated, residually finite group with the same profinite completion as $\Lambda_0$ is either a non-uniform lattice of covolume $v_0/24$ or is isomorphic to $\Gamma_0\times\mathbb{Z}/2\mathbb{Z}$. As $\Lambda_0$ is the only non-uniform lattice of covolume $v_0/24$ up to conjugacy in ${\rm{Isom}}(\mathbb{H}^3)$, we need only distinguish the profinite completions of $\Lambda_0$ and $\Gamma_0\times\mathbb{Z}/2\mathbb{Z}$. As above we can use Magma (see \cite[\S 5]{magma_calcs}) to show that $\Lambda_0$ and $\Gamma_0\times\mathbb{Z}/2\mathbb{Z}$ have different numbers of conjugacy classes of subgroups of index $8$, and so it follows from Lemma \ref{l:conj-classes} that $\widehat{\Lambda}_0\not\cong\widehat{\Gamma}_0 \times \mathbb{Z}/2\mathbb{Z}$.
\end{proof}
\section{Profinite rigidity of the Weeks manifold}\label{s:weeks_rigidity}
In this section we provide our second main example of a profinitely rigid arithmetic Kleinian group, namely $\Gamma_W$ (as described in \S \ref{weeks})
\begin{theorem}\label{main_weeks}
$\Gamma_W$ is profinitely rigid.
\end{theorem}
Throughout this section $\Delta$ is a finitely generated residually finite group with $\widehat{\Delta} \cong \widehat{\Gamma}_W$. We can quickly reduce consideration to the following situation.
\begin{lemma}\label{l: reduce_weeks}
With $\Delta$ as above, there exists a finite index subgroup $L<\Gamma_W$ with $\rho\colon\Delta \twoheadrightarrow L$.
\end{lemma}
\begin{proof}
From Example \ref{ex2}, together with the arithmetic description of $\Gamma_W$ in \S \ref{weeks} and Proposition \ref{reidwang}, we deduce that there exists a maximal order $\mathcal{O} \subset B_W$ with $\rho\colon\Delta \twoheadrightarrow L$ a finitely generated subgroup of $\Gamma_\mathcal{O}^1$. Also from \S \ref{weeks}, $\Gamma_\mathcal{O}^1$ contains $\Gamma_W$ as a normal subgroup of index $3$. We first claim that $L<\Gamma_W$. Indeed, if $L$ were not contained in $\Gamma_W$, then $L$ would map onto $\Gamma_\mathcal{O}^1/\Gamma_W$, which, from the above remark is isomorphic to $\mathbb{Z}/3\mathbb{Z}$. However, this is impossible as $L$ is a quotient of $\Delta$ and $H_1(\Delta,\mathbb{Z})\cong H_1(\Gamma_W,\mathbb{Z}) \cong {\Bbb Z}/5{\Bbb Z}\times {\Bbb Z}/5{\Bbb Z}$ (recall \S \ref{weeks}). Since $L < \Gamma_W$, the quotient $\mathbb{H}^3/L$ is a manifold. If $L$ were of infinite index then an application of ``half lives, half dies" (recall the proof of Lemma \ref{p:not-inf}) would imply that $L$ has infinite abelianization, which it can not since it is a quotient of $\Delta$.
\end{proof}
The proof of Theorem \ref{main_weeks} follows a similar strategy to that of the ``finite index case" of Theorem \ref{main_bianchi}; namely, we identify a particular fibered cover and combine the study of it with an analysis of low index subgroups. The next two subsections detail what we need in this direction.
\subsection{Subgroups of index $24$} \label{index23}
In this section $K$ denotes the trace-field of $\Gamma_W$ and $R_K$ its ring of integers (recall \S \ref{weeks}). It can be checked that $R_K$ contains two prime ideals of norm $23$, one corresponding to the ramified prime of norm $23$, which we denote by $\mathcal{Q}$, and a second unramified prime which we denote by $\mathcal{P}$. For both we have $\PSL(2,R_K/\mathcal{Q})\cong \PSL(2,R_K/\mathcal{P})\cong \PSL(2,{\Bbb F}_{23})$. We claim that, up to conjugacy, $\Gamma_W$ has two epimorphisms to $\PSL(2,{\Bbb F}_{23})$, and these arise as $\PSL(2,R_K/\mathcal{Q})$ and $\PSL(2,R_K/\mathcal{P})$. The existence of the epimorphisms is immediate from the inclusion of $\Gamma_W$ into $\Gamma_\mathcal{O}^1$ as a normal subgroup of index $3$, since $\PSL(2,{\Bbb F}_{23})$ is simple. For uniqueness, a theorem going back to Galois states that the minimal index of any proper subgroup of the simple group $\PSL(2,{\Bbb F}_{23})$ is $24$. Moreover, there is a unique conjugacy class of subgroups of index $24$. Thus every epimorphism $\Gamma_W\twoheadrightarrow \PSL(2,{\Bbb F}_{23})$ gives rise to a subgroup of index $24$ with normal core $\Omega$ such that $\Gamma_W/\Omega \cong \PSL(2,{\Bbb F}_{23})$. In \cite[\S 6]{magma_calcs} we provide the Magma calculations enumerating all subgroups of index $24$ in $\Gamma_W$ (up to conjugacy in $\Gamma_W$). From this we see that there $11$ such subgroups and two have normal core $\Omega$ such that $\abs{\Gamma_W/\Omega} =6072 = \abs{\PSL(2,{\Bbb F}_{23})}$. Moreover, the Magma routine also shows that these finite quotients are simple and hence must be isomorphic to $\PSL(2,{\Bbb F}_{23})$ (using the lists of simple groups of small order).
\subsection{A fibered cover of $M_W$}\label{fiberedcover}
We shall make use of an explicit cover of $M_W$ that is a genus $2$ surface bundle. The existence of a bundle cover of $M_W$ was exhibited by Button \cite{Butt}, but we require more detailed information about this cover. From the tables of \cite{Butt} we see that $M_W$ is commensurable with the fibered manifold $M=m289(7,1)$, which arises from surgery on the census manifold $m289$ from the SnapPy census \cite{CDW}. In fact, using the identification of certain of these census manifolds with knots in the tables through $9$ crossings \cite{CalDW}, one knows that the manifold $m289$ is homeomorphic to the complement of the knot $\mathcal{K}=6_2$ (shown below). Thus $M_W$ is commensurable with $0$--surgery on $S^3\setminus \mathcal{K}$ (the framing used by SnapPy is different from the standard one for the knot $\mathcal{K}$).
\begin{figure}[h]
\centering
\begin{tikzpicture}[scale=1]
\draw [thick] (-1.25,0) to [out=90,in=135] (-.25,.25);
\draw [thick] (.25,-.25) to [out=-45,in=-135] (1.5,-1);
\draw [thick] (.8,.6) to [out=135,in=35] (-.4,1.2);
\draw [thick] (-.7,0) to [out=-90,in=145] (-.4,-1.2);
\draw [thick] (.8,-.6) to [out=45,in=-45] (1.5,1);
\draw [thick] (.25,.25) to [out=-135,in=45] (-.25,-.25);
\draw [white , line width = 5pt] (-.25,.25) to [out=-45,in=135] (.25,-.25);
\draw [thick] (-.25,.25) to [out=-45,in=135] (.25,-.25);
\draw [white , line width = 5pt] (1.5,-1) to [out=45,in=-45] (.8,.6);
\draw [thick] (1.5,-1) to [out=45,in=-45] (.8,.6);
\draw [white , line width = 5pt] (-.4,1.2) to [out=-145,in=90] (-.7,0);
\draw [thick] (-.4,1.2) to [out=-145,in=90] (-.7,0);
\draw [white , line width = 5pt] (-.4,-1.2) to [out=-35,in=-135] (.8,-.6);
\draw [thick] (-.4,-1.2) to [out=-35,in=-135] (.8,-.6);
\draw [white , line width = 5pt] (1.5,1) to [out=135,in=45] (.25,.25);
\draw [thick] (1.5,1) to [out=135,in=45] (.25,.25);
\draw [white , line width = 5pt] (-.25,-.25) to [out=-135,in=-90] (-1.25,0);
\draw [thick] (-.25,-.25) to [out=-135,in=-90] (-1.25,0);
\end{tikzpicture}
\caption{}
\end{figure}
Since $\mathcal{K}$ is $2$--bridge (and hence alternating) and its Alexander polynomial is $t^4-3t^3+3t^2-3t+1$, $S^3\setminus \mathcal{K}$ fibers over the circle with fibre a once-punctured surface of genus $2$ (\cite{Kan}). Using \cite{Knot}, the monodromy, $\psi$ of this fibration can be described by the composition of Dehn twists $T_a\circ T_b\circ T_c\circ T_d^{-1}$ (see Figure 3 for the labelling of curves). Here, $T_\gamma$ denotes the right-handed twist in $\gamma$.
\begin{figure}[h]
\centering
\begin{tikzpicture} [scale=1.2]
\draw [semithick] (-3,0) to [out=90,in=170] (-1.5,1) to [out=-10,in=180] (0,.5)
to [out=0,in=190] (1.5,1) to [out=10,in=90] (3,0) to [out=-90,in=-10] (1.5,-1)
to [out=170,in=0] (0,-.5) to [out=180,in=10] (-1.5,-1) to [out=190,in=-90] (-3,0);
\filldraw (2.8,0) circle (1.5pt);
\draw [semithick] (1.6,-.25) arc [start angle=-90, end angle = 0,
x radius = 6.5mm, y radius = 4mm];
\draw [semithick] (1.6,-.25) arc [start angle=-90, end angle = -180,
x radius = 6.5mm, y radius = 4mm];
\draw [semithick] (1.6,.25) arc [start angle=90, end angle = 22,
x radius = 6.5mm, y radius = 4mm];
\draw [semithick] (1.6,.25) arc [start angle=90, end angle = 158,
x radius = 6.5mm, y radius = 4mm];
\draw [semithick] (-1.6,-.25) arc [start angle=-90, end angle = 0,
x radius = 6.5mm, y radius = 4mm];
\draw [semithick] (-1.6,-.25) arc [start angle=-90, end angle = -180,
x radius = 6.5mm, y radius = 4mm];
\draw [semithick] (-1.6,.25) arc [start angle=90, end angle = 22,
x radius = 6.5mm, y radius = 4mm];
\draw [semithick] (-1.6,.25) arc [start angle=90, end angle = 158,
x radius = 6.5mm, y radius = 4mm];
\draw [line width = .8pt] (-1.6,0) circle [x radius = 10mm, y radius = 5mm];
\draw node at (-2,-.65) {$a$};
\draw [line width = .8pt] (1.6,0) circle [x radius = 10mm, y radius = 5mm];
\draw node at (2.6,-.4) {$c$};
\draw [line width = .8pt] (0,.2) arc [start angle=90, end angle = 0,
x radius = 10mm, y radius = 2mm];
\draw [line width = .8pt] (0,.2) arc [start angle=90, end angle = 180,
x radius = 10mm, y radius = 2mm];
\draw [line width = .8pt, densely dashed] (0,-.2) arc [start angle=-90, end angle = 0,
x radius = 10mm, y radius = 2mm];
\draw [line width = .8pt, densely dashed] (0,-.2) arc [start angle=-90, end angle = -180,
x radius = 10mm, y radius = 2mm];
\draw node at (0,.3) {$b$};
\draw [line width = .8pt] (1.6,.25) arc [start angle=-90, end angle = 90,
x radius = 2mm, y radius = 3.8mm];
\draw [line width = .8pt, densely dashed] (1.6,.25) arc [start angle=-90, end angle = -270,
x radius = 2mm, y radius = 3.8mm];
\draw node at (1.6,1.2) {$d$};
\draw [line width = .8pt] (-1.6,.25) arc [start angle=-90, end angle = 90,
x radius = 2mm, y radius = 3.8mm];
\draw [line width = .8pt, densely dashed] (-1.6,.25) arc [start angle=-90, end angle = -270,
x radius = 2mm, y radius = 3.8mm];
\draw node at (-1.6,1.2) {$f$};
\draw [line width = .8pt] (1.6,-.25) arc [start angle=90, end angle = -90,
x radius = 2mm, y radius = 3.8mm];
\draw [line width = .8pt, densely dashed] (1.6,-.25) arc [start angle=90, end angle = 270,
x radius = 2mm, y radius = 3.8mm];
\draw node at (1.6,-1.2) {$e$};
\end{tikzpicture}
\caption{}
\end{figure}
One computes the action of $\psi$ on the homology of the fiber with respect to the basis $\{a,f,c,d\}$ is given by the matrix $A$ shown below.
\[ A = \begin{pmatrix} 0 & 1 & 1 & 1\cr -1 & 1 & 2 & 1\cr 0 & 0 & 2 & 1\cr 1 & 0 & -1 & 0\cr\end{pmatrix}~\hbox{and}~ A^6 = \begin{pmatrix} 18 & 17 & 88 & 57\cr 9 & 9 & 48 & 31\cr 14 & 12 & 66 & 43 \cr 3 & 2 & 9 & 6\cr \end{pmatrix}. \]
The fibration of $S^3\setminus \mathcal{K}$ extends to the surgered manifold $M$ and the action of the monodromy on the homology of the closed genus $2$ surface is again given by $A$. Using SnapPy \cite{CDW}, $M$ can be shown to be hyperbolic of volume approximately $3.77082945\ldots$.
We will be interested in the $6$--fold cyclic covering of $M$, which we denote by $M_6$. Our interest lies with the fact that $M_6$ arises as an index $24$ cover of the Weeks manifold. One could verify this by deriving presentations of the index $24$ subgroups of $\Gamma_W$, on the one hand, while on the other hand calculating a presentation of $\Gamma_6:=\pi_1(M_6)$ from its description as the $6$--fold cyclic cover of the surgered manifold $m289$; a package such as Magma could then verify that the groups are isomorphic, and Mostow Rigidity then assures us that the manifolds are the same. But such calculations would leave the reader in the dark as to why these facts are true, so instead we shall explain, with references, the structure that leads to this conclusion. The following lemma gathers the key facts from the preceding discussion and its proof contains the promised explanation.
\begin{lemma}\label{bundlecover}
Let $M$ denote the manifold obtained by $0$--surgery on $S^3\setminus \mathcal{K}$. Then
\begin{enumerate}
\item $S^3\setminus \mathcal{K}$ is fibered with fiber a once-punctured genus $2$ surface,
and $M$ is fibered with fiber a genus $2$ surface.
\item The action of the monodromy on the homology of the fiber for $S^3 \setminus \mathcal{K}$ is given by $A$ above.
\item $M_6$, the $6$--fold cyclic cover of $M$, is fibered with fiber a genus $2$ surface and $H_1(M_6,{\Bbb Z})\cong {\Bbb Z}\times {\Bbb Z}/5{\Bbb Z}\times {\Bbb Z}/55{\Bbb Z}$.
\item $M_6$ is a $24$--fold cover of $M_W$.
\item $\Gamma_6=\pi_1(M_6)$ lies in the conjugacy class of subgroup $l[1]$ of \cite[\S 6]{magma_calcs}.
\end{enumerate}
\end{lemma}
\begin{proof}
The proof of the first two items is contained in the preceding discussion. For the third part, $M_6$ is clearly fibered with fiber $\Sigma$ a once-punctured surface of genus $2$, and one calculates $H_1(M_6,\mathbb{Z})$ by taking the quotient of action of $A^6$ on $H_1(\Sigma,\mathbb{Z}) = \mathbb{Z}^4 = \<a,f,c,d\>$. (The torsion subgroup of $H_1(M_6,\mathbb{Z})$ has order $275$.) For the next part let $M={\Bbb H}^3/\Omega$, and note that by \cite{Butt} the lattice $\Omega$ is arithmetic and commensurable with $\Gamma_W$. However, $\Omega$ is not derived from a quaternion algebra. Indeed it can be checked using Snap \cite{snap} that its trace-field has degree $6$ and $\Omega^{(2)} <\Gamma_\mathcal{O}^1$. If $\Omega^{(2)}<\Gamma_W$ then, by volume considerations, the index would be $8$. However, a Magma calculation (see \cite[\S 6]{magma_calcs}) shows that $\Gamma_W$ has a unique conjugacy class of subgroups of index $8$ and the corresponding manifold is a rational homology $3$--sphere with first homology group ${\Bbb Z}/5{\Bbb Z}\times {\Bbb Z}/30{\Bbb Z}$; in particular it cannot be a fiber bundle over the circle. Thus $\Omega^{(2)}\cap \Gamma_W$ must have index $3$ in $\Pi^{(2)}$. The double cover of $S^3\setminus \mathcal{K}$ has first homology group ${\Bbb Z}\times {\Bbb Z}/11{\Bbb Z}$. Performing $0$--surgery on this manifold produces ${\Bbb H}^3/\Omega^{(2)}$. In particular, the first homology group is ${\Bbb Z}\times {\Bbb Z}/11{\Bbb Z}$, and so there is a unique homomorphism onto ${\Bbb Z}/3{\Bbb Z}$ whose kernel provides the cover $M_6$. Volume considerations show that the covering degree $M_6\to M_W$ is $24$. For the last part, by inspection of the lists of first homology groups in \cite[\S 6]{magma_calcs}, the only possibility for $\Gamma_6$ is the group $l[1]$.
\end{proof}
We will also need the following.
\begin{lemma}\label{inf_order}
Let ${\rm{Mod}}_g$ be the Mapping Class group of the closed orientable surface of genus $g$. Let $\eta\in {\rm{Mod}}_g$ and let $G(\eta) = S_g\rtimes_\eta\mathbb{Z}$ be the fundamental group of the bundle with holonomy $\eta$ and assume $b_1(G(\eta))=1$. If the image of $\eta\in {\rm{Mod}}_g$ under the natural homomorphism ${\rm{Mod}}_g\to{\rm{Sp}}(2g,\mathbb{Z})$ has infinite order and $d>1$, then $\widehat{G(\eta)}\not\cong\widehat{G(\eta^d)}$.
\end{lemma}
\begin{proof}
The congruence topology on ${\rm{Sp}}(2g,\mathbb{Z})$ induces the full profinite topology on abelian subgroups; see Segal \cite[Ch 10]{segal} (see also \cite{McR}). Thus, given $d$, there is an integer $n_0$ such that the images of $\eta$ and $\eta^d$ generate distinct cyclic subgroups of ${\rm{Sp}}(2g,\mathbb{Z}/n_0\mathbb{Z})$. As $b_1(G(\eta))=1$, we can now argue as in Lemma 2.5 of \cite{BRW}; we recall the details. The unique epimorphism $G(\eta)\to\mathbb{Z}$ defines a short exact sequence
\[ 1\to \widehat{S_g}\to \widehat{G(\eta)}\to \widehat{\mathbb{Z}}\to 1. \]
If $\Omega<S_g$ is a characteristic and of finite index, then the canonical map $S_g\to S_g/\Omega$ defines an epimorphism $\widehat{G(\eta)}\to G(\eta)/\Omega$. As $\widehat{\Omega}$ is normal in $\widehat{G(\eta)}$, the action of $\widehat{\mathbb{Z}}$ by conjugation on $\widehat{G(\eta)}$ descends to an outer action on $\widehat{S_g}/\widehat{\Omega}=S_g/\Omega$, defining a cyclic subgroup $C_\eta<{\rm{Out}}(G/\Omega)$. The righthand factor of $G(\eta)=S_g\rtimes_\eta\mathbb{Z}$ is dense in $\widehat{\mathbb{Z}}$, so the image of $\eta$ generates $C_\eta$. If $\Omega$ is the kernel of the canonical map $S_g\to H_1(S_g,\mathbb{Z}/n_0\mathbb{Z})$, then $C_\eta=C_\eta(n_0)$ is the cyclic group generated by the image of $\eta$ in ${\rm{Sp}}(2g,\mathbb{Z}/n_0\mathbb{Z})<{\rm{Out}}(\mathbb{Z}/n_0\mathbb{Z})^{2g}$. By construction, $C_\eta(n_0)$ is an invariant of $\widehat{G(\eta)}$ (rather than $G(\eta)$) and $\abs{C_{\eta^d}(n_0)} < \abs{C_\eta(n_0)}$. Thus $\widehat{G(\eta)}\not\cong\widehat{G(\eta^d)}$.
\end{proof}
\subsection{Proof of Theorem \ref{main_weeks}}
Let $\Gamma_6=\pi_1(M_6)$ be as in Lemma \ref{bundlecover}. Then $[\Gamma_W:\Gamma_6]=24$ and $\Gamma_6$ is the subgroup $l[1]$ described in Lemma \ref{bundlecover}(5). Denoting the core of $\Gamma_6$ in $\Gamma_W$ by $\Omega$, we have $\Gamma_6/\Omega\cong \PSL(2,{\F}_{23})$.
Let $L_6=\Gamma_6\cap L$ and $L_\Omega=\Omega\cap L$. From the discussion in \S \ref{congruence_quotients} we know that $L/L_\Omega\cong\Gamma_W/\Omega\cong \Delta/\rho^{-1}(\Omega)$ and that the epimorphism $L\to L/L_\Omega \cong \PSL(2,{\Bbb F}_{23})$ is the restriction of the epimorphism $\Gamma_W\to \Gamma_W/\Omega$. Note that $L\neq L_6$ since $b_1(L_6)>0$, and $L$ can only have finite abelian quotients (since it is a quotient of $\Delta$ and $\widehat{\Delta} \cong\widehat{\Gamma}_W$). It follows that $[L:L_6]=24$, since on the one hand $[L:L_6]\leq [\Gamma_W:\Gamma_6]=24$, while on the other hand, as remarked above, $24$ is the minimal index of a proper subgroup of $\PSL(2,{\Bbb F}_{23})$.
Now consider $\Delta_6=\rho^{-1}(L_6)<\Delta$. This has index $24$ in $\Delta$ and moreover by construction of $L$ (see \S \ref{congruence_quotients}) the normal core of $\Delta_6$ in $\Delta$ is $\rho^{-1}(\Omega)$. From \cite[\S 6]{magma_calcs}, and Proposition \ref{correspondence}, there is a unique subgroup (up to conjugacy in $\Delta$) of index $24$ with normal core having quotient $\PSL(2,{\Bbb F}_{23})$ and positive first Betti number. By construction, $\Delta_6$ is this unique subgroup and hence $\widehat{\Gamma}_6 \cong \widehat{\Delta}_6$.
Consider the epimorphism $\widehat{\Gamma}_6 \cong \widehat{\Delta}_6\to \widehat{L}_6$. Note that $b_1(L_6)=1$ as $b_1(\Gamma_6)=1$. By construction, $N={\mathbb{H}}^3/L_6$ is a closed hyperbolic $3$--manifold that fibers over the circle: it is a finite cover of the fibered manifold $M_6$. By Lemma \ref{bundlecover}, $M_6$ is a genus $2$ surface bundle over the circle; let $F<\Gamma_6$ be the fundamental group of the fiber. The fiber of $N$ has fundamental group $L_6\cap F$. When we take profinite completions, the epimorphism $\widehat{\Gamma}_6 \cong \widehat{\Delta}_6\to \widehat{L}_6$ restricts to an epimorphism $\widehat{F}\to \widehat{L_6\cap F}$ (because $F$ and $L_6\cap F$ are the respective kernels of the unique epimorphisms to $\mathbb{Z}$). It follows that $L_6\cap F$ has genus at most $2$, and hence $L_6\cap F= F$. In particular $N\to M_6$ is a cyclic cover (of degree $d$ say) and $L_6 = F\rtimes_{\psi^d}\mathbb{Z}$.
Now, since $\widehat{F}$ is Hopfian, we have that $\widehat{\rho}$ restricted to $\widehat{F}<\widehat{\Gamma}_6$ is injective. Elements of $\widehat{\Gamma}_6$ in the complement of $\widehat{F}$ project non-trivially to the right-hand factor of $\widehat{\Gamma}_6=\widehat{F}\rtimes\widehat{\mathbb{Z}}$ and hence map non-trivially under $\widehat{\rho}$ to the $\widehat{\mathbb{Z}}$ factor of $\widehat{L_6}= \widehat{F}\rtimes\widehat{\mathbb{Z}}$. We conclude that $\widehat{\rho}$ is injective on $\widehat{\Delta}_6$ and hence we have an isomorphism $\widehat{\Gamma_6}\cong \widehat{\Delta}_6\to \widehat{L}_6$. Thus $\widehat{F\rtimes_\psi\mathbb{Z}}\cong\widehat{F\rtimes_{\psi^d}\mathbb{Z}}$. Since the action of $\psi$ on $H_1(F,\mathbb{Z})$ is represented by a positive matrix, it has infinite order in ${\rm{Sp}}(4,\mathbb{Z})$, so Lemma \ref{inf_order} applies and we conclude that $d=1$; in other words $L_6=\Gamma_6$ and $\rho\colon\Delta_6\to \Gamma_6$ is an isomorphism. Finally, we have proved that $L_6$ has index $24$ in $L=\rho(\Delta)$, and by construction $\Delta_6$ has index $24$ in $\Delta$. Therefore $L=\Gamma_W$ (since $[\Gamma_W: \Gamma_6]=[\Gamma_W:L_6]=
24$) and $\rho\colon\Delta\to\Gamma_W$ is surjective. Thus we have a surjection $\widehat{\Gamma}_W\cong\widehat{\Delta}\to \widehat{\Gamma}_W$, and using the Hopf property again, we conclude that $\widehat{\rho}$ is injective, hence so is $\rho$. Thus $\rho\colon\Delta\to\Gamma_W$ is an isomorphism. $\sqcup\!\!\!\!\sqcap$
\subsection{Profinite rigidity of Kleinian groups containing $\Gamma_W$}\label{friendsofweeks}
As in the case of $\Gamma=\PSL(2,\mathbb{Z}[\omega])$, once the profinite rigidity of $\Gamma_W$ has been established we can deduce the profinite rigidity for other Kleinian groups. We only record two cases here. Recall from \S \ref{weeks} that $\Gamma_W$ is a normal subgroup of index $3$ in a group $\Gamma_\mathcal{O}^1$, which we now denote by $\Gamma_1$. Moreover, as noted in \S \ref{weeks}, there is a maximal group $\Gamma_\mathcal{O}$ containing $\Gamma_1$ with $\Gamma_\mathcal{O}/\Gamma_1\cong \mathbb{Z}/2\mathbb{Z} \times \mathbb{Z}/2\mathbb{Z}$.
\begin{theorem}\label{rigidO1}
$\Gamma_\mathcal{O}$ and $\Gamma_1$ are profinitely rigid.
\end{theorem}
\begin{proof}
We deal with $\Gamma_1$ first. As in the proof of Theorems \ref{t:rigid2} and \ref{t:rigid4}, if $\Delta$ is a residually finite group with $\widehat{\Delta}\cong \widehat{\Gamma}_1$, then $\Delta$ contains an index $3$ normal subgroup $\Omega$ with $\widehat{\Omega}\cong \widehat{\Gamma}_W$, and from the profinite rigidity of $\Gamma_W$ and Mostow Rigidity we deduce that $\Delta$ is either an arithmetic Kleinian group containing $\Gamma_W$ as a normal subgroup of index $3$, or else $\Delta\cong \Gamma_W\times \mathbb{Z}/3\mathbb{Z}$. The latter case can be excluded for this implies that $\Delta$ and $\Gamma_1$ surject $\mathbb{Z}/5\mathbb{Z}\times \mathbb{Z}/5\mathbb{Z}$, which they do not, because as was pointed out in \S \ref{weeks}, the orbifold $\mathbb{H}^3/\Gamma_1$ is obtained from $(3,0)$ Dehn surgery on $5_2$ and so has abelianization $\mathbb{Z}/3\mathbb{Z}$.
Thus we may assume that $\Delta$ is an arithmetic Kleinian group containing $\Gamma_W$. (We remark that this forces $\Delta$ to contain an element of finite order since $M_W$ is the minimal-volume closed (arithmetic) hyperbolic 3--manifold; and the only possible order for a torsion element is 3, since $\Gamma_W<\Delta$ is a normal subgroup of index $3$.) Because $\Gamma_1^{\rm{ab}}\cong \mathbb{Z}/3\mathbb{Z}$, we know that $H_1(\Delta,\mathbb{Z}/2\mathbb{Z})=0$, and since there is a unique maximal order in the invariant quaternion algebra (as remarked in \S \ref{weeks}) we can assume that $\Delta < \Gamma_1$. But then $[\Gamma_1:\Gamma_W]=[\Delta:\Gamma_W]=3$, so $\Delta=\Gamma_1$ as required.
We now deal with $\Gamma_\mathcal{O}$. Suppose that $\Delta$ is a finitely generated, residually finite group with $\widehat{\Delta} = \widehat{\Gamma}_{\mathcal{O}}$. The commutator subgroup of $\Gamma_{\mathcal{O}}$ is $\Gamma_1$, and ${\Gamma}_{\mathcal{O}}/\Gamma_1\cong \mathbb{Z}/2\mathbb{Z}\times \mathbb{Z}/2\mathbb{Z}$. Hence there is a unique normal subgroup of index $4$ in $\widehat{\Gamma}_{\mathcal{O}}$, which we denote by $\Omega$. By the first part of the proof, $\Delta_1:=\Omega\cap \Delta$ is isomorphic to $\Gamma_1= \Omega\cap {\Gamma}_{\mathcal{O}}$. $\Gamma_{\mathcal{O}}$ is maximal in the commensurability class of $\Gamma_1$ (see \cite[Ch 11]{MR}), so by Mostow Rigidity its action by conjugation on $\Gamma_1$ gives an isomorphism $\Gamma_{\mathcal{O}}\cong{\rm{Aut}}(\Gamma_1)$.
The action of $\Delta$ by conjugation on $\Delta_1$ defines a map $c\colon\Delta\to{\rm{Aut}}(\Gamma_1)\cong\Gamma_{\mathcal{O}}$ whose image contains $\Gamma_1$, the group of inner automorphisms. Thus we obtain a map $c'$ from $\Delta$ to the finite group ${\rm{Out}}(\Gamma_1)=\Gamma_{\mathcal{O}}/\Gamma_1$ with kernel $\Delta_1$. The restriction to $\Gamma_{\mathcal{O}}$ of $\widehat{c'}\colon \widehat{\Delta}\to {\rm{Out}}(\Gamma_1)=\Gamma_{\mathcal{O}}/\Gamma_1$ is the standard map. In particular it is surjective, so $c\colon\Delta\to\Gamma_{\mathcal{O}}$ is surjective. And since $c$ restricts to an isomorphism between subgroups of index $4$, it is an isomorphism.
\end{proof}
\begin{remark}
A slight adjustment to the second half of the above proof shows that if $\Delta$ is a finitely generated, profinitely rigid, centerless group with ${\rm{Out}}(\Delta)$ finite, then ${\rm{Aut}}(\Delta)$ is profinitely rigid.
\end{remark}
\subsection{Additional full-size examples}\label{full_sized}
$\Gamma_W$ denotes the fundamental group of the Weeks manifold.
\begin{theorem}\label{inf_many}
For all integers $r\geq 0$, the groups $\Gamma_W \times \mathbb{Z}^r$ are profinitely rigid.
\end{theorem}
\begin{proof}
Theorem \ref{main_weeks} shows that we can assume that $r\geq 1$. Suppose that $\Delta$ is a finitely generated residually finite group with $\widehat{\Delta}\cong \widehat{\Gamma_W \times \mathbb{Z}^r}\cong\widehat{\Gamma}_W \times \widehat{\mathbb{Z}}^r$. Since $\mathbb{Z}^r$ is central, any Zariski-dense representation of $\Gamma_W \times \mathbb{Z}^r$ into $\PSL(2,\C)$ must kill $\mathbb{Z}^r$. (It is important here to work with the centerless $\PSL(2,\C)$.) Hence there is a bijection between $\mathrm{X}_{\mathrm{zar}}(\Gamma_W,\mathbb{C})$ and $\mathrm{X}_{\mathrm{zar}}(\Gamma_W\times \mathbb{Z}^r,\mathbb{C})$. By Corollary \ref{weeks_galois_rigid}, $|\mathrm{X}_{\mathrm{zar}}(\Gamma\times \mathbb{Z}^r,\mathbb{C})|=3$. It follows that $\Gamma_W\times \mathbb{Z}^r$ is Galois rigid, and so by Theorem \ref{T1}, $\Delta$ is Galois rigid. From Galois rigidity, we get an epimorphism $\rho\colon\Delta \rightarrow L$ where $L<\Gamma_{\mathcal{O}^1}$ and $\mathcal{O}$ is a maximal order contained in the invariant quaternion algebra of $\Gamma_W$. Recall that we can conjugate $\Gamma_W$ into $\Gamma_{\mathcal{O}^1}$, in which case
$[\Gamma_{\mathcal{O}^1}:\Gamma_W]=3$. By construction, the kernel of $\widehat{\rho}\colon\widehat{\Delta}\to\widehat{L}$ contains $\{1\}\times \widehat{\mathbb{Z}^r}$. Thus $\hat{\rho}$ factors through the projection to the first factor of $\widehat{\Gamma_W}\times \widehat{\mathbb{Z}^r}$ and we obtain a continuous epimorphism $\widehat{\rho_0}\colon\widehat{\Gamma_W}\to \widehat{L}$. It follows that $\widehat{L}$ cannot have $\mathbb{Z}/3\mathbb{Z}$ as a quotient, and hence there is no $3$--torsion in $H_1(L,\mathbb{Z})$. The argument of Lemma \ref{l: reduce_weeks} now applies to show that $L<\Gamma_W$.
We may now run the argument in the proof of Theorem \ref{main_weeks} on $\widehat{\Gamma}_W\rightarrow \widehat{L} \hookrightarrow \widehat{\Gamma}_W$ to deduce that $L=\Gamma_W$ and $\widehat{\rho_0}$ is an isomorphism. Thus we obtain a short exact sequence
\begin{equation}\label{central}
1\to \Omega \to \Delta \to \Gamma_W\to 1,
\end{equation}
where $\Omega=\Delta \cap (\{1\} \times \widehat{\mathbb{Z}^r})$ is central in $\Delta$. Associated to any short exact sequence of groups $1\to G_1\to G_2\to G_3\to 1$ one has a 5--term exact sequence in homology with $\mathbb{Z}$--coefficients,
\[ H_2(G_3,\mathbb{Z})\to H_0(G_3, H_1(G_1,\mathbb{Z})) \to H_1(G_2,\mathbb{Z}) \to H_1(G_3,\mathbb{Z}) \to 0; \]
for central extensions the second term is simply $G_1$. In the case $G_3=\Gamma_W$, from \S \ref{weeks} we have $H_1(\Gamma_W,\mathbb{Z})=(\mathbb{Z}/5\mathbb{Z})^2$, and $H_2(\Gamma_W,\mathbb{Z})=H^1(\Gamma_W,\mathbb{Z})=0$ by Poincar\'e duality. Thus the 5--term sequence reduces to a short exact sequence
\begin{equation}\label{last-eq}
0\to \Omega \to H_1(\Delta,\mathbb{Z}) \to (\mathbb{Z}/5\mathbb{Z})^2\to 0,
\end{equation}
where $\Omega$ is torsion-free. But $\widehat{\Delta} = \widehat{\Gamma_W\times \mathbb{Z}^r}$, so $H_1(\Delta,\mathbb{Z}) = (\mathbb{Z}/5\mathbb{Z})^2 \times \mathbb{Z}^r$, by Lemma \ref{l:abel}. Thus $\Omega=\mathbb{Z}^r$ and (\ref{last-eq}) splits. It follows that the central extension (\ref{central}) also splits, and therefore $\Delta\cong \Gamma_W \times \mathbb{Z}^r$ as claimed.
\end{proof}
\bigskip
\noindent{\bf Appendix:}\\[\baselineskip]
Here we prove Lemma \ref{L4}, whose statement we repeat for the convenience of the reader.
\medskip
\noindent{\bf Lemma A.1.}~{\em If $K$, $K'$ are number fields and $\tau_f\colon V_{K'}^f \to V_K^f$ is an injective map with $K'_w \cong K_{\tau_f(w)}$ for all $w \in V_{K'}^f$, then $\tau_f$ is a bijection. }
\begin{proof}
As $K'_w \cong K_{\tau_f(w)}$ for all $w \in V_{K'}^f$, any prime that splits completely in $K$ must also split completely in $K'$. By \cite[p.~108, Cor to Thm 31]{Mar} and \cite[p.~164, Thm 5.5]{Jan}, we see that $K'$ is a subfield of the Galois closure $K_{\mathrm{gal}}$ of $K$ over $\mathbb{Q}$. Let $G = \mathrm{Gal}(K_{\mathrm{gal}}/\mathbb{Q})$, and let $H$ and $H'$ be the subgroups of $G$ that fix $K$ and $K'$ respectively. Let $\chi, \chi'$ be the permutation characters for the $G$ action on $G/H$ and $G/H'$. By \cite[p.~128, Prop 2.7]{Jan}, if $p$ is a rational prime that does not ramify in $K_{\mathrm{gal}}$ and $g \in G$ is the Frobenius automorphism of any prime of $K_{\mathrm{gal}}$ over $p$, then the inertia degrees of the primes over $p$ in $K, K'$ are given by the orders of the orbits of $\langle g \rangle$ acting on $G/H$ and $G/H'$. Restricting $\tau_f$ to $V_{K'}^p$, we get an injection $G/H' \to G/H$ of $\langle g \rangle$--sets. Thus $\chi'(g) \leq \chi(g)
$. By Chebotarev's density theorem, each $g \in G$ is the Frobenius automorphism for infinitely many unramified primes of $K_{\mathrm{gal}}$. Therefore,
\begin{equation}\label{Eq:CharEq0}
\chi'(g) \leq \chi(g)
\end{equation}
for every $g \in G$. For a $G$--set $Z$ with permutation character $\chi_Z$, the number of orbits of the action on $Z$ is $\frac{1}{\abs{G}} \sum_g \chi_Z(g)$. Hence
\[ \frac{1}{\abs{G}} \sum_g \chi(g) = \frac{1}{\abs{G}} \sum_g \chi'(g) = 1 \]
and $\sum_g (\chi(g) - \chi'(g)) = 0$. From (\ref{Eq:CharEq0}) we deduce $\chi = \chi'$. Thus $\abs{G/H} = \abs{G/H'}$ and $K,K'$ have the same degree over $\Q$, so $\tau_f$ is a bijection.
\end{proof}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,143 |
\section{Introduction}
After nearly seventy years since the first experimental confirmation, quantum field theory (QFT) has yet to fail a phenomenological test. Whether it is the correction to the magnetic moment of fermions or the energy splitting in hydrogen-like systems, QFT has always provided very precise predictions matching to less than a part in several trillions the most accurate of measurements.
Nonetheless, despite its success when compared to our observations, QFT is still lacking a proper mathematical definition: conceptual problems start from the fact that all calculations are done by expanding fields in terms of plane waves, which are not square integrable (and in fact they do not really exist); they continue with the fact that in such an expansion the coefficients are reinterpreted as a pair of creation/annihilation operators, which still lack a definition, and for that matter their set of commutation relations might make no sense \cite{sw}; and they end with the fact that for all these calculations we employ the so-called interaction picture, which has been demonstrated not to exist in general in the context of a Lorentz-covariant field theory at all \cite{h}. In the face of these issues, the fact that perturbative expansions do not converge, or that each of their terms is finite only up to a certain regularization or renormalization, looks like a minor problem indeed.
Of all these conceptual issues, the lack of some proper mathematical definition of the creation/annihilation operators was felt particularly by Schwinger, who took this unsatisfactory situation to prompt himself into finding a different formulation for field quantization: his efforts led him to the construction of the so-called source theory \cite{s}.
Nevertheless, again, this is not a solution: Schwinger's source theory is in fact a prototypical version of the path integral, whose measure has never been defined too.
More in general, if we were to go back to the roots of quantization, we would see that the first problem would be involving the use of plane waves, which are not square integrable and as such they can not represent a particle.
This point, however, may constitute a possible avenue for our way out. As it is, QFT might be too dramatic in its first assumption that only plane waves should be used, thus leaving out too much information, so that some of the lost information must be reintroduced, and this could be done through quantization. In other words, quantization fills the gaps left by too strong approximations enforced by having the particle described with plane waves.
Were this the case, any theory of fields employing only the particular solution given by the plane wave but later quantized through some subsidiary conditions should be replaceable by a theory of fields employing general solutions with no subsidiary condition to be implemented.
We do not know whether this is the case. On the other hand, some literature does exist which follows this path: a first example is the one given by Koba and Welton, who faced the treatment of the anomalous magnetic moment of the electron and the Lamb shift, respectively, in terms of semi-classical considerations \cite{k,w}; more systematic is the work of Barut and co-workers, Dowling above all, who study the electron in electrodynamic self-interaction, and with no references to quantization, recovering the above mentioned results, on anomalous magnetic moment and Lamb shift, in general \cite{bd1,bd2}; the deepening on Lamb shift and new results on spontaneous emission/absorption are also presented in \cite{Barut:1992cs, Barut:1983um, Barut:1986ih, Barut:1987hx}; a result on vacuum polarization is also given in \cite{Acikgoz:1993mu} and discussed with the same spirit.
In fact, the concept of vacuum polarization might come helpful in visualization. Because a plane wave describes a freely propagating point particle and quantization accounts for radiative processes involving virtual loops then much in the same way in which quantization fills the gaps left by working with plane waves all virtual loops fill the gaps left by working with point particles; in addition, the surrounding cloud of virtual loops gives an effective size to what would otherwise be a mere point particle.
In the perspective outlined above, any theory of point particles whose dynamics is corrected in terms of virtual loops should be replaceable by a theory of extended fields whose dynamics is comprehensive enough to contain the corrections attributed to the virtual loops.
Although the lack of any exact solution makes it difficult to know what are all the dynamical effects that could replace the corrections due to virtual processes, one of the possibilities that has been considered is zitterbewegung, as addressed by Hestenes \cite{h5}, and with more details by Recami and Salesi \cite{sr1,sr2}. Zitterbewegung is a dynamic effect due to the relative motion between left-handed and right-handed semi-spinor projections of spinors.
As such it cannot be present for a point particle, whose lack of size means lack of internal structures. Still, it can be, and in fact it is, present in general for extended fields, and therefore it does make sense to consider it as what might give rise to dynamical effects, among which some can be mimicking the corrections of virtual loops.
Consequently, having in mind the idea of reproducing quantum corrections, and following the hint that such an endeavour might be done in terms of zitterbewegung, it is wise to start from the most comprehensive dynamics.
In this paper we will do this, defining the most general dynamics for matter fields, deriving some of the possible consequences of zitterbewegung, and seeing what connection there can be with known quantum corrections.
\section{Fundamental Theoretical Generalities}
\subsection{Dirac Spinorial Field}
To begin, we recall that $\boldsymbol{\gamma}^{a}$ are Clifford matrices, from which $\left[\boldsymbol{\gamma}_{a}\!,\!\boldsymbol{\gamma}_{b}\right]\!=\!4\boldsymbol{\sigma}_{ab}$ and $2i\boldsymbol{\sigma}_{ab}\!=\!\varepsilon_{abcd}\boldsymbol{\pi}\boldsymbol{\sigma}^{cd}$ are the definitions of the $\boldsymbol{\sigma}_{ab}$ and the $\boldsymbol{\pi}$ matrix (this matrix is usually indicated as gamma with an index five, but since in the space-time this index has no meaning we use a notation with no index so to avoid confusion): given $\psi$ as a Dirac spinor field, we define the bi-linear quantities given by
\begin{eqnarray}
&M_{ab}\!=\!2i\overline{\psi}\boldsymbol{\sigma}_{ab}\psi
\end{eqnarray}
with
\begin{eqnarray}
&S^{a}\!=\!\overline{\psi}\boldsymbol{\gamma}^{a}\boldsymbol{\pi}\psi\\
&U^{a}\!=\!\overline{\psi}\boldsymbol{\gamma}^{a}\psi
\end{eqnarray}
as well as
\begin{eqnarray}
&\Theta\!=\!i\overline{\psi}\boldsymbol{\pi}\psi\\
&\Phi\!=\!\overline{\psi}\psi
\end{eqnarray}
and which, despite being written only with spinor fields, are all real tensors. From the metric we define the symmetric connection as usual with $\Lambda^{\sigma}_{\alpha\nu}$ and with it we define the spin connection $\Omega^{a}_{\phantom{a}b\pi}\!=\!\xi^{\nu}_{b}\xi^{a}_{\sigma}(\Lambda^{\sigma}_{\nu\pi}\!-\!\xi^{\sigma}_{i}\partial_{\pi}\xi_{\nu}^{i})$ in such a way that with the gauge potential $qA_{\mu}$ we can define
\begin{eqnarray}
&\boldsymbol{\Omega}_{\mu}
=\frac{1}{2}\Omega^{ab}_{\phantom{ab}\mu}\boldsymbol{\sigma}_{ab}
\!+\!iqA_{\mu}\boldsymbol{\mathbb{I}}\label{spinorialconnection}
\end{eqnarray}
needed to write the spinorial covariant derivative
\begin{eqnarray}
&\boldsymbol{\nabla}_{\mu}\psi\!=\!\partial_{\mu}\psi
\!+\!\boldsymbol{\Omega}_{\mu}\psi\label{spincovder}
\end{eqnarray}
in which for the moment no torsion is defined: the commutator of spinorial covariant derivatives can be used to justify the definitions of space-time and gauge curvature
\begin{eqnarray}
&R^{i}_{\phantom{i}j\mu\nu}\!=\!\partial_{\mu}\Omega^{i}_{\phantom{i}j\nu}
\!-\!\partial_{\nu}\Omega^{i}_{\phantom{i}j\mu}
\!+\!\Omega^{i}_{\phantom{i}k\mu}\Omega^{k}_{\phantom{k}j\nu}
\!-\!\Omega^{i}_{\phantom{i}k\nu}\Omega^{k}_{\phantom{k}j\mu}\\
&F_{\mu\nu}\!=\!\partial_{\mu}A_{\nu}\!-\!\partial_{\nu}A_{\mu}
\end{eqnarray}
which are again in the torsionless case. The Lagrangian we will consider is given according to the standard
\begin{eqnarray}
\nonumber
&\mathscr{L}\!=\!\frac{1}{4}(\partial W)^{2}\!-\!\frac{1}{2}M^{2}W^{2}
\!+\!R\!+\!\frac{1}{4}F^{2}-\\
&-i\overline{\psi}\boldsymbol{\gamma}^{\mu}\boldsymbol{\nabla}_{\mu}\psi
\!+\!XS^{\mu}W_{\mu}\!+\!m\Phi
\label{l}
\end{eqnarray}
with $R$ trace of the space-time curvature and $F^{2}$ square of the gauge curvature and where the generality we temporarily lost when we defined torsionless connections can now be restored by including torsion as an axial vector $W_{\mu}$ with curl given by $(\partial W)_{\mu\nu}$ for the sake of simplicity.
As it has been discussed first of all by Wigner and more recently by Lounesto and Cavalcanti \cite{L,Cavalcanti:2014wia}, spinor fields can be classified in two large classes, in terms of which a spinor such that $\Theta\!=\!\Phi\!=\!0$ is called \emph{singular} and it is the subject of many studies \cite{daSilva:2012wp, Ablamowicz:2014rpa, daRocha:2016bil, daRocha:2008we, Villalobos:2015xca, Cavalcanti:2014uta,daRocha:2013qhu}, while a spinor such that either $\Theta\!\neq\!0$ or $\Phi\!\neq\!0$ is called \emph{regular} and it is the center of attention of the present work: we can always write the most general regular spinor in terms of a generic complex Lorentz transformation $\boldsymbol{S}$ according to
\begin{eqnarray}
&\!\psi\!=\!\phi e^{-\frac{i}{2}\beta\boldsymbol{\pi}}
\boldsymbol{S}\left(\!\begin{tabular}{c}
$1$\\
$0$\\
$1$\\
$0$
\end{tabular}\!\right)
\label{spinor}
\end{eqnarray}
called \emph{polar form} \cite{Fabbri:2016msm}, and such that with it we have
\begin{eqnarray}
&\!\!M_{ab}\!=\!2i\overline{\psi}\boldsymbol{\sigma}_{ab}\psi
\!=\!2\phi^{2}(\cos{\beta}u^{j}s^{k}\varepsilon_{jkab}\!+\!\sin{\beta}u_{[a}s_{b]})
\end{eqnarray}
in terms of
\begin{eqnarray}
&\!S^{a}\!=\!\overline{\psi}\boldsymbol{\gamma}^{a}\boldsymbol{\pi}\psi\!=\!2\phi^{2}s^{a}\\
&\!U^{a}\!=\!\overline{\psi}\boldsymbol{\gamma}^{a}\psi\!=\!2\phi^{2}u^{a}
\end{eqnarray}
such that $u_{a}u^{a}\!=\!-s_{a}s^{a}\!=\!1$ and $u_{a}s^{a}\!=\!0$ and representing the velocity vector and the spin axial-vector as well as
\begin{eqnarray}
&\Theta\!=\!i\overline{\psi}\boldsymbol{\pi}\psi\!=\!2\phi^{2}\sin{\beta}\\
&\Phi\!=\!\overline{\psi}\psi\!=\!2\phi^{2}\cos{\beta}
\end{eqnarray}
being a scalar and a pseudo-scalar known as module and Yvon-Takabayashi angle and which are the only two real degrees of freedom of the spinor field. Because generally
\begin{eqnarray}
&\boldsymbol{S}\partial_{\mu}\boldsymbol{S}^{-1}\!=\!i\partial_{\mu}\theta\mathbb{I}
\!+\!\frac{1}{2}\partial_{\mu}\theta_{ij}\boldsymbol{\sigma}^{ij}
\end{eqnarray}
where $\theta$ is a generic complex phase and $\theta_{ij}\!=\!-\theta_{ji}$ are the six parameters of the Lorentz group, then we can define
\begin{eqnarray}
&\partial_{\mu}\theta_{ij}\!-\!\Omega_{ij\mu}\!\equiv\!R_{ij\mu}\label{R}\\
&\partial_{\mu}\theta\!-\!qA_{\mu}\!\equiv\!P_{\mu}\label{P}
\end{eqnarray}
being real tensors called \emph{tensorial connection} and \emph{gauge vector momentum} respectively, with which we have
\begin{eqnarray}
&\!\boldsymbol{\nabla}_{\mu}\psi\!=\!(\nabla_{\mu}\ln{\phi}\mathbb{I}
\!-\!\frac{i}{2}\nabla_{\mu}\beta\boldsymbol{\pi}
\!-\!iP_{\mu}\mathbb{I}\!-\!\frac{1}{2}R_{ij\mu}\boldsymbol{\sigma}^{ij})\psi
\label{decspinder}
\end{eqnarray}
and also
\begin{eqnarray}
&\nabla_{\mu}s_{i}\!=\!R_{ji\mu}s^{j}\label{ds}\\
&\nabla_{\mu}u_{i}\!=\!R_{ji\mu}u^{j}\label{du}
\end{eqnarray}
identically: from the commutator of the covariant derivatives we deduce that the curvatures are such that
\begin{eqnarray}
&\!\!\!\!\!\!\!\!R^{i}_{\phantom{i}j\mu\nu}\!=\!-(\nabla_{\mu}R^{i}_{\phantom{i}j\nu}
\!-\!\!\nabla_{\nu}R^{i}_{\phantom{i}j\mu}
\!\!+\!R^{i}_{\phantom{i}k\mu}R^{k}_{\phantom{k}j\nu}
\!-\!R^{i}_{\phantom{i}k\nu}R^{k}_{\phantom{k}j\mu})\label{Riemann}\\
\!\!\!\!&qF_{\mu\nu}\!=\!-(\nabla_{\mu}P_{\nu}\!-\!\nabla_{\nu}P_{\mu})\label{Maxwell}
\end{eqnarray}
telling that the tensors defined in (\ref{R}, \ref{P}) do not generate any curvature tensor that is not already generated by the spin connection and the gauge potential. The Lagrangian above gives rise to field equations that in the polar form are transcribed into the geometric field equations
\begin{eqnarray}
\nonumber
&\nabla_{k}R^{ka}_{\phantom{ka}a}g^{\rho\sigma}\!-\!\nabla_{i}R^{i\sigma\rho}
\!-\!\nabla^{\rho}R^{\sigma i}_{\phantom{\sigma i}i}\!+\!R_{ki}^{\phantom{ki}i}R^{k\sigma\rho}+\\
\nonumber
&+R_{ik}^{\phantom{ik}\rho}R^{k\sigma i}
\!-\!\frac{1}{2}R_{ki}^{\phantom{ki}i}R^{ka}_{\phantom{ka}a}g^{\rho\sigma}-\\
\nonumber
&-\frac{1}{2}R^{ika}R_{kai}g^{\rho\sigma}\!=\!\frac{1}{2}[M^{2}(W^{\rho}W^{\sigma}
\!\!-\!\!\frac{1}{2}W^{\alpha}W_{\alpha}g^{\rho\sigma})+\\
\nonumber
&+\frac{1}{4}(\partial W)^{2}g^{\rho\sigma}
\!-\!(\partial W)^{\sigma\alpha}(\partial W)^{\rho}_{\phantom{\rho}\alpha}+\\
\nonumber
&+\frac{1}{4}F^{2}g^{\rho\sigma}\!-\!F^{\rho\alpha}\!F^{\sigma}_{\phantom{\sigma}\alpha}-\\
\nonumber
&-\phi^{2}[(XW\!-\!\nabla\frac{\beta}{2})^{\sigma}s^{\rho}
\!+\!(XW\!-\!\nabla\frac{\beta}{2})^{\rho}s^{\sigma}-\\
\nonumber
&-P^{\sigma}u^{\rho}\!-\!P^{\rho}u^{\sigma}+\\
&+\frac{1}{4}R_{ij}^{\phantom{ij}\sigma}\varepsilon^{\rho ijk}s_{k}
\!+\!\frac{1}{4}R_{ij}^{\phantom{ij}\rho}\varepsilon^{\sigma ijk}s_{k}]]\label{ee}
\end{eqnarray}
with
\begin{eqnarray}
&\nabla^{2}P^{\mu}
\!-\!\nabla_{\sigma}\nabla^{\mu}P^{\sigma}\!=\!-2q^{2}\phi^{2}u^{\mu}\label{me}
\end{eqnarray}
and
\begin{eqnarray}
&\!\!\!\!\nabla^{2}(XW)^{\mu}\!-\!\nabla_{\alpha}\nabla^{\mu}(XW)^{\alpha}
\!+\!M^{2}XW^{\mu}\!=\!2X^{2}\phi^{2}s^{\mu}\label{se}
\end{eqnarray}
alongside to the matter field equations
\begin{eqnarray}
\nonumber
&\frac{1}{2}\varepsilon_{\mu\alpha\nu\iota}R^{\alpha\nu\iota}
\!-\!2P^{\iota}u_{[\iota}s_{\mu]}+\\
&+2(\nabla\beta/2\!-\!XW)_{\mu}\!+\!2s_{\mu}m\cos{\beta}\!=\!0\label{dep1}\\
\nonumber
&R_{\mu a}^{\phantom{\mu a}a}
\!-\!2P^{\rho}u^{\nu}s^{\alpha}\varepsilon_{\mu\rho\nu\alpha}+\\
&+2s_{\mu}m\sin{\beta}\!+\!\nabla_{\mu}\ln{\phi^{2}}\!=\!0\label{dep2}
\end{eqnarray}
specifying all the first-order derivatives of the module and the YT angle \cite{h1}, and which can be proven to be equivalent to the original Dirac spinor field equations \cite{Fabbri:2016laz}.
To see that, we start by considering the Lagrangian we have written in (\ref{l}) and then we vary it with respect to the spinor field, getting the field equations
\begin{eqnarray}
&i\boldsymbol{\gamma}^{\mu}\boldsymbol{\nabla}_{\mu}\psi
\!-\!XW_{\mu}\boldsymbol{\gamma}^{\mu}\boldsymbol{\pi}\psi\!-\!m\psi\!=\!0\label{D}
\end{eqnarray}
which we then multiply by $\boldsymbol{\gamma}^{a}\boldsymbol{\pi}$ and $\boldsymbol{\gamma}^{a}$ and by the conjugate spinor, splitting real and imaginary parts, to get the four real vectorial field equations given according to
\begin{eqnarray}
\nonumber
&i(\overline{\psi}\boldsymbol{\nabla}^{\alpha}\psi
\!-\!\boldsymbol{\nabla}^{\alpha}\overline{\psi}\psi)
\!-\!\nabla_{\mu}M^{\mu\alpha}-\\
&-XW_{\sigma}M_{\mu\nu}\varepsilon^{\mu\nu\sigma\alpha}\!-\!2mU^{\alpha}\!=\!0
\label{vr}\\
\nonumber
&\nabla_{\alpha}\Phi
\!-\!2(\overline{\psi}\boldsymbol{\sigma}_{\mu\alpha}\!\boldsymbol{\nabla}^{\mu}\psi
\!-\!\!\boldsymbol{\nabla}^{\mu}\overline{\psi}\boldsymbol{\sigma}_{\mu\alpha}\psi)+\\
&+2X\Theta W_{\alpha}\!=\!0\label{vi}\\
\nonumber
&\nabla_{\nu}\Theta\!-\!
2i(\overline{\psi}\boldsymbol{\sigma}_{\mu\nu}\boldsymbol{\pi}\boldsymbol{\nabla}^{\mu}\psi\!-\!
\boldsymbol{\nabla}^{\mu}\overline{\psi}\boldsymbol{\sigma}_{\mu\nu}\boldsymbol{\pi}\psi)-\\
&-2X\Phi W_{\nu}\!+\!2mS_{\nu}\!=\!0\label{ar}\\
\nonumber
&(\boldsymbol{\nabla}_{\alpha}\overline{\psi}\boldsymbol{\pi}\psi
\!-\!\overline{\psi}\boldsymbol{\pi}\boldsymbol{\nabla}_{\alpha}\psi)
\!-\!\frac{1}{2}\nabla^{\mu}M^{\rho\sigma}\varepsilon_{\rho\sigma\mu\alpha}+\\
&+2XW^{\mu}M_{\mu\alpha}\!=\!0\label{ai}
\end{eqnarray}
and in which we now plug the polar form (\ref{spinor}) obtaining
\begin{eqnarray}
\nonumber
&-\nabla_{\mu}\ln{\phi}M^{\mu\sigma}
\!+\!\frac{1}{2}(\frac{1}{2}\nabla_{\mu}\beta\!-\!XW_{\mu})M_{\pi\nu}
\varepsilon^{\pi\nu\mu\sigma}+\\
\nonumber
&+P^{\sigma}\Phi\!+\!\frac{1}{8}R^{\alpha\nu\rho}M_{\pi\kappa}
\varepsilon_{\alpha\nu\rho\mu}\varepsilon^{\pi\kappa\sigma\mu}-\\
&-\frac{1}{2}R_{\mu a}^{\phantom{\mu a}a}M^{\mu\sigma}\!-\!mU^{\sigma}\!=\!0\\
\nonumber
&-\nabla_{\sigma}\ln{\phi}\Phi\!+\!(\frac{1}{2}\nabla_{\sigma}\beta\!-\!XW_{\sigma})\Theta
\!-\!P^{\mu}M_{\mu\sigma}-\\
&-\frac{1}{4}R^{\alpha\nu\rho}\varepsilon_{\alpha\nu\rho\sigma}\Theta
\!-\!\frac{1}{2}R_{\sigma a}^{\phantom{\sigma a}a}\Phi\!=\!0\\
\nonumber
&\nabla_{\sigma}\ln{\phi}\Theta\!+\!(\frac{1}{2}\nabla_{\sigma}\beta\!-\!XW_{\sigma})\Phi+\\
\nonumber
&+\frac{1}{2}P^{\mu}M^{\pi\kappa}\varepsilon_{\pi\kappa\mu\sigma}
\!-\!\frac{1}{4}R^{\alpha\nu\rho}\varepsilon_{\alpha\nu\rho\sigma}\Phi+\\
&+\frac{1}{2}R_{\sigma a}^{\phantom{\sigma a}a}\Theta\!+\!mS_{\sigma}\!=\!0\\
\nonumber
&\frac{1}{2}\nabla_{\mu}\ln{\phi}M_{\pi\kappa}\varepsilon^{\pi\kappa\mu\sigma}
\!+\!(\frac{1}{2}\nabla_{\mu}\beta\!-\!XW_{\mu})M^{\mu\sigma}-\\
&-P^{\sigma}\Theta\!-\!\frac{1}{4}R^{\alpha\nu\rho}M^{\mu\sigma}\varepsilon_{\alpha\nu\rho\mu}
\!+\!\frac{1}{4}R_{\mu a}^{\phantom{\mu a}a}M_{\pi\kappa}\varepsilon^{\pi\kappa\mu\sigma}\!=\!0
\end{eqnarray}
as a straightforward substitution shows: the second and third, after inserting the bi-linear quantities, become
\begin{eqnarray}
\nonumber
&\frac{1}{2}\nabla_{\alpha}\ln{\phi^{2}}\cos{\beta}
\!-\!(\frac{1}{2}\nabla_{\alpha}\beta\!-\!XW_{\alpha})\sin{\beta}+\\
\nonumber
&+P^{\mu}(u^{\rho}s^{\sigma}\varepsilon_{\rho\sigma\mu\alpha}\cos{\beta}
\!+\!u_{[\mu}s_{\alpha]}\sin{\beta})+\\
&+\frac{1}{2}R_{\alpha\mu}^{\phantom{\alpha\mu}\mu}\cos{\beta}
\!+\!\frac{1}{4}R^{\rho\sigma\mu}\varepsilon_{\rho\sigma\mu\alpha}\sin{\beta}\!=\!0\\
\nonumber
&\frac{1}{2}\nabla_{\nu}\ln{\phi^{2}}\sin{\beta}
\!+\!(\frac{1}{2}\nabla_{\nu}\beta\!-\!XW_{\nu})\cos{\beta}+\\
\nonumber
&+P^{\mu}(u^{\rho}s^{\sigma}\varepsilon_{\rho\sigma\mu\nu}\sin{\beta}\!-\!u_{[\mu}s_{\nu]}\cos{\beta})-\\
&-\frac{1}{4}R^{\rho\sigma\mu}\varepsilon_{\rho\sigma\mu\nu}\cos{\beta}
\!+\!\frac{1}{2}R_{\nu\mu}^{\phantom{\nu\mu}\mu}\sin{\beta}\!+\!ms_{\nu}\!=\!0
\end{eqnarray}
and after diagonalization
\begin{eqnarray}
\nonumber
&\frac{1}{2}\varepsilon_{\mu\alpha\nu\iota}R^{\alpha\nu\iota}
\!-\!2P^{\iota}u_{[\iota}s_{\mu]}-\\
&-2XW_{\mu}\!+\!\nabla_{\mu}\beta\!+\!2s_{\mu}m\cos{\beta}\!=\!0\\
\nonumber
&R_{\mu a}^{\phantom{\mu a}a}
\!-\!2P^{\rho}u^{\nu}s^{\alpha}\varepsilon_{\mu\rho\nu\alpha}+\\
&+2s_{\mu}m\sin{\beta}\!+\!\nabla_{\mu}\ln{\phi^{2}}\!=\!0
\end{eqnarray}
in general. Conversely, from these and then considering the general identities given by the expressions
\begin{eqnarray}
&2\boldsymbol{\sigma}^{\mu\nu}u_{\mu}s_{\nu}\boldsymbol{\pi}\psi\!+\!\psi=0\\
&is_{\mu}\boldsymbol{\gamma}^{\mu}\psi\sin{\beta}
\!+\!s_{\mu}\boldsymbol{\gamma}^{\mu}\boldsymbol{\pi}\psi\cos{\beta}\!+\!\psi=0
\end{eqnarray}
it is possible to see that
\begin{eqnarray}
\nonumber
&i\boldsymbol{\gamma}^{\mu}\boldsymbol{\nabla}_{\mu}\psi
\!-\!XW_{\sigma}\boldsymbol{\gamma}^{\sigma}\boldsymbol{\pi}\psi\!-\!m\psi=\\
\nonumber
&=[i\boldsymbol{\gamma}^{\mu}P^{\rho}u^{\nu}s^{\alpha}\varepsilon_{\mu\rho\nu\alpha}+\\
\nonumber
&+P^{\iota}u_{[\iota}s_{\mu]}\boldsymbol{\gamma}^{\mu}\boldsymbol{\pi}
\!+\!P_{\mu}\boldsymbol{\gamma}^{\mu}-\\
&-is_{\mu}\boldsymbol{\gamma}^{\mu}m\sin{\beta}
\!-\!s_{\mu}\boldsymbol{\gamma}^{\mu}\boldsymbol{\pi}m\cos{\beta}\!-\!m\mathbb{I}]\psi\!=\!0
\end{eqnarray}
showing that when the spinor is in polar form these field equations are valid: as any spinor can always be written in polar form then also these field equations are valid in general. So (\ref{D}) is equivalent to (\ref{dep1}, \ref{dep2}) in general \cite{Fabbri:2017pwp}.
\subsection{Gauge Momentum}
As we already said, the objects (\ref{R}, \ref{P}) are real tensors and, because of the information content that can be deduced from their definition, they contain all information normally contained within the connection and the gauge potential, and thus we called them tensorial connection and gauge vector momentum, respectively: in particular we have that $R_{ijk}$ has a trace defined as
\begin{eqnarray}
&\!R_{a}\!=\!R_{ac}^{\phantom{ac}c}
\end{eqnarray}
and its completely antisymmetric part has dual as
\begin{eqnarray}
&\!\!B_{a}\!=\!\frac{1}{2}\varepsilon_{aijk}R^{ijk}
\end{eqnarray}
so that the non-completely antisymmetric traceless part
\begin{eqnarray}
&\!\!\!\!\Pi_{ijk}\!=\!R_{ijk}\!-\!\frac{1}{3}(R_{i}\eta_{jk}\!-\!R_{j}\eta_{ik})
\!-\!\frac{1}{3}\varepsilon_{ijka}B^{a}
\end{eqnarray}
is such that $\Pi_{ia}^{\phantom{ia}a}\!=0$ and $\Pi_{ijk}\varepsilon^{ijka}\!=0$ hold; instead, for $P_{a}$ we have irreducibility. However, it is possible to have $P_{a}$ written in terms of $R_{ijk}$ according to the expression
\begin{eqnarray}
\nonumber
&P^{\mu}\!=\!m\cos{\beta}u^{\mu}
\!-\!\frac{1}{2}(\nabla_{k}\beta\!-\!2XW_{k}\!+\!B_{k})s^{[k}u^{\mu]}-\\
&-\frac{1}{2}(\nabla_{k}\ln{\phi^{2}}\!+\!R_{k})s_{j}u_{i}\varepsilon^{kji\mu}\label{momentum}
\end{eqnarray}
although it is only a link between $P_{a}$ and the two vectorial parts of $R_{ijk}$ and not a link to the full tensor; this is clear, because the only occurrence of the full $R_{ijk}$ is within the field equations (\ref{ee}) but even there it is always either in derivatives or in products. Only
(\ref{dep1}, \ref{dep2}) contain the pure forms of $R_{a}$ and $B_{a}$ needed for (\ref{momentum}) to be expressed.
To see that, consider (\ref{dep1}, \ref{dep2}) in terms of $R_{a}$ and $B_{a}$
\begin{eqnarray}
&\!\!B_{\mu}\!-\!2P^{\iota}u_{[\iota}s_{\mu]}\!+\!2(\nabla\beta/2\!-\!XW)_{\mu}
\!+\!2s_{\mu}m\cos{\beta}\!=\!0\\
&\!R_{\mu}\!-\!2P^{\rho}u^{\nu}s^{\alpha}\varepsilon_{\mu\rho\nu\alpha}
\!+\!2s_{\mu}m\sin{\beta}\!+\!\nabla_{\mu}\ln{\phi^{2}}\!=\!0
\end{eqnarray}
and then contract the first by $u^{\mu}$ and $s^{\mu}$ and the second by $u^{\nu}s^{\alpha}\varepsilon_{\nu\alpha\mu\rho}$ eventually getting
\begin{eqnarray}
&Ps\!+\!\frac{1}{2}(\nabla\beta\!-\!2XW\!+\!B)u\!=\!0\\
&Pu\!+\!\frac{1}{2}(\nabla\beta\!-\!2XW\!+\!B)s\!-\!m\cos{\beta}\!=\!0\\
&\!P^{\rho}\!+\!Ps s^{\rho}\!-\!Pu u^{\rho}
\!+\!\frac{1}{2}(\nabla\ln{\phi^{2}}\!+\!R)_{\mu}s_{\alpha}u_{\nu}
\varepsilon^{\mu\alpha\nu\rho}\!=\!0
\end{eqnarray}
which are now easier to manipulate since in the last one $P^{\rho}$ appears isolated and the other occurrences $Ps$ and $Pu$ can be substituted in terms of the other two expressions given above: if the replacement is made then we obtain
\begin{eqnarray}
\nonumber
&P^{\rho}\!=\!\frac{1}{2}(\nabla\beta\!-\!2XW\!+\!B)u s^{\rho}
\!-\!\frac{1}{2}(\nabla\beta\!-\!2XW\!+\!B)s u^{\rho}+\\
&+m\cos{\beta}u^{\rho}\!-\!\frac{1}{2}(\nabla\ln{\phi^{2}}\!+\!R)_{\mu}s_{\alpha}u_{\nu}
\varepsilon^{\mu\alpha\nu\rho}
\end{eqnarray}
in general. Therefore (\ref{dep1}, \ref{dep2}) imply (\ref{momentum}) in general \cite{Fabbri:2018crr}.
\subsection{Velocity}
Expression (\ref{momentum}) gives the gauge momentum in terms of the module and the Yvon-Takabayashi angle, but also in terms of the velocity and the spin: because both momentum and spin are supposed to be constants of motion, it is interesting to invert it for the velocity. For this, define
\begin{eqnarray}
&m\cos{\beta}\!-\!\frac{1}{2}(\nabla\beta\!-\!2XW\!+\!B)_{k}s^{k}\!=\!X\\
&\frac{1}{2}(\nabla\beta\!-\!2XW\!+\!B)_{k}\!=\!Y_{k}\\
&-\frac{1}{2}(\nabla\ln{\phi^{2}}\!+\!R)_{k}\!=\!Z_{k}
\end{eqnarray}
in terms of which the momentum is written as
\begin{eqnarray}
&P^{a}\!=\!(X\eta^{ak}\!+\!Y^{k}s^{a}\!+\!Z_{i}s_{j}\varepsilon^{ijka})u_{k}\label{a}
\end{eqnarray}
in the form of a matrix containing only the spin applied to the velocity: when inverted it will give the velocity as a product of a specific spin-dependent matrix applied to the momentum. To get such inversion, have (\ref{a}) dotted into $Z^{i}s^{j}\varepsilon_{ijka}$ and $Z_{a}$ so to obtain
\begin{eqnarray}
\nonumber
&P^{a}Z^{i}s^{j}\varepsilon_{ijka}\!=\!XZ^{i}s^{j}u^{a}\varepsilon_{ijka}+\\
&+(Z^{2}\!+\!|Z\!\cdot\!s|^{2})u_{k}\!-\!Z\!\cdot\!u (Z_{k}\!+\!Z\!\cdot\!s s_{k})\label{b1}\\
&P\!\cdot\!Z\!+\!P\!\cdot\!s Z\!\cdot\!s\!=\!XZ\!\cdot\!u\label{b2}
\end{eqnarray}
and after having (\ref{b2}) substituted into (\ref{b1}) and the result substituted back into (\ref{a}) we finally end up with
\begin{eqnarray}
\nonumber
&u^{k}\!=\!(1\!+\!Z^{2}/X^{2}\!+\!|Z\!\cdot\!s|^{2}/X^{2})^{-1}[\eta^{ka}+\\
\nonumber
&+s^{a}s^{k}(1\!+\!|Z\!\cdot\!s|^{2}/X^{2})\!+\!Z^{a}Z^{k}/X^{2}+\\
&+(s^{a}Z^{k}\!+\!Z^{a}s^{k})Z\!\cdot\!s/X^{2}\!+\!Z_{i}s_{j}\varepsilon^{ijka}/X]P_{a}/X
\end{eqnarray}
giving the velocity as product of a specific spin-dependent matrix further applied onto the momentum in general.
More specifically we can introduce also
\begin{eqnarray}
&\zeta_{k}\!=\!Z_{k}/X
\end{eqnarray}
and write
\begin{eqnarray}
\nonumber
&u^{k}\!=\!(1\!+\!\zeta^{2}\!+\!|\zeta\!\cdot\!s|^{2})^{-1}[\eta^{ka}+\\
\nonumber
&+s^{a}s^{k}(1\!+\!|\zeta\!\cdot\!s|^{2})\!+\!\zeta^{a}\zeta^{k}+\\
&+(s^{a}\zeta^{k}\!+\!\zeta^{a}s^{k})\zeta\!\cdot\!s\!+\!\zeta_{i}s_{j}\varepsilon^{ijka}]P_{a}/X
\label{eq}
\end{eqnarray}
as the most compact form we can have again in general.
\section{Physical Effects}
\subsection{Internal Dynamics}
In the previous section, we have introduced the polar form of spinors in $4$ dimensions. Of course, one may also consider what the polar form would look like when time is a parameter, and so in $3$ dimensions: in such a case, a spinor would have two complex components, hence $4$ real components, and because the spinor transformation law would contain only $3$ rotations, up to $3$ of these components can be removed. This spinor in polar form would have a single degree of freedom, the module. Therefore, if the non-relativistic spinor ought be obtained as a limit of the general spinor, this limit must account for the fact that the Yvon-Takabayashi angle has to vanish beside the fact that the velocity spatial part has to vanish \cite{Fabbri:2016msm,Fabbri:2016laz}.
Because when the spatial part of the velocity is equal to zero but the Yvon-Takabayashi angle is different from zero we are still unable to obtain the full non-relativistic limit, then we must conclude that the Yvon-Takabayashi angle describes the motion of what remains even in rest frame, which has to be the intrinsic motion. That is, it is the motion describing the internal dynamics of a spinor.
We notice that in the definition of the non-relativistic limit, given by the $\beta\!\rightarrow\!0$ and $\vec{u}\!\rightarrow\!0$ above, there appears no gauge momentum (\ref{momentum}) at all: as a matter of fact, the momentum in non-relativistic limit is given by
\begin{eqnarray}
&E\!=\!m\!-\!(X\vec{W}\!-\!\frac{1}{2}\vec{B})\!\cdot\!\vec{s}\\
&\vec{P}\!=\!-(XW^{0}\!-\!\frac{1}{2}B^{0})\vec{s}
\!-\!\frac{1}{2}(\vec{\nabla}\ln{\phi^{2}}\!-\!\vec{R})\!\times\!\vec{s}
\end{eqnarray}
showing that the energy does not reduce to the mass and the spatial momentum does not vanish. The usual limit given by $P_{a}\!\rightarrow\!(m,\vec{0})$ can only be obtained by neglecting the spin content of the spinor, that is if the macroscopic approximation is also implemented for field distributions.
Again, this makes sense, because non-relativistic limit means small spatial momentum only if the momentum is free from spin contributions, and that is from the internal dynamics. Therefore, it is reasonable that such limit can be obtained in this way only by requiring that the internal dynamics be concealed inside the field distribution, as it would normally happen for macroscopic approximations.
Having obtained some insight from the non-relativistic limit, let us see what happens for velocities that are large in general. In this case we can use the expression of the velocity given by (\ref{eq}) in terms of gauge momentum and spin while depending on module and Yvon-Takabayashi angle in general. Considering that momentum and spin are constants of motion, module and Yvon-Takabayashi angle are the only variables. This tells us that spinorial fields can be seen as very peculiar types of fluid for which the velocity depends on density and internal dynamics.
Nonetheless, (\ref{eq}) is too complicated to get meaningful information. To simplify, we study limiting cases, and as special case we take $\zeta_{a}$ to be small so that we have
\begin{eqnarray}
&u^{k}\!\approx\!(\eta^{ka}\!+\!s^{a}s^{k}\!+\!\zeta_{i}s_{j}\varepsilon^{ijka})P_{a}/X
\end{eqnarray}
to the first-order perturbative. Notice that the $\zeta_{a}$ potential contains the gradient of the logarithm of the density, so it can be regarded as the de Broglie-Bohm quantum potential, in the first-order derivative form, and containing also the Yvon-Takabayashi angle contributions.
Differently from the non-relativistic case, based on the Schr\"{o}dinger equation, for which the quantum potential is second-order derivative, in this most general case, based on Dirac equations, the quantum potential is first-order derivative. And differently from the non-relativistic case, where only the module is present, in this most general of cases, both the module and the Yvon-Takabayashi angle give some contribution to the quantum potential.
As a consequence, we regard the $\zeta_{a}$ vector as the quantum potential in the most general form possible.
If we were to take the approximation of small $Y_{a}$ then
\begin{eqnarray}
&\!\!m\cos{\beta}u^{k}\!\approx\!\left(1\!+\!\frac{Y_{b}s^{b}}{m\cos{\beta}}\right)\!
(\eta^{ka}\!+\!s^{a}s^{k}\!+\!\zeta_{i}s_{j}\varepsilon^{ijka})P_{a}
\end{eqnarray}
and for small Yvon-Takabayashi angle
\begin{eqnarray}
&mu^{k}\!\approx\!(1\!+\!Y_{b}s^{b}/m)
(P^{k}\!+\!P_{a}s^{a}s^{k}\!-\!P_{a}\zeta_{i}s_{j}\varepsilon^{aijk})
\end{eqnarray}
with a contribution scaling the momentum in terms of the Yvon-Takabayashi angle plus a contribution changing the direction of the momentum in terms of both module and Yvon-Takabayashi angle. That is, as quantum potential.
Considering only the spatial part, cases of small spatial momentum allow us to better see the effects of the spin contributions. In such cases we have
\begin{eqnarray}
&\vec{u}\!\approx\!(1\!-\!\vec{Y}\!\cdot\!\vec{s}/m)\vec{Z}\!\times\!\vec{s}/m
\end{eqnarray}
with the spin contributions that are in fact explicit.
If torsion were negligible and $B_{a}\!=\!R_{a}\!=\!0$ \cite{Fabbri:2017pwp} then
\begin{eqnarray}
&\vec{u}\!\approx\!(1\!+\!\vec{\varsigma}\!\cdot\!\vec{\nabla}\beta/m)
\vec{\nabla}\ln{\phi^{2}}\!\times\!\vec{\varsigma}/m
\end{eqnarray}
having introduced $\vec{\varsigma}\!=\!\vec{s}/2$ as the usual expression of spin.
It is interesting to notice that in this case we can compute the magnetic moment obtaining the expression
\begin{eqnarray}
\nonumber
\vec{\mu}\!=\!\frac{1}{2}\!\int\!\vec{r}\!\times\!q\vec{U}dV=\\
\nonumber
=\frac{q}{2m}2\!\int\!(1\!+\!\vec{\varsigma}\!\cdot\!\vec{\nabla}\beta/m)
[\vec{r}\!\times\!(\vec{\nabla}\phi^{2}\!\times\!\vec{\varsigma}\,)]dV=\\
\nonumber
=\frac{q}{2m}2\!\int\!(1\!+\!\vec{\varsigma}\!\cdot\!\vec{\nabla}\beta/m)
2\phi^{2}\vec{\varsigma}dV=\\
=\frac{q}{2m}2\vec{\varsigma}\left(1\!+\!\langle\vec{\varsigma}\!\cdot\!\vec{\nabla}\beta/m\rangle\right)
\end{eqnarray}
and so that we can finally write
\begin{eqnarray}
&(g\!-\!2)/2\!\approx\!\langle\vec{\varsigma}\!\cdot\!\vec{\nabla}\beta/m\rangle
\label{correction}
\end{eqnarray}
in terms of the common form of the gyromagnetic factor.
To first order, this would agree with the $\alpha/2\pi$ term if on average the gradient of $\beta$ along the spin were equal to the fine-structure constant $\alpha$ over half Compton length, and this is remarkable since $\beta$ is generally expected to be of the order of the fine-structure constant $\alpha$ \cite{Fabbri:2018crr}.
On the other hand, going beyond order-of-magnitude evaluations requires the Yvon-Takabayashi angle $\beta$ to be known in terms of exact solutions and for the time being this task appears to be out of the possibilities.
More in general, the spin contributions are as in
\begin{eqnarray}
&\vec{u}\!\approx\!(1\!-\!\vec{Y}\!\cdot\!\vec{s}/m)\vec{Z}\!\times\!\vec{s}/m
\end{eqnarray}
showing that the correction to the magnetic moment will depend not only on the Yvon-Takabayashi angle but also on torsion and on the $B_{a}$ axial vector, altogether collected into the $Y_{a}$ axial vector. While the very presence of this term depends on the $\zeta_{a}$ vector, it is necessary for further corrections that the terms $X$ or $Y_{a}$ be present as well.
While the $\zeta_{a}$ vector is the quantum potential providing quantum mechanical effects, the $X$ or $Y_{a}$ terms are what provides quantum field theoretical corrections.
\subsection{Chirality}
When we consider again the non-relativistic limit, the first condition is that of boosting into the rest frame, and in this frame the assumption of rotating the spinor so to align its spin along the third axis is equivalent to require that in the polar form $\boldsymbol{S}$ be the identity: in this instance
\begin{eqnarray}
&\!\psi\!=\!\phi\left(\!\begin{tabular}{c}
$e^{\frac{i}{2}\beta}$\\
$0$\\
$e^{-\frac{i}{2}\beta}$\\
$0$
\end{tabular}\!\right)
\end{eqnarray}
in chiral representation or
\begin{eqnarray}
&\!\psi\!=\!\phi \sqrt{2}
\!\left(\!\begin{tabular}{c}
$\cos{\frac{\beta}{2}}$\\
$0$\\
$-i\sin{\frac{\beta}{2}}$\\
$0$
\end{tabular}\!\right)
\end{eqnarray}
in standard representation. The remaining condition demands that the Yvon-Takabayashi angle vanishes
\begin{eqnarray}
\!\psi\!=\!\phi \sqrt{2}
\!\left(\!\begin{tabular}{c}
$1$\\
$0$\\
$0$\\
$0$
\end{tabular}\!\right)
\end{eqnarray}
showing that the lower component is zero. This allows a single condition to represent the non-relativistic limit.
When the spinor is written in standard representation, its lower component is for this reason also called small component. As for the same spinor in chiral representation, the lower component is the right-handed component and the upper component is the left-handed component.
So the Yvon-Takabayashi angle, which is related to the small component, gives the phase opposition between the right-handed and left-handed components of spinors.
Because zitterbewegung effects are known to arise from the existence of the small component, or more in general from the interplay of the chiral components, then we can infer that zitterbewegung must be linked to the presence of the Yvon-Takabayashi angle quite generally.
It is also interesting to see that the bi-linear quantities $\Theta$ and $\Phi$ when written in terms of the left-handed and the right-handed components $L$ and $R$ have expressions given by $\Theta\!=\!i(L^{\dagger}R\!-\!R^{\dagger}L)$ and $\Phi\!=\!(L^{\dagger}R\!+\!R^{\dagger}L)$ so that, for such a reason, they assume a remarkably clear interpretation.
Considering that the Yvon-Takabayashi angle is related to the scalar $\Theta/\Phi$ while the module is $\Theta^{2}\!+\Phi^{2}$ we interpret the module and the Yvon-Takabayashi angle as the mean of the chiral components and the standard deviation from the mean of the chiral components, respectively.
This recovers the interpretation of the module and the Yvon-Takabayashi angle as describing an averaged material distribution with its internal structure.
\section{Applications}
\subsection{Anomaly}
Having the expression of the velocity (\ref{eq}), an application of some importance can be found for the electrodynamic potential. This potential has the form
\begin{eqnarray}
&\mathscr{V}\!=\!qU^{k}A_{k}
\end{eqnarray}
and consequently we can write
\begin{eqnarray}
\nonumber
&\mathscr{V}\!=\!2q\phi^{2}(1\!+\!\zeta^{2}\!+\!|\zeta\!\cdot\!s|^{2})^{-1}[P_{a}A^{a}+\\
\nonumber
&+P_{a}s^{a}A_{k}s^{k}(1\!+\!|\zeta\!\cdot\!s|^{2})\!+\!P_{a}\zeta^{a}A_{k}\zeta^{k}+\\
\nonumber
&+(P_{a}s^{a}A_{k}\zeta^{k}\!+\!P_{a}\zeta^{a}A_{k}s^{k})\zeta\!\cdot\!s+\\
&+P_{a}A_{k}\zeta_{i}s_{j}\varepsilon^{ijka}]/X
\label{edp}
\end{eqnarray}
quite straightforwardly. As (\ref{eq}) is the expression of the most general form of the velocity, then it is clear that the above is the most comprehensive electrodynamic potential density. Its integral over the volume must contain the complete information about all electrodynamic effects.
In the previous section we have seen that in some approximation, it is possible to find the magnetic moment correction (\ref{correction}), which should furnish the correct value in specific cases of Yvon-Takabayashi angle. Just the same, we also commented that this can be done only when exact solutions are found, and such task is difficult in general.
Nonetheless, since (\ref{edp}) is the electrodynamic potential developed in its most general form then it should contain full information about all electrodynamic effects.
In the simplest from for plane waves it reduces to
\begin{eqnarray}
&\mathscr{V}_{\mathrm{pw}}\!=\!2q\phi^{2}P_{a}A^{a}/m
\end{eqnarray}
which means that $\Delta\mathscr{V}\!=\!\mathscr{V}\!-\!\mathscr{V}_{\mathrm{pw}}$ contains the information about all effects arising from general solutions and which would not be obtained by analyses employing plane wave solutions. So $\Delta\mathscr{V}$ has the information found in non-zero Yvon-Takabayashi angles or in non-constant modules.
That $\Delta\mathscr{V}$ may encode electrodynamic effects not obtainable with the use of plane waves means that in it we might find all of the anomalous terms due to the radiative corrections normally arising from field quantization.
This idea is not a new suggestion, as anomalies to the gyromagnetic factor have already been studied by using affine structures or spinorial interactions \cite{Capozziello:2014gua,Cirilo-Lombardo:2014opa}.
\section{Conclusion}
In this paper, we have studied the Dirac spinor field in polar form giving the field equations, combining them as to get the explicit form of the momentum, and inverting it as to obtain the explicit form of the velocity: with such tools, we established that the Yvon-Takabayashi angle is what describes the internal dynamics, defined as relative motion between chiral parts, that this is connected to the effects of zitterbewegung, arising from spin contributions, and that both Yvon-Takabayashi angle and module come together to form the $\zeta_{a}$ vector, which is the most general form of quantum potential providing quantum mechanical effects; we also established that the Yvon-Takabayashi angle takes part in the $Y_{a}$ axial vector, which appears to encode the information about the anomalous terms that arise as quantum field theory corrections. Consequently, we have conjectured that the anomalous behaviour fields display might be obtained by considering solutions that are more general than the simplest mere plane waves.
We have shown that this might happen for the anomaly of the magnetic moment, and we have recalled that in the literature it has already been discussed that this may also happen in the case of the running of coupling constants for the renormalization group; other cases were exhibited in the literature mentioned in the introduction. The idea, expressed also in the introduction, that effects described by quantization in terms of plane wave solutions should also be described solely in terms of more general solutions has therefore some evidence, although the actual proof of this conjecture requires exact solutions, which we lack.
Our point in the present paper, however, was much less ambitious. What we wanted to do was merely to present a problem and conjecture a possible way out, that is that the problems of quantization may be altogether circumvented by an approach that does not involve quantization at all but which recovers its effects in terms of more general approaches involving fields displaying more general internal dynamics, and to this purpose, we provided the most general form for the Dirac spinor field theory.
This paper is intended to lay the grounds as a starting point for any future work that aims to face the problem of finding general solutions with internal dynamics which may recover the effects of quantization.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 6,436 |
<?php
namespace Pantheon\Terminus\UnitTests\Commands\Solr;
use Pantheon\Terminus\Commands\Solr\EnableCommand;
use Pantheon\Terminus\Models\Workflow;
use Pantheon\Terminus\UnitTests\Commands\CommandTestCase;
use Pantheon\Terminus\Models\Solr;
use Pantheon\Terminus\UnitTests\Commands\WorkflowProgressTrait;
/**
* Class EnableCommandTest
* Testing class for Pantheon\Terminus\Commands\Solr\EnableCommand
* @package Pantheon\Terminus\UnitTests\Commands\Solr
*/
class EnableCommandTest extends CommandTestCase
{
use WorkflowProgressTrait;
/**
* Tests the solr:enable command
*/
public function testEnableSolr()
{
$workflow = $this->getMockBuilder(Workflow::class)
->disableOriginalConstructor()
->getMock();
// workflow succeeded
$workflow->expects($this->once())->method('getMessage')->willReturn('successful workflow');
$this->solr = $this->getMockBuilder(Solr::class)
->disableOriginalConstructor()
->getMock();
$this->solr->expects($this->once())
->method('enable')
->willReturn($workflow);
$this->site->method('getSolr')->willReturn($this->solr);
$this->logger->expects($this->at(0))
->method('log')->with(
$this->equalTo('notice'),
$this->equalTo('successful workflow')
);
$this->command = new EnableCommand();
$this->command->setContainer($this->getContainer());
$this->command->setSites($this->sites);
$this->command->setLogger($this->logger);
$this->expectWorkflowProcessing();
$this->command->enable('mysite');
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 3,128 |
\section{Introduction}
The BFKL (Balitsky-Fadin-Kuraev-Lipatov) approach~\cite{BFKL} was formulated
in the momentum space. In this space the kernel of the BFKL equation was
calculated in the next-to-leading order (NLO) long ago, at first for the
forward scattering (i.e. for $t=0$ and colour singlet in the
$t$-channel)~\cite{Fadin:1998py} and then for any fixed (not growing with
energy) squared momentum transfer $t$ and any possible two-gluon
colour state in the $t$-channel~\cite{FF05}. Unfortunately, the NLO kernel is
rather complicated. In particular, the colour singlet kernel for
$t\neq 0$ is found in the NLO in the form of an intricate two-dimensional
integral.
In the most interesting for phenomenological applications case of colourless
particle scattering, the leading-order (LO) BFKL kernel has a remarkable
property~\cite{Lipatov:1985uk}: it can be taken in the M\"{o}bius
representation (i.e. in the space of functions vanishing at coinciding
transverse coordinates of Reggeons), where it turns out to be invariant
in regard to conformal transformations of these coordinates. Moreover, in the
coordinate space the M\"{o}bius representation (we will call it ``M\"{o}bius
form'') of the LO BFKL kernel coincides~\cite{Fadin:2006ha} with the kernel of
the colour dipole model~\cite{dipole}.
In the NLO the conformal invariance is violated in QCD by the running coupling.
One could hope that the M\"{o}bius form of the colour singlet NLO kernel is
quasi-conformal, i.e. conformal invariance is violated only by terms
proportional to the $\beta$-function. However, the direct transformation of
the colour singlet kernel found in Ref.~\cite{FF05} from momentum to
coordinate space with the restriction of M\"{o}bius representation gives a
kernel which is not quasi-conformal~\cite{Fadin:2007ee,Fadin:2007de,
Fadin:2007xy}.
But in the NLO kernel there is an
ambiguity~\cite{Fadin:2006ha,Kovchegov:2006wf}, analogous to the well known
ambiguity of the NLO anomalous dimensions, because it is possible to
redistribute radiative corrections between the kernel and the impact factors.
The ambiguity, discussed in details in Ref.~\cite{Fadin:2009za}, permits to
make transformations
\begin{equation}
\hat{\mathcal{K}}\rightarrow \hat{\mathcal{K}}
-\alpha_s[\hat{\mathcal{K}}^{(B)},\hat{U}]~\label{kernel transformation}
\end{equation}
conserving the LO kernel $\hat{\mathcal{K}}^{(B)}$ (which is fixed in our case
by the requirement of conformal invariance of its M\"{o}bius form) and
changing the NLO part of the kernel. Note that this transformation must
conserve the gauge invariance properties of the kernel, so that the operator
$\hat{U}$ must have in this respect the same properties as
$\hat{\mathcal{K}}^{(B)}$.
The NLO kernel calculated in Ref.~\cite{FF05} is defined according to the
prescriptions given in Ref.~\cite{Fadin:1998fv}. We will call it the
``standard kernel''. Recently it was shown~\cite{Fadin:2009gh} that there
exist an operator $\hat{U}$ such that the
transformation~(\ref{kernel transformation}) applied to this standard kernel
gives a kernel with quasi-conformal M\"{o}bius form, which agrees with the
form obtained in Ref.~\cite{Balitsky:2009xg} in the colour dipole approach.
It turns out that this form is quite simple. It is unbelievably simple in
comparison with the form of the standard kernel~\cite{FF05}. Evidently, the
question arose about the relation between these two forms.
This question is not trivial not only because the M\"{o}bius form is defined
in the coordinate space, whereas the standard kernel was calculated
in the momentum space.
Remind that the M\"{o}bius representation is defined on a special class of
functions. Therefore at the first sight it seemed impossible to reconstruct
the complete operator from its M\"{o}bius form. However, due to the gauge
invariance of the BFKL kernel, it is not so. It was shown~\cite{Fadin:2011jg}
for any gauge invariant two-particle operator that it is possible to restore
the complete operator from its M\"{o}bius form and the restoration is unique
up to terms which do not contribute to the operator matrix elements, because
of symmetry and gauge invariance of the wave functions.
Therefore, it is in principle possible to restore the complete BFKL kernel
from its quasi-conformal M\"{o}bius form. Since this form is quite simple,
one can hope for simplicity of the complete kernel in the momentum space too.
Evidently this kernel differs from the standard kernel found in
Ref.~\cite{FF05}, but is connected with the last one by the
transformation~(\ref{kernel transformation}). However, the direct restoration
is not easy. It includes the Fourier transformation of the M\"{o}bius form
from coordinate to momentum space and, although this form is very compact,
the transformation is intricate since it contains complicated integrals.
Instead, one can try to find the difference between the standard kernel and the
one restored from the quasi-conformal M\"{o}bius form. Our paper is devoted to
the solution of this problem. The difference under investigation is given by
the second term in the transformation~(\ref{kernel transformation}).
For the operator $\hat{U}$, both M\"{o}bius form and complete representation
in the momentum space are known now~\cite{Fadin:2011jg}. The same is true for
$\hat{\mathcal{K}}^{(B)}$. We are looking for the difference in the momentum
space. It can be found using for the calculation of the commutator
in the transformation~(\ref{kernel transformation}) both $\hat{U}$ and
$\hat{\mathcal{K}}^{(B)}$ in this space. Alternatively, it is possible to
calculate the commutator in the coordinate M\"{o}bius space and then to
restore its complete form in the momentum space using the method developed in
Ref.~\cite{Fadin:2011jg}. We use both these ways, on one side for
cross-checking the obtained result, on the other for a demonstration of the
efficiency of the method of restoration of complete operators from their
M\"{o}bius forms, developed in Ref.~\cite{Fadin:2011jg}.
The paper is organized as follows. In the next Section we calculate the
commutator in the transformation~(\ref{kernel transformation}) directly
in the momentum space. In Section~3 this commutator is calculated firstly in
the coordinate M\"{o}bius space and then the obtained result is used for
restoration of the complete form of the commutator in the momentum space. The
last Section contains our conclusions. The integrals used in the calculations
are presented in the Appendix.
\section{Direct calculation of the difference in momentum space}
\label{sec:direct}
We adopt the notation used in Ref.~\cite{Fadin:2011jg} and put the space-time
dimension $D$ equal to 4, so that states $|\vec{q}\rangle$ with definite
two-dimensional transverse Reggeon momentum $\vec{q}$ and states
$|\vec {r}\rangle$ with definite Reggeon impact parameter $\vec{r}$ are
normalized as follows:
\begin{equation}
\langle\vec{q}|\vec{q}^{\;\prime}\rangle=\delta(\vec{q}-\vec{q}^{\;\prime
})\;,\;\;\;\;\;\langle\vec{r}|\vec{r}^{\;\prime}\rangle=\delta(\vec{r}-\vec
{r}^{\;\prime})\;,\;\;\;\;\;
\langle\vec{r}|\vec{q}\rangle=\frac{e^{i\vec{q}\,\vec
{r}}}{2\pi}\;.\label{normalization}%
\end{equation}
As it was shown in Ref.~\cite{Fadin:2009gh}, the quasi-conformal
kernel $\hat{\mathcal{K}}^{QC}$ can be obtained from the kernel
calculated in Ref.~\cite{FF05} by the
transformation~(\ref{kernel transformation}), namely,
\begin{equation}
\hat{\mathcal{K}}^{QC} = \hat{\mathcal{K}}-\alpha_s[\hat{\mathcal{K}}^{(B)},
\hat{U}]~. \label{QC kernel}
\end{equation}
It is worthwhile to note here that the kernel $\hat{\mathcal{K}}$ is defined
in such a way that in the LO its M\"{o}bius form is conformal invariant.
Therefore one has (see Ref.~\cite{Fadin:2011jg} for details)
\begin{equation}
\langle\vec{q}_{1},\vec{q}_{2}|\hat{\mathcal{K}}|\vec{q}_{1}^{\;\prime},\vec
{q}_{2}^{\;\prime}\rangle=\delta(\vec{q}_{1}+\vec{q}_{2}-\vec{q}_{1}%
^{\;\prime}-\vec{q}_{2}^{\;\prime})\frac{1}{\vec{q}_{1}^{\,\,2}\vec
{q}_{2}^{\,\,2}}{K}(\vec{q}_{1},\vec{q}_{1}^{\;\prime};\vec{q})%
\;,\label{relation between kernels}%
\end{equation}
where $\vec{q}=\vec{q}_{1}+\vec{q}_{2}=\vec{q}_{1}^{\;\prime}+\vec
{q}_{2}^{\;\prime}$ and ${K}(\vec{q}_{1},\vec{q}%
_{1}^{\;\prime};\vec{q})$ is the symmetric kernel
\begin{equation}
{K}(\vec{q}_{1},\vec{q}_{1}^{\;\prime};\vec{q}) ={K}(\vec{q}_{1}^{\;\prime},
\vec{q}_{1};\vec{q})\;, \label{symmetry of the kernel}%
\end{equation}
defined in Ref.~\cite{Fadin:1998fv} and calculated in Ref.~\cite{FF05}.
Its real part ${K}_r$ satisfies the gauge invariance conditions
\begin{equation}
{K}_r(\vec{0},\vec{q}_{1}^{\;\prime};\vec{q}) ={K}_r(\vec{q}_{1},
\vec{0};\vec{q})
={K}_r(\vec{q},\vec{q}_{1}^{\;\prime};\vec{q}) ={K}_r(\vec{q}_{1},
\vec{q};\vec{q})\;. \label{gauge invariance of the kernel}%
\end{equation}
Our goal is to find in the momentum space an explicit form for the
commutator in Eq.~(\ref{QC kernel}). In this Section it is done using the known
expressions in this space for the LO kernel $\hat{\mathcal{K}}^{(B)}$ and the
operator $\hat{U}$.
The kernel $\hat{\mathcal{K}}^{(B)}$ can be presented as follows
\begin{equation}
\langle\vec{q}_{1},\vec{q}_{2}|{\hat{\mathcal{K}}}^{(B)}|\vec{q}%
_{1}^{\,\,\prime},\vec{q}_{2}^{\,\,\prime}\rangle=\delta(\vec{q}_{11^{\prime}%
}+\vec{q}_{22^{\prime}})\frac{\alpha_{s}N_{c}}{2\pi^{2}}\left[R(\vec{q}_{1},
\vec{q}_{2}; \vec{k})-\delta(\vec{k})\int d\vec{l} \; V(\vec{q}_{1},
\vec{q}_{2}; \vec{l}) \right]\;, \label{K B momentum}%
\end{equation}
where $\vec{k}=\vec{q}_{11^{\prime}}=-\vec{q}_{22^{\prime}}$
(here and below $\vec{a}_{ij^{\prime}}=\vec{a}_{i}-\vec{a}^{\;\prime}_{j},\;
\vec{a}_{ij}=\vec{a}_{i}-\vec{a}_{j}, \;\vec{a}_{i^{\prime}j^{\prime}}
=\vec{a}^{\;\prime}_{i}-\vec{a}^{\;\prime}_{j}$),
\begin{equation}
R(\vec{q}_{1},\vec{q}_{2}; \vec{k})= \frac{2}%
{\vec{k}^{\,2}}-2\frac{\vec{q}_{1}\vec{k}}{\vec{k}^{\,2}\vec{q}%
_{1}^{\,\,2}}+2\frac{\vec{q}_{2}\vec{k}}{\vec{k}^{\,2}\vec{q}_{2}^{\,\,2}%
} -2\frac{\vec{q}_{1}\vec{q}_{2}}{\vec{q}_{1}^{\,\,2}\vec{q}_{2}%
^{\,\,2}}\label{R momentum}
\end{equation}
and
\begin{equation}
V(\vec{q}_{1},\vec{q}_{2}; \vec{l})= \frac{2}{\vec{l}^{\,\,2}}%
-\frac{\vec{l}(\vec{l}-\vec{q}_{1})}{\vec{l}^{\,\,2}(\vec{l}-\vec{q}_{1})^{2}%
}-\frac{\vec{l}(\vec{l}-\vec{q}_{2})}{\vec{l}^{\,\,2}(\vec{l}-\vec{q}%
_{2})^{\,2}}\;. \label{V momentum}%
\end{equation}
Note that the term ${2}/{\vec{l}^{\,\,2}}$ in $V(\vec{q}_{1},\vec{q}_{2};
\vec{l})$ leads to the divergence of the integral over $d\vec{l}$ in the
second term of Eq.~(\ref{K B momentum}), which represents the virtual part of
the kernel. It is a well known infrared divergence which cancels with the
divergence coming from the term ${2}/{\vec{k}^{\,2}}$ in the part with
$R(\vec{q}_{1},\vec{q}_{2}; \vec{k})$ in Eq.~(\ref{K B momentum}) (real
part), when ${\hat{\mathcal{K}}}^{(B)}$ acts on some state. In the commutator
$[\hat{\mathcal{K}}^{(B)},\hat{U}]$ there are no problems with these
divergences at all, because they cancel separately in the virtual and real
parts.
The gauge invariance properties for $R$ look as follows:
\[
R(\vec{q}_{1},\vec{q}_{2}; \vec{q}_1)=R(\vec{q}_{1},\vec{q}_{2}; -\vec{q}_2)
=0, \;\;
\]
\begin{equation}
( \vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2} R(\vec{q}_{1},\vec{q}_{2};
\vec{k}))|_{\vec{q}_{1}=0}=(\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}
R(\vec{q}_{1},\vec{q}_{2}; \vec{k}))|_{\vec{q}_{2}=0}=0~.
\label{g i for R}
\end{equation}
An explicit form of the operator ${\hat{U}}$ in the momentum space was found
in Ref.~\cite{Fadin:2011jg}. Omitting terms which do not contribute to the
commutator in Eq.~(\ref{QC kernel}), we have
\[
\langle\vec{q}_{1},\vec{q}_{2}|\alpha_s{\hat{U}}|\vec{q}_{1}^{\,\,\prime}%
,\vec{q}_{2}^{\,\,\prime}\rangle=\delta(\vec{q}_{11^{\prime}}+\vec
{q}_{22^{\prime}})\frac{\alpha_s N_c}{4\pi^{2}}R_u(\vec{q}_1, \vec{q}_2;
\vec{k})
\]%
\begin{equation}
-\frac{\alpha_s\beta_0}{8\pi} \ln\left(\vec{q}_{1}^{\;2}
\vec{q}_{2}^{\;2}\right)\delta(\vec{q}_{11^{\prime}})\delta(\vec
{q}_{22^{\prime}})~,
\label{U momentum}
\end{equation}
where $\beta_0$ is the first coefficient of the Gell-Mann--Low function,
\begin{equation}
\beta_0 =\frac{11}{3}N_c -\frac{2}{3}n_f
\label{beta 0}
\end{equation}
and
\[
R_u(\vec{q}_1, \vec{q}_2; \vec{k})=
\frac{1}{\vec{q}_1^{\,\,2}}\ln\left(\frac{\vec{q}_{1}^{\;\prime\,2}
\vec{q}_{2}^{\;2}}
{\vec{k}^{\,2}\vec{q}^{\;2}}\right)+\frac{1}{\vec{q}_2^{\,\,2}}
\ln\left(\frac{\vec{q}_{2}^{\;\prime\,2}\vec{q}_{1}^{\;2}}
{\vec{k}^{\,2}\vec{q}^{\;2}}\right)+\frac{1}{\vec{k}^{\,2}}\ln\left(\frac
{\vec{q}_{1}^{\;\prime\,2}\vec{q}_{2}^{\;\prime\,2}}{\vec{q}_{1}^{\;2}
\vec{q}_{2}^{\;2}}\right)
\]
\begin{equation}
-2\frac{\vec q_1\vec k}{\vec{k}^{\,2}\vec{q}_1^{\,\,2}}
\ln\left(\frac{\vec{q}_{1}^{\;\prime\,2}}
{\vec{k}^{\,2}}\right)
+2\frac{\vec q_2\vec k}{\vec{k}^{\,2}\vec{q}_2^{\,\,2}}
\ln\left(\frac{\vec{q}_{2}^{\;\prime\,2}}
{\vec{k}^{\,2}}\right)
-2\frac{\vec q_1 \vec{q}_2}{\vec{q}_1^{\,\,2}\vec{q}_2^{\,\,2}}\ln
\left(\frac{\vec{q}^{\;2}}{\vec{k}^{\,2}}\right)~. \label{R u momentum}
\end{equation}
Note that $R_u$ has the same gauge invariance properties as $R$:
\[
R_u(\vec{q}_{1},\vec{q}_{2}; \vec{q}_1)=R_u(\vec{q}_{1},\vec{q}_{2};
-\vec{q}_2)=0, \;\;
\]
\begin{equation}
(\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}R_u(\vec{q}_{1},\vec{q}_{2};
\vec{k}))|_{\vec{q}_{1}=0}=(\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}
R_u(\vec{q}_{1},\vec{q}_{2}; \vec{k}))|_{\vec{q}_{2}=0}=0~.
\label{g i for R u}
\end{equation}
Indeed, these properties are required to conserve the gauge invariance in the
transformation~(\ref{kernel transformation}).
Another important property of $R_u$ is the absence of either infrared,
or ultraviolet non-integrable singularities, thus leading to
convergence of the integral
\begin{equation}
\int\frac{d\vec{k}_1d\vec{k}_2}{\pi} \delta(\vec{k}-\vec{k}_1-\vec{k}_2)
R_u(\vec{q}_1-\vec{k}_1, \vec{q}_2+\vec{k}_1; \vec{k}_2)=
-\ln\left(\frac{\vec{q}_{1}^{\;\prime\,2}}{\vec{q}^{\;2}}\right)
\ln\left( \frac{\vec{q}_{2}^{\;\prime\,2}}{\vec{q}^{\;2}}\right)~.
\label{int R u}
\end{equation}
The calculation of this integral and of the integrals appearing below is
described in the Appendix. The result~(\ref{int R u}) follows
from~(\ref{int phi ln}) and~(\ref{int phi a-l b-l}) with a subsequent
elementary integration over $l$.
Having Eqs.~(\ref{K B momentum}) and~(\ref{U momentum}), it is quite
straightforward to write the commutator $\left[\hat{\mathcal{K}}^{(B)},\hat
{U}\right]$ in the form
\[
\langle\vec{q}_{1},\vec{q}_{2}|\alpha_s\left[\hat{\mathcal{K}}^{(B)},\hat
{U}\right]|\vec{q}_{1}^{\,\,\prime}%
,\vec{q}_{2}^{\,\,\prime}\rangle=\delta(\vec{q}_{11^{\prime}}+\vec
{q}_{22^{\prime}})\frac{\alpha^2_s N^2_c}{8\pi^{3}}\left[ \frac{\beta_0}{2N_c}
\ln\left(\frac{\vec{q}_1^{\;2}\vec{q}_2^{\;2}}{\vec{q}_{1}^{\;\prime\,2}
\vec{q}_{2}^{\;2}}
\right) R(\vec{q}_1, \vec{q}_2; \vec{k})\right.
\]
\begin{equation}
\left.+ \int\frac{d\vec{l}}{\pi}\left(V(\vec{q}^{\;\prime}_1,
\vec{q}^{\;\prime}_2; \vec{l})-V(\vec{q}_1, \vec{q}_2; \vec{l})\right)\;
R_u(\vec{q}_1, \vec{q}_2; \vec{k})+
F(\vec{q}_1, \vec{q}_2; \vec{k})\right]~,\label{[] as sum}
\end{equation}
where
\[
F(\vec{q}_1, \vec{q}_2; \vec{k})=\int\frac{d\vec{k}_1d\vec{k}_2}{\pi}
\delta(\vec{k}-\vec{k}_1-\vec{k}_2){\cal F}(\vec{q}_1, \vec{q}_2; \vec{k}_1,
\vec{k}_2)~,\;\;{\cal F}(\vec{q}_1, \vec{q}_2; \vec{k}_1, \vec{k}_2)=
\]
\begin{equation}
=R(\vec{q}_1, \vec{q}_2; \vec{k}_1)R_u(\vec{q}_1-\vec{k}_1, \vec{q}_2
+\vec{k}_1; \vec{k}_{2})-R_u(\vec{q}_1, \vec{q}_2; \vec{k}_1)R(\vec{q}_1
-\vec{k}_1, \vec{q}_2+\vec{k}_1; \vec{k}_{2})~. \label{F definition}
\end{equation}
The infrared divergent pieces in the virtual parts entering the integral
over $d\vec l$ in Eq.~(\ref{[] as sum}) cancel, and one can easily obtain
(see~(\ref{int v}))
\begin{equation}
\int\frac{d\vec{l}}{\pi}\left(V(\vec{q}^{\;\prime}_1, \vec{q}^{\;\prime}_2;
\vec{l})-V(\vec{q}_1, \vec{q}_2; \vec{l})\right) =
\ln\left(\frac{\vec{q}_{1}^{\;\prime\,2}\vec{q}_{2}^{\;\prime\,2}}
{\vec{q}_1^{\;2}\vec{q}_2^{\;2}}\right)~. \label{virtual contribution}
\end{equation}
Unfortunately, the calculation of $F(\vec{q}_1, \vec{q}_2; \vec{k})$ is not
so easy, both because of the presence of a great number of terms in
${\cal F}(\vec{q}_1, \vec{q}_2; \vec{k}_1, \vec{k}_2)$ and of the complexity
of the integration. One of the reasons of this complexity is the singularity
of $R(\vec{q}_1, \vec{q}_2; \vec{k})$ at $\vec{k}^{\,2}=0$. Of course, this
singularity disappears in $F(\vec{q}_1, \vec{q}_2;\vec{k})$,
Eq.~(\ref{F definition}). To make this evident, let us write
\begin{equation}
R(\vec{q}_1, \vec{q}_2; \vec{k})=\frac{2}{\vec{k}^{\,2}}
+R_f(\vec{q}_1, \vec{q}_2; \vec{k}), \;\; R_f(\vec{q}_{1},\vec{q}_{2};
\vec{k})= -2\frac{\vec{q}_{1}\vec{k}}{\vec{k}^{\,2}\vec{q}%
_{1}^{\,\,2}}+2\frac{\vec{q}_{2}\vec{k}}{\vec{k}^{\,2}\vec{q}_{2}^{\,\,2}%
} -2\frac{\vec{q}_{1}\vec{q}_{2}}{\vec{q}_{1}^{\,\,2}\vec{q}_{2}%
^{\,\,2}}\;,\label{R s momentum}
\end{equation}
and divide $F(\vec{q}_1, \vec{q}_2; \vec{k})$, Eq.~(\ref{F definition}),
into three pieces:
\begin{equation}
F(\vec{q}_1, \vec{q}_2; \vec{k}) =\sum_{i=1}^3F_i(\vec{q}_1, \vec{q}_2;
\vec{k}),
\label{F division}
\end{equation}
where
\begin{equation}
F_1(\vec{q}_1, \vec{q}_2; \vec{k})=\int\frac{d\vec{k}_1}{\pi}R_f(\vec{q}_1,
\vec{q}_2; \vec{k}_1)R_u(\vec{q}_1-\vec{k}_1, \vec{q}_2+\vec{k}_1;
\vec{k}-\vec{k}_{1})\;,
\label{F1}
\end{equation}
\begin{equation}
F_2(\vec{q}_1, \vec{q}_2; \vec{k})=-\int\frac{d\vec{k}_1}{\pi}R_u(\vec{q}_1,
\vec{q}_2; \vec{k}_1)R_f(\vec{q}_1-\vec{k}_1, \vec{q}_2+\vec{k}_1;
\vec{k}-\vec{k}_{1})\;,
\label{F2}
\end{equation}
\begin{equation}
F_3(\vec{q}_1, \vec{q}_2; \vec{k})=\int\frac{d\vec{k}_1}{\pi}\frac{2}
{\vec{k}_1^{\,2}}\left(R_u(\vec{q}_1-\vec{k}_1, \vec{q}_2+\vec{k}_1;
\vec{k}-\vec{k}_{1})-R_u(\vec{q}_1, \vec{q}_2; \vec{k}-\vec{k}_1)\right)\;.
\label{F3}
\end{equation}
Now all the three pieces have no infrared singularities, the first two of them
because of the absence of singularities in the integrands, and the last one
because of the evident cancellation between the two terms with $R_u$ in
Eq.~(\ref{F3}) at $\vec{k}_1=0$. The integration of the first piece can
be performed with the help of
Eqs.~(\ref{int R u}),~(\ref{int vl2}),~(\ref{int vl3}) and~(\ref{int vl30})
and gives
\[
F_1(\vec{q}_{1}, \vec{q}_{2}; \vec{k})=
\left(\frac{\vec{q}_{1}\vec{q}_{2}}{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}}
-\frac{1}{\vec{q}_{1}^{\;2}}\right)\ln\left(\frac{\vec{q}^{\;2}\vec{q}_1^{\;2}}
{\vec{q}_{2}^{\;\prime\,2}\vec{k}^{\,2}}\right)
\ln\left(\frac{\vec{q}_{1}^{\;\prime\,2}\vec{q}_{2}^{\;2}}{\vec{q}^{\;2}
\vec{k}^{\,2}}\right)
+\frac{\vec{q}_{1}\vec{q}_{2}}{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}}
\ln\left(\frac{\vec{q}_1^{\;\prime\,2}}
{\vec{q}^{\;2}}\right)
\ln\left(\frac{\vec{q}_{2}^{\;\prime\,2}}
{\vec{q}^{\;2}}\right)
\]
\[
+4\left(\frac{[\vec{q}_{2}\times\vec{k}]}{\vec{q}_{2}^{\;2}\vec{k}^{\,2}}-
\frac{[\vec{q}_{1}\times\vec{k}]}{\vec{q}_{1}^{\;2}\vec{k}^{\,2}}\right)
[\vec{q}_{1}\times\vec{k}]
I_{\vec{k}, \vec{q}_{1}^{\;\prime}} +2\frac{[\vec{q}_{1}\times\vec{q}_{2}]^2}
{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}}I_{\vec{q}_{1}, \vec{q}_{2}}
\]
\begin{equation}
+2\frac{[\vec{q}_{1}\times\vec{q}_{2}][\vec{q}_{1}^{\;\prime}
\times\vec{q}_{2}^{\;\prime}]}
{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}}I_{\vec{q}_{1}^{\;\prime},
\vec{q}_{2}^{\;\prime}}+ \vec{q}_{1}\leftrightarrow -\vec{q}_{2}\;.
\label{F1 final}
\end{equation}
Here
\begin{equation}
I_{\vec p, \vec q}=
\int_{0}^{1}\frac{dx}{(\vec p +x\vec q)^{2}}\ln\left(\frac{\vec p^{\;2}}
{x^2\vec q^{\;2}}\right)~ \label{I p q 1}
\end{equation}
is the di-logarithmic function with high symmetry,
\begin{equation}
I_{\vec p,\vec q}=I_{-\vec p,-\vec q}=I_{\vec q, \vec p}=I_{\vec p,-\vec p
-\vec q}~.
\label{I definition}
\end{equation}
The representation exhibiting these properties~\cite{Fadin:2002hz} is
\begin{equation}
I_{\vec p,\vec q}=\int_{0}^{1}\int_{0}^{1}\int_{0}^{1}\frac{dx_{1}dx_{2}dx_{3}
\delta(1-x_{1}-x_{2}-x_{3})}{(\vec p^{\;2}x_{1}+\vec q^{\;2}x_{2}+(\vec p
+\vec q)^{2}x_{3})(x_{1}x_{2}+x_{1}x_{3}+x_{2}x_{3})}~.\label{I symmetric}
\end{equation}
Other useful representations are
\[
I_{\vec p,\vec q}=\int_{0}^{1}
\frac{dx}{a(1-x)+bx-c x(1-x)}\ln\left( \frac{a(1-x)+bx}{cx(1-x)}\right)
\]
\begin{equation}
=\int_{0}^{1}dx\int_{0}^{1}{dz}\;\frac{1}{cx(1-x)z+(b(1-x)+ax)(1-z)}~,
\label{integral I}
\end{equation}
where $a=\vec p^{\;2}, b=\vec q^{\;2}, c = (\vec p+\vec q)^{2} $.
Note that $F_1$ must turn into zero at $\vec{q}_{1}^{\;\prime}=0$ or
$\vec{q}_{2}^{\;\prime}=0$ due to the gauge invariance of $R_u$. It is easy
to see from Eq.~(\ref{F1 final}) that this property is fulfilled.
Unfortunately, neither $F_2$ nor $F_3$ possess such property. Moreover, the
separation~(\ref{F division}) destroys the good behaviour of $R(\vec{q}_1,
\vec{q}_2; \vec{k})$ in the ultraviolet region, so that the
integrals~(\ref{F2}) and~(\ref{F3}) diverge at large $\vec{k}_1^{\,2}$
and we have to introduce an ultraviolet cut-off $\Lambda^2$ for them.
The loss of gauge invariance and ultraviolet convergence of the integrals
makes them more complex. Using~(\ref{int vl3})--(\ref{int v3l0}) we obtain
\[
F_2(\vec{q}_{1}, \vec{q}_{2}; \vec{k})=
\frac{1}{\vec{q}_{1}^{\;2}}\left(\ln\left(\frac{\vec{q}_2^{\;2}}
{\vec{q}^{\;2}}\right)
\ln\left(\frac{\Lambda^4\vec{q}^{\;4}\vec{q}_{1}^{\;2}}
{\vec{q}_{1}^{\;\prime\,6}
\vec{q}_{2}^{\;\prime\,2}\vec{q}_2^{\;2}}\right)
+\ln\left(\frac{\vec{q}_1^{\;\prime\,2}}
{\vec{k}^{\,2}}\right)\ln\left(\frac{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}}
{\vec{k}^{\,2}
\vec{q}_{2}^{\;\prime\,2}}\right) \right)+\frac{\vec{q}_{1}\vec{q}_{2}}
{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}}
\]
\[
\times\left(\ln^2\left(\frac{\Lambda^2}
{\vec{q}^{\;2}}\right)-\ln\left(\frac{\vec{q}_1^{\;2}}
{\vec{q}^{\;2}}\right)\ln\left(\frac{\vec{q}_{2}^{\;2}}{\vec{q}^{\;2}}\right)
-\ln\left(\frac{\vec{q}_1^{\;\prime\,2}}
{\vec{q}^{\;2}}\right)\ln\left(\frac{\vec{q}_{1}^{\;\prime\,2}
\vec{q}_{2}^{\;\prime\,2}}
{\vec{q}^{\;4}}\right)
-\ln\left(\frac{\vec{k}^{\,2}}{\vec{q}_1^{\;\prime\,2}}\right)
\ln\left(\frac{\vec{k}^{\,2}\vec{q}_1^{\;\prime\,2}}{\vec{q}_1^{\;4}}
\right)\right)
\]
\[
+2\frac{\vec{q}_{1}\vec{k}}{\vec{q}_{1}^{\;2}\vec{k}^{\,2}}
\ln\left(\frac{\vec{q}^{\;2}}
{\vec{q}_2^{\;2}}\right)\ln\left(\frac{\vec{q}_{1}^{\;2}
\vec{q}_{2}^{\;\prime\,2}}
{\vec{q}_{1}^{\;\prime\,2}\vec{q}_{2}^{\;2}}\right)+4\left(\frac{[\vec{q}_{1}
\times\vec{k}]}
{\vec{q}_{1}^{\;2}\vec{k}^{\,2}}-
\frac{[\vec{q}_{2}\times\vec{k}]}{\vec{q}_{2}^{\;2}\vec{k}^{\,2}}-
\frac{[\vec{q}_{1}\times\vec{q}_2]}{\vec{q}_{2}^{\;2}\vec{q}_2^{\;2}}\right)
[\vec{q}_{1}\times\vec{k}]
I_{\vec{k}, \vec{q}_{1}^{\;\prime}}
\]
\begin{equation}
-2\left(\frac{[\vec{q}_{1}\times\vec{k}]}{\vec{q}_{1}^{\;2}\vec{k}^{\,2}}+
\frac{[\vec{q}_{2}\times\vec{k}]}{\vec{q}_{2}^{\;2}\vec{k}^{\,2}}+
\frac{[\vec{q}_{1}\times\vec{q}_2]}{\vec{q}_{2}^{\;2}\vec{q}_2^{\;2}}\right)
\Biggl([\vec{q}_{1}\times\vec{q}_{2}]I_{\vec{q}_{1}, \vec{q}_{2}}
-[\vec{q}_{1}^{\;\prime}\times\vec{q}_{2}^{\;\prime}]
I_{\vec{q}_{1}^{\;\prime}, \vec{q}_{2}^{\;\prime}}\Biggr)+ \vec{q}_{1}
\leftrightarrow -\vec{q}_{2}~. \label{F2 final}
\end{equation}
The result for $F_3(\vec{q}_1,\vec{q}_2; \vec{k})$ can be obtained
using Eqs.~(\ref{int vl2}),~(\ref{int s3l0})--(\ref{int s3ld}) and reads
\[
F_3(\vec{q}_{1}, \vec{q}_{2}; \vec{k})=
\frac{1}{\vec{q}_{1}^{\;2}}\left(\ln\left(\frac{\vec{q}^{\;2}}
{\vec{q}_2^{\;2}}\right)
\ln\left(\frac{\Lambda^4\vec{q}^{\;2}}{\vec{q}_{1}^{\;4}
\vec{q}_{2}^{\;2}}\right)-2\ln\left(\frac{\vec{q}_1^{\;\prime\,2}}
{\vec{k}^{\,2}}\right)\ln\left(\frac{\vec{q}_{1}^{\;\prime\,2}}
{\vec{q}_1^{\;2}}\right) \right)-\frac{\vec{q}_{1}\vec{q}_{2}}
{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}}
\]
\[
\times\left(\ln^2\left(\frac{\Lambda^2}
{\vec{q}^{\;2}}\right)-2\ln^2\left(\frac{\vec{q}_1^{\;2}}
{\vec{q}^{\;2}}\right)
-\ln\left(\frac{\vec{q}_1^{\;\prime\,2}}
{\vec{q}^{\;2}}\right)\ln\left(\frac{\vec{q}_{2}^{\;\prime\,2}}
{\vec{q}^{\;2}}\right)
+2\ln\left(\frac{\vec{k}^{\,2}}{\vec{q}_1^{\;2}}\right)
\ln\left(\frac{\vec{q}_1^{\;\prime\,2}}{\vec{q}_1^{\;2}}
\right)\right)
\]
\[
-4\frac{\vec{q}_{1}\vec{k}}{\vec{q}_{1}^{\;2}\vec{k}^{\,2}}
\ln\left(\frac{\vec{k}^{\,2}}
{\vec{q}_1^{\;\prime\,2}}\right)
\ln\left(\frac{\vec{q}_1^{\;\prime\,2}}{\vec{q}_1^{\;2}}
\right)-\frac{2}{\vec k^{\,2}}\ln^2\left(\frac{\vec{q}_1^{\;\prime\,2}}
{\vec{q}_1^{\;2}}
\right)
\]
\begin{equation}
+2\frac{[\vec{q}_{1}\times\vec{q}_2]}{\vec{q}_{1}^{\;2}\vec{q}_2^{\;2}}
\Biggl(2[\vec{q}_{1}\times\vec{k}]I_{\vec{k}_{1}, \vec{q}^{\;\prime}_{1}}
-[\vec{q}_{1}^{\;\prime}\times\vec{q}_{2}^{\;\prime}]
I_{\vec{q}_{1}^{\;\prime}, \vec{q}_{2}^{\;\prime}}\Biggr)
+ \vec{q}_{1}\leftrightarrow -\vec{q}_{2}\;. \label{F3 final}
\end{equation}
From the Eq.~(\ref{F division}) and the definitions~(\ref{F1})--(\ref{F3})
it follows
\[
F(\vec{q}_{1}, \vec{q}_{2}; \vec{k})=
\frac{2}{\vec{q}_{1}^{\;2}}\ln\left(\frac{\vec{q}_1^{\;2}}
{\vec{q}_1^{\;\prime\,2}}\right)
\ln\left(\frac{\vec{q}_{1}^{\;\prime\,2}\vec{q}_{2}^{\;2}}{
\vec{q}^{\;2}\vec{k}^{\,2}}\right)+\frac{2\vec{q}_{1}\vec{q}_{2}}
{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}}\ln\left(\frac{\vec{q}_1^{\;2}}
{\vec{q}_1^{\;\prime\,2}}\right)
\ln\left(\frac{\vec{k}^{\,2}}{
\vec{q}^{\;2}}\right)
\]
\[
+2\frac{\vec{q}_{1}\vec{k}}{\vec{q}_{1}^{\;2}\vec{k}^{\,2}}
\left(\ln\left(\frac{\vec{q}^{\;2}}
{\vec{q}_2^{\;2}}\right)\ln\left(\frac{\vec{q}_{1}^{\;2}
\vec{q}_{2}^{\;\prime\,2}}
{\vec{q}_{1}^{\;\prime\,2}\vec{q}_{2}^{\;2}}\right)
+2\ln\left(\frac{\vec{q}_{1}^{\;\prime\,2}}
{\vec{q}_1^{\;2}}\right)\ln\left(\frac{\vec{q}_{1}^{\;\prime\,2}}
{\vec{k}^{\,2}}\right)\right) -\frac{2}{\vec k^{\,2}}
\ln^2\left(\frac{\vec{q}_1^{\;\prime\,2}}{\vec{q}_1^{\;2}}
\right)
\]
\begin{equation}
-2\left(\frac{[\vec{q}_{1}\times\vec{q}_{2}]}{\vec{q}_{1}^{\;2}
\vec{q}_{2}^{\;2}}
+2\frac{[\vec{q}_{1}\times\vec{k}]}{\vec{q}_{1}^{\;2}\vec{k}^{\,2}}
\right)\left([\vec{q}_{1}\times\vec{q}_{2}]I_{\vec{q}_{1}, \vec{q}_{2}}
-[\vec{q}_{1}^{\;\prime}\times\vec{q}_{2}^{\;\prime}]I_{\vec{q}_{1}^{\;\prime},
\vec{q}_{2}^{\;\prime}}\right) +\vec q_1\leftrightarrow -\vec q_2~.
\label{F final}
\end{equation}
The definition~(\ref{F definition}) and the properties~(\ref{g i for R})
and~(\ref{g i for R u}) of $R$ and $R_u$, respectively, secure the gauge
invariance of $F$:
\[
F(\vec{q}_{1}, \vec{q}_{2}; \vec{q}_1)=F(\vec{q}_{1}, \vec{q}_{2}; -\vec{q}_2)
=0,
\]
\begin{equation}
( \vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2} F(\vec{q}_{1},\vec{q}_{2};
\vec{q}_1))|_{\vec{q}_{1}=0}=(\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}
F(\vec{q}_{1},\vec{q}_{2}; \vec{q}_1))|_{\vec{q}_{2}=0}=0~.
\label{g i for F}
\end{equation}
The fulfilment of these properties can be easily seen from Eq.~(\ref{F final}).
Finally, Eq.~(\ref{[] as sum}) together with
Eqs.~(\ref{R u momentum}),~(\ref{virtual contribution})
and~(\ref{F final}) gives
\[
\langle\vec{q}_{1},\vec{q}_{2}|\alpha_{s}[{\hat{\mathcal{K}}}^{(B)},
{\hat{U}}]|\vec{q}_{1}^{\,\,\prime
},\vec{q}_{2}^{\,\,\prime}\rangle=\delta(\vec{q}_{11^{\prime}}+\vec
{q}_{22^{\prime}})\frac{\alpha^2_{s}N^2_{c}}{8\pi^{3}}\left[ -\frac{\beta_0}
{2N_c}
R(\vec{q}_{1}, \vec{q}_{2}; \vec{k})\ln\left( \frac{\vec{q}_{1}^{\;\prime\,2}
\vec{q}_{2}^{\;\prime\,2}}%
{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;2}}\right) +
\right.
\]
\[
\left. + \frac{\vec{q}_{1}^{\;\prime\,2}}{\vec{q}_{1}^{\;2}\vec{k}^{\,2}}
\ln\left(\frac{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;\prime\,2}}{\vec{q}_{2}^{\;2}
\vec{q}_{1}^{\;\prime\,2}}\right)
\ln\left(\frac{\vec{q}_{2}^{\;2}\vec{q}_{1}^{\;\prime\,2}}{\vec{q}^{\;2}
\vec{k}^{\,2}}
\right) +\frac{\vec{q}_{2}^{\;\prime\,2}}{\vec{q}_{2}^{\;2}\vec{k}^{\,2}}
\ln\left(\frac{\vec{q}_{2}^{\;2}\vec{q}_{1}^{\;\prime\,2}}{\vec{q}_{1}^{\;2}
\vec{q}_{2}^{\;\prime\,2}}\right)
\ln\left(\frac{\vec{q}_{1}^{\;2}\vec{q}_{2}^{\;\prime\,2}}{\vec{q}^{\;2}
\vec{k}^{\,2}}
\right)\right.
\]
\begin{equation}
\left. -4\left(\frac{[\vec{q}_{1}\times\vec{q}_{2}]}{\vec{q}_{1}^{\;2}
\vec{q}_{2}^{\;2}}
+\frac{[\vec{q}_{1}\times\vec{k}]}{\vec{q}_{1}^{\;2}\vec{k}^{\,2}}
+\frac{[\vec{q}_{2}\times\vec{k}]}{\vec{q}_{2}^{\;2}\vec{k}^{\,2}}
\right)\left([\vec{q}_{1}\times\vec{q}_{2}]I_{\vec{q}_{1}, \vec{q}_{2}}
-[\vec{q}_{1}^{\;\prime}\times\vec{q}_{2}^{\;\prime}]I_{\vec{q}_{1}^{\;\prime},
\vec{q}_{2}^{\;\prime}}\right) \right]. \label{final commutator}
\end{equation}
\section{Use of M\"{o}bius space}
\label{sec:Mobius}
Since the result~(\ref{final commutator}) was derived by means of
lengthy and intricate calculations, we want to obtain it in a quite
independent way, starting from the M\"{o}bius forms of the kernel
${\hat{\cal K}}^{(B)}$ and of the operator $\hat{U}$, calculating their
commutator and restoring the complete commutator~(\ref{final commutator}) in
the momentum space from its M\"{o}bius form. Simultaneously, the efficiency
of the method of restoration developed in Ref.~\cite{Fadin:2011jg}
will be demonstrated. Here this alternative derivation is illustrated.
As it is known~\cite{Fadin:2006ha}, the M\"{o}bius form of the kernel
${\hat{\cal K}}^{(B)}$ coincides with the kernel of the colour dipole
model~\cite{dipole} and can be written as
\[
\langle\vec{r}_{1}\vec{r}_{2}|{\hat{\cal K}}^{(B)}_{M}|\vec{r}_{1}^{\;\prime}
\vec{r}%
_{2}^{\;\prime}\rangle=\frac{\alpha_{s}N_{c}}{2\pi^{2}}\int d\vec{r
}_0g(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{0})
\]
\begin{equation}
\times \Biggl[\delta(\vec{r}_{11^{\prime}})\delta(\vec
{r}_{2^{\prime}0})+\delta(\vec{r}_{1^{\prime}0})\delta(\vec
{r}_{22^{\prime}})-\delta(\vec{r}_{11^{\prime}})\delta({r}_{22^{\prime}%
})\Biggr]~, \label{K B M}%
\end{equation}
where
\begin{equation}
g(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{0})=g(\vec{r}_{2}, \vec{r}_{1},
\vec{r}_{0})=
\frac{\vec{r}_{12}^{\;2}}{\vec{r}_{10}^{\,\,2}\vec{r}_{20}^{\,\,2}}\;.
\label{g function}
\end{equation}
The M\"{o}bius form of the operator $U$ was found in Ref.~\cite{Fadin:2011jg}.
Omitting the term with $\hat{\cal K}^{(B)}$, which does not contribute to the
commutator in~(\ref{QC kernel}), one has
\[
\langle\vec{r}_{1}\vec{r}_{2}|\alpha_s\hat{U}_M|\vec{r}_{1}^{\;\prime}%
\vec{r}_{2}^{\;\prime}\rangle=\frac{\alpha_{s}N_{c}}{4\pi^{2}}
\Biggl[\delta(\vec{r}
_{11^{\prime}})V_1(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{2}^{\;\prime})
+\delta(\vec{r}_{22^{\prime}})V_1(\vec{r}_{2}, \vec{r}_{1},
\vec{r}_{1}^{\;\prime})
\]
\begin{equation}
+\frac{1}{\pi}V_3(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{1}^{\;\prime},
\vec{r}_{2}^{\;\prime})\Biggr]+\frac{\alpha_{s}\beta_{0}}{8\pi^{2}}
\Biggl[\delta(\vec{r}_{11^{\prime}})\left(\frac{1}{\vec{r}_{22^{\prime}}
^{\,\,2}}-\frac{1}{\vec{r}_{12^{\prime}}
^{\,\,2}} \right) +\delta(\vec{r}_{22^{\prime}})
\left(\frac{1}{\vec{r}_{11^{\prime}}
^{\,\,2}}-\frac{1}{\vec{r}_{21^{\prime}}
^{\,\,2}} \right)\Biggr]~,
\label{U coordinate}
\end{equation}
where
\begin{equation}
V_1(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{2}^{\;\prime})
=\frac{\vec{r}_{12}^{\,\,2}}{\vec{r}_{12'}^{\,\,2}\vec{r}
_{22'}^{\,\,2}}\ln\left(\frac{\vec{r}_{12}^{\,\,2}}{\vec{r}_{22'}^{\,\,2}}
\right) +\frac{1}{\vec{r}_{22'}^{\,\,2}}\ln\left( \frac{\vec
{r}_{22'}^{\,\,2}}{\vec{r}_{12'}^{\,\,2}}\right), \label{V1}
\end{equation}
\begin{equation}
V_3(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{1}^{\;\prime}, \vec{r}_{2}^{\;\prime})
=V_3(\vec{r}_{2}, \vec{r}_{1},\vec{r}_{2}^{\;\prime}, \vec{r}_{1}^{\;\prime})
= \frac{1}{\vec {r}_{1^{\prime}2^{\prime}}^{\,\, 2}}
\left[ \frac{2\vec{r}_{1 1^{\prime}}\vec{r}_{2 2^{\prime}}}
{\vec{r}_{1 1^{\prime}}^{\,\, 2}\vec{r}_{2 2^{\prime}}^{\,\, 2}}
- \frac{\vec{r}_{1 1^{\prime}}\vec{r}_{1 2^{\prime}}}
{\vec{r}_{1 1^{\prime}}^{\,\, 2}\vec{r}_{1 2^{\prime}}^{\,\, 2}}
- \frac{\vec{r}_{2 1^{\prime}}\vec{r}_{2 2^{\prime}}}
{\vec{r}_{2 1^{\prime}}^{\,\, 2}\vec{r}_{2 2^{\prime}}^{\,\, 2}}\right]~.
\label{V3}
\end{equation}
The treatment of the term with $\beta_0$ in $\hat{U}$ can be performed
quite easily in the momentum space (see Eq.~(\ref{[] as sum})), so that in the
following we will omit this term, denoting the remaining part of $\hat{U}$
as $\hat{U}^s$. With the notation~(\ref{K B M})--(\ref{V3}) the M\"{o}bius
form for the commutator $[\hat{\mathcal{K}}^{(B)},\hat{U}^s]$ can be presented
as
\[
\langle\vec{r}_{1}\vec{r}_{2}|\alpha_s[\hat{\mathcal{K}}^{(B)},\hat
{U}^s]_M|\vec{r}_{1}^{\;\prime}\vec{r}_{2}^{\;\prime}\rangle
=\frac{\alpha_s^2N_c^2}{8\pi^3}\left[\delta(\vec{r}_{11^{\prime}})
J(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{2}^{\;\prime})+\frac{1}{\pi}
F(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{1}^{\;\prime}, \vec{r}_{2}^{\;\prime})
\right.
\]
\begin{equation}
\left.
\frac{1}{\pi}I(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{1}^{\;\prime},
\vec{r}_{2}^{\;\prime})
\right] +1\leftrightarrow 2~, \label{mebius commutator}
\end{equation}
where $1\leftrightarrow 2$ means the substitution $\vec{r}_{1}\leftrightarrow
\vec{r}_{2}, \; \vec{r}_{1}^{\;\prime}\leftrightarrow \vec{r}_{2}^{\;\prime}$.
The first two terms in the square brackets in Eq.~(\ref{mebius commutator})
come from the term with $V_1(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{2}^{\;\prime})$
in Eq.~(\ref{U coordinate}) and are written as
\[
J(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{2}^{\;\prime}) =\int \frac{d\vec{r}_0}{\pi}
\left[g(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{0})V_1(\vec{r}_{1}, \vec{r}_{0},
\vec{r}_{2}^{\;\prime}) -V_1(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{0})
g(\vec{r}_{1}, \vec{r}_{0},\vec{r}_{2}^{\;\prime})\right.
\]
\begin{equation}
\left.-(g(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{0})-g(\vec{r}_{1},
\vec{r}_{2}^{\;\prime}, \vec{r}_{0}))V_1(\vec{r}_{1}, \vec{r}_{2},
\vec{r}_{2}^{\;\prime})\right] \label{J definition}
\end{equation}
and
\begin{equation}
F(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{1}^{\;\prime}, \vec{r}_{2}^{\;\prime})=
g(\vec{r}_{2}, \vec{r}_{1},\vec{r}_{1}^{\;\prime})V_1(\vec{r}_{1}^{\;\prime},
\vec{r}_{2},\vec{r}_{2}^{\;\prime})-V_1(\vec{r}_{1}, \vec{r}_{2},
\vec{r}_{2}^{\;\prime})
g(\vec{r}_{2}^{\;\prime}, \vec{r}_{1},\vec{r}_{1}^{\;\prime})\;.
\label{F coordinate}
\end{equation}
The last term in the square brackets in Eq.~(\ref{mebius commutator}) related
with $V_3$ is presented in the form
\[
I(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{1}^{\;\prime}, \vec{r}_{2}^{\;\prime})
=\int \frac{d\vec{r}_0}{\pi}
\left[\left(\frac{1}{\vec{r}^{\;2}_{10}}-\frac{\vec{r}_{10}\vec{r}_{20}}{\vec{r}^{\;2}_{10}\vec{r}^{\;2}_{20}}\right)
V_3(\vec{r}_{1}, \vec{r}_{0},\vec{r}_{1}^{\;\prime}, \vec{r}_{2}^{\;\prime})
+\left(\frac{1}{\vec{r}^{\;2}_{20}}-\frac{\vec{r}_{10}\vec{r}_{20}}{\vec{r}^{\;2}_{10}\vec{r}^{\;2}_{20}}\right)
\right.
\]
\[
\times \Biggl(
V_3(\vec{r}_{1}, \vec{r}_{0},\vec{r}_{1}^{\;\prime}, \vec{r}_{2}^{\;\prime})-
V_3(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{1}^{\;\prime}, \vec{r}_{2}^{\;\prime})
\Biggr)
-\frac{1}{\vec{r}^{\;2}_{02'}}\Biggl(
V_3(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{1}^{\;\prime}, \vec{r}_{0})
\frac{\vec{r}^{\;2}_{01'}}{\vec{r}^{\;2}_{1'2'}}
\]
\begin{equation}
\left.
-V_3(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{1}^{\;\prime}, \vec{r}_{2}^{\;\prime})
\Biggr)
-\frac{\vec{r}_{1'0}\vec{r}_{2'0}}{\vec{r}^{\;2}_{1'0}\vec{r}^{\;2}_{2'0}}V_3(\vec{r}_{1}, \vec{r}_{2},
\vec{r}_{1}^{\;\prime}, \vec{r}_{2}^{\;\prime})\right] ~.
\label{I definition bis}
\end{equation}
Note that each of the $J, F, I$ functions independently turns into zero
at $\vec{r}_{12}=0$.
In contrast to the function $F$, which is given explicitly
by Eq.~(\ref{F coordinate}), the functions $J$ and $I$ are expressed in terms
of the integrals~(\ref{J definition}) and~(\ref{I definition bis}),
respectively.
The integrals are not very intricate, although their calculation is
complicated by the ultraviolet divergences existing in separate terms.
The integrands in~(\ref{J definition}) and~(\ref{I definition bis}) are written in
such a way so as to make the cancellation evident. The results of the
integration (which can be performed by the method described in the Appendix)
are very simple:
\begin{equation}
J(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{2}^{\;\prime}) =\left(\frac{2(\vec{r}_{12'}
\vec{r}_{22'} )}{\vec{r}_{12'}^{\;2}\vec{r}_{22'}^{\;2}}
-\frac{1}{\vec{r}_{12'}^{\;2}}\right)
\ln\left(\frac{\vec{r}_{12'}^{\;2}}{\vec{r}_{12}^{\;2}}\right)
\ln\left(\frac{\vec{r}_{22'}
^{\;2}}{\vec{r}_{12}^{\;2}}\right)-\frac{1}{\vec{r}_{22'}^{\;2}}
\ln^2\left(\frac
{\vec{r}_{12'}^{\;2}}{\vec{r}_{12}^{\;2}}\right) \label{J coordinate}
\end{equation}
and
\[
I(\vec{r}_{1}, \vec{r}_{2},\vec{r}_{1}^{\;\prime},\vec{r}_{2}^{\;\prime})
=\frac{1}{\vec{r}_{1'2'}^{\;2}}\left(\frac{(\vec{r}_{11'}\vec{r}_{22'} )}
{\vec{r}_{11'}^{\;2}\vec{r}_{22'}^{\;2}}\ln\left(\frac{\vec{r}_{21'}^{\;2}
\vec{r}_{1'2'}^{\;2}}
{\vec{r}_{12}^{\;2}\vec{r}_{12'}^{\;2}}\right)+\frac{(\vec{r}_{11'}
\vec{r}_{12'} )}{\vec{r}_{11'}^{\;2}\vec{r}_{12'}^{\;2}}
\ln\left(\frac{\vec{r}_{12'}^{\;4}\vec{r}_{12}^{\;2}}
{\vec{r}_{11'}^{\;2}\vec{r}_{22'}^{\;4}}\right)\right.
\]
\begin{equation}
\left. +\frac{(\vec{r}_{22'}\vec{r}_{21'})}{\vec{r}_{22'}^{\;2}
\vec{r}_{21'}^{\;2}}
\ln\left(\frac{\vec{r}_{12}^{\;2}}{\vec{r}_{21'}^{\;2}}\right)
+\frac{(\vec{r}_{12'}
\vec{r}_{21'})}{\vec{r}_{12'}^{\;2}\vec{r}_{21'}^{\;2}}
\ln\left(\frac{\vec{r}_{11'}^{\;2}
\vec{r}_{22'}^{\;2}}{\vec{r}_{12}^{\;2}\vec{r}_{1'2'}^{\;2}}\right)\right)~.
\label{I coordinate}
\end{equation}
Note that the property of turning into zero at $\vec{r}_{12}=0$ is conserved
after integration. Thus, the M\"{o}bius form of the commutator given
by Eqs.~(\ref{mebius commutator}),~(\ref{F coordinate}),~(\ref{J coordinate})
and~(\ref{I coordinate}) is rather simple and does not contain special
functions.
Having this form one can find the complete commutator in the momentum space
$\langle\vec{q}_{1},\vec{q}_{2}|\alpha_s\left[\hat{\mathcal{K}}^{(B)},\hat
{U}\right]|\vec{q}_{1}^{\,\,\prime},\vec{q}_{2}^{\,\,\prime}\rangle$
according to the prescriptions of Ref.~\cite{Fadin:2011jg}. We write it in the
form
\[
\langle\vec{q}_{1},\vec{q}_{2}|\alpha_s\left[\hat{\mathcal{K}}^{(B)},\hat
{U}\right]|\vec{q}_{1}^{\,\,\prime}%
,\vec{q}_{2}^{\,\,\prime}\rangle=\delta(\vec{q}_{11^{\prime}}+\vec
{q}_{22^{\prime}}) \frac{\alpha^2_s N^2_c}{8\pi^{3}} \Biggl[\frac{\beta_0}
{4N_c} \ln\left(\frac{\vec{q}_1^{\;2}}{\vec{q}_{1}^{\;\prime\,2}}
\right) R(\vec{q}_1, \vec{q}_2; \vec{k})
\]
\begin{equation}
+ F(\vec{q}_{2}, \vec{q}_{2}; \vec{k})+J(\vec{q}_{2}, \vec{q}_{2}; \vec{k})
+I(\vec{q}_{2}, \vec{q}_{2}; \vec{k})+\vec{q}_1\leftrightarrow -\vec{q}_2
\biggr]~.
\label{restored commutator}
\end{equation}
Here $R(\vec{q}_1, \vec{q}_2; \vec{k})$ is given by
Eq.~(\ref{R momentum}) and
\[
F(\vec{q}_{1}, \vec{q}_{2}; \vec{k})=\frac{1}{\pi}<\int\frac{d\vec{r}_{11'}}
{2\pi}
\frac{d\vec{r}_{22'}}{2\pi}{d\vec{r}_{1'2'}}e^{-i(\vec{q}_1
\vec{r}_{11'}+\vec{q}_2\vec{r}_{22'}+\vec{k}\vec{r}_{1'2'})}F(\vec{r}_{1},
\vec{r}_{2},\vec{r}_{1}^{\;\prime}, \vec{r}_{2}^{\;\prime})>
\]
\[
=\left(\frac{1}{\vec{q}_{1}^{\;2}}\ln\left(\frac{\vec{q}_1^{\;2}}
{\vec{q}_1^{\;\prime\,2}}\right)
\ln\left(\frac{\vec{q}_2^{\;2}}{\vec{q}^{\;2}}\right)
+\frac{1}{2 \vec{k}^{\;2}}\ln\left(\frac{\vec{q}_1^{\;2}}
{\vec{q}_1^{\;\prime\,2}}\right)
\ln\left(\frac{\vec{q}_2^{\;2}}{\vec{q}_2^{\;\prime\,2}}\right)
-\frac{(\vec{q}_{1}\vec{k})}{\vec{q}_{1}^{\;2}
\vec{k}^{\;2}}\right.
\]
\[
\left.
\times
\left(\ln\left(\frac{\vec{q}_2^{\;\prime\,2}}
{\vec{q}_2^{\;2}}\right)\ln\left(\frac{\vec{q}_1^{\;\prime\,2}}
{\vec{k}^{\;2}}\right)+\ln\left(\frac{\vec{q}^{\;2}}
{\vec{q}_2^{\;2}}\right)\ln\left(\frac{\vec{q}_2^{\;2}\vec{q}_1^{\;\prime\,2}}
{\vec{q}_1^{\;2}\vec{q}_2^{\;\prime\,2}}\right)\right)
+\vec{q}_1\leftrightarrow -\vec{q}_2\right)+\frac{(\vec{q}_{1}\vec{q}_{2})}
{\vec{q}_{1}^{\;2}
\vec{q}_{2}^{\;2}}
\]
\[
\times\left(\ln\left(\frac{\vec{q}^{\;2}}{\vec{q}_1^{\;2}}\right)
\ln\left(\frac{\vec{q}_1^{\;\prime\,2}\vec{k}^{\;2}}
{\vec{q}_1^{\;2}\vec{q}_2^{\;2}}\right)
+\ln\left(\frac{\vec{q}_2^{\;\prime2}}{\vec{k}^{\;2}}\right)
\ln\left(\frac{\vec{q}^{\;2}}
{\vec{q}_1^{\;\prime2}}\right)\right)
-2
\frac{[\vec{q}_{1}\times\vec{q}_2]}{\vec{q}_{1}^{\;2}\vec{q}_2^{\;2}}
[\vec{q}_{1}\times\vec{k}]
I_{\vec{k}, \vec{q}_{1}^{\;\prime}}
\]
\begin{equation}
-2\left(\frac{[\vec{q}_{1}\times\vec{k}]}
{\vec{q}_{1}^{\;2}\vec{k}^{\;2}}+\frac{[\vec{q}_{2}\times\vec{k}]}
{\vec{q}_{2}^{\;2}\vec{k}^{\;2}}+
\frac{[\vec{q}_{1}\times\vec{q}_2]}{\vec{q}_{1}^{\;2}\vec{q}_2^{\;2}}\right)
\left([\vec{q}_{1}\times\vec{q}_2]I_{\vec{q}_{1}, \vec{q}_{2}}-
[\vec{q}_{1}^{\;\prime}\times\vec{q}_2^{\;\prime}]I_{\vec{q}_{1}^{\;\prime},
\vec{q}_{2}^{\;\prime}}\right)\;, \label{F momentum}
\end{equation}
\begin{equation}
J(\vec{q}_{2}, \vec{q}_{2}; \vec{k})=\frac{1}{\pi}<\int\frac{d\vec{r}_{11'}}
{2\pi}\frac{d\vec{r}_{22'}}{2\pi}{d\vec{r}_{1'2'}}e^{-i(\vec{q}_1
\vec{r}_{11'}+\vec{q}_2\vec{r}_{22'}+\vec{k}\vec{r}_{1'2'})}
\delta(\vec{r}_{11^{\prime}})J(\vec{r}_{1},\vec{r}_{2},\vec{r}_{2}^{\;\prime})>
\]
\[
\left(\frac{1}{\vec{q}_{2}^{\;2}}+2\frac{(\vec{q}_{2}\vec{k})}
{\vec{q}_{2}^{\;2}
\vec{k}^{\;2}}\right)\ln\left(\frac{\vec{q}_2^{\;2}}
{\vec{q}_2^{\;\prime2}}\right)
\ln\left(\frac{\vec{q}_2^{\;\prime\,2}}{\vec{k}^{\;2}}\right)
-\frac{1}{\vec{k}^{\;2}}
\ln^2\left(\frac{\vec{q}_2^{\;2}}
{\vec{q}_2^{\;\prime2}}\right)~, \label{J momentum}
\end{equation}
\[
I(\vec{q}_{1}, \vec{q}_{2}; \vec{k})=\frac{1}{\pi}<\int\frac{d\vec{r}_{11'}}
{2\pi}\frac{d\vec{r}_{22'}}{2\pi}{d\vec{r}_{1'2'}}e^{-i(\vec{q}_1
\vec{r}_{11'}+\vec{q}_2\vec{r}_{22'}+\vec{k}\vec{r}_{1'2'})}F(\vec{r}_{1},
\vec{r}_{2},\vec{r}_{1}^{\;\prime}, \vec{r}_{2}^{\;\prime})>
\]
\[
=
\frac{1}{2\vec{q}_{1}^{\;2}}\ln\left(\frac{\vec{q}_1^{\;\prime\,2}}
{\vec{q}^{\;2}}\right)\ln\left(\frac{\vec{q}_2^{\;\prime\,2}}
{\vec{q}^{\;2}}\right)+\frac{1}{2\vec{q}_{2}^{\;2}}
\left(\ln\left(\frac{\vec{q}_1^{\;\prime\,2}}
{\vec{q}^{\;2}}\right)\ln\left(\frac{\vec{q}_2^{\;\prime\,2}}
{\vec{q}^{\;2}}\right)\right.
\]
\[
\left.-2\ln\left(\frac{\vec{q}_1^{\;\prime\,2}}
{\vec{q}_1^{\;2}}\right)\ln\left(\frac{\vec{k}^{\;2}}
{\vec{q}_1^{\;2}}\right)\right)
-\frac{(\vec{q}_{1}\vec{q}_{2})}{\vec{q}_{1}^{\;2}
\vec{q}_{2}^{\;2}}\left(\ln\left(\frac{\vec{q}^{\;2}}{\vec{q}_1^{\;2}}\right)
\ln\left(\frac{\vec{q}_1^{\;\prime\,2}\vec{k}^{\;2}}
{\vec{q}_1^{\;2}\vec{q}_2^{\;2}}\right)
+\ln\left(\frac{\vec{q}_2^{\;\prime2}}{\vec{k}^{\;2}}\right)
\ln\left(\frac{\vec{q}^{\;2}}
{\vec{q}_1^{\;\prime2}}\right)\right)
\]
\begin{equation}
+2\frac{[\vec{q}_{1}\times\vec{q}_2]}{\vec{q}_{1}^{\;2}\vec{q}_2^{\;2}}
[\vec{q}_{1}\times\vec{k}]
I_{\vec{k}, \vec{q}_{1}^{\;\prime}}-2\frac{[\vec{q}_{1}
\times\vec{q}_2]}{\vec{q}_{1}^{\;2}
\vec{q}_2^{\;2}}[\vec{q}_{1}^{\;\prime}\times\vec{q}_2^{\;\prime}]I_{\vec{q}_{1}^{\;\prime}, \vec{q}_{2}^{\;\prime}}~. \label{I momentum}
\end{equation}
In these equalities the symbols $<.....>$ mean adding to the direct
Fourier transform terms
that depend only on $\vec{q}_{1}$ and $\vec{q}_{2}$ (and do not depend on
$\vec k$) and terms that are antisymmetric with respect to the substitution
$\vec{q}_1\leftrightarrow -\vec{q}_2$. These terms are fixed by the
requirement of the gauge invariance and the symmetry of the kernel,
according to Ref.~\cite{Fadin:2011jg}.
Equalities~(\ref{F momentum})--(\ref{I momentum}) can be derived using
formulas given in the Appendices of Ref.~\cite{Fadin:2007de} and of the
present paper. The substitution of these equalities
in Eq.~(\ref{restored commutator}) gives the same result
as Eq.~(\ref{final commutator}).
\section{Conclusion}
\label{sec:conclusion}
The simplicity of the M\"{o}bius form of the quasi-conformal NLO BFKL
kernel suggested to use just this form for finding the kernel in the
momentum space. The way to do that was not evident, and even the
possibility to do it seemed doubtful, because the M\"{o}bius form is defined
on a special class of functions in the coordinate space. However, it was
shown~\cite{Fadin:2011jg} that such possibility exists due to the gauge
invariance of the kernel and the way to obtain the kernel in the momentum
space from its M\"{o}bius form was elaborated. But technically obtaining
it turned out to be not easy.
In this paper we found in the momentum space the difference between the
standard BFKL kernel, defined according to the prescriptions given in
Ref.~\cite{Fadin:1998fv} and calculated in Ref.~\cite{FF05}, and the
quasi-conformal BFKL kernel. This difference turned out to be rather simple.
The most natural conclusion is that the simplicity of the M\"{o}bius form
of the quasi-conformal kernel is caused mainly by using the impact parameter
space. The other possibility is that the quasi-conformal kernel can be
written in simple form also in the transverse momentum space. If this is
true, the standard kernel of Ref.~\cite{FF05} could result itself in a much
simpler form. We plan to check this possibility using
both the representation of Ref.~\cite{FF05} and the representation in terms
of integrals in the transverse momentum space of Ref.~\cite{Fadin:2006zz}.
\vspace{0.5cm} {\textbf{{\Large Acknowledgments}}}
\vspace{0.5cm} V.S.F. thanks the Dipartimento di Fisica
dell'Universit\`{a} della Calabria and the Istituto Nazionale di
Fisica Nucleare (INFN), Gruppo Collegato di Cosenza, for warm hospitality
while part of this work was done and for financial support.
\newpage
\setcounter{equation}{0}
\defA.\arabic{equation}{A.\arabic{equation}}
\section*{Appendix}
The two-dimensional integrals of Section~\ref{sec:direct} were calculated
choosing appropriate integration vectors and performing firstly the
integration over azimuthal angles. It is convenient to make this integration
using ``helical'' vector components ``$\pm$'' instead of the
Cartesian ones ``$x,y$'', $a^{\pm}=a_x\pm i a_y$.
Denoting the integration vector as $\vec{l}$, we have
$l^\pm = le^{\pm i \phi}$, where $\phi$ is its azimuthal angle and $l$ is
its modulus. The integration over $\phi$ can be performed using
the representation $2(\vec{a}-\vec{l})(\vec{b}-\vec{l})=(a^+-l^+)(b^--l^-)
+(a^--l^-)(b^+-l^+)$
and the expansion of the integrands in positive or negative powers of
$l^\pm$ at various values of $l$. Thus one can easily obtain
\begin{equation}
\int_{-\pi}^{\pi}\frac{d\phi}{2\pi}\;\ln(\vec{a}-\vec{l})^2
=\theta(\vec{a}^{\;2}-\vec{l}^{\;2})\ln\vec{a}^{\;2}
+\theta(\vec{a}^{\;2}-\vec{l}^{\;2})\ln\vec{l}^{\;2}~,\label{int phi ln}
\end{equation}
\begin{equation}
\int_{-\pi}^{\pi}\frac{d\phi}{2\pi}\;\frac{1}{a^\pm-l^\pm}=
\frac{\theta(\vec{a}^{\;2}-\vec{l}^{\;2})}{a^\pm}~,\;\;\label{int phi pm}
\end{equation}
\begin{equation}
\int_{-\pi}^{\pi}\frac{d\phi}{2\pi}\;\frac{1}{(a^\pm -l^\pm)
(b^\mp -l^\mp)}=
\frac{\theta(\vec{a}^{\;2}-\vec{l}^{\;2})}{a^\pm b^\mp-\vec{l}^{\;2}}+
\frac{\theta(\vec{l}^{\;2}-\vec{b}^{\;2})}{\vec{l}^{\;2}-a^\pm b^\mp}~.
\label{int phi pm mp}
\end{equation}
In particular, one has from Eq.~(\ref{int phi pm mp})
\[\int_{-\pi}^{\pi}\frac{d\phi}{2\pi}\;\frac{2(\vec{l}
(\vec{a}-\vec{l}))}{\vec{l}^{\;2}
(\vec{a}-\vec{l})^{2}}=\theta(\vec{l}^{\;2}-\vec{a}^{\;2})\frac{-2}
{\vec{l}^{\;2}}~,\;\;
\]
\begin{equation}
\int_{-\pi}^{\pi}\frac{d\phi}{2\pi}\;\frac{2((\vec{a}-\vec{l}))
(\vec{b}-\vec{l}))}{(\vec{a}-\vec{l})^{2}(\vec{b}-\vec{l})^{2}}
=\left(\theta(\vec{a}^{\;2}-\vec{l}^{\;2})-\theta(\vec{b}^{\;2}-\vec{l}^{\;2})
\right)\left(\frac{1}{a^+b^--\vec{l}^{\;2}}
+\frac{1}{a^-b^+-\vec{l}^{\;2}}\right)~. \label{int phi a-l b-l}
\end{equation}
The result~(\ref{int R u}) follows from Eqs.~(\ref{int phi ln})
and~(\ref{int phi a-l b-l}) with the subsequent elementary integration over
$l$. Since the integral consists of several terms, which are not ultraviolet
convergent when taken separately, it is convenient to calculate them
introducing an ultraviolet cut-off $\Lambda$.
Using Eq.~(\ref{int phi a-l b-l}), one can also easily obtain
\begin{equation}
\int\frac{d{\vec{l}}}{\pi}\;\left(\frac{(\vec{l}(\vec{a}-\vec{l}))}
{\vec{l}^{\;2}
(\vec{a}-\vec{l})^{2}}-\frac{(\vec{l}(\vec{b}-\vec{l}))}{\vec{l}^{\;2}
(\vec{b}-\vec{l})^{2}}\right)=\ln\left(\frac{\vec{b}^{\;2}}{\vec{a}^{\;2}}
\right)~, \label{int v}
\end{equation}
that gives the result~(\ref{virtual contribution}).
Though we use the ultraviolet cut-off $\Lambda$ (which is supposed tending to
infinity) for separate integrals, it is possible to shift the integration
vectors in them, since these integrals have only logarithmic divergence.
Therefore, with an appropriate choice of $\vec{l}$, in all integrals of
Section~\ref{sec:direct} the integration over $\phi$ can be performed
using Eqs.~(\ref{int phi pm}) and~(\ref{int phi pm mp}). But sometimes it
is more convenient to use Eq.~(\ref{int phi ln}) as, for example, in the
integral
\begin{equation}
\int\frac{d{\vec{l}}}{\pi}\;\theta(\Lambda^2-\vec{l}^{\;2})
\frac{1}{(\vec{a}-\vec{l})^2}\ln\left(\frac{\vec{l}^{\;2}}{\vec{a}^{\;2}}
\right) =\int\frac{d{\vec{l}}}{\pi}\;\theta(\Lambda^2-\vec{l}^{\;2})
\frac{1}{\vec{l}^{\;2}}\ln\frac{(\vec{a}-\vec{l})^2}{\vec{a}^{\;2}} =
\frac{1}{2}\ln^2\left(\frac{\Lambda^2}{\vec{a}^{\;2}}\right)~. \label{int s1l}
\end{equation}
Using Eq.~(\ref{int phi pm mp}), we obtain:
\[
\int\frac{d \vec l}{\pi}\frac{1}{(\vec a- \vec 1)^+}
\frac{1}{(\vec b- \vec 1)^-}\ln\left(\frac{\vec l^{\;2}}{\mu^2}\right)
\theta(\Lambda^2-\vec l^{\;2})=
\frac12\ln\left(\frac{\Lambda^2}{(\vec a -\vec b)^{2}}\right)
\ln\left(\frac{\Lambda^2(\vec a -\vec b)^{2}}{\mu^4}\right)
\]
\begin{equation}
+\frac12\ln\left(\frac{(\vec a -\vec b)^{2}}{\vec b^{\;2}}\right)
\ln\left(\frac{(\vec a -\vec b)^{2}}{\vec a^{\;2}}\right)
+\frac{a^+b^--a^-b^+}{2}I_{\vec a,-\vec b }~, \label{int master}
\end{equation}
where $I_{\vec a,\vec b }$ is defined in Eq.~(\ref{I p q 1})
(see also Eqs.~(\ref{I symmetric}) and~(\ref{integral I})). In fact, all
integrals of Section~\ref{sec:direct} can be calculated using this one. In
particular, the integral~(\ref{int s1l}) can be obtained from
the integral~(\ref{int master}) as the limit $\vec b \rightarrow \vec a$ at
$\mu^2=\vec a^{\;2}$. The integrals~(\ref{int phi a-l b-l}) and~(\ref{int v})
also can be found using the part of the integral~(\ref{int master})
proportional to $\ln \mu^2$. We find also
\[
\int\frac{d\vec l}{\pi}\frac{2}{(\vec a-\vec l)^2}\frac{(\vec b-\vec l)}
{(\vec b-\vec l)^2}
\ln\left(\frac{\vec l^{\;2}}{\vec a^{\;2}}\right) =\frac{(\vec a-\vec b)}
{(\vec a-\vec b)^{2}}\ln\left(\frac{\vec a^{\;2}}{\vec b^{\;2}}\right)
\ln\left(\frac{(\vec a -\vec b)^{2}}{\vec a^{\;2}}\right)
\]
\begin{equation}
+2\frac{[(\vec a- \vec b)\times[\vec a \times\vec b]]}{(\vec a-\vec b)^2}
I_{\vec a, -\vec b}~,
\label{int vl2}
\end{equation}
\[
\int\frac{d\vec l}{\pi}\frac{(\vec{c}-\vec l)}{(\vec{c}-\vec l)^2}
\frac{2((\vec a-\vec l)(\vec b-\vec l))}{(\vec a -\vec l)^2(\vec b-\vec l)^2}
\ln\left(\frac{\vec l^{\;2}}{\mu^{2}}\right) =\frac{(\vec c -\vec b)}
{(\vec c -\vec b)^2}\left[\ln\left(\frac{\vec a^{\;2}}{\mu^{2}}\right)
\ln\left(\frac{(\vec c -\vec a)^{2}}{(\vec b -\vec a)^{2}}\right)\right.
\]
\[
\left.+\frac12\ln\left(\frac{c^{\;2}}{\vec a^{\;2}}\right)
\ln\left(\frac{(\vec a -\vec c)^{2}}{\vec b^{\;2}}\right)
-\frac12\ln\left(\frac{b^{\;2}}{\vec a^{\;2}}\right)
\ln\left(\frac{(\vec a -\vec b)^{2}}{\vec c^{\;2}}\right)
\right] +\frac{(\vec c -\vec a)}{(\vec c -\vec a)^2}
\left[\ln\left(\frac{\vec b^{\;2}}{\mu^{2}}\right)\right.
\]
\[
\left.
\times \ln\left(\frac{(\vec c -\vec b)^{2}}{(\vec a -\vec b)^{2}}\right)
+\frac12\ln\left(\frac{c^{\;2}}{\vec b^{\;2}}\right)
\ln\left(\frac{(\vec b -\vec c)^{2}}{\vec a^{\;2}}\right)
-\frac12\ln\left(\frac{a^{\;2}}{\vec b^{\;2}}\right)
\ln\left(\frac{(\vec b -\vec a)^{2}}{\vec c^{\;2}}\right)
\right]
\]
\[
+\left(\frac{[(\vec c -\vec b)\times[\vec a\times\vec b]]}
{(\vec c -\vec b)^2}+\frac{[(\vec c -\vec a)\times[\vec b\times\vec a]]}
{(\vec c -\vec a)^2}\right)I_{\vec a,-\vec b }
+\frac{[(\vec c -\vec b)\times[\vec c\times\vec a]]}
{(\vec c -\vec b)^2}I_{\vec c,-\vec a }
\]
\begin{equation}
+\frac{[(\vec c -\vec a)\times[\vec c\times\vec b]]}
{(\vec c -\vec a)^2}I_{\vec c,-\vec b }\label{int vl3}~.
\end{equation}
The result~(\ref{F1 final}) for $F_1(\vec{q}_1,\vec{q}_2; \vec{k})$ was
obtained using Eqs.~(\ref{int R u}),~(\ref{int vl2}) and~(\ref{int vl3}) with
its particular cases, such as
\[
\int\frac{d\vec l}{\pi}\frac{(\vec{a}-\vec l)}{(\vec{a}-\vec l)^2}
\frac{2(\vec l(\vec b-\vec l))}{\vec l^{\;2}(\vec b-\vec l)^2}
\ln\left(\frac{\vec l^{\;2}}{\mu^{2}}\right) =-\frac12
\frac{(\vec a -\vec b)}{(\vec a -\vec b)^2}\ln\left(\frac{\vec a^{\;2}}
{\vec b^{\;2}}\right)\ln\left(\frac{\vec a^{\;2}\vec b^{\;2}}{\mu^{4}}\right)
\]
\begin{equation}
-\frac12\frac{\vec a}{\vec a^{\;2}}\ln\left(\frac{\vec a^{\;2}\vec b^{\;2}}
{\mu^{4}}\right) \ln\left(\frac{(\vec a -\vec b)^{2}}{\vec b^{\;2}}\right)
-\frac{[\vec a\times[\vec a\times\vec b]]}{\vec a^{\;2}}I_{\vec a,-\vec b }~.
\label{int vl30}
\end{equation}
To obtain $F_2(\vec{q}_1,\vec{q}_2; \vec{k})$, Eq.~(\ref{F2 final}), we used
\[
\int\frac{d\vec l}{\pi}\theta(\Lambda^2-\vec l^{\;2})\;\frac{2((\vec a-\vec l)
(\vec b-\vec l))}{(\vec a-\vec l)^{2}(\vec b-\vec l)^2}
\ln\left(\frac{\vec l^{\;2}}{\mu^{2}}\right)
=\ln\left(\frac{\Lambda^2(\vec a -\vec b)^2}{\mu^4}\right)
\ln\left(\frac{\Lambda^2}{(\vec a -\vec b)^2}\right)
\]
\begin{equation}
+\ln\left(\frac{\vec a^{\;2}}{(\vec a -\vec b)^2}\right)
\ln\left(\frac{\vec b^{\;2}}{(\vec a -\vec b)^2}\right)~,
\label{int s2l}
\end{equation}
\[
\int\frac{d\vec l}{\pi}\;\frac{1}{\vec l^{\;2}}\frac{2((\vec a-\vec l)
(\vec b-\vec l))}{(\vec a-\vec l)^2(\vec b-\vec l)^2}
\ln\left(\frac{(\vec c-\vec l)^2}{\vec c^{\;2}}\right) =\frac{1}{\vec a^{\;2}
\vec b^{\;2}}\left[(\vec a\vec b)\left(\ln\left(\frac{(\vec c -\vec a)^{2}}
{\vec c^{\;2}}\right)\ln\left(\frac{(\vec c -\vec b)^{2}}{\vec c^{\;2}}\right)
\right.\right.
\]
\[
\left.\left.-\ln\left(\frac{(\vec c -\vec a)^{2}}{\vec c^{\;2}}\right)
\ln\left(\frac{(\vec a -\vec b)^{2}}{\vec a^{\;2}}\right)
-\ln\left(\frac{(\vec c -\vec b)^{2}}{\vec c^{\;2}}\right)
\ln\left(\frac{(\vec a -\vec b)^{2}}{\vec b^{\;2}}\right)\right)\right.
\]
\[
\left.
+2([\vec a\times\vec b][\vec a\times\vec c]) I_{\vec a, -\vec c}
+2([\vec a\times\vec b][\vec c\times\vec b])I_{\vec b, -\vec c}
\right.
\]
\begin{equation}
\left.+2([\vec a\times\vec b][(\vec a-\vec c)\times(\vec b-\vec c)])
I_{\vec a -\vec c, \vec c-\vec b }\right]~,
\label{int s3l}
\end{equation}
in particular,
\[
\int\frac{d\vec l}{\pi}\frac{1}{(\vec a-\vec l)^{2}}
\frac{2(\vec l(\vec b-\vec l))}{\vec l^{\;2}(\vec b-\vec l)^2}
\ln\left(\frac{\vec l^{\;2}}{\vec a^{\;2}}\right) =\frac{1}{\vec a^{\;2}
(\vec a-\vec b)^{2}}\left[(\vec a(\vec a-\vec b))\ln\left(\frac{\vec a^{\;2}}
{\vec a^{\;2}}\right)\ln\left(\frac{(\vec a -\vec b)^{2}}{\vec b^{\;2}}\right)
\right.
\]
\begin{equation}
\left.-2[\vec a \times\vec b]^2I_{\vec a -\vec b, -\vec a }\right]~,
\label{int s3l0}
\end{equation}
and Eq.~(\ref{int vl3}) with its particular cases~(\ref{int vl30}) and
\[
\int\frac{d\vec l}{\pi}\frac{\vec l}{\vec l^{\;2}}\frac{2((\vec a-\vec l)
(\vec b-\vec l))}{(\vec a -\vec l)^2(\vec b-\vec l)^2}
\ln\left(\frac{\vec l^{\;2}}{\mu^{2}}\right) =\frac{\vec b}{\vec b^{\;2}}
\left[\ln\left(\frac{\vec a^{\;2}}{\mu^{2}}\right)\ln\left(\frac{\vec a^{\;2}}
{(\vec b -\vec a)^{2}}\right)\right.
\]
\[
\left.+\frac12\ln\left(\frac{(\vec b -\vec a)^{2}}{\vec a^{\;2}}\right)
\ln\left(\frac{\vec a^{\;2}}{\vec b^{\;2}}\right)\right] +\frac{\vec a}
{\vec a^{\;2}}\left[\ln\left(\frac{\vec b^{\;2}}{\mu^{2}}\right)
\ln\left(\frac{\vec b^{\;2}}{(\vec a -\vec b)^{2}}\right)\right.
\]
\begin{equation}
\left.
+\frac12\ln\left(\frac{(\vec a -\vec b)^{2}}{\vec b^{\;2}}\right)
\ln\left(\frac{\vec b^{\;2}}{\vec a^{\;2}}\right)
\right]
+\left(\frac{[\vec b\times[\vec a\times\vec b]]}{\vec b^{\;2}}
+\frac{[\vec a\times[\vec b\times\vec a]]}{\vec a^{\;2}}\right)
I_{\vec a,-\vec b }~.
\label{int v3l0}
\end{equation}
The result~(\ref{F3 final}) for $F_3(\vec{q}_1,\vec{q}_2; \vec{k})$ can be
obtained using Eqs.~(\ref{int vl2}),~(\ref{int s3l0}),~(\ref{int v3l0}),
\begin{equation}
\int\frac{d\vec l}{\pi}\frac{1}{\vec l^{\;2}(\vec a-\vec l)^2}
\ln\left(\frac{(\vec b-\vec l)^{2}(\vec a-\vec b-\vec l)^{2}}{\vec b^{\;2}
(\vec a-\vec b)^{2}}\right) =\frac{1}{\vec a^{\;2}}\ln^2\left(\frac{(\vec a
-\vec b)^{2}}{\vec b^{\;2}}\right)\label{int s2ll}
\end{equation}
and
\[
\int\frac{d\vec l}{\pi}\;\frac{\theta(\Lambda^2-\vec l^{\;2})}
{(\vec c-\vec l)^2}\left(\frac{2((\vec a-\vec l)(\vec b-\vec l))}
{(\vec a-\vec l)^2(\vec b-\vec l)^2}-2\frac{2((\vec a-\vec c)
(\vec b-\vec c))}{(\vec a-\vec c)^2(\vec b-\vec c)^2}\right)
\ln\left(\frac{\vec l^{\;2}}{\mu^{2}}\right)
\]
\[
=\frac{((\vec a-\vec c)(\vec b-\vec c))}{(\vec a-\vec c)^2(\vec b-\vec c)^2}
\left[\ln\left(\frac{\Lambda^2}{(\vec a -\vec b)^{2}}\right)
\ln\left(\frac{\Lambda^2(\vec a -\vec b)^{2}}{\mu^4}\right)
+\ln\left(\frac{(\vec a -\vec b)^{2}}{\vec a^{\;2}}\right)
\ln\left(\frac{(\vec a -\vec b)^{2}}{\vec b^{\;2}}\right)\right.
\]
\[
\left.
-\ln\left(\frac{\Lambda^2}{(\vec a -\vec c)^{2}}\right)
\ln\left(\frac{\Lambda^2(\vec a -\vec c)^{2}}{\mu^4}\right)
- \ln\left(\frac{(\vec a -\vec c)^{2}}{\vec a^{\;2}}\right)
\ln\left(\frac{(\vec a -\vec c)^{2}}{\vec c^{\;2}}\right)
\right.
\]
\[
\left.
-\ln\left(\frac{\Lambda^2}{(\vec c -\vec b)^{2}}\right)
\ln\left(\frac{\Lambda^2(\vec c -\vec b)^{2}}{\mu^4}\right)
-\ln\left(\frac{(\vec c -\vec b)^{2}}{\vec c^{\;2}}\right)
\ln\left(\frac{(\vec c -\vec b)^{2}}{\vec b^{\;2}}\right)\right]
\]
\begin{equation}
+2\left(\frac{[(\vec a-\vec c)\times(\vec b-\vec c)]}
{(\vec a-\vec c)^2(\vec b-\vec c)^2}\left([\vec a\times\vec b]
I_{\vec a, -\vec b}-[\vec a\times\vec c]
I_{\vec a, -\vec c} -[\vec c\times\vec b] I_{\vec c, -\vec b} \right)\right)\;.
\label{int s3ld}
\end{equation}
Let us present also the integral
\[
\int\frac{d\vec l}{\pi}\frac{2}{(\vec a-\vec l)^2}\frac{l_i(b-\vec l)_j}
{\vec l^{\;2}(\vec b-\vec l)^2}
\ln\left(\frac{\vec l^{\;2}}{\vec a^{\;2}}\right)
=\frac{b_i(a-b)_j+(a-b)_i b_j-\delta_{ij}(\vec b(\vec a-\vec b))}
{2\vec b^{\;2}(\vec a-\vec b)^{2}}\ln\left(\frac{\vec a^{\;2}}
{\vec b^{\;2}}\right)
\]
\[
\times \ln\left(\frac{(\vec a -\vec b)^{2}}{\vec a^{\;2}}\right)
+\frac{b_ia_j-a_i b_j+\delta_{ij}(\vec a(\vec a-\vec b))}{2\vec a^{\;2}
(\vec a-\vec b)^{2}}\ln\left(\frac{\vec a^{\;2}}{\vec b^{\;2}}\right)
\ln\left(\frac{(\vec a -\vec b)^{2}}{\vec b^{\;2}}\right)
+\frac{I_{\vec a, -\vec b}}{(\vec a-\vec b)^2}
\]
\[
\times\left(\frac{1}{\vec b^{\;2}}
\left([\vec b\times[\vec a\times \vec b]]_i(\vec a- \vec b)_j
+(\vec a- \vec b)_i[\vec b\times[\vec a\times \vec b]]_j
-\delta_{ij}([\vec b\times[\vec a\times \vec b]](\vec a- \vec b))
\right)\right.
\]
\begin{equation}
\left.+\frac{1}{\vec a^{\;2}}\left( (\vec a- \vec b)_i[\vec a
\times[\vec a\times \vec b]]_j-[\vec a\times[\vec a\times \vec b]]_i(\vec a
- \vec b)_j -\delta_{ij}([\vec a\times[\vec a\times \vec b]](\vec a- \vec b))
\right)\right)
\label{int tl3}
\end{equation}
which is more general than the integral~(\ref{int vl30}) and can appear
in decompositions of the integrands for $F_i$ different from ours, and the
integrals
\[
\int\frac{d\vec l}{\pi}\frac{2}{\vec l^{\;2}}\left(\frac{(\vec a-\vec l)}{(\vec a-\vec l)^2}
\left(\frac{(\vec b-\vec l)}{(\vec b-\vec l)^2}-\frac{\vec b}{\vec b^{\;2}}
\right)\right)
\ln\left(\frac{\vec l^{\;2}}{\vec q^{\;2}}\right) =\frac{(\vec a\vec b)}
{\vec a^{\;2}\vec b^{\;2}}\ln\left(\frac{\vec a^{\;2}\vec b^{\;2}}
{\vec q^{\;4}}\right)\ln\left(\frac{\vec b^{\;2}}{(\vec a -\vec b)^{2}}\right)
\]
\begin{equation}
+\frac{2[\vec a\times\vec b]^{2}}{\vec a^{\;2}\vec b^{\;2}}
I_{\vec a, -\vec b}~,
\label{int sl3d}
\end{equation}
\[
\int\frac{d\vec l}{\pi}\frac{2}{\vec l^{\;2}}\left(\frac{(\vec b-\vec l)}
{(\vec b-\vec l)^2} -\frac{\vec b}{\vec b^{\;2}} \right)
\ln\left(\frac{(\vec a-\vec l)^2}{\vec l^{\;2}}\right)
=\frac{\vec b}{\vec b^{\;2}}\ln\left(\frac{\vec a^{\;2}}{\vec b^{\;2}}\right)
\ln\left(\frac{\vec b^{\;2}}{(\vec a -\vec b)^{2}}\right)
\]
\begin{equation}
+2\frac{[\vec b\times[\vec a \times\vec b]]}{\vec b^{\;2}}
I_{a,\vec b-\vec a }~,\label{int vl2d}
\end{equation}
which also can be useful.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,113 |
Q: How to test Java Bean Validation Annotation applicable to function input targets I'm adding Java Bean Validation custom constraints and trying to write tests that they get applied by the provider, but I can't figure out how to do that.
I found this post that gives an example of how to test such annotations inside a class and it works for testing the annotations added to predefined model objects.
However, now I also want to test function input annotation and this is where I'm stuck.
Here is an example:
// -- Annotation
@Constraint(validatedBy = IdValidator.class)
@Target({ElementType.METHOD, ElementType.FIELD, ElementType.ANNOTATION_TYPE, ElementType.PARAMETER})
@Retention(RetentionPolicy.RUNTIME)
public @interface Id{...}
// -- Validator
public class IdValidator implements ConstraintValidator<Id, String> {...}
// Now the way it's used (that I'm trying to test) is as follows:
public class MyController {
....
@Requestmapping(....)
@ResponseBody
public Response doStuff(@Id String personId) {....}
}
Is there a way to to write a test to check Validation Provider is able to apply custom validator in this way?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,310 |
<?php
/**
* @file
* Definition of Drupal\entity_test\Entity\EntityTest.
*/
namespace Drupal\entity_test\Entity;
use Drupal\Core\Entity\EntityNG;
use Drupal\Core\Entity\Annotation\EntityType;
use Drupal\Core\Annotation\Translation;
use Drupal\Core\Language\Language;
/**
* Defines the test entity class.
*
* @EntityType(
* id = "entity_test",
* label = @Translation("Test entity"),
* module = "entity_test",
* controllers = {
* "storage" = "Drupal\entity_test\EntityTestStorageController",
* "list" = "Drupal\entity_test\EntityTestListController",
* "access" = "Drupal\entity_test\EntityTestAccessController",
* "form" = {
* "default" = "Drupal\entity_test\EntityTestFormController"
* },
* "translation" = "Drupal\content_translation\ContentTranslationControllerNG"
* },
* base_table = "entity_test",
* fieldable = TRUE,
* field_cache = FALSE,
* entity_keys = {
* "id" = "id",
* "uuid" = "uuid",
* "bundle" = "type",
* },
* menu_base_path = "entity-test/manage/%entity_test"
* )
*/
class EntityTest extends EntityNG {
/**
* The entity ID.
*
* @var \Drupal\Core\Entity\Field\FieldInterface
*/
public $id;
/**
* The entity UUID.
*
* @var \Drupal\Core\Entity\Field\FieldInterface
*/
public $uuid;
/**
* The bundle of the test entity.
*
* @var \Drupal\Core\Entity\Field\FieldInterface
*/
public $type;
/**
* The name of the test entity.
*
* @var \Drupal\Core\Entity\Field\FieldInterface
*/
public $name;
/**
* The associated user.
*
* @var \Drupal\Core\Entity\Field\FieldInterface
*/
public $user_id;
/**
* Initialize the object. Invoked upon construction and wake up.
*/
protected function init() {
parent::init();
// We unset all defined properties, so magic getters apply.
unset($this->id);
unset($this->uuid);
unset($this->name);
unset($this->user_id);
unset($this->type);
}
/**
* Overrides Drupal\entity\Entity::label().
*/
public function label($langcode = Language::LANGCODE_DEFAULT) {
$info = $this->entityInfo();
if (isset($info['entity_keys']['label']) && $info['entity_keys']['label'] == 'name') {
return $this->getTranslation($langcode)->name->value;
}
else {
return parent::label($langcode);
}
}
/**
* {@inheritdoc}
*/
public static function baseFieldDefinitions($entity_type) {
$fields['id'] = array(
'label' => t('ID'),
'description' => t('The ID of the test entity.'),
'type' => 'integer_field',
'read-only' => TRUE,
);
$fields['uuid'] = array(
'label' => t('UUID'),
'description' => t('The UUID of the test entity.'),
'type' => 'uuid_field',
);
$fields['langcode'] = array(
'label' => t('Language code'),
'description' => t('The language code of the test entity.'),
'type' => 'language_field',
);
$fields['name'] = array(
'label' => t('Name'),
'description' => t('The name of the test entity.'),
'type' => 'string_field',
'translatable' => TRUE,
'property_constraints' => array(
'value' => array('Length' => array('max' => 32)),
),
);
$fields['type'] = array(
'label' => t('Type'),
'description' => t('The bundle of the test entity.'),
'type' => 'string_field',
'required' => TRUE,
// @todo: Add allowed values validation.
);
$fields['user_id'] = array(
'label' => t('User ID'),
'description' => t('The ID of the associated user.'),
'type' => 'entity_reference_field',
'settings' => array('target_type' => 'user'),
'translatable' => TRUE,
);
return $fields;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,288 |
export class ValidationOptions {
public undefinedAllowed:boolean = false;
public nullAllowed:boolean = false;
public stringEmptyAllowed:boolean = false;
public numberZeroAllowed:boolean = true;
public numberPositiveAllowed:boolean = true;
public numberNegativeAllowed:boolean = true;
public booleanFalseAllowed:boolean = true;
public arrayEmptyAllowed:boolean = false;
public objectEmptyAllowed:boolean = false;
}
export class ObjectUtils {
static Validate(obj:any, options?:ValidationOptions): boolean {
if(options === undefined) {
options = new ValidationOptions();
}
if(obj === undefined) {
return options.undefinedAllowed;
}
if(obj === null) {
return options.nullAllowed;
}
if(typeof obj === "number") {
if(obj === 0) {
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Q: How to include shortcodes and pdfs in WordPress search without plugin? I have a WordPress website using Avada theme. The website uses an inbuilt WordPress search tool which searches only posts. The website uses shortcode to add toggle/ accordion etc. I want to include search results from accordions/ tabs etc?
I need to include results from toggle/ accordions/ pdfs etc. It is fine if possible to include the page containing toggle?
Thanks
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\section{Introduction}
Currently, there are many ways to optically cool a substance. The most effective method of cooling micro and nano devices is the sideband method \cite{Wilson_2007,Genes_2008,Gigan_2006,SCHLIESSER_2008,Chan_2011,Teufel_2011}. This method shows great promise for using quantum effects (including coherent properties) in hybrid systems \cite{Machnes_2012,Aspelmeyer_2014} but first it is necessary to prepare the mechanical component in the ground state, i.e. cool it as much as possible. The sideband method is based on the connection of a mechanical device, that is, a resonator (target), with a microwave or optical resonator (auxiliary) \cite{Aspelmeyer_2014,Marquardt_2007}. Moreover, the frequency of the microwave or optical resonator should be high enough to be in the ground state at ambient temperature. Note that with modern technologies, the average number of photons in such a resonator can be very small \cite{Tian_2009,Wang_2011,Schmidt_2011,Triana_2015} and in theoretical calculations this value is usually chosen as $ n_{aux} = 0 $. The connection of a mechanical resonator with an electromagnetic field occurs through pressure by radiation.
Although this interaction is generally not linear, under real conditions the relationship can be considered linear. In theory, such a relationship is given in the form \cite{Aspelmeyer_2014,Marquardt_2007} $ {\hat {H}}_{int} = g(t) q x $, where $ g(t) $ is a communication parameter, $ x $ is a variable mechanical mode and $ q $ is a variable electromagnetic mode. Currently, a system with a Hamiltonian in the form of ${\hat{H}}={\hat{H}}_{t}+{\hat{H}}_{aux}+{\hat{H}}_{int}$ is being actively studied, where $ {\hat{H}}_{t} $ is the Hamiltonian of the target and $ {\hat{H}}_{aux} $ is the Hamiltonian of the electromagnetic field, with the presence of environment and losses in the cavity \cite{Wilson_2007,Teufel_2011,Wang_2011,Schmidt_2011,Triana_2015,Safavi_2013,Palomaki_2013}. The evolution of the density matrix in the presence of an environment is usually described using a master equation, for example, in the model of Ullersma-Caldeira-Leggett \cite{Ullersma_1966,Caldeira_1981}, and many theoretical results have been confirmed experimentally, which confirms the adequacy of the applied model. Currently, the study of such systems is mainly directed towards the optimal selection of the connecting function $ g(t) $ for maximum cooling of the target \cite{Wang_2011,Schmidt_2011,Triana_2015,Frank_2016}. For example, in \cite{Triana_2015} it was shown that with the optimal choice of the function $ g(t) $ taking into account non-Markov evolution processes of the system under consideration, the average value of the quantum number (phonon) of the target can be $n_t=10^{-3}$.
In spite of this, the problem of cooling charged mechanical resonators (charged targets) is relevant. Charged mechanical resonators can be not only simple mechanical systems, but also molecular ions, which expands the field of use of the results obtained in the work. So far, the cooling of such systems has not been implemented, and indeed, such studies are rarely found in the literature.
In this work, it will be shown that the mechanical charged resonator can be cooled to ultra-small quantum states, similar to the method of the sideband. As an example, we will consider the case of cooling a mechanical resonator without taking into account the external environment, where, with certain target parameters and an electromagnetic field, the target can be cooled to the ground quantum state. The main equation showing the average number of quantum states of $ n_t $ charged resonator has a surprisingly simple analytical form, which allows for the most complete analysis of the $ n_t $ quantity under investigation.
\section{The model and its solution}
Consider the interaction of a charged mechanical resonator (charged target) with a quantized electromagnetic field (auxiliary). The Schrodinger equation for such a system will be in the form ${\hat H}\Psi=i\hbar \frac{\partial \Psi}{\partial t}$, where the Hamiltonian ${\hat H}$ is
\begin{equation}
{\hat H}= \hbar\omega\left({\hat a}^{+} {\hat a}+\frac{1}{2}\right)+\hbar\Omega\left({\hat b}^{+} {\hat b}+\frac{1}{2}\right)+{\hat {H}}_{int},
\label{1}
\end{equation}
where $\omega$ is the frequency of the mechanical mode of the resonator, $ \Omega $ is the frequency of the electromagnetic mode, ${\hat a},{\hat b}$ are operators of annihilation of the mechanical and electromagnetic modes, respectively, and $ {\hat{H}}_{int} $ is the Hamiltonian responsible for the interaction of the electromagnetic field with a charged target. Obviously, in the case of interaction with a target having an electric charge of $ Z $ and a mass of $ M_t $, this interaction will be
\begin{equation}
{\hat {H}}_{int}= \frac{Z}{c M_t}\hat{\bf{ A}}\hat{\bf{ p}}+\frac{1}{2M_t}\left(\frac{Z}{c}\hat{\bf{ A}}\right)^2 ,
\label{2}
\end{equation}
where ${\hat{\bf{ A}}}$ is the vector potential of the electromagnetic field, and $\hat{\bf{ p}}$ is the moment operator of a mechanical resonator. Since a one-dimensional mechanical resonator is considered, $\hat{\bf{ p}}=-i\hbar \frac{\partial}{\partial x} {\bf i}$, where $ {\bf i} $ is the unit vector (mechanical resonator is one-dimensional). Since we are interested in the microwave or optical part of the electromagnetic spectrum, we can apply the dipole approximation, in which the coordinate representation $ {\hat{\bf{A}}} = \sqrt{\frac {4 \pi c^2} {\Omega V}}{\bf u} q $, where $ q $ is the field variable, $ V $ is the volume of space in which the electromagnetic field is located, and $ {\bf u} $ is the polarization of the electromagnetic field. Further, for convenience, we use a system of units, where $ \hbar = 1, M_t = 1, Z = 1 $. As a result, we must consider the Hamiltonian
\begin{equation}
{\hat{H}}= \frac{\Omega}{2}\left(q^2-\frac{\partial^2 }{\partial {q^2}}\right)+\frac{\omega}{2}\left(x^2-\frac{\partial^2 }{\partial {x^2}}\right)+\frac{\beta^2}{2}q^2-i\beta {\bf u i} \sqrt{\omega}q \frac{\partial }{\partial x} ,
\label{3}
\end{equation}
where $\beta=\sqrt{\frac{4\pi}{\Omega V}}$.
A similar Hamiltonian, but set to another problem, was considered in \cite{Makarov_SREP_2018} (see also \cite{Makarov_2017_adf,Makarov_2018_PRE}), where an analytical solution was found for the nonstationary Schrodinger equation with $ \beta \ll 1 $. Indeed, for a realistic microcavity or focal volume \cite{Tey_2008}, $ \beta $ takes values of the order of $ 10^{-5} - 10^{-3} $, and usually it is much less than even these values. We write out the basic equation for our case using the results \cite{Makarov_SREP_2018}.
The wave function of the system under consideration $ \Psi(x,q,t) $ will have the form
\begin{eqnarray}
\Psi(x,q,t)=\sum^{s_1+s_2}_{m_1=0}a_{m_1,s_1+s_2-m_1}(t)\Phi_{m_1}(x)\Phi_{s_1+s_2-m_1}(q),
\label{4}
\end{eqnarray}
where $ \Phi_{m}(z) $ are known wave functions of a harmonic oscillator in a state with a quantum number $ m $. In the case under consideration, $ s_1, s_2 $ are the quantum numbers of the initial states, respectively, for the charged mechanical resonator (charged target) and the electromagnetic field. As shown in \cite{Makarov_SREP_2018}, in equation (\ref{4}) a law of conservation of quantum numbers applies, $ s_1 + s_2 = m_1 + m_2 $, therefore in (\ref{4}) $ m_2 $ is replaced by $ m_2 = s_1 + s_2-m_1 $. The coefficient $ a_{m_1, s_1 + s_2-m_1}(t) $ is determined by the equation
\begin{eqnarray}
a_{m_1, m_2}(t)=\sum^{s_1+s_2}_{n=0}A^{s_1,s_2}_{n,s_1+s_2-n}A^{*{m_1,m_2}}_{n,s_1+s_2-n}e^{-i{\delta n}t},
\label{5}
\end{eqnarray}
where
\begin{eqnarray}
\delta = \beta \sqrt{\Omega}\left( \alpha + \epsilon \right), ~~\alpha = \sqrt{\frac{\omega}{\Omega}}\left(\epsilon \mp \sqrt{\epsilon^2+1} \right),~~ \epsilon =\frac{\Omega^2-\omega^2 +\beta^2 \Omega}{2\beta \sqrt{\Omega}\omega}.
\label{6}
\end{eqnarray}
In equation (\ref{6}) for $ \alpha $ with $ \epsilon>0 $, the upper sign, must be used; with $ \epsilon<0 $, the lower sign applies. As shown in \cite{Makarov_SREP_2018} with $ \beta \ll 1 $ coefficient $ \alpha \in (-1,1) $ and coefficient
\begin{eqnarray}
A^{s_1,s_2}_{n,m}=\frac{\alpha^{s_{2}+m}\sqrt{n!m!}(-1)^{s_2+m}i^{s_1-n}}{(1+\alpha^2)^{\frac{s_1+s_2}{2}}\sqrt{s_{1}!s_{2}!}}P^{(-(1+s_1+s_2), n-s_2)}_{m}\left(-\frac{2+\alpha^2}{\alpha^2} \right),
\label{7}
\end{eqnarray}
where $ P^{(b, c)}_{a}(x) $ is the Jacobi polynomial.
Thus, the solution of the Schrodinger equation for the problem in question has been found; next, we turn to the results of finding the average number of phonons $ n_t $ in such a system.
\section{Results}
The average number of quantum states (phonons) of a charged resonator will be
\begin{eqnarray}
n_t=\sum^{s_1+s_2}_{m_1=0}m_1|a_{m_1, s_1+s_2-m_1}(t)|^2.
\label{7}
\end{eqnarray}
Indeed, the probability amplitude to detect a charged resonator in the state $ |m_1\rangle $ will be $ a_{m_1} = \langle m_1|\Psi(x, q, t)\rangle $, and from (\ref{4}) it is easy to see that $ a_{m_1} =a_{m_1, s_1 + s_2-m_1}(t) $. Equation (\ref{7}) is the basis for further analysis. It should be added that the resulting equation for $ n_t $ depends on two parameters, $ \alpha $ with $ \delta t $, and $ n_t(\alpha,\delta t) = n_t (-\alpha,\delta t) $. In other words, the average number of phonons is an even function with respect to $ \alpha $, which allows us to use only $ \alpha \in (0,1) $ for further analysis.
We are interested in the maximum cooling of the charged resonator. As mentioned in the introduction, the number of photons in the resonator, under modern conditions, can be very small and in theoretical calculations one can choose a value tending to zero. This means that in our calculations we choose $ s_2 = 0 $, and $ s_1 $ can be changed arbitrarily, depending on the initial conditions of the problem. If we use the properties of the Jacobi polynomials $ P ^ {(b, c)} _ {a} (x) $, then equation (\ref {7}), for $ s_2 = 0 $, can be calculated analytically, leading to a simple equation
\begin{eqnarray}
n_t=s_1 \frac{1+\alpha^4+2\alpha^2\cos(\delta t)}{\left( 1+\alpha^2 \right)^2 }.
\label{8}
\end{eqnarray}
Figure \ref{fig_1} shows the graph of the function $ n_t/s_1 $ depending on the two parameters of the system in question: $ \alpha,\delta t $. It can be seen from figure \ref{fig_1} that for $ \alpha = 1$ and $\delta t=\pi $, the number of phonons $ n_t = 0 $; this result is also easily shown analytically. As an example, we present the results of calculations $n_t = n_t (\alpha, \delta t)$ in figure \ref{fig_2}, with $ s_2 = 1 $, and $ s_1 = (2; 5; 7; 10) $.
\begin{figure}[!h]
\center{\includegraphics[angle=0,width=0.7\textwidth, keepaspectratio]{fig_1}}
\caption[fig_1]{3D graph of the function $ n_t /s_1 $ as a function of the parameters $ \alpha, \delta t $}
\label{fig_1}
\end{figure}
\begin{figure}[!h]
\begin{minipage}[h]{0.49\linewidth}
\center{\includegraphics[angle=0,width=1\textwidth, keepaspectratio]{fig_2_a}} \\
\end{minipage}
\hfill
\begin{minipage}[h]{0.49\linewidth}
\center{\includegraphics[angle=0,width=1\textwidth, keepaspectratio]{fig_2_b}} \\
\end{minipage}
\hfill
\begin{minipage}[h]{0.49\linewidth}
\center{\includegraphics[angle=0,width=1\textwidth, keepaspectratio]{fig_2_c}} \\
\end{minipage}
\hfill
\begin{minipage}[h]{0.49\linewidth}
\center{\includegraphics[angle=0,width=1\textwidth, keepaspectratio]{fig_2_d}} \\
\end{minipage}
\caption[fig_2]{The dependence is $ n_t = n_t (\delta t) $ when $ s_2 = 1 $, and $ s_1 = (2; 5; 7; 10) $ (on charts it is bottom-up) and a fixed value of $ \alpha $. The dependence on $ s_1 $ on the presented graphs is such that when $ n_t (\delta t = 0) = s_1 $}
\label{fig_2}
\end{figure}
Figure \ref{fig_2} shows that the minimum value of $ n_t $ occurs as well as when $ s_2 = 0 $, i.e. with $ \alpha = 1, \delta t = \pi $, but $ min (n_t) = 1 $. Thus, the best cooling of a charged resonator occurs when $ s_2 = 0 $, and $ \alpha = 1, \delta t = \pi $.
Our data show that it is possible to cool the charged resonator to $ n_t = 0 $ (albeit without taking into account various losses) for certain system parameters. We also find the condition for the occurrence of such cooling. As shown, the maximum cooling occurs when $ s_2 = 0 $ and $ \alpha = 1 $; therefore, in further analysis, we assume that the cooling is the most intensive at $ \alpha \approx 1 $. For $ \beta \ll 1 $ and $ \alpha \approx 1 $, based on equation (\ref{6}), the condition $ | \epsilon | \ll 1 $ must be satisfied, and $ \Omega \approx \omega $, and also $ \delta \approx \beta \sqrt{\omega} $. In this case, $ \epsilon = \frac{\Delta \omega}{\beta \sqrt {\omega}} $, where $ \Delta \omega = \Omega- \omega $. As a result, we obtain $ \epsilon \approx \frac{\Delta \omega}{\delta} \ll 1 $, or more simply $ \Delta \omega \ll \delta $.
As a result, the maximum cooling of the resonator will be at $ s_2 = 0 $ when the condition $ \Delta \omega \ll \delta $ is satisfied.
\section{Conclusion}
The results show that there is a theoretical possibility of cooling a charged resonator to the zero quantum state $ n_t = 0 $. It should be added that our results did not take into account any external influences on the system under consideration, which of course would lead to different results. Despite this, the theoretical result reported on here is interesting and suggests that this line of research should be pursued, both theoretically and experimentally. The results obtained in the work have a surprisingly simple analytical form, which is especially true for equation (\ref{8}). The fact that the results were obtained in an analytical form allowed us to single out the fundamentals of the parameters of the system under consideration at which the maximum cooling of the charged resonator occurs. Physically, this means that with these optimal parameters of the system in question, energy exchange between the electromagnetic field and the charged resonator occurs with greatest efficiency, to the fact that the vibrational energy of a charged resonator transforms into the energy of an electromagnetic field, depending on the interaction time. Indeed, it is easy to draw such a conclusion, given that there is a preservation of quantum numbers $ s_1 + s_2 = m_1 + m_2 $ (see above or \cite{Makarov_SREP_2018}) and if we average this expression, we get $ s_1 + s_2 = n_t(\alpha, \delta t) + n_{aux} (\alpha, \delta t) $, where $ n_t $ is the average number of phonons in the system, and $ n_{aux} $ is the average number of photons in the system. Since $ s_1 + s_2 = const $, and $ n_t(\alpha, \delta t) $ has a dependence which is shown in equation (\ref{8}), then reducing the number of phonons by a certain amount increases the number of photons by the same value i.e. there is an exchange of energy. \\
{\bf Funding Information.} Grant of the President of the Russian Federation (No. MK-6289.2018.2)
\section*{References}
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{"url":"https:\/\/scholars.fiu.edu\/display\/pub214210","text":"## Extraction of $t$-slopes from experimental $\\gamma p\\rightarrow K^+\\Lambda$ and $\\gamma p\\rightarrow K^+\\Sigma_0$ cross section data Preprint\n\nFreese, Adam, Puentes, Daniel, Adhikari, Shankar et al. (2016). Extraction of $t$-slopes from experimental $\\gamma p\\rightarrow K^+\\Lambda$ and $\\gamma p\\rightarrow K^+\\Sigma_0$ cross section data .","date":"2022-05-29 09:27:56","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.8444608449935913, \"perplexity\": 6573.086817479261}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652663048462.97\/warc\/CC-MAIN-20220529072915-20220529102915-00742.warc.gz\"}"} | null | null |
{"url":"https:\/\/puzzling.stackexchange.com\/questions\/77898\/to-solve-a-simple-heist","text":"# To solve a simple heist\n\nIt happened by chance really. I was walking out from the bank when a man ran into me while rushing out. During our collision a piece of paper fell from his pocket.\n\nI quickly picked it up and tried to call out to him, but he had vanished. I then looked at the note and was instantly intrigued.\n\nIt said:\n\nBe where?\n\nI looked further down the note and saw something strange. It had a piece of music taped to it:\n\nBelow in a scribble it said:\n\nI'm not sure what all this meant. So I continued to look at the note.\n\nI turned the paper over and my heart sank. It was a blueprint of the very building I was in!\n\nBelow the blueprint was an object and strange text circled.\n\nWere these instructions on when to rob this bank?!\n\nI had to solve the puzzle so I could go to the police. But how to solve it?\n\nWhat time was the heist?\n\nWhat were they going to steal?\n\nHint 0:\n\nThe : in the music staff normally means repeat, it does not.\n\nThey were stealing\n\na diamond at 23:25.\n\nAABBAABA AABBAABB : AABBAABA AABBABAB\n\nAlpha Centauri A & B are a binary star system, hinting that we need to use binary to translate this.\n\nSo, convert to binary (A=0, B=1):\n00110010 00110011 : 00110010 00110101\n\n\u2022 Oh god, in my language we don't call that note B but H. That's not fair :) Dec 29, 2018 at 19:13","date":"2022-05-17 08:22:03","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.41728290915489197, \"perplexity\": 2630.088668286854}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2022-21\/segments\/1652662517018.29\/warc\/CC-MAIN-20220517063528-20220517093528-00752.warc.gz\"}"} | null | null |
{"url":"http:\/\/northtomom.blogspot.com\/2011\/","text":"## Wednesday, December 7, 2011\n\n### Let the Kids be Glad to Be Gay\n\nIn a recent article in the Toronto Star, I came across this breathtaking statement:\n\nBut there is a time and place for everything, said Rondo Thomas, of the Toronto-based Evangelical Association, but there is no \u201ctime and place\u201d in an 8-year-old\u2019s mind to try to make them conceptualize something beyond \u201ctying their shoes.\u201d\n\nReally. So this is what these religious \"leaders\" think of the intellectual capacities of eight-year-olds. It makes about as much sense as their claim that the McGuinty government's new anti-bullying legislation is tantamount to pro-gay education. Logic is clearly not Thomas's\u2014or his comrade in fanaticism, Charles McVety's\u2014strong suit. (But then these are the same kinds of people who believe that pre-marital sex leads to dancing.)\n\nBut even if it were true\u2014even if anti-bullying education of necessity raised the issue of homophobia (the word is mentioned in the new legislation) and other fears of difference, even it if raised it with eight-year-olds, or five-year-olds, for that matter\u2014so what? When you have teenagers like Jamie Hubley and countless others still taking their own lives after being bullied for being gay, clearly anti-bullying education must tackle homophobia. Kids are never too young to \"conceptualize\" hate. Or love. Which concept comes to predominate in their thinking about themselves, and others who may be different from them, depends to a large degree on the way in which they are raised and educated.\n\nSo yes, bring on the anti-bullying, pro-gay legislation. In fact, I urge schools to hold a special pro-LGBT assembly every year, for kids from grades kindergarten to 12. I humbly suggest that the theme song for such an assembly be this classic by Tom Robinson:\n\n(See also Separate Schools for LGBT Kids? and\u00a0 Breeding Tolerance: Is it Possible.)\n\n## Friday, November 11, 2011\n\n### Life in 21st-Century Classrooms: the Agenda\n\nI recently read a remarkable book entitled Life in Classrooms, first published in 1968, and reissued by Teachers College Press in 1990. Its author, Philip Jackson, was one of the first educational researchers to apply an ethnographic approach\u2014borrowed from anthropology and popularized through prominent studies of primates\u2014to the phenomena of schools and classrooms. The book is a methodological mishmash, but at its core are Jackson's reports on \"field visits\" he conducted over a period of two years to several elementary school classrooms in the University of Chicago Laboratory School.The book is full of astute observations about classroom life, most of which still apply today. I was struck, for instance, by an analogy that Jackson draws in the first chapter:\n\nThere is an important fact about a student's life that teachers and parents often prefer not to talk about . . . . This is the fact that young people have to be in school, whether they want to be or not. In this regard students have something in common with the members of two other of our social institutions that have involuntary attendance: prisons and mental hospitals.\n\nElaborating, Jackson writes:\n\n[T]he school child, like the incarcerated adult, is, in a sense a prisoner. He too must come to grips with the inevitability of his experience. He too must develop strategies for dealing with the conflict that frequently arises between his natural desires and interests on the one hand and institutional expectations on the other.\nJackson proceeds to discuss in some detail both the institutional exigencies of school, and the strategies that children come up with to cope with them. In his observations and interpretations of what he sees\u2014especially his reflections on classroom management, children's and teachers' attitudes towards school, and the power relations operating at the micro level in schools\u2014he anticipates Foucauldian studies of institutional life that began to emerge in humanities disciplines in the late seventies and early eighties.\n\nFor this reason\u2014or simply for the rich, troubling portrait of classroom life that Jackson offers\u2014I believe the book should be required reading for teachers' college students. But for the present purpose, what interests me is an image Jackson introduces in the first chapter and which he discusses in the introduction to the 1990 reissue of the book:\n\nI noted . . .how students propped their arms in the air by placing their left hands just above their right elbows when signaling the teacher's attention and I realized that that familiar posture was caused by the fact that the arm usually had to be held high for several seconds before the teacher noticed it . . . . Being heavy, the raised arm required support. The propped arm . . . was a reasonable response to the crowded conditions of classroom life. To my newly awakened interest in such matters, it stood as a symbol of those conditions.\nWhat's interesting about this passage, and the symbol of the propped arm, is how relevant it still is. In 1968, funding levels for education in both the US and Canada were much higher than they are today, yet large class sizes were the norm. Despite attempts by some provincial governments (Ontario, for example) to set caps on class size in primary grades, \"crowded conditions\" still obtain in most schools. My daughters' grade 7 class has 32 kids: arms are still being propped.\n\nI would argue, however, that overcrowding is not the most significant issue facing our schools today. It is now known, for instance, that small class size does not guarantee better outcomes for individual students. But the image of the propped arm got me thinking: what is its modern-day equivalent? What action or object epitomizes \"life in classrooms\" in the 21st century? When I thought about this question, one object immediately sprung to mind: the school agenda.\n\nMy daughters were issued their first agendas in Grade 2. The primary grade agendas, which cost five dollars a piece, were colourful weekly school calendars in ringed notebook format, containing all manner of information and trivia, as well as space for jotting down daily homework, an area for \"parent-teacher\" communication, and the all-important parent initial box. At first glance they looked fairly innocuous, and the girls were happy to have them. But my husband and I were surprised that our local school board, which issued the agendas, felt that seven-year-olds would need them. How much homework, how many deadlines or appointments, we wondered, would seven-year-olds have to keep track of? What issues would arise in Grade 2 that would require daily monitoring by parents (and thus daily initialing) or regular parent-teacher communication?\n\nThe reality, of course, is that second-graders do not need agendas. Neither do seventh-graders or even twelfth-graders. After all, most people over the age of 30 managed to get through their school years without them. Looked at another way, however, one could say that if today's school children require agendas, it is because the need for them has been created by the conditions of modern schooling and by the assumptions that underlie and give rise to these conditions. What are some of these assumptions? One is that children require and benefit from homework from early grades through high school, and that when it comes to schoolwork, quantity is more important than quality. (The abundance of evidence to the contrary has done little to shake this particular assumption.) Another is that children must be taught \"time management\" skills, the deeper assumption here being a blurring of the once distinct concepts of \"education\" and \"training,\" and the consequent belief that education should concern itself with preparing children to function in the corporate world from which such phrases such as \"time management\" hail. A third preconception driving the \"need\" for agendas is that constant monitoring and surveillance of the school-aged child's performance, by both parent and teacher, is necessary and desirable.\n\nTaken together, these assumptions give rise to the conditions that are symbolized by the agenda: not overcrowded classrooms, but overcrowded, over-scheduled, over-burdened young lives. The kids leading these lives are viewed\u00a0 less as children than as pre-adults who must be moulded into full-fledged adults capable of functioning in the \"real\" (read corporate) world.\n\nIt may seem as if I am (once again!) engaging in theoretical overreaching, but incidents that have occurred during the current school year\u2014my daughters' first in middle school\u2014lead me to think otherwise. For instance: the girls' math teacher told the kids on the first day of class that forgetting to bring their agendas to class was a detentionable offense, as significant as not completing homework. The message this warning was intended to send is that the para-curriculum or what Jackson calls the \"hidden curriculum\" (though these days it is not particularly well hidden)\u2014in other words behavioural or character lessons regarding organization, time-management, etc.\u2014are as important as the actual lessons being taught, in this case lessons about math.\n\nAnother incident involved an \"agenda check\" by the girls' homeroom teacher. Since parents are no longer required to initial agendas daily, this teacher decided that she would take a look at the kids' agendas to see if they were copying down homework reliably and legibly, as well as noting future assignments, important dates, etc. While flipping through J's agenda, the teacher noticed many doodles. She chided J for doodling in her agenda and told her to stop. J was mildly upset by this, as she is unused to being reprimanded by a teacher. (A year ago she would have been very upset, but middle school is teaching her to grow a thicker skin.) But more than anything, she was puzzled. \"Why can't I doodle in my agenda?\" she asked. \"Who owns my agenda?\"\n\nThe question of who owns the school-aged child's \"agenda\" is, I believe, worthy of further reflection by parents and educators alike.\n\n## Thursday, October 13, 2011\n\n### A Grade 7 Math Question\n\nThe other day my daughters were assigned a perplexing math question for homework. It was a question straight out of their Grade 7 math textbook, which is the French (immersion) version of Math Makes Sense 7. Math Makes Sense is a Trillium-approved, \"constructivist-lite\" math textbook series published by Pearson Education Canada, and widely used across the province of Ontario. Here is the question:\nUse a place value chart. Explain why you add one or more zeros to the end of a number that you multiply by 10, by 100, or by 1000. [translation mine]\nThe girls thought about it for a while. They understood that adding the zeros had something to do with the fact that you move the decimal place to the right when you multiply by 10, 100, or 1000,\u00a0 but they got stuck on that word \"why.\" Why do you move the decimal, thereby adding the zeros?\n\nNow, this type of question is not uncommon in the Math Make Sense series. Proponents of what is variously called \"discovery,\" \"constructivist\" or \"reform\" math would say it exemplifies the kind of challenging question that leads children into authentic mathematical \"discovery.\" But does it?\n\nThe problem with this question, and others of its ilk that we have encountered over the years with this series (and with Nelson Mathematics \u2014the \"competition\" to Math Makes Sense), is that the type of analytical reasoning needed\u00a0 to answer it adequately is not commonly taught in the contemporary math classroom. What the writers of the question are looking for is a kind of conceptual grasping, written in English. For instance, here is the answer provided in the back of the book:\nFor example: When I multiply a number by 10, it becomes 10 times bigger. In a place value chart, each digit of the number moves one position to the left. The digit 0 occupies the last position.\u00a0 [translation mine]\nFor a series that prides itself on furnishing teachers and students alike with a conceptual approach to mathematics, this answer is quite curious. It substitutes one mechanical trick\u2014adding zeros\u2014for another: moving the decimal place. But both tricks are answers to a \"how\" question, and not to the \"why\" question\u00a0 posed.\n\nThe inconvenient fact of the matter is that it is nearly impossible to answer the question in a way that is mathematically precise using English alone. A mathematically correct answer requires a mixture of notation (with which kids at this level are mostly unfamiliar) and English. In fact, it requires a proof like this one:\n\nA decimal number is written as $$a_k \\ldots a_3 a_2 a_1 a_0$$ (for some $$k$$)\nand represents the value $\\sum_{i\\ge 0}^k a_i 10^i.$\n\nSo $$10^d$$ is represented by a 1 followed by $$d$$ 0's.\n\nGiven a decimal number $$x$$ represented by $$a_k \\ldots a_3 a_2 a_1 a_0$$,\nwhat does the representation of $$10^dx$$ look like?\n\n$10^d x = 10^d (\\sum_{i \\ge 0}^k a_i 10^i) = \\sum_{i \\ge 0}^k a_i 10^{i+d} = (\\sum_{i \\ge d}^{k+d} a_{i-d}10^i) + \\sum_{0 \\le i < d} 0 \\cdot 10^i$\n\nSo $$10^dx$$ has the representation $$a_k \\ldots a_3 a_2 a_1 a_0$$\nfollowed by $$d$$ 0's,\nas we were required to show.*\n\nShow me the Grade 7 student who can \"discover\" that.\n\n*Proof courtesy of Prabhakar Ragde, professor of Computer Science at the University of Waterloo.\n\n## Friday, September 16, 2011\n\n### Summertime, and the reading is easy...\n\nWhen I was a child I had a reputation as a bookworm. I wore the label proudly, since at the time I was unaware of its negative connotations. I remember reading book after book around the pool during March break in Florida, stopping only to dip into the water when I got too hot. During the summer, I read constantly simply because I had the time: my parents, whose parenting philosophy could be summed up by the phrase \"benign neglect,\" did not feel the need to structure my summers.\n\nMy own parenting philosophy cannot be summed up as \"benign neglect.\" Like most parents of my generation, I constantly fight the urge to rein in my daughters' freedom and micromanage their lives. But there is one way in which I parent like my mother and father: I do not structure my kids' time during the summer. I eschew the role of \"camp counselor\" both at the cottage and in the city; I do not see it as my job, and I've found that when left to their own devices, my daughters come up with imaginative and engaging activities to fill their time.\n\nOne such activity is reading for pleasure. I don't know how many books J and E read this past summer, but I do know that I was constantly having to replenish their supply. I frequently caught sight of the two of them lounging on the sofa, deeply immersed in their books, and though I sometimes felt the urge to tell them to go outside and get some fresh air, I resisted. They would often make their way outside at some point anyway, but even if they hadn't, I'm not sure I could have justified interrupting their reading. Here's why: I knew that when school started in September, their reading for pleasure would come to a grinding, depressing halt.\n\nWhen school is in session, my daughters, like many school-aged kids, have very little time to read. Regular homework, extra-curricular activities, and socializing take up most of their free time. When they do find themselves with a spare moment, J and E\u2014who, like most kids, experience the school-year schedule as a grind\u2014are more likely to put on a DVD and collapse onto the sofa than to pick up a book.\n\nA more complicated and insidious impediment to reading for pleasure during the school year has to do with how reading is handled as an academic subject. In Ontario, the reading curriculum, as set forth in documents available on the Ministry of Education website, focuses on\ndeveloping the knowledge and skills that will enable students to become effective readers. An effective reader is one who not only grasps the ideas communicated in a text but is able to apply them in new contexts.\nNow, \"effective readers\" and people who read for pleasure are not mutually exclusive categories. And, to be fair, the curriculum document does acknowledge the importance of nurturing a love of reading:\n\nA well-balanced reading program will provide students with opportunities to read for the pleasure of discovering interesting information as well as for the pleasure of self-discovery . . . and for the sheer fun of it.\n\nBut in reality, the Ontario language curriculum and the pedagogies that support it are not particularly conducive to fun or pleasure. Both seem heavily informed by research into the mechanics of reading, drawn from cognitive science and psycholinguistics, as well as by myriad constructivist and reader response theories borrowed from disciplines such as sociocultural psychology and literary studies. The result is an emphasis on the process of reading, and the \"metacognitive\" strategies that children and adults use when learning to read or when actually reading.\n\nOne such strategy involves the making of connections. In their influential book, Mosaic of Thought, reading researchers Susan Zimmerman and Ellin Keene, (synthesizing insights from transactional\/reader response theory and cognitive science) outline three principal types of connections that competent readers make: \"text to self,\" \"text to world,\" and \"text to text.\" Other theorists\u2014such as Richard Anderson and P. David Pearson, in their seminal essay on Schema Theory\u2014have emphasized the importance of prior knowledge to the reading process. According to Schema Theory, competent readers activate their prior knowledge (organized into schemata) to draw inferences, make predictions or employ \"fix-up\" strategies when they read. These activities and strategies allow readers to assimilate unfamiliar material by comparing and integrating it with what they already know, thereby enabling comprehension and learning.\n\nIt is important to note that these theories of reading\u2014and the many others which inform reading curricula across North America*\u2014are essentially descriptive in nature: that is, they attempt to describe what actually happens in the minds (or more recently, in the brains) of readers while they read. But during the circuitous journey from university to teacher's college to classroom, descriptive theories invariably devolve into prescriptive practices. So, for example, educators deduce (not entirely logically) that if effective readers make \"text to text\" or \"text to self\" connections or use inference and prediction to aid in comprehension, then children should be taught to read in this manner. The resultant pedagogy can take some unexpected and occasionally counter-productive forms.\n\nSo in E's case, this emphasis on the supposed process by which efficient readers comprehend what they're reading backfired. The teacher's single-minded focus on what she referred to as \"metacognition\" actually prevented her from ascertaining who could read and comprehend simple chapter books and who could not. (According to my daughters, the outgoing kids would babble on about how the book reminded them of this and that, and would be rewarded for doing so, no matter how outlandish their answers.) Interestingly, E's teachers in the previous and following years chose not to use this method to assess reading ability; both recognized that E was a strong reader by evaluating her oral and written book reviews, and by asking her less scripted questions about the books she was reading.\n\nFortunately, these awkward moments with the Grade 2 teacher did not significantly affect E's attitude towards books or reading. But what worries me in retrospect is that they could have. They could easily have shaken E's confidence in her reading ability, thereby turning her off reading altogether. As it is, she learned that reading and discussing books in school (as opposed to at home) was not a pleasurable experience.\n\nUnfortunately, that impression persisted and was compounded by other aspects of the reading curriculum. In the later elementary years, for instance, literature circles became one of the main vehicles by which the reading portion of the language curriculum was fulfilled. Harvey Daniels, in his book on the topic, describes literature circles as \"a form of independent reading, structured as collaborative small groups, and guided by reader response principles in light of current comprehension research.\" In other words, a bit like a book club for kids, which sounds appealing. Indeed, it's difficult to object to the idea of students getting together in groups to discuss books; however, it seems that in the case of literature circles, somewhere between concept and execution, a vital ingredient got lost: fun.\n\nIn reality, literature circles are not kid versions of book clubs. Unlike adult book clubs, they are not self-organized. Most often, it is the teacher who chooses the books and the teacher who decides what types of activities the group will engage in. Typical (rotating) roles in literature circles include: \"Discussion Director,\" \"Passage Finder,\" \"Illustrator,\" \"Connector,\" \"Vocabulary Enricher,\" \"Investigator,\" and \"Summarizer.\" There is nothing particularly objectionable about any one of these roles taken individually, but I wonder how many adults would join a book club in which these sorts of activities were required. (I certainly wouldn't: I can't draw, for one thing!) It should come as no surprise, then, that kids are not enamoured of them either. Both of my daughters love to read, but neither of them enjoys literature circles. Too little choice, they say, and too much busy work, often sent home as homework.\n\nBut critiquing current practice is easy, especially for a parent like me; I don't need to worry about fulfilling curriculum requirements or engaging children in a classroom setting. The question that needs to be asked\u2014that I need to ask myself\u2014is, what would a reading program that strove to inculcate a love of reading look like? The conclusions I've come to as a result of thinking about this question are not easy to articulate. But my sense is that the current curriculum, while well-intentioned, focuses too much on notions of \"efficiency,\" \"mastery,\" and \"competence,\" and too little on concepts such as \"enjoyment\" or \"pleasure.\"\n\nWhat better reason to encourage or at least allow for pleasure reading during the school year? Doing so would not require a wholesale overhaul of the curriculum. Teachers could keep the literature circles, for instance, but make them truly student-directed. They could let students choose the books and determine the way in which the circle is organized. Let the children read and discuss in any way they see fit. But, most important, just let them read. Give students unfettered access to the school library, and set aside blocks of time daily for independent, no-strings-attached reading. In other words, import a bit of lazy summer reading into the school year. Perhaps in this way, educators\u2014with the help of supportive parents\u2014can begin to bridge the troubling chasm between reading for pleasure and reading for school .\n\n*For an overview of of these theories, see Lenses on Reading: An Introduction to Theories and Models, by Diane H. Tracey and Lesley Mandel Morrow.\n\n## Tuesday, July 26, 2011\n\nIt's mid-summer, we've been to the cottage and back, and my daughters have put Grade 6 graduation behind them. I, too, have tried to forget about it\u2014unsuccessfully. I've been brooding about the ceremony (held over four weeks ago), ruminating on aspects that gave me pause, caused me to wince or\u2014worse\u2014made me angry. I've hesitated to write about it here, but images of the event have persisted in my heat-addled brain, refusing to cede ground to more seasonally-appropriate thoughts. So here it is: my admittedly jaundiced take on one particular Grade 6 graduation and awards ceremony.\n\nIt began inauspiciously. Chairs were set up on the leafy lawn of the handsome, 90-year-old public school which my daughters, J and E, have attended since Grade 1. The setup looked pretty, as it always does when the school's Parent Association puts its collective mind (and applies its considerable financial muscle) to something. But just as parents, nannies, aunts, uncles and grandparents were piling out of their SUVs, it started to rain. Staff and parent organizers held out for several long minutes while freshly blow-dried hair wilted, and suits broke out in rain splotches. Finally, the principal called it, and guests were asked to bring their own chairs into the stifling, non-decorated gym.\n\nBut in the two and a half-hour ceremony that followed, physical discomfort on the part of guests was the least of the problems.\n\nThen the dispensing of awards began. There were prizes\u2014small wooden plaques with the recipient's name engraved on them\u2014in physical education, art, music, and French; there were also spirit, character and leadership awards (but, interestingly, given the emphasis on STEM in the TDSB, no science or math awards). I suspect I was not the only parent made uncomfortable by the way the awards were allocated and bestowed. In a misguided effort to be inclusive, several students were chosen to receive each award. So, for instance, the art prize was handed out to three students, the phys. ed. prize to four, and so on. While possibly a good idea in theory,* the result was that at least 70 per cent of the entire graduating class (of approximately 125 kids) received awards. That left a minority of kids who did not, which is far worse for the award-less than if only a few kids had been recognized. More troubling, regardless of their ostensible purpose, the awards seemed to celebrate the same types of kids. Art, music, and physical education plaques went to kids who were competent in those subjects, but who also\u2014perhaps more importantly\u2014demonstrated concomitant \"leadership qualities.\" In other words, with the exception of the honor roll certificates and a prize for highest academic achievement, the awards were in fact \"spirit\" awards\u2014validating kids for displaying the kind of meaningless \"school spirit\" I have critiqued elsewhere. So, the quiet, introverted, well-behaved kids, the ones who by default or by choice fall under the radar, were the ones who received nothing.\n\nGiven that I have twins in the same class who have completely different personalities, I feel I am uniquely positioned to understand the ramifications of such a system. Both my daughters made honour roll, but J also received an art award. Both she and her sister love art, but E is by far the better artist. She spends a great deal of her spare time creating and studying art, and has educated herself about technical matters not covered in the curriculum, such as shading and colour theory. But J is more outgoing, more obviously enthusiastic and less shy than E. J gets noticed, E does not. J gets the art prize, E does not. E was not upset (at least not overtly), but the irony was not lost on her or her sister. Both instinctively understood that the reward system favours a certain type of personality, irrespective of ability. The allocation of the actual \"character,\" \"leadership\" and \"spirit\" prizes reinforced my daughters' understanding of how the system works. These awards were given out to a specific type of kid: the extrovert who exhibits the requisite level of school-sanctioned enthusiasm\u2014at least outwardly.\n\n*One might ask, if inclusion is in fact the goal, why not go all the way, and reward each child for something he or she has achieved during elementary school (as this school in BC chose to do)? Or do we really believe that there are some children who have achieved nothing worthy of recognition?\n\n## Monday, June 27, 2011\n\n### Real-Life Problems and How to Solve Them: Grad\n\nRecently my daughter, J, has taken to writing an advice column\u2014she calls it Real-life Problems and How to Solve Them\u2014modelled on those she has seen in kids' magazines. She writes both questions and answers, and one particular question\u2014specifically, the answer she composed\u2014caught my eye. It has to do with the numerous, over-the-top events and celebrations taking place this week for her class's grade six graduation. Although J is looking forward to these events, her twin sister, E\u2014who is the kind of introvert that schools routinely overlook and can easily crush\u2014is not. J's advice is clearly directed towards her sister. It is not bad advice.\n\nQ: Everyone at my school is looking forward to the graduation festivities but me. They're always talking about the dresses and shoes they are planning to wear, but I don't even want to go. I know my friends will think I'm crazy, but I really want to just stay at home and read a book. \u2014 Ella, age 11\n\n## Monday, June 20, 2011\n\n### Smells Like School Spirit\n\nWhat the bourgeoisie has installed as its number-one, i.e. as its dominant ideological State apparatus, is the educational apparatus, which has in fact replaced in its functions the previously dominant ideological State apparatus, the Church. Louis Althusser\n\nThe other day, my daughter, E, woke up in a foul mood. She muttered something about it being the worst day of her life, then sullenly took her place at the breakfast table. She said she didn't want to go to school, but when my husband and I asked why, she was reluctant to to tell us. Finally, after some cajoling, she told us the reason: it was pajama day at school. E said the kids had been told to wear their pajamas to show school spirit. \"How does wearing pajamas at school show school spirit?\" she asked. \"And why do we have to show it, anyway?\"\n\nGood questions.\n\nThe official character education program at my daughters' school, Future Aces, is fairly innocuous. According to the program's website, the \"Aces\" part of the name is an acronym for:\n\nA Attitude, Ability, Action, Achieve\nC\nCo-operation, Courage, Confidence\nE\nEmpathy, Example, Education\nAt our school, these character traits or behavioral goals are inculcated by means of monthly assemblies in which students perform sketches or sing songs about the attribute of the month. As character education programs go, it is relatively harmless (especially compared to programs such as PBIS, which Chris Liebig has blogged about over at A Blog About School), but it also seems to have little effect on the kids, who can regularly be seen yawning and squirming during the assemblies.\n\nThere is, however, a parallel, less innocuous character education program in effect at my daughters' school, one that is part of what has been called the \"hidden curriculum.\" It involves regular exhortations to school spirit in the form of specially designated \"spirit days,\" house colour days (in this, our semi-private school has taken a page from private schools) and, yes, pajama days.\n\nThe dictionary lists as one of the many possible meanings of the word \"spirit,\" \"enthusiastic loyalty (school spirit).\" Most people would argue that enthusiastic loyalty to one's school, like loyalty to one's favourite sports team, is not in itself a bad thing. And the truth is, there are aspects of my daughters' school about which one could imagine both kids and parents being enthusiastic. (Its wonderful music program is one of them.) But the enthusiasm being encouraged by spirit days is not a considered enthusiasm; it is not a reasoned response to anything tangible. In fact, what is being exhorted (coerced, some might say) through spirit days is the kind of blind, general enthusiasm that precludes thought, or at least renders it superfluous: my school right or wrong. As such, spirit days are inimical to the school's stated goal of fostering independent, critical thinking. A more cynical person might even argue that spirit days constitute the principal means by which schools carry out their ideological function: in Althusserian terms, such events \"interpellate\" or \"hail\" children who, by responding appropriately\u2014i.e, with appropriate unthinking enthusiasm\u2014aid in their own construction as subjects (in this case, as proper, conformist school-children).\n\nYes, I know, it's only spirit day or pajama day or colour day. It is quite possible\u2014probable, even\u2014 that I am investing these events with too much meaning. But if they have no meaning, serve no deeper purpose, why do schools persist in proclaiming such days on a regular basis?\n\nPerhaps it's time for progressive educators and parents to think about alternatives to spirit days or, rather, to ask themselves what an alternative, more meaningful spirit day might look like. I don't have any definitive answers, but I can conceive of assemblies in which children would be encouraged to articulate reasons for their \"enthusiasm\" for their school, as well as reasons why they might not be enthusiastic. Too often teachers and parents solicit only the pre-conceived, positive responses they want from children, rather than being sincerely interested in hearing their views. An alternative \"school spirit\" would not be so far away in meaning or import from the kind of \"spirit\" that all schools claim to be interested in nourishing: the spirit of free and open inquiry.\n\n## Tuesday, June 14, 2011\n\n### By Our Fruits Our Children Shall Know Us\n\nA slightly different version of this piece was published in the Globe and Mail a couple of years ago. I was reminded of it recently when I bit into a sour, over-sized strawberry.\n\nOne thing has always puzzled me about my kids: They prefer vegetables to fruit. They willingly chow down on green beans, broccoli, asparagus, peas and cauliflower, salads of all types, carrots and cucumbers. But they look askance at the apples, oranges, berries and melons that I doggedly place in front of them. \"It doesn't taste good \" is their constant refrain. To me it tastes ... acceptable.\n\nPerhaps I do understand why my twin daughters don't like fruit: Most of it is tasteless. Vegetables may be tasteless too, but my children's expectations of them are lower and, like most parents, I dress veggies up with vinaigrettes or butter and salt to render them more palatable. But fruit is supposed to taste good as is. As an adult, I'm used to the fact that most of the time it does not. My children, who possess the enhanced taste buds of eight-year-olds, have not yet become accustomed to flavourless berries and melons.\n\nI wonder, then, why did I love fruit so much as a child? My parents were particular about their produce. Every Saturday, they shopped at the local Dominion for basics, but made a separate trip to stand-alone markets to buy fruits and vegetables. Even as a young child, I had a sense of seasonality, passed on from my parents. There were berries in spring and summer, along with pert plums, succulent peaches, and sweet and sour cherries from Ontario. Summer fruit, my parents called these. In the fall, we had bushels of russet and Macintosh apples. In winter, there were navel oranges and tart-sweet, white grapefruit. These were imported, but their quality was second to none because they were in season in the sunny place where they were grown.\n\nI also have fond memories of the gap year I spent in France. There, my palate first cottoned on to the reality that tomatoes are fruit. But it wasn't just tomatoes that blew me away. I remember biting into a plump russet apple, which the French called Reinette du Canada. I found the name amusing, doubly so when I realized that even these so-called Canadian apples tasted better in France. It was the eighties by this time, and I had noticed a decline in produce quality at home. Quite simply, everything tasted better in France. When I tell my husband this, he scoffs, as he does when I reminisce about the fruit I enjoyed as a child. \"Pure nostalgia,\" he says.\n\nIn an attempt to prove him wrong, I surf the Internet where I find evidence of a steady decline in the nutrient content of vegetables and fruits. I discover, for instance, that an apple today contains 55% less iron and 41 % less vitamin A than an apple from fifty years ago (see here and here). I email a professor of food science at the University of Georgia, Robert Shewfelt, who confirms that nutrition and flavour are linked since, for the most part, \"nutrition is optimal and flavour is optimal at the same time.\" So perhaps those bloated, mid-winter strawberries are as bad as they seem \u2014 nutritionally deficient\nand tasteless.\n\nWhat, then, can a parent of fruit-averse children do? According to Michael Pollan, author of In Defense of Food, one of the most subversive things we can do today is to plant a garden. I've always admired my elderly Greek-Canadian neighbour, who plants and harvests an impressive array of produce on her small North Toronto lot, but who knew she was such a radical?\n\nAs spring arrived this year, I began to wonder if I too could become a radical. My husband was skeptical, since I've rarely put trowel to dirt in my life, but as the days grew longer and the planting season approached, I resolved to try. I purchased books with titles such as The Gardener's A-Z Guide to Growing Organic Food and Fruits and Berries for the Home Garden. I took the plunge and planted strawberries, cucumbers, and tomatoes in pots; I dug up some sod and stuck two raspberry plants in the ground. I watered and waited. Summer arrived, along with unprecedented rain; I watered a little less and waited some more. I became disheartened when my raspberry plants inexplicably died.\n\nThen, seemingly out of nowhere, the small prickly beginnings of cucumbers appeared. The twins found a lone red strawberry amidst an abundance of runners and greenery. They shared it and pronounced it sweet! But my elation was short-lived. The strawberries stopped bearing, and the tiny cucumbers grew strangely misshapen, almost gourd-like. By summer's end, only my tomato plants were bearing well, and even they looked bedraggled and sad.\nToday, as I gaze upon what remains of my garden, and peer over at my neighbour's still-lush rows, I admit I'm tempted to throw in the trowel. But I suspect that next spring, hope will trump reality. I will begin my garden anew, spurred on by the thought that, even if it takes several seasons, even if I manage to produce a mere handful of red raspberries, I might just be able to bequeath to my children a memory of redolent, in-season fruit.\n\n## Thursday, June 2, 2011\n\n### Is It A Boy or a Girl?\n\nToday's guest blogger, Prabhakar Ragde, is a professor of Computer Science at the University of Waterloo in Ontario. In the 1990s, he and his wife, quietly and without fanfare, made the decision not to reveal the sexes of their two children. In this post, he reflects upon that decision and its repercussions in light of the unprecedented media frenzy surrounding the so-called \"genderless baby.\"\n\nIs It A Boy or a Girl?\n\nby Prabhakar Ragde\n\nTwitter is my link to the zeitgeist. It's where I learned of the Japanese earthquake and the death of Osama bin Laden. But I also learn about many less momentous events and situations, such as the one described in an article in the Toronto Star about a Toronto couple who weren't announcing the sex of their third child.\n\nThe article went \"viral\", exploding on both the Web and in traditional media, eliciting much ignorant reaction from anonymous readers and only slightly more nuanced expressions of concern from so-called experts. Back in my Twitterverse, some of my tweeps offered their own 140 characters of acerbic comment. I argued back, more confidently than usual, because I had something they didn't: empirical evidence. My wife and I had done the same thing, with the birth of our first child Arju in 1992, and again in 1995 with Zazuki (Zuki), and anyone who knows our teenagers knows how well they have turned out.\n\nWhy would we want a \"genderless baby\"? Well, we didn't, and neither did the Toronto couple. There are three related notions of gender here. The first is biological sex, for which people often use the word \"gender\" as a euphemism. The second is psychological gender, or gender identity \u2014 the sex with which a person self-identifies. The third is the social role assigned to a man or woman, leading to the quote, \"Gender is a social construct\". The correct answer to the question \"What is the baby's gender?\" is probably \"No one knows yet,\" for all babies. But what the question really is asking is \"What is the baby's biological sex?\"\n\nThe asker probably wants to know in order to fit the baby into a social role, and in doing so, change the nature of interaction. Although the asker will probably steadfastly deny that they would treat a boy baby and a girl baby differently, it's not hard to turn up peer-reviewed studies demonstrating otherwise. We cited a few of these in our birth announcement (we couldn't resist the conceit of including a bibliography).\n\nBut the largest influence on our children in the early years was, of course, their parents. We knew their biological sex. We'd grown up in an era when it was unusual for married women to work outside the home, or to keep their last names on being married. We're not self-aware or iron-willed enough to avoid our own gender biases, even if we wanted to completely eliminate them (which it's not clear we should, considering that our children have to live in a gender-biased world). So this was never about affecting the children, except indirectly in the examples we set as parents as they grew up. It was a minor bit of consciousness-raising among our immediate circle of family, friends, and acquaintances. Minor in the grand scheme of things, that is; it loomed fairly large for us at the time, despite the lack of media attention.\n\nFortunately, we have no traditions in this culture of routinely displaying the genitals of newborns, so the only things we needed to do were to avoid dressing our children in all pink or all blue, and avoid using words such as \"he\" or \"she\". It's not hard to do this in writing, especially if one is willing to adopt the singular \"they\" (for which there is historical precedent). Extemporaneous speech is another matter. I managed it by dint of furious concentration and much stammering, and it got easier with practice. If someone asked directly, we would briefly explain our stance, but otherwise we simply never corrected anyone's assumptions, except to spare them embarrassment. And people would make assumptions based on the flimsiest of evidence (even on whether they thought the children's names, which we made up, sounded like they referred to one sex or the other). I can remember four occasions when I slipped up and used a sex-specific pronoun to refer to Arju, but it was because the person I was talking to was using them. Twice I used \"he\", and twice I used \"she\".\n\nMy wife's parents, thousands of miles away, accepted our decision; mine, closer by, did not. Siblings and other family members we were close to were supportive. Among our friends, some were enthusiastic about the idea, and some were dubious, and it was sometimes surprising to us who took which stance. My wife and I are both professors in the same department, and our secretary reported mostly puzzlement among the academic staff. One woman was concerned that Arju would turn out gay (hard to see the logic in that one), and a few said that because they didn't know Arju's sex, they couldn't buy gifts (which we'd asked them not to do in the birth announcement, anyway). When we were out in public, we didn't make a point of bringing the topic up, but sometimes people would ask, and we'd gently explain. We never encountered any hostility; at worst, the subject would be abruptly dropped. More often, we got some clarifying questions, and maybe a nice expression of support.\n\nWe tried to choose sensible, attractive clothing in a range of colours, which meant choosing from both racks in the store, as pinks and pastels were on the girls' side, and other bright colours were on the boys' side. We gravitated towards toys that were not only fun but stimulated creativity and imagination; that meant a wide array, including both dolls and trucks, building blocks, and miniatures for role play both domestic and \"on the job\". (It probably helped that I did the cooking, while my wife mowed the lawn.)\n\nOur university, at the time, topped up salary during maternity leave, but only for thirteen weeks. We had a particular daycare in mind, but we hadn't put in an application before Arju was conceived, as we would have had to do to get a spot at three months. It was nearly a year later when a slot opened up. We'd visited several times in between, to keep up our visibility, and the director of the daycare had been one of those making an incorrect assumption about Arju's sex. So when we handed in the completed registration forms and the first cheque, we had to gently explain why we hadn't corrected her.\n\nWe didn't ask for special treatment, but the director clearly took the lesson to heart, and discussed it with the staff. Years later, my friend L attended a party where she overheard a conversation in which my children came up. One of the participants had been a worker at our daycare; she didn't know that L knew us and would report back. She said that attitudes had changed among the staff as a result of the situation; they thought about possible biases in their actions, and went about their jobs in a more thoughtful, introspective fashion. How long-lasting that effect was, it's impossible to say. But this is one way that progress occurs, through small, local changes.\n\nWhen Zazuki was born in 1995, no one blinked an eye when we said we were doing it again (and this time, we had put in a daycare application before conception!). Arju was by then a delightful, talkative creature, and any fears they might have had, had long since been put to rest. I kept the birth announcements up on my Web page for a while, though as the kids grew up, they seemed like old news, and I took them off. But history has a way of resurfacing.\n\nBack in 2011, my \"been there, done that\" tweets must have been noticed, because a reporter from Postmedia (which owns the National Post) e-mailed me requesting an interview. I refused phone contact but gave a short statement by e-mail. He wanted more detail, and I agreed to answer by e-mail the questions he would have asked over the phone. As a consequence of my being able to compose my responses, the article had fewer distortions than usual. What I hadn't expected were the phone calls from TV networks. I let them e-mail me, and turned them down. Apparently, the Toronto couple who started the latest furor did the same, as the woman explained in a rational and intelligent article written entirely in her own words. And with that, the attention of the world turned elsewhere.\n\nI wrote my answers to the reporter using gender-neutral language to refer to my children (as I have done here) to make a point, even though he had done his research on the Web (looking, perhaps, at their Facebook profile photos, in which it's fairly obvious) and figured out which pronouns he needed to use. His article highlights their sexes in the lede, so you can click through, if you wish, and find out for yourself. But before you do, ask yourself why that particular bit of information is so important. Really, it isn't. The desire to know, on the other hand, that is worth thinking about.\n\n## Monday, May 16, 2011\n\n### The New F Word and Kids\n\nRecently, I watched an episode of TVO's The Agenda with Steve Paikin on my computer. The episode was called \"The New 'F' Word,\" and I had missed it when it first aired in January 2011, despite the fact that a good friend of mine, Charlie Keil, a professor of Cinema Studies at the University of Toronto, was one of Paikin's guests on that occasion. The hour-long show consisted of a discussion of the recent decision by the Canadian Broadcast Standards Council to ban the Dire Straits song \"Money For Nothing\" from Canadian airwaves because of its repeated use of the anti-gay slur \"faggot.\"\n\nI was twenty minutes into the show when my 11-year-old daughter, J, walked into the room. Although she did not stay long, I am pleased that she walked in when she did, and saw the show's title displayed across the screen. The ensuing discussion\u2014which occurred while I paused the show, and which I have transcribed below\u2014served as a unplanned continuation of a conversation that began when she was quite young. (See here.) I'm not a big fan of the phrase \"teachable moment,\" but I do believe this was one.\n\nJ: Hey, what's Charlie doing on there?\n\nMe: He's a guest on the show.\n\nJ: \"The New 'F' Word\"? What's that? Charlie won't even say the old F word. [She's right! This is revealed near the end of the show, starting at 46:25.]\n\nMe: The new F word is \"faggot.\" It's a pejorative term for \"gay.\" Have you heard it?\n\nJ: No. Why are they talking about it?\n\nMe: Because a song containing the word has been banned from being played on the radio.\n\nJ: Good.\n\nMe: But it's complicated. In the song, the word is used satirically. Do you want to listen to the song?\n\nJ: Okay. [I find the the original version of \"Money for Nothing\" on YouTube and play it for her.]\n\nMe: So you see, the person saying the word in the song is a character. He doesn't represent the singer's views.\n\nJ: I still think he shouldn't say it. But I guess it's like in a story when there's a character you're not suppose to like, who says nasty things.\n\nMe: Yes.\n\nJ: Or like in Billy Elliot, when the miners say the old F word because that's the way they would talk in real life. [J and her twin sister have said they do not want to see the stage version of Billy Eliot because of the swearing.]\n\nMe: Sort of. But it's a bit different because the old F word isn't directed at a specific group. The new F word is worse because it targets gays.\n\nJ: Yes, but I still don't want to hear either the old or the new F word.\n\nMe: But do you understand why some people might object to the new F word being banned?\n\nJ: Sort of. But if it upsets people to hear it, maybe it should be banned.\n\nMe: But, J, in art\u2014in songs or stories or plays\u2014you have to be able to critique ideas or words, and to do that you have to be able to say them. When they're used in that context they're not necessarily hurtful.\n\nJ: Maybe you just think they're not hurtful when they're used that way because you're not gay.\n\nMe: Maybe.\n\n## Wednesday, April 27, 2011\n\n### Elections and Kids: Desperate Times\n\nThe first election I remember clearly was the federal election of 1972, in which Robert Stanfield ran as leader of the Progressive Conservatives against Pierre Trudeau. The reason I remember that political race in particular, has to do with my best friend Ann. I adored Ann and her family. Her father was an executive at Coca-Cola Canada, her mother a kindhearted former kindergarten teacher, and their airy suburban side-split (a literal mirror image of ours) overflowed with beautiful, happy children. Ann's parents seemed to be more particular about certain things than mine. For instance, they cared about brands. They drank Coke, never Pepsi (although after Ann's father transferred to Pepsi, this allegiance abruptly switched, something that puzzled me somewhat); they ate Kraft peanut butter, and Kraft macaroni and cheese. Store labels were not acceptable substitutes. Store brands were acceptable to my parents, and I complained about this state of affairs to my mother. She tried to explain that the products themselves were mostly the same, so it didn't really matter, but this explanation struck me as feeble. My parents clearly didn't understand the world the way Ann's family did.\n\nAnother brand Ann's family liked was the Progressive Conservative brand. Pierre Trudeau was anathema to her family, as he was to many of of our neighbours. The sign on Ann's front lawn was oversized\u2014and blue. I spent a lot of time with Ann's family during that election; I spent a lot of time with them in general. I fantasized about being part of their perfect family, although I was fond of my own less perfect version as well. So that June, I practically lived with Ann, which meant I spent a considerable amount of time riding around in her parents' station wagon, as her mother and father carried out their chores. I remember one day in particular when Ann, her younger sister and I were being driven around our neighbourhood by Ann's father. I don't recall how it started, but the three of us, sitting happily seatbelt-less in the back of the car, windows wide open and a warm wind whipping our hair, began to cheer every time we passed a Conservative sign, and boo when we passed a Liberal sign. (In our suburb, NDP signs were conspicuous by their absence.) Ann's father chuckled and smiled at us as we did this, and I remember the whole outing being a lot of fun.\n\nThat evening at the dinner table I told my parents what we had done. My father laughed; he said nothing. It wasn't until years later that I discovered my parents had never voted Conservative in their lives. My mother supported the NDP and my father was a Liberal, who occasionally voted NDP. Yet at the time, they felt no need to tell me this. They seemed unperturbed by my unbridled enthusiasm for Robert Stanfield and the Progressive Conservatives.\n\nFast forward almost 40 years. My 11-yr-old twin daughters are in the car with my husband and me. We're driving around our neighbourhood during a spring election, and my daughters are noting the preponderance of Conservative signs. We pass a Liberal sign and the girls begin to cheer. From that point on, they cheer every time we pass a Liberal sign. (In our suburb-in-the-city, NDP signs are conspicuous by their absence.) Sitting in the front seat listening to them, I realize that we have become Ann's family\u2014a family in which children are initiated at a young age into the world of partisan politics. I begin to think about how this situation came about, especially in light of what I thought were my beliefs about children, politics and indoctrination. (See here, for instance.) For the truth is, in retrospect, I admire my parents' restraint. I respect them for refusing to tell us what to think politically, for allowing us to mature, and figure out for ourselves where we stood on serious issues.\n\nIn the family that my husband and I have created, things are different. At the dinner table one evening not long ago, one of my daughters asked what the Conservative Party of Canada stood for. My husband answered flippantly, \"They stand for the rich getting richer and the poor getting poorer.\" My daughter seemed a bit shocked so I added, \"That's not how they would describe what they stand for,\" and I proceeded to try to explain the CPC in a manner more in keeping with how a supporter might explain it. I talked about how some people believe the role of government is to tax citizens who can afford to be taxed so as to provide services and programs to all people, but especially to those who would not be able to afford such services otherwise. I went on to explain that other politicians believe government should be as small as possible, and that taxes should be low, so people can decide for themselves what to to with their money. That was as much fair-mindedness as I could muster, and I couldn't help but add that the problem with the latter approach is that it disproportionally favours the rich.\n\nSo why have I chosen not to exercise the same degree of restraint during this election that my parents exercised effortlessly throughout every election that occurred during my childhood?\n\nSo perhaps we can be forgiven for being a little more forthright with our children about the current political scene in Canada. The truth is, as a parent, I'm afraid: afraid that should Harper achieve his majority, the Canada I knew as a child\u2014a Canada which my parents and Ann's took for granted, one which poured money into schools, health care, and social programs, and taxed its citizens progressively in order to do so\u2014will cease to exist as my children grow into adults. Perhaps, in other words, desperate times call for desperate parenting.\n\n## Saturday, April 16, 2011\n\n### The Toronto Homework Policy\n\nThis review of the 2008 TDSB homework policy was written last spring as a guest post for Sara Bennett's StopHomework site. (Sara Bennett, along with Nancy Kalish, is the author of The Case Against Homework.) The post can still be found there, but the site is largely inactive, as Sara has moved on to other pursuits. It remains, however, a great resource for research and lively discussions on the topic of homework. My post originally appeared in two parts\u2014because it's long!\u2014but I'm posting it here as one piece, on the third anniversary of the enactment of the TDSB homework policy. As a result of my review, the homework situation at my daughters' school did improve somewhat (see here). But old habits die hard, and much room for improvement remains.\n\nThe Toronto Homework Policy\n: A Parent's Perspective\n\nOn a recent Saturday morning, my 10-year-old daughter emerged from the basement on the verge of tears: \u201cThe temple\u2019s collapsed,\u201d she announced. Though it sounded dire, she was speaking not of an actual building, but of the model of an ancient Greek temple she and a classmate had constructed out of cardboard the previous week. They had piled on the white paint, and the structure had simply buckled under the weight. Later that day I glanced out the window to see my two daughters turning cartwheels on the back lawn while my husband diligently sawed wooden cylinders into pillars for the new temple. It was a brilliant spring day, and soon my husband would finish his task and call my reluctant daughter in out of the sunshine to start rebuilding the temple. What is wrong with this picture?\n\nFrom the perspective of a homework skeptic, many things: arts and crafts busywork, weekend homework, parental involvement. But the main problem is that I live in Toronto, and my children attend public school in a board which in 2008 enacted one of the most progressive, \u201cfamily friendly\u201d homework policies in North America. So what happened?\n\nWhen I read the news in early 2008 that the Toronto District School Board was re-evaluating its homework policy, my heart did a little happy dance. At the time, my twin daughters were in third grade. Although we had not yet experienced homework overload, the prospect of a reformed homework policy thrilled me because the following year my daughters were due to enter mid-elementary French immersion, a program renowned for its heavy workload both inside and outside the classroom. Suddenly there was hope that French immersion would provide a qualitatively (as opposed to quantitatively) different experience for my daughters, with enrichment enabled not by means of extra work, but simply through learning the curriculum in a second language.\n\nThe TDSB\u2014the largest school board in Canada, serving approximately 250,000 students\u2014appeared to have done its homework, so to speak, on homework. Spurred on by parent Frank Bruni and sympathetic Trustee, Josh Matlow, the board reviewed and eventually rewrote its homework policy, approving a new family-friendly version on April 16, 2008. The new policy (available online here) re-defines \u201ceffective\u201d homework, promotes \u201cdifferentiated\u201d assignments and removes punitive consequences for incomplete work. It virtually eliminates homework in the early elementary years, and mandates substantial decreases for all other grades. But perhaps the most progressive feature of the Toronto policy is its recognition of the deleterious effect of homework on family life. It stipulates that homework should not be assigned on scheduled holidays or \u201cdays of significance,\u201d and that \u201ctime spent on homework should be balanced with the importance of personal and family wellness . . . .\u201d\n\nMy excitement back in 2008 was not unfounded: this was a good policy. So why two years later am I complaining about my children's homework?\n\nBefore I attempt to answer this question, I should note that many parents I\u2019ve spoken to have indeed noticed a decrease in their children\u2019s homework. But my experience\u2014and that of other French immersion parents I've consulted\u2014has been that teachers continue to assign homework inconsistent with the new policy. On curriculum night in September 2008, the Grade 4 teacher warned parents to expect a difficult year. She explained that the nature of \u201cmid-immersion\u201d\u2014its compression compared to immersion programs starting in Kindergarten\u2014made it necessary to work the children particularly hard. (There was scant mention of the new homework policy, no hint that the program might have to be adjusted in order to comply with it.)\n\nShe was not kidding. On a nightly basis, students were expected to review copious notes from class, practice spelling words, complete math and grammar sheets, and study for tests (two per week). In addition, there were projects to be completed outside of class. Although my daughters loved learning in French and their grades remained strong, they were unaccustomed to a such a heavy workload. They began to show signs of stress (read, meltdowns) almost immediately. By Christmas, they were proclaiming their hatred for school; I prepared to pull them out of French immersion. After the holidays, homework eased up\u2014marginally, but enough to convince me I would not be irreparably harming my daughters by keeping them in the program.\n\nGrade five was initially better. On curriculum night, the teacher professed her dislike of homework; as a parent herself, she understood how busy today's children are. Yet this teacher is renowned within the school as a kind of project queen. Every year, her students (or their parents) produce extraordinary projects in science and social studies, which are displayed on designated days to the other students and teachers in the school. And sure enough, it was the projects\u2014spaced inconsistently and piled on top of regular homework\u2014that nearly did us in. Three of them were clumped together in the space of five weeks in the spring term when, as my daughter put it, kids have \u201chad it with the torture of school.\u201d To be fair, the teacher allocated class time to the projects, but often project time encroached on core subjects such as math and grammar, so more homework came home in those subjects. Moreover, class time was not allocated to the building of temples or eyeballs or machines; parents were responsible for supplying materials, and were expected to provide space and time at home for their children to complete all of the arts and crafts components. As a result, my daughters had little choice but to spend multiple weekends\u2014including \u201cdays of significance\u201d and holidays, such as Passover, Easter, Mother's Day and Victoria Day\u2014working on various elements of assigned projects.\n\nFrustrated and confused by the contradiction between the new policy and the homework we were experiencing, I decided to do a little investigating. I asked several people\u2014the principal of my daughters' school, the superintendent of our particular school district, and my local school Trustee\u2014a simple question: Is the homework policy a set of voluntary guidelines, or is it binding? The answer, it turns out, is not simple. Howard Goodman, school Trustee for my area, summed up the confusion when he answered: \u201csomewhere in between.\u201d Both he and John Chasty, the area Superintendent, insisted that schools are expected to comply with the new policy, and that responsibility for implementation lies with principals and teachers. However, as Goodman reminded me in an email, the TDSB is \u201ca highly decentralized organization which works hard to be responsive to . . . local conditions.\u201d In other words, the board tolerates a certain latitude in the interpretation of its policies in order to empower schools and teachers to respond flexibly to the needs of students.\n\nI began to wonder whether the TDSB counts French immersion\u2014along with other enrichment programs such as gifted classes\u2014as a local condition necessitating a \u201cliberal\u201d interpretation of the homework policy. Not so, according to Lyn Gaetz, principal of my daughters' school. The new recommendations, Gaetz told me, were well received by teachers at the school. She explained that she meets with the teaching staff yearly to discuss the policy and to monitor its implementation. No program is exempt, but Gaetz did acknowledge the challenges the school has faced reducing homework in French immersion.\n\nMy sense from talking to teaching staff is that most of them\u2014French-immersion teachers included\u2014believe they are complying with the new policy. And returning to the document itself, I see how this belief is enabled by a discernible vagueness of wording. For example, in reference to the early elementary years, the policy notes the \u201cstrong connection between reading to or with elementary children every day . . . and student achievement\u201d and goes on to encourage regular reading at home, among other family activities. One would be hard pressed to object to such a recommendation, but its lack of specificity allows for some bizarre interpretations. The teacher of a third-grader I know seems to have interpreted it as an endorsement of reading logs. As followers of stophomework are well aware, reading logs are a discredited form of homework which often instill in children a loathing rather than a love of reading. Yet so convinced is this teacher of the value of reading logs that she instructs her students to complete them during major holidays, such as Christmas, a demand clearly in conflict with the new policy.\n\nAnother troubling area of vagueness is the section on homework in the later elementary years. Time guidelines for these pivotal grades (3-6) are conspicuous by their absence. And the one directive specified\u2014namely,\u201cHomework may begin to take the form of independent work\u201d\u2014is so vague it barely counts as a directive at all. I suspect it is commonly interpreted to mean projects, since projects are considered a more creative, engaging form of homework than, say, drill work. This may be true, although, as most parents know, many projects are comprised of arts and crafts-type busywork. Even the most educationally valid projects are labour-intensive, especially when they are assigned as group endeavours, which adds an element of scheduling chaos to the mix. And when projects are used as the principle means of covering the curriculum, as they seemed to be for much of the spring term in my daughters' class . . . well, before you know it you have temples collapsing and tearful children rebuilding them in dark basements on brilliant spring afternoons.\n\nWhich leads back to the initial question: what went wrong? Has the Toronto policy failed to achieve true homework reform? One could argue that my experience with French immersion is atypical, and that it renders invalid any answer I might offer to such a question. But one could also reasonably view French immersion as a kind of microcosm of elementary education in Ontario, a system characterized by an over-stuffed curriculum (the phrase \u201cmile wide and inch deep\u201d comes to mind) and an over-reliance on standardized tests as a measure of quality. In French immersion, as elsewhere in the system, homework overload and curriculum are inextricably intertwined. To paraphrase blogger Fred Baumgarten, who has written about this interconnection on his blog Homework Headaches, when you pull at the thread of one, you inevitably catch the other, and the whole overwrought educational fabric threatens to unravel.\n\nBut issues of curriculum are beyond the scope of this post. With respect to the homework policy itself, ambiguous language and inconsistent enforcement notwithstanding, I regard the April 2008 revisions as a huge step in the right direction. I applaud Frank Bruni for instigating them. The TDSB also deserves credit for taking the issue of homework overload seriously enough to review the research and change the policy. However, the last two years have taught me some crucial lessons. Policies\u2014even well-meaning, progressive ones\u2014must be seen as works in progress, in continual need of re-evaluation. More importantly, I have learned that passivity\u2014my own in particular\u2014is part of the problem. A change in practice does not flow seamlessly from a change in policy. It is up to all of us to remain vigilant and advocate for the the ultimate stakeholders in any educational system: children.\n\n## Friday, April 15, 2011\n\n### All-Day Kindergarten\n\nRecently, I wrote a guest post for Activekidsclub.com, a website devoted to promoting active, outdoor play for children. The post concerns all-day kindergarten, a program which the government here in Ontario has committed to providing for all 4- and 5-year-olds in the province by the year 2014. You can check it out here.\n\n## Wednesday, March 23, 2011\n\n### Letter to a science teacher\n\nIf I haven't blogged for a while, it's because my husband and I and the twins have recently taken two trips: one pleasurable, to Quebec for March break, and the other an all too familiar (figurative) journey to school-project-hell and back. I've written about how and why I dislike school projects previously on this blog, so I won't repeat my arguments here. What I do want to do is share a letter I wrote earlier in the school year (November 2010) to my daughters' science teacher, outlining my concerns about a project he assigned. I was worried that this was to be the first of many unreasonable, pedagogically questionable projects, and after last year's experience with project overload, I decided to see if I could nip the problem in the bud by detailing my concerns to the teacher. This is the letter (with minor deletions and changes to protect the innocent)I wrote:\n\nDear Mr. X:\n\nWe are the parents of J and E in your Grade 6 science class. J and E are enjoying science so far, but we do have a few concerns about the previous project and the one new one that has recently been assigned. While the girls were working on the habitat project, a number of issues and problems arose that we would like to share with you in the hope that these problems can be addressed, and possibly resolved in time to make a difference for the second project.\n\nThe scope of the habitat project: We found that the scope of the habitat project was a little too broad. We believe that 11-year-olds do not possess the developmental tools to take a large subject and circumscribe it so that it becomes a workable, doable project. Since our daughters seemed initially to be at a loss regarding how to limit their topic, we directed them to ask you for more details regarding what was expected. They told us that your response to this, and to most questions, was: \u201cwhat do you think?\u201d While we understand the pedagogical goal inherent in this type of response, it was not particularly helpful to our daughters, since what they were looking for was specific guidance on the amount and type of work expected. Like many of the students in your grade six class, our daughters are hard-working, high-achieving kids who have been taught to strive to meet expectations. When the expectations are not clear (and unfortunately, the Rubrics section for this assignment on your blog was blank), they feel disoriented. Since no specific guidance on the number of plants and animals to be covered was provided, we took the liberty of giving our daughters some suggestions. Both girls chose to cover more than 15 animals, and since they also felt (although, again, they were not sure) that they should write several sentences about each animal (along with the required classification), the project took them a very long time to complete at home. The girls spent the better part of two consecutive weekends and multiple evenings working on this project. This made it a very stressful endeavour both for them, and for all of us as a family, as very little non-project activities could be planned for those weekends\/evenings.\n\nTime in class: We do realize that you gave the class . . . time at the computers to work on this project, and we are thankful for that. However, you may not realize that many of your students\u2014our girls included\u2014are not proficient on computers, as they do not use them regularly, either at home or anywhere else. So, for instance, when they research topics at school, they know how to save the information, but they do not know how to print it, at least not in the manner you suggested: i.e., by first copying the information into a Word document. They would need step-by-step guidance on how to do this, in order to learn to do it efficiently, or at all, and they have not received such guidance.\n\nThere are two other reasons why the time given to work on the project in class was not particularly productive for J and E: they cannot type with any degree of speed or fluidity, and they have not been taught how to transfer their work to a memory stick, so that they can bring it home for final formatting, etc. So any actual written work done on the computers at school had to be repeated at home, which was not a productive use of time, and greatly added to the hours required to complete this project.\n\nBibliography: We believe that the requirement of a bibliography at this grade or age, should be accompanied not simply by a reference to [board] guidelines (which are clearly geared towards high school students, and are confusing and out of date, to boot), but by detailed instruction on why one includes a bibliography, how to put together a bibliography, where to find copyright information in a book, or an encyclopedia, etc. The [board] guide, for instance, does not include a single sample entry for a Wikipedia article, despite the fact that Wikipedia is the source children use most frequently for research projects. The girls told us that you commented that students should include authors in their bibliographies. Given this requirement, perhaps during the research phase, you could direct the children, not only to the computer, but also to books, where they will indeed find actual authors, and where the copyright information is more straightforward. In any case, we believe that, since classifications were requested for every plant and animal mentioned in this project, the required bibliography was beyond the ability of most 11-year-old children to complete independently.\n\nI should add, that J and E enjoyed working on their projects, and they both learned a lot. They are also excited about the topics they have chosen for their next project, In general, they are enthusiastic about science class, and we don't want that enthusiasm to dampen because of problems they encountered while working on the first project. That is why we have decided\u2014respectfully, and in the spirit of constructive dialogue\u2014to bring these concerns to your attention.\n\nWe are available to discuss these issues further, either in person, via email or by telephone.\n\nSincerely,\n\nJ and E's parents\n\nI delivered this letter (not without some trepidation) to Mr X's mailbox, and a few days later he called me. I was nervous about speaking to him because on past occasions, teachers with whom I've raised concerns or to whom I have written notes such as this one, have sometimes become defensive, which leads to an unproductive exchange. (I should note that I accept my share of responsibility for failed communications; it is quite possible that the way I spoke or worded my written messages rubbed the teachers the wrong way. This is something I'm continually working on.) But I needn't have been nervous. Mr. X was extremely gracious and receptive to my concerns. He tried to address them all individually, noting, for example, that he thought my daughters' class had been taught how to create proper bibliographies in grades 4 and 5 (which was not the case). He also admitted that he'd made some assumptions about the kids' knowledge of computers and research that he should probably not have made. We ended the conversation amicably, with him assuring me that he would try to do things differently for future projects.\n\nThe good news is, he did change things\u2014a lot! Since I wrote that letter, for instance, there have been no more take-home projects. All work in science is now done in class, and it includes a balance of research and group experiments, such as designing a small electric car! Since I did not want to overload Mr X, I had not even mentioned in my letter another of my concerns: namely, the lack of experiments in what was, after all, a science class. But the letter seemed to propel him to re-think everything, and now the class is completely different. At the beginning of the year, the girls were complaining that in science class, they were either plopped in front of the computer (researching) or watching boring movies. They actually disliked the class intensely and, in fact, instructed me to change the second sentence of my letter from \"J and E are enjoying science class so far\" to \"J and E are enjoying science so far.\" Now it is, hands-down, their favourite class. They especially enjoy the experiments, and one of them has even expressed a new interest in becoming a scientist or at least in continuing to find out \"how things work.\"\n\nAnd the bad news? Mr. X only teaches the girls science. Their core teacher, Mr. Y., is the one who assigned the social studies project that resulted in our recent journey to homework hell. The project displayed all of the problems I detailed in my letter to Mr. X, and then some (for instance, a completely useless \"artistic\" component). So what to do? Do I write a similar letter to the core teacher? Ask to meet with him? It's tricky because he is my daughters' main teacher, and I've already had to approach him concerning several other, non-project related issues (such as incompressible math questions!). I do not want to alienate him or stress him out, but I also do not want him to assign another project such as the one we just suffered through. The girls learned next to nothing from it, which is perhaps the most important reason I object to context-less, single-focus research projects for 11-yr-olds (and is itself the topic for another post!). Any ideas regarding what my next steps, should be would be greatly appreciated.\n\n## Tuesday, March 8, 2011\n\n### Semi-Private Schools\n\nA long time ago\u2014in another life, it seems\u2014my husband and I found ourselves looking to buy a house in the greater Boston area. During this ultimately fruitless period of house-hunting, our real estate agent accompanied us on numerous expeditions to quaint urban neighbourhoods and not-so-quaint neighbouring suburbs. In her attempt to sell us on a particular house, the agent would invariably say something like, \"The local school is wonderful\u2014very high test scores.\" We were surprised by this, because we were quite obviously childless, and had never expressed the slightest interest in children or schools. We informed our agent that proximity to schools, good or bad, did not matter to us, that we were more concerned with proximity to decent restaurants and bookstores. She ignored us and continued to rattle off test scores of the schools close to the houses we viewed. It must be an American thing, we figured, something to do with the inequitable way schools are funded in the US. We were pretty certain school test scores were not of equal concern to house-hunters in Canada.\n\nIt took us many years\u2014during which we became parents to twins, moved in a panic back to Canada, and slowly realized that babies grow up to be kids who eventually attend school\u2014to realize how wrong we were.\n\nRecently, the Toronto Star published the results of an investigation into fundraising disparities among public schools in certain boards within the province of Ontario. I was not surprised that our neighbourhood junior school was one of the highest-ranked schools in Toronto in terms of money raised through fundraising. I was, however, somewhat taken aback to discover that during the year being studied (2008-2009), our school raised\u2014through a combination of school and parent fundraising\u2014$252,072 more than the elementary school at the bottom of the Toronto School Board list. (See full report here.) What is a person who believes in public education to make of such an obscene discrepancy? How is it even possible? Two explanations spring to mind: first, the government of Ontario no longer adequately funds public education, and has not done so since the Harris years, despite promises by the governing liberals to amend the flawed funding formula introduced by the conservatives; second, perhaps as a result of its decision to continue underfunding education, the government has chosen not to set limits on fundraising by, for instance, restricting the uses to which parent-raised money can be put. In my daughters' school, some of the money raised by the parent association goes to programs such as \"Scientists in the School\" and \"Learning Through the Arts''\u2014curriculum-related programs whose presence in a school should not be tied to the availability of parent-generated funds. But the fundraising issue is also part of a larger picture of education in Ontario (and elsewhere in North America) that has emerged in the last few decades. Specifically, it is an integral part of a cycle of inequity in which (as my Boston real estate agent understood) standardized tests scores also figure prominently. Before the advent of standardized tests such as EQAO in Ontario, inequality among schools\u2014including differences in parental involvement and fundraising\u2014no doubt existed, but standardized testing has amplified existing differences through its direct effect on real estate values. A high-scoring school drives up surrounding property values, which leads to parents-of-means moving into the neighbourhood, contributing time, energy and money to the school, which in turn leads to even higher test scores . . . and on and on the cycle goes. The result is the creation of a tripartite system of schooling in Ontario, consisting of public, private, and what people in my neighbourhood jokingly refer to as \"semi-private\" schools. In a semi-private school, private money, to the tune of more than a million dollars for some high schools in the province, is funnelled into the public school, making up for any deficiencies caused by inadequate government funding. Well-heeled parents who contribute the money are thrilled to save the$28,000 in private school fees. Indeed, for these parents it's a fantastic deal. For the parents of students attending the province's truly public schools, not so much.","date":"2018-07-20 16:20:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 2, \"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\": 0.2960931956768036, \"perplexity\": 2390.724309930114}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-30\/segments\/1531676591718.31\/warc\/CC-MAIN-20180720154756-20180720174756-00235.warc.gz\"}"} | null | null |
\section*{Conflict of interest}
\bibliographystyle{spphys}
\section{Introduction}
Twitter is what's happening around the world. The platform strives to keep users informed with relevant and healthy content at a global level. In the context of online information overload, it is extremely important for both producers of Tweets to reach the right (target) audience, and for consumers to be recommended the most relevant content (normally generated by the hundreds or thousands of people they follow).
Twitter's Home timeline, the default starting point for most Twitter users, displays a stream of Tweets from accounts the user has chosen to follow on Twitter. Users can decide if they want Tweets to be displayed in a reverse chronological order, or if they want them algorithmically ranked. In the latter case, every Tweet is scored, with the score provided by a predictive model and indicating how interesting and engaging the content would be for the user.
Over the years, Twitter has been, and continues to be, a great motivation and source for a number of research works studying user behavior in social media platforms, and the effect of these platforms on the society as a whole. Some of the most prominent works analyzed topics such as the role of Twitter in the social communication \cite{murthy2018}, how to use a global ``mood'' on Twitter to predict the stock market movements \cite{bollen2011}, how to detect events \cite{weng2011}, or influenza pandemics by using content available on Twitter \cite{aramaki2011}, or even analyzing mental health issues of Twitter users \cite{coppersmith2014,odea2015}.
However, it is not so often that we see work addressing or analyzing challenges of the core Twitter task, i.e., delivering relevant content to users. For instance, back in 2010~\cite{duan2010}, researchers introduced a learning to rank method to distinguish relevant from irrelevant Tweets that integrated information about the authority of the Tweet creator (``publisher''), and features of the Tweet. In \cite{chen2012}, a collaborative ranking method was proposed. The method accounted for user historical preferences of the content (i.e., Tweets), social relationships between users, as well as authority of the Tweet creator, and the quality of the Tweet content. Then, in~\cite{demaio2019}, a deep learning ranking algorithm that incorporated a time dimension, in order to rank tweets accounting for the time of day when a user is active on Twitter was evaluated. Earlier work~\cite{pennacchiotti2012} suggested a recommendation approach based on similarity between active user's Tweets and their friends' Tweets. With the Home timeline being one of the core products on the platform, the quality of the ranking model is of extreme importance, as it determines the quality of the user experience. People have more conversations and are more likely to come back to the platform when the timeline is optimized to show the most relevant Tweets first. Quality itself is a very personal concept that is hard to define. In this work, engagement is considered as a proxy for quality, i.e., the user interacts with the content if they value it. The goal of this paper is to invite and facilitate a broader research community to explore, within the scope of their interest, the task of delivering / recommending / ranking content such as Tweets, with all its peculiarities.
The task falls within the scope of engagement prediction / user satisfaction, which is omnipresent in recommender systems~\cite{Knijnenburg2012Explaining}. However, the feedback that is received from users on the displayed content is only implicit (as users do not exclusively rate Tweet relevance on their timeline). The lack of explicit feedback makes this an even harder task. In an effort to: a)~address the lack of large public datasets for user engagement prediction, and b)~advance the state-of-the-art in user recommendations with implicit feedback, we release a public dataset of 160 million samples from Twitter's Home timeline, split almost equally between positive and negative examples. To the best of our knowledge, this is the largest public dataset released by a social network platform. The dataset is shared with the RecSys community in the form of a challenge~\footnote{\url{http://www.recsyschallenge.com/2020/}}, paying special attention to user privacy. The release is compliant with existing privacy laws, since a)~the dataset is only includes public Tweets, and b)~any Tweets that are removed by users on the platform are removed from the dataset shortly thereafter. As a result, the size of the training dataset shrinks over time compared to the original release. In the context of the challenge, four different types of interaction (engagement) are considered: Like, Reply, Retweet and Quote Tweet. These interactions are described more elaborately in Section \ref{problem-def}.
The main contributions of the paper are:
\begin{itemize}
\item A concise \textit{problem definition} of delivering, recommending or ranking Tweets as the engagement prediction, i.e., a binary classification or a ranking problem.
\item A \textit{set of challenges}, general and specific ones for the presented task.
\item A \textit{set of state-of-the-art approaches}, serving as a starting point for the future research endeavors in dealing with the defined task.
\item And finally, a \textit{detailed description of the publicly released dataset}.
\end{itemize}
The rest of the paper is organized as follows, in section \ref{problem-def}, the problem is defined, section \ref{key-challenges} introduces the key challenges, section \ref{sota} provides an overview of the state of the art approaches, while in section \ref{challenge}, we describe the RecSys Challenge 2020 as an instrument for inviting the research community to participate in this compelling task facilitated by the publicly released large-scale dataset. The paper is concluded with a summary and the future directions in section \ref{conc}.
\section{Problem definition} \label{problem-def}
Recommender systems optimize for different objectives in different contexts. In an online marketplace, for example, the number of product views or clicks might be the target variable, while in display advertising, conversions might be considered. In many of these cases, the recommender system does not directly optimize for the business objective, e.g., revenue or user retention, but rather a proxy metric like the ones mentioned above. At Twitter, we are mostly interested in engagements on the Home timeline, which is where users see a stream of Tweets from accounts they have chosen to follow, as well as content produced by users outside of their immediate network that is considered relevant to their interests.
\\
In general, engagement prediction can be formulated either as a binary classification problem (i.e., will the user engage with the content or not), or as a ranking problem (e.g., is the user more likely to engage with this content in comparison to other content candidates). In the former case, predictions will be pointwise, meaning that each candidate will have its own score, normally ranging between 0 and 1 in order to correspond to the probability of engagement. In the latter case, the approaches can be pointwise, pairwise or listwise \cite{liu2009learning}. \\
Let $\mathcal{U}$ be a set of users and $\mathcal{I}$ a set of items. For each user $u \in \mathcal{U}$ we aim to discover a total ordering over $\mathcal{I}$, where $i \succ_u i'$ implies that $i$ is preferred to $i'$ for $u$. The goal is to learn a ranking function $f$, defined such that
$f : \mathcal{U} \times \mathcal{I} \to \mathbb{R}$ preserves the preference order as much as possible. That is, given a user $u$, for all $i \succ_u i'$, we want $f$ to satisfy $f(u, i) \succ_u f(u, i')$.
In pointwise approaches, each item is assumed to have an ordinal score. Ranking can, then, be formulated as a regression problem in which the absolute value of each item is estimated as an absolute quantity. Such techniques do not consider the interdependency across items. In pairwise approaches, the ranked list is decomposed into a set of item pairs. Ranking is, therefore, considered as the classification of pairs of items, such that the classifier is trained by minimizing the number of misorderings in ranking. Listwise approaches take the entire ranked list of items for each query as a training instance. As a direct consequence, these approaches are able to differentiate items from different queries, and consider their position in the output ranked list at the training stage.
\section{Key Challenges}\label{key-challenges}
Recommending the most relevant Tweet for the user's timeline turns out to be a difficult problem to tackle at scale. This section summarizes the main challenges that must be addressed in this endeavour.
\subsection{Sampling} Every day, hundreds of millions of users log in to Twitter to engage with the existing content or to create new content. Using the total number of Tweets that have been created since the launch of the platform\footnote{https://twitter.com/jack/status/20} would be intractable and massively expensive computationally, therefore modelers need to consider their sampling strategy carefully. As an example, it is often reasonable to sample candidates from the most recent past (limited to a fixed time window).
\subsection{Label Imbalance} Users tend to engage with only a fraction of the Tweets displayed on their timeline. This translates to a problem of class imbalance, especially when there is no negative downsampling performed as part of the training pipeline.
\subsection{Social Graph} The social follow graph, i.e., the graph that contains the information about which user follows whom, provides very valuable contextual features for the engagement prediction task at hand. Previous work on smaller datasets has demonstrated performance gains by leveraging this graph structure between users~\cite{monti2017geometric}. Given the hundreds of millions of users that are active on Twitter, such approaches are not as straightforward or even feasible to adopt. Some users might like a certain author more than others, and storing such information makes the problem quadratic in the number of users.
\subsection{Language} The language on Twitter is much less formal and loosely defined. It is not uncommon for users to Tweet in multiple languages, sometimes within the same Tweet. This makes the use of pre-trained language models, such as Word2Vec~\cite{mikolov2013distributed,mikolov2013efficient,mikolov2013linguistic} and BERT~\cite{devlin2018bert}, more challenging.
Even the use of hashtags (used to categorize a Tweet by topic) might be difficult to interpret and process. As an example, consider hashtags created by concatenating multiple words.
\subsection{Data Shift} The conversation on Twitter can change rapidly. Novel hashtags might be trending as a response to real-world events, or the same ones might mean different things at different times. Trained models might become stale very quickly. One way that this problem can be mitigated is described in \cite{Shiebler2018FightingRA}: by introducing embeddings (e.g., at the user or content level) that are trained more often than the rest of the models.
\subsection{Engineering considerations} Finally we must consider the trade-offs between model capacity (to what extent the model is able to correctly predict the preferences of all the users) and model size, which increases resources, utilization and latency. Given the real time nature of Twitter, the speed of prediction is a key factor for any production model.
\subsection{Metrics vs Intrinsic Value} It is also worth noting that the (personal) intrinsic value of a recommendation and the metric used to measure it might diverge. While engagement is the main metric used in industry, it might not fully represent the quality of engagement~\cite{Knijnenburg2012Explaining}. One example of this would be people replying to very polarized content with inflammatory comments to express their disagreement. While the engagement metric went up, the user probably found negative value in the recommendation.
\section{State-of-the-art}\label{sota}
We are now going to describe some of the state of the art techniques that have been adopted for recommendation problems.
\subsection{CTR model architectures}
The Neural Collaborative Filtering (CF) model for implicit feedback (only available feedback is engagement) was proposed in~\cite{he2017neural}. Each user and item is initially represented as a sparse input and embedded to a latent representation. This is achieved via a fully connected layer that projects the sparse representation to a dense vector. The user embedding and item embedding are then fed to a multi-layer neural architecture, to map the latent vectors to prediction scores. Each layer of the neural CF layers can be customized to discover certain latent structures of user–item interactions. The final output layer is the predicted score and a standard log loss is used for the optimization. Even though this work claims that content features can be used to represent users and items to address the cold-start problem, in fact, only their corresponding identities are used as input features in the form of one-hot encodings.
The Wide and Deep model~\cite{cheng2016wide} consists of a wide component and a deep component. The former is a generalized linear model that handles cross-product transformations / interactions between binary features. The deep component is a feed forward neural network that handles sparse, high-dimensional categorical features, by first embedding them into a low-dimensional and dense real-valued vector (of dimensionality O(10) to O(100)), and concatenates those with the continuous features. Continuous real-valued features are normalized to $[0, 1]$ by mapping a feature value $x$ to its cumulative distribution function $P(X \leq x)$, divided into $n_q$ quantiles. The normalized value is $\frac{i-1}{n_q-1}$ for values in the $i^{th}$ quantile. Quantile boundaries are computed during data generation.
The Deep FM model~\cite{guo2017deepfm} emphasizes both low- and high-order feature interactions by combining the power of factorization machines (FMs) for recommendations and deep learning for feature learning. In other words, this model consists of an FM component and a deep component. The FM component is described by:
\begin{equation*}
y_{FM} = \langle w, x \rangle + \sum_{j_1=1}^d \sum_{j_2=j_1+1}^d \langle V_i, V_j \rangle x_{j_1} \cdot x_{j_2}
\end{equation*}
where $d$ is the dimensionality of the input vector, while $x_{j_1}$, $x_{j_2}$ are the vector representations of fields $j_1$ and $j_2$, respectively (field here corresponds to a categorical or continuous variable). $V_i$ and $V_j \in \mathbb{R}$ are latent factors that allow the model to learn a representation whenever $i$ or $j$ appear in the data record, removing the constraint that both features $i$ and $j$ need to be present to train the parameter of their interaction. In the above equation, $\langle w, x \rangle$ reflects the importance of order-1 features, while the second term represents the impact of order-2 feature interactions.
The deep component is a standard feed-forward neural network. In this network structure, the embeddings of the different fields/categories are all of the same size $k$. Furthermore, the latent feature vectors ($V$) serve as learned network weights and are used to compress the input field vectors into the embedding vectors. It is worth highlighting that the FM and deep components share the same feature embedding, which brings two important benefits: 1)~it learns both low- and high-order feature interactions from raw features; 2)~there is no need for expertise feature engineering of the input, as required in Wide \& Deep model.
The Deep \& Cross network~\cite{wang2017deep} explicitly applies feature crossings at each layer and learns highly non-linear interactions of bounded degrees. In contrast to the wide-and-deep model, which hinges on a proper choice of cross features, this approach does not require manual feature engineering and has low computational cost. Similar to~\cite{naumov2019deep}, an embedding is obtained for each category, where a category can be e.g., a country. At each layer $x_{l+1}$, feature crossing is guaranteed by multiplying its input $x_l$ with the original feature vector $x_0$, leading to a highest polynomial degree of $l+1$ for an $l$-layer cross network. At the last layer, the output of the cross network and a deep network are concatenated and a standard log loss with regularisation is used. The cross network shares the spirit of parameter sharing as the factorization machine model and further extends it to a deeper structure, while the number of parameters in a cross network only grows linearly with the input dimension.
xDeepFM~\cite{lian2018xdeepfm} can learn certain bounded-degree feature interactions explicitly, while implicitly learning arbitrary low- and high-order feature interactions. The embedding layer in this model operates in a similar manner with the Deep \& Cross and DeepFM models, in the sense that each field is embedded in a vector of the same dimensionality. One observation made by~\cite{lian2018xdeepfm} is that the Deep \& Cross network learns a special type of high-order feature interactions, where each hidden layer is a scalar multiple of $x_0$. On the contrary, in each layer of xDeepFM higher order interactions are computed using the Hadamard product between $X_0$ and $X_l$. In particular, for hidden layer $l$ with $H_l$ feature vectors, the output is calculated as:
\begin{equation*}
X_{h,*}^l = \sum_{i=1}^{H_{l-1}}\sum_{j=1}^m W_{ij}^{k,l}\big(X_{i,*}^{l-1} \circ X_{j,*}^0 \big)
\end{equation*}
where $m$ is the number of fields/categories (e.g., user id, interests, gender etc.), $1 \leq h \leq H_l$ and $W^{k,l} \in \mathbb{R}^{H_{l-1} \times m}$ is the parameter matrix for the $h$-th feature vector. Hence, the output of the $l$-th layer is also a matrix $X_l \in \mathbb{R}^{H_l \times D}$, with $D$ being the dimensionality of the field embedding.
In the DLRM system~\cite{naumov2019deep}, each categorical feature is represented by an embedding vector of the same dimension $D$, as previously done in Deep \& Cross and xDeepFM models. Unlike other architectures, the continuous features are transformed by a multi-layer perceptron (MLP), which yields a dense representation of the same length as the embedding vectors.
The model also computes second-order interactions of different features explicitly, following the intuition for handling sparse data in FMs~\cite{koren2009matrix}, optionally passing them through MLPs. This is done by computing the dot product between all pairs of embedding vectors and processed dense features. These dot products are concatenated with the original processed dense features and post-processed with another MLP (the top or output MLP), and fed into a sigmoid function to yield a click probability. DLRM interacts embeddings in a structured way that mimics factorization machines to significantly reduce the dimensionality of the model by only considering cross-terms produced by the dot-product between pairs of embeddings in the final MLP.
\subsection{Hashing}
In \cite{chapelle2015simple}, feature hashing to regulate the size of the CTR model is used. The idea behind hashing is to reduce the number of values a feature can take by projecting it to a lower dimensional space. This is a commonly used strategy for ID features and there are two major strategies for hashing. The first approach hashes each feature $f_i$ into a $d_{f_i}$ dimensional space and concatenate the codes, resulting in $\sum_i d_{f_i}$. The alternative approach hashes all features into the same space, when a different hash function can be used for each feature.
A collision between two frequent values can lead to a degradation in the log likelihood. An interesting point made in~\cite{chapelle2015simple} is regarding using multiple hash functions. This is to alleviate the potential issue of degradation, in a similar manner that the Bloom filter operates~\cite{bloom1970space}, by replicating each value using different hash functions.
\subsection{Handling continuous features}
Normalization is considered an important step for continuous features and the approach used in~\cite{cheng2016wide} mapped features to the $[0,1]$ range by splitting their cumulative distribution function into $n_q$ quantiles. The quantile boundaries are computed during data generation. \cite{chapelle2015simple} also uses quantization of the continuous features before feeding them to the prediction model. Even though it is not exactly a normalization approach, Facebook's DLRM transforms the continuous features using a multi-layer perceptron (MLP), in order to yield a dense representation of the same length as the embedding vectors used for the categorical features.
\section{RecSys 2020 Challenge}\label{challenge}
Twitter has partnered with ACM RecSys to sponsor the 2020 challenge.
The task of the challenge is user engagement prediction, as described in Section \ref{problem-def}.
As part of the challenge, participants are invited to train a model on the data that is publicly released and to beat the baseline that is made available.
\subsection{Dataset description}
An engagement dataset is openly released\footnote{\url{http://recsys-twitter.com/}}. The dataset comprises of (approximately) 160 million possible engagements sampled over one week. Another 40 million are sampled from the following week and split evenly in half for validation and testing.
\subsubsection{Input features}
The dataset features are described in detail in Table~\ref{tab:feature_list}.
The features are divided into three separate feature groups: \textit{user-}, \textit{Tweet-} and \textit{engagement features}. There are two instantiations of \textit{user features}, one for the author (producer) and one for the reader (consumer) of the Tweet. \textit{Tweet features} group all the attributes describing the original Tweet, that is possibly engaged with by the consumer. Finally, the \textit{engagement features} contain all the details of the engagement itself.
\begin{table}[H]
\resizebox{\linewidth}{!}{%
\begin{tabular}{|c| l | l | l |}
\hline
~ & \multicolumn{1}{c|}{\textbf{Feature name}} & \multicolumn{1}{c|}{\textbf{Feature type} } & \multicolumn{1}{c|}{\textbf{Feature description} } \\
\hline
{\multirow{5}{*}{\textbf{User features}}} & userId & \textit{string} & User identifier (hashed) \\
& follower count & \textit{int} & Number of followers of the user \\
& following count & \textit{int} & Number of accounts this user is following \\
& is verified & \textit{bool} & Is the account verified? \\
& account creation timestamp in ms & \textit{int} & Unix timestamp (in seconds) of the creation time of the account \\
\hline
{\multirow{7}{*}{\textbf{Tweet features}}} & tweetId & \textit{string} & Tweet identifier (hashed) \\
& presentMedia & \textit{list[string]} & Tab-separated list of media types; media type can be in (Photo, Video, Gif) \\
& presentLinks & \textit{list[string]} & Tab-separated list of links included in the tweet (hashed) \\
& presentDomains & \textit{list[string]} & Tab-separated list of domains (e.g. twitter.com) included in the tweet (hashed)
\\ & tweetType & \textit{string} & Tweet type, can be either Retweet, Quote, Reply, or Toplevel \\
& language & \textit{string} & Identifier corresponding to inferred language of the tweet \\
& tweet timestamp& \textit{int} & Unix timestamp, in seconds of the creation time of the Tweet \\
& tweet tokens & \textit{list[int]} & Ordered list of Bert ids corresponding to Bert tokenization of Tweet text \\
& tweet hashtags & \textit{list[string]} & Tab-separated list of hashtags present in the tweet \\
\hline
\multirow{5}{*}{\textbf{Engagement features}} & reply engagement timestamp & \textit{int} & Unix timestamp, in seconds, of the Reply engagement if one exists \\
& retweet engagement timestamp & \textit{int} & Unix timestamp, in seconds, of the Retweet engagement if one exists \\
& quote engagement timestamp & \textit{int} & Unix timestamp, in seconds, of the Quote engagement if one exists \\
& like engagement timestamp & \textit{int} & Unix timestamp, in seconds, of the Like engagement if one exists
\\
& engageeFollowsEngager & \textit{bool} & Does the account of the engaged tweet author follow the account that \\
& & & has made the engagement? \\ \hline
\end{tabular}}
\caption{List of features provided for the challenge dataset}
\label{tab:feature_list}
\end{table}
\subsubsection{User privacy}
Previous dataset releases have disclosed private information in a anonymized format. However, user- and item anonymization has proven ineffective to linkage attacks (e.g., \cite{Sweeney1997GuaranteeingAW} and \cite{Narayanan2006HowTB}) that de-anonymize the data by joining with publicly available datasets on seemingly innocuous features.
In contrast, this dataset contains public Tweets that are accessible via the Twitter API\footnote{https://developer.twitter.com/}. No private information is disclosed. To further increase the difficulty of misusing the dataset, we took extra steps described in following sections.
The goal was to provide a dataset that is useful and stimulating for researchers, while at the same time preventing anyone from learning private information about users.
\paragraph{Developer policy and IRB} The use of the dataset is subject to Twitter's Developer Policy\footnote{https://developer.twitter.com/en/developer-terms/policy}, which, among other things, restricts ``off-Twitter matching'' to data that has been directly provided by the person and/or public data. This would prevent researchers from using the dataset to conduct inference attacks on private datasets.
We also encourage researchers to discuss any use of the dataset outside the context of the RecSys Challenge with their local ethics review board, if available. For instance, we note that U.S. Academic and Government researchers would be required to apply for Institutional Review Board approval to use the dataset. Most uses of the dataset would qualify for ``exempt review'' under Category 4 of federal IRB guidelines (``Secondary research for which consent is not required: Secondary research uses of identifiable private information or identifiable biospecimens'').
\paragraph{Creating pseudo-negative features} For public profiles, all Tweet engagements are public. This made it easy for us to create the first half of the dataset, i.e., the positive examples of engagements. We also wanted to give examples of negative interactions (i.e., this user did \emph{not} engage with this item), but disclosing this information will create a privacy leak. Negative examples are items the user might have seen but not engaged with. However, a set of such examples would reveal what content was seen by users---this is private information. To get around this, we created the pseudo-negative dataset as follows: for each user we considered all the Tweets that were created by their followers in the considered timeframe and removed the positive examples (i.e., the Tweets that were engaged with). We sampled from the set of remaining Tweets, which does not distinguish between negative examples (items the user saw and did not engage with) and items the user did not see, thereby effectively protecting this private information.
\paragraph{Scrubbing deleted content}
\label{gdpr}A novel challenge we encountered in the creation of the dataset was to keep it continuously synced with the Twitter platform. This means that if a user deletes a Tweet and/or their profile (or makes it private), this has to be reflected in the dataset. While we acknowledge that a shared and static dataset is fundamental for the reproducibility of the research, we wanted to prioritize user's choice.
The way we are solving this problem is by keeping the dataset on the website constantly updated. A change in the system will be promptly reflected in the publicly available dataset. The challenge competitors are required to make the necessary changes in the data they have already downloaded in order to keep them compliant. This requirement is also reflected in Twitter's Developer Policy. To facilitate the task, we are also going to provide a list of the deleted user/Tweet ids.
This process makes the whole dataset dynamic (including the validation and test set, used for scoring). Given the size of the dataset, it is very unlikely that the scrubbing will results in a meaningful reduction in dataset size.
\paragraph{Handling text features}
Since providing raw Tweet text could make the reconstruction of the dataset immediate, we tokenized the text and provided the IDs of each token according to BERT. Special attention was given to links. We provide both the hash of the full link and the hash of the top level domain.
On top of this, we also hashed user- and Tweet identifiers, preventing researchers from simply looking up the Tweet text via the Twitter API.
While we recognize that de-anonymization and hence the full reconstitution of the dataset is possible (and likely), we take this measure as a means to discourage misuse of the dataset. It is worth noting again that the original dataset was public already, hence any de-anonymization would not reveal any private information.
\subsubsection{User sampling}
We took some extra precautions to make sure that the set of users that have positive engagements is similar to the set of users that had negative/no engagements. This is to give researchers more insight into specific users' histories. Separate uniform sampling for positives and negatives at Twitter's scale would have led to mostly disjoint user sets.
\subsection{Metrics}
In the following the two metrics used to evaluate the performance of a model are presented.
\subsubsection{Relative Cross Entropy}
Relative Cross Entropy (RCE) corresponds to the improvement of a prediction relative to the straw man, or the naive prediction, measured in cross entropy (CE). The naive prediction corresponds to the case that does not take into account the user and Tweet features, e.g., it always predicts the average (observed) CTR of the training set. Suppose the average CE of the naive prediction is $CE_{naive}$ and average CE of the prediction to be evaluated is $CE_{pred}$, then RCE is defined as $(CE_{naive} - CE_{pred}) \times 100/CE_{naive}$. Note that the lower the CE the better the quality of the predictions, so the higher the RCE. The benefit of using RCE is that we can obtain a confident estimate of whether the model is under or over performing the naive prediction.
\subsubsection{Area Under the Precision-Recall Curve (PR-AUC)}
Recall is equivalent to the true positive rate (or sensitivity) in a classification problem, while precision is the same as positive predictive value. Reviewing both precision and recall is particularly useful in cases there is an imbalance in the observations between two classes. The Area Under the Precision-Recall Curve (PR-AUC) is a commonly used evaluation metric and is more sensitive than AUC on skewed data.
\subsection{Baseline}
In this section, we will describe a simple baseline model architecture that works with the provided data format. It mainly constitutes of the following feature embedding and prediction components.
\subsubsection{Numeric Features}
For a numeric feature $num_i$ (e.g., follower count), we compute $n_q$ quantiles based on its cumulative distribution function and create $n_q + 1$ buckets, $(-\infty, q_i^1), ..., [-q_i^j, q_i^{j+1}), ... , [-q_i^{n_q}, +\infty)$, where $q_i^j$ denotes the $j^{th}$ quantile. Note that we also reserve an extra bucket for missing feature values. The feature $num_i$ is then bucketized into a one-hot encoded representation $e_i \in \mathbb{R}^{n_q+2}$, where $e_i^j = 1$ if $num_i$ falls into the $j^{th}$ bucket.
\subsubsection{Categorical Features}
For categorical features $c_i$ (e.g., Tweet language), we one-hot encode them in $ \mathbb{R}^{N_{c_i}+1}$ where $N_{c_i}$ denotes the cardinally of the vocabulary. We reserve one extra dimension for out-of-vocabulary item. Binary features are considered a type of categorical features.
\subsubsection{Id Features}
For ID features $id_i$, the vocabulary is either unknown or of extremely large cardinality (e.g. user or Tweet id), we choose to hash $id_i$ into number, and then mod it into a given number of buckets.
\subsubsection{Tweet Text}
The text of the Tweet is tokenized and embedded using Chaos Free Recurrent Neural Network (CFRNN)~\cite{cfrnn}, chosen for computational efficiency and stability. In the datset we release the list of integers $(s_1, ..., s_l)$ corresponding to the index of the token in the embedding.
\subsubsection{Model}
For Numeric, Categorical, and ID Features, the corresponding one-hot encodings are converted to dense representations of size 16, then concatenated along with Tweet feature embedding (embedding size 16). The obtained feature vector is then fed into a 3 layer multi-layer perceptron (hidden state size is 128, 64 and 32, activation function is leaky ReLU) to do the final predictions, in which the output size of the model is 4, corresponding to 4 engagement classes (Like, Reply, Retweet, and Retweet with comment). Since these four types of engagement are not mutually exclusive we use a sigmoid rather than softmax as the activation function in the final layer.
For the baseline, $n_q$ is set to 49, resulting in 50 buckets in total. We use the huber loss~\cite{huber} for each class and the model is trained with Adam optimizer~\cite{adam} and learning rate 0.001 for 1 million steps.
\section{Conclusion and Future Directions}\label{conc}
In this paper we have provided an overview of a rather challenging task: predicting user engagement with Tweets. To this end, a detailed and formal problem definition is provided, a set of concrete, tangible challenges faced within a real-world environment, and a set of the state-of-the-art approaches to motivate and further explore the task. We described the RecSys Challenge and the publicly released large-scale dataset, being an invitation for the broader research community to tackle this task. Finally, we also provided details of an exemplary, baseline approach developed upon the described dataset.
As we briefly addressed in the paper, various challenges emerge when delivering content such as Tweets to a user, but surely, there are numerous to be further explored. For instance, contextual information can enrich the recommendation model in order to deliver more appropriate content according to the contextual situation of a user at hand \cite{adomavicius2011context}, which was also shown in \cite{demaio2019} by simply accounting for the time of day when the user is active. Furthermore, we mentioned that engagement is used as a proxy for content quality, but this does not have to be the case, a user could engage with a certain content even when they dislike it (this even being the reason for engaging with it). Certainly, the models are tuned to accurately predict the next Tweet a user is likely to engage with, but other issues should as well be considered, such as, serendipity, diversity, coverage, etc., with a goal to truly comprehend the relevance of the delivered content. To tackle these and many other challenges, to broaden the knowledge-base, it is crucial that the data describing such user-content interactions is available for practitioners and researchers. Therefore, this paper is only a step forward to making a stronger cooperation between industry and academia. | {
"redpajama_set_name": "RedPajamaArXiv"
} | 144 |
{"url":"https:\/\/www.physicsforums.com\/threads\/physics-power-and-motion.347685\/","text":"# Homework Help: Physics - power and motion\n\n1. Oct 21, 2009\n\n### deathnote93\n\n[SOLVED] Physics - power and motion\n\n1. The problem statement, all variables and given\/known data A body is moving in a straight line under the influence of a source of constant power. Show that its displacement in time t is proportional to t3\/2\n\n2. Relevant equations\nP = F.v\n\nF = dp\/dt, p=mv\n\nv = dx\/dt\n\n3. The attempt at a solution Absolutely no idea where to begin, sorry. I'm not even sure what 'source of constant power' means here - is the force constant or the velocity?\n\nLast edited: Oct 21, 2009\n2. Oct 21, 2009\n\n### Troels\n\nFirst it helps to notice that the motion is one-dimensional; so you can drop the vectors.\n\nUnlike motion under constant force, motion under constant power is not uniformly accelerating, because it takes more and more energy to achieve the same velocity increase. Neither the Force nor the velocity are constant in time, but their product is.\n\nYou must have, in other words:\n\n$$F(t)v(t)=P_0$$\n\nwhere P0 is a constant. Inserting $$F=m dv\/dt$$, you get:\n\n$$mv(t)\\frac{dv(t)}{dt}=P_0$$\n\nWhich is a seperateable ODE which gives you v which can then be integrated to give x.\n\nGood luck and Have Fun :)\n\nLast edited: Oct 21, 2009\n3. Oct 21, 2009\n\n### deathnote93\n\nAh, using integration was the last thing on my mind when I was trying this problem in my head.\n\nGot it now, thanks a LOT!\n\nShare this great discussion with others via Reddit, Google+, Twitter, or Facebook","date":"2018-07-18 13:07:04","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 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\": 0.71449214220047, \"perplexity\": 1135.1290961451327}, \"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-2018-30\/segments\/1531676590169.5\/warc\/CC-MAIN-20180718115544-20180718135544-00174.warc.gz\"}"} | null | null |
Hit Mania Champions 2013 è una compilation di artisti vari facente parte della serie Hit Mania e uscita nei negozi il 19 marzo 2013, raggiungendo la posizione numero 4 in classifica.
La compilation è mixata dal dj Mauro Miclini.
Tracce
Note
Voci correlate
Hit Mania
Musica dance
Musica elettronica
Musica house
Collegamenti esterni | {
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{"url":"https:\/\/mathoverflow.net\/questions\/39968\/can-the-minimal-index-of-a-subfactor-take-all-values-in-4cos2pi-nn-3-4-5\/39982","text":"# Can the minimal index of a subfactor take all values in {4cos^2(pi\/n);n=3,4,5,\u2026} u [4,infinity]?\n\nGiven a subfactor $N\\to M$ and a conditional expectation $E:M\\to N$, there is a numerical invariant $Ind(E)$ associated to to this situation, called the index of $E$. The possible values of $Ind(E)$ are restricted to the set {$4\\cdot \\cos^2(\\pi\/n);n=3,4,5,...$} $\\cup$ $[4,\\infty]$.\n\nThe minimal conditional expectation is the one that minimizes the value of $Ind(E)$. The minimal index of the subfactor is then defined to be the index of its minimal conditional expectation.\n\nCan the minimal index take all values in {$4\\cdot \\cos^2(\\pi\/n);n=3,4,5,...$} $\\cup$ $[4,\\infty]$? In other words, given a real number in the above set, is there a subfactor whose minimal index is that real number?\n\nRemark: If the factors are of type $II_1$, there is another preferred conditional expectation: the one that is compatible with the traces. The corresponding index is called the Jones index. This is not the index I care about. Jones' index agrees with the minimal index in the case of irreducible subfactors, but not in general.\nJones' index is known to take all the above values. But the subfactors used in the construction are not irreducible (and one can also check that their minimal index is different from their Jones index).\n\n-\n\nThere is an irreducible Temperley-Lieb subfactor at every allowed index. For $n\\geq 3$, it has index $4\\cos^2(\\pi\/n)$ and principal graph $A_{n-1}$ (in fact all subfactors of index less than $4$ are irreducible), and for every $r\\geq 4$, it has index $r$ and principal graph $A_\\infty$. Doesn't that do the job by your remark?\nEvery standard invariant that arises from a finite index type $II_1$ subfactor also arises as the standard invariant of a type $III$ subfactor, and vice versa. See Izumi's paper \"On type II and type III principal graphs of subfactors.\" \u2013\u00a0Dave Penneys Sep 26 '10 at 20:16","date":"2016-04-30 13:23:39","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\": 0.9186512231826782, \"perplexity\": 170.99849475361532}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2016-18\/segments\/1461860111865.15\/warc\/CC-MAIN-20160428161511-00095-ip-10-239-7-51.ec2.internal.warc.gz\"}"} | null | null |
Q: Nothing I Do Will Disable Access Warnings I've tried all of the suggestions....
*
*Access Options, Confirm settings for action queries
*VBA SetWarnings off and back on
*The db is in a trusted folder location
No matter what I do I still get warnings / confirmation popups in Access when running my forms and macros. What else could be overriding my settings?
Is it possible for group policies to override my settings? How do I find those policy settings?
We have a process that we run in Access 2x monthly, and it takes too long because the users have to click Yes on all the popups. I've tried the obvious tricks, but nothing prevents these confirmation popups.
A: If you are using VBA to execute SQL, then try to use:
CurrentDb.Execute strSQL
instead of
DoCmd.RunSQL strSQL
where strSQL represents the SQL statement you are trying to execute.
This automatically suppresses the normal warnings from running a query, but will still give you other errors, such as when the SQL is invalid. You can also use this to get certain items from the query, such as the amount of records affected by the query, that you otherwise cannot get from the docmd statement.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,429 |
Un Wiki estructurado proporciona una manipulación parecida a una base de datos de campos almacenados en páginas, y por lo general ofrecen una extracción y presentación del lenguaje o margen con la funcionalidad algo similar a SQL
Introducción
Wikis típicamente son usados como compartido whiteboards que permite a usuarios para añadir, quitar, o de otra manera corregir todo el contenido muy rápida y fácilmente. La facilidad de interacción y operación hace a una wiki plana un instrumento eficaz para la escritura de colaboración y para compartir el conocimiento.
Los sistemas de base de datos no son muy idóneos para mantener el contenido en colaboración, pero ellos contienen datos sumamente estructurados, ofrecen un fácil reporte, y soportan el volumen de trabajo.
Un Wiki estructurado combina los beneficios de la aparentemente contradicción de los mundos de las wikis planas y los sistemas de base de datos. Esto le da un entorno de base de datos de colaboración, donde el conocimiento puede ser compartido libremente, y donde la estructura puede ser añadida como se necesite. En un wiki estructurado, los usuarios pueden crear Wikis muy específica a sus necesidades.
Comparando las wikis planas, los sistemas de Base de datos y un wiki estructurado
Motores de Wikis estructuradas
TWiki
JotSpot
Trac, para el caso especial de venta de entradas (ticketing en inglés), pero permite tipos de entradsss flexibles para cualquier contenido.
Véase también
Wiki
Wiki semántica
MediaWiki
Dynamic Page List, una extensión de la cual permite la generación de lostas de completas preguntas en categorías, Wikipedia:namespaces, y/o artículos.
OmegaWiki (formalmente WiktionaryZ) construye una base de datos especializada en MediaWiki.
Enlaces externos
Wiki estructurada artículo de la Web de co-desarrollo TWiki
Wikis
en:Wiki application#Structured wikis | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,957 |
La communauté de communes du Haut Cabardès est une communauté de communes française, située dans le département de l'Aude et la région Languedoc-Roussillon.
Histoire
Composition
Elle regroupe les 14 communes suivantes :
Compétences
Notes et références
Voir aussi
Article connexe
Intercommunalités de l'Aude
Lien externe
Haut Cabardès | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,443 |
Butiksstängningslagen var en lag som reglerade detaljhandelns öppettider.
Lagar reglerande butikers öppettider antogs 1900 i Norge och 1904 i Storbritannien, Österrike-Ungern 1908 och Danmark 1910. I Sverige försökte man länge på frivillig väg reglera butikernas öppettider, men 1904 lade riksdagsledamöterna Jakob Byström och Karl Gustaf Karlsson fram en motion om regleringar även i Sverige. De förnyades under de följande åren och 1906 beslutade regeringen att begära förslag om lagbestämmelser inom området och 1909 lades förslaget fram och antogs 6 juni 1909. Lagen kallades vanligen Butiksstängningslagen och bestämde att butikerna på vardagar inte fick öppna tidigare än klockan 7.00, och måste stängas senast 20.00. Matvarubutiker och tobaksaffärer fick dock öppna redan klockan 6.00, och tobaksaffärer fick hålla öppet ända till 21.00. Öppettiderna på söndagar reglerades tidigare om lagstiftningen om sabbatsbrott, men genom en ny lag 6 juni 1912 inkluderades även ett förbud mot öppna butiker på söndagar och helgdagar. Lokala ordningsstadgor tillät även strängare reglering av öppettiderna.
30 maj 1919 antogs en ny lag som reglerade öppettiderna än hårdare. Handelsbodarna hade tillstånd att hålla öppet 8.00-19.00, under perioden 10–23 december fick dock butikerna ha öppet till 20.00. Butiker där mejeriprodukter, ägg, margarin, konditori- eller bagerivaror såldes fick lov att öppna redan 7.00 och ha öppet en timme senare än övriga butiker. På söndagar eller helgdagar fick inga handelsbodar, rak- eller frisersalonger, eller badhus hållas öppna. Butiker för färska livsmedel som ovan eller frukt och levande blommor fick dock ha öppet på söndagarna 7.00–10.00. Specialtillstånd kunde utfärdas.
Undantagna från bestämmelsen var kaféer eller restauranger med förtäring på stället, försäljning av biljetter för järnväg och båtar, apotek och försäljning av "för motorfordons framföring nödvändiga varor".
Butiksstängningslagen av 22 juni 1939 innehöll i stort sett samma bestämmelser som 1919 års lag.
Från 1967 tilläts alla butiker ha öppet till 20.00, och 1972 fick Sverige helt fria öppettider. Men frågan om begränsning av öppettiderna har dock återkommit tid efter annan. 1975 tillsattes en affärstidskommitté i riksdagen, och ett beslut om reglering av öppettiderna stöddes då av alla riksdagspartier utom Folkpartiet och Moderaterna. Någon sådan reglering kom dock aldrig till, men frågan återkom politiskt fler gånger fram till 1991. Fram till slutet av 1990-talet drev Handelsanställdas förbund frågan om reglering av öppettiderna.
Källor
Svensk rättshistoria | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,567 |
Using an app like Waze has huge benefits when navigating traffic situations. However, many things can go wrong especially if the app tells you go a route that everyone else is taking to avoid a traffic situation or when others purposely report an "accident" when there is no accident just to route traffic out of their neighborhoods. What a wonderful time to be alive!
Verizon was recently a victim of a data breach that affected six million customers. What makes this breach different was that it was caused by one of Verizon's third-party partners accidentally misconfigured an Amazon S3 cloud based data repository, which was set to "public". A great example of why third-party security is so important to businesses.
If you have an Apple iOS device you should update to iOS 10.3.3 ASAP. You should also update your Android device if you so happen to have a vulnerable one of the listed Android devices as well (see this page for more info). This update fixes a very serious vulnerability in the Broadcom wifi chip on the device. The researchers that discovered this vulnerability discussed (at the BlackHat conference in Las Vegas last week) how they were able to take over a vulnerable device all through a wifi connection.
Did you know that if you have an Amazon Echo device you can use it to make voice calls and send messages to other Echo owners? Sounds great, except that by default Amazon needs access to your entire contact list to see who else is an Amazon Echo owner which allows everyone to be able to call each other. This is fine except, how many of your contacts to you "really" know? Many times we put temporary contacts or have people in our contact list that we really don't want to talk to again (old bosses?). Unfortunately, Amazon doesn't allow you to choose who you want to connect with…it's all or nothing.
This entry was posted in Podcast Episodes and tagged Amazon, Amazon Echo, Android, Apple, Blackhat, DEF CON, home security, iOS, Ring, Verizon, Waze. Bookmark the permalink. | {
"redpajama_set_name": "RedPajamaC4"
} | 8,231 |
Q: VS Addin - How to add controls to IDE I have created a VS addin.
I'd like to interact with the user by adding a dropbox above the Solution Explorer.
I can't figure out how to do that.
Thanks!
A: See a code example of "VSSDK IDE Sample: Combo Box", it covers:
*
*Adding a Drop Down Combo to Visual Studio and handling it
*Adding an
*Index Combo to Visual Studio and handling it Adding a MRU Combo to
*Visual Studio and handling it Adding a Dynamic Combo to Visual
*Studio and handling it Controling the programmatic name of the
combo box commands by placing the commands within a menu ("Tools"
in our case) of the main menu bar.
I believe you should start investigating VSIP (Visual Studio Integration SDK and Package information:
*
*Visual Studio Integration SDK
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,487 |
/* Author: Michael Ferguson, Ioan Sucan, E. Gil Jones */
#include <ros/ros.h>
#include <moveit_simple_controller_manager/action_based_controller_handle.h>
#include <moveit_simple_controller_manager/gripper_controller_handle.h>
#include <moveit_simple_controller_manager/follow_joint_trajectory_controller_handle.h>
#include <pluginlib/class_list_macros.h>
#include <algorithm>
#include <map>
namespace moveit_simple_controller_manager
{
class MoveItSimpleControllerManager : public moveit_controller_manager::MoveItControllerManager
{
public:
MoveItSimpleControllerManager() : node_handle_("~")
{
if (!node_handle_.hasParam("controller_list"))
{
ROS_ERROR_STREAM_NAMED("manager", "No controller_list specified.");
return;
}
XmlRpc::XmlRpcValue controller_list;
node_handle_.getParam("controller_list", controller_list);
if (controller_list.getType() != XmlRpc::XmlRpcValue::TypeArray)
{
ROS_ERROR("Parameter controller_list should be specified as an array");
return;
}
/* actually create each controller */
for (int i = 0; i < controller_list.size(); ++i)
{
if (!controller_list[i].hasMember("name") || !controller_list[i].hasMember("joints"))
{
ROS_ERROR_STREAM_NAMED("manager", "Name and joints must be specifed for each controller");
continue;
}
try
{
std::string name = std::string(controller_list[i]["name"]);
std::string action_ns;
if (controller_list[i].hasMember("ns"))
{
/* TODO: this used to be called "ns", renaming to "action_ns" and will remove in the future */
action_ns = std::string(controller_list[i]["ns"]);
ROS_WARN_NAMED("manager", "Use of 'ns' is deprecated, use 'action_ns' instead.");
}
else if (controller_list[i].hasMember("action_ns"))
action_ns = std::string(controller_list[i]["action_ns"]);
else
ROS_WARN_NAMED("manager", "Please note that 'action_ns' no longer has a default value.");
if (controller_list[i]["joints"].getType() != XmlRpc::XmlRpcValue::TypeArray)
{
ROS_ERROR_STREAM_NAMED("manager", "The list of joints for controller " << name << " is not specified as an "
"array");
continue;
}
if (!controller_list[i].hasMember("type"))
{
ROS_ERROR_STREAM_NAMED("manager", "No type specified for controller " << name);
continue;
}
std::string type = std::string(controller_list[i]["type"]);
ActionBasedControllerHandleBasePtr new_handle;
if (type == "GripperCommand")
{
new_handle.reset(new GripperControllerHandle(name, action_ns));
if (static_cast<GripperControllerHandle*>(new_handle.get())->isConnected())
{
if (controller_list[i].hasMember("parallel"))
{
if (controller_list[i]["joints"].size() != 2)
{
ROS_ERROR_STREAM_NAMED("manager", "Parallel Gripper requires exactly two joints");
continue;
}
static_cast<GripperControllerHandle*>(new_handle.get())
->setParallelJawGripper(controller_list[i]["joints"][0], controller_list[i]["joints"][1]);
}
else
{
if (controller_list[i].hasMember("command_joint"))
static_cast<GripperControllerHandle*>(new_handle.get())
->setCommandJoint(controller_list[i]["command_joint"]);
else
static_cast<GripperControllerHandle*>(new_handle.get())
->setCommandJoint(controller_list[i]["joints"][0]);
}
if (controller_list[i].hasMember("allow_failure"))
static_cast<GripperControllerHandle*>(new_handle.get())->allowFailure(true);
ROS_INFO_STREAM_NAMED("manager", "Added GripperCommand controller for " << name);
controllers_[name] = new_handle;
}
}
else if (type == "FollowJointTrajectory")
{
new_handle.reset(new FollowJointTrajectoryControllerHandle(name, action_ns));
if (static_cast<FollowJointTrajectoryControllerHandle*>(new_handle.get())->isConnected())
{
ROS_INFO_STREAM_NAMED("manager", "Added FollowJointTrajectory controller for " << name);
controllers_[name] = new_handle;
}
}
else
{
ROS_ERROR_STREAM_NAMED("manager", "Unknown controller type: " << type.c_str());
continue;
}
if (!controllers_[name])
{
controllers_.erase(name);
continue;
}
/* add list of joints, used by controller manager and moveit */
for (int j = 0; j < controller_list[i]["joints"].size(); ++j)
controllers_[name]->addJoint(std::string(controller_list[i]["joints"][j]));
}
catch (...)
{
ROS_ERROR_STREAM_NAMED("manager", "Caught unknown exception while parsing controller information");
}
}
}
virtual ~MoveItSimpleControllerManager()
{
}
/*
* Get a controller, by controller name (which was specified in the controllers.yaml
*/
virtual moveit_controller_manager::MoveItControllerHandlePtr getControllerHandle(const std::string& name)
{
std::map<std::string, ActionBasedControllerHandleBasePtr>::const_iterator it = controllers_.find(name);
if (it != controllers_.end())
return static_cast<moveit_controller_manager::MoveItControllerHandlePtr>(it->second);
else
ROS_FATAL_STREAM_NAMED("manager", "No such controller: " << name);
return moveit_controller_manager::MoveItControllerHandlePtr();
}
/*
* Get the list of controller names.
*/
virtual void getControllersList(std::vector<std::string>& names)
{
for (std::map<std::string, ActionBasedControllerHandleBasePtr>::const_iterator it = controllers_.begin();
it != controllers_.end(); ++it)
names.push_back(it->first);
ROS_INFO_STREAM_NAMED("manager", "Returned " << names.size() << " controllers in list");
}
/*
* This plugin assumes that all controllers are already active -- and if they are not, well, it has no way to deal
* with it anyways!
*/
virtual void getActiveControllers(std::vector<std::string>& names)
{
getControllersList(names);
}
/*
* Controller must be loaded to be active, see comment above about active controllers...
*/
virtual void getLoadedControllers(std::vector<std::string>& names)
{
getControllersList(names);
}
/*
* Get the list of joints that a controller can control.
*/
virtual void getControllerJoints(const std::string& name, std::vector<std::string>& joints)
{
std::map<std::string, ActionBasedControllerHandleBasePtr>::const_iterator it = controllers_.find(name);
if (it != controllers_.end())
{
it->second->getJoints(joints);
}
else
{
ROS_WARN_NAMED("manager", "The joints for controller '%s' are not known. Perhaps the controller configuration is "
"not loaded on the param server?",
name.c_str());
joints.clear();
}
}
/*
* Controllers are all active and default -- that's what makes this thing simple.
*/
virtual moveit_controller_manager::MoveItControllerManager::ControllerState
getControllerState(const std::string& name)
{
moveit_controller_manager::MoveItControllerManager::ControllerState state;
state.active_ = true;
state.default_ = true;
return state;
}
/* Cannot switch our controllers */
virtual bool switchControllers(const std::vector<std::string>& activate, const std::vector<std::string>& deactivate)
{
return false;
}
protected:
ros::NodeHandle node_handle_;
std::map<std::string, ActionBasedControllerHandleBasePtr> controllers_;
};
} // end namespace moveit_simple_controller_manager
PLUGINLIB_EXPORT_CLASS(moveit_simple_controller_manager::MoveItSimpleControllerManager,
moveit_controller_manager::MoveItControllerManager);
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,532 |
Discuss about disease named Bulimia Nervosa?
This study mainly focuses on the psychology disorder which is named as Bulimia Nervosa and it take place when anyone consume huge amount of food during a short span of time (Dryer, Tyson & Kiernan, 2012). It usually happens to those who have the habit of eating very frequently without break and also they do not have a control over their eating. The individuals who are suffering from this disorder can actually consume 3400 calories in less than an hour which totals up with sum of 20000 calories in the full eight hours.
This disorder if properly taken care of by the individual at correct time by the cat of self realization, then the problem can get solved because it is not that much a serious issue as compared to other diseases (Guillaume et al., 2012). So, if the individuals can make effort to maintain a diet plan in their everyday life and continuous exercise and yoga can make the individual feel better and can easily get rescued by this disease in a short span of time.
It is therefore advisable to take care of health at the wisest manner so that no problem as such occurs due to any unhealthy habits. Eating four times a day will keep the individuals fresh and fine which will ultimately help them in the long course of time which will certainly be helpful for the individuals as well as their family who are obviously concerned towards the person possessing the psychology disorder.
Individuals suffering from bulimia cannot stop eating till it reaches the physical discomfort level.
It is often found that these people tend to go kitchen at a regular interval of time in a secret manner so that no one else can know the fact that they eat this much for every short period of time.
Weight actually does not matter for these individuals because they tend in eating large variety of food at regular interval without any break (Hannon-Engel, 2011).
It can be noticed that the individuals often feel like fasting but actually they overeat during that time even which ultimately serves no purpose.
The main symptoms to judge the individuals suffering from this disease can be that they often go washroom after having the meals and can be seen vomiting due to overeating.
These individuals can often see taking diet pills like laxatives, diuretics so that their appetite can be controlled in a sufficient course of time.
As it is already mentioned in the above point that the individuals actually vomit after eating so they cover up the smell by the usage of mouth fresher (Zweig & Leahy, 2012).
Exercise is used in an excessive manner by these individual so that they can control over the eating habits by practicing yoga as well as aerobics specially.
One of the symptoms that can be noticed is that scars can be seen on the body of the individuals due to sticking of fingers which is actually on the portion of throat which provokes vomiting in an excessive manner.
The cheeks of these individuals usually become puffy because of the excessive vomiting which happens due to overeating.
Another symptom which can be noticed from the person suffering from bulimia is that the teeth become yellow due to acid in the stomach (Judd, 2011).
A unique symptom of this disease is that it is found that men as well as women are normal weight even if they consume large amount of food at regular interval of time.
The first step towards success of getting over anything is that to admit the fact of suffering from a specific disease and then make efforts to minimise it as far as possible at individual manner. If the person feels that he is gaining weight, then efforts need to be taken to reduce the habit of eating at regular intervals of time (Masheb & White, 2012). It should be kept in mind that the food should be ate in control rather than have a habit to eat every single moment of time and suffering due to that.
The person suffering from this disorder should feel free to talk to their dear ones who are a good listener so that they will understand the problem and make the person feel good that he is not alone at these tough times (Pedersen, Lunn, Katznelson & Poulsen, 2012). It can pose problem because most people feel ashamed to discuss about the fact that they suffer from bulimia but it is not at all wise to keep it as a secret which will actually serves no purpose.
The individuals who are suffering from this disease should always make effort to stay away from people or place which tempts them to eat more because that actually poses them danger and feel unsecure of the habit of eating more at a time (Robinson & Nicholls, n.d.). If the friend circle is supportive and the places visited are under control, then the individual can easily get over this habit at a gradual pace of time which will help them in the long run.
It is recommended that professional help can be more useful for these individuals because they can seek the professional guide as they will know more than the common man as they are more specialized and experienced in this particular field (Schmidt & Treasure, 2012).
This study has completely based on the disease named Bulimia Nervosa which actually happens due to overeating and having a bad habit at eating in regular intervals of time without break. There are various signs and symptoms that have been recognized in the study and the main symptoms is gaining in weight as well as vomiting due to eating at continuous mode.
It is recommended to share this problem with friends as well as parents so that to take the support of them to get rescue at a faster pace of time. Professional guidance can also solve the problem to the great extent as they have the edge of experience to handle with this disorder to make the individual deal a normal life.
4. Judd, S. (2011). Eating disorders sourcebook. Detroit, Mich.: Omnigraphics, Inc.
7. Robinson, P., & Nicholls, D. Critical care for anorexia nervosa.
8. Schmidt, U., & Treasure, J. (2012). Getting Better Bit(e) by Bit(e). Hoboken: Taylor and Francis.
9. Zweig, R., & Leahy, R. (2012). Treatment plans and interventions for bulimia and binge-eating disorder. New York: Guilford Press. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,577 |
The Project Gutenberg Etext of MM. and Bebe, by Gustave Droz, v2
#11 in our series The French Immortals Crowned by the French Academy
#2 in our series by Gustave Droz
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MONSIEUR, MADAME AND BEBE
By GUSTAVE DROZ
BOOK 2.
CHAPTER XIII
THE BLUE NOTE-BOOK
Toward midnight mamma made a sign to me with her eyes, and under cover of
a lively waltz we slipped out of the drawing-room. In the hall the
servants, who were passing to and fro, drew aside to let us go by them,
but I felt that their eyes were fixed upon me with the curiosity which
had pursued me since the morning. The large door giving on to the park
was open, although the night was cool, and in the shadow I could make out
groups of country folk gathered there to catch a glimpse of the
festivities through the windows. These good people were laughing and
whispering; they were silent for a moment as we advanced to ascend the
staircase, but I once more felt that I was the mark of these inquisitive
looks and the object of all these smiles. The face of mamma, who
accompanied me, was much flushed, and large tears were flowing from her
eyes.
How was it that an event so gay for some was so sad for others?
When I think over it now I can hardly keep my countenance. What silly
terrors at that frightful yet charming moment! Yet, after all, one
exaggerates things a great deal.
On reaching the first floor mamma stopped, choking, took my head in her
hands, and kissed me on the forehead, and exclaimed, "Valentine!" I was
not greatly moved by this outburst, knowing that mamma, since she has
grown a little too stout, has some difficulty in getting upstairs.
I judged, therefore, that the wish to take breath for a moment without
appearing to do so had something to do with this sudden halt.
We entered the nuptial chamber; it was as coquettish as possible,
refreshing to the eye, snug, elegant, and adorned with fine Louis XVI
furniture, upholstered in Beauvais tapestry. The bed, above all, was a
marvel of elegance, but to tell the truth I had no idea of it till a week
later. At the outside it seemed to me that I was entering an austere-
looking locality; the very air we breathed appeared to me to have
something solemn and awe-striking about it.
"Here is your room, child," said mamma; "but first of all come and sit
here beside me, my dear girl."
At these words we both burst into tears, and mamma then expressed herself
as follows:
"The kiss you are giving me, Valentine, is the last kiss that I shall
have from you as a girl. Your husband--for Georges is that now--"
At these words I shuddered slightly, and by a singular freak of my brain
pictured to myself Monsieur Georges--Georges--my husband--in a cotton
night cap and a dressing-gown. The vision flashed across my mind in the
midst of the storm. I saw him just as plainly as if he had been there.
It was dreadful. The nightcap came over his forehead, down to his
eyebrows, and he said to me, pressing my hand; "At last, Valentine; you
are mine; do you love me? oh! tell me, do you love me?" And as his
head moved as he uttered these words, the horrible tuft at the end of his
nightcap waggled as an accompaniment.
"No," I said to myself, "it is impossible for my husband to appear in
such a fashion; let me banish this image--and yet my father wears the
hideous things, and my brother, who is quite young, has them already.
Men wear them at all ages, unless though--" It is frightful to relate,
but Georges now appeared to me with a red-and-green bandanna handkerchief
tied round his head. I would have given ten years of my life to be two
hours older, and hurriedly passed my hand across my eyes to drive away
these diabolical visions.
However, mamma, who had been still speaking all the time, attributing
this movement to the emotion caused by her words, said, with great
sweetness:
"Do not be alarmed, my dear Valentine; perhaps I am painting the picture
in too gloomy colors; but my experience and my love render this duty
incumbent upon me."
I have never heard mamma express herself so fluently. I was all the more
surprised as, not having heard a word of what she had already said, this
sentence seemed suddenly sprung upon me. Not knowing what to answer,
I threw myself into the arms of mamma, who, after a minute or so, put me
away gently, saying, "You are suffocating me, dear."
She wiped her eyes with her little cambric handkerchief, which was damp,
and said, smilingly:
"Now that I have told you what my conscience imposed on me, I am strong.
See, dear, I think that I can smile. Your husband, my dear child, is a
man full of delicacy. Have confidence; accept all without misgiving."
Mamma kissed me on the forehead, which finished off her sentence, and
added:
"Now, dear one, I have fulfilled a duty I regarded as sacred. Come here
and let me take your wreath off."
"By this time," I thought, "they have noticed that I have left the
drawing-room. They are saying, 'Where is the bride?' and smiling,
'Monsieur Georges is getting uneasy. What is he doing? what is he
thinking? where is he?'"
"Have you tried on your nightcap, dear?" said mamma, who had recovered
herself; "it looks rather small to me, but is nicely embroidered. Oh, it
is lovely!"
And she examined it from every point of view.
At that moment there was a knock at the door. "It is I," said several
voices, among which I distinguished the flute-like tones of my aunt
Laura, and those of my godmother. Madame de P., who never misses a
chance of pressing her two thick lips to some one's cheeks, accompanied
them. Their eyes glittered, and all three had a sly and triumphant look,
ferreting and inquisitive, which greatly intimidated me. Would they also
set about fulfilling a sacred duty?
"Oh, you are really too pretty, my angel!" said Madame de P., kissing me
on the forehead, after the moist fashion peculiar to her, and then
sitting down in the large Louis XVI armchair.
My maid had not been allowed to undress me, so that all of them, taking
off their gloves, set to work to render me this service. They tangled
the laces, caught their own lace in the hooks, and laughed heartily all
the while.
"It is the least that the oldest friend of the family," --she loved to
speak of herself as such-- "should make herself useful at such a moment,"
muttered Madame de P., holding her eyeglass in one hand and working with
the other.
I passed into a little boudoir to complete my toilette for the night,
and found on the marble of the dressing-table five or six bottles of
scent, tied up with red, white, and blue ribbons--an act of attention on
the part of my Aunt Laura. I felt the blood flying to my head; there was
an unbearable singing in my ears. Now that I can coolly weigh the
impressions I underwent, I can tell that what I felt above all was anger.
I would have liked to be in the farthest depths of the wildest forest in
America, so unseemly did I find this curious kindness which haunted me
with its attentions. I should have liked to converse a little with
myself, to fathom my own emotion somewhat, and, in short, to utter a
brief prayer before throwing myself into the torrent.
However, through the open door, I could hear the four ladies whispering
together and stifling their outbursts of laughter; I had never seen them
so gay. I made up my mind. I crossed the room, and, shaking off the
pretty little white slippers which my mother had embroidered for me,
jumped into bed. I was not long in finding out that it was no longer my
own narrow little bed. It was immense, and I hesitated a moment, not
knowing which way to turn. I felt nevertheless a feeling of physical
comfort. The bed was warm, and I do not know what scent rose from its
silken coverlet. I felt myself sink into the mass of feathers, the
pillows, twice over too large and trimmed with embroidery, gave way as it
were beneath me, burying me in a soft and perfumed abyss.
At length the ladies rose, and after giving a glance round the room,
doubtless to make sure that nothing was lacking, approached the bed.
"Good-night, my dear girl," said my mother, bending over me.
She kissed me, carried her handkerchief, now reduced to a wet dab, to her
eyes, and went out with a certain precipitation.
"Remember that the old friend of the family kissed you on this night, my
love," said Madame de P., as she moistened my forehead.
"Come, my little lamb, good-night and sleep well," said my aunt, with her
smile that seemed to issue from her nose. She added in a whisper: "You
love him, don't you? The slyboots! she won't answer! Well, since you
love him so much, don't tell him so, my dear. But I must leave you; you
are sleepy. Goodnight."
And she went away, smiling.
At length I was alone. I listened; the doors were being closed, I heard
a carriage roll along the road; the flame of the two candles placed upon
the mantelshelf quivered silently and were reflected in the looking-
glass.
I thought about the ceremony of that morning, the dinner, the ball.
I said to myself, clenching my fists to concentrate my thoughts: "How was
Marie dressed? She was dressed in--dressed in--dressed in--" I repeated
the words aloud to impart more authority to them and oblige my mind to
reply; but do what I would, it was impossible for me to drive away the
thought that invaded my whole being.
"He is coming. What is he doing? Where is he? Perhaps he is on the
stairs now. How shall I receive him when he comes?"
I loved him; oh! with my whole soul, I can acknowledge it now; but I
loved him quite at the bottom of my heart. In order to think of him I
went down into the very lowest chamber of my heart, bolted the door, and
crouched down in the darkest corner.
At last, at a certain moment, the floor creaked, a door was opened in the
passage with a thousand precautions, and I heard the tread of a boot--a
boot!
The boot ceased to creak, and I heard quite close to me, on the other
side of the wall, which was nothing but a thin partition, an armchair
being rolled across the carpet, and then a little cough, which seemed to
me to vibrate with emotion. It was he! But for the partition I could
have touched him with my finger. A few moments later I could distinguish
the almost imperceptible sound of footsteps on the carpet; this faint
sound rang violently in my head. All at once my breathing and my heart
both stopped together; there was a tap at the door. The tapping was
discreet, full of entreaty and delicacy. I wanted to reply, "Come in,"
but I had no longer any voice; and, besides, was it becoming to answer
like that, so curtly and plainly? I thought "Come in" would sound
horribly unseemly, and I said nothing. There was another tap. I should
really have preferred the door to have been broken open with a hatchet or
for him to have come down the chimney. In my agony I coughed faintly
among my sheets. That was enough; the door opened, and I divined from
the alteration in the light shed by the candles that some one at whom I
did not dare look was interposing between them and myself.
This some one, who seemed to glide across the carpet, drew near the bed,
and I could distinguish out of the corner of my eye his shadow on the
wall. I could scarcely restrain my joy; my Captain wore neither cotton
nightcap nor bandanna handkerchief. That was indeed something. However,
in this shadow which represented him in profile, his nose had so much
importance that amid all my uneasiness a smile flitted across my lips.
Is it not strange how all these little details recur to your mind? I did
not dare turn round, but I devoured with my eyes this shadow representing
my husband; I tried to trace in it the slightest of his gestures; I even
sought the varying expressions of his physiognomy, but, alas! in vain.
I do not know how to express in words all that I felt at that moment; my
pen seems too clumsy to write my sensations, and, besides, did I really
see deep into my heart?
Do men comprehend all this? Do they understand that the heart requires
gradual changes, and that if a half-light awakens, a noon-day blaze
dazzles and burns? It is not that the poor child, who is trembling in
a corner, refuses to learn; far from that, she has aptitude, good-will,
and a quick and ready intelligence; she knows she has reached the age at
which it is necessary to know how to read; she rejects neither the
science nor even the teacher. It is the method of instruction that makes
her uneasy. She is afraid lest this young professor, whose knowledge is
so extensive, should turn over the pages of the book too quickly and
neglect the A B C.
A few hours back he was the submissive, humble lover, ready to kneel down
before her, hiding his knowledge as one hides a sin, speaking his own
language with a thousand circumspections. At any moment it might have
been thought that he was going to blush. She was a queen, he a child;
and now all at once the roles are changed; it is the submissive subject
who arrives in the college cap of a professor, hiding under his arm an
unknown and mysterious book. Is the man in the college cap about to
command, to smile, to obtrude himself and his books, to speak Latin, to
deliver a lecture?
She does not know that this learned individual is trembling, too; that he
is greatly embarrassed over his opening lesson, that emotion has caused
him to forget his Latin, that his throat is parched and his legs are
trembling beneath him. She does not know this, and I tell you between
ourselves, it is not her self-esteem that suffers least at this
conjecture. She suffers at finding herself, after so many signatures,
contracts, and ceremonies-still a charming child, and nothing more.
I believe that the first step in conjugal life will, according to the
circumstances accompanying it, give birth to captivating sympathies or
invincible repulsion. But to give birth to these sympathies, to strike
the spark that is to set light to this explosion of infinite gratitude
and joyful love--what art, what tact, what delicacy, and at the same time
what presence of mind are needed.
How was it that at the first word Georges uttered my terrors vanished?
His voice was so firm and so sweet, he asked me so gayly for leave to
draw near the fire and warm his feet, and spoke to me with such ease and
animation of the incidents of the day. I said to myself, "It is
impossible for the least baseness to be hidden under all this."
In presence of so much good-humor and affability my scaffolding fell to
pieces. I ventured a look from beneath the sheets: I saw him comfortably
installed in the big armchair, and I bit my lips. I am still at a loss
to understand this little fit of ill-temper. When one is reckoning on a
fright, one is really disappointed at its delaying itself. Never had
Georges been more witty, more affectionate, more well-bred; he was still
the man of the day before. He must really have been very excited.
"You are tired out, I am certain, darling," he said.
The word "darling" made me start, but did not frighten me; it was the
first time he had called me so, but I really could not refuse him the
privilege of speaking thus. However it may be, I maintained my reserve,
and in the same tone as one replies, "No thanks, I don't take tea,"
I answered:
"Oh, yes! I am worn out."
"I thought so," he added, approaching the bed; "you can not keep your
eyes open; you can not even look at me, my dear little wife."
"I will leave you," continued he. "I will leave you; you need repose."
And he drew still more closely to me, which was not natural. Then,
stretching out his hand, which I knew was white and well cared for:
"Won't you give me a little shake of the hand, dear? I am half asleep,
too, my pretty little wife." His face wore an expression which was
alarming, though not without its charm; as he said this, I saw clearly
that he had lied to me like a demon, and that he was no more sleepy than
I was.
However that may be, I was guilty of the fault, the carelessness that
causes disaster, of letting him take my hand, which was straying by
chance under the lace of the pillows.
I was that evening in a special condition of nervous sensibility, for at
this contact a strange sensation ran through me from head to foot. It
was not that the Captain's hand had the softness of satin--I believe that
physical sensations, in us women, have causes directly contrary to those
which move men; for that which caused me such lively emotion was
precisely its firmness. There was something strong, manly, and powerful
about it. He squeezed my hand rather strongly.
My rings, which I have a fancy for wearing all at once, hurt me, and--
I really should not have believed it--I liked it very much, perhaps too
much. For the first time I found an inexplicable, an almost
intoxicating, charm in this intimate contact with a being who could have
crushed me between his fingers, and that in the middle of the night too,
in silence, without any possibility of help. It was horribly delicious.
I did not withdraw my hand, which he kissed, but lingeringly. The clock
struck two, and the last sound had long since died away when his lips
were still there, quivering with rapid little movements, which were so
many imperceptible kisses, moist, warm, burning. I felt gleams of fire
flashing around me. I wished to draw away my hand, but could not; I
remember perfectly well that I could not. His moustache pricked me, and
whiffs of the scent with which he perfumed it reached me and completed my
trouble. I felt my nostrils dilating despite myself, and, striving but
in vain to take refuge in my inmost being, I exclaimed inwardly: "Protect
me, Lord, but this time with all your might. A drop of water, Lord; a
drop of water!" I waited--no appreciable succor reached from above. It
was not till a week afterward that I understood the intentions of
Providence.
"You told me you were sleepy," I murmured, in a trembling voice. I was
like a shipwrecked person clutching at a floating match-box; I knew quite
well that the Captain would not go away.
"Yes, I was sleepy, pet," said Georges, approaching his face to mine;
"but now I am athirst." He put his lips to my ear and whispered softly,
"Athirst for a kiss from you, love."
This "love" was the beginning of another life. The spouse now appeared,
the past was fleeing away, I was entering on the future. At length I had
crossed the frontier; I was in a foreign land. Oh! I acknowledge--for
what is the use of feigning?--that I craved for this love, and I felt
that it engrossed me and spread itself through me. I felt that I was
getting out of my depth, I let go the last branch that held me to the
shore, and to myself I repeated: "Yes, I love you; yes, I am willing to
follow you; yes, I am yours, love, love, love!"
"Won't you kiss your husband; come, won't you?"
And his mouth was so near my own that it seemed to meet my lips.
"Yes," said I.
.............................
August 7th, 185- How many times have I not read through you during the
last two years, my little blue note-book! How many things I might add as
marginal notes if you were not doomed to the flames, to light my first
fire this autumn! How could I have written all this, and how is it that
having done so I have not dared to complete my confidences! No one has
seen you, at any rate; no one has turned your pages. Go back into your
drawer, dear, with, pending the first autumn fire, a kiss from your
Valentine.
NOTE.--Owing to what circumstances this blue note-book, doomed to the
flames, was discovered by me in an old Louis XVI chiffonnier I had just
bought does not greatly matter to you, dear reader, and would be out of
my power to explain even if it did.
CHAPTER XIV
THE BLUE NOTE-BOOK AGAIN
Only to think that I was going to throw you into the fire, poor dear!
Was I not foolish? In whom else could I confide? If I had not you,
to whom could I tell all those little things at which every one laughs,
but which make you cry!
This evening, for instance, I dined alone, for Georges was invited out;
well, to whom else can I acknowledge that when I found myself alone,
face to face with a leg of mutton, cooked to his liking, and with the
large carving-knife which is usually beside his plate, before me, I began
to cry like a child? To whom else can I admit that I drank out of the
Bohemian wine-glass he prefers, to console me a little?
But if I were to mention this they would laugh in my face. Father
Cyprien himself, who nevertheless has a heart running over with kindness,
would say to me:
"Let us pass that by, my dear child; let us pass that by."
I know him so well, Father Cyprien; while you, you always listen to me,
my poor little note-book; if a tear escapes me, you kindly absorb it and
retain its trace like a good-hearted friend. Hence I love you.
And, since we are tete-a-tete, let us have a chat. You won't be angry
with me for writing with a pencil, dear. You see I am very comfortably
settled in my big by-by and I do not want to have any ink-stains. The
fire sparkles on the hearth, the street is silent; let us forget that
George will not return till midnight, and turn back to the past.
I can not recall the first month of that dear past without laughing and
weeping at one and the same time.
How foolish we were! How sweet it was! There is a method of teaching
swimming which is not the least successful, I am told. It consists in
throwing the future swimmer into the water and praying God to help him.
I am assured that after the first lesson he keeps himself afloat.
Well, I think that we women are taught to be wives in very much the same
fashion.
Happy or otherwise--the point is open to discussion marriage is a
hurricane--something unheard-of and alarming.
In a single night, and without any transition, everything is transformed
and changes color; the erst while-cravatted, freshly curled, carefully
dressed gentleman makes his appearance in a dressing-gown. That which
was prohibited becomes permissible, the code is altered, and words
acquire a meaning they never had before, et cetera, et cetera.
It is not that all this is so alarming, if taken the right way--a woman
with some courage in her heart and some flexibility in her mind supports
the shock and does not die under it; but the firmest of us are amazed at
it, and stand open-mouthed amid all these strange novelties, like a
penniless gourmand in the shop of Potel and Chabot.
They dare not touch these delicacies surrounding them, though invited to
taste. It is not that the wish or the appetite is lacking to them, but
all these fine fruits have been offered them so lately that they have
still the somewhat acid charm of green apples or forbidden fruit. They
approach, but they hesitate to bite.
After all, why complain? What would one have to remember if one had
entered married life like an inn, if one had not trembled a little when
knocking at the door? And it is so pleasant to recall things, that one
would sometimes like to deck the future in the garments of the past.
It was, I recollect, two days after the all-important one. I had gone
into his room, I no longer remember why--for the pleasure of going in,
I suppose, and thereby acting as a wife. A strong desire is that which
springs up in your brain after leaving church to look like an old married
woman. You put on caps with ribbons, you never lay aside your cashmere
shawl, you talk of "my home"--two sweet words--and then you bite your
lips to keep from breaking out into a laugh; and "my husband," and "my
maid," and the first dinner you order, when you forget the soup. All
this is charming, and, however ill at ease you may feel at first in all
these new clothes, you are quite eager to put them on.
So I had gone into the dressing-room of my husband, who, standing before
the glass, very lightly clad, was prosaically shaving.
"Excuse me, dear," said he, laughing, and he held up his shaving-brush,
covered with white lather. "You will pardon my going on with this. Do
you want anything?"
"I came, on the contrary," I answered, "to see whether you had need of
anything;" and, greatly embarrassed myself, for I was afraid of being
indiscreet, and I was not sure whether one ought to go into one's
husband's room like this, I added, innocently, "Your shirts have buttons,
have they not?"
"Oh, what a good little housewife I have married! Do not bother yourself
about such trifles, my pet. I will ask your maid to look after my
buttons," said he.
I felt confused; I was afraid of appealing too much of a schoolgirl in
his eyes. He went on working his soap into a lather with his shaving-
brush. I wanted to go away, but I was interested in such a novel fashion
by the sight of my husband, that I had not courage to do so. His neck
was bare--a thick, strong neck, but very white and changing its shape at
every movement--the muscles, you know. It would have been horrible in a
woman, that neck, and yet it did not seem ugly to me. Nor was it
admiration that thus inspired me; it was rather like gluttony. I wanted
to touch it. His hair, cut very short--according to regulation--grew
very low, and between its beginning and the ear there was quite a smooth
white place. The idea at once occurred to me that if ever I became brave
enough, it was there that I should kiss him oftenest; it was strange,
that presentiment, for it is in fact on that little spot that I--
He stopped short. I fancied I understood that he was afraid of appearing
comical in my eyes, with his face smothered in lather; but he was wrong.
I felt myself all in a quiver at being beside a man--the word man is
rather distasteful to me, but I can not find another, for husband would
not express my thoughts--at being beside a man in the making of his
toilette. I should have liked him to go on without troubling himself;
I should have liked to see how he managed to shave himself without
encroaching on his moustache, how he made his parting and brushed his
hair with the two round brushes I saw on the table, what use he made of
all the little instruments set out in order on the marble-tweezers,
scissors, tiny combs, little pots and bottles with silver tops, and a
whole arsenal of bright things, that aroused quite a desire to beautify
one's self.
I should have liked him while talking to attend to the nails of his
hands, which I was already very fond of; or, better still, to have handed
them over to me. How I should have rummaged in the little corners, cut,
filed, arranged all that.
"Well, dear, what are you looking at me like that for?" said he,
smiling.
I lowered my eyes at once, and felt that I was blushing. I was uneasy,
although charmed, amid these new surroundings. I did not know what to
answer, and mechanically I dipped the tip of my finger into the little
china pot in which the soap was being lathered.
"What is the matter, darling?" said he, approaching his face to mine;
"have I offended you?"
I don't know what strange idea darted through my mind, but I suddenly
took my hand from the pot and stuck the big ball of lather at the end of
my finger on the tip of his nose. He broke out into a hearty laugh, and
so did I; though I trembled for a moment, lest he should be angry.
"So that's the way in which you behave to a captain in the lancers? You
shall pay for this, you wicked little darling;" and, taking the shaving
brush in his hand, he chased me round the room. I dodged round the
table, I took refuge behind the armchair, upsetting his boots with my
skirt, getting the tongs at the same time entangled in it. Passing the
sofa, I noticed his uniform laid out--he had to wait on the General that
morning--and, seizing his schapska, I made use of it as a buckler. But
laughter paralyzed me, and besides, what could a poor little woman do
against a soldier, even with a buckler?
He ended by catching me--the struggle was a lovely one. It was all very
well for me to scream, as I threw my head backward over the arm by which
he clasped me; I none the less saw the frightful brush, like a big
snowball, at the end of a little stick, come nearer and yet nearer.
But he was merciful; he was satisfied with daubing a little white spot on
my chin and exclaiming, "The cavalry have avenged themselves."
Seizing the brush in turn, I said to him roguishly, "Captain, let me
lather your face," for I did so want to do that.
In answer, he held his face toward me, and, observing that I was obliged
to stand on the tips of my toes and to support myself a little on his
shoulder, he knelt down before me and yielded his head to me.
With the tip of my finger I made him bend his face to the right and the
left, backward and forward, and I lathered and lathered, giggling like a
schoolgirl. It amused me so to see my Captain obey me like a child;
I would have given I don't know what if he had only had his sword and
spurs on at that moment. Unfortunately, he was in his slippers. I
spread the lather over his nose and forehead; he closed his eyes and put
his two arms round me, saying:
"Go on, my dear, go on; but see that you don't put any into my mouth."
At that moment I experienced a very strange feeling. My laughter died
away all at once; I felt ashamed at seeing my husband at my feet and at
thus amusing myself with him as if he were a doll.
I dropped the shaving-brush; I felt my eyes grow moist; and, suddenly,
becoming more tender, I bent toward him and kissed him on the neck, which
was the only spot left clear.
Yet his ear was so near that, in passing it, my lips moved almost in
spite of myself, and I whispered:
"Don't be angry, dear," then, overcome by emotion and repentance,
I added: "I love you, I do love you."
"My own pet!" he said, rising suddenly. His voice shook.
What delightful moments these were! Unfortunately, oh! yes, indeed,
unfortunately, he could not press his lathered face to mine!
"Wait a little," he exclaimed, darting toward the washbasin, full of
water, "wait an instant!"
But it seemed as if it took him a week to wash it off.
CHAPTER XV
MY WIFE GOES TO A DANCE
Madame--Ah! it is so nice of you to come home early! (Looking at the
clock.) A quarter to six. But how cold you are! your hands are frozen;
come and sit by the fire. (She puts a log on the fire.) I have been
thinking of you all day. It is cruel to have to go out in such weather.
Have you finished your doubts? are you satisfied?
Monsieur--Quite well satisfied, dear. (Aside.) But I have never known my
wife to be so amiable. (Aloud, taking up the bellows.) Quite well
satisfied, and I am very hungry. Has my darling been good?
Madame--You are hungry. Good! (Calling out.) Marie, call into the
kitchen that your master wants to dine early. Let them look after
everything--and send up a lemon.
Monsieur--A mystery?
Madame--Yes, Monsieur, I have a little surprise for you, and I fancy that
it will delight you.
Monsieur--Well, what is the surprise?
Madame--Oh! it is a real surprise. How curious you look! your eyes are
glittering already. Suppose I were not to tell you anything?
Monsieur--Then you would vex me very much.
Madame--There, I don't want to vex you. You are going to have some
little green oysters and a partridge. Am I good?
Monsieur--Oysters and a partridge! You are an angel. (He kisses her.)
An angel. (Aside.) What on earth is the matter with her? (Aloud.) Have
you had visitors to-day?
Madame--I saw Ernestine this morning, but she only stayed a moment. She
has just discharged her maid. Would you believe it, that girl was seen
the night before last dressed up as a man, and in her master's clothes,
too! That was going too far.
Monsieur--That comes of having confidential servants. And you just got a
sight of Ernestine?
Madame--And that was quite enough, too. (With an exclamation.) How
stupid I am! I forgot. I had a visit from Madame de Lyr as well.
Monsieur--God bless her! But does she still laugh on one side of her
mouth to hide her black tooth?
Madame-How cruel you are! Yet, she likes you very well. Poor woman!
I was really touched by her visit. She came to remind me that we--
now you will be angry. (She kisses him and sits down beside him.)
Monsieur--Be angry! be angry! I'm not a Turk. Come, what is it?
Madame--Come, we shall go to dinner. You know that there are oysters and
a partridge. I won't tell you--you are already in a bad temper.
Besides, I all but told her that we are not going.
Monsieur--(raising his hands aloft)--I thought so. She and her evening
may go to the dogs. What have I done to this woman that she should so
pester me?
Madame--But she thinks she is affording you pleasure. She is a charming
friend. As for me, I like her because she always speaks well of you.
If you had been hidden in that cabinet during her visit, you could not
have helped blushing. (He shrugs his shoulders.) "Your husband is so
amiable," she said to me, "so cheery, so witty. Try to bring him; it is
an honor to have him." I said, "Certainly," but without meaning it, you
know. But I don't care about it at all. It is not so very amusing at
Madame de Lyr's. She always invites such a number of serious people. No
doubt they are influential people, and may prove useful, but what does
that matter to me? Come to dinner. You know that there is a bottle left
of that famous Pomard; I have kept it for your partridge. You can not
imagine what pleasure I feel in seeing you eat a partridge. You eat it
with such a gusto. You are a glutton, my dear. (She takes his arm.)
Come, I can hear your rascal of a son getting impatient in the dining-
room.
Monsieur--(with a preoccupied air)--Hum! and when is it?
Madame--When is what?
Monsieur--The party, of course.
Madame--Ah! you mean the ball--I was not thinking of it. Madame de Lyr's
ball. Why do you ask me that, since we are not going? Let us make
haste, dinner is getting cold . . . . This evening.
Monsieur--(stopping short)--What! this party is a ball, and this ball is
for this evening. But, hang it! people don't invite you to a ball like
that. They always give notice some time beforehand.
Madame--But she sent us an invitation a week ago, though I don't know
what became of the card. I forgot to show it to you.
Monsieur--You forgot! you forgot!
Madame--Well, it is all for the best; I know you would have been sulky
all the week after. Come to dinner.
They sat down to table. The cloth was white, the cutlery bright, the
oysters fresh; the partridge, cooked to perfection, exhaled a delightful
odor. Madame was charming, and laughed at everything. Monsieur unbent
his brows and stretched himself on the chair.
Monsieur--This Pomard is very good. Won't you have some, little dear?
Madame--Yes, your little dear will. (She pushes forward her glass with a
coquettish movement.)
Monsieur--Ah! you have put on your Louis Seize ring. It is a very pretty
ring.
Madame--(putting her hand under her husband's nose)--Yes; but look--see,
there is a little bit coming off.
Monsieur--(kissing his wife's hand)--Where is the little bit?
Madame--(smiling)--You jest at everything. I am speaking seriously.
There--look--it is plain enough! (They draw near once another and bend
their heads together to see it.) Don't you see it? (She points out a
spot on the ring with a rosy and slender finger.) There! do you see now
--there?
Monsieur--That little pearl which--What on earth have you been putting on
your hair, my dear? It smells very nice--You must send it to the
jeweller. The scent is exquisite. Curls don't become you badly.
Madame--Do you think so? (She adjusts her coiffure with her white hand.)
I thought you would like that scent; now, if I were in your place I
should--
Monsieur--What would you do in my place, dear?
Madame--I should--kiss my wife.
Monsieur--(kissing her)--Well, I must say you have very bright ideas
sometimes. Give me a little bit more partridge, please. (With his mouth
full.) How pretty these poor little creatures look when running among the
corn. You know the cry they give when the sun sets?--A little gravy.--
There are moments when the poetic side of country life appeals to one.
And to think that there are barbarians who eat them with cabbage. But
(filling his glass) have you a gown ready?
Madame--(with innocent astonishment.)--What for, dear?
Monsieur--Why, for Madame de Lyr's--
Madame--For the ball?--What a memory you have--There you are still
thinking of it--No, I have not--ah! yes, I have my tarletan, you know;
but then a woman needs so little to make up a ball-room toilette.
Monsieur--And the hairdresser, has he been sent for?
Madame--No, he has not been sent for; but I am not anxious to go to this
ball. We will settle down by the fireside, read a little, and go to bed
early. You remind me, however, that, on leaving, Madame de Lyr did say,
"Your hairdresser is the same as mine, I will send him word." How stupid
I am; I remember now that I did not answer her. But it is not far, I can
send Marie to tell him not to come.
Monsieur--Since this blessed hairdresser has been told, let him come and
we will go and--amuse ourselves a little at Madame de Lyr's. But on one
condition only; that I find all my dress things laid out in readiness on
my bed with my gloves, you know, and that you tie my necktie.
Madame--A bargain. (She kisses him.) You are a jewel of a husband. I am
delighted, my poor dear, because I see you are imposing a sacrifice upon
yourself in order to please me; since, as to the ball itself, I am quite
indifferent about it. I did not care to go; really now I don't care to
go.
Monsieur--Hum. Well, I will go and smoke a cigar so as not to be in your
way, and at ten o'clock I will be back here. Your preparations will be
over and in five minutes I shall be dressed. Adieu.
Madame--Au revoir.
Monsieur, after reaching the street, lit his cigar and buttoned up his
great-coat. Two hours to kill. It seems a trifle when one is busy, but
when one has nothing to do it is quite another thing. The pavement is
slippery, rain is beginning to fall--fortunately the Palais Royal is not
far off. At the end of his fourteenth tour round the arcades, Monsieur
looks at his watch. Five minutes to ten, he will be late. He rushes
home.
In the courtyard the carriage is standing waiting.
In the bedroom two unshaded lamps shed floods of light. Mountains of
muslin and ribbons are piled on the bed and the furniture. Dresses,
skirts, petticoats, and underpetticoats, lace, scarfs, flowers, jewels,
are mingled in a charming chaos. On the table there are pots of pomade,
sticks of cosmetic, hairpins, combs and brushes, all carefully set out.
Two artificial plaits stretch themselves languishingly upon a dark mass
not unlike a large handful of horsehair. A golden hair net, combs of
pale tortoise-shell and bright coral, clusters of roses, sprays of white
lilac, bouquets of pale violets, await the choice of the artist or the
caprice of the beauty. And yet, must I say it? amidst this luxury of
wealth Madame's hair is undressed, Madame is uneasy, Madame is furious.
Monsieur--(looking at his watch)--Well, my dear, is your hair dressed?
Madame--(impatiently)--He asks me whether my hair is dressed? Don't you
see that I have been waiting for the hairdresser for an hour and a half?
Can't you see that I am furious, for he won't come, the horrid wretch?
Monsieur--The monster!
Madame--Yes, the monster; and I would advise you not to joke about it.
There is a ring. The door opens and the lady's-maid exclaims, "It is he,
Madame!"
Madame--It is he!
Monsieur--It is he!
The artist enters hurriedly and bows while turning his sleeves up.
Madame--My dear Silvani, this is unbearable.
Silvani--Very sorry, very, but could not come any sooner. I have been
dressing hair since three o'clock in the afternoon. I have just left the
Duchesse de W., who is going to the Ministry this evening. She sent me
home in her brougham. Lisette, give me your mistress's combs, and put
the curling-tongs in the fire.
Madame--But, my dear Silvani, my maid's name is not Lisette.
Silvani--You will understand, Madame, that if I had to remember the names
of all the lady's-maids who help me, I should need six clerks instead of
four. Lisette is a pretty name which suits all these young ladies very
well. Lisette, show me your mistress's dress. Good. Is the ball an
official one?
Madame--But dress my hair, Silvani.
Silvani--It is impossible for me to dress your hair, Madame, unless I
know the circle in which the coiffure will be worn. (To the husband,
seated in the corner.) May I beg you, Monsieur, to take another place?
I wish to be able to step back, the better to judge the effect.
Monsieur--Certainly, Monsieur Silvani, only too happy to be agreeable to
you. (He sits down on a chair.)
Madame--(hastily)--Not there, my dear, you will rumple my skirt. (The
husband gets up and looks for another seat.) Take care behind you, you
are stepping on my bustle.
Monsieur--(turning round angrily)--Her bustle! her bustle!
Madame--Now you go upsetting my pins.
Silvani--May I beg a moment of immobility, Madame?
Monsieur--Come, calm yourself, I will go into the drawing-room; is there
a fire there?
Madame--(inattentively)--But, my dear, how can you expect a fire to be in
the drawing-room?
Monsieur--I will go to my study, then.
Madame--There is none there, either. What do you want a fire in your
study for? What a singular idea! High up, you know, Silvani, and a dash
of disorder, it is all the rage.
Silvani--Would you allow a touch of brown under the eyes? That would
enable me to idealize the coiffure.
Monsieur--(impatiently)--Marie, give me my top-coat and my cap. I will
walk up and down in the anteroom. (Aside.) Madame de Lyr shall pay for
this.
Silvani--(crimping)--I leave your ear uncovered, Madame; it would be a
sin to veil it. It is like that of the Princesse de K., whose hair I
dressed yesterday. Lisette, get the powder ready. Ears like yours,
Madame, are not numerous.
Madame--You were saying--
Silvani--Would your ear, Madame, be so modest as not to listen?
Madame's hair is at length dressed. Silvani sheds a light cloud of
scented powder over his work, on which he casts a lingering look of
satisfaction, then bows and retires.
In passing through the anteroom, he runs against Monsieur, who is walking
up and down.
Silvani--A thousand pardons, I have the honor to wish you good night.
Monsieur--(from the depths of his turned-up collar) Good-night.
A quarter of an hour later the sound of a carriage is heard. Madame is
ready, her coiffure suits her, she smiles at herself in the glass as she
slips the glove-stretchers into the twelve-button gloves.
Monsieur has made a failure of his necktie and broken off three buttons.
Traces of decided ill-humor are stamped on his features.
Monsieur--Come, let us go down, the carriage is waiting; it is a quarter
past eleven. (Aside.) Another sleepless night. Sharp, coachman; Rue de
la Pepiniere, number 224.
They reach the street in question. The Rue de la Pepiniere is in a
tumult. Policemen are hurriedly making way through the crowd. In the
distance, confused cries and a rapidly approaching, rumbling sound are
heard. Monsieur thrusts his head out of the window.
Monsieur--What is it, Jean?
Coachman--A fire, Monsieur; here come the firemen.
Monsieur--Go on all the same to number 224.
Coachman--We are there, Monsieur; the fire is at number 224.
Doorkeeper of the House--(quitting a group of people and approaching the
carriage)--You are, I presume, Monsieur, one of the guests of Madame de
Lyr? She is terror-stricken; the fire is in her rooms. She can not
receive any one.
Madame--(excitedly)--It is scandalous.
Monsieur--(humming)--Heart-breaking, heartbreaking! (To the coachman.)
Home again, quickly; I am all but asleep. (He stretches himself out and
turns up his collar.) ( Aside.) After all, I am the better for a well-
cooked partridge.
CHAPTER XVI
A FALSE ALARM
Every time I visit Paris, which, unhappily, is too often, it rains in
torrents. It makes no difference whether I change the time of starting
from that which I had fixed upon at first, stop on the way, travel at
night, resort, in short, to a thousand devices to deceive the barometer-
at ten leagues from Paris the clouds begin to pile up and I get out of
the train amidst a general deluge.
On the occasion of my last visit I found myself as usual in the street,
followed by a street porter carrying my luggage and addressing despairing
signals to all the cabs trotting quickly past amid the driving rain.
After ten minutes of futile efforts a driver, more sensible than the
others, and hidden in his triple cape, checks his horses. With a single
bound I am beside the cab, and opening, the door with a kind of frenzy,
jump in.
Unfortunately, while I am accomplishing all this on one side, a
gentleman, similarly circumstanced, opens the other door and also jumps
in. It is easy to understand that there ensues a collision.
"Devil take you!" said my rival, apparently inclined to push still
farther forward.
I was about to answer him, and pretty sharply, too, for I hail from the
south of France and am rather hotheaded, when our eyes met. We looked
one another in the face like two lions over a single sheep, and suddenly
we both burst out laughing. This angry gentleman was Oscar V., that dear
good fellow Oscar, whom I had not seen for ten years, and who is a very
old friend of mine, a charming fellow whom I used to play with as a boy.
We embraced, and the driver, who was looking at us through the window,
shrugged his shoulders, unable to understand it all. The two porters,
dripping with water, stood, one at each door, with a trunk on his
shoulder. We had the luggage put on the cab and drove off to the Hotel
du Louvre, where Oscar insisted on dropping me.
"But you are travelling, too, then?" said I to my friend, after the
first moments of expansion. "Don't you live in Paris?"
"I live in it as little as possible and have just come up from Les
Roches, an old-fashioned little place I inherited from my father, at
which I pass a great deal of the year. Oh! it is not a chateau; it is
rustic, countrified, but I like it, and would not change anything about
it. The country around is fresh and green, a clear little river flows
past about forty yards from the house, amid the trees; there is a mill in
the background, a spreading valley, a steeple and its weather-cock on the
horizon, flowers under the windows, and happiness in the house. Can I
grumble? My wife makes exquisite pastry, which is very agreeable to me
and helps to whiten her hands. By the way, I did not tell you that I am
married. My dear fellow, I came across an angel, and I rightly thought
that if I let her slip I should not find her equal. I did wisely. But I
want to introduce you to my wife and to show you my little place. When
will you come and see me? It is three hours from Paris--time to smoke a
couple of cigars. It is settled, then--I am going back to-morrow morning
and I will have a room ready for you. Give me your card and I will write
down my address on it."
All this was said so cordially that I could not resist my friend's
invitation, and promised to visit him.
Three or four days later, Paris being empty and the recollection of my
old companion haunting me, I felt a strong desire to take a peep at his
conjugal felicity and to see with my own eyes this stream, this mill,
this steeple, beside all which he was so happy.
I reached Les Roches at about six in the evening and was charmed at the
very first glance. Oscar's residence was a little Louis Quinze chateau
buried in the trees; irregularly built, but charmingly picturesque. It
had been left unaltered for a century at least, and everything, from the
blackened mansard roofs with their rococo weather-cocks, to the bay
windows with their tiny squares of glass and the fantastic escutcheon
over the door, was in keeping. Over the thick tiles of the somewhat
sunken roof, the rough-barked old chestnuts lazily stretched their
branches. Creepers and climbing roses wantoned over the front, framing
the windows, peeping into the garrets, and clinging to the waterspouts,
laden with large bunches of flowers which swayed gently in the air. Amid
all these pointed roofs and this profusion of verdure and trees the blue
sky could only be caught a glimpse of here and there.
The first person I saw was Oscar, clad in white from head to foot, and
wearing a straw hat. He was seated on an enormous block of stone which
seemed part and parcel of the house, and appeared very much interested in
a fine melon which his gardener had just brought to him. No sooner had
he caught sight of me than he darted forward and grasped me by the hand
with such an expression of good-humor and affection that I said to
myself, "Yes, certainly he was not deceiving me, he is happy." I found
him just as I had known him in his youth, lively, rather wild, but kind
and obliging.
"Pierre," said he to the gardener, "take this gentleman's portmanteau to
the lower room," and, as the gardener bestirred himself slowly and with
an effort, Oscar seized the portmanteau and swung it, with a jerk, on to
the shoulders of the poor fellow, whose legs bent under the weight.
"Lazybones," said Oscar, laughing heartily. "Ah! now I must introduce
you to my little queen. My wife, where is my wife?"
He ran to the bell and pulled it twice. At once a fat cook with a red
face and tucked-up sleeves, and behind her a man-servant wiping a plate,
appeared at the ground-floor windows. Had they been chosen on purpose?
I do not know, but their faces and bearing harmonized so thoroughly with
the picture that I could not help smiling.
"Where is your mistress?" asked Oscar, and as they did not answer
quickly enough he exclaimed, "Marie, Marie, here is my friend George."
A young girl, fair as a lily, appeared at a narrow, little window, the
one most garlanded by, flowers, on the first floor. She was clad in a
white dressing-gown of some particular shape; I could not at first make
out. With one hand she gathered its folds about her, and with the other
restrained her flowing hair. Hardly had she seen me when she blushed,
somewhat ashamed, no doubt, at having been surprised in the midst of her
toilet, and, giving a most embarrassed yet charming bow; hurriedly
disappeared. This vision completed the charm; it seemed to me that I had
suddenly been transported into fairy-land. I had fancied when strapping
my portmanteau that I should find my friend Oscar installed in one of
those pretty, little, smart-looking houses, with green shutters and gilt
lightning-conductor, dear to the countrified Parisian, and here I found
myself amid an ideal blending of time-worn stones hidden in flowers,
ancient gables, and fanciful ironwork reddened by rust. I was right in
the midst of one of Morin's sketches, and, charmed and stupefied, I stood
for some moments with my eyes fixed on the narrow window at which the
fair girl had disappeared.
"I call her my little queen," said Oscar, taking my arm. "It is my wife.
Come this way, we shall meet my cousin who is fishing, and two other
friends who are strolling about in this direction, good fellows, only
they do not understand the country as I do--they have on silk stockings
and pumps, but it does not matter, does it? Would you like a pair of
slippers or a straw hat?
I hope you have brought some linen jackets. I won't offer you a glass of
Madeira--we shall dine at once. Ah! my dear fellow, you have turned up
at the right moment; we are going to taste the first melon of the year
this evening."
"Unfortunately, I never eat melons, though I like to see others do so."
"Well, then, I will offer you consolation by seeking out a bottle of my
old Pomard for you. Between ourselves, I don't give it to every one; it
is a capital wine which my poor father recommended to me on his deathbed;
poor father, his eyes were closed, and his head stretched back on the
pillow. I was sitting beside his bed, my hand in his, when I felt it
feebly pressed. His eyes half opened, and I saw him smile. Then he said
in a weak, slow, and the quavering voice of an old man who is dying: 'The
Pomard at the farther end--on the left--you know, my boy--only for
friends.' He pressed my hand again, and, as if exhausted, closed his
eyes, though I could see by the imperceptible motion of his lips that he
was still smiling inwardly. Come with me to the cellar," continued
Oscar, after a brief silence, "at the farther end to the left, you shall
hold the lantern for me."
When we came up from the cellar, the bell was ringing furiously, and
flocks of startled birds were flying out of the chestnut-trees. It was
for dinner. All the guests were in the garden. Oscar introduced me in
his off-hand way, and I offered my arm to the mistress of the house to
conduct her to the dining-room.
On examining my friend's wife, I saw that my first impression had not
been erroneous--she was literally a little angel, and a little angel in
the shape of a woman, which is all the better. She was delicate, slender
as a young girl; her voice was as thrilling and harmonious as the
chaffinch, with an indefinable accent that smacked of no part of the
country in particular, but lent a charm to her slightest word. She had,
moreover, a way of speaking of her own, a childish and coquettish way of
modulating the ends of her sentences and turning her eyes toward her
husband, as if to seek for his approbation. She blushed every moment,
but at the same time her smile was so bewitching and her teeth so white
that she seemed to be laughing at herself. A charming little woman!
Add to this a strange yet tasteful toilette, rather daring, perhaps,
but suiting this little queen, so singular in herself. Her beautiful
fair hair, twisted up apparently at hazard, was fixed rather high up on
the head by a steel comb worn somewhat on one side; and her white muslin
dress trimmed with wide, flat ruches, cut square at the neck, short in
the skirt, and looped up all round, had a delicious eighteenth-century
appearance. The angel was certainly a trifle coquettish, but in her own
way, and yet her way was exquisite.
Hardly were we seated at table when Oscar threw toward his little queen
a rapid glance, but one so full of happiness and-why should I not say it?
--love that I experienced a kind of shiver, a thrill of envy,
astonishment, and admiration, perhaps. He took from the basket of
flowers on the table a red rose, scarcely opened, and, pushing it toward
her, said with a smile:
"For your hair, Madame."
The fair girl blushed deeply, took the flower, and, without hesitation,
quickly and dexterously stuck it in her hair, high up on the left, just
in the right spot, and, delightedly turning round to each of us, repeated
several times, amid bursts of laughter, "Is it right like that?"
Then she wafted a tiny kiss with the tips of her fingers to her husband,
as a child of twelve would have done, and gayly plunged her spoon into
the soup, turning up her little finger as she did so.
The other guests had nothing very remarkable about them; they laughed
very good-naturedly at these childish ways, but seemed somewhat out of
place amid all this charming freedom from restraint. The cousin, above
all, the angler, with his white waistcoat, his blue tie, his full beard,
and his almond eyes, especially displeased me. He rolled his r's like an
actor at a country theatre. He broke his bread into little bits and
nibbled them as he talked. I divined that the pleasure of showing off
a large ring he wore had something to do with this fancy for playing with
his bread. Once or twice I caught a glance of melancholy turned toward
the mistress of the house, but at first I did not take much notice of it,
my attention being attracted by the brilliant gayety of Oscar.
It seemed to me, however, at the end of a minute or so, that this young
man was striving in a thousand ways to engage the attention of the little
queen.
The latter, however, answered him in the most natural way in the world,
neither betraying constraint nor embarrassment. I was mistaken,
no doubt. Have you ever noticed, when you are suddenly brought into the
midst of a circle where you are unacquainted, how certain little details,
matters of indifference to every one else, assume importance in your
eyes? The first impression is based upon a number of trifles that catch
your attention at the outset. A stain in the ceiling, a nail in the
wall, a feature of your neighbor's countenance impresses itself upon your
mind, installs itself there, assumes importance, and, in spite of
yourself, all the other observations subsequently made by you group
around this spot, this nail, this grimace. Think over it, dear reader,
and you will see that every opinion you may have as to a fact, a person,
or an object has been sensibly influenced by the recollection of the
little trifle that caught your eye at the first glance. What young girl
victim of first impressions has not refused one or two husbands on
account of a waistcoat too loose, a cravat badly tied, an inopportune
sneeze, a foolish smile, or a boot too pointed at the toe?
One does not like admitting to one's self that such trifles can serve as
a base to the opinion one has of any one, and one must seek attentively
in order to discover within one's mind these unacknowledged germs.
I recollect quite well that the first time I had the honor of calling on
Madame de M., I noticed that one of her teeth, the first molar on the
right, was quite black. I only caught a glimpse of the little black
monster, such was the care taken to hide it, yet I could not get this
discovery out of my head. I soon noticed that Madame de M. made
frightful grimaces to hide her tooth, and that she took only the smallest
possible mouthfuls at table to spare the nervous susceptibilities of the
little monster.
I arrived at the pitch of accounting for all the mental and physical
peculiarities of Madame de M. by the presence of this slight blemish,
and despite myself this black tooth personified the Countess so well that
even now, although it has been replaced by another magnificent one, twice
as big and as white as the bottom of a plate, even now, I say, Madame de
M. can not open her mouth without my looking naturally at it.
But to return to our subject. Amid all this conjugal happiness, so
delightfully surrounded, face to face with dear old Oscar, so good, so
confiding, so much in love with this little cherub in a Louis XV dress,
who carried grace and naivete to so strange a pitch, I had been struck by
the too well combed and foppish head of the cousin in the white
waistcoat. This head had attracted my attention like the stain on the
ceiling of which I spoke just now, like the Countess's black tooth, and
despite myself I did not take my eyes off the angler as he passed the
silver blade of his knife through a slice of that indigestible fruit
which I like to see on the plates of others, but can not tolerate on my
own.
After dinner, which lasted a very long time, we went into the garden,
where coffee had been served, and stretched ourselves out beatifically,
cigar in mouth. All was calm and silent about us, the insects had ceased
their music, and in an opaline sky little violet clouds were sleeping.
Oscar, with a happy air, pointed out to me the famous mill, the quiet
valley, and farther on his loved stream, in which the sun, before
setting, was reflecting itself amid the reeds. Meanwhile the little
queen on her high heels flitted round the cups like a child playing at
party-giving, and with a thousand charming touches poured out the boiling
coffee, the odor of which blended deliciously with the perfume of the
flowers, the hay, and the woods.
When she had finished she sat down beside her husband, so close that her
skirt half hid my friend, and unceremoniously taking the cigar from his
lips, held it at a distance, with a little pout, that meant, "Oh, the
horrid thing!" and knocked off with her little finger the ash which fell
on the gravel. Then she broke into a laugh, and put the cigar back
between the lips of her husband held out to her.
It was charming. Oscar was no doubt accustomed to this, for he did not
seem astonished, but placed his hand on his wife's shoulder, as one would
upon a child's, and, kissing her on the forehead, said, "Thanks, my
dear."
"Yes, but you are only making fun of me," said the young wife, in a
whisper, leaning her head against her husband's arm.
I could not help smiling, there was so much coaxing childishness and
grace in this little whispered sentence. I do not know why I turned
toward the cousin who had remained a little apart, smoking in silence.
He seemed to me rather pale; he took three or four sudden puffs, rose
suddenly under the evident influence of some moral discomfort, and walked
away beneath the trees.
"What is the matter with cousin?" said Oscar, with some interest.
"What ails him?"
"I don't know," answered the little queen, in the most natural manner in
the world, "some idea about fishing, no doubt."
Night began to fall; we had remained as I have said a long time at table.
It was about nine o'clock. The cousin returned and took the seat he had
occupied before, but from this moment it seemed to me that a strange
constraint crept in among us, a singular coolness showed itself. The
talk, so lively at first, slackened gradually and, despite all my efforts
to impart a little life to it, dragged wretchedly. I myself did not feel
very bright; I was haunted by the most absurd notions in the world;
I thought I had detected in the sudden departure of the cousin, in his
pallor, in his embarrassed movements, the expression of some strong
feeling which he had been powerless to hide. But how was it that that
adorable little woman with such a keen intelligent look did not
understand all this, since I understood it myself? Had not Oscar,
however confiding he might be, noted that the departure of the cousin
exactly coincided with the kiss he had given his wife? Were these two
blind, or did they pretend not to see, or was I myself the victim of an
illusion? However, conversation had died away; the mistress of the
house, singular symptom, was silent and serious, and Oscar wriggled in
his chair, like a man who is not altogether at ease. What was passing in
their minds?
Soon we heard the clock in the drawing-room strike ten, and Oscar,
suddenly rising, said: "My dear fellow, in the country it is Liberty
Hall, you know; so I will ask your permission to go in--I am rather tired
this evening. George," he added to me, "they will show you your room; it
is on the ground floor; I hope that you will be comfortable there."
Everybody got up silently, and, after bidding one another good-night in
a somewhat constrained manner, sought their respective rooms. I thought,
I must acknowledge, that they went to bed rather too early at my
friend's. I had no wish to sleep; I therefore examined my room, which
was charming. It was completely hung with an old figured tapestry framed
in gray wainscot. The bed, draped in dimity curtains, was turned down
and exhaled that odor of freshly washed linen which invites one to
stretch one's self in it. On the table, a little gem dating from the
beginning of the reign of Louis XVI, were four or five books, evidently
chosen by Oscar and placed there for me. These little attentions touch
one, and naturally my thoughts recurred to the dear fellow, to the
strange incident of the evening, to the vexations and tortures hidden,,
perhaps, by this apparent happiness. I was ridiculous that night--
I already pitied him, my poor friend.
I felt quite touched, and, full of melancholy, went and leaned against
the sill of the open window. The moon had just risen, the sky was
beautifully clear, whiffs of delicious perfumes assailed my nostrils.
I saw in the shadow of the trees glowworms sparkling on the grass, and,
in the masses of verdure lit up mysteriously by the moon, I traced
strange shapes of fantastic monsters. There was, above all, a little
pointed roof surmounted by a weathercock, buried in the trees at about
fifty paces from my window, which greatly interested me. I could not in
the obscurity make out either door or windows belonging to this singular
tower. Was it an old pigeon-house, a tomb, a deserted summer-house?
I could not tell, but its little pointed roof, with a round dormer
window, was extremely graceful. Was it chance or an artist lull of taste
that had covered this tower with creepers and flowers, and surrounded it
with foliage in such capricious fashion that it seemed to be hiding
itself in order to catch all glances? I was gazing at all this when I
heard a faint noise in the shrubbery. I looked in that direction and I
saw--really, it was an anxious moment--I saw a phantom clad in a white
robe and walking with mysterious and agitated rapidity. At a turning of
the path the moon shone on this phantom. Doubt was impossible; I had
before my eyes my friend's wife. Her gait no longer had that coquettish
ease which I had noticed, but clearly indicated the agitation due to some
strong emotion.
I strove to banish the horrible suspicion which suddenly forced itself
into my mind. "No," I said to myself, "so much innocence and beauty can
not be capable of deception; no doubt she has forgotten her fan or her
embroidery, on one of the benches there." But instead of making her way
toward the benches I noticed on the right, the young wife turned to the
left, and soon disappeared in the shadow of the grove in which was hidden
the mysterious turret.
My heart ached. "Where is she going, the hapless woman?" I exclaimed to
myself. "At any rate, I will not let her imagine any one is watching
her." And I hurriedly blew out my candle. I wanted to close my window,
go to bed, and see nothing more, but an invincible curiosity took me back
to the window. I had only been there a few minutes when I plainly
distinguished halting and timid footsteps on the gravel. I could see no
one at first, but there was no doubt that the footsteps were those of a
man. I soon had a proof that I was not mistaken; the elongated outline
of the cousin showed up clearly against the dark mass of shrubbery.
I should have liked to have stopped him, the wretch, for his intention
was evident; he was making his way toward the thicket in which the little
queen had disappeared. I should have liked to shout to him, "You are a
villain; you shall go no farther." But had I really any right to act
thus? I was silent, but I coughed, however, loud enough to be heard by
him.
He suddenly paused in his uneasy walk, looked round on all sides with
visible anxiety, then, seized by I know not what impulse, darted toward
the pavilion. I was overwhelmed. What ought I to do? Warn my friend,
my childhood's companion? Yes, no doubt, but I felt ashamed to pour
despair into the mind of this good fellow and to cause a horrible
exposure. "If he can be kept in ignorance," I said to myself, "and then
perhaps I am wrong--who knows? Perhaps this rendezvous is due to the
most natural motive possible."
I was seeking to deceive myself, to veil the evidence of my own eyes,
when suddenly one of the house doors opened noisily, and Oscar--Oscar
himself, in all the disorder of night attire, his hair rumpled, and his
dressing-gown floating loosely, passed before my window. He ran rather
than walked; but the anguish of his heart was too plainly revealed in the
strangeness of his movements. He knew all. I felt that a mishap was
inevitable. "Behold the outcome of all his happiness, behold the bitter
poison enclosed in so fair a vessel!" All these thoughts shot through my
mind like arrows. It was necessary above all to delay the explosion,
were it only for a moment, a second, and, beside myself, without giving
myself time to think of what I was going to say to him, I cried in a
sharp imperative tone:
"Oscar, come here; I want to speak to you."
He stopped as if petrified. He was ghastly pale, and, with an infernal
smile, replied, "I have no time-later on."
"Oscar, you must, I beg of you--you are mistaken."
At these words he broke into a fearful laugh.
"Mistaken--mistaken!"
And he ran toward the pavilion.
Seizing the skirt of his dressing-gown, I held him tightly, exclaiming:
"Don't go, my dear fellow, don't go; I beg of you on my knees not to go."
By way of reply he gave me a hard blow on the arm with his fist,
exclaiming:
"What the devil is the matter with you?"
"I tell you that you can not go there, Oscar," I said, in a voice which
admitted of no contradiction.
"Then why did not you tell me at once."
And feverishly snatching his dressing-gown from my grasp, he began to
walk frantically up and down.
CHAPTER XVII
I SUP WITH MY WIFE
That evening, which chanced to be Christmas Eve, it was infernally cold.
The snow was falling in heavy flakes, and, driven by the wind, beat
furiously against the window panes. The distant chiming of the bells
could just be heard through this heavy and woolly atmosphere. Foot-
passengers, wrapped in their cloaks, slipped rapidly along, keeping close
to the house and bending their heads to the wintry blast.
Enveloped in my dressing-gown, and tapping with my fingers on the window-
panes, I was smiling at the half-frozen passers-by, the north wind, and
the snow, with the contented look of a man who is in a warm room and has
on his feet comfortable flannel-lined slippers, the soles of which are
buried in a thick carpet. At the fireside my wife was cutting out
something and smiling at me from time to time; a new book awaited me on
the mantelpiece, and the log on the hearth kept shooting out with a
hissing sound those little blue flames which invite one to poke it.
"There is nothing that looks more dismal than a man tramping through the
snow, is there?" said I to my wife.
"Hush," said she, lowering the scissors which she held in her hand; and,
after smoothing her chin with her fingers, slender, rosy, and plump at
their tips, she went on examining the pieces of stuff she had cut out.
"I say that it is ridiculous to go out in the cold when it is so easy to
remain at home at one's own fireside."
"Hush."
"But what are you doing that is so important?"
"I--I am cutting out a pair of braces for you," and she set to work
again. But, as in cutting out she kept her head bent, I noticed, on
passing behind her, her soft, white neck, which she had left bare that
evening by dressing her hair higher than usual. A number of little downy
hairs were curling there. This kind of down made me think of those ripe
peaches one bites so greedily. I drew near, the better to see, and I
kissed the back of my wife's neck.
"Monsieur!" said Louise, suddenly turning round.
"Madame," I replied, and we both burst out laughing.
"Christmas Eve," said I.
"Do you wish to excuse yourself and to go out?"
"Do you mean to complain?"
"Yes, I complain that you are not sufficiently impressed by the fact of
its being Christmas Eve. The ding-ding-dong of the bells of Notre Dame
fails to move you; and just now when the magic-lantern passed beneath the
window, I looked at you while pretending to work, and you were quite
calm."
"I remain calm when the magic-lantern is going by! Ah! my dear, you are
very severe on me, and really--"
"Yes, yes, jest about it, but it was none the less true that the
recollections of your childhood have failed."
"Now, my dear, do you want me to leave my boots out on the hearth this
evening on going to bed? Do you want me to call in the magic-lantern
man, and to look out a big sheet and a candle end for him, as my poor
mother used to do? I can still see her as she used to entrust her white
sheet to him. 'Don't make a hole in it, at least,' she would say. How
we used to clap our hands in the mysterious darkness! I can recall all
those joys, my dear, but you know so many other things have happened
since then. Other pleasures have effaced those."
"Yes, I can understand, your bachelor pleasures; and, there, I am sure
that this Christmas Eve is the first you have passed by your own
fireside, in your dressing-gown, without supper; for you used to sup on
Christmas Eve."
"To sup, to sup."
"Yes, you supped; I will wager you did."
"I have supped two or three times, perhaps, with friends, you know; two
sous' worth of roasted chestnuts and--"
"A glass of sugar and water."
"Oh, pretty nearly so. It was all very simple; as far as I can
recollect. We chatted a little and went to bed."
"And he says that without a smile. You have never breathed a word to me
of all these simple pleasures."
"But, my dear, all that I am telling you is strictly true. I remember
that once, however, it was rather lively. It was at Ernest's, and we had
some music. Will you push that log toward me? But, never mind; it will
soon be midnight, and that is the hour when reasonable people--"
Louise, rising and throwing her arms around my neck, interrupted me with:
"Well, I don't want to be reasonable, I want to wipe out all your
memories of chestnuts and glasses of sugar and water."
Then pushing me into my dressing-room she locked the door.
"But, my dear, what is the matter with you?" said I through the keyhole.
"I want ten minutes, no more. Your newspaper is on the mantelpiece; you
have not read it this evening. There are some matches in the corner."
I heard a clatter of crockery, a rustling of silk my wife mad?
Louise soon came and opened the door.
"Don't scold me for having shut you up," she said, kissing me. "Look how
I have beautified myself? Do you recognize the coiffure you are so fond
of, the chignon high, and the neck bare? Only as my poor neck is
excessively timid, it would have never consented to show itself thus if
I had not encouraged it a little by wearing my dress low. And then one
must put on full uniform to sup with the authorities."
"To sup?"
"Certainly, to sup with you; don't you see my illuminations and this
table covered with flowers and a heap of good things? I had got it all
ready in the alcove; but you understand that to roll the table up to the
fire and make a little toilette, I wanted to be alone. Come, Monsieur,
take your place at table. I am as hungry as a hunter. May I offer you a
wing of cold chicken?"
"Your idea is charming, but, dear, really I am ashamed; I am in my
dressing-gown."
"Take off your dressing-gown if it incommodes you, Monsieur, but don't
leave this chicken wing on my hands. I want to serve you myself." And,
rising, she turned her sleeves up to the elbow, and placed her table
napkin on her arm.
"It is thus that the waiters at the restaurant do it, is it not?"
"Exactly; but, waiter, allow me at least to kiss your hand."
"I have no time," said she, laughing, sticking the corkscrew into the
neck of the bottle. "Chambertin--it is a pretty name; and then do you
remember that before our marriage (how hard this cork is!) you told me
that you liked it on account of a poem by Alfred de Musset? which, by
the way, you have not let me read yet. Do you see the two little
Bohemian glasses which I bought expressly for this evening? We will
drink each other's health in them."
"And his, too, eh?"
"The heir's, poor dear love of an heir! I should think so. And then I
will put away the two glasses against this time next year; they shall be
our Christmas Eve glasses? Every year we will sup like this together,
however old we may get."
"But, my dear, how about the time when we have no longer any teeth?"
"Well, we will sup on good strong soups; it will be very nice, all the
same. Another piece, please, with some of the jelly. Thanks."
As she held out her plate I noticed her arm, the outline of which was
lost in lace.
"Why are you looking up my sleeve instead of eating?"
"I am looking at your arm, dear. You are charming, let me tell you, this
evening. That coiffure suits you so well, and that dress which I was
unacquainted with."
"Well, when one seeks to make a conquest--"
"How pretty you look, pet!"
"Is it true that you think me charming, pretty, and a pet this evening?
Well, then," lowering her eyes and smiling at her bracelets, "in that
case I do not see why--"
"What is it you do not see, dear?"
"I do not see any reason why you should not come and give me just a
little kiss."
And as the kiss was prolonged, she said to me, amid bursts of laughter,
her head thrown back, and showing the double row of her white teeth:
"I should like some pie; yes, some brie! You will break my Bohemian
glass, the result of my economy. You always cause some mishap when you
want to kiss me. Do you recollect at Madame de Brill's ball, two days
before our marriage, how you tore my skirt while waltzing in the little
drawing-room?"
"Because it is difficult to do two things at once-to keep step and to
kiss one's partner."
"I recollect, too, when mamma asked how my skirt had got torn, I felt
that I was blushing up to my ears. And Madame D., that old jaundiced
fairy, who said to me with her Lenten smile, 'How flushed you are
tonight, my dear child!' I could have strangled her! I said it was the
key of the door that had caught it. I looked at you out of the corner of
my eye; you were pulling your moustache and seemed greatly annoyed--you
are keeping all the truffles for yourself; that is kind--not that one;
I want the big black one there in the corner-it was very wrong all the
same, for--oh! not quite full--I do not want to be tipsy--for, after all,
if we had not been married--and that might have happened, for you know
they say that marriages only depend on a thread. Well, if the thread had
not been strong enough, I should have remained a maid with a kiss on my
shoulder, and a nice thing that would have been."
"Bah! it does not stain."
"Yes, Monsieur, it does, I beg your pardon. It stains so much that there
are husbands, I believe, who even shed their blood to wash out such
little stains."
"But I was joking, dear. Hang it!--don't you think--yes, certainly, hang
it!"
"Ah! that's right, I like to see you angry. You are a trifle jealous,
dear--oh! that is too bad; I asked you for the big black one, and you
have gone and eaten it."
"I am sorry, dear; I quite forgot about it."
"It was the same at the Town Hall, where I was obliged to jog your elbow
to make you answer 'Yes' to the Mayor's kind words."
"Kind!"
"Yes, kind. I thought him charming. No one could have been more
graceful than he was in addressing me. 'Mademoiselle, will you consent
to accept for your husband that great, ugly fellow standing beside you?'"
(Laughing, with her mouth full.) "I wanted to say to him, 'Let us come to
an understanding, Mr. Mayor; there is something to be said on either
side.' I am choking!"--she bursts out laughing-- "I was wrong not to
impose restrictions. Your health, dear! I am teasing you; it is very
stupid. I said 'Yes' with all my heart, I can assure you, dear, and I
thought the word too weak a one. When I think that all women, even the
worst, say that word, I feel ashamed not to have found another." Holding
out her glass: "To our golden wedding--will you touch glasses?"
"And to his baptism, little mamma."
In a low voice: "Tell me--are you sorry you married me?"
Laughing, "Yes." Kissing her on the shoulder, "I think I have found the
stain again; it was just there."
"It is two in the morning, the fire is out, and I am a little--you won't
laugh now? Well, I am a little dizzy."
"A capital pie, eh?"
"A capital pie! We shall have a cup of tea for breakfast tomorrow, shall
we not?"
CHAPTER XVIII
FROM ONE THING TO ANOTHER
SCENE.--The country in autumn--The wind is blowing without--MADAME,
seated by the fireside in a large armchair, is engaged in needlework
--MONSIEUR, seated in front of her, is watching the flames of the
fire--A long silence.
Monsieur--Will you pass me the poker, my dear?
Madame--(humming to herself)--"And yet despite so many fears." (Spoken.)
Here is the poker. (Humming.) "Despite the painful----"
Monsieur--That is by Mehul, is it not, my dear? Ah! that is music--I saw
Delaunay Riquier in Joseph. (He hums as he makes up the fire.) "Holy
pains." (Spoken.) One wonders why it does not burn, and, by Jove! it
turns out to be green wood. Only he was a little too robust--Riquier.
A charming voice, but he is too stout.
Madame--(holding her needlework at a distance, the better to judge of the
effect)--Tell me, George, would you have this square red or black? You
see, the square near the point. Tell me frankly.
Monsieur--(singing) "If you can repent." (Spoken without turning his
head.) Red, my dear; red. I should not hesitate; I hate black.
Madame--Yes, but if I make that red it will lead me to-- (She reflects.)
Monsieur--Well, my dear, if it leads you away, you must hold fast to
something to save yourself.
Madame--Come, George, I am speaking seriously. You know that if this
little square is red, the point can not remain violet, and I would not
change that for anything.
Monsieur--(slowly and seriously)--My dear, will you follow the advice of
an irreproachable individual, to whose existence you have linked your
fate? Well, make that square pea-green, and so no more about it. Just
look whether a coal fire ever looked like that.
Madame--I should only be too well pleased to use up my pea-green wool; I
have a quantity of it.
Monsieur--Then where lies the difficulty?
Madame--The difficulty is that pea-green is not sufficiently religious.
Monsieur--Hum! (Humming.) Holy pains! (Spoken.) Will you be kind enough
to pass the bellows? Would it be indiscreet to ask why the poor pea-
green, which does not look very guilty, has such an evil reputation? You
are going in for religious needlework, then, my dear?
Madame--Oh, George! I beg of you to spare me your fun. I have been
familiar with it for a long time, you know, and it is horribly
disagreeable to me. I am simply making a little mat for the
confessional-box of the vicar. There! are you satisfied? You know what
it is for, and you must understand that under the present circumstances
pea-green would be altogether out of place.
Monsieur--Not the least in the world. I can swear to you that I could
just as well confess with pea-green under my feet. It is true that I am
naturally of a resolute disposition. Use up your wool; I can assure you
that the vicar will accept it all the same. He does not know how to
refuse. (He plies the bellows briskly.)
Madame--You are pleased, are you not?
Monsieur--Pleased at what, dear?
Madame--Pleased at having vented your sarcasm, at having passed a jest on
one who is absent. Well, I tell you that you are a bad man, seeing that
you seek to shake the faith of those about you. My beliefs had need be
very fervent, principles strong, and have real virtue, to resist these
incessant attacks. Well, why are you looking at me like that?
Monsieur--I want to be converted, my little apostle. You are so pretty
when you speak out; your eyes glisten, your voice rings, your gestures--
I am sure that you could speak like that for a long time, eh? (He kisses
her hand, and takes two of her curls and ties them under hey chin.) You
are looking pretty, my pet.
Madame--Oh! you think you have reduced me to silence because you have
interrupted me. Ah! there, you have tangled my hair. How provoking you
are! It will take me an hour to put it right. You are not satisfied
with being a prodigy of impiety, but you must also tangle my hair. Come,
hold out your hands and take this skein of wool.
Monsieur--(sitting down on a stool, which he draws as closely as possible
to Madame, and holding up his hands) My little Saint John!
Madame--Not so close, George; not so close. (She smiles despite
herself.) How silly you are! Please be careful; you will break my wool.
Monsieur--Your religious wool.
Madame--Yes, my religious wool. (She gives him a little pat on the
cheek.) Why do you part your hair so much on one side, George? It would
suit you much better in the middle, here. Yes, you may kiss me, but
gently.
Monsieur--Can you guess what I am thinking of?
Madame--How do you imagine I could guess that?
Monsieur--Well, I am thinking of the barometer which is falling and of
the thermometer which is falling too.
Madame--You see, cold weather is coming on and my mat will never be
finished. Come, let us make haste.
Monsieur--I was thinking of the thermometer which is falling and of my
room which faces due north.
Madame--Did you not choose it yourself? My wool! Good gracious! my
wool! Oh! the wicked wretch!
Monsieur--In summer my room with the northern aspect is, no doubt, very
pleasant; but when autumn comes, when the wind creeps in, when the rain
trickles down the windowpanes, when the fields, the country, seem hidden
under a huge veil of sadness, when the spoils of our woodlands strew the
earth, when the groves have lost their mystery and the nightingale her
voice--oh! then the room with the northern aspect has a very northern
aspect, and--
Madame--(continuing to wind her wool)--What nonsense you are talking!
Monsieur--I protest against autumns, that is all. God's sun is hidden
and I seek another. Is not that natural, my little fairhaired saint, my
little mystic lamb, my little blessed palmbranch? This new sun I find in
you, pet--in your look, in the sweet odor of your person, in the rustling
of your skirt, in the down on your neck which one notices by the lamp-
light when you bend over the vicar's mat, in your nostril which expands
when my lips approach yours--
Madame--Will you be quiet, George? It is Friday, and Ember week.
Monsieur--And your dispensation? (He kisses her.) Don't you see that
your hand shakes, that you blush, that your heart is beating?
Madame--George, will you have done, sir? (She pulls away her hand,
throws herself back in the chair, and avoids her husband's glance.)
Monsieur--Your poor little heart beats, and it is right, dear; it knows
that autumn is the time for confidential chats and evening caresses, the
time for kisses. And you know it too, for you defend yourself poorly,
and I defy you to look me in the face. Come! look me in the face.
Madame--(she suddenly leans toward hey husband, the ball of wool rolling
into the fireplace, the pious task falling to the ground. She takes his
head between her hands)--Oh, what a dear, charming husband you would be
if you had--
Monsieur--If I had what? Tell me quickly.
Madame--If you had a little religion. I should only ask for such a
little at the beginning. It is not very difficult, I can assure you.
While, now, you are really too--
Monsieur--Pea-green, eh?
Madame--Yes, pea-green, you great goose. (She laughs frankly.)
Monsieur--(lifting his hands in the air)--Sound trumpets! Madame has
laughed; Madame is disarmed. Well, my snowwhite lamb, I am going to
finish my story; listen properly, there, like that--your hands here, my
head so. Hush! don't laugh. I am speaking seriously. As I was saying
to you, the north room is large but cold, poetic but gloomy, and I will
add that two are not too many in this wintry season to contend against
the rigors of the night. I will further remark that if the sacred ties
of marriage have a profoundly social significance, it is--do not
interrupt me--at that hour of one's existence when one shivers on one's
solitary couch.
Madame--You can not be serious.
Monsieur--Well, seriously, I should like the vicar's mat piously spread
upon your bed, to keep us both warm together, this very evening. I wish
to return as speedily as possible to the intimacy of conjugal life. Do
you hear how the wind blows and whistles through the doors? The fire
splutters, and your feet are frozen. (He takes her foot in his hands.)
Madame--But you are taking off my slipper, George.
Monsieur--Do you think, my white lamb, that I am going to leave your poor
little foot in that state? Let it stay in my hand to be warmed. Nothing
is so cold as silk. What! openwork stockings? My dear, you are rather
dainty about your foot-gear for a Friday. Do you know, pet, you can not
imagine how gay I wake up when the morning sun shines into my room. You
shall see. I am no longer a man; I am a chaffinch; all the joys of
spring recur to me. I laugh, I sing, I speechify, I tell tales to make
one die of laughter. Sometimes I even dance.
Madame--Come now! I who in the morning like neither noise nor broad
daylight--how little all that suits!
Monsieur--(suddenly changing his tone)--Did I say that I liked all that?
The morning sun? Never in autumn, my sweet dove, never. I awake, on the
contrary full of languor and poesy; I was like that in my very cradle.
We will prolong the night, and behind the drawn curtain, behind the
closed shutter, we will remain asleep without sleeping. Buried in
silence and shadow, delightfully stretched beneath your warm eider-down
coverlets, we will slowly enjoy the happiness of being together, and we
will wish one another good-morning only on the stroke of noon. You do
not like noise, dear. I will not say a word. Not a murmur to disturb
your unfinished dream and warn you that you are no longer sleeping; not a
breath to recall you to reality; not a movement to rustle the coverings.
I will be silent as a shade, motionless as a statue; and if I kiss you--
for, after all, I have my weaknesses--it will be done with a thousand
precautions, my lips will scarcely brush your sleeping shoulder; and if
you quiver with pleasure as you stretch out your arms, if your eye half
uncloses at the murmur of my kiss, if your lips smile at me, if I kiss
you, it would be because you would like me to, and I shall have nothing
to reproach myself with.
Madame--(her eyes half closed, leaning back in hey armchair, her head
bent with emotion, she places her hands before his mouth. In a low
voice)--Hush, hush! Don't say that, dear; not another word! If you knew
how wrong it was!
Monsieur--Wrong! What is there that is wrong? Is your heart of marble
or adamant, that you do not see that I love you, you naughty child? That
I hold out my arms to you, that I long to clasp you to my heart, and to
fall asleep in your hair? What is there more sacred in the world than to
love one's wife or love one's husband? (Midnight strikes.)
Madame--(she suddenly changes hey expression at the sound, throws her
arms round her husband, and hurriedly kisses him thrice)--You thought I
did not love you, eh, dear? Oh, yes! I love you. Great baby! not to
see that I was waiting the time.
Monsieur--What time, dear?
Madame--The time. It has struck twelve, see. (She blushes crimson.)
Friday is over. (She holds out her hand for him to kiss.)
Monsieur--Are you sure the clock is not five minutes fast, love?
CHAPTER XIX
A LITTLE CHAT
MADAME F----- MADAME H------
(These ladies are seated at needlework as they talk.)
Madame F--For myself, you know, my dear, I fulfil my duties tolerably,
still I am not what would be called a devotee. By no means. Pass me
your scissors. Thanks.
Madame H--You are quite welcome, dear. What a time those little squares
of lace must take. I am like yourself in respect of religion; in the
first place, I think that nothing should be overdone. Have you ever-
I have never spoken to any one on the subject, but I see your ideas are
so in accordance with my own that--
Madame F--Come, speak out, dear; you trust me a little, I hope.
Madame H--Well, then, have you--tell me truly--ever had any doubts?
Madame F--(after reflecting for a moment)--Doubts! No. And you?
Madame H--I have had doubts, which has been a real grief to me. Heavens!
how I have wept.
Madame F--I should think so, my poor dear. For my own part, my faith is
very strong. These doubts must have made you very unhappy.
Madame H--Terribly so. You know, it seems as if everything failed you;
there is a vacancy all about you--I have never spoken about it to my
husband, of course--Leon is a jewel of a man, but he will not listen to
anything of that kind. I can still see him, the day after our marriage;
I was smoothing my hair--broad bands were then worn, you know.
Madame F--Yes, yes; they were charming. You will see that we shall go
back to them.
Madame H--I should not be surprised; fashion is a wheel that turns.
Leon, then, said to me the day after our wedding: "My dear child, I shall
not hinder you going to church, but I beg you, for mercy's sake, never to
say a word to me about it."
Madame F--Really, Monsieur H. said that to you?
Madame H--Upon my honor. Oh! my husband is all that is most--or, if you
prefer it, all that is least--
Madame F--Yes, yes, I understand. That is a grief, you know. Mine is
only indifferent. From time to time he says some disagreeable things to
me on the question, but I am sure he could be very easily brought back to
the right. At the first illness he has, you shall see. When he has only
a cold in the head, I notice the change. You have not seen my thimble?
Madame H--Here it is. Do not be too sure of that, dear; men are not to
be brought back by going "chk, chk" to them, like little chickens. And
then, though I certainly greatly admire the men who observe religious
practices, you know me well enough not to doubt that--I think, as I told
you, that nothing should be exaggerated. And yourself, pet, should you
like to see your husband walking before the banner with a great wax taper
in his right hand and a bouquet of flowers in his left?
Madame F--Oh! no, indeed. Why not ask me at once whether I should like
to see Leon in a black silk skull cap, with cotton in his ears and a holy
water sprinkler in his hand? One has no need to go whining about a
church with one's nose buried in a book to be a pious person; there is a
more elevated form of religion, which is that of--of refined people, you
know.
Madame H--Ah! when you speak like that, I am of your opinion. I think,
for instance, that there is nothing looks finer than a man while the host
is being elevated. Arms crossed, no book, head slightly bowed, grave
look, frock coat buttoned up. Have you seen Monsieur de P. at mass?
How well he looks!
Madame F--He is such a fine man, and, then, he dresses so well. Have you
seen him on horseback? Ah! so you have doubts; but tell me what they
are, seeing we are indulging in confidences.
Madame H--I can hardly tell you. Doubts, in short; about hell, for
instance, I have had horrible doubts. Oh! but do not let us speak about
that; I believe it is wrong even to think of it.
Madame F--I have very broad views on that point; I never think about it.
Besides, my late confessor helped me. "Do not seek too much," he always
said to me, "do not try to understand that which is unfathomable." You
did not know Father Gideon? He was a jewel of a confessor; I was
extremely pleased with him. Not too tedious, always discreet, and, above
all, well-bred. He turned monk from a romantic cause--a penitent was
madly in love with him.
Madame H--Impossible!
Madame F--Yes, really. What! did you not know about it? The success of
the monastery was due to that accident. Before the coming of Father
Gideon it vegetated, but on his coming the ladies soon flocked there in
crowds. They organized a little guild, entitled "The Ladies of the
Agony." They prayed for the Chinese who had died without confession,
and wore little death's heads in aluminum as sleeve-links. It became
very fashionable, as you are aware, and the good fathers organized, in
turn, a registry for men servants; and the result is that, from one thing
leading to another, the community has become extremely wealthy. I have
even heard that one of the most important railway stations in Paris is
shortly to be moved, so that the size of their garden can be increased,
which is rather restricted at present.
Madame H--As to that, it is natural enough that men should want a place
to walk in at home; but what I do not understand is that a woman, however
pious she may be, should fall in love with a priest. It is all very
well, but that is no longer piety; it is--fanaticism. I venerate
priests, I can say so truly, but after all I can not imagine myself--you
will laugh at me--ha, ha, ha!
Madame F--Not at all. Ha, ha, ha! what a child you are!
Madame H--(working with great briskness)--Well, I can not imagine that
they are men--like the others.
Madame F--(resuming work with equal ardor)--And yet, my dear, people say
they are.
Madame H--There are so many false reports set afloat. (A long silence.)
Madame F--(in a discreet tone of voice)--After all, there are priests who
have beards--the Capuchins, for instance.
Madame H--Madame de V. has a beard right up to her eyes, so that counts
for nothing, dear.
Madame F--That counts for nothing. I do not think so. In the first
place, Madame de V.'s beard is not a perennial beard; her niece told me
that she sheds her moustaches every autumn. What can a beard be that can
not stand the winter? A mere trifle.
Madame H--A mere trifle that is horribly ugly, my dear.
Madame F--Oh! if Madame de V. had only moustaches to frighten away
people, one might still look upon her without sorrow, but--
Madame H--I grant all that. Let us allow that the Countess's moustache
and imperial are a nameless species of growth. I do not attach much
importance to the point, you understand. She has a chin of heartbreaking
fertility, that is all.
Madame F--To return to what we were saying, how is it that the men who
are strongest, most courageous, most manly--soldiers, in fact--are
precisely those who have most beard?
Madame H--That is nonsense, for then the pioneers would be braver than
the Generals; and, in any case, there is not in France, I am sure, a
General with as much beard as a Capuchin. You have never looked at a
Capuchin then?
Madame F--Oh, yes! I have looked at one quite close. It is a rather
funny story. Fancy Clementine's cook having a brother a Capuchin--an
ex-jeweller, a very decent man. In consequence of misfortunes in
business--it was in 1848, business was at a stand-still--in short,
he lost his senses--no, he did not lose his senses, but he threw himself
into the arms of Heaven.
Madame H--Oh! I never knew that! When? Clementine--
Madame F--I was like you, I would not believe it, but one day Clementine
said to me: "Since you will not believe in my Capuchin, come and see me
tomorrow about three o'clock; he will be paying a visit to his sister.
Don't have lunch first; we will lunch together." Very good. I went the
next day with Louise, who absolutely insisted upon accompanying me, and I
found at Clementine's five or six ladies installed in the drawing-room
and laughing like madcaps. They had all come to see the Capuchin.
"Well," said I, as I went in, when they all began to make signs to me and
whisper, "Hush, hush!" He was in the kitchen.
Madame H--And what was he like?
Madame F--Oh! very nice, except his feet; you know how it always gives
one a chill to look at their feet; but, in short, he was very amiable.
He was sent for into the drawing-room, but he would not take anything
except a little biscuit and a glass of water, which took away our
appetites. He was very lively; told us that we were coquettes with our
little bonnets and our full skirts. He was very funny, always a little
bit of the jeweller at the bottom, but with plenty of good nature and
frankness. He imitated the buzzing of a fly for us; it was wonderful.
He also wanted to show us a little conjuring trick, but he needed two
corks for it, and unfortunately his sister could only find one.
Madame H--No matter, I can not understand Clementine engaging a servant
like that.
Madame F--Why? The brother is a guarantee.
Madame H--Of morality, I don't say no; but it seems to me that a girl
like that can not be very discreet in her ways.
Madame F--How do you make that out?
Madame H--I don't know, I can not reason the matter out, but it seems to
me that it must be so, that is all, . . . besides, I should not like
to see a monk in my kitchen, close to the soup. Oh, mercy! no!
Madame F--What a child you are!
Madame H--That has nothing to do with religious feelings, my dear; I do
not attack any dogma. Ah! if I were to say, for instance--come now, if I
were to say, what now?
Madame F--In point of fact, what really is dogma?
Madame H--Well, it is what can not be attacked. Thus, for instance,
a thing that is evident, you understand me, is unassailable, . . . or
else it should be assailed, . . in short, it can not be attacked. That
is why it is monstrous to allow the Jewish religion and the Protestant
religion in France, because these religions can be assailed, for they
have no dogma. I give you this briefly, but in your prayer-book you will
find the list of dogmas. I am a rod of iron as regards dogmas. My
husband, who, as I said, has succeeded in inspiring me with doubts on
many matters--without imagining it, for he has never required anything of
me; I must do him that justice--but who, at any rate, has succeeded in
making me neglect many things belonging to religion, such as fasting,
vespers, sermons, . . . confession.
Madame F--Confession! Oh! my dear, I should never have believed that.
Madame H--It is in confidence, dear pet, that I tell you this. You will
swear never to speak of it?
Madame F--Confession! Oh! yes, I swear it. Come here, and let me kiss
you.
Madame H--You pity me, do you not?
Madame F--I can not pity you too much, for I am absolutely in the same
position.
Madame H--You, too! Good heavens! how I love you. What can one do, eh?
Must one not introduce some plan of conciliation into the household,
sacrifice one's belief a little to that of one's husband?
Madame F--No doubt. For instance, how would you have me go to high mass,
which is celebrated at my parish church at eleven o'clock exactly? That
is just our breakfast time. Can I let my husband breakfast alone? He
would never hinder me from going to high mass, he has said so a thousand
times, only he has always added, "When you want to go to mass during
breakfast time, I only ask one thing--it is to give me notice the day
before, so that I may invite some friends to keep me company."
Madame H--But only fancy, pet, our two husbands could not be more alike
if they were brothers. Leon has always said, "My dear little chicken--"
Madame F--Ha! ha! ha!
Madame H--Yes, that is his name for me; you know how lively he is. He
has always said to me, then, "My dear little chicken, I am not a man to
do violence to your opinions, but in return give way to me as regards
some of your pious practices." I only give you the mere gist of it; it
was said with a thousand delicacies, which I suppress. And I have agreed
by degrees, . . . so that, while only paying very little attention to
the outward observances of religion, I have remained, as I told you, a
bar of iron as regards dogmas. Oh! as to that, I would not give way an
inch, a hair-breadth, and Leon is the first to tell me that I am right.
After all, dogma is everything; practice, well, what would you? If I
could bring Leon round, it would be quite another thing. How glad I am
to have spoken to you about all this.
Madame F--Have we not been chattering? But it is half-past five, and I
must go and take my cinchona bark. Thirty minutes before meals, it is a
sacred duty. Will you come, pet?
Madame H--Stop a moment, I have lost my thimble again and must find it.
ETEXT EDITOR'S BOOKMARKS:
But she thinks she is affording you pleasure
Do not seek too much
First impression is based upon a number of trifles
Sometimes like to deck the future in the garments of the past
The heart requires gradual changes
End of this Project Gutenberg Etext of TMonsieur, Madame, and Bebe, v2
by Gustave Droz
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\section{Introduction}
After almost five decades since being hypothesized \cite{Englert:1964et}, experimental evidence for the Higgs boson has appeared. The Large Hadron Collider's (LHC) ATLAS and CMS collaborations have reported discovery of an even-integer-spin particle with production and decay modes approximately consistent with those of the Standard Model Higgs. ATLAS \cite{ATLASdiscovery} has reported discovery of such a particle with a best fit mass of $126.5$~GeV at $5.0\sigma$, while CMS \cite{CMSdiscovery} finds a particle with a mass of $125.3\pm0.6$~GeV with $4.9\sigma$ significance. Furthermore, the Tevatron's CDF and D\O\ Collaborations have each observed a statistically significant excess among events in their $b\bar{b}+W/Z$ channels \cite{TevatronBB,TEVNPH:2012ab,CDF:2012cn}, corresponding to a rate that is roughly consistent with that predicted from the production and decay of a 125 GeV Standard Model Higgs boson. The current body of evidence strongly supports the conclusion that these experiments are observing the scalar particle that is responsible for electroweak symmetry breaking.
The quest for the Higgs, however, does not end with discovery. Increasingly precise measurements of this particle's production cross sections, decay widths, and mass can provide potentially valuable probes of physics beyond the Standard Model. For example, within many extensions of the Standard Model (most notably supersymmetry \cite{Dimopoulos:1981zb}) there is not one Higgs boson but several. These Higgs bosons can mix among each other, leading to modified couplings relative to those predicted by the Standard Model. Furthermore, in the Standard Model and extensions thereof, the Higgs' effective couplings to gluons and photons are induced only at the loop-level \cite{Ellis:1975ap} and thus can be potentially sensitive to the presence of new charged or colored particles with significant couplings to the Higgs. In such scenarios, the particle that is currently being observed at the LHC and Tevatron may be similar to the Higgs boson of the Standard Model, but with modfied couplings. Precision measurements of the production rates and branching fractions of this particle could thus provide a new window through which to potentially reveal new physics. For recent studies of 2011 Tevatron and LHC results within the context of a supersymmetric Higgs sector, see Refs.~\cite{Kane:2011kj,Hall:2011aa,Baer:2011ab,Feng:2011aa,Heinemeyer:2011aa,Arbey:2011ab,Draper:2011aa, Carena:2011aa,Ellwanger:2011aa,Akula:2011aa,Kadastik:2011aa,Gunion:2012zd,King:2012is,Cao:2012fz,Aparicio:2012iw,Christensen:2012ei,Ajaib:2012vc,Brummer:2012ns}.
In this article, we study the rates observed by the LHC and Tevatron experiments in various Higgs search channels and compare them to those predicted for a Standard Model Higgs boson. We allow the ratios of widths to Standard Model particles to vary, and consider the effect this has on the global fit to the ATLAS, CMS and CDF/D\O\ observations across all channels. As had been noted in a number of similar studies of previous LHC and Tevatron data~\cite{Giardino:2012ww,Carena:2012xa,Carena:2012gp,Blum:2012ii,Bonne:2012im,Chang:2012tb,Carmi:2012zd,Azatov:2012ga,Wang:2012gm,Azatov:2012wq,Akeroyd:2012ms,Espinosa:2012vu,Arhrib:2012yv,Barroso:2012wz,Bellazzini:2012tv,Klute:2012pu,Draper:2012xt,Gabrielli:2012hd,Ellis:2012rx}, these observations appear to favor an increased rate of $h\to\gamma\gamma$, as well as reduced production cross sections in the channels that rely on $gg \to h$, as compared to the Standard Model predictions. Incorporating all publicly available data, we quantify this preference, and find that, neglecting possible systematic effects, increasing $\Gamma(h\to \gamma \gamma)$ by a factor of approximately three while decreasing $\Gamma(h \to g g)$ by a factor of two very significantly improves the quality of the global fit, reducing the $\chi^2$ by 9.9.
The remainder of this paper is structured as follows. In Sec.~\ref{datasec}, we summarize the current status of Higgs searches at the LHC and Tevatron and compare the observed rates to those predicted for a Standard Model Higgs boson. In Sec.~\ref{widthcouplings}, we use these data to constrain the decay widths and effective couplings of the particle being observed. In Sec.~\ref{scenarios}, we discuss classes of beyond the Standard Model physics scenarios which could help to alleviate the tension between the observed and predicted rates. We find that the statistical tension between the theory and experiment can be significantly relieved by the addition of new particles with both color and electric charge, and large couplings to the Higgs. An obvious realization of such a particle is the stop squark of supersymmetry. We find that a light ($m_{\tilde{t_1}} \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 300$ GeV) and highly mixed stop can provide a good fit to these observations.
\section{Higgs Boson Searches At The LHC and Tevatron}
\label{datasec}
Standard Model Higgs production at both the LHC and the Tevatron is dominated by gluon-gluon fusion, made possible by the effective coupling induced by a top quark loop \cite{Spira:1995rr,Anastasiou:2008tj,deFlorian:2009hc}. Smaller but not insignificant rates for Higgs production also occur through vector boson fusion (VBF) and through Higgs production in association with a $W$ or $Z$. At $\sqrt{s}=7(8)$~TeV, the cross-sections for these channels are \cite{Dittmaier:2012vm}:
\begin{eqnarray}
\sigma(g g \to h) & = & 15.3 \pm 2.3~(19.5\pm 2.9)~\mbox{pb}, \\
\sigma(p p \to j j h) & = & 1.21 \pm 0.03~(1.56\pm0.05)~\mbox{pb}, \nonumber \\
\sigma(p p\to Wh) & = & 0.57 \pm 0.02~(0.70\pm0.03)~\mbox{pb}, \nonumber \\
\sigma(p p\to Zh) & = & 0.32\pm0.02~(0.39\pm0.02)~\mbox{pb}. \nonumber
\end{eqnarray}
While the cross sections for the later three processes are much smaller than those from gluon-gluon fusion, the additional very forward jets present in VBF events and the high transverse momentum typical among associated production events makes these channels important for LHC Higgs studies, especially in the case of $h \rightarrow \gamma \gamma$.
For a mass of $m_h \approx 125$ GeV, as reported by the high resolution $\gamma \gamma$ and $ZZ\to 4\ell$ channels at CMS and ATLAS \cite{ATLASdiscovery,CMSdiscovery}, the Standard Model Higgs boson is predicted to decay primarily to $b\bar{b}$ (58\%) and $W^+W^-$ (22\%), with smaller branching fractions to $gg$ (8.5\%), $\tau^+ \tau^-$ (6.4\%), $c\bar{c}$ (2.7\%), $ZZ$ (2.7\%), and $\gamma \gamma$ (0.22\%) \cite{Dittmaier:2012vm}. Higgs decays to $gg$ are completely invisible at hadron colliders, due to the very large QCD multi-jet background. However, as the loop-induced coupling responsible for this branching ratio also sets the $gg\to h$ cross-section, it is of critical importance to LHC and Tevatron Higgs phenomenology.
Higgs searches have been conducted in many channels by the LHC and Tevatron experimental collaborations. Some Higgs decay modes (such as $h\rightarrow W^+W^-$, $ZZ$, and $\tau^+\tau^-$) yield signatures that are significantly distinctive to study independently of the production mechanism. Others (such as $h\rightarrow b\bar{b}$) can only be identified above backgrounds when observed in conjunction with an additional gauge boson, and thus rely on those events in which the Higgs is produced in association with an additional $W$ or $Z$ \cite{CMSbb}.
\begin{figure*}[t]
\centering
\includegraphics[angle=0.0,width=6.0in]{data.pdf}
\caption{The observed rates in various Higgs search channels at the LHC and Tevatron, compared to those predicted from a 125 GeV Standard Model Higgs boson. The Standard Model Higgs boson provides a somewhat poor global fit ($\chi^2=19.67$, over 17-1 degrees-of-freedom). If the Higgs' decay widths to gluons and photons are modified to their best fit values (0.66 and 2.5 times the Standard Model values, respectively), the global fit improves significantly to $\chi^2=9.77$, over 17-3 degrees-of-freedom.}
\label{data}
\end{figure*}
\begin{figure}[!]
\centering
\includegraphics[angle=0.0,width=3.4in]{fit-width-update.pdf}
\caption{The 68\%, 95\% and 99\% confidence level contours of the global fit to the data summarized in Fig.~\ref{data}, allowing the decay widths of the Higgs to photons and gluons to vary while keeping all others fixed. We have also fixed the mass of the Higgs to 125 GeV. The best fit point (shown as a cross) yields an overall $\chi^2$ of 9.77, over 17-3 degrees-of-freedom. In contrast, the Standard Model case, located at (1,1) in this figure, yields $\chi^2=19.67$, and at face value appears to be disfavored over the best-fit case at the 99\% confidence level.}
\label{fitwidth}
\end{figure}
Further search strategies focus on events which primarily result from VBF (such as the $pp\rightarrow jjh \rightarrow jjWW$ channel studied by the CMS Collaboration \cite{CMSvbfWW}). ATLAS and CMS have applied a number of different sets of cuts on their searches for $h \rightarrow \gamma \gamma$ \cite{ATLASdiscovery,CMSdiscovery,Chatrchyan:2012tw,ATLAS:2012ad,CMSvbf}, which have the effect of focusing on and isolating different Higgs production mechanisms. However, while the event selection cuts designed to zero in on VBF events are very effective, the large ratio of cross sections means that even a small survival probability of gluon-gluon fusion events can have large consequences. Using MadGraph5~\cite{Alwall:2011uj} combined with Pythia6~\cite{Sjostrand:2006za} to simulate Higgs production and PGS4~\cite{PGS} for detector simulation, we match the $\sim 80\%$ purity of the $p p \rightarrow \gamma\gamma j j$ ``di-jet tight'' selection criteria as reported by the CMS experiment \cite{CMSvbf} and find that (at $\sqrt{s} = 8$~TeV) it is sensitive to the following approximate combination:
\begin{equation}
\left[0.02 \, \sigma(gg\to h)+\sigma(pp\to j j h) \right] \times \mbox{BR}(h\to \gamma\gamma).
\end{equation}
The ATLAS $p p \to hX \to \gamma\gamma X$ \cite{ATLAS:2012ad} originally reported in the July 4, 2012 announcement has been updated \cite{ATLASVBF} to a two-photon plus two-jet search which isolates a combination of cross sections similar to that the CMS result. These simulations agree well with the results found in Ref~\cite{Giardino:2012ww}. Note that we do not include errors on the admixture of gluon fusion and VBF contributions when performing our global fits in Sec.~\ref{widthcouplings}.
In Fig.~\ref{data} we present a summary of the Higgs search results, shown as a comparison between the observed rate in a given channel and the rate predicted for a 125 GeV Standard Model Higgs boson, updated using the most recent ATLAS results \cite{ATLASupdate}. Standing out among these results is that all six measurements of channels involving $h\rightarrow \gamma \gamma$ have been observed at higher rates than predicted, possibly suggesting an enhanced Higgs decay width to photons. It is also the case that the rates observed in six out of the seven measurements dominated by gluon-gluon fusion ($h \rightarrow W^+ W^-$, $ZZ$, $\tau^+ \tau^-$) are lower than predicted, possibly favoring a reduced Higgs width to gluons. We find that the Standard Model Higgs without any modified widths provides a somewhat poor fit to the combined data ($\chi^2=19.67$, over 17-1 degrees-of-freedom). In the following section, we will consider variations to the Higgs' widths and couplings and discuss how the quality of this fit might be improved.
\section{Constraining The Decay Widths and Couplings of the Higgs}
\label{widthcouplings}
In this section, we consider departures from the Higgs widths and couplings predicted by the Standard Model, and discuss how such variations impact the global fit to the data shown in Fig.~\ref{data}. As an initial point of departure, we focus on modifications to the Higgs decay widths to photons and gluons. As these couplings are only induced at loop-level, contributions from new physics are more likely to significantly influence these channels than those which occur through tree-level couplings.
In Fig.~\ref{fitwidth}, we show the results of a global fit to the data summarized in Fig.~\ref{data}. We have allowed the decay widths of the Higgs to $\gamma \gamma$ and $gg$ to vary, while holding all other decay widths fixed to the Standard Model prediction. We find that the best fit occurs for values of $\Gamma_{h \rightarrow \gamma\gamma}$ and $\Gamma_{h \rightarrow gg}$ that are approximately 2.5 and 0.66 times their Standard Model values, yielding an overall fit of $\chi^2=9.77$ over 17-3 degrees-of-freedom. Modifying these two loop-induced decay widths can potentially improve the global fit by $\Delta \chi^2 = 9.9$ relative to the Standard Model, while only adding two additional degrees-of-freedom. If this data is taken at face value, without consideration of any unknown systematics or other uncertainties, this fit favors modified values of $\Gamma_{h \rightarrow \gamma\gamma}$ and $\Gamma_{h \rightarrow gg}$ over those of the Standard Model at the 99\% confidence level.
Taking this approach a step further, we have performed a global fit varying not only $\Gamma_{h \rightarrow \gamma\gamma}$ and $\Gamma_{h \rightarrow gg}$, but also the Higgs decay widths to $W^+W^-$/$ZZ$, $b\bar{b}$, and $\tau^+ \tau^-$.\footnote{As custodial $SU(2)$ forces us to require $\Gamma_{h \rightarrow WW}/\Gamma^{\rm SM}_{h \rightarrow WW}=\Gamma_{h \rightarrow ZZ}/\Gamma^{\rm SM}_{h \rightarrow ZZ}$, we chose to not vary these two widths independently of each other.} The best fit set of values found in this scan further improves the global fit to $\chi^2=8.49$. As the addition of these three new free parameters only improves the fit beyond the previous exercise by $\Delta \chi^2=1.3$, there does not appear to be any support for modifications of the tree-level couplings of the Higgs. We have also considered the possibility that the Higgs has a significant decay width to invisible final states, but find that this does not improve the global fit.
\section{Signs of Light Stops?}
\label{scenarios}
In this section, we discuss some of the types of physics beyond the Standard Model that may lead to improvements to the global fit, as discussed above. As outlined in the previous section, we focus on new particles which modify the loop-induced Higgs widths to photons and gluons. In general, new charged particles can lead to modifications to $\Gamma_{h\rightarrow \gamma\gamma}$ while new colored particles can alter $\Gamma_{h\rightarrow gg}$.
\begin{figure}[t]
\centering
\includegraphics[angle=0.0,width=3.4in]{contours.pdf}
\caption{The 68\%, 95\% and 99\% confidence level contours of the global fit to the data (as shown in Fig.~\ref{fitwidth}), compared to the range of decay widths of the Higgs that can result from the addition of new charged particles, or new charged and colored particles (top partners, defined as particles with Higgs coupling, charge, and color equal to that of the top quark).}
\label{contours}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[angle=0.0,width=3.4in]{Rt.pdf}
\caption{The quality of the global fit as a function of the parameter $R_t \equiv y_t/y^{\rm SM}_t$. The miniumum near $R_t = -0.85$ (corresponding to the point marked by a star on Fig.~\ref{contours}) provides a significantly better fit to the data than the Standard Model case.}
\label{Rt}
\end{figure}
In Fig.~\ref{contours}, we show how classes of new particles impact these decay widths. Additional charged particles simply alter the width to photons, moving vertically in the plane of the figure. Particles with both charge and QCD color, however, alter both of these widths simultaneously. As a concrete and well motivated class of examples, we consider generic partners of the top quark (particles with Higgs coupling, charge, and color equal to that of the top quark). As can be seen in the figure, the inclusion of a top partner particle can potentially lead to a good fit to the data, falling within the approximate 68\% confidence region.
Specializing to a top partner, such a particle's effect on the Higgs decay widths can be parameterized by its modification of the effective Higgs-top-top coupling: $R_t \equiv y_t/y^{\rm SM}_t$ (see, for example, Ref.~\cite{Giardino:2012ww}). In Fig.~\ref{Rt}, we show how the quality of the global fit changes with this parameter. The best fit is found for $R_t \simeq -0.85$, with $\chi^2=10.03$ over 17-2 degrees-of-freedom (corresponding to the point marked with a star shown along the top partner contour in Fig.~\ref{contours}). Once again, this marks a considerable improvement over the fit to the Standard Model ($\Delta \chi^2 =9.6$). Also shown as a shaded region in Fig.~\ref{Rt} is the 95\% confidence region around this best fit value of $R_t$.
The best known and most well motivated example of a top partner are the supersymmetric partners of the top quark, known as stops. The presence of stops modifies $R_t$ at leading order~\cite{Blum:2012ii} from its Standard Model value of one as follows:
\begin{equation}
R_t \approx 1+ \frac{m^2_t}{4} \bigg[\frac{1}{m^2_{\tilde{t}_1}}+\frac{1}{m^2_{\tilde{t}_2}}-\frac{(A_t-\mu \cot \beta)^2}{m^2_{\tilde{t}_1} m^2_{\tilde{t}_2}}\bigg],
\end{equation}
where $m_{\tilde{t}_{1,2}}$ are the masses of the two stops, $A_t$ is the top trilinear coupling, $\mu$ is the higgsino mass parameter, and $\tan \beta$ is the ratio of the vacuum expectation values of the two higgs doublets of the MSSM. To achieve the negative values of $R_t$ as are favored by our fit, we must require that the mixing term dominates. In this limit, and for $A_t \gg \mu \cot \beta$, this reduces to $R_t \sim 1 - (m^2_t A^2_t /4 m^2_{\tilde{t}_1} m^2_{\tilde{t}_2})$. Thus to achieve the desired values of $R_t$, we are required to consider light and highly mixed stops.
\begin{figure}[t]
\centering
\includegraphics[angle=0.0,width=3.4in]{stops.pdf}
\caption{$R_t$ as a function of the mass of the lightest stop, for several values of $A_t$. Here, we have assumed that $A_t \gg \mu \cot \beta$ and $m_{Q_3}=m_{U_3}$. The shaded horizontal band represents the range of $R_t$ that is favored by the global fit, as shown in Fig.~\ref{Rt}. For values of $A_t \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 2$ TeV and $m_{\tilde{t}_1} \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 300$ GeV, the favored range of $R_t$ can be accommodated.}
\label{stops}
\end{figure}
In Fig.~\ref{stops}, we show the value of $R_t$ as a function of the mass of the lightest stop, for several values of $A_t$. Here, we have assumed that $A_t \gg \mu \cot \beta$ and that the parameters $m_{Q_3}$ and $m_{U_3}$, appearing in the diagonal entries of the stop mass matrix, are equal to each other. We find that for values of $A_t \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 2$ TeV and $m_{\tilde{t}_1} \mathrel{\raise.3ex\hbox{$<$\kern-.75em\lower1ex\hbox{$\sim$}}} 300$ GeV, values of $R_t$ consistent with those favored by the global fit to the Higgs data can be accommodated. While stops below 103.5~GeV are excluded by LEP-II~\cite{LEPstops}, the current searches from ATLAS~\cite{ATLASstops} and CMS~\cite{CMSstops} do not yet cover the full range of light stops, especially in the regime where the stop, top, and missing energy particle are nearly degenerate.
Such light and highly mixed stops may lead to problems with charge and color-breaking vacua. An approximate condition for metastability is sometimes given as \cite{Kusenko:1996jn}
\begin{equation}
A_t^2 +3\mu^2 \lesssim 7.5 (m_{\tilde{t}_1}^2+m_{\tilde{t}_2}^2).
\end{equation}
For the values of $R_t$ favored in our analysis, this condition is only met for stops lighter than the top. One could also consider models with particle content beyond the MSSM that may modify this requirement (see also Ref.~\cite{Reece:2012gi}).
\section{Conclusions}
\label{conclusions}
The discovery of the Higgs boson by the ATLAS and CMS experiments ushers in an exciting and much anticipated era in particle physics. The observed production mechanisms and decays (especially to pairs of electroweak gauge bosons) are sufficient to state, with reasonable certainty, that this new particle is intimately tied to the mechanism of electroweak symmetry breaking. The next task at hand is to ascertain whether the characteristics of this Higgs are consistent with those predicted by the Standard Model.
By considering the results of all reported Higgs channels from the LHC and Tevatron detector collaborations, we find that the observed rates of Higgs production followed by decays into $WW$, $ZZ$, and $\tau\tau$ are uniformly low compared to the Standard Model expectation. Furthermore, the rates observed in Higgs channels ending in $\gamma\gamma$ final states are uniformly high. This upward deviation is more pronounced after the application of selection criteria designed to isolate vector boson fusion and associated production diagrams over gluon-gluon fusion.
To accommodate this combination of observed rates, our fit favors a Higgs decay width to photons that is enhanced by a factor of approximately three, and a width to gluons that is suppressed by a factor of two, relative to the predictions of the Standard Model. As the widths of the Higgs to photons and gluons are set by loop-level interactions, these can be particularly sensitive to the presence of physics beyond the Standard Model. And although we must be cautious not to over-interpret these results, at face value, assuming no unknown systematics, we find that the best fit values for the Higgs widths to photons and gluons is preferred over those predicted by the Standard Model at greater than 99\% confidence. In contrast, our global fit finds no significant preference for modified widths to any of the Higgs' tree-level decay widths, such as $WW$, $ZZ$, $\tau\tau$, or $b\bar{b}$.
To alter the loop-level decays in a way that significantly improves the global fit, we postulate new particles with large couplings to the Higgs and that are charged under both QCD and electromagnetism. The most obvious candidates are partners of the top partner, such as stops within the context of supersymmetry. We find that the presence of such a particle can easily modify the Standard Model Higgs widths in such a way to come within approximately one standard deviation of the observed values. More specifically, we find that good fits to the data can be obtained for light ($\lesssim 300$~GeV) and well-mixed ($A_t \mathrel{\raise.3ex\hbox{$>$\kern-.75em\lower1ex\hbox{$\sim$}}} 2$ TeV) stops.
The full $\sim 20$~fb$^{-1}$ data set that is anticipated from the LHC by the end of 2012 will greatly reduce the experimental errors associated with the Higgs widths that drive our global fit. Furthermore, future consideration of systematic effects could plausibly modify the results under discussion here. With these caveats in mind, however, it appears that the presence the experimental evidence favors a Standard Model Higgs boson with modified decay widths to photons and gluons, suggesting the presence of new, strongly coupled physics present well below the TeV scale.
\section*{Aknowledgements}
The authors would like to thank Patrick Fox, Graham Kribs, Joseph Lykken, Tilman Plehn, Nausheen Shah, Alessandro Strumia, Carlos Wagner, and Marcela Carena for their useful advice on this project.
MRB and DH are supported by the US Department of Energy.
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Juries announced for the 2019 Sony World Photography Awards
The World Photography Organization is delighted to announce the juries of the 2019 Sony World Photography Awards, the world's most diverse photography competition. To mark the occasion, a selection of entries to the Open competition is released, and the Chair of Judges offers advice to potential entrants.
The World Photography Organization today also publishes the opening list of participating countries for 2019's National Awards. A longstanding part of the Sony World Photography Awards, the National Awards provide the unique opportunity for entrants of all levels, from more than 50 countries, to be recognized in an internationally renowned photography competition.
Now in its 12th year, the Sony World Photography Awards celebrate the finest contemporary photography from the past 12 months across all genres of the medium. All entries are free at www.worldphoto.org/swpa.
Expert juries seeking the world's finest contemporary photography
The 2019 juries specially selected by the World Photography Organization are all experts working across the photographic industry. This year's panel, who will judge series across 10 categories of the Professional competition, consists of industry leaders from across the world:
Erin Barnett, Director of Exhibitions and Collections, International Center of Photography (USA)
Brendan Embser, Editor, Aperture (USA)
Olivier Laurent, Foreign Photo Editor, The Washington Post (USA)
Emma Lewis, Assistant Curator, Tate (UK)
Isabella van Marle, Head of Exhibitor Relations, Unseen (The Netherlands)
Chair: Mike Trow, editor, photographer, producer (UK)
Chair Mike Trow comments:
"It is an honor to be asked to be Chair of Judges for the Sony World Photography Awards' Professional competition. What makes the Sony World Photography Awards so exciting is the range of subjects and global reach of the stories and images selected. My advice for entrants is to choose your categories carefully and believe in your story. Show how you see the world and avoid cliché. Photography techniques and styles are getting more adventurous and dynamic, so technical excellence is also necessary. To impress this world-leading panel of judges will take your best output, and the ability to edit your work so it is coherent, dynamic and beautiful."
The Open competition, judged on a single image across 10 categories, will be chaired by Rebecca McClelland, Photography Director & Head of Art Production for Saatchi Saatchi & Prodigious (UK), who we are delighted will also chair the Youth and National Awards competitions.
This year's Student competition will also be judged by 3 further leading judges from the international photography industry - Jason Baron, Creative Head of Photography, BBC Creative (UK); Bruno Bayley, Managing Editor, Magnum Photos (UK) and Jeff Hamada, Founder & Editor, BOOOOOOOM (Canada)
For full biographies of each judge please go to:
https://www.worldphoto.org/sony-world-photography-awards/judges
Inspiring images from across the Open competition
Following a record-breaking number of entries in 2018*, the 2019 Sony World Photography Awards have already received thousands of diverse and exceptional images from across the world. While the Professional competition is judged on a series of images, the new images released today are all entries from across the 10 categories of the Open competition. Judged upon a single image, the submissions include Marco Gaiotti's (Italy) image of icebergs stuck in frozen water in Svalbard (Landscape category), Pedro Luis Ajuriaguerra Saiz's (Spain) image showing a diver in motion outside the Guggenheim Bilbao, Christy Lee Rogers' (USA) and images of swirling images of people entangled in water (both Motion category).
Worldwide recognition for all levels through the National Awards and Sony Grant
Each year the Sony World Photography Awards celebrate and reward photographers of all abilities, recognizing stunning bodies of work in the Professional and Student competitions and the world's best single images across the categories of the Open and Youth competitions. The National Awards celebrate local photographic talent from more than 50 countries. The range of eligible countries and prizes can be found below, with more being added in the coming months: https://www.worldphoto.org/2019-national-awards
The Awards' winning and shortlisted photographers can enjoy worldwide recognition and exposure in addition to cash prizes, the latest digital imaging equipment from Sony and inclusion in global exhibitions.
Award-winners can also secure a Sony Grant to fund future photographic projects. Multiple grants of $7,000 (USD) will be awarded to selected winners of the Professional competition and multiple grants of $3,500 (USD) will be given to selected shortlisted photographers in the Student competition to work together on a new photographic commission set by Sony and the World Photography Organization.
Forthcoming deadlines and announcements
The closing dates for the 2019 Sony World Photography Awards are:
November 30, 2018 (13.00 GMT) – Student competition
January 4, 2019 (13.00 GMT) – Open, Youth and National Award competitions
January 11 , 2019 (13.00 GMT) – Professional competition
The Open and Youth shortlist for the Awards will be announced on February 5, 2019. The Open and National Awards winners will be announced February 26, 2019. The Professional and Student competitions' shortlist will be announced on April 2, 2019. The Photographer of the Year, Overall Open, Student, Youth, and Professional category winners will be announced April 17, 2019. The 2019 Sony World Photography Awards exhibition will run from April 18 - May 6, 2019, at Somerset House, London.
Labels: Awards, Competition, International, Lifestyle, Photography | {
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{"url":"http:\/\/www.maths.usyd.edu.au\/s\/scnitm\/zhangou-SYD-UNSWJointColloquium-F-001","text":"SMS scnews item created by Zhou Zhang at Tue 27 Nov 2012 0018\nType: Seminar\nModified: Tue 27 Nov 2012 0024; Tue 27 Nov 2012 1637; Thu 29 Nov 2012 1225\nDistribution: World\nExpiry: 18 Dec 2012\nCalendar1: 30 Nov 2012 1400-1500\nCalLoc1: Chemistry Lecture Theatre 2\nAuth: zhangou@cpe-124-183-243-228.lns13.ken.bigpond.net.au (zhouz) in SMS-WASM\n\n# SYD-UNSW Joint Colloquium : Friedlander -- Active Scalar Equations and a Geodynamo Model\n\nThis is the first talk of the colloquium double-header.\n\nThe second talk is announced at\n\nhttp:\/\/www.maths.usyd.edu.au\/s\/scnitm\/zhangou-SYD-UNSWJointColloquium-J\n\n*************************************************************\n\nSpeaker: Prof. Susan Friedlander (USC)\n\nhttp:\/\/cams.usc.edu\/~susanfri\/\n\nTime: Friday, Nov. 30, 2--3PM\n\nRoom: Chemistry Lecture Theatre 2\n\nLunch plan: meet near Level 2 entrance to Carslaw Building around 1PM.\nThe lunch would be at Law Annex Cafe with reservation at 1:05PM.\n\nPlease keep in mind that our schedule would be a little tight with\nAlgebra Seminar finishing at 1PM and the first colloquium talk starting\nat 2:05PM.\n\nRefreshments would be provided by the school in the common room on\nLevel 7 around 4PM.\n\n-----------------------------------------------\n\nTitle: Active Scalar Equations and a Geodynamo Model\n\nAbstract: we discuss an advection-diffusion equation that has\nbeen proposed by Keith Moffatt as a model for the Geodynamo.\nEven though the drift velocity can be strongly singular, we\nprove that the critically diffusive PDE is globally well-posed.\nWe examine the nonlinear instability of a particular steady\nstate and use continued fractions to construct a lower bound\non the growth rate of a solution. This lower bound grows as\nthe inverse of the diffusivity coefficient. In the Earth\u2019s\nfluid core this coefficient is expected to be very small.\nThus the model does indeed produce very strong Geodynamo action.\n\nThis work is joint with Vlad Vicol.\n\n-----------------------------------------------\n\nJoint Colloquium web site:\n\nhttp:\/\/www.maths.usyd.edu.au\/u\/SemConf\/JointColloquium\/index.html\n\n\nIf you are registered you may mark the scnews item as read.\nSchool members may try to .","date":"2017-12-14 06:30:23","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.303475558757782, \"perplexity\": 7951.769343248754}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2017-51\/segments\/1512948541253.29\/warc\/CC-MAIN-20171214055056-20171214075056-00500.warc.gz\"}"} | null | null |
Lázár Miklós, Lederer (Nyíregyháza, 1886. március 22. – Baden bei Wien, 1968. november 15.) magyar újságíró, szerkesztő, országgyűlési képviselő.
Élete
Apja, Lederer Ignác (1859–1905) gyógyszerész volt, akit Krúdy Gyula is gyakran emlegetett írásaiban, édesanyja Klein Kornélia (1864–1910) volt. A középiskola alsó osztályait szülővárosában végezte, majd családjával a fővárosba költözött. Érettségi után 1905-ben a Magyar Hírlap munkatársa lett, és társadalmi, törvényszéki, rendőri és színházi tudósításokat írt. Elbeszéléseit és verseit főképp A Hét közölte, de több önálló kötete is megjelent. Beutazta Nyugat-Európát, és Londonban a Daily Expressnél dolgozott. Hazatérése után a Pesti Napló munkatársa, utóbb politikai rovatvezetője és vezércikkírója lett. A liberális irányzat követője volt. Az első világháború idején, 1914 és 1917 között haditudósítóként dolgozott. Tudósításai megjelentek a Pesti Napló, Az Est, a Pester Lloyd és a Berliner Tagesblatt című lapokban. 1917-ben katonaként szolgált az orosz fronton. Elbeszéléseinek témáit főként ekkor szerzett élményeiből merítette. 1917 és 1919 között a Déli Hírlap felelős szerkesztője volt. 1919-ben először Szegedre ment, majd márciusban Bécsben csatlakozott Bethlen István Tanácsköztársaság elleni szervezkedéséhez. 1919-ben a kormányt támogatta írásaival. 1921-től A Reggel című hetente megjelenő liberális szellemű politikai lap alapító főszerkesztő-tulajdonosa volt. Az 1920-as évek második felében és az 1930-as években bekapcsolódott a politika életbe. 1925-ben a demokratikus blokk listáján a székesfővárosi törvényhatósági bizottság tagjává választották. 1930 és 1936 között pártonkívüliként a Tokaji kerület országgyűlési képviselője volt és ebbéli minőségében kezdeményezője az azóta is megrendezésre kerülő Tokaji Szüreti Napoknak. Munkatársa volt a Magyar városok monográfiája sorozat szerkesztőségének. Műveit németre és angolra fordították. 1948-ban elhagyta az országot, az Egyesült Államokban telepedett le. A Szabad Európa Rádió megalakulása után annak New York-i munkatársa lett. 1954 novembere és 1957 között a müncheni szerkesztőségben dolgozott. 1952 és 1954 az Amerikai Magyarság című lapot szerkesztette. Halála előtt Ausztriába költözött és Baden bei Wienben hunyt el.
Házastársa Radó Kornélia volt, Radó Izrael és Hoffmann Berta lánya, akivel 1915. december 31-én Budapesten, a Józsefvárosban kötött házasságot.
Művei
Varieté (1914)
Fronton (1915)
Magyarok dalolnak (1916)
A magyar háború (1933)
Jegyzetek
Források
Magyar újságírók
Magyar írók
1886-ban született személyek
1968-ban elhunyt személyek
Nyíregyháziak
Emigráns magyarok
Családi nevüket magyarosított személyek | {
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{"url":"https:\/\/math.stackexchange.com\/questions\/linked\/53752","text":"3k views\n\n### Is the class of cardinals totally ordered?\n\nIn a Wikipedia article http:\/\/en.wikipedia.org\/wiki\/Aleph_number#Aleph-one I encountered the following sentence: \"If the axiom of choice (AC) is used, it can be proved that the class of cardinal ...\n1k views\n\n### A question about cardinal arithmetics without the Axiom of Choice\n\nIs multiplication of infinite cardinals defined in ZF without Choice?\n2k views\n\n370 views\n\n### If $\\kappa$ is a cardinal, $\\aleph$ is any $\\aleph$-number, and if $\\kappa\\leq\\aleph$ then $\\kappa$ can be well ordered as well.\n\nI'm having trouble understanding the statement: If $\\kappa$ is a cardinal, $\\aleph$ is any $\\aleph$-number, and if $\\kappa\\leq\\aleph$ then $\\kappa$ can be well ordered as well. I understand the ...\n189 views\n\n### Injections from all ordinals into a set $X$\n\nWe are working in $\\mathsf{ZF}$. Let $X$ be a set. Let $A$ be the class of all injections $f: \\alpha \\to X$ for arbitrary ordinals $\\alpha$. I am quite sure that, in fact, $A$ is a set, since if not,...\n67 views\n\n### If AC is false, does that mean there exist a set $A$ which has different cardinality from any ordinals?\n\nIf a set $A$ has the same cardinality as an ordinal $\\alpha$, then there exists a bijection $f:\\alpha\\to A$, so $A$ is indexed by $\\alpha$ and hence well-ordered. Therefore a choice function \\$g:\\...\n53 views\n\n### Relation between limit ordinals and alephs. [duplicate]\n\nI was wondering what the relation is between a limit ordinal and the alephs. Are all limit ordinals alephs and if so can it be proven.","date":"2019-07-21 06:18:58","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\": 0.9911210536956787, \"perplexity\": 271.39414510300384}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 5, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195526931.25\/warc\/CC-MAIN-20190721061720-20190721083720-00313.warc.gz\"}"} | null | null |
\section{Introduction}
\IEEEPARstart{S}HOPPING in the brick-and-mortar stores takes considerable time to purchase one satisfactory item of clothing because it usually requires entering a store to try on several clothing candidates. In contrast, shopping online is expected to be a much faster purchase journey because the process of finding the products with relevant items is facilitated by the online searching and recommendation technology. However, although the usage rate of online shopping is rapidly increasing, it is still overshadowed by the brick-and-mortar stores because e-commerce platforms cannot provide sufficient information for
the consumers. Among many promising approaches~\cite{Viton2017,BeautyGlow2019,fashion1,fashion2,fashion3,fashion4,DressAttention2019,DressBest2018} bridging the gap between online and offline shopping, virtual try-on is regarded as the key technology for the online fashion industry to burgeon, as well as a feasible work to bridge the gap between online and offline shopping.
\begin{figure}[t]
\includegraphics[width=8cm]{firstpic_1.jpg}
\centering
\caption{Examples of our template-free try-on image synthesis (TF-TIS) network, which takes only the in-shop clothing image and user image as input without a defined pose. Our goal is to generate a realistic try-on image according to the synthesized pose from the referential clothing image, which can reduce the cost of hiring photographers. No other work has achieved this.}
\label{fig:7}
\end{figure}
To realize virtual try-on services, a recent line of studies has used clothing warping to transform the in-shop clothing and then paste the warped clothing on user images~\cite{CPVTON2018,Viton2017}, which preserves the details of clothes, including patterns and decorative designs. Nonetheless, the quality of the results significantly decreases when an occlusion (e.g., the user's arm is in front of the chest, obscuring the garment in the source image) or a dramatic pose change (e.g., the limbs are from wide-opened to crossed) occurs. To solve these challenging cases, our previous work~\cite{FashionOn} introduced a semantic-guided method, which uses the semantic parsing to learn the relationship between different poses. However, the clothing part of the virtual try-on results still has some artifacts (i.e., missing details, such as small buttons and local inconsistency, such as distorted plaid), which are important for a try-on service. Moreover, although the state-of-the-art virtual try-on applications~\cite{VirtuallyTryOn, FitMe} have demonstrated the try-on results in arbitrary poses (including our previous work~\cite{FashionOn}), they require users to assign the target poses instead of directly recommending the suitable poses based on the clothing style. Therefore, to create a convenient and practical virtual try-on service, a virtual try-on application that automatically synthesizes a suitable pose corresponding to the target clothing is desirable.
Based on the above observations, in this paper, we propose a novel virtual try-on network, namely, the template-free try-on image synthesis (TF-TIS) framework, for synthesizing high-quality try-on images with automatically synthesized poses. In addition, Fig.~\ref{fig:7} illustrates examples of the template-free virtual try-on. Given a source user image and an in-shop clothing, the goal is to first synthesize the target pose automatically, which is further leveraged to generate the try-on image. Fig. \ref{fig:arc1} presents the TF-TIS framework comprising four modules: 1)
\textit{cloth2pose}, which synthesizes a suitable pose from the in-shop clothing (Column 3 in Fig. \ref{fig:7}), 2) the \textit{pose-guided parsing translator}, which translates the source pose to semantic segmentation according to the synthesized poses (Column 4 in Fig. \ref{fig:7}), 3) \textit{segmentation region coloring}, which renders the clothing and human information on the semantic segmentation (Column 5 in Fig. \ref{fig:7}), and 4) \textit{salient region refinement}, which polishes the important regions, such as faces and logos (the last column in Fig. \ref{fig:7}).
Specifically, given an in-shop clothing for try-on, we first aim to synthesize a suitable corresponding pose represented as keypoints\footnote{The poses are specified by keypoints, which contain 18 keypoints and each keypoint represents one human body joint.}. One of the basic approaches is to use the images of mannequins wearing corresponding in-shop clothes as the target poses. However, some in-shop clothes may not have corresponding images. Another approach is to cluster the in-shop clothes first and assign the most frequent poses in the cluster as the target pose. Nevertheless, such an approach highly depends on the clustering results, whereas rare/unseen clothes may not find the appropriate poses. Therefore, we propose a novel cloth2pose network to directly learn the relationship between the in-shop garment and the target pose, which leverages the deep features from the pretrained model and then uses the regressor to fit the joint map (i.e., keypoints). To the best of our knowledge, this is the first work to generate suitable poses for corresponding in-shop clothes.
Afterward, given a source user image and a synthesized pose, the goal is to synthesize a realistic try-on image. An intuitive method to tackle difficult cases of the body occlusion or the dramatic pose transfer is offering the body parsing information to the current try-on networks. However, this method is not compatible with existing try-on models because most of the previous try-on works have focused on directly warping the clothing item and pasting the warped clothing onto the users.
Therefore, the \textit{pose-guided parsing translator} is proposed by constructing a deep convolutional network to transform a pose into a semantic segmentation form to guide the learning of the next stage. Semantic segmentation plays a critical role in solving difficult cases. For example, limb parsing provides information for solving dramatic pose changes, whereas limb and clothing parsing offer clues addressing body occlusion issues.
Moreover, to present realistic try-on images to users, we color the transformed semantic segmentation with the appearance of the a human and clothes by using a conditional generative adversarial network (CGAN) in \textit{segmentation region coloring}. Finally, \textit{salient region refinement} focuses on two salient regions for try-on services (i.e., face and clothing) and improves these regions with details to achieve better virtual try-on images. For clothing refinement, We constructed a detail-retaining network, which adopts two encoders to extract relatively important features and global and local discriminators to retain the consistency of images, especially clothing.
Our previous work is called FashionOn\cite{FashionOn}. We have made several changes in this work, and the contributions are summarized as follows.
\begin{itemize}
\item We designed a new pose synthesis framework, which directly learns the relationship between in-shop clothes and try-on poses to synthesize a suitable try-on pose. The automatically synthesized poses can facilitate a user-friendly platform without the extra effort of uploading a target pose and exhibit better virtual try-on results to attract customers. To the best of our knowledge, TF-TIS is the first virtual try-on network to provide a suitable pose for the corresponding clothing image.
\item We redesigned the clothing refinement generator (Section III-D-2) composed of two distinct encoders due to the unsatisfying results caused by the same parameters of the encoder for the two input features of in-shop clothing and warped coarse clothing. One is to encode the warped coarse clothing and we integrate it with the very detailed features of the in-shop clothing extracted from the other encoder. In addition, we adopted a UNet-like architecture to avoid losing the warped coarse clothing information, such as shape and color.
\item To enhance the consistency of the generated image, which improves the quality of pictures, we proposed global and local discriminators for our ClothingGAN (Section III-D-2). With the local discriminator, the generator is forced to synthesize the image with more natural details. Using the global discriminator, the generator is forced to generate a realistic picture.
\end{itemize}
\section{Related work}
\subsection{Virtual Try-on}
Existing virtual try-on approaches can be roughly categorized into 3D-based methods (e.g., 3D body shape) and 2D-based methods (e.g., clothing warping). We first introduce these two approaches and then compare them with TF-TIS.
\subsubsection{3D-based Try-on}
To generate more realistic results, numerous approaches~\cite{ClothCap, LearningSSS, GarNet} have used users' 3D body shape measurements and 4D sequence (e.g., video) to offer more information.
For example, with high-resolution videos, Pons-Moll \textit{et al.} \cite{ClothCap} first captured the geometry of clothing on a body to obtain a rough body meshes and then aligned the defined clothing templates to garments of the input scans again to generate more realistic and body-fitting clothes.
Given the high cost of physics-based simulation to accurately drape a 3D garment on a 3D body, Gundogdu \textit{et al.} \cite{GarNet} implemented 3D cloth draping using neural networks. Specifically, they used a PointNet-like model to derive the user information and encoded the garment meshes to obtain the point-wise, patch-wise, and global features for the fitted garment estimation.
In summary, although 3D-based approaches can produce try-on videos, the collection of measurement data can be costly, requiring extensive manual labeling or expensive equipment. Therefore, many scholars have resorted to using rich 2D images, which can be easily found online, to achieve the virtual try-on task. Moreover, the proposed TF-TIS only requires a source image and an in-shop clothing to a synthesize try-on image with the suitable pose.
\subsubsection{2D-based Try-on} To synthesize the try-on images, it is necessary to transform the in-shop clothing to fit users' poses.
Therefore, spline-based approaches are introduced to achieve this task. Among them, thin plate spline (TPS)~\cite{TPS} has been widely adopted and predominates in the nonrigid transfer of images instead of direct generation using neural networks.
For example, Han \textit{et al.} \cite{Viton2017} presented an image-based virtual try-on network (VITON) that warps in-shop clothes through TPS and cascades a refinement network to generate the warped-clothing details with the coarse-grained person image. However, some details are still missed due to the refinement network~\cite{Viton2017}.
To correct deficiencies in \cite{Viton2017}, Wang \textit{et al.} \cite{CPVTON2018} constructed a two-stage framework, CP-VTON, combining the generated person with the warped clothes through a generated composition mask without adopting the refinement network.
Moreover, Zheng \textit{et al.} advanced CP-VTON \cite{CPVTON2018} and proposed Virtually Trying on New Clothing with Arbitrary Poses (VTNCAP)\cite{VirtuallyTryOn} by adopting a bidirectional GAN and an attention mechanism, which take the place of the generated composition mask in CP-VTON, to focus more on the clothing region.
Nevertheless, they still neglected that the facial region is also an important factor to determine the quality of the virtual try-on task and cannot preserve the detailed clothing information (e.g., pleats and shadows) to follow the human poses.
In contrast, TF-TIS was developed as a semantic segmentation-based method that avoids these issues. In addition, TF-TIS preserves the comprehensive details of the in-shop clothes (e.g., patterns and texture) and the realistic human appearance (e.g., hair color and facial features) in accordance with human poses and different body shapes. The visual comparison between TF-TIS and the other mentioned methods is illustrated in Fig.~\ref{fig:result1}.
\begin{figure*}[t]
\includegraphics[width=\textwidth]{rebuttal_img/arc_all_in_one.jpg}
\centering
\caption{\textbf{Training overview}. Stage I (\textit{cloth2pose}) exploits the correlation of clothes and poses and synthesizes a pose from the in-shop clothes via sequential convolution blocks. Stage II (\textit{pose-guided parsing translator}) transfers the human semantic segmentation to $M_g$ according to $M^{'}_s$, $M_c$, and $P$. Stage III (\textit{segmentation region coloring}) fills clothing information and the user's appearance into the segmentation to synthesize a realistic try-on image $I_g$. Stage IV (\textit{salient region refinement}) consists of two parts: FacialGAN and ClothingGAN. FacialGAN generates high-frequency details as a residual output and directly adds on the facial region of $I_g$. ClothingGAN extracts the fine information from the in-shop clothing image and uses the features for the details of the clothes $C_r$.}
\label{fig:arc1}
\end{figure*}
\subsection{Pose Transfer}
Research on human pose transfer~\cite{Ma2017PoseGP,Multistage,UnsuperPose,DeformableGAN,UnsuperPIG,DenseIntrinsic,ProgressivePoseAttn} has trended recently, as copious applications are planned in the future.
The process of human pose transfer comprises two stages: pose estimation and image generation. The first stage can be divided into two categories (i.e., keypoints estimation \cite{Openpose,LCR,SinglePose} and human semantic parsing~\cite{HumanSemanticParsing,Gong2018InstanceLevelHP,Beyond}).
For example, Hidalgo \textit{et al.}\cite{SinglePose} trained a single-stage network through multi-task learning to determine the keypoints of the whole body simultaneously. Gong \textit{et al.}\cite{Gong2018InstanceLevelHP} used the part grouping network (PGN) to reformulate instance-level human parsing as two twinned sub-tasks that can be jointly learned and mutually refined.
For the second stage, with the advances of GANs, image generation has received considerable attentions and has been widely adopted \cite{Multistage, UnsuperPIG, ProgressivePoseAttn} to generate realistic images. Among the existing pose transfer research, most constructed novel architectures and successfully transferred the pose of the given human image based on the human joint points.
For instance, Ma \textit{et al.}\cite{Ma2017PoseGP} separated this task into two stages: pose integration, which generates initial but blurry images, and image refinement, which refines images by training a refinement network in an adversarial way. Siarohin \textit{et al.}\cite{DeformableGAN} used geometric affine transformation to mitigate the misalignment problem between different poses.
However, most of the previous works did not extend the application of pose transfer to explore virtual try-on. By infusing pose transfer, virtual try-on services provide consumers with more chances to realize their appearance in trying on new clothes in multiple aspects and induce them to buy clothes. Hence, by converting semantic segmentation, TF-TIS seamlessly integrates virtual try-on with pose transfer to generate multi-view try-on images for customers.
\subsection{Cross-modal Learning}
The association between different fields has been studied and exploited recently~\cite{LocalizeSource, VisuallyIndicatedSounds, AudioVisualSceneAnalysis,DeepCMProjection, cross1, IdentityAware, RecurrentResidualFusion,cross2} (e.g., the cross-modal matching between audio and visual signals~\cite{LocalizeSource, VisuallyIndicatedSounds, AudioVisualSceneAnalysis}, image and text~\cite{DeepCMProjection,IdentityAware,RecurrentResidualFusion}).
Castrej{\'o}n \textit{et al.}~\cite{AlignedCMRepresentations} used the identical network architecture with different weights as encoders to extract low-level features from different modalities (e.g., sketches and natural images) and then inputted them into the shared cross-modal representation network to learn the representation for scenes. Tae-Hyun \textit{et al.}~\cite{Speech2Face} attempted to learn voice-face correlation in a self-supervised manner (i.e., directly capturing the dominant facial traits of the person correlated with the input speech instead of synthesizing the face from the attributes).
Inspired by the cross-modal learning, which can find hidden details from an inconspicuous part of data or align the embedding from one domain to another, we used a similar concept to learn the correlation between clothes and human poses to synthesize the image of the virtual try-on with a suitable pose for the user from the corresponding clothing.
\begin{table}[bt]
\centering
\caption{Notation Table}
\begin{tabular}{c|c}
\textbf{Symbols} & \textbf{Definitions} \\
\hline \hline
$C_t$ & in-shop clothing image \\
$M_c$ & in-shop clothing mask \\
$C_d$ & detailed clothing representation\\
$C_w$ & warped-clothing representation\\
$C_r$ & refined clothing \\
\hline
$P$ & keypoint tensor (suitable pose) \\
$F$ & in-shop clothing feature map \\
\hline
$I_s$ & source user image \\
$I^{'}_s$ & source user image without clothes\\
$I_t$ & target user image \\
$I_g$ & generated try-on image \\
\hline
$M_s$ & source body semantic segmentation \\
$M^{'}_s$ & source body semantic segmentation without clothing part\\
$M_g$ & generated body semantic segmentation \\
$M_t$ & target body semantic segmentation \\
\hline
$M^{fg}_i$ & foreground channels of $M_i$, $i \in s, g, t$ \\[3pt]
$M^{face}_i$ & facial channels of $M_i$, $i \in s, g, t$ \\[3pt]
$M^{clothing}_i$ & clothing channels of $M_i$, $i \in s, g, t$ \\[3pt]
$I^{face}_i$ & facial part of $I_i$, $i \in s, g, t$ \\[3pt]
$I^{clothing}_i$ & clothing part of $I_i$, $i \in s, g, t$ \\[3pt]
\hline
$d$ & high-frequency residual face details \\
$N_{c2p}$ & number of convolution blocks in \textit{cloth2pose} \\
$\otimes$ & pixel-wise multiplication \\
\end{tabular}
\label{tab:table2}
\end{table}
\section{Proposed Method}
As illustrated in Fig.~\ref{fig:arc1}, given an in-shop clothing image $C_t$ and a source user image $I_s$, the goal of TF-TIS is to generate the try-on image $I_g$ with an automatically synthesized suitable pose such that the personal appearance and the clothing texture are retained. To achieve this goal, we developed a four-stage framework in TF-TIS: (I) \textit{cloth2pose}, which derives a suitable pose $P$ based on the in-shop clothing $C_t$ by exploiting the correlation between poses and clothes; (II) the \textit{pose-guided parsing translator}, which transforms the body semantic segmentation $M_s$ into a new one, $M_g$, according to the derived pose; (III) \textit{segmentation region coloring}, which takes $I_s$, $C_t$, and $M_g$ as input and synthesizes a coarse try-on image $I_g$ by rendering the personal appearance and clothing information into the segmentation regions; and (IV) \textit{salient region refinement}, which refines the salient but blurry regions of the try-on result $I_g$, generated from the last stage (i.e, FacialGAN refines facial regions and ClothingGAN refines clothing regions). To clarify the definition of each symbol, we created a table (Table \ref{tab:table2}) to illustrate this clearly.
\subsection{Cloth2pose}
A virtual try-on service usually requires three inputs~\cite{VirtuallyTryOn,CPVTON2018, FitMe}: 1) a user image, 2) an in-shop clothing image, and 3) a target pose. One potential improvement is to automatically generate the target pose according to the in-shop clothing because it reduces the users' efforts. Moreover, a suitable pose better demonstrates the in-shop clothing, which may stimulate consumption. For example, plain T-shirts in a sideways pose can mostly show muscle lines. To synthesize the target pose directly from in-shop clothing, \textit{cloth2pose} uses pairs of in-shop clothes and mannequin photos on the online shopping site for training. Specifically, \textit{cloth2pose} first derives keypoints of mannequin photos by existing models, e.g.,~\cite{Openpose,pose1,pose2}.\footnote{We use OpenPose model~\cite{Openpose}, a 2D pose estimation model pre-trained on large-scale human pose datasets (COCO~\cite{MSCOCO} and MPII~\cite{PoseEstimation2014}) in our experiment. The following keypoints are used: nose, eyes, ears, neck, shoulders, elbows, wrists, hips, knees, and ankles.} Let $x_k$ denote the 2D position of the $k^{th}$ keypoint on the image ($I_t$). Because it is difficult to regress the clothing features to a single point, we converted the keypoint position $x_k$ into the pose map $P_k$ by applying a 2D Gaussian distribution for each keypoint. The values at position $p \in {R}^2$ in $P_k$ are defined as follows:
\begin{small}
\begin{equation
P_k(p) = \exp{\left( -\frac{\lVert p - x_k\lVert^2_2}{\sigma^2} \right)},
\end{equation}
\end{small}where $\sigma$ determines the spread of the peak. After constructing the 2D keypoint map for each keypoint, we stack all the 2D keypoint maps together as a keypoint tensor, denoted as $P$.
Afterward, \textit{cloth2pose} extracts features of the in-shop clothes by using the first 10 layers of VGG-19~\cite{VGG19}, denoted as $\phi_0$. Let $C_t$ denote the image of in-shop clothing. The clothing feature map $F$ is obtained as $\phi_0(C_t)$. Here, \textit{cloth2pose} exploits a progressive refinement architecture as illustrated in Fig.~\ref{fig:arc1}. Specifically, at the first block, the network produces a set of keypoint information only from the clothing feature map: $P^1 = \phi_1(F)$, where $\phi_1$ refers to the first convolutional block. For the succeeding convolutional blocks, we employed five convolutional layers with a $7\times7$ kernel and two with a $1\times1$ kernel to generate the keypoint tensor, and each layer is followed by a ReLU. The convolutional block takes the concatenation of $F$ and the prediction from the previous block as input to predict the refined keypoint tensor:
\begin{small}
\begin{equation}
P^i = \phi_i(F, P^{i-1}), \forall 2 \leq i \leq N_{c2p},
\end{equation}
\end{small}where $\phi_i$ represents the $i^{th}$ convolutional block and $N_{c2p}$ is the number of total convolutional blocks in \textit{cloth2pose}.
An intuitive choice for the loss function is the $L_2$ distance between the keypoint tensors extracted from the pose estimation model ($P$) and that estimated from \textit{cloth2pose} ($P^{N_{c2p}}$), i.e., $\lVert P^{N_{c2p}}-P \lVert^2_2$. However, only using the $L_2$ loss is likely to generate many responses in various locations for one joint. Given this condition, we employed the sparsity constraint to limit the number of candidates. Therefore, if the model predicts several candidates for one keypoint, the nonjoint area is penalized less by L2 loss than by L1 loss. The final loss is as follows:
\begin{equation
\mathcal{L}_{c2p} = \sum_{i \in N_{c2p}}\lVert P^i-P \lVert^2_2 \ + \ \lambda\lVert P^{N_{c2p}} \lVert_1,
\end{equation}
where $\lambda=0.00008$ is the hyperparameter for striking a balance between multiple candidates and keypoint vanishing.
If the value of $\lambda$ is too high (e.g., $\lambda = 0.001$), then the output $P^{N_{c2p}}$ is without any candidates. Conversely, if the value is too low (e.g., $\lambda = 0.00001$), then the sparsity constraint becomes ineffective, and the output still has more than one candidate.
\subsection{Pose-guided Parsing Translator}
Showing the corresponding area of each body part explicitly, the human body segmentation are employed to synthesize realistic human images. Accordingly, the goal of the \textit{pose-guided parsing translator} is to translate the source body semantic segmentation $M_{s}$ to the target body semantic segmentation $M_{t}$ according to the target pose $P$. We first used the PGN\cite{Gong2018InstanceLevelHP}, which is pretrained on the Crowd Instance-level Human Parsing dataset, to produce semantic parsing labels. The labels contain 20 categories, including left-hand, top clothes, and face. Afterward, to precisely map each item to the new position according to the pose $P$, we used one-hot encoding to constitute a 20-channel tensor $M \in {R}^{{20}\times{W}\times{H}}$, where each channel is a binary mask representing one category.
Due to the unnecessity of the clothing channel of $M_{s}$, we replace it with the original in-shop clothing mask $M_c$. This replacement facilitates offering the in-shop clothing shape to realize the virtual try-on service.
Adapted from pix2pix\cite{Phillip2017pix2pix}, the \textit{pose-guided parsing translator} consists of two downsampling layers, nine residual blocks, and two upsampling layers. Convolutional layers and highway connections, concatenating the input and the output of the corresponding block, are composed in each residual block. The objective of the translator $G_t$ adopts a CGAN as follows:
\begin{small}
\begin{equation
\begin{split}
\mathcal{L}^{G_t}_{GAN}(G_t, D_t) &= \mathbb{E}_{M_{in}, M_t}[log\textit{D}(M_{in}, M_t)] \\
&+ \mathbb{E}_{M_{in}}[log(1-\textit{D}(M_{in}, G_t(M_{in}))],
\end{split}
\end{equation}
\end{small}where $G_t$ minimizes the objective against $D_t$ that maximizes it (i.e., $\arg \min_{G_t} \max_{D_t} \mathcal{L}^{G_t}_{GAN}(G_t, D_t)$) and $M_{in}$ represents the concatenation of $M^{'}_s$, $P$, and $M_c$.
To accurately differentiate each pixel as the corresponding channel, we integrate a pixel-wise binary cross-entropy loss of the $G_t$, denoted as $\mathcal{L}^{G_t}_{BCE}$, with our CGAN objective, and the discriminator stays the same:
\begin{small}
\begin{equation
\begin{split}
\mathcal{L}&^{G_t}_{BCE}(G_t) = \\
&-\sum_{n_c} M_t\log(G_t(M_{in}))+(1-M_t)\log(1-G_t(M_{in})),
\end{split}
\end{equation}\end{small}where $n_c$ denotes the total number of channels of human parsing masks. In summary, the objective of the \textit{pose-guided parsing translator} is derived as follows:
\begin{equation
\arg \min_{G_t} \max_{D_t} \mathcal{L}^{G_t}_{GAN}(G_t, D_t)+ \lambda_{bce}\mathcal{L}^{G_t}_{BCE}(G_t).
\end{equation}
\subsection{Segmentation Region Coloring}
Having obtained the target semantic segmentation from the previous stage, the \textit{segmentation region coloring} aims to synthesize a coarse try-on result by rendering information into the segmentation regions, denoted as $M_g = G_t(M_{in})$.
Given the great success of applying GANs in various image generation tasks, we adopte the architecture of CGAN\cite{cGAN} to synthesize results. Specifically, we propose a coloring generator $G_{c}$ rendering the personal information into the body semantic segmentation $M_g$ according to $I_{s}$ and $C_{t}$ (i.e., the appearance of the source person and in-shop clothing texture). Because it is difficult to derive a significant number of training images, we traine our network to change the source person. To avoid supplying $G_c$ with the clothing information, we remove the clothing information from $I_{s}$. In other words, we take as input 1) the in-shop clothing $C_{t} \in {R}^{{3}\times{W}\times{H}}$, 2) the source person image without clothing information $I^{'}_{s}\in {R}^{{3}\times{W}\times{H}}$, and 3) the target semantic segmentation $M_g \in {R}^{20\times{W}\times{H} }$ for $G_{c}$.
Fig.~\ref{fig:arc1} illustrates the architecture of TF-TIS. We adopted the UNet architecture with highway connections, combining the input and processed information. Highway connections were employed to avoid the vanishing gradient \cite{GradDesDiff}.
Six residual blocks were implemented between the encoder and the decoder of $G_c$. For each residual block, two convolutional layers and ReLU were stacked to integrate $M_g$, $I^{'}_{s}$, and $C_{t}$ from small local regions to broader regions so that the appearance information of $I^{'}_{s}$ and $C_{t}$ can be extracted.
Because the background information is less important and easily distracts the generator from synthesizing try-on images, we filtered out it to force $G_{c}$ to concentrate on generating the correct human part of the image rather than the whole image. Specifically, the background information of the generation result $I_{g}= G_{c}(C_{t},I^{'}_{s},M_{g})$ is filtered out with $M^{fg}_g$ and so is the ground truth $I_{t}$ with $M^{fg}_t$, where $M^{fg}_g$ and $M^{fg}_t$ represent $M_g$ and $M_{t}$ without the background channel, respectively.
Afterward, a global structural information and other low-frequency features are obtained from calculating the L1 distance function:
\begin{equation
\mathcal{L}^{G_c}_{L1}=\sum_{W}\sum_{H} \left\lVert I_{g} \otimes M^{fg}_g -I_{t} \otimes M^{fg}_t\right\lVert_{1},
\end{equation}
where $\otimes$ represents the pixel-wise multiplication.
For the discriminator, we constructed the coloring discriminator $D_{c}$ against $G_{c}$ to distinguish two pairs: one including $I_t$ and $I_s$, and the other including $I_g$ and $I_s$. With the additional real image $I_s$, $D_c$ impels $G_{c}$ to generate more realistic images. Moreover, because this is a binary classification problem (i.e., the image is real or fake), we employed the binary cross-entropy loss as the GAN loss to compare the generated images:
\begin{equation
\mathcal{L}^{G_c}_{GAN} = \mathcal{L}_{BCE}(D_{c}(G_{c}(C_{t},I^{'}_{s}, M_g), I_{s}),1),
\end{equation}
\begin{equation
\begin{split}
\mathcal{L}^{D_c}_{GAN} &= \mathcal{L}_{BCE}(D_{c}(G_{c}(C_{t},I^{'}_{s},M_g),I_{s}),0)\\
&+\mathcal{L}_{BCE}(D_{c}(I_{t},I_{s}),1),
\end{split}
\end{equation}
where $G_{c}$ attempts to deceive $D_{c}$ to recognize the synthesized image as a real image; thus, the goal of $\mathcal{L}_{BCE}$ in $\mathcal{L}^{G_c}_{GAN}$ is equal to $1$.
In contrast, because $D_c$ must classify the generated or real images correctly, the goals of $\mathcal{L}_{BCE}$ in $\mathcal{L}^{D_c}_{GAN}$ are equal to $0$ and $1$, respectively. In summary, the overall loss function of \textit{segmentation region coloring} is as follows:
\begin{equation
\mathcal{L}^{G_c} = \mathcal{L}^{G_c}_{GAN} +\lambda \mathcal{L}^{G_c}_{L1}.
\end{equation}
\subsection{Salient Region Refinement}
Because users care most about the characteristics of products, the performance of the virtual try-on service is highly dependent on the saliency of the synthesized image, for example, users (e.g., facial details or body shape), clothing features (e.g., button or bow tie), and 3D physics (e.g., pleat and shadows). Hence, in the fourth stage, we proposed two networks to refine the facial and clothing regions separately.
\subsubsection{FacialGAN}
Modeling faces and hair is challenging but essential in synthesizing try-on images. To simplify this complicated work, our network generates residual face details instead of the whole face. Precisely, for the facial refinement network $G_{rf}$, we adjusted the model of the \textit{segmentation region coloring} ($G_c$) to the facial refinement task by excluding the fully connected layer to avoid losing input details during compression. To force $G_{rf}$ to concentrate on facial details, $M^{face}_g$ and $M^{face}_s$ were introduced to filter out the facial region from $I_{g}$ and $I_{s}$, respectively, where $M^{face}_g$ denotes the parsing channels representing the head (including the face, neck, and hair).
As such, $G_{rf}$ generates the high-frequency details as the residual output $d=G_{rf}(I^{face}_g, I^{face}_s)$, where $I^{face}_g = I_{g}\otimes M^{face}_g$ and $I^{face}_s = I_{s}\otimes M^{face}_s$. After processing images through $G_{rf}$, the fine-tuned result is obtained by adding $d$ to $I_{g}$.
In addition, inspired by \cite{EnhanceNet,perception1}, the perceptual loss was exploited to produce images that have a similar feature representation even though the pixel-wise accuracy is not high.
Let $(d+I_g)^{face}$ and $I^{face}_t$ denote the regions within $M^{face}_g$ of $(d+I_g)$ and $I_t$, respectively.
In addition to calculating the loss pixel-wise $\left\lVert (d+I_g)^{face} -I^{face}_t \right\lVert_{1}$, we computed the perceptual loss by mapping both $(d+I_g)^{face}$ and $I^{face}_t$ into the perceptual feature space through the different layers ($\phi_{i}$) of the VGG-19 model.
This additional loss allows the model to reconstruct the details and edges better.
\begin{small}
\begin{equation
\begin{split}
\mathcal{L}^{G_{rf}}_{vgg}((d+I_{g})&^{face}, I^{face}_t) \\
&= \sum_{i} \lambda_{i}\left\lVert \phi_{i}((d+I_{g})^{face}) - \phi_{i}(I^{face}_t)\right\lVert_{1},
\end{split}
\end{equation}
\end{small}where $\phi_{i}$ represents the feature map retrieved from the $i^{th}$ layer in the pretrained VGG-19 model\cite{VGG19}. Furthermore, like previous stages, we integrated the GAN loss as follows:
\begin{small}
\begin{equation
\mathcal{L}^{G_{rf}}_{GAN} =
\mathcal{L}_{BCE}(D_{rf}((d+I_g)^{face}, I^{face}_s), 1)
\end{equation}
\end{small}
\begin{small}
\begin{equation
\begin{split}
\mathcal{L}^{D_{rf}}_{GAN} &= \mathcal{L}_{BCE} (D_{rf}(I^{face}_s, (d+I_g)^{face}), 0)\\
&+\mathcal{L}_{BCE}(D_{rf}(I^{face}_s, I^{face}_t), 1).
\end{split}
\end{equation}
\end{small}
The overall loss function of FacialGAN is as follows:
\begin{small}
\begin{equation
\begin{split}
\mathcal{L}^{G_{rf}} &=
\lambda_{f1}\mathcal{L}^{G_{rf}}_{GAN}\\
&+\lambda_{f2}\mathcal{L}^{G_{rf}}_{vgg}((d+I_g)^{face}, I^{face}_t)\\
&+\lambda_{f3}\sum_{W}\sum_{H} \left\lVert (d+I_g)^{face} -I^{face}_t \right\lVert_{1}\\
&+\lambda_{f4}\sum_{W}\sum_{H} \left\lVert (d+I_g) \otimes M^{fg}_g - I_t \otimes M^{fg}_t\right\lVert_{1},
\end{split}
\end{equation}
\end{small}
where $\lambda_i$ denotes the weight of the corresponding loss.
\subsubsection{ClothingGAN}
\label{clothingGAN}
Most state-of-the-art virtual try-on networks \cite{Viton2017, CPVTON2018, M2E, VirtuallyTryOn} preserve detailed clothing information by fusing the prewarped clothes onto the try-on images directly. However, these approaches encounter the problems of limbs occlusion or incorrect warping patterns of clothes. To solve these problems, in our previous work (FashionOn)\cite{FashionOn}, we implemented the virtual try-on framework by 1) transforming the human pose into the semantic segmentation form through $G_t$, 2) coloring the clothing textures and human appearance through $G_c$, and 3) processing images through refinement networks.
Although FashionOn fills in most clothing information back, some tiny but important details (e.g., neckline or button) are missing and the generated images are not sufficient realistic. Hence, we modify the previous clothing refinement generator and construct a new one ($G_{rc}$) to retrieve clothing features directly from the in-shop clothing $C_{t}$ and render them into the clothing region of $I_{g}$. Inputting the concatenation of the in-shop clothing and warped clothing into the Clothing UNet in our previous work~\cite{FashionOn} improved the details, but the generated clothing region still lacks fined details, such as the neckline and buttons. The unsatisfactory results are caused by the same parameters of the encoder for the two input features of in-shop clothing and warped clothing. Moreover, the subtle difference in the details is neglected by the discriminator.
Based on these observations, the proposed ClothingGAN $G_{rc}$ contains four parts: (a) detail encoder ($E_D$), (b) warped-clothing encoder ($E_W$), (c) decoder ($Dec$), and (d) context discriminator ($D_{rc}$). The generator exploits detailed information on in-shop clothing and warped clothing obtained from $E_D$ and $E_W$, respectively, which are then input into $Dec$ to generate an image of refined clothing. Next, $D_{rc}$, which consists of the local and the global discriminators, differentiates whether the refined clothing is real or fake by comparing the local and global consistency with real images.
\textbf{Detail Encoder ($E_D$).}
The objective of $E_D$ is to learn the detailed and neglected information (i.e., missing information in the previous stage) from an in-shop clothing image ($C_t$). To extract detailed visual features, we use seven convolutional layers followed by an instance normalization (IN) layer\cite{improvedtexture} together with LeakyReLU\cite{LeakyReLU} as the activation function, which is more than $E_W$ because detailed information is required from the original in-shop clothing, such as texture and logos. After training, $E_D$ can generate a detailed clothing representation, denoted as $C_d = E_D(C_t)$, which is further employed by the decoder to complement the details and synthesize the refined clothing.
\textbf{Warped-Clothing Encoder ($E_W$).}
As depicted in Fig.~\ref{fig:arc1}, we use the UNet architecture to encode $I^{clothing}_g = I_{g} \otimes M^{clothing}_g$, where $M^{clothing}_g\in {R}^{{W}\times{H}}$ is the clothing part of $M_{g}$. The encoder includes five downsampling convolutional layers with kernel=5, and each layer is followed by an IN layer with LeakyReLU. Each layer of the UNet encoder is connected to the corresponding layer of the UNet decoder through highway connections to produce high-level features. Finally, we obtain the warped-clothing representation $C_w = E_w(I^{clothing}_g)$. In the following section, we present how the outputs of $E_D$ and $E_W$ have been further employed in the decoder network.
\textbf{Decoder ($Dec$).}
To generate refined clothing via the decoder, we concatenate the encoded features $C_d$ and $C_w$ obtained from $E_D$ and $E_W$, respectively, as input. From layer to layer in the decoder, we first derive the features obtained from the previous layer and the precomputed feature maps at $E_W$ connected through a highway connection. Next, we upsample the feature map with the $2\times2$ bicubic operation. After upsampling, a $3\times3$ convolutional and ReLU operation are applied. Using the highway connections with $E_W$ allows the network to align the detailed clothing features with the warped-clothing features obtained by the UNet encoder ($E_W$). In other words, the generator can be written as follows:
\begin{small}
\begin{equation}
\label{eq:15}
\begin{split}
C_r &= G_{rc}(C_{t}, I^{clothing}_g) \\
&= Dec(E_D(C_{t}), E_W(I^{clothing}_g)).
\end{split}
\end{equation}
\end{small}
To bridge the difference between the refined clothing $C_{r}$ and the target clothing region $I^{clothing}_t = I_{t} \otimes M^{clothing}_t$, where $M^{clothing}_t$ represents the clothing channel of $M_{t}$, we introduced the L1 loss ($ \mathcal{L}^{G_{rc}}_{L_1}$) and the perceptual loss ($\mathcal{L}^{G_{rc}}_{vgg}$) to refine the clothing as follows:
\begin{small}
\begin{equation
\label{eq:12}
\mathcal{L}^{G_{rc}}_{L_{1}}(C_{r},I^{clothing}_t)=\sum_{W}\sum_{H} \left\lVert C_{r}-I^{clothing}_t\right\lVert_{1},
\end{equation}
\end{small}
\begin{small}
\begin{equation
\label{eq:11}
\mathcal{L}^{G_{rc}}_{vgg}(C_{r},I^{clothing}_t)=\sum_{i=1}^{5} \lambda_{i}\left\lVert
\phi_{i}(C_{r})-\phi_{i}(I^{clothing}_t)\right\lVert_{1},
\end{equation}
\end{small}where $\phi_{i}(C)$ represents the feature map of the clothing $C$ of the $i^{th}$ layer in the VGG-19 model \cite{VGG19}. By exploiting the L1 loss instead of L2 loss here, we address the problems of blurry generated images. To further avoid the misalignment, the refined clothing $C_{r}$ is integrated into $I_{g}$, where the clothing region is removed, to synthesize a refined human $I_{rg} = C_{r} \otimes M^{clothing}_g + I_{g} \otimes (1-M^{clothing}_g)$. The parsing mask $M^{clothing}_g$ is used to select the clothing region, which facilitates the process of excluding limbs in front of the clothing when fusing the clothing. The loss for the refined clothing try-on is defined as follows:
\begin{small}
\begin{equation
\label{eq:13}
\mathcal{L}^{G_{rc}}_{fullbody}(I_{rg},I_{t})=\sum_{W}\sum_{H} \left\lVert I_{rg}-I_{t}\right\lVert_{1}.
\end{equation}
\end{small}
\textbf{Context Discriminator.}
To make the refined clothing more realistic, we also employed the GAN loss $ \mathcal{L}^{G_{rc}}_{GAN}$ by adopting the context discriminator comprising the global and the local discriminators that classify the refined clothing as real or fake by comparing the local and the global consistency with real images. Both discriminators are based on a convolutional network that compresses the images into small feature tensors. A fully connected layer is applied to the concatenation of the output feature tensors and predicts a constant value between $1$ and $0$, which represents the probability that the refined clothing is real.
The global discriminator takes as input the image in which we create a bounding box of the clothing part from the result and resizes it, using bilinear interpolation, to $128\times128$. It consists of five two-stride convolutional layers with kernel=5 and a fully connected layer that outputs a 1024-dimensional vector. The local discriminator follows a similar pattern, except the last two single-stride convolutional layers have a kernel=3 and an input size of $64\times64$. The input of the local discriminator is generated by randomly sampling $16\times16$ from the bounding box and resizing it to $64\times64$.
After deriving the outputs from the global and the local discriminators, we build a fully connected layer, followed by a sigmoid function to process the concatenation of two vectors (a 2048-dimensional vector). The output value ranges from 0 to 1, representing the probability that the refined clothing is real, rather than generated. The GAN loss is defined as follows:
\begin{small}
\begin{equation
\mathcal{L}^{G_{rc}}_{GAN}= \mathcal{L}_{BCE}(D_{rc}(r^f_{local}, r^f_{global}), 1),
\end{equation}
\end{small}
\begin{small}
\begin{equation
\begin{split}
\mathcal{L}^{D_{rc}}_{GAN}&= \mathcal{L}_{BCE}(D_{rc}(r^f_{local}, r^f_{global}), 0) \\
& +\mathcal{L}_{BCE}(D_{rc}(r^t_{local}, r^t_{global}), 1),
\end{split}
\end{equation}
\end{small}where $r$ is the resized result, the subscript $global$ or $local$ denoted the whole or sub-sampled result, respectively, and the superscript $t$ or $f$ means that result is true or fake (generated), respectively. The overall loss function of the ClothingGAN is defined as follows:
\begin{small}
\begin{equation
\label{eq:14}
\mathcal{L}^{G_{rc}} =
\mathcal\lambda_{c1} \mathcal{L}^{G_{rc}}_{vgg} +
\lambda_{c2} \mathcal{L}^{G_{rc}}_{L_{1}} +
\lambda_{c3} \mathcal{L}^{G_{rc}}_{fullbody} +
\lambda_{c4}\mathcal{L}^{G_{rc}}_{GAN},
\end{equation}
\end{small}where $\lambda_{ci}\ (i=1,2,3,4)$ denotes the weight of the corresponding loss.
\section{Experiments}
The datasets and implementation are detailed here. Afterward, we conduct qualitative and quantitative experiments with the state-of-the-art method and our previous work FashionOn~\cite{FashionOn} to demonstrate the effectiveness of TF-TIS.
\begin{figure*}[t]
\includegraphics[width=\textwidth]{rebuttal_img/Fig4.jpg}
\centering
\caption{\textbf{Visual detail comparison.} To compare the details of generated images between different models, we excluded the \textit{cloth2pose} module from our network. The leftmost three columns are the input, and the rest of the columns are the output of different models and the local enlargement of them. Our TF-TIS has the best performance regarding details, such as the neckline of polo shirts and clothing pattern, and retains global and local consistency.}
\label{fig:result1}
\end{figure*}
\subsection{Dataset}
To train and evaluate the proposed TF-TIS, a dataset containing two different poses and one clothing image for each person is required. Still, most of the existing datasets provide either only one pose for each person with the corresponding clothing image~\cite{Viton2017,CPVTON2018} or multiple poses for each person but without clothing images\cite{DeepFashion2016}. Therefore, we collected a new large-scale dataset containing $10,895$ in-shop clothes with the corresponding images of mannequins wearing in-shop clothes in two different poses.\footnote{Please refer to the images in \url{https://github.com/fashion-on/FashionOn.github.io}.}
In addition, the DeepFashion dataset \cite{DeepFashion2016}, with a size of $288\times192$, is also adopted to broaden the diversity of the data. After removing the incomplete image pairs and wrapping one in-shop clothing and two human images into each triplet, $11,283$ triplets were created. Finally, we randomly split the dataset into the training set and the testing set with $9,590$ and $1,693$ triplets, respectively.
\begin{figure}[bth]
\includegraphics[width=8cm]{retrieval_2.jpg}
\centering
\caption{\textbf{Pose retrieval examples of \textit{cloth2pose}.} We queried our training dataset by comparing the features extracted via \textit{cloth2pose} to all clothing features in the dataset. For each query, we present the top five retrieved samples. To focus on the pose information, we eliminate the human information, such as skin or hair color. The leftmost two columns are input clothes and the translated parsing, generated via Stage II from the derived pose. Although some examples are not like the query, it still shows that we could easily find results visually close to the query.}
\label{fig:retrieval}
\end{figure}
\subsection{Implementation Details}
\textbf{Cloth2pose.} We initialize the first 10 layers with that of the VGG-19~\cite{VGG19} and fine-tune them to generate a set of clothing feature maps $F$ from the information on the in-shop clothing. For the following convolutional blocks, each contains five convolutional layers with a $7\times7$ kernel and two with a $1\times1$ kernel. Each layer is followed by a ReLU. In this stage, we apply $N_{c2p} = 4$ to the number of the convolutional blocks.
\textbf{Pose-guided Parsing Translator.} Based on the framework of ResNet, we implement two downsampling layers, nine residual blocks, and followed by two upsampling layers. Specifically, we construct two single-stride convolutional layers with a $3\times3$ kernel and one highway connection, combining the input and the output of each corresponding residual block.
\textbf{Segmentation Region Coloring.} The architecture is composed of the encoder and decoder with six residual blocks between them. Except for the last residual block and one fully connected layer, each block contains two single-stride convolutional layers with a $3\times3$ kernel, one downsampling two-stride convolutional layer with a $3\times3$ kernel. The number of filters of all convolutional layers linearly increases and decreases, respectively, for the encoder and decoder.
\textbf{Salient Region Refinement.} The generator of FacialGAN ($G_{rf}$) is similar to $G_c$ but without the fully connected layer. In addition, $G_{rf}$ has four residual blocks containing two convolutional layers and one downsampling convolutional layer. For ClothingGAN, the generator ($G_{rc}$) comprises two different encoders and one decoder. The detail encoder ($E_D$) consists of four downsampling convolutional layers and three convolutional layers, and the warped-clothing encoder ($E_W$) consists of four downsampling convolutional layers and one convolutional layer. All downsampling convolutional layers have a $4\times4$ kernel and a $2\times2$ stride, and other convolutional layers have a $3\times3$ kernel and a $1\times1$ stride. Both kinds of convolutional layers are followed by the IN layer and LeakyReLU. The decoder ($Dec$) consists of five $3\times3$ convolutional layers, and each layer is followed by one upsampling layer, one IN layer, and one ReLU.
For the context discriminator ($D_{rc}$), we adopt two discriminators: (1) the global discriminator, which consists of four downsampling convolutional layers and outputs a 1024-dimensional vector representing the global consistency, and (2) the local discriminator, which consists of three downsampling convolutional layers and outputs a 1024-dimensional vector representing the local consistency. A fully connected layer and sigmoid function are applied to the concatenation of the two vectors to differentiate whether the image is real or generated.
We used Adam \cite{Adam} with $\beta_1 = 0.5$ and $\beta_2 = 0.999$ as the optimizer for all stages. The learning rates of the \textit{pose-guided parsing translator} and the other stages are 2e-4 and 2e-5, respectively.
\subsection{Qualitative Results}
\begin{figure}[bt]
\includegraphics[width=8.5cm]{result2_5.jpg}
\centering
\caption{\textbf{Qualitative results sampled from our testing dataset.} For every example (six images as a group) we show from left to right is: the input clothing, the generated segmentation image with the synthesized pose from TF-TIS, the try-on result with the synthesized pose, the generated segmentation image with the defined pose in our dataset, the try-on result with the defined pose, and the real try-on image.}
\label{fig:result2}
\end{figure}
Several try-on results are depicted in Fig~\ref{fig:result1}, \ref{fig:result2}, \ref{fig:AdaIN}, and \ref{fig:mmbig}.
\subsubsection{Evaluation of Virtual Try-on}
As Fig.~\ref{fig:result1} reveals, we compare TF-TIS with the state-of-the-art clothing warping-based method (VTNCAP~\cite{VirtuallyTryOn}) and our previous work (FashionOn~\cite{FashionOn}), which adopts a coarse-to-fine strategy. In addition, because CP-VTON does not include the pose transfer, we combine the state-of-the-art pose transfer method GFLA~\cite{gfla} with CP-VTON~\cite{CPVTON2018} as an additional baseline (GFLA+CP-VTON). The results indicate that all methods accomplish the task of virtual try-on with arbitrary poses. However, the results of VTNCAP and GFLA+CP-VTON contain some artifacts, while the results of FashionOn lose some details and local consistency. Several cases are worth mentioning and listed below.
\noindent \textbf{Neglecting Tiny but Essential Details.}
Fig.~\ref{fig:result1} illustrates that the ClothingGAN ($G_{rc}$) does generate detailed information. From the left to the right, the results are from the state-of-the-art works (VTNCAP, GFLA+CP-VTON, and FashionOn) and ablation studies for $G_{rc}$ in TF-TIS (TF-TIS without $G_{rc}$, TF-TIS without the local discriminator, and TF-TIS). The approaches without $G_{rc}$ (two encoders), including VTNCAP, GFLA+CP-VTON, and FashionOn, fail at the erroneous neckline and the small button, as revealed in Rows 1 and 3. The neckline and the small button on the clothing image by FashionOn are neglected because FashionOn uses only one encoder to extract the information of the concatenation of the in-shop clothing and warped clothing, which degrades the focus of both images. In contrast, the local discriminator of TF-TIS discerns tiny clothing details and the global discriminator is applied to retain the consistency of the entire image. As a result, TF-TIS generates the neckline and small button based on more comprehensive information of the warping clothing, which generates an appearance that is closer to the in-shop clothing images.
\noindent\textbf{Wrong Warping Pattern.} As depicted in Row 2 in Fig.~\ref{fig:result1}, FashionOn and TF-TIS successfully resolve the wrong warping pattern problems of VTNCAP. Because warping clothes through TPS \cite{TPS} only considers the deformation of clothes in two dimensions, the warped clothes are unrealistic. Although in Rows 1 and 2 GFLA+CP-VTON preserves the neckline and the button and generates smooth plaid, GFLA+CP-VTON misses the shade and makes the clothes an average color in Row 4. In Row 6, GFLA+CP-VTON mistakes the red pocket as being on the right side. In contrast, we predict the warped-clothing mask based on the in-shop clothing mask and the warped body segmentation, which consider the correlation between body parts. Moreover, the proposed TF-TIS retains the consistency of clothes, such as the pattern shape, which makes the plaid shirts more realistic because we adopt global and local discriminators to discern the clothing details and to retain consistency.
\noindent\textbf{Average Face.} The VTNCAP often synthesizes an average face as depicted in the fourth column of Fig.~\ref{fig:result1}, because it simply uses the whole body as a mask and renders the human information into it. In contrast, we treat human parsing using 18 channels and render the information for each body part into the corresponding region, which is more specific for every part. Additionally, our works employs the FacialGAN to refine the facial part, making it more distinctive, instead of synthesizing the average faces.
\noindent\textbf{Clothing Color Degradation.} In the second, fourth, fifth and sixth rows in Fig.~\ref{fig:result1}, the clothing color of the results derived by VTNCAP changes from the color of the in-shop clothing. In contrast, FashionOn and TF-TIS successfully preserve the color of the in-shop clothing, which is important in virtual try-on services.
\begin{figure}[bth]
\includegraphics[width=8.5cm]{rebuttal_img/AdaIN_vs_IN.jpg}
\centering
\caption{Visual comparison of AdaIN~\cite{AdaIN} and IN~\cite{improvedtexture} for ClothingGAN.}
\label{fig:AdaIN}
\end{figure}
\noindent\textbf{Human Limbs Occlusion.} Rows 5 and 6 in Fig.~\ref{fig:result1} reveal that the proposed TF-TIS can solve the human limbs occlusion problems in VTNCAP. Rather than simply warping it through TPS, we simultaneously warp the clothing and the body segmentation, then render the human appearance and the clothing information sequentially. Hence, $G_c$ can easily render the appearance based on all semantic segmentation, preserving the natural correlation between clothes and humans.
\noindent\textbf{Dropping the Detailed Logo.} In Fig.~\ref{fig:result1}, the rightmost two columns are the ablation study for the local discriminator within the context discriminator. Row 4 shows that the local discriminator generates the full logo. The ``PARIS'' logo is evident with almost all five characters, using the local discriminator in the rightmost column. Without the local discriminator, it only generates three characters.
\noindent\textbf{Comparison of AdaIN and IN for $G_{rc}$.} We replace the IN layer in the two encoders of the ClothingGAN with an adaptive instance normalization layer (AdaIN) to evaluate whether AdaIN helps preserve the clothing details in Fig.~\ref{fig:AdaIN}. Equation~\ref{eq:15} for AdaIN becomes the following:
\begin{small}
\begin{equation}
\begin{split}
C_r = Dec(&(1-\gamma)E_W(I^{clothing}_g)\\
&+\gamma AdaIN(E_W(I^{clothing}_g),E_D(C_{t}))),
\end{split}
\end{equation}
\end{small}where $\gamma$ is a hyperparameter for the content-style trade-off. We used $\gamma=0.25$ and 0.75 to evaluate the difference and demonstrated the visual comparison in Fig.~\ref{fig:AdaIN}.
\begin{small}
\begin{equation
{AdaIN}(x,y)=\alpha(y){\left( \frac{x-\mu(x)}{\alpha(x)} \right)}+\mu(y),
\end{equation}
\end{small}where $x$ represents the content input, $y$ is the style input, and $\alpha(y)$ denotes the standard deviation of $y$. The AdaIN simply scales the normalized content input with $\alpha(y)$ and shifts it using $\mu(y)$. Fig.~\ref{fig:AdaIN} reveals that AdaIN tends to generate global features for the clothing information and fails to generate robust details. For example, as presented in Row 4, AdaIN fails to synthesize the robust edge of the suspenders. Moreover, as displayed in Row 5, AdaIN tends to generate the blurry flowers. When increasing the hyperparameter $\gamma$ to contain a higher proportion of features from $C_t$, the $G_{rc}$ adopting AdaIN generates more robust but still blurrier results than using IN.
\begin{figure*}[bth!]
\includegraphics[width=0.93\textwidth]{mmbig.jpg}
\centering
\caption{Qualitative results sampled from the testing dataset.}
\label{fig:mmbig}
\end{figure*}
\subsubsection{Evaluation of cloth2pose}
Because none of the previous research can generate the target poses according to the in-shop clothes, we evaluate the performance of \textit{cloth2pose} by determining whether \textit{cloth2pose} can learn the relationship between the in-shop clothing and try-on pose. Specifically, in the testing phase, given the in-shop clothing, we use \textit{cloth2pose} to derive the synthesized pose and generate the translated parsing (second column in Fig.~\ref{fig:retrieval}). Afterward, we compute the L2 distances between the in-shop clothing feature and all clothing features in the training dataset and retrieve top five try-on poses results with the smallest clothing distance. The in-shop clothing features are extracted by using the first 10 layers of the VGG-19~\cite{VGG19}.
Fig.~\ref{fig:retrieval} presents several examples. The retrieved results reveal that the synthesized poses are very close to some real poses in the top five results (e.g., the fourth sample in Row 2, the first sample in Row 5, and the first sample in Row 6). Moreover, our retrieved examples also demonstrate that different poses should be synthesized in accordance with the in-shop clothing to better present the clothing. For example, T-shirts, like the clothes in Rows 1 to 3, are demonstrated in the front views to show the logo or with one hand in the pocket to show the muscles. However, the camisole tops in Rows 4 to 6, are demonstrated with people standing sideways to show their body shapes, facing the right or left.
Moreover, the qualitative results of our testing dataset are presented in Fig.~\ref{fig:result2}, and indicate that our model can synthesize a better pose to display clothing. For each example, we present the input clothing ($C_t$), the user ($I_s$), the translated human parsing with the synthesized pose via the \textit{cloth2pose} module and the generated image, the human parsing with the defined pose and the generated image, and the ground truth image of the defined pose. Although appearing a little different from the image with the defined pose, the \textit{cloth2pose} results capture the key information about the human, such as the direction they face. Moreover, we synthesize suitable poses for clothes. For instance, 1) in Row 3, we derive the pose in the front view to show the pattern of the clothing and 2) in Row 4 to 5, we synthesize the sideways pose to show the upper arms and shoulders of people. Therefore, our model understands the relation between clothes and poses and can synthesize better poses to present better try-on results, which induces users to buy clothes.
\subsection{Quantitative Results}
Because the structural similarity (SSIM)~\cite{SSIM} and inception score (IS)~\cite{ImprovedGAN} are fairly standard metrics that focus on the overall quality of the generated image instead of the pixel-wise comparison, we calculated them for the reconstruction of the try-on results in our dataset.
The SSIM measures the similarity by comparing the generated images against the original images in the structural information, whereas IS provides scores to indicate whether the generated results are visually diverse and semantically meaningful.
Compared with the other virtual try-on systems (i.e., VTNCAP, CP-VTON, GFLA+CP-VTON, and FashionOn), our method outperforms them in terms of SSIM and IS, as revealed in Table~\ref{tab:table1}.
Moreover, TF-TIS outperforms VTNCAP and CP-VTON in terms of IS by $18.9\%$ and $8.14\%$, respectively. Additionally, the comparison in term of SSIM indicates that TF-TIS exceeds VTNCAP and CP-VTON by $19.8\%$ and $11.5\%$, respectively. Although TF-TIS only surpasses the results of FashionOn within 1\% in both metrics, the result complements the important details and the local and global consistency that FashionOn lacks, as demonstrated in Fig~\ref{fig:result1}.
\begin{table}[t]
\centering
\caption{\textbf{Comparison of the virtual try-on testing dataset.} We randomly sampled 1300 data from the testing dataset.}
\begin{tabular}{l|c c}
\textbf{Method} & \textbf{IS} & \textbf{SSIM} \\
\hline \hline
VTNCAP~\cite{VirtuallyTryOn} & 2.5874 $\pm$ 0.0965 & 0.7282 \\
CP-VTON \cite{CPVTON2018} & 2.8495 $\pm$ 0.0832 & 0.7824 \\
GFLA \cite{gfla} + CP-VTON \cite{CPVTON2018} & 3.0266 $\pm$ 0.1740 & 0.8070 \\
FashionOn (w/o refine) & 3.0679 $\pm$ 0.1247 & 0.8689 \\
FashionOn (w/ refine)~\cite{FashionOn} & 3.0693 $\pm$ 0.1560 & 0.8724\\
TF-TIS (Ours) & \textbf{3.0777 $\pm$ 0.1143} & \textbf{0.8725} \\
\hline
Real Data & 3.2350 $\pm$ 0.1282 & 1 \\
\end{tabular}
\footnotesize Note: IS: inception score; SSIM: structural similarity. The higher the score, the better the result.
\label{tab:table1}
\end{table}
\textbf{Runtime.} We evaluated the efficiency of the proposed TF-TIS by separately reporting the running time of the four modules. The results of the runtime were conducted on a NVIDIA 1080-Ti GPU and were averaged with 2000 randomly selected image sets. The runtime of each module is as follows: \textit{cloth2pose} (1.3 ms), \textit{pose-guided parsing translator} (2.6 ms), \textit{segmentation region coloring} (3.1 ms), and \textit{salient region refinement} ($G_{rf}$: 1.9 ms, $G_{rc}$: 2.6 ms). The results indicate that the proposed TF-TIS not only reduces the cost of hiring photographers but also provides a real-time try-on service for fashion e-commerce platforms.
\section{Conclusion and Future work}
In this paper, we present a part-level learning network (TF-TIS) for virtual try-on service with automatically synthesized poses. The previous work requires a user-specified target pose for try-on. In contrast, TF-TIS precisely generates try-on images with the poses synthesized from the clothing characteristics, which better demonstrates the clothes. The experimental results indicate that TF-TIS significantly outperforms the state-of-the-art virtual try-on approaches on various clothing types, is better in term of being lifelike in appearance, and recommends poses that induce customers to buy clothes. Moreover, as shown in the experiments, TF-TIS captures the relation between clothes and poses to synthesize better poses to present users with better try-on results. In addition, by proposing the global and the local discriminators in the clothing refinement network, TF-TIS retains consistency of images and preserves critical human information and clothing characteristics. Therefore, TF-TIS resolves many challenging problems (e.g., generating tiny but essential details and preserving detailed logos). In the future, we plan to extend our approach to learn how different garment sizes deform on a real body in images using transfer training from 3D human model methods.
\section*{Acknowledgments}
This work was supported in part by the Ministry of Science and Technology of Taiwan under Grants MOST-109-2221-E-009-114-MY3, MOST-109-2218-E-009-025, MOST-109-2221-E-009-097, MOST-109-2218-E-009-016, MOST-109-2223-E-009-002-MY3, MOST-109-2218-E-009-025 and MOST-109-2221-E-001-015, in part by the National Natural Science Foundation of China under Grant 61772043, in part by the Fundamental Research Funds for the Central Universities, and in part by the Beijing Natural Science Foundation under Contract 4192025.
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\fi
\bibliographystyle{IEEEtran}
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On Artprice.com's Standardized Marketplace, you can buy or sell artworks by QIAN Dexiang.
Updated on 23 Apr 2019: QIAN Dexiang (1953) is an artist born in 1953 The oldest auction result ever registered on the website for an artwork by this artist is a painting sold in 2013, at Beijing Hanhai Art Auction Co.Ltd., and the most recent auction result is a drawing-watercolor sold in 2017. Artprice.com's price levels for this artist are based on 3 auction results. Especially: painting, drawing-watercolor. | {
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Rogovići je naselje u Republici Hrvatskoj, u sastavu općine Kaštelir-Labinci, Istarska županija.
Stanovništvo
Prema popisu stanovništva iz 2001. godine, naselje je imalo 90 stanovnika te 28 obiteljskih kućanstava.
Prema popisu stanovništva 2011. godine naselje je imalo 101 stanovnika.
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SCHOTT solutions no. 2/2014 > Energy
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Cable feedthroughs from SCHOTT are tested carefully and meet high standards. They are designed to last for decades. Photo: SCHOTT/H.-R. Schulz
SCHOTT EPAs were recently installed at Forsmark 3, a nuclear reactor north of Stockholm. The EPAs for this project were designed and stringently tested to provide the higher standards that were key to plant operator Forsmarks Kraftgrupp's (a company of the Vattenfall Group) upgrading of the reactor, which included modified safety scenarios. The EPAs were designed to withstand submerged conditions under 13 meters of water for at least 30 days, together with pressures of up to 8.3 bar, and temperatures up to 185 degrees Celsius. In addition, the radiological exposure of the EPA during a severe accident scenario had to reach 1.7 MGy at a dose rate of 2360 Gy/h. SCHOTT Nuclear Safety Division General Manager Thomas Fink said: "Our EPAs have been thoroughly tested and meet a number of high standards, which means that the EPAs will last for the Forsmark 3 life extension of 30 years." EPAs are also a small part of the total cost of a new nuclear plant. Fink: "In terms of new build, the cost of these safety-critical components is only 0.1 % of the entire budget, which is a small investment for a very significant increase in safety in a key part of the reactor."
Specially developed cable feedthroughs from SCHOTT were installed in the nuclear reactor Forsmark 3 located just north of Stockholm. Photo: Vattenfall
Vattenfall has stated that the modernization of its plants in Sweden will mean longer operational lives for the reactors. Torbjörn Wahlborg, Chairman of the Board for Forsmarks Kraftgrupp AB, said: "Vattenfall is currently conducting the most extensive modernization program in the history of Swedish nuclear power, and the company is planning to invest SEK 16 billion (1.75 billion euros) over a five-year period between 2013 and 2017. From a technical standpoint, the modernization process will lay the way to operating these plants for many more decades to come."
There will undoubtedly be a significant market for nuclear reactor components for many years to come, both in retrofitting and new build, and the ongoing global aim is likely to be for increased safety standards, both after Fukushima and beyond. SCHOTT is leading the way with its glass-to-metal EPAs that withstand severe accident conditions, and this places the company at the forefront of nuclear component manufacture due to proactive application of stringent safety standards. <
SCHOTT website
Electrical Penetration Assemblies for Nuclear Power Plants
joe.hale@us.schott.com
Global Home › SCHOTT solutions › Meeting the highest standards | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,902 |
Shannon Chan-Kent, née le , est une actrice et chanteuse canadienne.
Carrière
Shannon a surtout fait des doublages de séries animées ainsi que de dessins animés. Elle a fait des apparitions dans des séries et a fait plusieurs films dont Spectacular! un de ses premiers grands rôles.
Filmographie
Téléfilms
2007 : : Kelsey Leung
2009 : Spectacular! : Janet
2012 : Un pacte mortel : Zoe
2019 : Un rôle sur mesure pour Noël (Holiday Date) : Megan
Série de téléfilms
2015-2020 : Enquêtes gourmandes (The Gourmet Detective)
2015 : Enquêtes gourmandes : Meurtre au menu (The Gourmet Detective) de Scott Smith
2015 : Enquêtes gourmandes : Meurtre quatre étoiles (The Gourmet Detective: A Healthy Place to Die) de Scott Smith
2017 : Enquêtes gourmandes : Festin mortel (The Gourmet Detective: Eat, Drink & Be Buried) de Mark Jean
Séries télévisées
2007 : Dans la peau de Ian : Grace Lum
2007 : Samurai Girl : Gardienne
2009 : The Troop : Shellie
2010 : Life Unexpected : Brynn
2020-2021 : Good Trouble : Ruby (13 épisodes)
2021 : You : Kiki
Doublage
Cinéma
2002 : Madeline et le Roi : Chloe
2006 : La traversée du temps : Miyuki Konno
2017 : Barbie la magie des dauphins : Isla
2008 : Barbie et la magie de Noël : Ann et Nan
2010 : Barbie et le secret des sirènes : Deandra
2010 : Barbie et la magie de la mode : Delphine
Télévision
2003-2004 : Le Secret de Sabrina
2006 : Death Note : Misa Amane
2006-2007 : Pucca : Chief
2007 : Gundam 00 : Christina Sierra
2010 : Care Bears to the Rescue
2010 : My Little Pony : Pinkie Pie (chant), Silver Spoon
Notes et références
Liens externes
Naissance à Vancouver
Naissance en septembre 1988
Actrice canadienne de doublage
Actrice canadienne de télévision | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,047 |
In tassonomia la diagnosi è una descrizione scritta di una specie o di altri taxa che la valga a distinguere da tutte le altre. In particolare, una descrizione scritta, originariamente in latino, e pubblicata.
Il termine, dal latino diagnōsis, attraverso il greco antico διάγνωσις (diágnōsis, da διαγιγνώσκειν: diaghignóskein, capire), è la procedura di ricondurre a una categoria un fenomeno o un gruppo di fenomeni, dopo averne considerato ogni aspetto. Il processo diagnostico sfrutta in qualche modo concetti riconducibili al teorema di Bayes, intuitivamente o esplicitamente.
Inizialmente Aristotele utilizzò alcuni caratteri animali, particolarmente utili, per la diagnosi tassonomica, ovvero per decidere a quale raggruppamento appartenesse una determinata creatura, questo diversamente da Platone, che aveva teorizzato l'individuazione di un unico rappresentativo carattere per l'assegnazione ad una determinata categoria dei viventi.
Note
Voci correlate
Tassonomia
Altri progetti
Tassonomia | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,199 |
On March 24, 2010 I attended a lecture presented by hornist Lin Foulk entitled New Standards: Women in Orchestras in the 21st Century. Lin Foulk is the horn instructor at Western Michigan University, and she has done extensive research on women in music. As you can imagine, I was thrilled to learn that she would be giving this lecture. Lin Foulk is a great role model for any woman brass musician– she is an exceptional musician, articulate speaker, and genuinely nice person.
I took notes on her lecture, which I will share (if I can read my handwriting!), but you can also find many of Foulk's quotes and sources on her website under PDF Downloads.
By WWII, women began filling the empty spots in the orchestras/bands of the men who had left for war. While this was a big step, the men regained their positions after returning from the war.
By the 1960s-70s, a second wave of feminism occurred, which resulted in more inclusion of women in orchestras. This was achieved by publically announcing vacancies in orchestras and holding blind auditions.
Foulk presented many graphs and statistics that described the relationship of women in the orchestra to the well being of the ensemble.
In an orchestra with 10% or less women, the women generally keep a low profile and behave accordingly to the orchestra norm.
An orchestra in transition (10-40% women) will develop gender boundaries and cross-group stereotyping, which results in conflict.
In a balanced orchestra (40-60% women) will form inter-group relationships and allow both gender goups to feel legitimate.
Foulk discussed many other topics such as women conductors and composers. She also engaged in a question and answer session with the audience. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,991 |
Sydney Sierota (Echosmith)
LA siblings Echosmith are slowly taking over the indie pop world and are about to head to Austin for South by Southwest. If you haven't heard them yet, don't worry, you will (sooner if you scroll your mouse down and press play on that video). Sydney, Jamie, Noah, and Graham Sierota come together to produce an original and refreshing sound that is much-needed in today's music industry
We had the chance to chat with Sydney about the origins of the band's sound, their upcoming trip to Austin, and her place as a potential role model for young girls.
And while you're reading, be sure to check out the video below of an acoustic version of their song, "Tell Her You Love Her".
Echosmith's sound definitely stands out from the sea of indie pop bands that are around these days – did you set out to make something different?
When you're doing anything, I think it's really important to try and do something that's unique because there's already a million people doing exactly what you want to do – so you might as well do something that's unique to you and that people have to come to you to get. People love taking the easy way and doing the same kind of music as this person or that person and we really wanted to be original. And at the same time, Echosmith's sound really came together in a very natural way for us – in the writing process and the recording process – it was what we wanted to do and make music just like that. It took us some time to get to that point, but we finally got there and knew that it was right.
The "other" bands that you do seem to share some musical heritage with are 20 and 30 years old – how/when did you get introduced to those artists?
Our dad is a music producer and songwriter and musician himself, so we grew up in a very artsy family with our mom being a painter, and they were both really into 80s New Wave, so we grew up hearing that as well as artists like The Strokes. We kind of took from that and thought it was very different and original sounding and we loved it – and we still love it, obviously. We grew up hearing it and then finally it sort of clicked as to why we really loved it.
You were interviewed for a piece about International Women's Day – and you had some pretty strong feelings about how female artists are portrayed in the music industry. Do you think it's important to try and represent yourself a certain way, to act as a role model for your younger fans?
Yeah, that's really important with what I'm doing to be aware of everything that I do, everything I say, everything I wear, everything I sing about – and even the way I act. People are watching – people are always watching – and you just have to be aware of that. And it's not 'people are always watching me that's so great' – it's more of the fact that people are always watching me and how can I show them that this is how life should be – or this how I think life should be. And I think even having joy is really important and that can show through, even in how you dress. I remember looking at these magazines and every single picture of a girl was all about 'how much skin can I show without getting too much crap for it?' – and that's disappointing to me.
When I was younger, while I never dressed like that, I did think for a second that if I did dress like that that more guys would like me – things like that. I think that's what girls get caught up in so much and I feel like this industry keeps telling girls that that's the way to do it, that's the way to get a good guy, and to get attention in general and just look pretty. I think that we should use clothes as a statement – and not be taking them off as a statement. I feel really strongly about that. I think the modesty thing is cool – it doesn't hurt to be more modest. And you're never going to see anyone have something bad written about them for being modest. I really feel like whether I am a role model or not to people, I want to live a life that can be [looked up to], and live the life that I enjoy and feel strongly about and passionate about. [Click here to read Sydney's interview with Alter The Press]
What was the first really difficult thing you had to learn about being out on tour?
We had to learn that nothing's going to be perfect – whether the entire band and the crew have to share one hotel room, to not being able to take a really nice shower, to all the things that can happen before a show – you just have to accept it and make some fun out of it.
You're set to play 6 showcases next week – in what free time you do have, are there any other bands you're looking forward to seeing?
Yeah, we are playing with some cool bands at some of our showcases like Aloe Blacc, and it will be really cool to see some friends like Tori Kelly and For The Foxes, as well as a few other people we met at Warped Tour [last year]. And I know that Coldplay is playing, but I doubt I can get in to that, but we'll see – they're my favorite band in the world.
[Note: If anyone from Coldplay is reading this (hey, I can dream, right?) – Sydney Sierota, add her to your guest list. Thanks!]
What can we expect from Echosmith over the next few months, between SXSW and Warped Tour?
We'll be doing some more touring. Some one-offs in LA and some college shows and radio shows. We're really excited about what's to come and to see how this year goes.
Thanks Sydney!
If you're in Austin for SXSW, be sure to check out Echosmith at the following showcases:
Sun 3/9 — 4:30pm — SB Proects Sunday Funday – Banger's (79 Rainey St.)
Wed 3/12 — 11pm — Home Grown: LA – The Velveeta Room (521 E 6th St.)
Thi 3/13 — 4pm — Cricket/Muve Music Acoustic Performance (4th and Congress)
Thu 3/13 — 9:20pm — Live from Live Nation Labs – (1100 E 5th St)
Sat 3/15 — 2pm — Dickies Road House SXSW Showcase – Banger's (79 Rainey St.)
Sat 3/15 — TBD — Stubbs Evening SXSW Showcase – Stubbs, Indoor Stage (801 Red River St.)
Chief Of All The Things at Emo At Heart
Josh is the co-founder and editor-in-chief of Emo At Heart.
5 years ago in Interviews by Josh | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,333 |
Q: How to include top comment per each post as a separate collection in meteor? If I have a collection of posts when entering a view of a posts collection, and each of these posts has a collection of comments on them, how could I list the top comment for each post along side of them?
E.g:
this.route('postsList', {
path: '/:posts',
waitOn: function() {
return Meteor.subscribe('posts');
},
data: function() {
return Posts.find({});
}
});
And then I'm iterating through the collection of posts on a page.
{{#each posts}}
{{> postItem}}
{{/each}}
<template name="postItem">
{{title}}
{{topComment}}
</template>
I'd like to put the top comment for each post item.
How can I do this with my templates/subscriptions/publications?
Posts and comments are separate collections.
If it were an embedded collection I could see the ease of use but how to deal with separate collections?
If I published a recent comment type of publication how could I subscribe to it for each post as the most recent one? Or am I thinking the wrong way here?
A: If you insist on having two totally separated collections, you would get into problems with efficient database queries. What you could do is to have something like recentComment field in your posts collection. Should this field point to id of the most recent comment related to the given post, you could alter your posts subscription to include the recent comments as well:
Meteor.publish('posts', function() {
var listOfIds = _.pluck(Posts.find({}, {fields: recentComment}).fetch(), 'recentComment');
return [
Posts.find(),
Comments.find({_id:{$in:listOfIds}})
];
});
Note that this solution is not fully reactive but it's good enough in most cases.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,788 |
goog.module('grrUi.client.virtualFileSystem.fileTextViewDirective');
goog.module.declareLegacyNamespace();
/**
* Controller for FileTextViewDirective.
*
* @constructor
* @param {!angular.Scope} $scope
* @param {!grrUi.core.apiService.ApiService} grrApiService
* @ngInject
*/
const FileTextViewController = function(
$scope, grrApiService) {
/** @private {!angular.Scope} */
this.scope_ = $scope;
/** @private {!grrUi.core.apiService.ApiService} */
this.grrApiService_ = grrApiService;
/** @type {!grrUi.client.virtualFileSystem.fileContextDirective.FileContextController} */
this.fileContext;
/** @type {?string} */
this.fileContent;
/** @type {string} */
this.encoding = 'UTF_8';
/** @export {number} */
this.page = 1;
/** @export {number} */
this.pageCount = 1;
/** @private {number} */
this.chunkSize_ = 10000;
this.scope_.$watchGroup(['controller.fileContext.clientId',
'controller.fileContext.selectedFilePath',
'controller.fileContext.selectedFileVersion'],
this.onContextChange_.bind(this));
this.scope_.$watch('controller.encoding', this.onEncodingChange_.bind(this));
this.scope_.$watch('controller.page', this.onPageChange_.bind(this));
};
/**
* Handles changes to the clientId and filePath.
*
* @private
*/
FileTextViewController.prototype.onContextChange_ = function() {
var clientId = this.fileContext['clientId'];
var filePath = this.fileContext['selectedFilePath'];
if (angular.isDefined(clientId) && angular.isDefined(filePath)) {
this.fetchText_();
}
};
/**
* Handles changes to the encoding.
* @param {number} page
* @param {number} oldPage
* @private
*/
FileTextViewController.prototype.onPageChange_ = function(page, oldPage) {
if (this.page !== oldPage){
this.fetchText_();
}
};
/**
* Handles page changes.
* @param {string} encoding
* @param {string} oldEncoding
* @private
*/
FileTextViewController.prototype.onEncodingChange_ = function(encoding, oldEncoding) {
if (this.encoding !== oldEncoding) {
this.fetchText_();
}
};
/**
* Fetches the file content.
*
* @private
*/
FileTextViewController.prototype.fetchText_ = function() {
var clientId = this.fileContext['clientId'];
var filePath = this.fileContext['selectedFilePath'];
var fileVersion = this.fileContext['selectedFileVersion'];
var offset = (this.page - 1) * this.chunkSize_;
var url = 'clients/' + clientId + '/vfs-text/' + filePath;
var params = {};
params['encoding'] = this.encoding;
params['offset'] = offset;
params['length'] = this.chunkSize_;
if (fileVersion) {
params['timestamp'] = fileVersion;
}
this.grrApiService_.get(url, params).then(function(response) {
this.fileContent = response.data['content'];
var total_size = response.data['total_size'];
this.pageCount = Math.ceil(total_size / this.chunkSize_);
}.bind(this), function() {
this.fileContent = null;
}.bind(this));
};
/**
* FileTextViewDirective definition.
* @return {angular.Directive} Directive definition object.
*/
exports.FileTextViewDirective = function() {
return {
restrict: 'E',
scope: {},
require: '^grrFileContext',
templateUrl: '/static/angular-components/client/virtual-file-system/file-text-view.html',
controller: FileTextViewController,
controllerAs: 'controller',
link: function(scope, element, attrs, fileContextController) {
scope.controller.fileContext = fileContextController;
}
};
};
/**
* Name of the directive in Angular.
*
* @const
* @export
*/
exports.FileTextViewDirective.directive_name = 'grrFileTextView';
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,378 |
Several years ago, I was strolling through the garden center and I stumbled across a tarragon plant in the herb section. This is not an herb I have ever used a lot of and I had no idea what uses for tarragon I might come across but I bought a plant for the backyard herb garden. Little did I know that tarragon would THRIVE in my backyard! I never researched how to grow tarragon…I just stuck it in the dirt and lucked out. And wouldn't you know it but tarragon is a perennial here in the south. It gets bigger every year! I have given some to neighbors and dried some in the dehydrator. I decided to make this easy tarragon vinegar recipe and thought I would share it with you. Of course, I still have more tarragon than I need so I started researching other uses for tarragon as well. I will include those below in case you have as much tarragon in the garden as I do!
Tarragon is a perennial plant in the daisy and sunflower family. It has narrow aromatic leaves that are used primarily as a culinary herb. In my own garden, the tarragon that I planted about 4 years ago is still coming back every year. I honestly have so much tarragon I don't even know what to do with it all! It is incredibly easy to grow so I thought I would share a few tips on how to grow tarragon so you can try a few new recipes.
Start them indoors around April or before your area's last expected frost date. Sow about four to six seeds per small pot. Make sure you use only moist, composted potting soil. Cover the seeds lightly and keep them in indirect sunlight at room temperature. Once they are about an inch tall, you can thin the plants and leave only the strongest and healthiest tarragon plants in the pot. Transplant them into the garden once they are about 4 to 6 inches tall or leave them in the pot for an indoor kitchen garden.
Tarragon herb plants should be grown in full sun. Space them approximately 18 inches apart in well-drained, fertile soil. They will, however, grow fairly well in poor soil and can handle drought better than over watering. Add a bit of mulch in the fall for added nutrients and they should regrow in the spring with no issues. Check out my post on how to prepare your garden for winter to learn more about keeping your garden healthy during the colder months.
Make sure you trim the plants regularly to keep them healthy and strong. A nice pair of herb trimmers or small pruning shears are an essential gardening tool in my yard! Check out the video below to learn how to grow tarragon.
Relieve a toothache by chewing on tarragon leaves. They have a mild numbing effect which may take away some of the pain.
Drink tarragon tea before bed to help promote sleep. It is also good for relieving anxiety. Tarragon tea for weight loss is also a very popular health benefit you might enjoy. You can buy tarragon capsules if you prefer.
Make a vinaigrette with fresh tarragon. Minced tarragon leaves added to olive oil and lemon juice makes a delicious and light salad dressing recipe or marinade.
Make tarragon oil. Great to rub on fish before you put it on the grill. My tarragon vinegar recipe below is great on salads.
Use it in a cocktail. Add sprigs of tarragon to homemade lemonade. Drink as is or add a shot of vodka for a lemon/tarragon cocktail.
There are tons of health benefits of tarragon. Check out The Free Range Life for more of them.
This tarragon vinegar recipe is incredibly easy to make. Try not to spray your tarragon plant with any pesticides and use only organic fertilizers. Whenever you use a fresh herb to make oils and vinegar, you should store them in a glass bottle. Plastic really isn't good for long term storage of this type of product.
This tarragon vinegar recipe is quick and easy and makes a delicious addition to homemade salad dressing!
Wash the tarragon leaves and pat dry.
Pour the vinegar into a decorative glass jar and add an additional sprig or two of tarragon to the jar for decoration.
Store in a cool dark place for up to 6 months.
What other uses for tarragon have you found? | {
"redpajama_set_name": "RedPajamaC4"
} | 1,253 |
Q: Images from folder on sd card I have an application that needs to read images from a folder created by the application on the sdcard(sdcard/"foldername"/"filename.jpg". I have no idea what the names of the files are because the user specifies the names of the files. I need to read the images from the folder and make something like the default image viewer. Im thinking read them into a grid view first but 1) cant figure out how to dynamically read them from a folder 2) how would I implement the image options like the default viewer? If there was a way to open the default viewer on a certain folder that would help.
any input would be amazing been working on it for a while.
Thanks
A: Here's how you can get a list of folders off of the memory card:
String state = Environment.getExternalStorageState();
if(state.contentEquals(Environment.MEDIA_MOUNTED) || state.contentEquals(Environment.MEDIA_MOUNTED_READ_ONLY))
{
String homeDir = Environment.getExternalStorageDirectory();
File file = new File(homeDir);
File[] directories = file.listFiles();
}
else
{
Log.v("Error", "External Storage Unaccessible: " + state);
}
This code is from the top of my head, so some syntax may be off a bit, but the general idea should work. You can use something like this to filter down the folders to only folders that contain images:
FileFilter filterForImageFolders = new FileFilter()
{
public boolean accept(File folder)
{
try
{
//Checking only directories, since we are checking for files within
//a directory
if(folder.isDirectory())
{
File[] listOfFiles = folder.listFiles();
if (listOfFiles == null) return false;
//For each file in the directory...
for (File file : listOfFiles)
{
//Check if the extension is one of the supported filetypes
//imageExtensions is a String[] containing image filetypes (e.g. "png")
for (String ext : imageExtensions)
{
if (file.getName().endsWith("." + ext)) return true;
}
}
}
return false;
}
catch (SecurityException e)
{
Log.v("debug", "Access Denied");
return false;
}
}
};
Then, change the first example to:
File[] directories = file.listFiles(filterForImageFolders);
That should return only directories that contain images. Hopefully this helps some!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,883 |
Oh God: Ron Paul Plans Paultard Convention
As of now, your Wonkette still plans on attending the Republican National Convention in the Twin Cities in early September, because why not. But now there may be something interesting to do! That's right: Ron Paul will hold aRival Convention, same time, same cities, with every Paultard in the world. We're all over this shit.
Paul was denied a speaking spot at the "real" GOP Convention, which would make sense, since he doesn't support John McCain. Well, in truth, none of the Republicans "support" John McCain, but most are willing to run with it as long as he's in favor of Perpetual War, and McCain has indicated, every now and then, that he likes that. And Ron Paul stayed flat.
A Paul campaign aide said the Texas congressman hopes to pack about 11,000 supporters into the Williams Arena at the University of Minnesota on Sept. 2, which coincides with the second day of the Republican National Convention at the Xcel Energy Center in neighboring St. Paul.
Lovely. And in case either you forgot or we haven't mentioned it at all, the big Ron Paul March on Washington is scheduled for July 12. Here's the predictably comical "trailer":
Should Wonkette hold a barbecue for Paultard Pilgrims that week? BRING YR NUNCHUKS & SABRES.
Ron Paul plans his own convention [Pittsburgh Tribune-Review]
Ron Paul Revolution March July 12 2008 [YouTube] | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,733 |
Lauri Veijalainen: People have a lot to say on Stockmann and I absolutely love it!
Posted on 30.11.2017 by RES
Lauri Veijalainen, the CEO of Stockmann Group, visited the Faculty of Management at the University of Tampere as a guest lecturer on the Leadership for Change (LFC) lecture series on 23 November 2017. The lectures are organized and facilitated by the LFC master's students. Here, you can read a report of the event written by Team Veijalainen.
LFC Student Iida Jokinen and Lauri Veijalainen preparing for the lecture
At his visit to the Faculty of Management, Lauri Veijalainen spoke to a full room about what it has been like to lead a widely known company through difficult times and how his early ties to Russia have affected his career.
Before joining Stockmann, Veijalainen worked for 15 years in Russia at Skanska and IKEA. Professor Veli-Pekka Tynkkynen (University of Helsinki), an expert in the field of Russian studies and childhood friend of Veijalainen, commented on the lecture.
"Tell good news immediately and bad news even faster"
The Stockmann Group is a listed Finnish retail corporation with three business divisions: Stockmann Retail, Lindex and Real Estate. The Helsinki flagship store is the largest department store in the Nordics.
Stockmann, founded in 1862, has gone through rough times and stayed in the headlines while going through significant strategic changes. These developments, beginning in the year 2014 and still going on, have roots in the increasing challenges of the online market and the different customer groups. Veijalainen has been a part of the Stockmann group since 2010 and became its CEO in early 2016.
Lauri Veiijalainen speaking in the UTA Faculty of Managament
Recently Stockmann gained a great deal of publicity by selling its food business to S-Group. Lauri Veijalainen comments:
All this has been very emotional, probably the most emotional transaction I have ever been in. Because every single person in Finland knows Stockmann, every second person has an opinion about it, and every third person knows better than me how to run the business. And that is a big advantage for us because creating emotions in retail is absolutely the key.
People are the most important resource for Stockmann and any reaction is seen as good publicity, as it shows that people care about the company. Customer behaviour is monitored closely because to keep up with the fast-changing world, reaction time is utmost essential. Nothing lasts forever, and that is why business needs to keep moving forward and evolving.
Veijalainen spoke honestly about the challenges and sacrifices the company and he have had to make to adapt to the evolving environment. His top priority has always been people: customers and employees come first. As CEO he tries to create a feeling of togetherness and an open, transparent working culture together with Stockmann employees to lead the change.
In Stockmann, the managers must understand where the numbers come from but also be available in-store, in the field, for both the employees and the customers. Veijalainen argued that he has put a lot of effort into developing internal communications and finding the right people for the right spots. One of his maxims is to tell good news immediately and bad news even faster. Withholding information doesn't make a leader more powerful. Veijalainen said that he believes in motivation through open communication:
Thank you is small but sometimes very important effort and recognition on a job well done. It's used so infrequently in companies it could be used a lot better.
In Stockmann, he asks people to challenge decisions to improve and find defaults early; a culture that is adopted from his former employer IKEA. At the lecture, Veijalainen said that he seeks to live by his own principles. Therefore, he also responds, personally, to each and every customer feedback sent to him.
Similarly, in stores, every Stockmann customer shall be provided a special experience. Restaurants and cafés attract people. One particular example are the roof top terraces in the Helsinki and Tallinn stores. They have become 'must-see places' and places for social media moments, especially for tourists. While a Finnish customer spends 30 euros on average, a Chinese tourist buys ten times more. Hence, Stockmann is one of the first shops accepting Alipay, the most common virtual banking among Chinese.
There is an inspector for everything in Russia
When the first Stockmann store opened in Russia in 1993, even Vladimir Putin, as then Head of International Relations and Anatoli Sobchak, Mayor of St. Petersburg didn't want to miss the event.
Lauri Veijalainen told about the early 1990s when Stockmann's grocery store was opened in St. Petersburg
This is how Russia has changed through the last 25 years. Although Stockmann is not doing direct business in Russia anymore, Veijalainen saw all the cultural changes first hand. He grew up as son of Finnish embassy staff in the former Soviet Union and came back after the Iron Curtain fell to work as CFO of IKEA Russia and Administrative and Finance Director of Skanska Russia.
On account of this, he is aware of the many differences in Russian and Finnish company and social cultures. And he emphasizes the importance of knowing the local culture.
He told the audience about an encounter with a Russian ecological inspector during his time at Skanska, which made clear what he means by this:
He said that I'm going to give you a fine and I said, 'Ok, for what?' And he said, 'Yeah, because your snow is dirty.' It was March-April in Moscow. There is just pollution and snow melts, you know it's dirty. I think the fee was ten euros or four to five-hundred rubles but it was the principle. We argued 10-15 minutes and then I took my last card out and said, 'I'm really sorry but it is not my fault', and he had the best answer: 'It's not my fault either'.
Veijalainen paid the fine and got a receipt. This story is a good example of the differences in legislation and norms between cultures. In Russia, form is often more important than content. By knowing this, major trouble can be avoided. Another option to avoid obstacles in Russia is to build up networks and identify the goalkeepers. Veli-Pekka Tynkkynen commented that this behaviour has roots in Soviet times, when knowing people was essential to facilitate life.
Veli-Pekka Tynkkynen commented on the lecture
Tynkkynen later challenged his childhood friend by asking about the government interference in business. Veijalainen's answer might be surprising; there is almost no interferences as retail is not as important to the Russian economy as for example the energy sector.
Following the rules is the most important to not fall from grace by Russian authorities. Of course, one has to take into account that Russia is a culturally diverse country. Each city and region is unique. Moscow, is in Veijalainen's sight, the fast-moving capital to make money, St. Petersburg is much calmer and a place of culture, while Kazan and its region is an Islamic part of Russia. All these factors and different identities have to be taken into consideration when making successful business in Russia.
Experiencing these differences, challenges and fast developments, Lauri Veijalainen has had a great school for now leading the change at the Stockmann Group.
We want to thank Lauri Veijalainen for sharing his expertise with the audience, and also thank Veli-Pekka Tynkkynen for commenting and challenging the lecturer.
Team Veijalainen: Gabriel Esber, Iida Jokinen, Aline Mayr, Eve Vuorela and Zheng Zhao
Team Veijalainen (Zheng, Iida, Gabriel, Eve and Aline) was in charge of organizing the lecture by Lauri Veijalainen. Great job!
This entry was posted in lectures, Yleinen by RES. Bookmark the permalink.
1 thought on "Lauri Veijalainen: People have a lot to say on Stockmann and I absolutely love it!"
Lauri Veijalainen on 30.11.2017 at 15:05 said:
Excellent blog! Thanks for hosting me.
All the best, Lauri | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,682 |
\section{Introduction}
The binarity fraction (BF) of early type stars is poorly known
because traditional spectroscopic searches are undermined by
their wide spectral lines. Furthermore, there is evidence that
the BF depends on the stellar density. However, the cluster
multiplicity studies carried out so far can cover only a
limited range of density and age. This prompted us to estimate
the binarity fraction of a representative, volume-limited
sample of early-type field stars.
We designed a survey able to detect at $\sim$10$\sigma$
level an M4-type companion at the mean distance of our sample
($\sim$200\,pc) down to 0.4\,arcsec separation from 100\,Myr
old A-type primary. The companions around B-type stars will be
younger ($\sim$10\,Myr), brighter, and easier to detect. Most
importantly, the physical nature of the candidate companions
is verified by their common proper motion. Our goal is to
compare the properties such as BF and mass ratio of the
multiple stars in the field and in different star-forming
regions. The target accuracy for the BF is 3-5\%.
\section{Sample Selection}
The sample stars were selected according to the following
criteria:
\begin{itemize}
\item spectral types - only BA; most of the stars are B8-A0
because bluer stars are rare and too bright, while redder
stars become too faint to make it into the sample
\item field stars only - the known OB-association members
listed in de Zeeuw et al. (1999) were excluded from the
sample
\item distance $\leq$300\,pc from the Sun as measured by
HIPPARCOS; at the maximum distance the telescope's diffraction
limit of 0.07\,arcsec corresponds to $\sim$21\,A.U. (for
comparison Shatsky \& Tokovinin 2002 probed separations of
45-900\,A.U.)
\item proper motions $>$27\,mas\,yr$^{-1}$ allowing us to
confirm physical companion candidates taking observations at
two epoch separated by 1-2\,yr
\item apparent $V$=5-6\,mag, so the targets are suitable NACO
reference sources even under poor weather, and at the same
time they do not saturate the detector
\item Dec$\leq$+15\,deg, i.e. the targets are visible from the
VLT
\end{itemize}
The final sample consists of 308 stars (136 of them are known
binaries from the Washington catalog), the average distance
is 114\,pc and the median distance is 104\,pc.
\section{Observations}
The observations were carried out with NAOS--CONICA (Nasmyth
Adaptive Optics System -- Near-Infrared Imager and Spectrograph)
at the ESO VLT over the last two years. The pixel scale was
27.03\,mas\,px$^{-1}$, giving 27.7$\times$27.7 arcsec field
of view. Each target was observed at 9 different position on the
detector, collecting total of $\sim$7.5\,min of integration in
$K_S$ or in the intermediate band filter IB\_2.18.
The data reduction includes sky subtraction, flat-fielding,
aligning and combination of the images into a single frame.
\section{Current Status}
As of Mar 2006 we have observed 257 objects from our sample.
We carried out a visual inspection of the combined frames
(with and without PSF subtraction of the target stars), and
{\bf we found 195 companion candidates around 117 sample targets}.
The second epoch observations with $\geq$2\,yr baseline of the
first 16 targets started during ESO Period 77.
\section{Analysis: Modeling the Survey}
To estimate the sensitivity and the completeness of the survey we
have created a Monte-Carlo simulation that takes into account all
available information for the survey stars. The model input
parameters are:
\begin{itemize}
\item the known distances, spectral types and absolute
luminosities for all primaries
\item adopted binarity fraction of 30\%
\item secondary star mass - randomly sampled from the Kroupa
IMF; preferences in the mass ratio of the two components will
be included in the future
\item secondary star's spectral type and absolute magnitude -
calculated from the mass
\item orbital periods - randomly generated from the Duquennoy \&
Mayor (1991) distribution
\item major sxis - calculated from the Kepler's law and the
period
\item random ellipticity and random orbital inclination
\item visibility criterion based on the magnitude difference and
the angular separation between the primary and the companion -
based on the available observations
\end{itemize}
The model predicts: the distributions of periods, angular
separations, magnitude differences and spectral types for the
detected binaries. The simulations indicate that we will detect
$\sim$2/3 of the physical companions.
\acknowledgements
We are grateful to our colleagues from the ESO-Paranal Science
Operations Department who carried out these observation in
Service Mode.
\logbook{09/05/2006}{09/05/2006}{...}
\adressehermes
\end{document}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,163 |
The *Directory Server* (DS) JSON-RPC server (from now on referred to as
the *server*) publish a number of methods to be used by nodes in order
for them to be a part of an Anond overlay network. The server provides
the central functionality provided by the *Public Directory Server* as
introduced in the paper [Anonymous overlay network supporting
authenticated routing](Schlegel-Wong-3.pdf) by Schlegel and Wong.
The following methods are made available:
* [*publish-node*](#user-content-21-method-publish-node)
* [*unpublish-node*](#user-content-22-method-unpublish-node)
* [*still-published-nodes*](#user-content-23-method-still-published-nodes)
* [*get-random-nodes*](#user-content-24-method-get-random-nodes)
* [*reserve-oa*](#user-content-25-method-reserve-oa)
* [*get-network-topology*](#user-content-26-method-get-network-topology) (experimental API)
The server conforms to the [JSON-RPC 2.0
Specification](http://www.jsonrpc.org/specification) and uses HTTP
over SSL as its transport mechanism.
All incoming HTTP requests to the server must be signed using a
[Hash based Message Authentication
Code](http://en.wikipedia.org/wiki/Hash-based_message_authentication_code)
(HMAC). To bootstrap the HMAC signing scheme each node initially
publish its public signing key with a non-signed call to the
*publish-node* method.
The result returned from the call to the *publish-node* method is a
unique `node-id` (and more) and the node must send it along with each
HTTP request it sends to the server as a `Node-ID` HTTP header. After
this initial call the DS knows about the association between a
specific `node-id` and a public signing key.
From here on the node must sign all HTTP requests it sends to the
server and to do this it calculates a HMAC from the hashed body of the
HTTP request and sends this HMAC along with the HTTP request as a
base64 encoded `Content-HMAC` HTTP header.
There is more to this. Please read on.
## 2) The JSON-RPC methods
In this section each server method's input parameters and the result
it produces are specified using [JSON schema](http://json-schema.org)
specifications. Do not be scared. If your are new to JSON schema you
can sift through something like [Understanding JSON
Schema](http://spacetelescope.github.io/understanding-json-schema)
instead of digging into the opaque JSON schema specification.
<!------------------------------------------------------------------------->
### 2.1) Method: *publish-node*
The *publish-node* method is used to publish a node in an Anond
overlay network.
#### Params:
```json
{
"$schema": "http://json-schema.org/draft-03/schema#",
"name": "publish-node params",
"type": "object",
"properties": {
"public-key": {
"description":
"A base64 encoded public signing key as defined in the
NaCl library (http://nacl.cr.yp.to/sign.html), i.e. a
key suitable for a signature scheme based on
Curve25519 in Edwards form and SHA-512.",
"type": "string",
"media": {
"binaryEncoding": "base64",
}
}
},
"required": ["public-key"]
}
```
#### Result:
```json
{
"$schema": "http://json-schema.org/draft-03/schema#",
"name": "publish-node result",
"type": "object",
"properties": {
"ds-id": {
"description":
"A 'ds-id' is an id in the same number space as the
node ids. DS picks this id in order to identify itself
when it communicates with nodes using Anond's
encrypted and bit oriented protocol used for node
tunnel establishment etc. Read more about this bit
protocol in ds-node-udp-protocol.md.",
"type": "number",
"minimum": 0,
"maximum": 2147483647
},
"node-id": {
"description":
"If a node does not specify 'Content-HMAC' and
'Node-ID' HTTP headers in the HTTP request the DS will
allocate a new unique 'node-id' which the node should
use to identify itself. If a node specify these HTTP
headers the 'node-id' will be the same as specified in
the 'Node-ID' header.",
"type": "number",
"minimum": 0,
"maximum": 2147483647
},
"shared-key": {
"description":
"A base64 encoded shared stream key as defined in the
NaCl library (http://nacl.cr.yp.to/stream.html),
i.e. a key suitable for encryption based on a
Salsa20/20 encryption scheme. This shared stream key
is used by the node when it communicates with the DS
using Anond's encrypted bit oriented protocol used for
node tunnel establishment etc. Read more about this
bit protocol in ds-node-udp-protocol.md",
"type": "string",
"media": {
"binaryEncoding": "base64",
}
},
"node-ttl": {
"description":
"A node must republish itself within this number of
milli-seconds or else it will be purged from the Anond
overlay network. A node republish itself with a new
call to the 'publish-node' method and is then provided
with a new shared stream key but it also has the
opportunity to renegotiate a new public signing key
with the DS. Note: If a node has forgotten its secret
signing key (the companion to the public signing key)
it has to call 'publish-node' again without any
'Content-HMAC' and 'Node-ID' HTTP headers in the HTTP
request, i.e. and it will get a new 'node-id'. The old
'node-id' will not be given to any other node until a
server defined grace period has passed (~two weeks).",
"type": "number",
"minimum": 900000
}
}
}
```
#### Error codes:
`JSONRPC_PARSE_ERROR` (-32700), `JSONRPC_INVALID_REQUEST` (-32600),
`JSONRPC_METHOD_NOT_FOUND` (-32601), `JSONRPC_INVALID_PARAMS` (-32602),
`JSONRPC_INTERNAL_ERROR` (-32603)
#### Example:
First we generate a set of signing keys using `anond -j` and we also
assign a number of environment variables and a `curl` configuration
file to make life easier:
```
$ bin/anond -j public.key secret.key
$ cat public.key
vOFs8Jc3zA8Rvax4Ot+kzNHIdcJZhvGAhioc/WoxD4Q=
$ cat secret.key
o0XiiJxdw8akZdg/NbXYyIT7HgEU4iSajoeI2OaDZxi84WzwlzfMDxG9rHg636TM0ch1wlmG8YCGKhz9ajEPhA==
$ PK=`cat public.key`
$ SK=`cat secret.key`
$ cat > curlrc
url="https://127.0.0.1:6700/jsonrpc"
header="Content-Type: application/json"
insecure
request="POST"
<Ctrl-D>
```
Then we issue an initial non-signed call to the *publish-node* method
without any `Node-ID` and `Content-HMAC` HTTP headers:
```
$ BODY='{"jsonrpc": "2.0", "method": "publish-node", "params": {"public-key": "'${PK}'"}, "id": 1}'
$ curl --config curlrc --data "${BODY}"
{
"jsonrpc": "2.0",
"result": {
"ds-id": 472742719,
"node-id": 22,
"shared-key": "3CMFFX1G1ExgdNhYwB+JgCJ0A+VydTga9G5ZKEevXqw=",
"node-ttl": 10800000
},
"id": 1
}
```
In return we got a `ds-id`, `node-id` and a `shared-key`. The
`node-ttl` tell us that we need to republish the node within 3 hours
or else the node will be purged from the Anond overlay network.
To republish a node we call *publish-node* again and get a new
`shared-key` back as a side effect.
All method calls except for the initial call to *publish-node*
**must** be signed and the `Node-ID` and `Content-HMAC` HTTP headers
**must** be specified in the HTTP request. In this example we generate
a HMAC with `anond -l`:
```
$ echo -n ${BODY} > body.dat
$ HMAC=`bin/anond -l secret.key body.dat`
$ echo ${HMAC}
oSV4N9labmCL0dVzetktQtGbCikSA2Sl936bBEW39LUVJYqzVyQ+N9bSlNpKNre7vc6Ydq6DuMNg/2MHNiz/Ap8Hte3CRA/Cb997Esw+2MJpxF4Cgx9ekSxCHnh+7UcT4BeHQ3zRbLxjYlS7tv8UiTGKmt0+ygsffitoWF36e5k=
```
Then we republish the node:
```
$ curl --config curlrc -H "Node-ID: 22" -H "Content-HMAC: ${HMAC}" --data "${BODY}"
{
"jsonrpc": "2.0",
"result": {
"ds-id": 472742719,
"node-id": 22,
"shared-key": "oxEQBqIOnbzJPKJJqbjgKoknw3R7Pvythvu8DEL052Q=",
"node-ttl": 10800000
},
"id": 1
}
```
We got a new `shared-key` back and could have specified a new
`public-key` in the call in order to renegotiate a new public signing
key, but this is left out as an exercise to the reader.
<!------------------------------------------------------------------------->
### 2.2) Method: *unpublish-node*
The *unpublish-node* method is used to unpublish a node in an Anond
overlay network.
#### Params:
No parameters are needed, i.e. the `node-id` specified in the `Node-ID`
HTTP header in the HTTP request will be unpublished.
#### Result:
```json
{
"$schema": "http://json-schema.org/draft-03/schema#",
"name": "unpublish-node result",
"description": "Signals if the unpublish was successful or not.",
"type": "boolean"
}
```
#### Error codes:
`JSONRPC_PARSE_ERROR` (-32700),`JSONRPC_INVALID_REQUEST` (-32600),
`JSONRPC_METHOD_NOT_FOUND` (-32601), `JSONRPC_INVALID_PARAMS`
(-32602), `JSONRPC_INTERNAL_ERROR` (-32603)
#### Example:
```
$ BODY='{"jsonrpc": "2.0", "method": "unpublish-node"}, "id": 1}'
$ curl --config curlrc -H "Node-ID: 22" -H "Content-HMAC: ${HMAC}" --data "${BODY}"
{
"jsonrpc": "2.0",
"result": true,
"id": 1
}
```
> The HMAC is calculated as seen in the
[*publish-node*](#user-content-example) example. The
[`curlrc`](#user-content-example) file is also defined there.
<!------------------------------------------------------------------------->
### 2.3) Method: *still-published-nodes*
The *still-published-nodes* method takes an array of `node-ids` as
input and returns an array of all `node-ids` that still are published.
#### Params:
```json
{
"$schema": "http://json-schema.org/draft-03/schema#",
"name": "still-published-nodes params",
"description":
"An array of node-ids to be checked if they still are published.",
"type": "array",
"items": {
"type": "number",
"minimum": 0,
"maximum": 2147483647
},
"uniqueItems": true
}
```
#### Result:
```json
{
"$schema": "http://json-schema.org/draft-03/schema#",
"name": "still-published-nodes result",
"description": "An array of still published node-ids.",
"type": "array",
"items": {
"type": "number",
"minimum": 0,
"maximum": 2147483647
},
"uniqueItems": true
}
```
#### Error codes:
`JSONRPC_PARSE_ERROR` (-32700), `JSONRPC_INVALID_REQUEST` (-32600),
`JSONRPC_METHOD_NOT_FOUND` (-32601), `JSONRPC_INVALID_PARAMS`
(-32602), `JSONRPC_INTERNAL_ERROR` (-32603)
#### Example:
```
$ BODY='{"jsonrpc": "2.0", "method": "still-published-nodes"}, "params": [1281, 3410, 52]}, "id": 1}'
$ curl --config curlrc -H "Node-ID: 22" -H "Content-HMAC: ${HMAC}" --data "${BODY}"
{
"jsonrpc": "2.0",
"result": [1281, 3410],
"id": 1
}
```
> The HMAC is calculated as seen in the
[*publish-node*](#user-content-example) example. The
[`curlrc`](#user-content-example) file is also defined there.
<!------------------------------------------------------------------------->
### 2.4) Method: *get-random-nodes*
The *get-random-nodes* method returns an array of random `node-ids`
which are suitable to be used as neighbour nodes.
#### Params:
```json
{
"$schema": "http://json-schema.org/draft-03/schema#",
"name": "get-random-nodes params",
"description": "The number of random node-ids to return.",
"type": "number",
"minimum": 1,
"required": true
}
```
#### Result:
```json
{
"$schema": "http://json-schema.org/draft-03/schema#",
"name": "get-random-nodes result",
"description": "A random set of node-ids.",
"type": "array",
"items": {
"type": "number",
"minimum": 0,
"maximum": 2147483647
},
"uniqueItems": true
}
```
#### Error codes:
`DS_JSONRPC_TOO_FEW_NODES`(3), `DS_JSONRPC_TOO_MANY_NODES`(5),
`JSONRPC_PARSE_ERROR` (-32700), `JSONRPC_INVALID_REQUEST` (-32600),
`JSONRPC_METHOD_NOT_FOUND` (-32601), `JSONRPC_INVALID_PARAMS`
(-32602), `JSONRPC_INTERNAL_ERROR` (-32603)
#### Example:
```
$ BODY='{"jsonrpc": "2.0", "method": "get-random-nodes"}, "params": 5}, "id": 1}'
$ curl --config curlrc -H "Node-ID: 22" -H "Content-HMAC: ${HMAC}" --data "${BODY}"
{
"jsonrpc": "2.0",
"result": [21212, 23121, 44439, 3882, 81819],
"id": 1
}
```
> The HMAC is calculated as seen in the
[*publish-node*](#user-content-example) example. The
[`curlrc`](#user-content-example) file is also defined there.
<!------------------------------------------------------------------------->
### 2.5) Method: *reserve-oa*
The *reserve-oa* method reserves an *Overlay Address* (OA), i.e. an
ipv6-address from the ipv6 subnet annexed by Anond, to be the node's
address on the Anond overlay network. A node typically registers
several OAs and pick new ones now and then to improve node anonymity.
#### Params:
```json
{
"$schema": "http://json-schema.org/draft-03/schema#",
"name": "reserve-oa params",
"description": "A random OA, i.e. a random ipv6-address in a given range.",
"type": "string",
"required": true
}
```
#### Result:
```json
{
"$schema": "http://json-schema.org/draft-03/schema#",
"name": "reserve-oa result",
"description": "Signals if the reservation was successful or not."
"type": "boolean"
}
```
#### Error codes:
`DS_JSONRPC_PERMISSION_DENIED` (1), `DS_JSONRPC_UNKNOWN_NODE` (2),
`JSONRPC_PARSE_ERROR` (-32700), `JSONRPC_INVALID_REQUEST` (-32600),
`JSONRPC_METHOD_NOT_FOUND` (-32601), `JSONRPC_INVALID_PARAMS`
(-32602), `JSONRPC_INTERNAL_ERROR` (-32603)
#### Example:
```
$ BODY='{"jsonrpc": "2.0", "method": "reserve-oa"}, "params": "fe80::230:48ff:fe33:bc33"}, "id": 1}'
$ curl --config curlrc -H "Node-ID: 22" -H "Content-HMAC: ${HMAC}" --data "${BODY}"
{
"jsonrpc": "2.0",
"result": true,
"id": 1
}
```
> The HMAC is calculated as seen in the
[*publish-node*](#user-content-example) example. The
[`curlrc`](#user-content-example) file is also defined there.
<!------------------------------------------------------------------------->
### 2.6) Method: *get-network-topology*
The *get-network-topology* method can only be called if the
experimental API has been enabled in the DS. This method uses the bit
oriented [DS-Node UDP protocol](ds-node-udp-protocol.md) to extract
the neighbour nodes and routing entries from each node in the Anond
overlay network and then generates a global network topology. The
experimental API enables functionality which obviously defeats the
purpose of Anond but it is nice for experimentation and development
purposes.
> You have been duly warned.
#### Params:
No parameters are needed.
#### Result:
```json
{
"$schema": "http://json-schema.org/draft-03/schema#",
"name": "get-network-topology result",
"description": "The network topology."
"type": "array",
"items": {
"description":
"An array of all nodes and their neighbour nodes and
routing entries.",
"type": "object",
"properties": {
"node-id": {
"type": "number",
"minimum": 1,
"maximum": 2147483647
},
"na": {
"description":
"The node address (NA), i.e. the node's
external/outside ipv4-address."
"type": "string"
},
"neighbours": {
"type": "array",
"items": {
"description": "An array of all neighbour nodes.",
"type": "object",
"properties": {
"node-id": {
"type": "number",
"minimum": 1,
"maximum": 2147483647
},
"na": {
"description":
"The node address (NA), i.e. the
node's external/outside published
ipv4-address.",
"type": "string"
},
"path-cost": {
"description":
"The path cost to reach this node.",
"type": "number",
"minimum": 0,
"maximum": 65535
},
"incoming-neighbour": {
"description":
"Signals if this node is chosen as an
explicit neighbour node, or if another
node has chosen this node as its
neighbour node.",
"type": "boolean"
}
}
}
},
"route-entries": {
"type": "array",
"items": {
"description":
"An array of route entries, i.e. which nodes
has to be traversed in order to send data
from this node to all other nodes in the the
Anond overlay network.",
"type": "object",
"properties": {
"path-cost": {
"description":
"The path cost to travel this route."
"type": "number",
"minimum": 0,
"maximum": 65535
},
"route": {
"type": "array",
"items": {
"description":
"An array of node-ids constituting
the route.",
"type": "number",
"minimum": 0,
"maximum": 65535
}
}
}
}
}
}
}
```
#### Error codes:
`JSONRPC_PARSE_ERROR` (-32700), `JSONRPC_INVALID_REQUEST` (-32600),
`JSONRPC_METHOD_NOT_FOUND` (-32601), `JSONRPC_INVALID_PARAMS`
(-32602), `JSONRPC_INTERNAL_ERROR` (-32603)
#### Example:
```
$ BODY='{"jsonrpc": "2.0", "method": "get-network-topology", "id": 1}'
$ curl --config curlrc --data "${BODY}"
{
"jsonrpc": "2.0",
"result": [
{
"node-id": 2,
"na": "127.0.0.1:50009",
"neighbours": [
{
"node-id": 3,
"na": "127.0.0.1:50010",
"path-cost": 509,
"incoming-neighbour": true
},
{
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"incoming-neighbour": true
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...
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"id": 1
}
```
> No HMAC is needed when calling methods in the experimental API. The
[`curlrc`](#user-content-example) file is as seen in the
[*publish-node*](#user-content-example) example.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,137 |
Q: Which planets of our solar system could be discovered from another solar system with present technology? Recently many exoplanets have been found orbiting nearby stars. Assume there is a civilization with identical technology residing in a nearby (< 100 light years) solar system. Could they discover Earth or other planets orbiting the Sun?
A: They probably could, but if they were identical, as they would have the understanding of gravity, doppler effect, spectrography, and it's applications to chemical analysis to spot us.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,004 |
class TechnologiesController < ApplicationController
def index
@technology_header = Header.select('technology').first.technology
@technology_text = Text.select('technology_text').first.technology_text
@technologies = Technology.order 'position ASC'
respond_to do |format|
format.html # index.html.erb
format.xml { render :xml => @technologies }
end
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,377 |
\section{} \label{appendix:asy-expectations}
Our approach in this section is quite similar to that of St{\o}ve and Tj{\o}stheim in \cite[Supp. Material]{stove2012convolution}. In fact, we use the same approximation these authors introduce in \cite[Supp. Material, Proof of Theorem 2]{stove2012convolution}; suppose $n$ rows, given by the index set $\Ib = \{i_j\}_{j=1}^n$, where $i_j \in \{1,2,\dots,N\}$, are removed from the multiple regression model \eqref{eq:samnple-mult-reg}, with $n \in \{1,2,3,4\}$. Denote by $\alphabh[I]$ the OLS estimator associated with this reduced multiple regression model.
We use the reduced OLS estimator to approximate $K_{ij}(\alphabh[])$ by $K_{ij}(\alphabh[I])$ in the expectations in this section, where we recall that $K_{ij}$ is given by \eqref{eq:K-ij}. Intuitively speaking, this change can be ignored asymptotically because removing a very small finite number of rows from the multiple regression model \eqref{eq:samnple-mult-reg} has an asymptotically negligible effect on the convergence as the sample size $N \to \infty$.
Denote by
\begin{align*}
f_{\epsb[i_1 i_2 \dots i_n-]}(\eb[i_1 i_2 \dots i_n-])
= \prod_{\substack{i=1 \\ i \notin \{i_1, i_2, \dots, i_n\} }}^N f_{\epsb[i]}(\eb[i]), \quad \quad \quad \quad
d\eb[i_1, i_2, \dots, i_n-]= \prod_{\substack{i=1 \\ i \notin \{i_1, i_2, \dots, i_n\} }}^N d\eb[i].
\end{align*}
This notation allows us to write expressions such as
\begin{align*}
\int g(e_1,e_2,\dots,e_N) \prod_{i=1}^N f_{\eps[i]}(e_i) \prod_{i=1}^N d e_i,
\end{align*}
in a form that is more convenient for the analysis in this section, namely,
\begin{align*}
\int g(e_1,e_2,\dots,e_N) f_{\epsb[12-]}(\eb[12-])f_{\eps[1]}(e_1)f_{\eps[2]}(e_2) d\eb[12-] de_1 de_2,
\end{align*}
where $g$ is some arbitrary function. The lemmas in this section hold under assumptions (A), (B), (C), (D), (E), and (F) given in Section \ref{subsec:Assumptions}.
\begin{lemma}[] \label{lem:E-K-11-leading-order}
As $h \to 0$, it holds that
\begin{align*}
E[K_h(y - \Xb[1]^T\alphab[] - \eps[1])] \sim h \fY(y) + h^3 \frac{\mu_K}{2} \fY''(y).
\end{align*}
\begin{proof}
Performing the change of variables $-r = (y - \xb[1]^T \alphab[] - \eps[1])/h$, using the fact that $K$ is a symmetric function, Taylor expanding with respect to $h$, and making use of \eqref{eq:int-K}, \eqref{eq:int-K-defs}, and \eqref{eq:fYk-conv-form}, we find that
\begin{align*}
& E[K_h(y - \Xb[1]^T \alphab[] - \eps[1])] \\
& = \int \cdots \int K_h(y - \xb[1]^T \alphab[] - e_1) f_{\eps[1]}(e_1) f_{\Xb[1]}(\xb[1]) \ d\xb[1] de_1 \\
& = h \int \cdots \int K(r) f_{\eps[1]}(y - \xb[1]^T \alphab[] + hr) f_{\Xb[1]}(\xb[1]) \ d\xb[1] dr \\
& \sim h \int \cdots \int K(r) (f_{\eps[1]}(y - \xb[1]^T \alphab[]) + hr f_{\eps[1]}'(y - \xb[1]^T \alphab[]) + \frac{(hr)^2}{2}f_{\eps[1]}''(y - \xb[1]^T \alphab[])) f_{\Xb[1]}(\xb[1]) \ d\xb[1] dr \\
& = h \int (f_{\eps[1]}(y - \xb[1]^T \alphab[]) + h^2 \frac{\mu_K}{2}f_{\eps[1]}''(y - \xb[1]^T \alphab[])) f_{\Xb[1]}(\xb[1]) \ d\xb[1] \\
& = h E[f_{\eps[]}(y - \Xb[1]^T \alphab[])] + h^3 \frac{\mu_K}{2} E[f_{\eps[]}''(y - \Xb[1]^T \alphab[])] \\
& = h \fY(y) + h^3 \frac{\mu_K}{2} \fY''(y).
\end{align*}
\end{proof}
\end{lemma}
\begin{lemma} \label{lem:E-feps-2}
As $N \to \infty$, it holds that
\begin{align*}
E[f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))]
& \sim \fY(y)
+ N^{-1} \frac{\sigmaeps^2}{2} \sum_{p_1,p_2=0}^J E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii[N]^{-1} \Xb[1])_{p_2}] \\
& \times E[f_{\eps[]}''(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[2] - \Xb[1])_{p_2}].
\end{align*}
\begin{proof}
Taylor expanding $f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))$ with respect to $\alphabh[I]$ about $\alphab[]$, taking the expectation, and using \eqref{eq:yt-simple}, we get
\begin{align*}
E[f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))]
&= E[f_{\eps[]}(y - \Xb[1]^T \alphab[])]
+ \sum_{p_1=0}^J E[(\alphabh[I]-\alphab[])_{p_1}] E\bigg[\frac{\partial f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))}{\partial(\alphabh[I])_{p_1}} \bigg|_{\alphabh[I]=\alphab[]}\bigg] \\
& + \frac{1}{2} \sum_{p_1,p_2=0}^J E[(\alphabh[I]-\alphab[])_{p_1}(\alphabh[I]-\alphab[])_{p_2}] E\bigg[\frac{\partial^2 f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))}{\partial(\alphabh[I])_{p_1}\partial(\alphabh[I])_{p_2}} \bigg|_{\alphabh[I]=\alphab[]}\bigg] \\
& + \frac{1}{6} \sum_{p_1,p_2,p_3=0}^J E[(\alphabh[I]-\alphab[])_{p_1}(\alphabh[I]-\alphab[])_{p_2}(\alphabh[I]-\alphab[])_{p_3} \frac{\partial^3 f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))}{\partial(\alphabh[I])_{p_1}\partial(\alphabh[I])_{p_2}\partial(\alphabh[I])_{p_3}} \bigg|_{\alphabh[I]=\zetab[]}\bigg],
\end{align*}
where $\zetab[] = \alphab[] + c(\alphabh[I] - \alphab[])$ for $c \in (0,1)$. The first order term vanishes since $E[(\alphabh[I]-\alphab[])_{p_1}]=0$. Note that once the derivatives in the first and second order terms are evaluated at $\alphab[I] = \alphab[]$ they become independent of $\alphab[I]$ since they only depend on $\Xb[1]$ and $\Xb[2]$, and these covariate observations are not present in $\alphab[I]$. On the other hand, in the remainder term, $\zetab[]$ does depend on $\alphab[I]$. However, at leading order as $N \to \infty$, $\alphabh[I] \sim \alphab[]$, which means that $\zetab[] \sim \alphab[]$. Thus,
\begin{align*}
E[f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))]
&\sim \fY(y)
+ \frac{1}{2} \sum_{p_1,p_2=0}^J E[(\alphabh[I]-\alphab[])_{p_1}(\alphabh[I]-\alphab[])_{p_2}] E\bigg[\frac{\partial^2 f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))}{\partial(\alphabh[I])_{p_1}\partial(\alphabh[I])_{p_2}} \bigg|_{\alphabh[I]=\alphab[]}\bigg] \\
& + \frac{1}{6} \sum_{p_1,p_2,p_3=0}^J E[(\alphabh[I]-\alphab[])_{p_1}(\alphabh[I]-\alphab[])_{p_2}(\alphabh[I]-\alphab[])_{p_3}] E\bigg[\frac{\partial^3 f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))}{\partial(\alphabh[I])_{p_1}\partial(\alphabh[I])_{p_2}\partial(\alphabh[I])_{p_3}} \bigg|_{\alphabh[I]=\alphab[]}\bigg],
\end{align*}
where we used \eqref{eq:fY-conv-form} for the leading-order term. Next, we can approximate $\alphabh[I]$ by $\alphabh[]$ as $N\to \infty$, and then use \eqref{eq:OLS-estimator-Phi} and the independence of the error observations to get
\begin{align*}
E[(\alphabh[I]-\alphab[])_{p_1}(\alphabh[I]-\alphab[])_{p_2}]
& \sim E[(\alphabh[]-\alphab[])_{p_1}(\alphabh[]-\alphab[])_{p_2}] \\
&= E\bigg[(N^{-1} \Phii[N]^{-1} \sum_{_1i=1}^N \Xb[i_1]\eps[i_1])_{p_1}(N^{-1} \Phii[N]^{-1} \sum_{i_2=1}^N \Xb[i_2]\eps[i_2])_{p_2}\bigg] \\
&\sim N^{-2} \sum_{i_1,i_2=1}^N E[\eps[i_1]\eps[i_2]]E[(\Phii[N]^{-1} \Xb[i_1])_{p_1}(\Phii[N]^{-1} \Xb[i_2])_{p_2}] \\
&= N^{-2} \sum_{i=1}^N E[\eps[i]\eps[i]] E[(\Phii[N]^{-1} \Xb[i])_{p_1}(\Phii[N]^{-1} \Xb[i])_{p_2}] \\
&= N^{-1} \sigmaeps^2 E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii[N]^{-1} \Xb[1])_{p_2}].
\end{align*}
Next, evaluating the derivative, and then retaining the leading-order term gives
\begin{align*}
E\bigg[\frac{\partial^2 f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))}{\partial(\alphabh[I])_{p_1}\partial(\alphabh[I])_{p_2}}\bigg|_{\alphabh[]=\alphab[]}\bigg]
& \sim E[f_{\eps[]}''(y - \Xb[1]^T \alphab[I]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[2] - \Xb[1])_{p_2}] \\
& \sim E[f_{\eps[]}''(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[2] - \Xb[1])_{p_2}].
\end{align*}
Finally,
\begin{align*}
E[(\alphabh[I]-\alphab[])_{p_1}(\alphabh[I]-\alphab[])_{p_2}(\alphabh[I]-\alphab[])_{p_3}]
&\sim E[(\alphabh[]-\alphab[])_{p_1}(\alphabh[]-\alphab[])_{p_2}(\alphabh[]-\alphab[])_{p_3}] \\
&= N^{-3} \sum_{i_1,i_2,i_3=1}^N E[\eps[i_1]\eps[i_2]\eps[i_3]] E[(\Phii[N]^{-1}\Xb[i_1])_{p_1}(\Phii[N]^{-1} \Xb[i_2])_{p_2}(\Phii[N]^{-1} \Xb[i_3])_{p_3}] \\
&\sim N^{-3} \sum_{i=1}^N E[\eps[i]\eps[i]\eps[i]] E[(\Phii[N]^{-1} \Xb[i])_{p_1}(\Phii[N]^{-1} \Xb[i])_{p_2}(\Phii[N]^{-1} \Xb[i])_{p_3}] \\
&= N^{-2} E[\eps[]^3] E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii[N]^{-1} \Xb[1])_{p_2}(\Phii[N]^{-1} \Xb[1])_{p_3}] \\
&= O(N^{-2}),
\end{align*}
where we used Lemma \ref{lem:E-prod-PhiiN-Xi} for the last equality, which shows that the remainder is controlled.
\end{proof}
\end{lemma}
\begin{lemma} \label{lem:E-K-12}
As $N \to \infty$ and $h\to 0$, it holds that
\begin{align*}
E[K_{12}(\alphabh[])]
& \sim h \fY(y) + h^3 \frac{\mu_K}{2} \fY''(y) \\
& + h N^{-1} \frac{\sigmaeps^2}{2} \sum_{p_1,p_2=0}^J E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii[N]^{-1} \Xb[1])_{p_2}] E[f_{\eps[]}''(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[2] - \Xb[1])_{p_2}].
\end{align*}
\begin{proof}
Setting $\Ib = \{1,2\}$, we have that at leading-order,
\begin{align*}
E[K_{12}(\alphabh[])]
\sim E[K_{12}(\alphabh[I])]
& = \int \cdots \int K_h(\yt(\alphabh[I],\xb[1],\xb[2]) - e_2) f_{\eps[2]}(e_2) f_{\epsb[12-]}(\eb[12-]) f_{\Xb[]}(\xb[]) \ d\xb[] d\eb[12-] de_2.
\end{align*}
Performing the change of variables $-r = \yt(\alphabh[I],\xb[1],\xb[2]) - e_2)/h$, and using the fact that $K$ is a symmetric function, we have
\begin{align*}
E[K_{12}(\alphabh[])]
& \sim h \int \cdots \int K(r) f_{\eps[2]}(\yt(\alphabh[I],\xb[1],\xb[2]) + hr) f_{\epsb[12-]}(\eb[12-]) f_{\Xb[]}(\xb[]) \ d\xb[] d\eb[12-] dr.
\end{align*}
Next, Taylor expanding with respect to $h$ about $0$, and using \eqref{eq:int-K} and \eqref{eq:int-K-defs} gives
\begin{align*}
E[K_{12}(\alphabh[])]
& \sim h \int \cdots \int (f_{\eps[2]}(\yt(\alphabh[I],\xb[1],\xb[2])) + h^2 \frac{\mu_K}{2} f_{\eps[2]}''(\yt(\alphabh[I],\xb[1],\xb[2])) f_{\epsb[2-]}(\eb[2-]) f_{\Xb[]}(\xb[]) \ d\xb[] d\eb[12-] \\
& \sim h E[f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))] + h^3 \frac{\mu_K}{2} E[f_{\eps[]}''(\yt(\alphabh[I],\Xb[1],\Xb[2]))].
\end{align*}
The expression for the first expectation on the right hand side is provided in Lemma \ref{lem:E-feps-2}. On the other hand, taking the leading-order approximation $\alphab[]$ of $\alphabh[I]$, for the other expectation, we have that
\begin{align*}
E[f_{\eps[]}''(\yt(\alphabh[I],\Xb[1],\Xb[2]))]
= E[f_{\eps[]}''(y - \Xb[1]^T \alphab[] + (\Xb[2] - \Xb[1])^T(\alphabh[I] - \alphab[]))]
\sim E[f_{\eps[]}''(y - \Xb[1] \alphab[])] = \fY''(y),
\end{align*}
where we used \eqref{eq:fYk-conv-form} for the last equality.
\end{proof}
\end{lemma}
\begin{corollary} \label{cor:E-K-12-sqr}
As $N \to \infty$ and $h\to 0$, it holds that
\begin{align*}
E[K_{12}(\alphabh[])]^2
& \sim h^2 \fY^2(y) + h^4 \mu_K \fY(y) \fY''(y) \\
& + h^2 N^{-1} \sigmaeps^2 \fY(y) \sum_{p_1,p_2=0}^J E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii[N]^{-1} \Xb[1])_{p_2}] E[f_{\eps[]}''(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[2] - \Xb[1])_{p_2}].
\end{align*}
\end{corollary}
\begin{lemma} \label{lem:E-K-12-sqrt-internal}
As $N \to \infty$ and $h\to 0$, it holds that
\begin{align*}
E[K_{12}^2(\alphabh[])]
& \sim h \sigma_{K} \fY(y) + h^3 \frac{\sigma_{K,2}}{2} \fY''(y) \\
& + hN^{-1} \frac{\sigmaeps^2 \sigma_K}{2} \sum_{p_1,p_2=0}^J E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii[N]^{-1} \Xb[1])_{p_2}] E[f_{\eps[]}''(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[2] - \Xb[1])_{p_2}].
\end{align*}
\begin{proof}
Setting $\Ib = \{1,2\}$, we have that at leading-order
\begin{align*}
E[K_{12}^2(\alphabh[])]
\sim E[K_{12}^2(\alphabh[I])]
& = \int \cdots \int K_h^2(\yt(\alphabh[I],\xb[1],\xb[2]) - e_2) f_{\eps[2]}(e_2) f_{\epsb[2-]}(\eb[2-]) f_{\Xb[]}(\xb[]) \ d\xb[] d\eb[12-] de_2.
\end{align*}
Performing the change of variables $-r = \yt(\alphabh[I],\xb[1],\xb[2]) - e_2)/h$, and using the fact that $K$ is a symmetric function, we have
\begin{align*}
E[K_{12}^2(\alphabh[])]
& \sim h \int \cdots \int K^2(r) f_{\eps[2]}(\yt(\alphabh[I],\xb[1],\xb[2]) + hr) f_{\epsb[2-]}(\eb[2-]) f_{\Xb[]}(\xb[]) \ d\xb[] d\eb[12-] dr.
\end{align*}
Next, Taylor expanding with respect to $h$ about $0$, and using \eqref{eq:int-K} and \eqref{eq:int-K-defs} gives
\begin{align*}
E[K_{12}^2(\alphabh[])]
& \sim h \sigma_{K} \int \cdots \int (f_{\eps[2]}(\yt(\alphabh[I],\xb[1],\xb[2])) + h^2 \mu_K \frac{1}{2} f_{\eps[2]}''(\yt(\alphabh[I],\xb[1],\xb[2])) f_{\epsb[2-]}(\eb[2-]) f_{\Xb[]}(\xb[]) \ d\xb[] d\eb[12-] \\
& \sim h \sigma_{K} E[f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))] + h^3 \frac{\sigma_{K,2}}{2} E[f_{\eps[]}''(\yt(\alphabh[I],\Xb[1],\Xb[2]))].
\end{align*}
Leading-order expressions for these expectations were already derived in Lemma \ref{lem:E-K-12}.
\end{proof}
\end{lemma}
\begin{lemma} \label{lem:E-K-1213}
As $N \to \infty$ and $h\to 0$, it holds that
\begin{align*}
E[K_{12}(\alphabh[])K_{13}(\alphabh[])]
& \sim h^2 E[f_{\eps[]}^2(y - \Xb[1]\alphab[])].
\end{align*}
\begin{proof}
Setting $\Ib = \{1,2,3\}$, we have that at leading-order,
\begin{align*}
E[K_{12}(\alphabh[])K_{13}(\alphabh[])]
\sim E[K_{12}(\alphabh[I])K_{13}(\alphabh[I])]
& = \int \cdots \int K_h(\yt(\alphabh[I],\xb[1],\xb[2]) - e_2)K_h(\yt(\alphabh[I],\xb[1],\xb[3]) - e_3) \\
& \times f_{\eps[2]}(e_2)f_{\eps[3]}(e_3) f_{\epsb[23-]}(\eb[23-]) f_{\Xb[]}(\xb[]) \ d\xb[] d \eb[23-] de_2 de_3.
\end{align*}
Performing the change of variables $-r_1 = \yt(\alphabh[I],\xb[1],\xb[2]) - e_2)/h$ and $-r_2 = \yt(\alphabh[I],\xb[1],\xb[3]) - e_3)/h$, we get
\begin{align*}
E[K_{12}(\alphabh[])K_{13}(\alphabh[])]
& \sim h^2 \int \cdots \int K(r_1)K(r_2) f_{\eps[2]}(\yt(\alphabh[I],\xb[1],\xb[2]) + hr_1)f_{\eps[3]}(\yt(\alphabh[I],\xb[1],\xb[3]) + hr_2) \\
& \times f_{\epsb[23-]}(\eb[23-]) f_{\Xb[]}(\xb[]) \ d\xb[] d \eb[23-] dr_1 dr_2.
\end{align*}
Taylor expanding with respect to $h$ about $0$, and using \eqref{eq:int-K}, gives
\begin{align*}
E[K_{12}(\alphabh[])K_{13}(\alphabh[])]
& \sim h^2 \int \cdots \int f_{\eps[2]}(\yt(\alphabh[I],\xb[1],\xb[2]))f_{\eps[3]}(\yt(\alphabh[I],\xb[1],\xb[3])) f_{\epsb[23-]}(\eb[23-]) f_{\Xb[]}(\xb[]) \ d\xb[] d \eb[23-] \\
& = h^2 E[f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[3]))].
\end{align*}
Finally, taking the leading-order approximation $\alphab[]$ of $\alphabh[I]$, we have that
\begin{align*}
E[K_{12}(\alphabh[])K_{13}(\alphabh[])]
& \sim h^2 E[f_{\eps[]}(y - \Xb[1]\alphab[])f_{\eps[]}(y - \Xb[1]\alphab[])]
= h^2 E[f_{\eps[]}^2(y - \Xb[1]\alphab[])].
\end{align*}
\end{proof}
\end{lemma}
\begin{lemma} \label{lem:E-K-1232}
As $N \to \infty$ and $h\to 0$, it holds that
\begin{align*}
E[K_{12}(\alphabh[])K_{32}(\alphabh[])]
\sim h^2 \int_{R} f_{\eps[]}(y - \xb[1]^T \alphab[]) f_{\Xb[1]}(\xb[1]) f_{\Xb[3]}(\xb[3]) \ d\xb[1] d\xb[3].
\end{align*}
where the region of integration is $R =\{(\xb[1],\xb[3]) : (\xb[1] - \xb[3])^T \alphab[]=0\}$.
\begin{proof}
We have $E[K_{12}(\alphabh[])K_{32}(\alphabh[])] = E[K_{12}(\alphab[])K_{32}(\alphab[])] + (E[K_{12}(\alphabh[])K_{32}(\alphabh[])]-E[K_{12}(\alphab[])K_{32}(\alphab[])])$, which corresponds to the decomposition approach used in \cite{stove2012convolution}. In that work, it was established that the first term dominates asymptotically so it suffices to consider
\begin{align*}
E[K_{12}(\alphab[])K_{32}(\alphab[])]
& \sim \int \cdots \int K_h(y - \xb[1]^T \alphab[] - e_2)K_h(y - \xb[3]^T \alphab[] - e_2) f_{\eps[2]}(e_2) f_{\Xb[1]}(\xb[1]) f_{\Xb[3]}(\xb[3]) \ d\xb[1] d\xb[3] de_2.
\end{align*}
Performing the change of variables $-r_1 = (y - \xb[1]^T \alphab[] - e_2)/h$, using the fact that $K$ is a symmetric function, and taking the leading-order Taylor approximation of $f_{\eps[2]}(y - \xb[1]^T \alphab[] + hr_1)$, we find that
\begin{align*}
E[K_{12}(\alphab[])K_{32}(\alphab[])]
& = h \int \cdots \int K(r_1)K\bigg(\frac{(\xb[1] - \xb[3])^T \alphab[]}{h} - r_1\bigg) f_{\eps[2]}(y - \xb[1]^T \alphab[] + hr_1) f_{\Xb[1]}(\xb[1]) f_{\Xb[3]}(\xb[3]) \ d\xb[1] d\xb[3] dr_1 \\
& \sim h \int \cdots \int K(r_1)K\bigg(\frac{(\xb[1] - \xb[3])^T \alphab[]}{h} - r_1\bigg) f_{\eps[2]}(y - \xb[1]^T \alphab[]) f_{\Xb[1]}(\xb[1]) f_{\Xb[3]}(\xb[3]) \ d\xb[1] d\xb[3] dr_1 \\
& = h^2 \int \cdots \int h^{-1} K^*\bigg(\frac{(\xb[1] - \xb[3])^T \alphab[]}{h}\bigg) f_{\eps[2]}(y - \xb[1]^T \alphab[]) f_{\Xb[1]}(\xb[1]) f_{\Xb[3]}(\xb[3]) \ d\xb[1] d\xb[3] \\
& \sim h^2 \int_{R} f_{\eps[]}(y - \xb[1]^T \alphab[]) f_{\Xb[1]}(\xb[1]) f_{\Xb[3]}(\xb[3]) \ d\xb[1] d\xb[3],
\end{align*}
as $h \to 0$, up to a constant factor that is irrelevant to the asymptotic analysis, where $K^*(a)$ is the Gaussian function given by the convolution $K^*(a) = \int K(b)K(a-b)db$. See \cite[Supp. Material, Proof of Theorem 3] {stove2012convolution} for the analogous approach in the case of the Nadaraya-Watson-enhanced convolution estimator; in particular, the expression above has a correspondence to the second term in \cite[(19)] {stove2012convolution}.
\end{proof}
\end{lemma}
\begin{lemma} \label{lem:E-feps-2-feps-4}
As $N \to \infty$, it holds that
\begin{align*}
E[f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))f_{\eps[]}(\yt(\alphabh[I],\Xb[3],\Xb[4]))]
& \sim \fY^2(y)
+ N^{-1} \sigmaeps^2 \sum_{p_1,p_2=0}^J E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii [N]^{-1} \Xb[1])_{p_2}] \\
& \times (\fY(y) E[f_{\eps[]}''(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[2] - \Xb[1])_{p_2}] \\
& + E[f_{\eps[]}'(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}]E[f_{\eps[]}'(y - \Xb[3]^T \alphab[]) (\Xb[4] - \Xb[3])_{p_2}]).
\end{align*}
\begin{proof}
Taylor expanding with respect to $\alphabh[I]$ at about $\alphab[]$, and taking the expectation, we get
\begin{align*}
& E[f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))f_{\eps[]}(\yt(\alphabh[I],\Xb[3],\Xb[4]))] \\
&\sim E[f_{\eps[]}(y - \Xb[1]^T \alphab[])] E[f_{\eps[]}(y - \Xb[3]^T \alphab[])] \\
&+ \sum_{p_1=0}^J E[(\alphabh[I]-\alphab[])_{p_1}] E\bigg[\frac{\partial f_{\eps[]}(\yt(\alphabh[],\Xb[1],\Xb[2]))f_{\eps[]}(\yt(\alphabh[I],\Xb[3],\Xb[4]))}{\partial(\alphabh[I])_{p_1}} \bigg|_{\alphabh[I]=\alphab[]}\bigg] \\
& + \frac{1}{2} \sum_{p_1,p_2=0}^J E[(\alphabh[I]-\alphab[])_{p_1}(\alphabh[I]-\alphab[])_{p_2}] E\bigg[\frac{\partial^2 f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))f_{\eps[]}(\yt(\alphabh[I],\Xb[3],\Xb[4]))}{\partial(\alphabh[I])_{p_1}\partial(\alphabh[I])_{p_2}} \bigg|_{\alphabh[I]=\alphab[]}\bigg],
\end{align*}
where we have ignored the remainder since it can be neglected as shown in \eqref{lem:E-feps-2}. Now, $E[f_{\eps[]}(y - \Xb[1]^T \alphab[])] E[f_{\eps[]}(y - \Xb[3]^T \alphab[])] = \fY^2(y)$. The first order term vanishes since $E[(\alphabh[I] - \alphab[])p_1] = 0$. It was shown in \eqref{lem:E-feps-2} that $E[(\alphabh[I]-\alphab[])_{p_1}(\alphabh[I]-\alphab[])_{p_2}] = N^{-1} \sigmaeps^2 E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii [N]^{-1} \Xb[1])_{p_2}]$. Next,
\begin{align*}
\frac{\partial^2 f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))f_{\eps[]}(\yt(\alphabh[I],\Xb[3],\Xb[4]))}{\partial(\alphabh[I])_{p_1}\partial(\alphabh[I])_{p_2}}\bigg|_{\alphabh[I]=\alphab[]}
&= f_{\eps[]}''(y - \Xb[1]^T \alphab[]) f_{\eps[]}(y - \Xb[3]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[2] - \Xb[1])_{p_2} \\
& + f_{\eps[]}'(y - \Xb[1]^T \alphab[]) f_{\eps[]}'(y - \Xb[3]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[4] - \Xb[3])_{p_2} \\
& + f_{\eps[]}'(y - \Xb[1]^T \alphab[]) f_{\eps[]}'(y - \Xb[3]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_2}(\Xb[4] - \Xb[3])_{p_1} \\
& + f_{\eps[]}(y - \Xb[1]^T \alphab[]) f_{\eps[]}''(y - \Xb[3]^T \alphab[]) (\Xb[4] - \Xb[3])_{p_1}(\Xb[4] - \Xb[3])_{p_2}.
\end{align*}
Therefore
\begin{align*}
& E\bigg[\frac{\partial^2 f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))f_{\eps[]}(\yt(\alphabh[I],\Xb[3],\Xb[4]))}{\partial(\alphabh[I])_{p_1}\partial(\alphabh[I])_{p_2}}\bigg|_{\alphabh[I]=\alphab[]}\bigg] \\
&= 2 (E[f_{\eps[]}''(y - \Xb[1]^T \alphab[]) f_{\eps[]}(y - \Xb[3]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[2] - \Xb[1])_{p_2}] \\
& + E[f_{\eps[]}'(y - \Xb[1]^T \alphab[]) f_{\eps[]}'(y - \Xb[3]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[4] - \Xb[3])_{p_2}]) \\
&= 2 (\fY(y) E[f_{\eps[]}''(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[2] - \Xb[1])_{p_2}] \\
& + E[f_{\eps[]}'(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}]E[f_{\eps[]}'(y - \Xb[3]^T \alphab[]) (\Xb[4] - \Xb[3])_{p_2}]).
\end{align*}
\end{proof}
\end{lemma}
\begin{lemma} \label{lem:E-K-1234}
As $N \to \infty$ and $h\to 0$, it holds that
\begin{align*}
E[K_{12}(\alphabh[])K_{34}(\alphabh[])]
& \sim h^2 \fY^2(y)
+ h^4 \mu_K^2 \fY(y) \fY''(y) \\
& + h^2 N^{-1} \sigmaeps^2 \sum_{p_1,p_2=0}^J E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii[N]^{-1} \Xb[1])_{p_2}] \\
& \times (\fY(y) E[f_{\eps[]}''(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[2] - \Xb[1])_{p_2}] \\
& + E[f_{\eps[]}'(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}]E[f_{\eps[]}'(y - \Xb[3]^T \alphab[]) (\Xb[4] - \Xb[3])_{p_2}]).
\end{align*}
\begin{proof}
Setting $\Ib = \{1,2,3,4\}$, we have that at leading-order,
\begin{align*}
E[K_{12}(\alphabh[])K_{34}(\alphabh[])] \sim E[K_{12}(\alphabh[I])K_{34}(\alphabh[I])]
& = \int \cdots \int K_h(\yt(\alphabh[I],\xb[1],\xb[2]) - e_2)K_h(\yt(\alphabh[I],\xb[3],\xb[4]) - e_4) \\
& \times f_{\eps[2]}(e_2) f_{\eps[4]}(e_4) f_{\epsb[1234-]}(\eb[1234-]) f_{\Xb[]}(\xb[]) \ d\xb[] d\eb[1234-] de_2 de_4.
\end{align*}
Performing the change of variables $-r_1 = \yt(\alphabh[I],\xb[1],\xb[2]) - e_2)/h$ and $-r_2 = \yt(\alphabh[I],\xb[3],\xb[4]) - e_4)/h$, we get
\begin{align*}
E[K_{12}(\alphabh[])K_{34}(\alphabh[])]
& \sim h^2 \int \cdots \int K(r_1)K(r_2) f_{\eps[2]}(\yt(\alphabh[I],\xb[1],\xb[2]) + hr_1) f_{\eps[4]}(\yt(\alphabh[I],\xb[3],\xb[4]) + hr_2) \\
& \times f_{\epsb[1234-]}(\eb[1234-]) f_{\Xb[]}(\xb[]) \ d\xb[] d\eb[1234-] dr_1 dr_2.
\end{align*}
Taylor expanding with respect to $h$ about $0$, and using \eqref{eq:int-K} and \eqref{eq:int-K-defs}, we get
\begin{align*}
E[K_{12}(\alphabh[])K_{34}(\alphabh[])]
& \sim h^2 \int \cdots \int f_{\eps[2]}(\yt(\alphabh[I],\xb[1],\xb[2]))f_{\eps[4]}(\yt(\alphabh[I],\xb[3],\xb[4])) f_{\epsb[1234-]}(\eb[1234-]) f_{\Xb[]}(\xb[]) \ d\xb[] d\eb[1234-] \\
&+ h^4 \frac{\mu_K^2}{2} \int \cdots \int (f_{\eps[2]}(\yt(\alphabh[I],\xb[1],\xb[2]))f_{\eps[4]}''(\yt(\alphabh[I],\xb[3],\xb[4]))
+ f_{\eps[2]}''(\yt(\alphabh[I],\xb[1],\xb[2])) f_{\eps[4]}(\yt(\alphabh[I],\xb[3],\xb[4]))) \\
& \times f_{\epsb[1234-]}(\eb[1234-]) f_{\Xb[]}(\xb[]) \ d\xb[] d\eb[1234-] \\
& = h^2 E[f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))f_{\eps[]}(\yt(\alphabh[I],\Xb[3],\Xb[4]))] \\
& + h^4 \frac{\mu_K^2}{2}(E[f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))f_{\eps[]}''(\yt(\alphabh[I],\Xb[3],\Xb[4]))] + E[f_{\eps[]}''(\yt(\alphabh[I],\Xb[1],\Xb[2]))f_{\eps[]}''(\yt(\alphabh[I],\Xb[3],\Xb[4]))]) \\
& = h^2 E[f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))f_{\eps[]}(\yt(\alphabh[I],\Xb[3],\Xb[4]))]
+ h^4 \mu_K^2 E[f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))f_{\eps[]}''(\yt(\alphabh[I],\Xb[3],\Xb[4]))].
\end{align*}
By Lemma \ref{lem:E-feps-2-feps-4}, we have that at leading-order,
\begin{align*}
E[f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))f_{\eps[]}(\yt(\alphabh[I],\Xb[3],\Xb[4]))]
& \sim \fY^2(y)
+ N^{-1} \sigmaeps^2 \sum_{p_1,p_2=0}^J E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii[N]^{-1} \Xb[1])_{p_2}] \\
& \times (\fY(y) E[f_{\eps[]}''(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[2] - \Xb[1])_{p_2}] \\
& + E[f_{\eps[]}'(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}]E[f_{\eps[]}'(y - \Xb[3]^T \alphab[]) (\Xb[4] - \Xb[3])_{p_2}]).
\end{align*}
Finally, taking the leading-order approximation $\alphab[]$ of $\alphabh[I]$, and using \eqref{eq:fYk-conv-form}, we have
\begin{align*}
E[f_{\eps[]}(\yt(\alphabh[I],\Xb[1],\Xb[2]))f_{\eps[]}''(\yt(\alphabh[I],\Xb[3],\Xb[4]))]
\sim E[f_{\eps[]}(y - \Xb[1] \alphab[])]E[f_{\eps[]}''(y - \Xb[3] \alphab[])]
= \fY(y) \fY''(y).
\end{align*}
\end{proof}
\end{lemma}
\section{} \label{sec:appendix-E-PhiiN-inv-sqr-ord-mag}
For convenience, throughout this section we often use the notation $\lesssim$ to represent an expression that holds up an asymptotically irrelevant constant factor as $N \to \infty$. For example, instead of writing $5N^{-1} + 3N^{-2} \le C N^{-1}$ for $C>0$ as $N \to \infty$, we write $5N^{-1} + 3N^{-2} \lesssim N^{-1}$ as $N \to \infty$. Before proving Lemma \ref{lem:E-prod-PhiiN-Xi}, which is the primary objective of this section, we need several lemmas.
\begin{lemma} \label{lem:E-A-v}
Let $\Ab[] \in \mathbb{R}^{(J+1)\times (J+1)}$ be a matrix such that $E[|(\Ab[])_{ij}|^r]$ and $E[|(\Ab[])_{kl}|^r]$ have the same scaling with respect to $N$, and let $\vb[] \in \mathbb{R}^{J+1}$ be a vector such that $E[|(\vb[])_{j}|^r]$ and $E[|(\vb[])_{l}|^r]$ have the same scaling with respect to $N$, as $N \to \infty$, for $r \ge 1$ and $i,j,k,l \in \{0,1,\dots,J\}$. In particular, assume that
\begin{align}
E[(\Ab[])_{ij}^4] & \lesssim E[(\Ab[])_{11}^4], \label{eq:A-ij-le} \\
E[(\vb[])_{j}^4] & \lesssim E[(\vb[])_{1}^4], \label{eq:v-j-le}
\end{align}
as $N \to \infty$, for $i,j \in \{0,1,\dots,J\}$. Then, as $N \to \infty$,
\begin{align*}
E[(\Ab[]\vb[])_i^2] \lesssim E[(\Ab[])_{11}^4]^{1/2} E[(\vb[])_{1}^4]^{1/2}.
\end{align*}
\begin{proof}
The $i$-th element of the vector $\Ab[]\vb[]$ is $(\Ab[]\vb[])_i = \sum_{j=0}^J (\Ab[])_{ij} (\vb[])_{j}$. Therefore,
\begin{align*}
E[(\Ab[]\vb[])_i^2]
= E\bigg[\bigg(\sum_{j=0}^J (\Ab[])_{ij} (\vb[])_{j}\bigg)^2\bigg]
= \sum_{j=0}^J \sum_{k=0}^J E[(\Ab[])_{ij} (\Ab[])_{ik} (\vb[])_{j} (\vb[])_{k}].
\end{align*}
Then, we use the Cauchy-Schwarz inequality, followed by \eqref{eq:A-ij-le} and \eqref{eq:v-j-le}, to get
\begin{align*}
E[(\Ab[]\vb[])_i^2]
& \le \sum_{j=0}^J \sum_{k=0}^J E[((\Ab[])_{ij} (\Ab[])_{ik})^2]^{1/2} E[((\vb[])_{j} (\vb[])_{k})^2]^{1/2} \\
& \le \sum_{j=0}^J \sum_{k=0}^J E[(\Ab[])_{ij}^4]^{1/4} E[(\Ab[])_{ik}^4]^{1/4} E[(\vb[])_{j}^4]^{1/4} E[(\vb[])_{k}^4]^{1/4} \\
& \lesssim (J+1)^2 E[(\Ab[])_{11}^4]^{1/2} E[(\vb[])_{1}^4]^{1/2} \\
& \lesssim E[(\Ab[])_{11}^4]^{1/2} E[(\vb[])_{1}^4]^{1/2}.
\end{align*}
\end{proof}
\end{lemma}
\begin{proposition} \label{prop:E-S_N}
Let $Z_1,Z_2,\dots,$ be i.i.d positive random variables, and let $S_N = \sum_{i=1}^N Z_i$. Suppose $E[Z_1^r] < \infty$ for $r \ge 2$,. Then, as $N \to \infty$, $E[|S_N|^r]$ scales as
\begin{align*}
E[|S_N|^r] \sim C N^r,
\end{align*}
for $C > 0$.
\begin{proof}
Since $Z_i$ is positive, there exists $C_1 > 0$ such that $Z_i \ge C_1$. Then $(\sum_{i=1}^N Z_i)^r \ge C_1^r N^r=C_2N^r$, and so $E[|S_N|^r] \ge C_2N^r$.
Denote by $\mu_Z = E[Z_i]$ and $\sigma_Z^2 = \Var[Z_i]$, and note that by Minkowski's inequality,
\begin{align*}
E\bigg[\bigg|\frac{S_N}{\sigma_Z \sqrt{N}}\bigg|^r\bigg]^{1/r}
& = E\bigg[\bigg|\frac{S_N-N\mu_Z}{\sigma_Z \sqrt{N}} + \frac{N\mu_Z}{\sigma_Z \sqrt{N}}\bigg|^r\bigg]^{1/r} \\
& \le E\bigg[\bigg|\frac{S_N-N\mu_Z}{\sigma_Z \sqrt{N}}\bigg|^r\bigg]^{1/r} + \frac{\sqrt{N}\mu_Z}{\sigma_Z}.
\end{align*}
Then, by Theorem \cite[Thm. 5.1]{gut2013probability}, there exists $C_3 > 0$ such that as $N \to \infty$,
\begin{align*}
E\bigg[\bigg|\frac{S_N-N\mu_Z}{\sigma_Z \sqrt{N}}\bigg|^r\bigg]^{1/r} \le C_3 E[|W|^r]^{1/r},
\end{align*}
where $W \sim \mathcal{N}(0,1)$. This means that means that as $N \to \infty$,
\begin{align*}
E\bigg[\bigg|\frac{S_N}{\sigma_Z \sqrt{N}}\bigg|^r\bigg]^{1/r}
\le C_3 E[|W|^r]^{1/r} + \frac{\sqrt{N}\mu_Z}{\sigma_Z}
\le C_4 \sqrt{N}.
\end{align*}
Therefore,
\begin{align*}
E[|S_N|^r]
& \le C_5 N^r.
\end{align*}
Since $C_2 N^r \le E[S_N] \le C_5 N^r$, we have $E[|S_N|^r] \sim C N^r$.
\end{proof}
\end{proposition}
\begin{lemma} \label{lem:E-Det-r}
Let $r \ge 1$. As $N \to \infty$,
\begin{align*}
E[\Det^{r}(\Xb[]^T\Xb[])] \sim C N^{r(J+1)},
\end{align*}
for $C > 0$.
\begin{proof}
The formula for the determinant of a matrix can be written as a summation that includes a term given by the product of the diagonals of the matrix. It suffices to consider only this term since the scaling with respect to $N$ is common to all of the terms in summation. Therefore, since the diagonal of $\Xb[]^T \Xb[]$ is $[N,\sum_{i=1}^N \X[i1]^2,\sum_{i=1}^N \X[i2]^2\dots,\sum_{i=1}^N \X[iJ]^2]^T
$, we have that as $N \to \infty$, the determinant scales as
\begin{align*}
E[\Det^r(\Xb[]^T \Xb[])]
\sim C_1 E\bigg[\bigg(N \prod_{j=1}^J \sum_{i=1}^N \X[ij]^2\bigg)^r\bigg]
= C_1 N^r \prod_{j=1}^J E\bigg[\bigg(\sum_{i=1}^N \X[ij]^2\bigg)^r\bigg],
\end{align*}
where we used the fact that $\sum_{i=1}^N \X[ij]^2$ is independent of $\sum_{i=1}^N \X[ik]^2$ for $j \neq k$ in the last line. Then, by Proposition \ref{prop:E-S_N}, $E[(\sum_{i=1}^N \X[ij]^2)^r] \sim C_2 N^r$, and thus
\begin{align*}
E[\Det^r(\Xb[]^T \Xb[])]
\sim C_3 N^r \prod_{j=1}^J N^r
= C_3 N^{r(J+1)}.
\end{align*}
\end{proof}
\end{lemma}
\begin{corollary} \label{cor:E-Det-neg-r}
Let $r \ge 1$. As $N \to \infty$,
\begin{align*}
E[\Det^{-r}(\Xb[]^T \Xb[])]
\sim C N^{-r(J+1)}.
\end{align*}
\begin{proof}
Define $g(x) = x^{-1}$, and note that $\Det^r(\Xb[]^T \Xb[])$, is a non-negative random variable. Taylor expanding $g(\Det^r(\Xb[]^T \Xb[]))$ at $\Det^r(\Xb[]^T \Xb[]) = E[\Det^r(\Xb[]^T \Xb[])]$ and then taking the leading-order approximation of its expectation, we have that
\begin{align*}
E[\Det^{-r}(\Xb[]^T \Xb[])]
= E[g(\Det^r(\Xb[]^T \Xb[]))]
\sim g(E[\Det^{r}(\Xb[]^T \Xb[])])
= E[\Det^{r}(\Xb[]^T \Xb[])]^{-1},
\end{align*}
as $N \to \infty$.
The result follows by Lemma \ref{lem:E-Det-r}.
\end{proof}
\end{corollary}
\begin{corollary} \label{cor:E-Adj-r}
Let $r,s \ge 1$. As $N \to \infty$,
\begin{align*}
E[(\Adj^r(\Xb[]^T\Xb[])_{ij})^s] \sim C N^{rsJ},
\end{align*}
for $C > 0$ and $i,j \in \{0,1,\dots,J\}$.
\begin{proof}
The adjugate matrix of $\Xb[]^T\Xb[]$ is the transpose of its cofactor matrix. The elements of the cofactor matrix are themselves determinants of $J\times J$ submatrices formed from the elements of the $(J+1)\times(J+1)$-dimensional matrix $\Xb[]^T\Xb[]$. In particular, the scaling derived in Lemma \ref{lem:E-Det-r} applies to these determinants. Denote by $\Xb[J-]$ the matrix $\Xb[]$ in the multiple regression model \eqref{eq:samnple-mult-reg} with the $J$-th column removed. Then $\Adj^r(\Xb[]^T\Xb[])_{ij}$ scales like $\Det^r(\Xb[J-]^T\Xb[J-])$ with respect to $N$, which in turn means that $(\Adj^r(\Xb[]^T\Xb[])_{ij})^s$ scales like $\Det^{rs}(\Xb[J-]^T\Xb[J-])$. To be precise,
\begin{align*}
E[\Adj^r(\Xb[]^T\Xb[])_{ij}]
\sim C_1 E[\Det^{sr}(\Xb[J-]^T\Xb[J-])]
\sim C_2 N^{rs J},
\end{align*}
where we used Lemma \eqref{lem:E-Det-r} with the $(J+1)\times(J+1)$-dimensional matrix $\Xb[]^T\Xb[]$ replaced by the $J\times J$-dimensional matrix $\Xb[J-]^T\Xb[J-]$ to obtain the final expression.
\end{proof}
\end{corollary}
\begin{corollary} \label{cor:E-Det-Adj-r}
Let $r,s \ge 1$. As $N \to \infty$,
\begin{align*}
E[((\Xb[]^T \Xb[])^{-r})_{11}^s] \lesssim N^{-rs}.
\end{align*}
\begin{proof}
Since $(\Xb[]^T \Xb[])^{-r} = \Det^{-r}(\Xb[]^T \Xb[])\Adj^r(\Xb[]^T \Xb[])$, using properties of the determinant and adjugate matrix, followed by the Cauchy-Schwarz inequality, we get
\begin{align*}
E[((\Xb[]^T \Xb[])^{-r})_{11}^s]
& = E[(\Det^{-r}(\Xb[]^T \Xb[]) \Adj^r(\Xb[]^T \Xb[]))_{11}^s] \\
& = E[\Det^{-sr}(\Xb[]^T \Xb[]) (\Adj^r(\Xb[]^T \Xb[]))_{11}^s] \\
& \le E[(\Det^{-2sr}(\Xb[]^T \Xb[])]^{1/2} E[\Adj^r(\Xb[]^T \Xb[]))_{11}^{2s}]^{1/2}.
\end{align*}
Now, by Corollary \eqref{cor:E-Det-neg-r}, $E[(\Det^{-2sr}(\Xb[]^T \Xb[])]^{1/2} \sim N^{sr(J+1)}$, and by
Corollary \eqref{cor:E-Adj-r}, $E[\Adj^r(\Xb[]^T \Xb[]))_{11}^{2s}]^{1/2} \sim N^{rsJ}$. Hence,
\begin{align*}
E[((\Xb[]^T \Xb[])^{-r})_{11}^s]
\lesssim N^{-rs(J+1)} N^{rsJ}
= N^{-rs}.
\end{align*}
\end{proof}
\end{corollary}
\begin{lemma} \label{lem:X-T-X-inv-X-1-ineq}
As $N \to \infty$, it holds that
\begin{align*}
E[((\Xb[]^T \Xb[])^{-1}\Xb[1])_{i}^k] = O(N^{-k}).
\end{align*}
\begin{proof}
By Lemma \ref{lem:E-A-v},
\begin{align*}
E[((\Xb[]^T \Xb[])^{-1}\Xb[1])_{i}^k]
\lesssim E[((\Xb[]^T \Xb[])^{-1})_{11}^{2k}]^{1/2} E[(\Xb[1])_{1}^{2k}]^{1/2}
\lesssim E[((\Xb[]^T \Xb[])^{-1})_{11}^{2k}]^{1/2},
\end{align*}
since $\Xb[1]$ is independent of $N$. Then applying \eqref{cor:E-Det-Adj-r} with $(r,s) = (1,2k)$, we get
\begin{align*}
E[((\Xb[]^T \Xb[])^{-1})_{11}^{2k}]^{1/2}
\lesssim
(N^{-2k})^{1/2}
= N^{-k}.
\end{align*}
\end{proof}
\end{lemma}
\begin{lemma} \label{lem:E-prod-PhiiN-Xi}
As $N\to \infty$, it holds that
\begin{align*}
E\bigg[\prod_{i=1}^k (\Phii[N]^{-1} \Xb[1])_{p_i}\bigg] = O(1),
\end{align*}
for $k \in \mathbb{N}$.
\begin{proof}
Since all the elements of the vector $\Phii[N]^{-1} \Xb[1]$ have the same order of magnitude with respect to $N$,
\begin{align*}
E\bigg[\prod_{i=1}^k (\Phii[N]^{-1} \Xb[1])_{p_i}\bigg]
\lesssim E[(\Phii[N]^{-1} \Xb[1])_{p_1}^k].
\end{align*}
Therefore, recalling \eqref{eq:Phi}, and then using \eqref{lem:X-T-X-inv-X-1-ineq}, we have that
\begin{align*}
E\bigg[\prod_{i=1}^k (\Phii[N]^{-1} \Xb[1])_{p_i}\bigg]
\lesssim E[(\Phii[N]^{-1} \Xb[1])_{p_1}^k]
= N^k E[((\Xb[]^T \Xb[])^{-1} \Xb[1])_{p_1}^k]
\lesssim N^k N^{-k}
= 1.
\end{align*}
\end{proof}
\end{lemma}
\section{Introduction}
The standard approach to estimating the unknown probability density function of a random variable $Y$ is kernel density estimation, a nonparametric statistical technique which can be traced back to the pioneering works of Rosenblatt \cite{rosenblatt1956} and Parzen \cite{parzen1962estimation} over fifty years ago. Conventional kernel density estimation involves estimating the density $\fY$ of $Y$ using the Rosenblatt–Parzen density estimator
\begin{align} \label{eq:kde}
\fhY(y) = \frac{1}{hN} \sum_{i=1}^N K_h(y - Y_i),
\end{align}
where the set $\{Y_i\}_{i=1}^N$ is a sample of $N$ i.i.d observations of $Y$, $K_h(\cdot) = K(h^{-1}(\cdot))$ with $K$ being some kernel function, and $h>0$ is the bandwidth. Recently, there has been a lot of interest in another type of density estimator known as a convolution estimator. A convolution estimator can be employed when $Y$ is related to a set of covariates through a regression model such as
\begin{align} \label{eq:reg-nonlin}
Y = m(X) + \varepsilon,
\end{align}
where $m$ is a regression function, the covariate vector $X$ and the error $\varepsilon$ are independent, and $\varepsilon$ has mean zero and finite variance. The naming convention arises due to the fact that the probability distribution of a summation of random variables can be expressed in terms of a convolution. Estimating the density $\fY$ of $Y$ with a convolution estimator involves first estimating the underlying regression function $m$.
Escancianoa and Jacho-Ch{\'a}vez \cite{escanciano2012n} used the Nadaraya–Watson estimator to estimate the underlying regression function, and established asymptotic normality of their convolution estimator. M\"uller \cite{muller2012estimating} approached the problem in terms of an arbitrary estimator for the underlying regression function, and showed that the convolution estimator can achieve the optimal parametric convergence rate $\sqrt{N}$. St{\o}ve and Tj{\o}stheim \cite{stove2012convolution}, who also employed the Nadaraya–Watson estimator for the underlying regression function, derived explicit expressions for the asymptotic bias and variance of their convolution estimator, and proved that the mean square error (MSE) converges as $O(N^{-1})$.
Li and Tu \cite{li2016n} estimated the underlying regression function using nonlinear least squares, and investigated important topics such as endogeneity and robustness to misspecification in the regression function, along with proving the $\sqrt{N}$-consistency and asymptotic normality of their convolution estimator.
It is also worth mentioning that both St{\o}ve and Tj{\o}stheim, and Li and Tu, considered the case when the error can be heteroskedastic. Some other relatively recent works featuring convolution estimators are \cite{schick2004root,schick2007root,saavedra1999rate,saavedra2000estimation}.
By exploiting special structure of $Y$ in \eqref{eq:reg-nonlin}, convolution estimators can achieve $\sqrt{N}$-consistency, and thereby converge much faster than the conventional kernel density estimator which is only $\sqrt{Nh}$-consistent. That said, the convergence of convolution estimators is tied to the convergence of the estimator for the underlying regression model; Muller \cite{muller2012estimating} showed that $\sqrt{N}$-consistency requires plugging in an efficient regression function estimator, while St{\o}ve and Tj{\o}stheim found that their Nadaraya-Waton-based convolution estimator suffers from the curse of dimensionality \cite{stove2012convolution,li2016n}.
A powerful feature of convolution estimators, which has been largely unexplored in existing works on this topic, is that they provide a convenient mechanism by which additional covariate observations, above and beyond the covariate observations in the complete case dataset (response variable and associated covariates) used to define the regression model \eqref{eq:reg-nonlin}, can be incorporated into the estimation process in a straightforward fashion. Denote by $N$ the number of observations in the complete case sample, and by $M$ the number of observations in an additional sample featuring the covariates only. In all of the aforementioned works, apart from M\"uller \cite{muller2012estimating}, the total number of covariate observations matches the number of observations of the response variable, that is, $M=0$. However, there is no reason why the total number of covariate observations can't be larger than the number of observations of the response variable, that is, $M > 0$. This is very interesting as it raises the possibility of enhancing density estimates of a response variable without needing more response observations; instead, additional observations of the covariates can be used to enhance the density estimates.
M\"uller \cite{muller2012estimating} investigated a scenario involving a dataset in which some of the response observations are missing at random, while all of the covariate observations are present. Another interpretation of this situation is that one is in possession of a complete case sample featuring $N$ observations of a response variable and an associated set of covariates, along with an additional sample featuring $M$ observations of the covariates only. This is the perspective we take in this paper.
While the convergence of convolution estimators with respect to $N$ has been established as discussed above, the convergence with respect to $M$ is an open question. This is an important question; it would be useful to know just how effective the incorporation of additional covariate observations is in terms of enhancing the accuracy of density estimates, since often-times in practical applications it can be difficult if not downright impossible to obtain more observations of a response variable, while at the same time it can be very straightforward to obtain more observations of the covariates. For instance, we may want to estimate the density of a response variable that is difficult to measure due to time and/or cost constraints. Since this variable is challenging to measure, it is quite possible that only a small sample of measurements is available. On the other hand, we may find it easy to take or obtain a large number of measurements of other variables that are correlated with the difficult to measure response. To improve the accuracy of estimates of the density of the difficult to measure response, one can incorporate the abundant auxiliary covariate information using a convolution estimator.
In fact, our work in this paper was inspired by an application of this type from the field of ophthalmology. Measurement of the axial length of the human eye has historically been confined to specialist practice areas of ophthalmology, most notably for cataract and refractive surgery. It has not been measured routinely beyond this due to the high cost of biometric devices which are capable of measuring axial length precisely. Axial length has recently emerged as the most important clinical parameter required for the medical management of myopia, a condition associated with excessive eye growth and consequential ocular tissue damage and disease. New treatments are available to limit eye growth in children at risk of progressive myopia, but the clinicians tasked with prescribing and monitoring the efficacy of such treatments do not typically have access to the expensive specialised biometry devices. Consequently, accurate estimates of the axial length distribution in human populations are required to better understand, treat and monitor this and other ocular diseases. Datasets featuring axial length information are limited and small, whereas datsets orders of magnitude greater in size featuring measurements of ocular parameters such as refractive error, corneal radius, and age are readily available.
The convolution estimator proposed in this paper is based on the ordinary least squares estimator (OLS) estimator for an underlying multiple regression framework. This is in contrast to previous works on convolution estimators which have generally considered nonlinear regression functions and utilized nonlinear estimators such as the Nadaraya-Watson estimator. Nononparametric methods such as the Nadaraya-Watson estimator are afflicted by the curse of dimensionality; their convergence scales badly as the dimensionality of the covariates increases \cite{gyorfi2006distribution}. The authors of \cite{stove2012convolution} note that their Nadaraya-Watson based estimator is not suitable for covariate vectors with more than three dimensions for this reason. On the other hand, the convergence of the OLS estimator for a multiple regression model is independent of the number of covariates, since multiple regression is an additive model. Therefore, it is reasonable to expect that the curse of dimensionality will not be an issue for a convolution estimator that employs the OLS estimator for the underlying regression model. Note that by 'multiple regression', we mean a regression model that is linear in the parameters but potentially non-linear in the covariates, such as polynomial regression.
The key issues to consider when deciding on an underlying regression framework for a convolution estimator are (i) the level of nonlinearity present in the data, and (ii) the dimensionality of the covariates. If the data is highly nonlinear with low-dimensional covariate vectors, a Nadaraya-Watson-based estimator for a nonlinear regression model is a strong choice. On the other hand, if the data can be be well fit by a linear model, possibly after some non-linear transformations, or if one wants to use covariate vectors that span many dimensions, the OLS estimator and multiple regression may be a better choice.
Another aspect of convolution density estimators worth highlighting is that they are considerably more computationally expensive than the Rosenblatt–Parzen density estimator, since an evaluation with a convolution estimator requires two summations over the sample data instead of one, for each point on the evaluation grid. This can lead to high computational costs, so techniques for accelerating the computation of convolution estimator evaluations are desirable.
Our focus in this work is on establishing the theoretical and computational foundations of the multiple regression-enhanced convolution estimator. Our applied work on the estimating the distribution of the axial length of the human eye using this estimator will be reported in a future ophthalmology-focused research article. The main contributions of this work are as follows.
\begin{enumerate}
\item We derive the asymptotically optimal bandwidth for the multiple regression-enhanced convolution estimator, and show that it can be related to the asymptotically optimal bandwidth for the classical Rosenblatt–Parzen density estimator. In particular, the dependence of the optimal bandwidth on both $N$ and $M$ is established.
\item We show that the MSE of the multiple regression-enhanced convolution estimator converges as $O(N^{-1})$ irrespective of the dimensionality of the covariates, which means that it is not afflicted by the curse of dimensionality.
\item We resolve the question on the convergence of convolution estimators with respect to the number of covariate observations in the additional sample, by showing that for a large fixed $N$, the MSE converges as $O(M^{-4/5})$ towards an $O(N^{-1})$ constant. In other words, the accuracy improvement achievable through the incorporation of additional covariate observations eventually saturates at a level that is dependent on the number of complete case samples used in the underlying multiple regression model.
\item We develop a Fast Gauss Transform-based algorithm that substantially reduces the amount of computational time needed to perform convolution density estimator evaluations.
\end{enumerate}
This paper is structured as follows. In Section \ref{sec:MRBCDE}, we define the multiple regression-enhanced convolution density estimator and state some assumptions that are necessary for the mathematical analysis of the estimator, while also introducing some notational conventions.
In Section \ref{sec:theoretical-analysis}, we present our theoretical analysis which involves deriving the asymptotic bias and variance of the convolution estimator.
In Section \ref{sec:bandwidth-selection}, we derive the asymptotically optimal bandwidth for the convolution estimator, in particular showing how it depends on both $N$ and $M$. Moreover, we derive the rate of convergence of the MSE of the convolution estimator with respect to both $N$ ad $M$.
In Section \ref{sec:numerical-implementation}, we consider numerical implementation of the convolution estimator. We propose a computational algorithm that incorporates the high-performance C++ library FIGTree \cite{morariu2008automatic}. This library combines the (Improved) Fast Gauss Transform \cite{greengard1991fast} and Approximate Nearest Neighbor searching \cite{arya1993approximate} to reduce the computational complexity of Gauss transform evaluations.
In Section \ref{sec:numerical-simulations}, we perform a series of numerical simulations to gain an understanding of the convolution estimator's performance and investigate the potential reduction in MISE through the incorporation of additional covariate observations.
The paper ends with some concluding remarks in Section \ref{sec:concluding-remarks}. Appendix \ref{appendix:asy-expectations} features the asymptotic analysis of expectations that arise during the derivation of the asymptotic bias and variance. Appendix \ref{sec:appendix-E-PhiiN-inv-sqr-ord-mag} contains some technical proofs that are required to establish the order of magnitude of a specific term that arises in the bias and variance.
\section{Multiple regression-enhanced convolution density estimator} \label{sec:MRBCDE}
Without loss of generality, we assume that the multiple regression model that we are interested in, which is linear in the parameters but potentially nonlinear in the covariates, has if necessary been converted to a multiple linear regression model by variable transformations. Thus, let $\{(\Y[i], \Xb[i])\}_{i=1}^N$ be a sample of $N$ i.i.d. complete case observations of a random vector $(Y,X)$, where $Y$ is related to the $J$-dimensional covariate vector $X$ through the following multiple regression model
\begin{align} \label{eq:pop-mult-reg}
Y = X^T \alphab[] + \varepsilon.
\end{align}
Here, $\alphab[] = [\alphaa[0],\alphaa[1],\dots,\alphaa[J]]^T$, with $\alphaa[0] \neq 0$ for $i \in \{0,1,\dots,J\}$, is the vector of regression coefficients, and the first element of the covariate vector $X = [1,X_1,X_2,\dots,X_J]^T$ is defined to be one for convenience. The assumptions on the error $\varepsilon$ will be specified later. We are interested in estimating the probability density function $\fY$ of $Y$.
Let $\{\Xb[i]\}_{i=N+1}^L$, where $L = N+M$, be an additional sample of $M$ i.i.d. observations of the covariate vector $X$ only. While we could estimate $\fY$ directly using kernel density estimation applied to the $N$ observations of $Y$, instead we will leverage both the regression model \eqref{eq:pop-mult-reg} and the full set of $L$ covariate observations to provide more accurate density estimates than those given by the conventional approach.
The multiple regression model associated with the complete case dataset $\{(\Y[i], \Xb[i])\}_{i=1}^N$ is
\begin{align} \label{eq:samnple-mult-reg}
\Yb = \Xb[]\alphab[] + \epsb[],
\end{align}
where
\begin{align*}
\Yb
=
\begin{bmatrix}
\Y[1] \\
\Y[2] \\
\vdots \\
\Y[N]
\end{bmatrix},
\quad
\Xb[]
=
\begin{bmatrix}
\Xb[1]^T \\
\Xb[2]^T \\
\vdots \\
\Xb[N]^T
\end{bmatrix}
=
\begin{bmatrix}
1 & \X[11] & \dots & \X[1J] \\
1 & \X[21] & \dots & \X[2J] \\
\vdots & \vdots & \ddots & \vdots \\
1 & \X[N1] & \dots & \X[NJ] \\
\end{bmatrix}
\quad
\alphab[]
=
\begin{bmatrix}
\alpha_0 \\
\alpha_1 \\
\vdots \\
\alpha_J
\end{bmatrix},
\quad
\epsb[]
=
\begin{bmatrix}
\eps[1] \\
\eps[2] \\
\vdots \\
\eps[N]
\end{bmatrix}.
\end{align*}
Denote by $\alphabh[] = [\alphah[0],\alphah[1],\dots,\alphah[J]]^T$ the OLS estimator for the coefficient vector $\alphab[]$. The OLS estimator is given by \cite[4.4]{greene2003econometric}
\begin{align} \label{eq:OLS-estimator}
\alphabh[] = \alphab[] + (\Xb[]^T \Xb[])^{-1} \Xb[]^T \epsb[].
\end{align}
Denote by
\begin{align} \label{eq:Phi}
\Phii[N] := N^{-1} \Xb[]^T \Xb[],
\end{align}
and note that since $\Xb[]^T \epsb[] = \sum_{i=1}^N \Xb[i]\eps[i]$, the OLS estimator can be expressed as
\begin{align} \label{eq:OLS-estimator-Phi}
\alphabh[] = \alphab[] + N^{-1} \Phii[N]^{-1} \sum_{i=1}^N \Xb[i]\eps[i].
\end{align}
The residual vector is $\epsbh = \Yb - \Xb[]\alphabh[] = [\epsh[0],\epsh[1],\dots,\epsh[N]]^T$. Since $\Y[]$ is the sum of random variables, its density can be written as a convolution. Denoting by $\feps$ the error density, and by $F$ the covariate distribution, it holds that \cite{muller2012estimating}
\begin{align} \label{eq:fY-conv-form}
\fY(y) = \int \feps(y - \xb[]^T \alphab[]) F(d\xb[]) = E[\feps(y - X^T \alphab[])].
\end{align}
More generally, it holds that
\begin{align} \label{eq:fYk-conv-form}
\fY^{(k)}(y) = E[\feps^{(k)}(y - X^T \alphab[])],
\end{align}
where $\fY^{(k)}$ is the $k$-th derivative of $\fY$. The OLS estimator and the full set of $L$ covariate observations can be used to estimate the right hand side of \eqref{eq:fY-conv-form}:
\begin{align} \label{eq:fY-approx}
\fY(y) \approx \frac{1}{L} \sum_{i=1}^L \feps(y - \Xb[i]^T \alphabh[]).
\end{align}
Next, the residuals and conventional kernel density estimation can be used to estimate $\feps(y)$:
\begin{align} \label{eq:feps-approx}
\feps(y) \approx \frac{1}{N} \sum_{i=1}^N K_h(y-\epsh[i]),
\end{align}
Using \eqref{eq:fY-approx} and \eqref{eq:feps-approx}, we define the multiple regression-enhanced convolution estimator $\fhY$ by
\begin{align} \label{eq:fhY}
\fhY(y) = \frac{1}{hNL} \sum_{i=1}^L \sum_{j=1}^N K_h(y - \Xb[i]^T \alphabh[]-\epsh[j]).
\end{align}
This is the form of the estimator we use for computation. For the mathematical analysis, it is convenient to work with a slightly different expression for the estimator. Noting that $\epsh[j] = \y[j] - \Xb[j]^T \alphabh[]$, it is straightforward to show that $\fhY$ can be written as
\begin{align} \label{eq:fhY-convenient}
\fhY(y)
&= \frac{1}{hNL} \sum_{i=1}^L \sum_{j=1}^N K_h(y - \Xb[i]^T \alphab[] - \eps[j] + (\Xb[j] - \Xb[i])^T(\alphabh[] - \alphab[])).
\end{align}
\subsection{Notation} \label{subsec:Notation}
We introduce a function $\yt$ for notational convenience:
\begin{align} \label{eq:yt}
\yt(\gammab,\Xb[i],\Xb[j]) &:= y - \Xb[i]^T \alphab[] + (\Xb[j] - \Xb[i])^T(\gammab - \alphab[])
\end{align}
Note that $\yt$ reduces to a particularly simple form in certain cases, that is,
\begin{align} \label{eq:yt-simple}
\yt(\gammab,\Xb[i],\Xb[j]) = y - \Xb[i]^T \alphab[], \quad \quad \text{for} \ \gammab = \alphab[], \ \text{or} \ \Xb[i] = \Xb[j].
\end{align}
Using \eqref{eq:yt}, the convolution estimator \eqref{eq:fhY-convenient} can be written as
\begin{align} \label{eq:fhY-rewritten}
\fhY(y)
&= \frac{1}{hNL} \sum_{i=1}^L \sum_{j=1}^N K_h(\yt(\alphabh[],\Xb[i],\Xb[j]) - \eps[j]).
\end{align}
While we have explicitly defined the elements of the random vectors encountered above, for convenience we denote by $(\vb[])_i$ the $i$-th element of a random vector $\vb[]$, since this makes it easier to work with more complicated random vectors. Similarly, we denote by $(\Ab[])_{ij}$ the $(i,j)$-th element of a random matrix $\Ab[]$. By an abuse of notation, since $\vb[]$ is random not deterministic, we write $\partial f(\vb[])/\partial(\vb[])_{i}$ for the partial derivative of a function $f$ with respect to the $i$-th element of its vector-valued argument.
We make the following definitions for convenience.
\begin{align}
K_{ij}(\gammab) &:= K_h(\yt(\gammab,\Xb[i],\Xb[j])-\eps[j]), \label{eq:K-ij} \\
\Cov_{ijkl}(\gammab) & := \Cov[K_h(\yt(\gammab,\Xb[i],\Xb[j]) - \eps[j]),K_h(\yt(\gammab,\Xb[k],\Xb[l]) - \eps[l])]. \label{eq:Cov-ijkl}
\end{align}
Also, we denote by
\begin{align} \label{eq:int-K-defs}
\mu_K := \int r^2 K(r) dr, \quad \quad \sigma_K := \int K^2(r) dr, \quad \quad \sigma_{K,2} := \int r^2 K^2(r) dr.
\end{align}
\subsection{Assumptions} \label{subsec:Assumptions}
\begin{asu} \label{asu:K}
$K(y) = (2\pi)^{-1/2} e^{-\frac{1}{2}y^2}$.
\end{asu}
\begin{asu} \label{asu:second-moments}
$E[Y^2] < \infty$, and $E[X_i^2] < \infty$, for $i=1,\dots,J$.
\end{asu}
\begin{asu} \label{asu:iid}
The $N$ observations in dataset $\{(\Y[i],\Xb[i])\}_{i=1}^N$, and the $M$ observations in the dataset $\{\Xb[i]\}_{i=N+1}^L$ are independent and identically distributed.
\end{asu}
\begin{asu} \label{asu:error}
$E[\epsb[]|\Xb[]] = 0$, $E[\epsb[i]^2|\Xb[]] = \sigmaeps^2$, and $E[\epsb[i]\epsb[j]|\Xb[]] = 0$ for $i \neq j$.
\end{asu}
\begin{asu} \label{asu:diff}
The error density $\feps$ is four times differentiable.
\end{asu}
\begin{asu} \label{asu:n-sqr-h}
The bandwidth $h = h(N)$ behaves as $\lim_{N \to \infty} h(N) = 0$, and $\lim_{N \to \infty} h(N)N^2 = \infty$.
\end{asu}
Assumption \ref{asu:K} means that we are restricting to the Gaussian kernel function. We have restricted the kernel to the Gaussian function because our computational implementation of the multiple regression-enhanced convolution estimator is based on the Fast Gauss Transform. Due to Assumption \ref{asu:K}, all derivatives of $K$ are bounded. Moreover,
\begin{align} \label{eq:int-K}
\int K(r) dr = \int r^2 K(r) dr = 1, \quad \quad \int r K(r) dr = 0, \quad \quad \int r K^2(r) dr = 0.
\end{align}
Assumptions \ref{asu:second-moments}, \ref{asu:iid}, and \ref{asu:error} are standard conditions for multiple regression. Assumption \ref{asu:diff} is a standard regularity condition that ensures well-defined Taylor expansions. Assumption \ref{asu:n-sqr-h} is analogous to the usual assumption in conventional kernel density estimation that ensures the variance converges to zero as $N \to \infty$.
\section{Theoretical Analysis} \label{sec:theoretical-analysis}
We begin by deriving the asymptotic bias and variance of the multiple regression-enhanced convolution estimator.
\begin{theorem}[] \label{thm:bias-asy}
Under assumptions (A), (B), (C), (D), (E), and (F), the asymptotic bias of the multiple regression-enhanced convolution estimator is
\begin{align} \label{eq:bias}
\Bias[\fhY(y)]
& = O(h^2) + O\bigg(\frac{1}{N}\bigg).
\end{align}
\begin{proof}
First,
\begin{align*}
E[\fhY(y)]
&= \frac{1}{hNL} \sum_{i=1}^L \sum_{j=1}^N E[K_h(\yt(\alphabh[],\Xb[i],\Xb[j]) - \eps[j])] \\
&= \frac{1}{hNL} \bigg(\sum_{i=1}^N \sum_{j=1}^N + \sum_{i=1}^L \sum_{\substack{j=1 \\ j\neq i}}^N\bigg) E[K_h(\yt(\alphabh[],\Xb[i],\Xb[j]) - \eps[j])] \\
&= \frac{N}{hNL} E[K_h(\yt(\alphabh[],\Xb[1],\Xb[1]) - \eps[1])]
+ \frac{N(N-1)+ MN}{hNL} E[K_h(\yt(\alphabh[],\Xb[1],\Xb[2]) - \eps[2])] \\
&= \frac{1}{hL} E[K_h(y - \Xb[1]^T \alphab[] - \eps[1])]
+ \frac{L-1}{hL} E[K_h(\yt(\alphabh[],\Xb[1],\Xb[2]) - \eps[2])],
\end{align*}
where we used \eqref{eq:yt-simple} for the first expression on the last line.
Then, by Lemma \ref{lem:E-K-11-leading-order},
\begin{align*}
\frac{1}{hL} E[K_h(y - \Xb[1]^T \alphab[] - \eps[1])]
\sim \frac{1}{hL} (h \fY(y) + h^3 \frac{\mu_K}{2} \fY''(y))
= \frac{1}{L} (\fY(y) + h^2 \frac{\mu_K}{2} \fY''(y)).
\end{align*}
Next, by Lemma \ref{lem:E-K-12}, as $N \to \infty$,
\begin{align*}
\frac{L-1}{hL} E[K_h(\yt(\alphabh[],\Xb[1],\Xb[2]) - \eps[2])]
& \sim \frac{L-1}{L}\bigg(\fY(y) + h^2 \frac{\mu_K}{2} \fY''(y) + N^{-1} \frac{\sigmaeps^2}{2} \sum_{p_1,p_2=0}^J E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii[N]^{-1} \Xb[1])_{p_2}] \\
& \times E[f_{\eps[]}''(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[2] - \Xb[1])_{p_2}]\bigg).
\end{align*}
Combining these results, we have that
\begin{align*}
E[\fhY(y)]
& \sim \fY(y) + h^2 \frac{\mu_K}{2} \fY''(y)
+ N^{-1} \frac{\sigmaeps^2}{2} \sum_{p_1,p_2=0}^J E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii[N]^{-1} \Xb[1])_{p_2}] \\
& \times E[f_{\eps[]}''(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}(\Xb[2] - \Xb[1])_{p_2}].
\end{align*}
Now, $E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii[N]^{-1} \Xb[1])_{p_2}] = O(1)$ as $N \to \infty$ by Lemma \ref{lem:E-prod-PhiiN-Xi}. Therefore, the result follows since $\Bias[\fhY(y)] = E[\fhY(y)] - \fY(y)$.
\end{proof}
\end{theorem}
Before we derive the variance, we need a couple of lemmas. These lemmas, which hold under assumptions (A), (B), (C), (D), (E), and (F) given in Section \ref{subsec:Assumptions}, provide leading-order expressions for terms that arise when we perform a decomposition of the variance in Theorem \ref{thm:var-asy}. It transpires that only the terms $\Cov_{ijkl}(\alphabh[])$ for $(i,j,k,l) \in \{(1,2,1,2),(1,2,1,3),(1,2,3,2),(1,2,3,4)\}$ are important asymptotically. Expressions for these terms are derived in Appendix \ref{appendix:asy-expectations}. The remaining $\Cov_{ijkl}(\alphabh[])$ terms can be handled in a similar fashion so we omit the repetitive derivations.
\begin{lemma} \label{lem:Cov-1212}
It holds that
\begin{align*}
\Cov_{1212}(\alphabh[])
\sim O(h).
\end{align*}
\begin{proof}
Since $\Cov_{1212}(\alphabh[]) = E[K_{12}^2(\alphabh[])] - E[K_{12}(\alphabh[])]^2$, by Lemma \ref{lem:E-K-12-sqrt-internal} and Corollary \ref{cor:E-K-12-sqr}, it holds that
\begin{align*}
\Cov_{1212}(\alphabh[])
\sim h \sigma_{K} \fY(y).
\end{align*}
\end{proof}
\end{lemma}
\begin{lemma} \label{lem:Cov-1213}
It holds that
\begin{align*}
\Cov_{1213}(\alphabh[])
\sim O(h^2).
\end{align*}
\begin{proof}
Since $\Cov_{1213}(\alphabh[]) = E[K_{12}(\alphabh[])K_{13}(\alphabh[])] - E[K_{12}(\alphabh[])]^2$, by Lemma \ref{lem:E-K-1213} and Corollary \ref{cor:E-K-12-sqr},
\begin{align*}
\Cov_{1213}(\alphabh[])
\sim h^2 (E[f_{\eps[]}^2(y - \Xb[1]\alphab[])] - \fY^2(y)).
\end{align*}
\end{proof}
\end{lemma}
\begin{lemma} \label{lem:Cov-1232}
It holds that
\begin{align*}
\Cov_{1232}(\alphabh[])
\sim O(h^2).
\end{align*}
\begin{proof}
Since $\Cov_{1232}(\alphabh[]) = E[K_{12}(\alphabh[])K_{32}(\alphabh[])] - E[K_{12}(\alphabh[])]^2$, by Lemma \ref{lem:E-K-1232} and Corollary \ref{cor:E-K-12-sqr},
\begin{align*}
\Cov_{1232}(\alphabh[])
\sim h^2 \bigg(\int_{R} f_{\eps[]}(y - \xb[1]^T \alphab[]) f_{\Xb[1]}(\xb[1]) f_{\Xb[3]}(\xb[3]) \ d\xb[1] d\xb[3] - \fY^2(y)\bigg),
\end{align*}
where the region of integration is $R =\{(\xb[1],\xb[3]) : (\xb[1] - \xb[3])^T \alphab[]=0\}$.
\end{proof}
\end{lemma}
\begin{lemma} \label{lem:Cov-1234}
It holds that
\begin{align*}
\Cov_{1234}(\alphabh[])
& \sim O(h^2 N^{-1}).
\end{align*}
\begin{proof}
Since $\Cov_{1234}(\alphabh[]) = E[K_{12}(\alphabh[])K_{34}(\alphabh[])] - E[K_{12}(\alphabh[])]^2$, by Lemma \ref{lem:E-K-1234} and Corollary \ref{cor:E-K-12-sqr},
\begin{align*}
\Cov_{1234}(\alphabh[])
& \sim h^2 N^{-1} \sigmaeps^2 \sum_{p_1,p_2=0}^J E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii[N]^{-1} \Xb[1])_{p_2}] \\
& \times E[f_{\eps[]}'(y - \Xb[1]^T \alphab[]) (\Xb[2] - \Xb[1])_{p_1}]E[f_{\eps[]}'(y - \Xb[3]^T \alphab[]) (\Xb[4] - \Xb[3])_{p_2}].
\end{align*}
\end{proof}
\end{lemma}
\begin{lemma} \label{lem:Cov-ijkl-1122}
It holds that
\begin{align*}
\Cov_{1122}(\alphabh[])
& = 0.
\end{align*}
\begin{proof}
By \eqref{eq:yt-simple},
\begin{align*}
\Cov_{1122}(\alphabh[])
&= E[K_{11}(\alphabh[])K_{22}(\alphabh[])] - E[K_{11}(\alphabh[])]^2 \\
&= E[K_h(y - \Xb[1]^T \alphab[] - \eps[1]) K_h(y - \Xb[2]^T \alphab[] - \eps[2])] - E[K_h(y - \Xb[1]^T \alphab[] - \eps[1])]^2 \\
&= 0.
\end{align*}
\end{proof}
\end{lemma}
The following lemmas relate to the covariances that turn out to be asymptotically negligible when the variance gets decomposed.
\begin{lemma} \label{lem:Cov-ijkl-1231}
It holds that
\begin{align*}
\Cov_{ijkl}(\alphabh[])
& \sim O(h^2 N^{-1}),
\end{align*}
where $(i,j,k,l) \in \{(1,2,3,1),(1,2,2,1),(1,1,2,3),(2,1,1,3)\}$.
\end{lemma}
\begin{lemma} \label{lem:Cov-ijkl-1111}
It holds that
\begin{align*}
\Cov_{ijkl}(\alphabh[])
& \sim
\begin{cases}
O(h), & \quad \quad (i,j,k,l) = (1,1,1,1), \\
O(h^2), & \quad \quad (i,j,k,l) \in \{(1,1,1,2),(1,1,2,1)\}.
\end{cases}
\end{align*}
\end{lemma}
Now we are in a position to derive the asymptotic variance.
\begin{theorem}[] \label{thm:var-asy}
Under assumptions (A), (B), (C), (D), (E), and (F), the asymptotic variance of the multiple regression-enhanced convolution estimator is
\begin{align} \label{eq:variance}
\Var[\fhY(y)]
& = O\bigg(\frac{1}{hNL}\bigg) + O\bigg(\frac{1}{L}\bigg) + O\bigg(\frac{1}{N}\bigg).
\end{align}
\begin{proof}
The variance of the convolution estimator \eqref{eq:fhY-rewritten} can be decomposed as
\begin{align} \label{eq:var-proof-start}
\begin{split}
& \Var[\fhY(y)] \\
& = \frac{1}{(hNL)^2} \sum_{i=1}^L \sum_{j=1}^N \sum_{k=1}^L \sum_{l=1}^N \Cov_{ijkl}(\alphabh[]) \\
& = \frac{1}{(hNL)^2} \bigg(\sum_{i=1}^N \sum_{j=1}^N \sum_{k=1}^L \sum_{l=1}^N + \sum_{i={N+1}}^L \sum_{j=1}^N \sum_{k=1}^L \sum_{l=1}^N\bigg) \Cov_{ijkl}(\alphabh[]) \\
& = \frac{1}{(hNL)^2} \bigg(\sum_{i=1}^N \sum_{j=1}^N \sum_{k=1}^N \sum_{l=1}^N + \sum_{i=1}^N \sum_{j=1}^N \sum_{k=N+1}^L \sum_{l=1}^N + \sum_{i={N+1}}^L \sum_{j=1}^N \sum_{k=1}^N \sum_{l=1}^N + \sum_{i={N+1}}^L \sum_{j=1}^N \sum_{k=N+1}^L \sum_{l=1}^N\bigg) \Cov_{ijkl}(\alphabh[]) \\
& = \frac{1}{(hNL)^2} \bigg(\sum_{i=1}^N \sum_{j=1}^N \sum_{k=1}^N \sum_{l=1}^N + 2 \sum_{i={N+1}}^L \sum_{j=1}^N \sum_{k=1}^N \sum_{l=1}^N + \sum_{i={N+1}}^L \sum_{j=1}^N \sum_{k=N+1}^L \sum_{l=1}^N\bigg) \Cov_{ijkl}(\alphabh[]).
\end{split}
\end{align}
These sets of summations can be expressed as follows:
\begin{align} \label{eq:var-decomp-sum-A-B-C}
\begin{split}
\sum_{i=1}^N \sum_{j=1}^N \sum_{k=1}^N \sum_{l=1}^N \Cov_{ijkl}(\alphabh[]) \ & = \sum_{i=1}^{12} c_i^{(A)} \Psi_i^{(A)}, \\
\sum_{i={N+1}}^L \sum_{j=1}^N \sum_{k=1}^N \sum_{l=1}^N \Cov_{ijkl}(\alphabh[]) \ & = \sum_{i=1}^{5} c_i^{(B)} \Psi_i^{(B)}, \\
\sum_{i={N+1}}^L \sum_{j=1}^N \sum_{k=N+1}^L \sum_{l=1}^N \Cov_{ijkl}(\alphabh[]) \ & = \sum_{i=1}^{5} c_i^{(C)} \Psi_i^{(C)},
\end{split}
\end{align}
with
\begin{alignat*}{4}
&
c_{1}^{(A)} \sim N, & \quad \Psi_{1}^{(A)} = \Cov_{1111}(\alphabh[]), & \quad \quad
c_{1}^{(B)} \sim MN, & \quad \Psi_{1}^{(B)} = \Cov_{1222}(\alphabh[]), \\
&
c_{2}^{(A)} \sim N^2, & \quad \Psi_{2}^{(A)} = \Cov_{1212}(\alphabh[]), & \quad \quad
c_{2}^{(B)} \sim MN^2, & \quad \Psi_{2}^{(B)} = \Cov_{1223}(\alphabh[]), \\
&
c_{3}^{(A)} \sim N^3, & \quad \Psi_{3}^{(A)} = \Cov_{1213}(\alphabh[]), & \quad \quad
c_{3}^{(B)} \sim MN^3, & \quad \Psi_{3}^{(B)} = \Cov_{1234}(\alphabh[]), \\
&
c_{4}^{(A)} \sim 2N^3, & \quad \Psi_{4}^{(A)} = \Cov_{1231}(\alphabh[]), & \quad \quad
c_{4}^{(B)} \sim MN^2, & \quad \Psi_{4}^{(B)} = \Cov_{1232}(\alphabh[]), \\
&
c_{5}^{(A)} \sim N^3, & \quad \Psi_{5}^{(A)} = \Cov_{1232}(\alphabh[]), & \quad \quad
c_{5}^{(B)} \sim MN^2, & \quad \Psi_{5}^{(B)} = \Cov_{1233}(\alphabh[]), \\
& c_{6}^{(A)} \sim N^2, & \quad \Psi_{6}^{(A)} = \Cov_{1221}(\alphabh[]), & \quad \quad
c_{1}^{(C)} \sim MN, & \quad \Psi_{1}^{(C)} = \Cov_{1212}(\alphabh[]), \\
& c_{7}^{(A)} \sim 2N^2, & \quad \Psi_{7}^{(A)} = \Cov_{1112}(\alphabh[]), & \quad \quad
c_{2}^{(C)} \sim MN^2, & \quad \Psi_{2}^{(C)} = \Cov_{1213}(\alphabh[]), \\
& c_{8}^{(A)} \sim 2N^2, & \quad \Psi_{8}^{(A)} = \Cov_{1121}(\alphabh[]), & \quad \quad
c_{3}^{(C)} \sim M^2N, & \quad \Psi_{3}^{(C)} = \Cov_{1232}(\alphabh[]), \\
& c_{9}^{(A)} \sim N^4, & \quad \Psi_{9}^{(A)} = \Cov_{1234}(\alphabh[]), & \quad \quad
c_{4}^{(C)} \sim M^2N^2, & \quad \Psi_{4}^{(C)} = \Cov_{1234}(\alphabh[]), \\
& c_{10}^{(A)} \sim N^2, & \quad \Psi_{10}^{(A)} = \Cov_{1122}(\alphabh[]), \\
& c_{11}^{(A)} \sim N^3, & \quad \Psi_{11}^{(A)} = \Cov_{1123}(\alphabh[]), \\
& c_{12}^{(A)} \sim N^3, & \quad \Psi_{12}^{(A)} = \Cov_{2113}(\alphabh[]),
\end{alignat*}
It is worth noting that this is somewhat of a generalization of similar decompositions in \cite[Supp. Material]{stove2012convolution}. For example, the set $\{c_{i}^{(A)} \Psi_{i}^{(A)}\}_{i=1}^{12}$ can be related to the set $\{(S_i)\}_{i=1}^{12}$ in that work; see also \cite{saavedra2000estimation}. It suffices to evaluate the terms $\{\Psi_{i}^{(A)}\}_{i=1}^{12}$ since these terms correspond to equivalent terms in the sets $\{\Psi_{i}^{(B)}\}_{i=1}^{5}$ and $\{\Psi_{i}^{(C)}\}_{i=1}^{4}$. To be specific,
\begin{alignat*}{5}
& \Psi_{2}^{(A)} = \Psi_{1}^{(C)}, \quad \quad & \Psi_{3}^{(A)} = \Psi_{2}^{(C)}, \quad \quad & \Psi_{5}^{(A)} = \Psi_{4}^{(B)} = \Psi_{3}^{(C)}, \quad \quad & \Psi_{8}^{(A)} = \Psi_{1}^{(B)}, \\
& \Psi_{9}^{(A)} = \Psi_{3}^{(B)} = \Psi_{4}^{(C)}, \quad \quad & \Psi_{11}^{(A)} = \Psi_{5}^{(B)}, \quad \quad & \Psi_{12}^{(A)} = \Psi_{2}^{(B)}.
\end{alignat*}
By accounting for these correspondences, and using \eqref{eq:var-proof-start} and \eqref{eq:var-decomp-sum-A-B-C}, we can write the variance in the following form:
\begin{align} \label{eq:var-sum-d-Psi}
\Var[\fhY(y)]
= \sum_{i=1}^{12} d_{i} \Psi_{i}^{(A)},
\end{align}
where at leading-order,
\begin{alignat*}{8}
d_{1} &= \frac{c_{1}^{(A)}}{(hNL)^{2}} \ &\sim& \ \frac{1}{h^2NL^2}, \quad \quad & d_{7} &= \frac{2c_{7}^{(A)} }{(hNL)^{2}} \ &\sim& \ \frac{2}{h^2L^2}, \\
d_{2} &= \frac{(c_{2}^{(A)} + c_{1}^{(C)})}{(hNL)^{2}} \ &\sim& \ \frac{1}{h^2NL}, \quad \quad & d_{8} &= \frac{2(c_{8}^{(A)} + c_{1}^{(B)})}{(hNL)^{2}} \ &\sim& \frac{2}{h^2 NL}, \\
d_{3} &= \frac{(c_{3}^{(A)} + c_{2}^{(C)})}{(hNL)^{2}} \ &\sim& \ \frac{1}{h^2L}, \quad \quad & d_{9} &= \frac{(c_{9}^{(A)} + 2c_{3}^{(B)} + c_{4}^{(C)})}{(hNL)^{2}} \ &\sim& \ \frac{1}{h^2}, \\
d_{4} &= \frac{2 c_{4}^{(A)}}{(hNL)^{2}} \ &\sim& \ \frac{2N}{h^2L^2}, \quad \quad & d_{10} &= \frac{c_{10}^{(A)}}{(hNL)^{2}} \ &\sim& \ \frac{1}{h^2L^2}, \\
d_{5} &= \frac{(c_{5}^{(A)} + 2c_{4}^{(B)} + c_{3}^{(C)})}{(hNL)^{2}} \ &\sim& \ \frac{1}{h^2N}, \quad \quad & d_{11} &= \frac{(c_{11}^{(A)} + 2c_{5}^{(B)})}{(hNL)^{2}} &\sim& \ \frac{1}{h^2L}, \\
d_{6} &= \frac{c_{6}^{(A)}}{(hNL)^{2}} \ &\sim& \ \frac{1}{h^2L^2}, \quad \quad & d_{12} &= \frac{(c_{12}^{(A)} + 2c_{2}^{(B)})}{(hNL)^{2}} &\sim& \ \frac{1}{h^2L}.
\end{alignat*}
Then, combing these expressions with the results for $\{\PsiA[i]\}_{i=1}^{12}$ derived in Lemmas \ref{lem:Cov-1212}, \ref{lem:Cov-1213}, \ref{lem:Cov-1232}, \ref{lem:Cov-1234}, \ref{lem:Cov-ijkl-1122}, \ref{lem:Cov-ijkl-1231}, and \ref{lem:Cov-ijkl-1111}, we find that
\begin{alignat*}{8}
d_{1} \PsiA[1] &\sim \frac{1}{h^2NL^2} O(h) \ &=& \ O\bigg(\frac{1}{hNL^2}\bigg), \quad \quad & d_{7} \PsiA[7] &\sim \frac{2}{h^2 L^2}O(h^2) \ &=& \ O\bigg(\frac{1}{L^2}\bigg), \\
d_{2} \PsiA[2] &\sim \frac{1}{h^2NL} O(h) \ &=& \ O\bigg(\frac{1}{hNL}\bigg), \quad \quad & d_{8} \PsiA[8] &\sim \frac{2}{h^2 NL}O(h^2) \ &=& \ O\bigg(\frac{1}{NL}\bigg), \\
d_{3} \PsiA[3] &\sim \frac{1}{h^2L}O(h^2) \ &=& \ O\bigg(\frac{1}{L}\bigg), \quad \quad & d_{9} \PsiA[9] &\sim \frac{1}{h^2}O(h^2N^{-1}) \ &=& \ O\bigg(\frac{1}{N}\bigg), \\
d_{4} \PsiA[4] &\sim \frac{2 N}{h^2L^2}O(h^2N^{-1}) \ &=& \ O\bigg(\frac{1}{L^2}\bigg), \quad \quad & d_{10} \PsiA[10] &\sim \frac{1}{h^2L^2} \cdot 0 \ &=& \ 0, \\
d_{5} \PsiA[5] &\sim \frac{1}{h^2N}O(h^2) \ &=& \ O\bigg(\frac{1}{N}\bigg), \quad \quad & d_{11} \PsiA[11] &\sim \frac{1}{h^2L}O(h^2N^{-1}) \ &=& \ O\bigg(\frac{1}{NL}\bigg), \\
d_{6} \PsiA[6] &\sim \frac{1}{h^2L^2}O(h^2N^{-1}) \ &=& \ O\bigg(\frac{1}{NL^2}\bigg), \quad \quad & d_{12} \PsiA[12] &\sim \frac{1}{h^2L}O(h^2N^{-1}) \ &=& \ O\bigg(\frac{1}{NL}\bigg).
\end{alignat*}
where we used the fact that $E[(\Phii[N]^{-1} \Xb[1])_{p_1}(\Phii[N]^{-1} \Xb[1])_{p_2}] = O(1)$ as $N \to \infty$ by Lemma \ref{lem:E-prod-PhiiN-Xi} for $\PsiA[9]$. Finally, as $N\to \infty$ and $h \to 0$, four of these terms are seen to dominate, that is to say,
\begin{align*}
\Var[\fhY(y)]
& \sim d_{2} \PsiA[2] + d_{3} \PsiA[3] + d_{5} \PsiA[5] + d_{9} \PsiA[9] \\
& = O\bigg(\frac{1}{hNL}\bigg) + O\bigg(\frac{1}{L}\bigg) + O\bigg(\frac{1}{N}\bigg) + O\bigg(\frac{1}{N}\bigg).
\end{align*}
\end{proof}
\end{theorem}
\begin{corollary}[] \label{cor:asy-mse}
Under assumptions (A), (B), (C), (D), (E), and (F), the asymptotic MSE of the multiple regression-enhanced convolution estimator is
\begin{align} \label{eq:mse}
\MSE[\fhY(y)]
&= \Bias[\fhY(y)]^2 - \Var[\fhY(y)]
= O(h^4) + O\bigg(\frac{1}{hNL}\bigg) + O\bigg(\frac{1}{L}\bigg) + O\bigg(\frac{1}{N}\bigg).
\end{align}
\end{corollary}
\begin{corollary}[] \label{cor:asy-mise}
Under assumptions (A), (B), (C), (D), (E), and (F), the asymptotic mean integrated square error (MISE) of the multiple regression-enhanced convolution estimator is
\begin{align} \label{eq:mise}
\MISE[\fhY]
&= \int \Bias[\fhY(y)]^2 - \Var[\fhY(y)] dy
= O(h^4) + O\bigg(\frac{1}{hNL}\bigg) + O\bigg(\frac{1}{L}\bigg) + O\bigg(\frac{1}{N}\bigg).
\end{align}
\end{corollary}
Note that in the case when there is no additional covariate information, that is when $M = 0$ and thus $L = N$, the mean square error reduces to $\MSE[\fhY(y)] = O(h^4) + O(h^{-1}N^{-2}) + O(N^{-1})$, which recovers a result established by St{\o}ve and Tj{\o}stheim \cite[Eq. (23)]{stove2012convolution}, and Escancianoa and Jacho-Ch{\'a}vez \cite[Eq. (3.4)]{escanciano2012n}.
Since $L = M+N$ appears in the variance \eqref{eq:variance} but not in the bias \eqref{eq:bias}, the presence of the supplemental sample of $M$ covariate observations leads to a direct reduction in the variance, but not the bias. However, the additional sample does have an indirect effect on the bias, since the asymptotically optimal bandwidth $h$ depends on $M$, as we show in the next section. Thus, the presence of the $M$ covariate observations in the additional sample ultimately leads to a reduction in the MSE and MISE through both the bias and the variance.
The effect that the amount of additional auxiliary data supplied to the convolution estimator has on the variance is clear from \eqref{eq:variance}. In the absence of additional auxiliary data, that is, when $M=0$ and thus $L=N$, we get
\begin{align} \label{eq:Var-C-0}
\Var[\fhY(y)]
& = O\bigg(\frac{1}{hN^2}\bigg) + O\bigg(\frac{1}{N}\bigg).
\end{align}
When the number of additional covariate observations is on the order of the number of complete case observations, that is, when $M = O(N)$, we get
\begin{align} \label{eq:Var-C-times}
\Var[\fhY(y)]
& = O\bigg(\frac{1}{hN(N+M)}\bigg) + O\bigg(\frac{1}{(N+M)}\bigg) + O\bigg(\frac{1}{N}\bigg),
\end{align}
which is just a rewritten version of the general expression \eqref{eq:variance} that makes the dependence on $M$ explicit. Finally, when the number of additional auxiliary data observations is much larger than the number of complete case observations, that is, when $N$ is large and fixed while $M \to \infty$, the first two terms in \eqref{eq:variance} vanish and we are left with
\begin{align} \label{eq:Var-C-gg-1}
\Var[\fhY(y)]
& = O\bigg(\frac{1}{N}\bigg).
\end{align}
So, the presence of additional auxiliary data guarantees a reduction in the asymptotic variance.
No matter how many additional covariate observations are incorporated into the convolution estimator, however, the variance can't be reduced beyond $O(N^{-1})$ because $d_{5} \PsiA[5]$ and $d_{9} \PsiA[9]$ in \eqref{eq:var-sum-d-Psi} do not depend on the additional sample of $M$ covariate observations. This $O(N^{-1})$ uncertainty arises because the $O(N^{-1})$ uncertainty present in the underlying OLS estimator, which we recall was defined \eqref{eq:OLS-estimator} with respect to $N$ complete case observations, ultimately propagates into uncertainty in the convolution estimator.
Since the variance can't be reduced beyond $O(N^{-1})$, a saturation phenomenon arises. Eventually, as more and more additional auxiliary data observations are supplied to the convolution estimator, the improvement in accuracy will become completely negligible and the variance will saturate at $\Var[\fhY(y)] \sim d_{5} \PsiA[5] + d_{9} \PsiA[9]$.
\section{Bandwidth Selection and Convergence of the MSE with respect to $N$ and $M$} \label{sec:bandwidth-selection}
\begin{lemma}[] \label{lem:hopt}
The asymptotically optimal bandwidth $\hopt$ for the multiple regression-enhanced convolution estimator is
\begin{align} \label{eq:hopt}
\hopt = \bigg(\frac{\sigma_{K}}{\mu_K^2 \int (\fY''(y))^2 dy} \bigg)^{1/5} \frac{1}{(NL)^{1/5}}.
\end{align}
\begin{proof}
The MISE \eqref{eq:mise} depends on $h$ through the $h^2 \mu_K \fY''(y)/2$ term in the bias and the $d_2 \Psi_2^{(A)}$ term in the variance. Thus, the bandwidth $h$ which minimizes the MISE solves the following equation.
\begin{align*}
0
&= \frac{\partial}{\partial h}\bigg(h^4 \frac{\mu_K^2}{4} \int (\fY''(y))^2 dy + \frac{\sigma_{K}}{hNL} \bigg)
= h^3 \mu_K^2 \int (\fY''(y))^2 dy - \frac{\sigma_{K}}{h^2NL}.
\end{align*}
\end{proof}
\end{lemma}
\begin{corollary}[] \label{cor:hopt-RP}
The asymptotically optimal bandwidth $\hopt$ for the multiple regression-enhanced convolution estimator can be written as
\begin{align} \label{eq:hopt-RP}
\hopt = \hoptRP L^{-1/5},
\end{align}
where
\begin{align*}
\hoptRP
&= \bigg(\frac{\sigma_{K}}{\mu_K^2 \int (\fY''(y))^2 dy} \bigg)^{1/5} \frac{1}{N^{1/5}},
\end{align*}
is the asymptotically optimal bandwidth for the Rosenblatt–Parzen density estimator \eqref{eq:kde} \cite[Eq. 3.21]{silverman1986density}.
\end{corollary}
Since $\fY''(y)$ is unknown, the optimal bandwidth formulas \eqref{eq:hopt} and \eqref{eq:hopt-RP} are not directly applicable. However, there are numerous methods in the literature for estimating $\hoptRP$, such as cross-validation \cite{rudemo1982empirical}, Silverman's rule of thumb \cite{silverman1986density}, and the plug-in approach of Sheather and Jones \cite{sheather1991reliable}. In any, case due to Corollary \ref{cor:hopt-RP}, we can employ an established means of choosing $\hoptRP$ and then scale it by $L^{-1/5}$ to obtain an estimate of the optimal bandwidth $\hopt$ for the convolution estimator.
In the case when there is no additional covariate information, that is when $M = 0$ and thus $L = N$, convolution estimators already allow for reduced bias in comparison to the classical kernel density estimator \eqref{eq:kde}. For convolution estimators, $\hopt = O(N^{-2/5})$ which implies that $\Bias[\fhY(y)] = O(\hopt^2) + O(N^{-1}) = O(N^{-4/5}) + O(N^{-1}) = O(N^{-4/5})$. For the classical kernel density estimator, on the other hand, $\hoptRP = O(N^{-1/5})$ which implies that $\Bias[\fhY(y)] = O((\hoptRP)^2) \sim O(N^{-2/5})$. The incorporation of additional covariate observations into the convolution estimator allows for an even greater reduction in bias, since the $h^2$ term in the bias in this case is $O((NL)^{-2/5})$ which is smaller than the $O(N^{-4/5})$ that arises in the usual case of the convolution estimator with no additional covariate information.
With the asymptotically optimal bandwidth in hand, we are now in a position to quantify the reduction in the MSE as the number of complete case observations $N$ increases, and the number of additional covariate observations $M$ increases.
\begin{lemma} \label{lem:MSE-N-infty}
For a fixed $M$, at the asymptotically optimal bandwidth, the MSE of the multiple regression-enhanced convolution estimator decays as $O(N^{-1})$ as $N \to \infty$.
\begin{proof}
As $N \to \infty$, we have that $\hopt = (NL)^{-1/5} \sim N^{-2/5}$, and thus
\begin{align*}
\MSE[\fhY(y)]
& = O(\hopt^4) + O\bigg(\frac{1}{\hopt NL}\bigg) + O\bigg(\frac{1}{L}\bigg) + O\bigg(\frac{1}{N}\bigg) \\
& = O\bigg(\frac{1}{N^{8/5}}\bigg) + O\bigg(\frac{N^{2/5}}{N^2}\bigg) + O\bigg(\frac{1}{N}\bigg) \\
& = O\bigg(\frac{1}{N^{8/5}}\bigg) + O\bigg(\frac{1}{N^{8/5}}\bigg) + O\bigg(\frac{1}{N}\bigg) \\
& = O\bigg(\frac{1}{N}\bigg).
\end{align*}
\end{proof}
\end{lemma}
\begin{lemma} \label{lem:MSE-M-infty}
For a large fixed $N$, at the asymptotically optimal bandwidth, the MSE of the multiple regression-enhanced convolution estimator decays as $O(M^{-4/5})$ towards an $O(N^{-1})$ constant as $M \to \infty$.
\begin{proof}
For $N$ large, we have that
\begin{align*}
\MSE[\fhY(y)]
& = O(\hopt^4) + O\bigg(\frac{1}{\hopt NL}\bigg) + O\bigg(\frac{1}{L}\bigg) + O\bigg(\frac{1}{N}\bigg) \\
& = O\bigg(\frac{1}{(NL)^{4/5}}\bigg) + O\bigg(\frac{(NL)^{1/5}}{NL}\bigg) + O\bigg(\frac{1}{L}\bigg) + O\bigg(\frac{1}{N}\bigg) \\
& = O\bigg(\frac{1}{(NL)^{4/5}}\bigg) + O\bigg(\frac{1}{L}\bigg) + O\bigg(\frac{1}{N}\bigg).
\end{align*}
The $O(N^{-1})$ third term acts as the saturation threshold since it doesn't change with $M$. Now, with $N$ fixed, the first term on the last line decays as $O((NL)^{-4/5}) = O((N^2 + NM)^{-4/5}) = O(M^{-4/5})$. This term dominates the second term on the last line which decays as $O(L^{-1}) = O((N + M)^{-1}) = O(M^{-1})$.
\end{proof}
\end{lemma}
It is worth highlighting the fact that the convergence of the MSE of the multiple regression-enhanced convolution estimator with respect to $N$ and $M$ is independent of the dimensionality of the covariates. This is in contrast to the case of Nadaraya-Watson which suffers from the curse of dimensionality \cite[Sec. 6]{stove2012convolution}. Therefore, if the data is well-fit by a multiple regression model, possibly after some variable transformations, and the dimensionality of the covariate vector is significant, it is recommended to use the multiple regression-enhanced convolution estimator to achieve $O(N^{-1})$ convergence.
By characterizing the decay of the MSE with respect to $M$, Lemma \ref{lem:MSE-M-infty} resolves the question regarding precisely how much the accuracy of density estimates can be enhanced through the incorporation of additional covariate information into a convolution estimator. Since increasing the number of complete case observations causes the MSE to decay as $O(N^{-1})$ towards zero, while increasing number of additional covariate observation causes the MSE to decay as $O(M^{-4/5})$ towards an $O(N^{-1})$ constant, we see that supplying more covariate observations to the convolution estimator is not quite as effective as supplying more complete case observations. However, supplying additional covariate observations can still provide a very significant performance improvement, as will be demonstrated in the numerical simulations in Section \ref{sec:numerical-simulations}. This is good news because in many in practical applications it may be difficult if not downright impossible to obtain additional response observations, whereas it can often be very straightforward to obtain large amounts of additional covariate data.
\section{Efficient Computational Implementation} \label{sec:numerical-implementation}
Since the expression for the multiple regression-enhanced convolution estimator \eqref{eq:fhY} involves summing over the entire set of $L$ covariate observations, and also the set of $N$ residuals from the multiple regression model \eqref{eq:samnple-mult-reg}, the evaluation of the convolution estimator can be very time consuming in comparison to the evaluation of the classical density estimator \eqref{eq:kde} which features only a single summation over the set of response observations. In particular, in the case of large datasets, or in applications involving cross-validation or bootstrapping, the computational costs can become prohibitive.
To reduce computational times, in this section we present a Fast Gauss Transform (FGT)-based acceleration algorithm. This algorithm utilizes the high-performance C++ library FIGTree \cite{morariu2008automatic} which combines the (Improved) Fast Gauss Transform \cite{greengard1991fast} with Approximate Nearest Neighbor searching \cite{arya1993approximate} to efficiently evaluate the Gauss transform. The Gauss transform $G$ is defined as
\begin{align} \label{eq:FGT}
G_h(y_v,\{x_i\}_{i=1}^L,\{q_i\}_{i=1}^L) = \sum_{i=1}^L q_i e^{-((y_v-x_i)/h)^2},
\end{align}
where $\{q_j\}_{i=1}^L$ is a set of coefficients, $\{y_v\}_{v=1}^V$ is a set of target points, and $\{x_i\}_{i=1}^L$ is a set of source points.
The computational complexity involved in directly evaluating this expression at the $V$ target points is $O(VL)$. The FGT reduces the computational complexity to $O(V+L)$.
We need to evaluate the convolution estimator \eqref{eq:fhY} at the set of $V$ target points:
\begin{align} \label{eq:fhY-target-points}
\fhY(y_v) = \frac{1}{hNL} \sum_{i=1}^L \sum_{j=1}^N K_h(y_v - \Xb[i]^T \alphabh[]-\epsh[j]).
\end{align}
Since this expression features an extra summation compared to \eqref{eq:FGT}, a naive computational implementation results in a complexity of $O(VLN)$. By employing the FGT, the complexity can be reduced to $O(VN + L)$. With the complexity now scaling linearly with the number of complete case observations and additional covariate observations, the algorithm proposed in this section is particularly effective for accelerating evaluations when the number of additional covariate observations is potentially orders of magnitude larger than the number of complete cases observations, such as in the case of a difficult to measure response variable.
FIGTree cannot be used directly for the computation of \eqref{eq:fhY-target-points}, since FIGTree accelerates the evaluation of the single summation in \eqref{eq:FGT}. Therefore, we need to transform \eqref{eq:fhY} into a single summation expression. This can be achieved by stacking the set of $N$ residuals and $V$ target points into a single set of $VN$ artificial target points. Define the artificial target points $\{z_k\}_{k=1}^{VN}$ by
\begin{align} \label{eq:z-k}
z_k = y_{\floor{(p-1)/N}+1} - \hat{\varepsilon}_{((p-1 \bmod N)+1}, \quad \quad p = 1, \dots, VN.
\end{align}
Then \eqref{eq:fhY-target-points} can be rewritten as
\begin{align} \label{eq:fhY-target-points-figtree}
\fhY(y_v) = \frac{1}{\sqrt{2\pi}}\frac{1}{hNL} \sum_{p=(v-1)N+1}^{vN} G_h(z_p,\{\Xb[i]^T \alphabh[]\}_{i=1}^L,\{1\}_{i=1}^L).
\end{align}
Now, $\{G_h(z_k,\{\Xb[i]^T \alphabh[]\}_{i=1}^L,\{1\}_{i=1}^L)\}_{k=1}^{VN}$ can be evaluated using FIGTree. Once the evaluations have been performed at the artificial target points, the convolution estimator evaluations at the actual target points $\{y_v\}_{v=1}^V$ can be recovered using \eqref{eq:fhY-target-points-figtree}; see Algorithm \ref{alg:cnv-est}.
In Table \ref{tab:comp-times}, we present computational times for the evaluation of the multiple regression-enhanced convolution estimator \eqref{eq:fhY}, using a variety of approaches in the case of $N=100$ complete case observations and $V = 50$ target points, as a progressively larger number of additional covariate observations is supplied to the estimator. The evaluation approaches are as follows.
\begin{enumerate}[(i)]
\item Naive (R): Evaluating the density estimator using for loops in R.
\item Naive (C++): Evaluating the density estimator using for loops in C++.
\item FFT: Stacking the $L$ variables in the set $\{\Xb[i]^T \alphab[]\}_{i=1}^L$ and the $N$ variables in the set $\{\epsh[j]\}_{i=1}^N$ in \eqref{eq:fhY-target-points} into a single set of length $LN$, and then evaluating the density estimator using the \texttt{density()} function in R, which employs the Fast Fourier Transform (FFT) to accelerate computations \cite{deng2011density}.
\item FGT: The FGT acceleration technique presented above.
\end{enumerate}
Naive evaluation in R is very slow, which is to be expected since R is an interpreted language. For $M=12800$ additional covariate observations, naive evaluation in C++ is about $33$ times faster than naive evaluation in R. Performing the evaluations using the FFT-accelerated \texttt{density()} function in R is about $1.8$ times faster than performing the evaluations using Naive (C++).
At $M=100$, the FGT acceleration technique introduced above is approximately $159, 5.9$, and $4.6$ times faster than performing the evaluations using Naive (R), Naive (C++), and the FFT, respectively. As $M$ increases, the acceleration becomes even more pronounced; at $M=12800$, the FGT acceleration technique is approximately $2722, 81$, and $41$ times faster than performing the evaluations using Naive (R), Naive (C++), and the FFT, respectively. At this value of $M$, the FGT evaluation takes about 22 milliseconds whereas the FFT evaluation takes almost $1$ second.
Note that this algorithm can easily be adapted to other types of convolution estimators such as the Nadaraya-Watson-based convolution estimator; we can simply replace the set $\{\Xb[i]^T \alphabh[]\}_{i=1}^L$ with the analogous set $\{\hat{m}(\Xb[i])\}_{i=1}^L$, where $\hat{m}$ is the Nadaraya-Watson estimator of the nonlinear regression function $m$ in \eqref{eq:reg-nonlin}.
\renewcommand{\arraystretch}{1.2}
\begin{table}[htb!]
\footnotesize
\centering
\begin{tabular}[t]{|l|rrrrrrrrr|}
\hline
M & 0 & 100 & 200 & 400 & 800 & 1600 & 3200 & 6400 & 12800 \\
\hline
Naive (R) & 4.86e-01 & 1.10e+00 & 1.39e+00 & 2.30e+00 & 4.23e+00 & 7.72e+00 & 1.53e+01 & 2.96e+01 & 6.01e+01 \\
Naive (C++) & 1.97e-02 & 4.12e-02 & 5.65e-02 & 9.21e-02 & 1.60e-01 & 2.87e-01 & 5.13e-01 & 9.62e-01 & 1.78e+00 \\
FFT & 6.98e-03 & 3.18e-02 & 1.87e-02 & 3.65e-02 & 6.27e-02 & 1.16e-01 & 2.17e-01 & 4.31e-01 & 8.99e-01 \\
FGT & 4.99e-03 & 6.98e-03 & 6.02e-03 & 6.98e-03 & 6.03e-03 & 7.92e-03 & 9.99e-03 & 1.49e-02 & 2.21e-02 \\
\hline
\end{tabular}
\caption{\label{tab:comp-times} Computational times (seconds) for evaluations of the multiple regression-enhanced convolution estimator as a progressively larger number of additional covariate observations are supplied to the estimator, using a variety of evaluation approaches. The FGT-based algorithm is much faster than the other approaches in all cases, with the reduction in computational times becoming even more pronounced as $M$ increases.}
\end{table}
\begin{algorithm}[htb]
\caption{Multiple regression-enhanced convolution estimator \label{alg:cnv-est}}
\begin{enumerate}
\item Fit a regression model to the complete case dataset $\{(\Y[i], \Xb[i])\}_{i=1}^N$ to obtain the OLS estimator $\alphabh[]$ and the residuals $\{\epsh[j]\}_{i=1}^N$.
\item Combine the $N$ covariate observations in the complete case dataset and the $M$ covariate observations in the additional dataset into a single dataset $\{\Xb[i]\}_{i=1}^L$.
\item Generate the artificial target points $\{z_k\}_{k=1}^{VN}$ using \eqref{eq:z-k}.
\item Compute $\{G_h(z_k,\{\Xb[i]^T \alphabh[]\}_{i=1}^L,\{1\}_{i=1}^L)\}_{k=1}^{VN}$ by supplying the covariate observations $\{\Xb[i]\}_{i=1}^L$ and the artificial target points $\{z_k\}_{k=1}^{VN}$ to FIGTree which evaluates \eqref{eq:FGT}.
\item Convert $\{G_h(z_k,\{\Xb[i]^T \alphabh[]\}_{i=1}^L,\{1\}_{i=1}^L)\}_{k=1}^{VN}$ into density estimate evaluations at the actual target points $\{\fhY(y_v)\}_{v=1}^V$ using \eqref{eq:fhY-target-points-figtree}.
\end{enumerate}
\end{algorithm}
\section{Numerical Simulations} \label{sec:numerical-simulations}
In this section we present some numerical simulations to investigate the accuracy of the multiple regression-enhanced convolution estimator. Let $\{y_v\}_{v=1}^V$ be a uniformly spaced set of target points, where $V = 128$. To compute the MISE of Rozenblatt-Parzen estimator and the multiple regression-enhanced convolution estimator we use a reference solution $\fY$ which is obtained by estimating the density of the response variable with the Rozenblatt-Parzen estimator using a very large sample size, $N = 10^6$.
Denote by $\fhY^{(p)}(y_v)$ a realization of an estimate given by a density estimator at the target point $y_v$. We approximate the integrated square error (ISE) of the realization $\fhY^{(p)}$ by the Riemann sum
\begin{align*}
\ISE[\fhY^{(p)}]
&= (v_2-v_1)\sum_{v=2}^V (\fhY^{(p)}(y_v) - \fY(y_v))^2,
\end{align*}
where $P = 500$. The MISE of the estimator $\fhY$ is then approximated by
\begin{align*}
\MISE[\fhY]
&= \frac{1}{P} \sum_{p=1}^P \ISE[\fhY^{(p)}].
\end{align*}
Denote by $\fhY^{(RP)}$ the Rosenblatt–Parzen density estimator, and by $\fhY^{(MR)}$ the multiple regression-enhanced convolution estimator. Since the performance of convolution estimators with respect to the number of complete cases observations $N$ has already been analyzed in papers such as \cite{stove2012convolution,li2016n,escanciano2012n,muller2012estimating}, in this work we are more concerned with the performance with respect to the number of additional covariate observations $M$. In particular, we are interested in investigating by how much $\MISE[\fhY^{(MR)}]$ can be reduced as the number of additional covariate observations supplied to the convolution estimator increases. Denote by $\tau = M/N$ the ratio of additional covariate observations to complete case observations.
For the convolution estimator, we use bandwidths given by the asymptotically optimal bandwidth formula \eqref{eq:hopt-RP}, which involves scaling the corresponding asymptotically optimal bandwidths for the Rozenblatt-Parzen estimator. For the Rozenblatt-Parzen estimator itself, we use the Sheather-Jones method of bandwidth selection \cite{sheather1991reliable}.
\subsection{Single peaked negatively skewed distribution}
\begin{figure}[ht!]
\centering
\captionsetup[subfloat]{labelformat=empty}
\begin{tabular}{ccc}
\subfloat[(i) $\tau=0$]{\includegraphics[scale=0.6]{Skewed_Dist_0-crop}} \hspace{2em}
\subfloat[(ii) $\tau=4$]{\includegraphics[scale=0.6]{Skewed_Dist_4-crop}} \\
\subfloat[(iii) $\tau=16$]{\includegraphics[scale=0.6]{Skewed_Dist_16-crop}} \hspace{2em}
\subfloat[(iv) $\tau=64$]{\includegraphics[scale=0.6]{Skewed_Dist_64-crop}}
\end{tabular}
\caption{Typical realizations of estimates for the density of the response variable $Y$ in \eqref{eq:mult-reg-skewed} given by the Rosenblatt–Parzen density estimator $\fhY^{(RP)}$ (dotted line), and the multiple regression-enhanced convolution estimator $\fhY^{(MR)}$ (dashed line). The black solid line is the true density $\fY$. \label{fig:typ-real-skew}}
\end{figure}
Consider the regression model
\begin{align} \label{eq:mult-reg-skewed}
Y = \alpha_0 + \alpha_1 X_1 + \alpha_2 X_2 + \eps[],
\end{align}
where $(\alpha_0,\alpha_1,\alpha_2) = (1,3,3)$, with
\begin{align*}
X_1 \sim \beta(5,1), \quad \quad
X_2 \sim \mathcal{N}(7,0.05), \quad \quad
\eps[] & \sim \mathcal{N}(0,0.1).
\end{align*}
The density $\fY$ of $Y$ is in this case is single-peaked and negatively skewed. In Figure \ref{fig:typ-real-skew}, we plot $\fY$ along with typical realizations of density estimates given by $\fhY^{(RP)}$ and $\fhY^{(MR)}$, when the complete case dataset features $N=100$ observations, while the ratio of additional covariate observations supplied to the convolution estimator progressively increases, $\tau \in \{0,4,16,64\}$. In subplot (i), both $\fhY^{(RP)}$ and $\fhY^{(MR)}$ provide poor estimates, particularly in the under-smoothed tail region.
The estimate $\fhY^{(MR)}$ for $\tau=0$ is no better than $\fhY^{(RP)}$. If anything, its actually worse since it is bimodal when the true distribution is unimodal. However, as $\tau$ increases, $\fhY^{(MR)}$ approaches the true distribution; the problematic tail region gets smoothed out and $\fhY^{(MR)}$ becomes unimodal. At $\tau = 64$, $\fhY^{(MR)}$ provides a very accurate estimate of $\fY$.
In Table \ref{tab:skew} we report MISE results for both $\fhY^{(RP)}$ and $\fhY^{(MR)}$. Three complete cases samples sizes are considered, $N \in \{50,100,200\}$. For each sample size, we compute the MISE when the ratio of additional covariate observations to complete case observations is $\tau \in \{0,2,4,8,16,32,64,128,256,512\}$. Note that for all complete case sample sizes, $\MISE[\fhY^{(MR)}] \approx \MISE[\fhY^{(RP)}]$ when $\tau = 0$. Now consider, for example, the $N=200$ case. By the time $\tau = 128$, $\MISE[\fhY^{(MR)}]$ is about $23$ times smaller than $\MISE[\fhY^{(RP)}]$, which demonstrates that a very substantial reduction in MISE is achievable through the incorporation of additional covariate observations. The change in the MISE as $\tau$ increase from $\tau = 128$ to $\tau = 512$ is negligible as saturation has occurred by this stage.
To understand the difference between incorporating additional covariate observations versus incorporating additional complete case observations in the context of MISE reduction, consider the loglog plot in Figure \ref{fig:skewed-mise-cvg}. The $\fhY^{(MR)}$ MISE results from Table \ref{tab:skew} for the case of $N=50$ are represented by the dashed line in this plot. The fact that $\MISE[\fhY^{(MR)}]$ is slightly higher at $\tau = 256$ compared to $\tau = 128$ and $\tau = 512$ is just a numerical artefact of the convergence flat-lining once the saturation threshold has been reached. As $M$ increases, this MISE is converging asymptotically as $O(M^{-4/5})$ towards an $O(N^{-1})$ constant. The solid black line is the MISE of $\fhY^{(MR)}$ when the initial dataset of $N=50$ complete case observations is supplemented with a progressively larger number of additional complete case observations as opposed to additional covariate observations. This line decays asymptotically as $O(N^{-1})$ towards zero. The corresponding $\fhY^{(RP)}$ MISE result is also shown for reference as a dotted line. Clearly, incorporating additional complete cases observations is more effective than incorporating additional covariate observations. Nevertheless, incorporating more covariate observations still allows for a substantial reduction in the MISE, which is very useful in situations in which obtaining more covariate observations is straightforward while obtaining more complete cases observations may be impossible.
\begin{figure}[h]
\centering
\includegraphics[scale=0.85]{Skewed_MISE_Cvg_N_5e1-crop}
\caption{The convergence of the MISE as additional observations are supplied to the convolution estimator. The $\MISE[\fhY^{(MR)}]$ results from Table \ref{tab:skew} in which $N=50$ complete case observations are supplemented by an additional $M = \tau N$ covariate observations are represented by the dashed line. The solid black line is the MISE of $\fhY^{(MR)}$ when the convolution estimator is supplied with a progressively larger number of additional complete case observations. The corresponding $\MISE[\fhY^{(RP)}]$ result is the dotted line.
\label{fig:skewed-mise-cvg}}
\end{figure}
\renewcommand{\arraystretch}{1.4}
\setlength{\tabcolsep}{.4em}
\begin{table}[ht]
\footnotesize
\centering
\begin{tabular}[t]{|l|c|cccccccccc|}
\hline
& $\MISE[\fhY^{(RP)}]$ & \multicolumn{10}{c|}{$\MISE[\fhY^{(MR)}]$} \\
& & $\tau=0$ & $\tau=2$ & $\tau=4$ & $\tau=8$ & $\tau=16$ & $\tau=32$ & $\tau=64$ & $\tau=128$ & $\tau=256$ & $\tau=512$ \\
\hline
$N=50$ & $3.49\text{e-}02$ & $3.57\text{e-}02$ & $1.38\text{e-}02$ & $9.50\text{e-}03$ & $5.85\text{e-}03$ & $4.02\text{e-}03$ & $3.02\text{e-}03$ & $2.28\text{e-}03$ & $1.87\text{e-}03$ & $1.96\text{e-}03$ & $1.80\text{e-}03$ \\
$N=100$ & $2.04\text{e-}02$ & $1.98\text{e-}02$ & $7.33\text{e-}03$ & $4.76\text{e-}03$ & $3.06\text{e-}03$ & $2.04\text{e-}03$ & $1.44\text{e-}03$ & $1.21\text{e-}03$ & $9.01\text{e-}04$ & $9.27\text{e-}04$ & $8.71\text{e-}04$ \\
$N=200$ & $1.16\text{e-}02$ & $9.61\text{e-}03$ & $3.82\text{e-}03$ & $2.44\text{e-}03$ & $1.62\text{e-}03$ & $1.01\text{e-}03$ & $7.84\text{e-}04$ & $6.03\text{e-}04$ & $5.17\text{e-}04$ & $4.69\text{e-}04$ & $4.36\text{e-}04$ \\
\hline
\end{tabular}
\caption{\label{tab:skew} MISE results for the Rozenblatt-Parzen estimator and the multiple regression-enhanced convolution estimator for the density of the response variable in the regression model \eqref{eq:mult-reg-skewed}.}
\end{table}
\subsection{Response variable with multimodal distribution}
Next, we consider the case of a response variable with a multimodal distribution,
\begin{align} \label{eq:mult-reg-multimodal}
Y = \alpha_0 + \alpha_1 X + \eps[],
\end{align}
where $(\alpha_0,\alpha_1) = (4,1.5)$, $\eps[] \sim \mathcal{N}(0,4)$, and the density $\fX$ of $X$ is given by
\begin{align*}
\fX(y) = \sum_{i=1}^4 w_i \Psi_i(y;\mu_i,\sigma_i),
\end{align*}
with $(w_1,w_2,w_3,w_4) = (0.2,0.2,0.4,0.2)$, $(\mu_1,\mu_2,\mu_3,\mu_4) = (-4,4,14,21)$, $(\sigma_1,\sigma_2,\sigma_3,\sigma_4) = (3,2,2,2)$. In Figure \ref{fig:typ-real-multimodal}, we plot $\fY$ along with typical realizations of density estimates given by $\fhY^{(RP)}$ and $\fhY^{(MR)}$, for a complete case dataset of size $N=100$, while the ratio of additional covariate observations supplied to the convolution estimator progressively increases, $\tau \in \{0,4,16,64\}$. The Rozenblatt-Parzen $\fhY^{(RP)}$ completely fails to resolve two of the modes of this distribution and severely underestimates the largest mode. At $\tau=0$, the convolution estimator $\fhY^{(MR)}$ manages to pick out three modes, although the magnitudes of these modes and the general shape of the density is not so accurate. However, as additional covariate observations are incorporated, the convolution estimator provides an increasingly accurate representation of the true density.
In Table \ref{tab:mult-reg-multimodal}, we report MISE results for both $\fhY^{(RP)}$ and $\fhY^{(MR)}$. Three complete cases samples sizes are considered, $N \in \{50,100,200\}$. Once again a very substantial substantial reduction in MISE is observed, with $\MISE[\fhY^{(MR)}]$ about $35$ times smaller than $\MISE[\fhY^{(RP)}]$ at $\tau = 128$ and $N=200$.
\begin{figure}[h]
\centering
\captionsetup[subfloat]{labelformat=empty}
\begin{tabular}{ccc}
\subfloat[(i) $\tau=0$]{\includegraphics[scale=0.6]{Multimodal_Dist_0-crop}} \hspace{2em}
\subfloat[(ii) $\tau=4$]{\includegraphics[scale=0.6]{Multimodal_Dist_4-crop}} \\
\subfloat[(iii) $\tau=16$]{\includegraphics[scale=0.6]{Multimodal_Dist_16-crop}} \hspace{2em}
\subfloat[(iv) $\tau=64$]{\includegraphics[scale=0.6]{Multimodal_Dist_64-crop}}
\end{tabular}
\caption{Typical realizations of estimates for the density of the response variable $Y$ in \eqref{eq:mult-reg-multimodal} given by the Rosenblatt–Parzen density estimator $\fhY^{(RP)}$ (dotted line), and the multiple regression-enhanced convolution estimator $\fhY^{(MR)}$ (dashed line). The black solid line is the true density $\fY$. \label{fig:typ-real-multimodal}}
\end{figure}
\renewcommand{\arraystretch}{1.4}
\setlength{\tabcolsep}{.4em}
\begin{table}[ht]
\footnotesize
\centering
\begin{tabular}[t]{|l|c|cccccccccc|}
\hline
& $\MISE[\fhY^{(RP)}]$ & \multicolumn{10}{c|}{$\MISE[\fhY^{(MR)}]$} \\
& & $\tau=0$ & $\tau=2$ & $\tau=4$ & $\tau=8$ & $\tau=16$ & $\tau=32$ & $\tau=64$ & $\tau=128$ & $\tau=256$ & $\tau=512$ \\
\hline
$N=50$ & $2.43\text{e-}03$ & $1.89\text{e-}03$ & $8.08\text{e-}04$ & $5.35\text{e-}04$ & $3.28\text{e-}04$ & $2.24\text{e-}04$ & $1.59\text{e-}04$ & $1.22\text{e-}04$ & $1.03\text{e-}04$ & $8.76\text{e-}05$ & $8.83\text{e-}05$ \\
$N=100$ & $1.54\text{e-}03$ & $1.05\text{e-}03$ & $4.19\text{e-}04$ & $2.64\text{e-}04$ & $1.71\text{e-}04$ & $1.09\text{e-}04$ & $7.34\text{e-}05$ & $6.06\text{e-}05$ & $5.23\text{e-}05$ & $4.94\text{e-}05$ & $4.22\text{e-}05$ \\
$N=200$ & $9.15\text{e-}04$ & $5.90\text{e-}04$ & $2.17\text{e-}04$ & $1.35\text{e-}04$ & $8.45\text{e-}05$ & $5.57\text{e-}05$ & $3.92\text{e-}05$ & $2.98\text{e-}05$ & $2.70\text{e-}05$ & $2.29\text{e-}05$ & $2.37\text{e-}05$ \\
\hline
\end{tabular}
\caption{MISE results for the Rozenblatt-Parzen estimator and the multiple regression-enhanced convolution estimator for the density of the response variable in the regression model \eqref{eq:mult-reg-corr-cov}. \label{tab:mult-reg-multimodal}}
\end{table}
\subsection{Multiple regression with correlated covariates and non-Gaussian error}
Next we consider a multiple regression model where the covariates are correlated and the error is non-Gaussian,
\begin{align} \label{eq:mult-reg-corr-cov}
Y = \alpha_0 + \alpha_1 X_1 + \alpha_2 X_2 + \alpha_3 X_3 + \eps[],
\end{align}
where $(\alpha_0,\alpha_1,\alpha_2,\alpha_3) = (1,1,2,0.5)$, the covariates are distributed as
\begin{align*}
X_1 \sim \beta(2,5), \quad \quad
X_2 \sim \mathcal{N}(6,4), \quad \quad X_3 \sim t_{6}, \quad \quad
\eps[] & \sim \text{Skew-Normal}(\xi,\omega,\alpha),
\end{align*}
and the correlation matrix for the covariates is defined as
\begin{align*}
\Corr(X)
&=
\begin{bmatrix}
1 & 0.2 & 0.5, \\
0.2 & 1 & 0.3, \\
0.5 & 0.3 & 1
\end{bmatrix}.
\end{align*}
The parameters for the skew normal error distribution are set to $(\xi,\omega,\alpha) = (-\omega \sqrt{2 \alpha^2/((1+\alpha^2)\pi)},1,3)$, where $\xi$ has been chosen to ensure that the error has mean zero. In Figure \ref{fig:typ-real-corr-cov}, we plot $\fY$ along with typical realizations of density estimates given by $\fhY^{(RP)}$ and $\fhY^{(MR)}$.
MISE results for both estimators are presented in Table \ref{tab:corr-cov-non-neg-error}. This time $\MISE[\fhY^{(MR)}]$ is about $32$ times smaller than $\MISE[\fhY^{(RP)}]$ at $\tau = 128$ and $N=200$.
\begin{figure}[ht!]
\centering
\captionsetup[subfloat]{labelformat=empty}
\begin{tabular}{ccc}
\subfloat[(i) $\tau=0$]{\includegraphics[scale=0.6]{CorrCov_Dist_0-crop}} \hspace{2em}
\subfloat[(ii) $\tau=4$]{\includegraphics[scale=0.6]{CorrCov_Dist_4-crop}} \\
\subfloat[(iii) $\tau=16$]{\includegraphics[scale=0.6]{CorrCov_Dist_16-crop}} \hspace{2em}
\subfloat[(iv) $\tau=64$]{\includegraphics[scale=0.6]{CorrCov_Dist_64-crop}}
\end{tabular}
\caption{Typical realizations of estimates for the density of the response variable $Y$ in \eqref{eq:mult-reg-corr-cov} given by the Rosenblatt–Parzen density estimator $\fhY^{(RP)}$ (dotted line), and the multiple regression-enhanced convolution estimator $\fhY^{(MR)}$ (dashed line). The black solid line is the true density $\fY$. \label{fig:typ-real-corr-cov}}
\end{figure}
\renewcommand{\arraystretch}{1.4}
\setlength{\tabcolsep}{.4em}
\begin{table}[htb!]
\footnotesize
\centering
\begin{tabular}[t]{|l|c|cccccccccc|}
\hline
& $\MISE[\fhY^{(RP)}]$ & \multicolumn{10}{c|}{$\MISE[\fhY^{(MR)}]$} \\
& & $\tau=0$ & $\tau=2$ & $\tau=4$ & $\tau=8$ & $\tau=16$ & $\tau=32$ & $\tau=64$ & $\tau=128$ & $\tau=256$ & $\tau=512$ \\
\hline
$N=50$ & $3.66\text{e-}03$ & $4.94\text{e-}03$ & $1.87\text{e-}03$ & $1.24\text{e-}03$ & $7.73\text{e-}04$ & $4.51\text{e-}04$ & $2.81\text{e-}04$ & $1.92\text{e-}04$ & $1.39\text{e-}04$ & $1.27\text{e-}04$ & $1.08\text{e-}04$ \\
$N=100$ & $2.04\text{e-}03$ & $2.83\text{e-}03$ & $1.08\text{e-}03$ & $6.72\text{e-}04$ & $3.92\text{e-}04$ & $2.25\text{e-}04$ & $1.41\text{e-}04$ & $9.56\text{e-}05$ & $7.04\text{e-}05$ & $5.55\text{e-}05$ & $5.45\text{e-}05$ \\
$N=200$ & $1.15\text{e-}03$ & $1.51\text{e-}03$ & $5.81\text{e-}04$ & $3.58\text{e-}04$ & $2.02\text{e-}04$ & $1.17\text{e-}04$ & $7.16\text{e-}05$ & $4.83\text{e-}05$ & $3.58\text{e-}05$ & $3.01\text{e-}05$ & $2.64\text{e-}05$ \\
\hline
\end{tabular}
\caption{MISE results for the Rozenblatt-Parzen estimator and the multiple regression-enhanced convolution estimator for the density of the response variable in the regression model \eqref{eq:mult-reg-corr-cov}. \label{tab:corr-cov-non-neg-error}}
\end{table}
\section{Concluding remarks} \label{sec:concluding-remarks}
In this work, we have proposed a convolution estimator for enhancing the accuracy of estimates of the density of a response variable in a sample of $N$ complete case observations, by using an additional sample of $M$ covariate observations. While previous works on convolution estimators have modelled the relationship between the response variable and the covariates using nonlinear regression models, in this paper a multiple regression model was employed. Unlike Nadaraya-Watson-based convolution estimators that suffer from the curse of dimensionality, we showed that the convergence of the multiple regression-enhanced convolution estimator is independent of the dimensionality of the covariates, which is due to the fact that the convergence of the underlying OLS estimator is also dimension independent.
The case of $M$ additional covariate observations is a generalization of the usual convolution estimator setting considered in the literature. The usual setting involves estimating the density of a response variable using a sample of $N$ complete case observations of a response variable and an associated set of covariates. By setting $M = 0$, we recover this case. Indeed, we showed that $\MSE[\fhY(y)] = O(h^4) + O((hNL)^{-1}) + O(L^{-1}) + O(N^{-1})$ reduces to $\MSE[\fhY(y)] = O(h^4) + O((hN^2)^{-1}) + O(N^{-1})$ when $M=0$, which recovers previous results in the literature \cite[Eq. (23)]{stove2012convolution}and \cite[Eq. (3.4)]{escanciano2012n}.
By deriving the asymptotic MSE and the asymptotically optimal bandwidth, we resolved the question on the convergence of convolution estimators with respect to the size of the additional sample of $M$ covariate observations. That is, we proved that for a large fixed $N$, at the asymptotically optimal bandwidth, the MSE converges as $O(M^{-4/5})$ towards an $O(N^{-1})$ constant. We also showed that for a fixed $M$, the MSE converges as $O(N^{-1})$. This means that supplying the convolution estimator with additional covariate observations is not quite as effective as supplying it with more complete cases observations. Crucially, however, in many practical applications it can be difficult if not downright impossible to obtain more observations of a response variable, while at the same time it can be very straightforward to obtain more observations of the covariates.
Numerical simulations confirmed the existence of the saturation phenomena predicted by the theory, whereby the MISE converges as $O(M^{-4/5})$ towards an $O(N^{-1})$ constant, as opposed to zero, as the size of the additional sample increases, whereas it converges as $O(N^{-1})$ towards zero as the size of the complete case sample increases. Moreover numerical simulations demonstrated that even if the MISE of the multiple regression-enhanced convolution estimator is greater than that of the Rosenblatt-Parzen density estimator on the complete case dataset, by supplying the convolution estimator with additional covariate observations, its MISE can be made about $20$ to $35$ times smaller than the MISE of the Rosenblatt-Parzen before the accuracy improvement saturates.
The evaluation of the multiple regression-enhanced convolution estimator is an order of magnitude more computationally expensive than the evaluation of the Rosenblatt–Parzen density estimator. To reduce computational costs, we developed a FGT-based acceleration algorithm that draws on the high performance C++ library FIGTree. Simulations showed that this algorithm dramatically outperformed a variety of alternative evaluation approaches. In particular, it was demonstrated to be up to $40$ times faster than FFT-based acceleration, with the reduction in computational times becoming even more pronounced as $M$ increases.
In terms of future research directions, some interesting topics include heteroscedasticity, segmented regression-enhanced convolution estimators, and transfer learning for cases when the $N$ complete case observations and $M$ additional covariate observations have different distributions. Convolution estimators in the presence of heteroscedasticity have been considered in works such as \cite{stove2012convolution} and \cite{li2016n}. As we have demonstrated in this paper, however, to obtain fast convergence rates of $O(N^{-1})$ and $O(M^{-4/5})$ that are independent of the covariate dimensionality, the underlying regression function estimator has to be immune to the curse of dimensionality. Thus, we anticipate the method chosen to model heteroscedastic errors also needs to be immune to the curse of dimensionality if one wants to main these fast convergence rates.
In cases where the data is not well fit by a multiple regression model, it might still be possible to fit it with a piecewise multiple regression model, as opposed to employing a fully nonlinear regression model. For example, in situations where the data is well fit by a multiple regression model in several segments, each with a different, albeit constant, error variance, a multidimensional segmented regression model could be employed \cite{diakonikolas2020efficient,liu1997segmented}. Since segmented regression involves partitioning the data into several segments, each with its own dedicated OLS estimator, we conjecture that a segmented regression-enhanced convolution estimator could retain the fast dimension independent convergence rates of the multiple regression-enhanced convolution estimator. Of course, segmented regression itself can be subject to the curse of dimensionality if too many segments are used, but this can be avoided by placing an upper bound on the number of partitions \cite{liu1997segmented}.
Transfer learning is another very interesting avenue for future research with regards to convolution estimators. To the best of our knowledge this topic is completely unexplored. Transfer learning has been demonstrated to be very effective at utilizing labelled information from a source domain to enhance the performance of a model in a separate target domain with little or no labelled data \cite{day2017survey,zhuang2020comprehensive,pan2009survey}. Our work in this paper was concerned with the case where the $N$ complete case observations and the $M$ additional covariate observations came from the same distribution. There are many practical applications in which the complete case observations and additional covariate observations could come from different, yet closely, related distributions. We expect that incorporating transfer learning capabilities into convolution estimators could provide a significant improvement in accuracy in situations such as these.
\section{The Elsevier article class}
\section*{}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,388 |
\section{Introduction}
Let $a_1,a_2,\ldots,a_n$ denote $n$ positive numbers. Let $A_n$ be their arithmetic mean, $\frac{\sum_ka_k}{n}$, and $G_n$ be their geometric mean, $\sqrt[n]{\Pi_ka_k}$. We shall prove the following inequalities.
\begin{equation}\label{eq:xia1}
\frac{G_n}{A_n} \leq (\frac{n-\sum_{k=1}^mr_k}{n-m})^{1-\frac{m}{n}}(\Pi_{k=1}^mr_k)^{\frac{1}{n}}\leq 1 \: .
\end{equation}
if we know $a_k=A_nr_k$ ($1\leq k\leq m\leq n$) for instance.
\begin{equation}\label{eq:xia2}
\frac{G_n}{A_n} \leq \frac{1}{(1-\frac{m}{n})\Pi_{k=1}^mr_k^{\frac{-1}{n-m}}+\frac{1}{n}\sum_{k=1}^mr_k} \leq 1
\end{equation}
if we know $a_k=G_nr_k$ ($1\leq k\leq m \leq n$) for instance.
S. H. Tung~\cite{Tung} obtained the following lower bound for $A_n-G_n$ in terms of the smallest value, $a$, and the largest value, $A$.
\begin{equation}\label{eq:Tung0}
A_n-G_n \geq \frac{(\sqrt{A}-\sqrt{a})^2}{n} \: .
\end{equation}
We will see that our results are better than this bound.
\section{Proof of the inequalities}
Suppose we know the first $m$ numbers, $a_1,a_2,\ldots,a_m$. We can construct the following inequality.
\begin{eqnarray}
&& \sum_{k=1}^m\lambda_k^{n-m}a_k + \frac{1}{\Pi_{i=1}^m\lambda_i}\sum_{k=m+1}^n a_k \\
= && \sum_{k=1}^m(\lambda_k^{n-m}-\frac{1}{\Pi_{i=1}^m\lambda_i})a_k + \frac{1}{\Pi_{i=1}^m\lambda_i}\sum_{k=1}^n a_k \label{eq:step2}\\
\geq && n\sqrt[n]{\Pi_ka_k}
\end{eqnarray}
Suppose we know $a_k=A_nr_k$ ($k=1,2,\ldots,m$). Inserting them into Eq.~\ref{eq:step2}, we obtain an upper bound of $G_n/A_n$ as a function of $\lambda_1,\lambda_2,\ldots,\lambda_m$.
\begin{equation}
\frac{G_n}{A_n} \leq \frac{1}{n}\sum_{k=1}^m\lambda_k^{n-m}r_k+\frac{1}{\Pi_{k=1}^m\lambda_k}\frac{n-\sum_{k=1}^mr_k}{n} \equiv f(\lambda_1,\lambda_2,\ldots,\lambda_m) \: .
\end{equation}
$\frac{\partial f}{\partial \lambda_i}=0$ ($i=1,2,\ldots,m$) gives the best choice, namely,
\begin{eqnarray}
&& \lambda_i = \left(\frac{n-\sum_{k=1}^mr_k}{n-m}\Pi_{k=1}^mr_k^{\frac{1}{n-m}}\right)^{\frac{1}{n}}r_i^{\frac{-1}{n-m}} \: , \: (i=1,2,\ldots,m)
\end{eqnarray}
and hence the bound in Ineq.~\ref{eq:xia1}.
Similarly suppose we know $a_k=G_nr_k$ ($k=1,2,\ldots,m$). Inserting them into Eq.~\ref{eq:step2}, we obtain an upper bound of $G_n/A_n$ as a function of $\lambda_1,\lambda_2,\ldots,\lambda_m$.
\begin{equation}
\frac{G_n}{A_n} \leq \frac{1}{\Pi_{k=1}^m\lambda_k(1-\frac{1}{n}\sum_{k=1}^mr_k\lambda_k^{n-m})+\frac{1}{n}\sum_{k=1}^mr_k} \equiv g(\lambda_1,\lambda_2,\ldots,\lambda_m) \: .
\end{equation}
$\frac{\partial f}{\partial \lambda_i}=0$ ($i=1,2,\ldots,m$) gives the best choice, namely,
\begin{eqnarray}
&& \lambda_i = r_i^{\frac{-1}{n-m}} \: , \: (i=1,2,\ldots,m)
\end{eqnarray}
and hence the bound in Ineq.~\ref{eq:xia2}.
For comparison, Tung's inequality can be written into the following form,
\begin{equation}
\frac{G_n}{A_n} \leq 1 - \frac{1}{n}(\sqrt{r_1}-\sqrt{r_2})^2 \:, \label{eq:tung1}
\end{equation}
if we know $A=A_nr_1$ and $a=A_nr_2$ with $0<r_2\leq 1\leq r_1 \leq n-r_2$, or
\begin{equation}
\frac{G_n}{A_n} \leq \frac{1}{1+\frac{1}{n}(\sqrt{r_1}-\sqrt{r_2})^2} \:, \label{eq:tung2}
\end{equation}
if we know $A=G_nr_1$ and $a=G_nr_2$ with $0<r_2\leq 1\leq r_1$. We can show that our results are better than these bounds using simple calculus. For illustration, different upper bounds of $G_n/A_n$ as a function of $r_2$ with $r_1=5$ and $n=10$ are compared in Fig.~\ref{fig:comparison}.
\begin{figure}
\includegraphics[width=0.45\textwidth]{Cs1.pdf}
\includegraphics[width=0.45\textwidth]{Cs2.pdf}
\caption{\label{fig:comparison}
Comparison of different upper bounds of $G_n/A_n$ as a function of $r_2$ for $r_1=5$ and $n=5$. The left diagram correponds to the case of $A=A_nr_1$ and $a=A_nr_2$ while the right diagram corresponds to the case of $A=G_nr_1$ and $a=G_nr_2$. The results derived from Ref.~\cite{Tung} are represented by the red-dashed curves (namely, Ineq.~\ref{eq:tung1} and ~\ref{eq:tung2}. The black curves represent our results
(namely, Ineq.~\ref{eq:xia1} and ~\ref{eq:xia2}).
}
\end{figure}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 5,700 |
Die Rhakotis der Hamburg-Amerikanischen Packetfahrt-Actien-Gesellschaft (Hapag) war ein 1928 von der Deutschen Werft in Hamburg als San Francisco gebautes Kombischiff mit Dieselantrieb. Es war das erste neue Schiff für den gemischten Passagier- und Frachtdienst zur US-amerikanischen Pazifikküste. 1935 wurde sie in Rhakotis umbenannt, als es vor allem zur Westküste Südamerikas im Einsatz kam.
Bei Kriegsbeginn befand sich das Schiff in Callao und wurde dort aufgelegt. 1940 wurde die Rhakotis nach Japan überführt.
Am 1. Januar 1943 scheiterte der Versuch der Rhakotis, als Blockadebrecher das besetzte Frankreich zu erreichen.
Geschichte
Seit 1923 betrieb die Hapag zusammen mit der inzwischen mit der DADG fusionierten DDG Kosmos und den zur Harriman-Gruppe gehörenden United American Lines einen Gemeinschaftsdienst zur Pazifikküste der USA und Kanadas. Als erstes Schiff setzte die Hapag den Frachter Sachsen ein, dem bald sein Schwesterschiff Hessen folgte. Im Sommer 1926 wurden auch diese beiden Schiffe abgezogen, so dass die Linie eine reine Frachtlinie wurde, auf der auch Frachter der Bremer Roland-Linie liefen. Die Trennung von Harriman im Sommer 1926 und die Fusion der Hapag mit Austral-Kosmos-Stinnes und die Eingliederung der Roland-Linie in den Norddeutschen Lloyd führten zu einer Auflösung der bisherigen Gemeinschaftsdienste und zu einer Konkurrenz der deutschen Großreedereien auf den Westküstenlinien.
Um ihre Position auszubauen, bestellte die Hapag bei der ihr nahestehenden Deutschen Werft in Hamburg-Finkenwärder zwei Kombischiffe mit Dieselantrieb für bis zu 48 Fahrgäste in zwei Klassen, 9000 t Tragfähigkeit und einer Dienstgeschwindigkeit von 13 Knoten. Bei der Planung waren Helgoland und Westerland als Namen vorgesehen, tatsächlich kamen sie aber als San Francisco und Los Angeles zu Wasser. Noch während des Baus erteilte die Hapag weitere Aufträge für zwei etwas größere Schiffe an die Deutsche Werft (Seattle) und den Bremer Vulkan (Portland). Zum Abschluss der Serie lieferte die Deutsche Werft mit der wieder etwas kleineren Oakland noch ein fünftes Schiff.
Die San Francisco eröffnete den Dienst am 10. März 1928 und bis Juli waren die vier ersten Schiffe im Einsatz. Zur Verstärkung und Sicherstellung der dreiwöchentlichen Abfahren mussten allerdings auch andere Schiffe herangezogen werden. So wurden die Motorschiffe Ramses (7983 BRT, 13 kn, bis 33 Fahrgäste, DDG Kosmos), Duisburg dann umbenannt Heidelberg (7389 BRT, 13,5 kn, bis 37 Fahrgäste, DADG), Münsterland (6408 BRT, 12 kn, bis 18 Fahrgäste, Ostasiendienst Hapag) und das ehemalige Stinnesschiff Emil Kirdorf (5695 BRT, 12 kn, bis 74 Fahrgäste) auf der Route eingesetzt. Auch die Frachter Sachsen und Hessen kehrten für zwei Jahre auf die USA-Westküstenroute zurück.
1930 erhielt der Dienst eine weitere Verstärkung durch die 8300 BRT großen, 14,5 kn schnellen Turbinenschiffe Tacoma und Vancouver, die wieder von der Deutschen Werft geliefert wurden. Einen ernsthaften Angriff auf die Position der Hapag hatte es nicht gegeben, aber die politische Lage führte zu einem rückläufigen Verkehr mit den USA und Kanada. Dazu hatte die staatliche Regulierung innerdeutsche Konkurrenzkämpfe unmöglich gemacht. Um dem politisch gewollten und erfolgversprechenden südamerikanischen Westküstendienst aufzuwerten, setzte die Hapag 1935 die Schwesterschiffe San Francisco und Los Angeles auf diese Route um. Wie schon die Frachtschiffe Spreewald und Odenwald erhielten auch die beiden Kombischiffe traditionelle DDG-Kosmos-Namen. Aus der San Francisco wurde jetzt die Rhakotis. Den Namen des Stadtteils für die Ägypter im alten Alexandria hatte schon 1907 ein 6982 BRT großes, von Blohm & Voss geliefertes Kombischiff der DDG Kosmos geführt.
Eine Fahrt der Rhakotis im Frieden war ihr Einsatz zum Ende der Irrfahrt der St. Louis. Als das unglückliche Schiff am 19. Juni 1939 wieder in Antwerpen einlief, übernahm die Rhakotis die 511 Juden des Schiffes, die Frankreich und Großbritannien aufnehmen wollten. Die Rhakotis wurde dafür vorbereitet, um der St. Louis die Rückfahrt in die USA und die Durchführung einer devisenbringenden Vergnügungsreise zu ermöglichen. Sie brachte die Asylsuchenden am 20. nach Boulogne-sur-Mer und am 21. nach Southampton.
Kriegsschicksal der Rhakotis
Bei Kriegsbeginn 1939 befand sich das Schiff im peruanischen Hafen Callao, der vom Deutschen Reich als geeigneter Versorgungspunkt angesehen wurde. Neben der Rhakotis sammelten sich dort auch die Kombischiffe Leipzig (5898 BRT, 1938) und München (5619 BRT, 1936) des NDL und die Hermonthis (4833 BRT, 1935) und Monserrate (5578 BRT, 1938) der Hapag, die dort 1941 verloren gingen. Die Entscheidung, Callao in Peru für einen sicheren Stützpunkt zu halten, erwies sich als falsch, denn der 1939 neugewählte Präsident Prado wandte sich zunehmend den westlichen Alliierten zu. Die Rhakotis entkam dem Wechsel der politischen Situation, als sie am 16. Mai 1940 erst nach Antofagasta lief und von dort weiter nach Japan, wo sie am 29. Juni in Yokohama eintraf.
1942 wurde die Rhakotis als Blockadebrecher nach Europa eingesetzt und verließ am 27. September Yokohama, um ab dem 15. Oktober in Singapur beladen zu werden. Sie lud Rohkautschuk, Zinn, Zink, Fette, Reis, Tee, Chinarinde, Kokosöl und Perlen. Sie verlegte dann nach Batavia, von wo sie am 5. November endgültig nach Europa auslief. Am 18. November traf sie den Hilfskreuzer Michel im Indischen Ozean. Um das Kap der Guten Hoffnung lief sie in den Atlantik, wo sie am 12. Dezember ein Rettungsboot mit drei Mann entdeckte, die 36 Tage in ihrem Boot verbracht hatten. Einer der drei starb nach der Bergung. Die beiden Überlebenden der City of Cairo überlebten auch die Selbstversenkung der Rhakotis, als diese kurz vor ihrem Ziel von der britischen Luftaufklärung erfasst wurde, die den Leichten Kreuzer HMS Scylla an den Blockadebrecher heranführte. Am 1. Januar 1943 versenkte die eigene Besatzung die Rhakotis bei Annäherung des Kreuzers circa 200 Seemeilen nordwestlich von Kap Finisterre auf . Die Besatzung rettete sich in vier Rettungsboote. Von den zur Suche eingesetzten U-Booten fand U 410 am 2. Januar zwei der Boote und brachte 80 Mann einschließlich der beiden geretteten Engländer am folgenden Tag nach St. Nazaire. Die beiden anderen Boote erreichten die spanische Küste.
Schicksal der Hapag-Kombischiffe des Nordamerika-Westküstendienst
Literatur
E. Goos, E. Gräber: Die Motorschiffe "San Francisco" und "Los Angeles". In: Zeitschrift des Vereines deutscher Ingenieure, 72. Jahrgang, Nr. 41 (13. Oktober 1928), S. 1450–1456.
R. Hughes: Flagship to Murmansk, Futura Publications (1975) pp. 108-12 (Hughes war Offizier an Bord HMS Scylla).
Arnold Kludas: Die Geschichte der deutschen Passagierschiffahrt Bd. IV Vernichtung und Wiedergeburt 1914 bis 1930, Schriften des Deutschen Schiffahrtsmuseum, Band 21
Arnold Kludas: Die Geschichte der deutschen Passagierschiffahrt Bd. V Eine Ära geht zu Ende 1930 bis 1990, Schriften des Deutschen Schiffahrtsmuseum, Band 22
Weblinks
Blockadebrecher
Untergang der Rhakotis
Seekrieg "1.1.– 7.2.1943 Biskaya"
Einzelnachweise
Schiff (Hamburg-Amerikanische Packetfahrt-Actien-Gesellschaft)
Motorschiff
Passagierschiff (Deutschland)
Frachtschiff (Deutschland)
Deutsche Werft
Schiffsverlust im Zweiten Weltkrieg
Schiffsverlust 1943 | {
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Jeziorowy ogórek kiszony – regionalny produkt spożywczy, charakterystyczny dla gminy Kalisz Pomorski (powiat drawski). 7 maja 2008 wpisany na polską listę produktów tradycyjnych (zgłaszającym był Miejsko-Gminny Ośrodek Kultury w Kaliszu Pomorskim).
Historia
Receptura wywodzi się z lat 60. XX wieku, kiedy to oddział Rejonowej Spółdzielni Ogrodniczo-Pszczelarskiej w Kaliszu Pomorskim rozpoczął produkcję przetworów owocowo-warzywnych (głównie ogórków kiszonych) na potrzeby Ludowego Wojska Polskiego oraz stacjonujących licznie na Pomorzu Zachodnim jednostek Armii Radzieckiej. Odbiorcą produktów spółdzielni były też zakłady karne, a niewielkie nadwyżki wysyłano na eksport do Niemiec. Ogórki zakiszano w stulitrowych beczkach, które umieszczano w wodach jeziora Młyńskiego. Zwyczaj ten zaczerpnięto z głównej siedziby spółdzielni w Szczecinku, gdzie zapoczątkowali go pracownicy wywodzący się z Podlasia, głównie z Białej Podlaskiej. W maju 1984 kaliski oddział spółdzielni uległ likwidacji, a część beczek pozostała wówczas pod wodą i zdarzają się przypadki wypływania niektórych z nich na powierzchnię. Tradycje wyrobu produktu wznowiono w 2005 w ramach akcji promowania miasta.
Corocznie w lipcu organizowane jest w Kaliszu Pomorskim Święto Ogórka, mające na celu popularyzację tego warzywa i regionu.
Charakterystyka
Ogórki są podłużne i cylindryczne (wymiary: długość 8-15 cm, średnica od 3-5 cm), mają gładką powierzchnię z niewielkimi brodawkami. Barwa zewnętrzna jest oliwkowozielona, a na przekroju nieco jaśniejsza. Ogórki winny być twarde i pełne w środku, a smak charakterystyczny dla ogórka kiszonego z wyczuwalnym zapachem użytych przypraw – kopru, czosnku i chrzanu. Ogórki w beczkach układane są tak wysoko, aby dekiel po zabiciu przyciskał je, a zalana woda sięgała poziomu miejsca, w którym będzie się on znajdował. Istotna jest także stała temperatura przechowywania produktu w wodach naturalnego akwenu, czyli 3-4°C. Beczki pozostawać muszą pod wodą minimum przez okres trzech miesięcy.
Zobacz też
ogórek kołobrzeski
ogórek konserwowy
Przypisy
Kuchnia pomorska
Polskie produkty tradycyjne
Kalisz Pomorski (gmina)
Przetwory z warzyw
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"redpajama_set_name": "RedPajamaWikipedia"
} | 1,576 |
{"url":"https:\/\/electrical.codidact.com\/categories\/26\/tags","text":"Q&A\n\nAnalog \u00d7\u00a03\nrelay \u00d7\u00a02\nEMC \u00d7\u00a02\nLC-network \u00d7\u00a02\ndatasheet \u00d7\u00a02\noutput \u00d7\u00a02\nop-amp \u00d7\u00a02\npower-supply \u00d7\u00a02\nerrors \u00d7\u00a01\nsensor \u00d7\u00a01\nH-Bridge \u00d7\u00a01\nMOSFET \u00d7\u00a01\ndiode \u00d7\u00a01\nthyristor \u00d7\u00a01\ntriac \u00d7\u00a01\ndiac \u00d7\u00a01\nhigh-voltage \u00d7\u00a01\nsafety \u00d7\u00a01\nbattery \u00d7\u00a01\nsplit-phase \u00d7\u00a01\ntransformer \u00d7\u00a01\nbaudrate \u00d7\u00a01\ncomponent \u00d7\u00a01\nmodule \u00d7\u00a01\nreliability \u00d7\u00a01\nquality \u00d7\u00a01\nbode-plot \u00d7\u00a01\nrlc-filter \u00d7\u00a01\ncan-bus \u00d7\u00a01\ntermination \u00d7\u00a01\nPi-Filter \u00d7\u00a01\nc \u00d7\u00a01\ncapacitor \u00d7\u00a01\nlife-time \u00d7\u00a01","date":"2020-08-14 05:58:17","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\": 0.9273754358291626, \"perplexity\": 6858.441312073675}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-34\/segments\/1596439739177.25\/warc\/CC-MAIN-20200814040920-20200814070920-00288.warc.gz\"}"} | null | null |
#include "BenchmarkTimer.hpp"
#include "PerformanceReporter.hpp"
#include "ResultReporter.hpp"
#include "Reference.hpp"
#include <Tensile/hip/HipUtils.hpp>
#include <csignal>
#include <cstddef>
namespace Tensile
{
namespace Client
{
static_assert(BenchmarkTimer::clock::is_steady, "Clock must be steady.");
BenchmarkTimer::BenchmarkTimer(po::variables_map const& args, Hardware const& hardware)
: m_numWarmups(args["num-warmups"].as<int>())
, m_syncAfterWarmups(args["sync-after-warmups"].as<bool>())
, m_numBenchmarks(args["num-benchmarks"].as<int>())
, m_numEnqueuesPerSync(args["num-enqueues-per-sync"].as<int>())
, m_minFlopsPerSync(args["min-flops-per-sync"].as<size_t>())
, m_numSyncsPerBenchmark(args["num-syncs-per-benchmark"].as<int>())
, m_hardware(hardware)
, m_numEnqueuesPerSolution(m_numEnqueuesPerSync * m_numSyncsPerBenchmark)
, m_useGPUTimer(args["use-gpu-timer"].as<bool>())
, m_sleepPercent(args["sleep-percent"].as<int>())
, m_timeInSolution(0)
, m_totalGPUTime(0)
{
}
bool BenchmarkTimer::needMoreBenchmarkRuns() const
{
return m_numBenchmarksRun < m_numBenchmarks;
}
void BenchmarkTimer::preBenchmarkRun() {}
void BenchmarkTimer::postBenchmarkRun()
{
m_numBenchmarksRun++;
}
void BenchmarkTimer::preProblem(ContractionProblem const& problem)
{
m_problem = problem;
}
void BenchmarkTimer::postProblem() {}
void BenchmarkTimer::preSolution(ContractionSolution const& solution)
{
m_numEnqueuesInSolution = 0;
m_timeInSolution = double_millis::zero();
ContractionSolution::ProjectedPerformance pp
= solution.projectedPerformance(m_problem, m_hardware);
m_solution = solution;
m_reporter->report(ResultKey::Tile0Granularity, pp.granularities.tile0Granularity);
m_reporter->report(ResultKey::Tile1Granularity, pp.granularities.tile1Granularity);
m_reporter->report(ResultKey::CuGranularity, pp.granularities.cuGranularity);
m_reporter->report(ResultKey::WaveGranularity, pp.granularities.waveGranularity);
m_reporter->report(ResultKey::TotalGranularity, pp.granularities.totalGranularity);
m_reporter->report(ResultKey::NumCus, perf.CUs);
m_reporter->report(ResultKey::TilesPerCu, pp.granularities.tilesPerCu);
m_reporter->report(ResultKey::MemReadBytes, pp.staticModel.memReadBytes);
m_reporter->report(ResultKey::MemWriteBytes, pp.staticModel.memWriteBytesD);
}
void BenchmarkTimer::postSolution()
{
double timePerEnqueue_us
= double_micros(m_timeInSolution).count() / m_numEnqueuesInSolution;
ContractionSolution::ProjectedPerformance pp
= m_solution.projectedPerformance(m_problem, m_hardware);
double gflops = m_problem.flopCount() / (timePerEnqueue_us) / 1000.0;
int tiles = pp.granularities.tilesPerCu * perf.CUs;
int usedCus = std::min(tiles, perf.CUs);
double gflopsPerCu = gflops / usedCus;
m_reporter->report(ResultKey::TimeUS, timePerEnqueue_us);
m_reporter->report(ResultKey::SpeedGFlopsPerCu, gflopsPerCu);
m_reporter->report(ResultKey::SpeedGFlops, gflops);
m_timeInSolution = double_millis::zero();
m_numEnqueuesInSolution = 0;
}
bool BenchmarkTimer::needMoreRunsInSolution() const
{
return m_numEnqueuesInSolution < m_numEnqueuesPerSolution;
}
size_t BenchmarkTimer::numWarmupRuns()
{
return m_numWarmups;
}
void BenchmarkTimer::setNumWarmupRuns(size_t count)
{
if(count < m_numWarmups)
throw std::runtime_error(concatenate(
"Expected at least", m_numWarmups, " warmup runs, got ", count, "."));
}
void BenchmarkTimer::preWarmup() {}
void BenchmarkTimer::postWarmup() {}
void BenchmarkTimer::validateWarmups(std::shared_ptr<ContractionInputs> inputs,
TimingEvents const& startEvents,
TimingEvents const& stopEvents)
{
if(m_syncAfterWarmups && (stopEvents->size() > 0) && (stopEvents->back().size() > 0))
HIP_CHECK_EXC(hipEventSynchronize(stopEvents->back().back()));
}
size_t BenchmarkTimer::numSyncs()
{
return m_numSyncsPerBenchmark;
}
void BenchmarkTimer::setNumSyncs(size_t count)
{
m_numSyncsInBenchmark = count;
}
void BenchmarkTimer::preSyncs() {}
void BenchmarkTimer::postSyncs() {}
size_t BenchmarkTimer::numEnqueuesPerSync()
{
size_t enqueuesByFlops = 0;
if(m_minFlopsPerSync > 0)
{
size_t flopsInProblem = m_problem.flopCount();
enqueuesByFlops = CeilDivide(m_minFlopsPerSync, flopsInProblem);
}
return std::max<size_t>(m_numEnqueuesPerSync, enqueuesByFlops);
}
void BenchmarkTimer::setNumEnqueuesPerSync(size_t count)
{
m_curNumEnqueuesPerSync = count;
}
void BenchmarkTimer::preEnqueues()
{
if(!m_useGPUTimer)
{
// Synchronize before timer so warmup runs are not included in benchmark time
HIP_CHECK_EXC(hipDeviceSynchronize());
m_startTime = clock::now();
}
}
void BenchmarkTimer::postEnqueues(TimingEvents const& startEvents,
TimingEvents const& stopEvents)
{
if(!m_useGPUTimer)
{
HIP_CHECK_EXC(hipDeviceSynchronize());
m_endTime = clock::now();
}
}
void BenchmarkTimer::validateEnqueues(std::shared_ptr<ContractionInputs> inputs,
TimingEvents const& startEvents,
TimingEvents const& stopEvents)
{
double_millis totalTime(0.0);
if(m_useGPUTimer)
{
HIP_CHECK_EXC(hipEventSynchronize(stopEvents->back().back()));
for(size_t i = 0; i < startEvents->size(); i++)
{
float enqTime = 0.0f;
HIP_CHECK_EXC(hipEventElapsedTime(
&enqTime, startEvents->at(i).front(), stopEvents->at(i).back()));
totalTime += double_millis(enqTime);
}
}
else
{
totalTime = double_millis(m_endTime - m_startTime);
}
m_timeInSolution += totalTime;
m_totalGPUTime += totalTime;
m_numEnqueuesInSolution += startEvents->size();
if(m_sleepPercent > 0)
{
auto sleepTime = totalTime * (m_sleepPercent / 100.0);
std::this_thread::sleep_for(sleepTime);
}
}
void BenchmarkTimer::finalizeReport() {}
int BenchmarkTimer::error() const
{
return 0;
}
} // namespace Client
} // namespace Tensile
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,922 |
Delt invites 28, fraternity interest rises
Jenn Lepore
Increased numbers from this year's fraternity rush and the introduction of a new fraternity next year means that Greek life is on the rise.
Forty men rushed for each fraternity chapter this semester, according to Greek life adviser Courtney McKenna. That's "higher than usual," she said.
Delta Tau Delta invited 28 students to join their fraternity, according to McKenna. Tau Kappa Epsilon invited 27, and Sigma Phi Epsilon asked 15 to join.
"Spring rush always has the largest numbers," said Matt Hudak, vice president of recruitment for the Interfraternity Council and member of SigEp.Hudak said the numbers had to do with freshman recruits–more freshmen rush during their second semester.
Heightened interest will be met with a new fraternity, Pi Kappa Phi, next fall, according to McKenna.
A new sorority, Pi Beta Phi, will also be inaugurated next fall. Kappa Delta, another sorority, is expected to colonize in the fall of 2012.
This year's number of Greek lifers is upward of 800, Hudak said, a large increase from just four years ago.
"My freshman year, seven percent of the campus was Greek," he said. "As of last year, 14 percent of the campus was Greek."
Quinnipiac vice president and chief of staff supports Hamden's new police chief
Quinnipiac community responds to new law requiring high schools to offer African American, Black, Latino and Puerto Rican studies
Quinnipiac gears its website to community members
Turning Point USA trying to influence SGA elections
A new class takes stage
A coalition of progressive groups — and some Quinnipiac students — call for Biden to cancel student debt
A challenging semester for professors | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,219 |
The History of Bad Ideas – Episode 269: Nepotism with Fighting!
Emanating from their studio in Cincinnati, Ohio, The History of Bad Ideas sees hosts Jason, Jeff and Blake talk about all things geeky on their podcast. Whether it's rumours of the latest comic book movies, debating who really is the worst villain of all time, discussing the latest comic issues or just wondering about life […]
Tags : HOBI, Hulk Hogan, Oscars, Podcast, The History of Bad Ideas, Umbrella Academy
'Ultimate Warrior: Always Believe' Blu-ray Review
by Chris Cummings
Whether you're a wrestling fan or not, it's likely you have heard of a few names from the world of professional wrestling. Hulk Hogan. The Rock. "Stone Cold" Steve Austin. Oh, and The Ultimate Warrior. A household name, and a massive wrestling superstar in the 1980's and 1990's, Warrior (Formerly Jim Hellwig) had a career […]
Category : Reviews, TV
Tags : Always Believe, Blu-ray, Dana Warrior, DVD, Hall of Fame, Hulk Hogan, Jim Hellwig, Review, The Ultimate Warrior, Triple H, Ultimate Warrior: Always Believe, Warrior, WWE, WWF
WWE RAW Results & Review (24.02.14)
We are twenty-four hours removed from WWE Elimination Chamber, the show that ended with Randy Orton retaining his WWE World Title in a chamber match against five opponents, including Daniel Bryan. Tonight has a lot of hype about it, with the launch of the WWE Network in the US, and the return of Hulk Hogan […]
Tags : Hulk Hogan, Raw, Review, TV, WWE, WWE Raw | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,281 |
At Love Your Melon, we celebrate National Superhero Day (April 28, 2018) by honoring the thousands of children undergoing cancer treatment across the world. These children are our superheroes, and we want to do everything we can to make a difference in their lives. Join us this week as we celebrate and encourage as many superheroes as possible!
To celebrate our biggest Superhero Day yet, we're launching our new vending machine initiative! This week, the first of ten Beanie Giving Machines will be unveiled in children's hospitals across the country, providing children battling cancer the opportunity to receive a knit beanie and have a fun, positive experience during treatment. Our goal is to put a giving machine at every children's hospital across the world.
Starting April 26, 2018, we are celebrating our superheroes by visiting hospitals across the world to hand out knit beanies and make them feel like the superheroes they are! Our Campus Crew Members will also host various Superhero Adventures to provide therapeutic experiences for children and families battling cancer.
We're teaming up with three of our nonprofit partners to maximize our impact for this year's Superhero Day! We will be joining Make-A-Wish to make five wishes come true for children battling cancer, preparing and serving 25 dinners for families at Ronald McDonald Houses across the United States and providing 100 Pinky Swear Orange Envelopes to financially assist families undergoing childhood cancer treatment.
Follow along on Superhero Day with our social stream! | {
"redpajama_set_name": "RedPajamaC4"
} | 4,983 |
/*
* File: UdpTransport.hpp
* Author: Steven Bidny
*
* Created on May 22, 2012, 12:57 PM
*/
#if !defined(UDPTRANSPORT_HPP)
#define UDPTRANSPORT_HPP
/*- HEADER FILES -------------------------------------------------------------*/
// System Headers
#include <string>
#include <cstdlib>
#include <sstream>
#include <stdint.h>
// Third-party Headers
#define BOOST_SYSTEM_NO_LIB
#include <boost/functional/hash.hpp>
#include <boost/date_time/posix_time/posix_time.hpp>
#include <boost/asio.hpp>
#include <boost/bind.hpp>
#include <boost/lexical_cast.hpp>
#include <boost/interprocess/detail/os_thread_functions.hpp>
// Other Headers
#include "ITransport.hpp"
/*- NAMESPACES ---------------------------------------------------------------*/
namespace gelf4cplus
{
namespace transport
{
using std::string;
/*- CONSTANTS ----------------------------------------------------------------*/
const uint16_t DISABLE_CHUNKING = 0; ///< Constant used to disable chunking.
const uint16_t DEFAULT_CHUNK_SIZE = 1024; ///< The default size of chunks.
const int DEFAULT_GRAYLOG2_PORT = 12201; ///< The default Graylog2 port.
const string DEFAULT_GRAYLOG2_HOST = "localhost"; ///< The default Graylog2 host.
/*- CLASSES ------------------------------------------------------------------*/
/**
* This class defines a UDP transport for use with the GELF appender.
*/
class UdpTransport : public ITransport
{
public:
// Constructors & Destructor
/**
* The default constructor.
* @param aDstHost A destination host name.
* @param aDstPort A destination port.
* @param aMaxChunkSize The maximum size of each chunk.
*/
UdpTransport(const string &aDstHost = "localhost",
const int &aDstPort = DEFAULT_GRAYLOG2_PORT,
const uint16_t &aMaxChunkSize = DEFAULT_CHUNK_SIZE) :
m_maxChunkSize(aMaxChunkSize)
{
// Build the id string using the IP, PID, and TID
std::ostringstream ss;
ss << boost::asio::ip::host_name() <<
boost::interprocess::detail::get_current_process_id() <<
boost::interprocess::detail::get_current_thread_id();
m_threadId = ss.str();
// Set up the Boost Asio stuff
boost::asio::ip::udp::resolver resolver(m_service);
boost::asio::ip::udp::resolver::query query(boost::asio::ip::udp::v4(),
aDstHost,
boost::lexical_cast<string>(aDstPort));
m_endpoint = *resolver.resolve(query);
m_socket = new boost::asio::ip::udp::socket(m_service, m_endpoint.protocol());
}
/**
* A virtual destructor in case someone wants to derive from this class.
*/
virtual ~UdpTransport()
{
delete m_socket;
m_socket = NULL;
}
// Methods
/**
* Gets the maximum chunk size.
* @return The maximum chunk size.
*/
virtual uint16_t maxChunkSize()
{
return m_maxChunkSize;
}
/**
* Sets the maximum chunk size.
* @param aValue The new maximum chunk size.
*/
virtual void maxChunkSize(const uint16_t &aValue)
{
m_maxChunkSize = aValue;
}
/**
* Sends a message using this transport.
* @param aMessage The message to send.
*/
virtual void send(const string &aMessage)
{
size_t length = aMessage.length();
if (m_maxChunkSize != DISABLE_CHUNKING &&
length > m_maxChunkSize)
{
size_t chunkCount = (length / m_maxChunkSize) + 1;
string messageId;
generateMessageId(messageId);
for (size_t i = 0; i < chunkCount; ++i)
{
string messageChunkPrefix;
createChunkedMessagePart(messageId, i, chunkCount, messageChunkPrefix);
size_t skip = i * m_maxChunkSize;
// Send the message chunk to the UDP endpoint
m_socket->async_send_to(boost::asio::buffer(messageChunkPrefix + aMessage.substr(skip, m_maxChunkSize)),
m_endpoint, boost::bind(&UdpTransport::handler, this));
}
}
else
{
// Send the message to the UDP endpoint
m_socket->async_send_to(boost::asio::buffer(aMessage), m_endpoint, boost::bind(&UdpTransport::handler, this));
}
}
protected:
// Constant Static Members
const static uint8_t MAX_HEADER_SIZE = 8; ///< Maximum message ID size.
// Members
uint16_t m_maxChunkSize; ///< The maximum chunk size.
boost::asio::ip::udp::endpoint m_endpoint; ///< The Boost endpoint.
boost::asio::ip::udp::socket *m_socket; ///< The Boost socket.
boost::asio::io_service m_service; ///< The Boost IO service.
string m_threadId; ///< The thread ID.
// Methods
/**
* Creates the prefix for the specific chunk.
* @param aMessageId The unique ID of this message.
* @param anIndex This chunk index.
* @param aChunkCount The total chunk count.
* @param aResult The resultant chunk.
*/
virtual void createChunkedMessagePart(const string &aMessageId,
const size_t &anIndex,
const size_t &aChunkCount,
string &aResult)
{
// Chunked GELF ID: 0x1e 0x0f (identifying this message as a chunked GELF message)
aResult.push_back(0x1e);
aResult.push_back(0x0f);
// Message ID: 8 bytes
aResult += aMessageId;
// Sequence Number: 1 byte (The sequence number of this chunk)
aResult.push_back((char) anIndex);
// Total Number: 1 byte (How many chunks does this message consist of in total)
aResult.push_back((char) aChunkCount);
}
/**
* Generates a unique 8-byte message ID by hashing the host name, process
* ID, thread ID, and time.
* @param aMessageId The resultant message ID
*/
virtual void generateMessageId(string &aMessageId)
{
// Create the "unique" string using the host name and current time
std::ostringstream ss;
ss << m_threadId << boost::posix_time::microsec_clock::universal_time();
// Create a hash of the "unique" string using Boost
size_t hash = boost::hash<string>()(ss.str());
// Return a byte array of the hash using std::string
aMessageId.assign((char*) &hash, MAX_HEADER_SIZE);
}
/**
* Dummy handler for the Boost async_send_to()
*/
virtual void handler()
{
}
};
} // namespace transport
} // namespace gelf4cplus
#endif // #if !defined(UDPTRANSPORT_HPP)
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,363 |
\section{Introduction}\label{intro}
Recent advances in controlling ultra cold atoms
lead to the realization
of truly one dimensional systems, and the
study of many-body effects therein. Important benchmarks,
such as the Tonks-Girardeau gas \cite{paredes,weiss} and
the Mott transition in one dimension\cite{stoeferle}, have been achieved
by trapping bosonic atoms in tight tubes formed by an optical
lattice potential. Novel transport properties of
one dimensional lattice bosons have been studied using these
techniques\cite{fertig}.
More recently, a strongly interacting one dimensional Fermi gas
was realized using similar trapping methods\cite{Moritz}.
Interactions between the fermion atoms
were
controlled
by tuning a Feshbach resonance
in these experiments.
On the theory side, numerous proposals were given for realizing
a variety of different phases in ultra cold Fermi systems
\cite{theory1, theory2},
Bose-Fermi mixtures\cite{cazalilla,mathey,mathey2,pinaki}, as well as
Bose-Bose mixtures\cite{isacsson}.
In [\onlinecite{kuklov}],
commensurate mixtures in higher dimensions were studied.
In this paper we explore the behavior of ultracold atomic mixtures, confined
to one-dimensional (1D)
motion in an optical lattice, that exhibit different types of
commensurability, by which
we mean that the atomic densities and/or the inverse
lattice spacing have an integer ratio.
Commensurable fillings arise naturally in many ultracold atom systems, because
the external trap potential approximately corresponds to a sweep of the
chemical potential through the phase diagram, and therefore passes through points
of commensurability.
At these
points the system can
develop an energy gap, which fixes
the density commensurability over a spatially extended volume.
This was demonstrated in
the celebrated Mott insulator
experiment by Greiner et al.\cite{greiner},
where Mott phases with integer filling
occurred in shell-shaped regions in the atom trap.
These gapped phases gave rise to the
well-known signature in the time-of-flight images\cite{trap},
and triggered the endeavor of `engineering' many-body states
in optical lattices.
Further examples include the recently created density-imbalanced
fermion mixtures \cite{fmixtures} in which the development of
a balanced, i.e. commensurate,
mixture at the center of the trap is observed.
In 1D, this phenomenon is of particular importance,
because it is the only effect that can lead to the opening of
a gap, for a system with short-range interactions.
In contrast to higher dimensional systems, where, for instance, pairing
can lead to a state with an energy gap,
in 1D only discrete symmetries
can be broken, due to the importance of fluctuations.
Orders that correspond to a
continuous symmetry can, at most, develop quasi long range order (QLRO),
which refers to a state in which an order parameter $O(x)$
has a correlation function with algebraic scaling,
$\langle O(x) O(0)\rangle \sim |x|^{-(2-\alpha)}$, with
a positive scaling exponent $\alpha$.
Due to its importance in solid state physics, the most thoroughly
studied commensurate 1D system is the SU(2) symmetric
system of spin-1/2 fermions.
This system develops a spin gap for attractive
interaction and remains gapless
for repulsive interaction, as
can be seen from a second order RG calculation.
However, the assumed symmetry between the
two internal spin states, which is natural in solid state systems,
does not generically occur
in Fermi-Fermi mixtures (FFMs)
of ultra-cold atoms, where
the `spin' states are in fact different
hyperfine states of the atoms.
An analysis of the generic system
is therefore highly called for.
Furthermore, we will extend this analysis
to both Bose-Fermi (BFMs) and Bose-Bose mixtures (BBMs), as well
as to the dual commensurability, in which the charge field, and not
the spin field, exhibits commensurate filling, as will be explained
below.
The main results of this paper are the phase
diagrams shown in Fig. \ref{FFMfig}--\ref{BFM2fig}.
We find that both attractive and repulsive interactions
can open an energy gap.
For FFMs the entire phase diagram is gapped, except for
the repulsive SU(2) symmetric regime (cp. [\onlinecite{theory2}]), for
BFMs or BBMs the bosonic liquid(s) need(s) to be close to the
hardcore limit, otherwise the system remains gapless.
Furthermore, we find a rich structure of quasi-phases,
including charge and spin density wave order (CDW, SDW),
singlet and triplet pairing (SS, TS), polaron pairing \cite{mathey,mathey2},
and a supersolid phase, which is the first example
of a supersolid phase in 1D.
These results are derived within a Luttinger liquid (LL)
description, which treats bosonic and fermionic liquids on equal footing.
This paper is organized as follows: In Section \ref{class} we classify the
different types of commensurate mixtures that can occur,
and in Section \ref{effact} we discuss the effective action of the
mixtures with the most relevant commensurability term.
In Section \ref{RG} we discuss the set of renormalization group
equations for such systems, and in Section \ref{FFM}, \ref{BBM}, and \ref{BFM},
we apply these results to Fermi-Fermi, Bose-Bose, and Bose-Fermi mixtures,
respectively. In Section \ref{detect}, we discuss the experimental
detectibility, and in Section \ref{conclude} we conclude.
\section{Classification of commensurate mixtures}\label{class}
We will now classify the
types of commensurability that can occur in
a system with short-ranged density-density
interaction.
We consider Haldane's representation \cite{Haldane}
of the densities for the two species:
\begin{eqnarray}
n_{1/2} & = & [\nu_{1/2} + \Pi_{1/2}] \sum_{m} e^{2 m i\Theta_{1/2}}
\end{eqnarray}
$\nu_1$ and $\nu_2$ are the densities of the two liquids,
$\Pi_{1/2}(x)$ are the low-k parts (i.e. $k\ll 1/\nu$) of the density fluctuations;
the fields
$\Theta_{1/2}(x)$ are given by
$\Theta_{1/2}(x)
= \pi \nu_{1/2} x + \theta_{1/2}(x)$,
with $\theta_{1/2}(x)=\pi \int^x dy \Pi_{1/2}(y)$.
These expressions hold for both bosons and fermions.
If we use this representation in a density-density
interaction
term $U_{12}\int dx n_1(x)n_2(x)$,
we generate to lowest order a term of the shape $U_{12}\int dx \Pi_1(x)\Pi_2(x)$,
but in addition an infinite number of nonlinear terms, corresponding
to all harmonics in the representation.
However, only the terms for which the linear terms ($2 \pi m_{1/2} \nu_{1/2}x$) cancel,
can drive a phase transition.
For a continuous system this happens for $m_1 \nu_1 - m_2\nu_2=0$,
whereas for a system on a lattice we have the condition $m_1 \nu_1 - m_2\nu_2=m_3$,
where $m_1$,$m_2$ and $m_3$ are integer numbers.
In general, higher integer numbers correspond to terms
that are less relevant, because the scaling dimension
of the non-linear term scales quadratically with these integers.
We are therefore lead to consider
small integer ratios between
the fillings and/or the lattice if present.
In [\onlinecite{mathey2}], we considered
two cases of commensurabilities: a Mott insulator transition
coupled to an incommensurate liquid,
and a fermionic liquid at
half-filling coupled to an incommensurate bosonic liquid.
In both cases
the
commensurability occurs between one species and the
lattice, but does not involve the second species.
In this
paper we
consider the
two most relevant, i.e. lowest order,
cases which exhibit a commensurability that
involves both species.
The first case is the case
of equal filling $\nu_1=\nu_2$,
the second is the case
of the total density being unity, i.e. $\nu_1+\nu_2=1$,
where the densities $\nu_{1}$ and $\nu_2$ themselves are incommensurate.
The first case can drive the system to a spin-gapped state, the
second to a charge gapped state.
We will determine in which parameter
regime these transitions occur,
and what type of QRLO the
system exhibits in the
vicinity of the transition.
These two cases can be mapped onto each other via a dual mapping, which
enables us to study only one
case and then infer the results for the
second by using this mapping.
We will
write out our discussion
for the case of equal filling
and merely state
the corresponding results for
complementary filling.
\begin{figure}
\includegraphics[width=7cm]{fig1.eps}
\caption{\label{FFMfig}
Phase diagram of a commensurate FFM or a
BBM of
hardcore bosons (with the replacement $TS_z{\rightarrow} SS$),
in terms of the interaction $U_{12}$
and the parameter $z=|v_1-v_2|/(v_1+v_2)$.
For both attractive and repulsive
interactions a spin gap opens, except for
$z=0$ and positive interaction.
In the attractive regime, a FFM or a BBM shows
either singlet pairing or CDW order, or a coexistence of these phases.
For repulsive interaction
these mixtures show SDW ordering, with FFMs and BBMs
showing subdominant triplet or singlet pairing, respectively,
for a large range of $z$.
In the gapless
regime, a FFM shows degenerate
SDW and CDW order,
and a BBM shows SF with subdominant
CDW, i.e. supersolid behavior.
For very large positive values of $U_{12}$ the system undergoes
phase separation (PS); for very large negative values it collapses (CL).
}
\end{figure}
\section{Effective action}\label{effact}
The action of a two-species mixture
with equal filling
in bosonized form is given by:
\begin{eqnarray}\label{S}
S & = & S_{0,1} + S_{0,2} + S_{12} +S_{int}.
\end{eqnarray}
The terms $S_{0,j}$, with $j=1,2$, are given by
\begin{eqnarray}
S_{0, j} & = & \frac{1}{2\pi K_{j}} \int d^2r \Big( \frac{1}{v_{j}}(\partial_{\tau} \theta_{j})^2 + v_{j} (\partial_x \theta_{j})^2 \Big)
\end{eqnarray}
Each of the two types of atoms, regardless of being bosonic or fermionic,
are characterized by a Luttinger parameter $K_{1/2}$ and a velocity $v_{1/2}$.
Here we integrate over ${\bf r}=(v_0 \tau, x)$,
where we defined the
energy scale $v_0=(v_1+v_2)/2$.
The term $S_{12}$ describes the acoustic coupling between the two species, and is bilinear:
\begin{eqnarray}
S_{12} & = & \frac{U_{12}}{\pi^2} \int d^2r \partial_x \theta_1 \partial_x \theta_2
+ \frac{V_{12}}{\pi^2} \int d^2r \partial_\tau \theta_1 \partial_\tau \theta_2.
\end{eqnarray}
The second term is created during the RG flow; its prefactor therefore has the initial value
$V_{12}(0)=0$.
We define $S_0 = S_{0,1} + S_{0,2} + S_{12}$, which is
the diagonalizable part of the action.
$S_{int}$
corresponds
to the non-linear coupling between the two liquids,
which we study within an RG approach:
\begin{eqnarray}
S_{int} & = & \frac{2 g_{12}}{(2 \pi \alpha)^2} \int d^2r
\cos(2 \theta_1 - 2\theta_2).
\label{Sint}
\end{eqnarray}
This
bosonized
description
applies to a BBM, a
BFM, and a FFM.
Depending on which of these mixtures
we want to describe we either
construct bosonic or fermionic operators
according to Haldane's contruction \cite{Haldane}:
\begin{eqnarray}
f/b & = & [\nu_0 + \Pi]^{1/2} \sum_{m\, odd / even} e^{m i\Theta} e^{i\Phi}.
\end{eqnarray}
$\nu_0$ is the zero-mode of the density,
$\Phi(x)$ is the phase field, which is the conjugate field of the density fluctuations $\Pi(x)$.
The action for a mixture with complementary
filling, $\nu_1+\nu_2 =1$, is of the form $S_0+S'_{int}$, where
the interaction $S'_{int}$ is given by:
\begin{eqnarray}
S'_{int} & = & \frac{2 g_{12}}{(2 \pi \alpha)^2} \int d^2r
\cos(2 \theta_1 + 2\theta_2).
\label{Sint'}
\end{eqnarray}
To map the action in Eq. (\ref{S}) onto
this system we use
the mapping: $\theta_2{\rightarrow} -\theta_2$, $\phi_2{\rightarrow} -\phi_2$, and
$g_{12}{\rightarrow} -g_{12}$, which evidently
maps a mixture with complementary filling and attractive (repulsive)
interaction and onto a mixture with equal
filling with repulsive (attractive) interaction.
\section{Renormalization group}\label{RG}
To study the action given in Eq. (\ref{S}),
we perform an RG calculation along the lines of the
treatment of the sine-Gordon model in [\onlinecite{SG}].
In our model,
a crucial modification
arises: the linear combination
$\theta_1 - \theta_2$, that appears
in the non-linear term, is not proportional to an eigenmode
of $S_0$, and therefore the RG flow does not affect only one
separate sector of the system, as in an
SU(2)-symmetric system.
The RG scheme that we use here
proceeds as follows:
First, we diagonalize $S_0$ through the transformation
\begin{eqnarray}
\label{diag1}
\theta_1 & = & B_1 \tilde{\theta}_1 + B_2\tilde{\theta}_{2},\\
\theta_2 & = & D_1 \tilde{\theta}_1 + D_2\tilde{\theta}_{2}.\label{diag2}
\end{eqnarray}
The coefficients
$B_{1/2}$ and
$D_{1/2}$ are given in the Appendix.
The fields
$\tilde{\theta}_{1/2}$ are the eigenmode
fields with velocities
$\tilde{v}_{1/2}$ (see Appendix).
\begin{figure}
\includegraphics[width=7cm]{fig2.eps}
\caption{\label{BBMfig}
Phase diagram of a BBM with the first
species being in the hardcore limit,
in
terms of $U_{12}$, and
the Luttinger parameter of the second species ($K_2$), at
the fixed
velocity ratio $|v_1-v_2|/(v_1+v_2)=0.5$.
For large repulsive interaction the system undergoes phase separation (PS), for large attractive
interaction the system collapses (CL).
In the regime below the thick line the system opens
a gap, i.e. if species 2 is close to the hardcore limit.
However, for larger values of $K_2$,
the gapless phase is restored.
Close to the
transition, the properties of the hardcore bosons,
are affected by the RG flow, leading
to supersolid behavior.
}
\end{figure}
As the next step,
we introduce an energy cut-off $\Lambda$ on
the fields $\tilde{\theta}_{1/2}$
according to
$\omega^2/\tilde{v}_{1/2} + \tilde{v}_{1/2}k^2 < \Lambda^2$.
We shift this cut-off by an amount $d\Lambda$,
and correct for this shift up to second order in $g_{12}$.
At first order, only $g_{12}$
is affected, its
flow equation is given by:
\begin{eqnarray}
\frac{d g_{12}}{d l} & = & \Big(2 - K_1 - K_2 -
\frac{2}{\pi}\frac{U_{12} + V_{12} v_1 v_2}{v_1 + v_2}\Big)
g_{12}, \label{RG_g12}
\end{eqnarray}
with $dl=d\Lambda/\Lambda$.
At second order several terms are created that are
quadratic in the original fields $\theta_1$ and $\theta_2$.
We undo the diagonalization, Eq. (\ref{diag1}) and (\ref{diag2}), and absorb
these terms into the parameters of the action, which
concludes the RG step.
By iterating this procedure we obtain these flow equations
at second order
in $g_{12}$:
\begin{eqnarray}
\frac{d K_{1/2}}{d l} & = & - \frac{g_{12}^2}{16\pi^2}
\Big(2+\Big(\frac{v_2}{v_1}+\frac{v_1}{v_2}\Big)\Big)\label{RG_K1}\\
\frac{d v_1}{d l} & = & v_1\frac{g_{12}^2}{16\pi^2}
\Big(\frac{v_2}{v_1}-\frac{v_1}{v_2}\Big)\label{RG_v1}\\
\frac{d v_2}{d l} & = & v_2 \frac{g_{12}^2}{16\pi^2}
\Big(\frac{v_1}{v_2}-\frac{v_2}{v_1}\Big)\label{RG_v2}\\
\frac{d U_{12}}{d l} & = & - \frac{g_{12}^2}{8\pi} (v_1+v_2)\label{RG_U12}\\
\frac{d V_{12}}{d l} & = & - \frac{g_{12}^2}{8\pi} (1/v_1+1/v_2)\label{RG_V12}
\end{eqnarray}
A similar set of equations has been
derived in [\onlinecite{theory2}] for a FFM
in non-bosonized form.
The difference between our result and the result
in [\onlinecite{theory2}]
is the
renormalization of the velocities,
that we find
here,
which
is due to different types of expansions:
In [\onlinecite{theory2}] only one-loop contributions
are taken into account, whereas here we use a
cumulant expansion in $g_{12}$, which at second
order includes contributions that are two-loop for the
renormalization of the velocities.
These contributions, which
would integrate to zero for equal velocities,
as can be
seen from Eqns. \ref{RG_v1} and
\ref{RG_v2}, leads to the discrepancy between the expansion
in the number of loops and the cumulant expansion,
and
gives
a small quantitative correction of the velocities.
As mentioned before, the advantage of the current
approach is that the QRLO of the
system can be directly determined from the
resulting renormalized parameters, and that
the same action can be used to study BBMs and BFMs.
The system of differential equations, Eqns. (\ref{RG_g12})
to (\ref{RG_V12}),
can show two types of qualitative behavior:
The coefficient $g_{12}$ of the non-linear term (\ref{Sint}) can
either flow to zero, i.e. $S_{int}$ is irrelevant, or it diverges,
leading to the formation of an energy gap.
In the first case, the system flows to
a fixed point that is described
by a renormalized diagonalizable
action of the type $S_0$, from which the quasi-phases can be
determined.
When $S_{int}$ is relevant,
we
introduce the fields \cite{review}
$\theta_{\rho/\sigma} = \frac{1}{\sqrt{2}}(\theta_1 \pm \theta_2)$,
which define the charge and the spin sector of the system.
In this regime,
these sectors
decouple.
Each of the
two sectors is characterized by a Luttinger parameter
and a velocity, $K_{\rho/\sigma}$ and $v_{\rho/\sigma}$,
which are related to the original parameters
in $S_0$ in a
straightforward
way.
Using the numerical solution of the flow equations,
we find that $K_\sigma\rightarrow 0$, as can be expected for
an ordering of the nature of a spin gap, leaving $K_\rho$
the only parameter characterizing the QLRO in this phase.
In order to determine
the QLRO in the system we determine
the scaling
exponents of various order parameters.
For that purpose, we use the
bosonization representation of these order parameters,
which contain the fields $\theta_{1/2}$
and $\phi_{1/2}$, and use the diagonalization, Eqs. (\ref{diag1})
and (\ref{diag2}),
for the fields $\theta_{1/2}$, as well
as the dual transformation for the fields $\phi_{1/2}$:
\begin{eqnarray}\label{dualdiag1}
\phi_1 & = & C_1 \tilde{\phi}_1 + C_2\tilde{\phi}_{2},\\
\phi_2 & = & E_1 \tilde{\phi}_1 + E_2\tilde{\phi}_{2}.\label{dualdiag2}
\end{eqnarray}
The coefficients $C_{1/2}$ and $D_{1/2}$ are given in the Appendix.
Since the order parameters are now written in terms
of the eigenfields $\tilde{\theta}_{1/2}$
and $\tilde{\phi}_{1/2}$, the correlation functions can be
evaluated in a straight forward manner.
The scaling exponents are given by various quadratic
expressions of the parameters in Eqs. (\ref{diag1}), (\ref{diag2}),
(\ref{dualdiag1}), and (\ref{dualdiag2}).
In [\onlinecite{mathey2}], we give an extensive list
of correlation functions, which can be
transferred to the system considered here, with the
formal replacement: $\beta_{1/2}\rightarrow B_{1/2}$,
$\gamma_{1/2}\rightarrow C_{1/2}$, $\delta_{1/2}\rightarrow D_{1/2}$,
and $\epsilon_{1/2}\rightarrow E_{1/2}$.
The order parameter with the largest positive scaling exponent
shows the dominant order, whereas other orders with positive
exponent are subdominant.
\begin{figure}
\includegraphics[width=7cm]{fig3.eps}
\caption{\label{BFM1fig}
Phase diagram of a BFM with hardcore bosons,
in terms of the interaction $U_{12}$
and the parameter $z=|v_1-v_2|/(v_1+v_2)$.
For both attractive and repulsive
interactions a spin gap opens, except for
$z=0$ and positive interaction.
In the attractive regime,
a BFM shows either CDW order or polaron pairing;
for repulsive interaction
BFMs show SDW ordering.
In the gapless
regime,
a BFM shows CDW order for the fermions
and SF for the bosons.
For very large positive values of $U_{12}$ the system undergoes
phase separation (PS); for very large negative values it collapses (CL).
}
\end{figure}
\section{Fermi-Fermi mixtures}\label{FFM}
We will now apply this procedure to the different types of mixtures.
For a FFM
we find that the system always develops a
gap, with the exception of the repulsive SU(2) symmetric regime
(cp. [\onlinecite{theory2}]).
To determine the QLRO
we introduce the following operators \cite{review, giamarchi}:
\begin{eqnarray}
O_{SS} &=& \sum_{\sigma, \sigma'} \tilde{\sigma} f_{R,\sigma} \delta_{\sigma,\sigma'}
f_{L, 3-\sigma'},\\
O_{TS}^a& =& \sum_{\sigma, \sigma'} \tilde{\sigma} f_{R,\sigma} \sigma^a_{\sigma,\sigma'}
f_{L, 3-\sigma'},\\
O_{CDW}& =& \sum_{\sigma, \sigma'} f_{R,\sigma}^\dagger \delta_{\sigma,\sigma'}
f_{L,\sigma'},\\
O_{SDW}^a& =& \sum_{\sigma, \sigma'} \tilde{\sigma} f_{R,\sigma}^\dagger
\sigma^a_{\sigma,\sigma'}
f_{L,\sigma'},
\end{eqnarray}
with $\sigma, \sigma'=1,2$, $\tilde{\sigma}=3-2\sigma$, and $a=x,y,z$.
In the gapless SU(2) symmetric regime,
both CDW and SDW show QLRO, with both scaling exponents of the
form $\alpha_{SDW/CDW}=1-K_\rho$\cite{review}, which
shows that these
orders are algebraically degenerate.
Within
the gapped regime
the scaling exponents of
these operators are given by $\alpha_{SS,TS_z}= 2 - K_\rho^{-1}$ and
$\alpha_{CDW,SDW_z} = 2 - K_\rho$.
As discussed in [\onlinecite{giamarchi}], the
sign of $g_{12}$ determines
whether CDW or SDW$_z$, and SS or TS$_z$ appears.
In Fig. \ref{FFMfig},
we show the phase diagram based on these results.
In addition to these phases we indicate the appearance
of the Wentzel-Bardeen instability, shown as phase separation
for repulsive
interaction and collapse for attractive interaction.
We will now use the dual mapping
to obtain the
phase diagram of a FFM with complementary
filling from Fig. \ref{FFMfig}.
Under this mapping, the attractive and repulsive regimes
are exchanged with the following replacements:
$CDW{\rightarrow} SDW_z$, $SDW_z{\rightarrow} CDW$, $SS,TS_z{\rightarrow} SDW$, and $SDW{\rightarrow} SS$.
Note that the gapless regime
is now on the attractive side,
with degenerate CDW and SS pairing.
\section{Bose-Bose mixtures}\label{BBM}
For BBMs we proceed in the
same way as for FFMs.
We introduce the following set of order parameters:
\begin{eqnarray}
O_{CDW}&=&b_1^\dagger b_1 +b^\dagger_2 b_2,\\
O_{SS}&=&b_1 b_2,\\
O_{SDW_z}&=&b_1^\dagger b_1 -b^\dagger_2 b_2,\\
O_{SDW_x}&=&b_1^\dagger b_2 +b^\dagger_2 b_1,\\
O_{SDW_y}&=&-i (b_1^\dagger b_2 -b^\dagger_2 b_1),
\end{eqnarray}
and
in addition the superfluid (SF) order
parameters $b_1$ and $b_2$.
In Fig. \ref{FFMfig}
we show the phase
diagram of a mixture of a BBM of hardcore bosons, which
is almost identical to the one of a FFM.
The phase diagram of the mixture with complementary
filling, as obtained from the dual mapping, is
also of the same form as its fermionic equivalent,
with the
exception of the gapless regime, in which BBMs show
supersolid behavior (coexistence
of SF and CDW order), and with the replacement $TS_z{\rightarrow} SS$.
In Fig. \ref{BBMfig}, we
show the phase diagram of
a mixture of hardcore bosons (species 1) and bosons in the intermediate
to hardcore regime (species 2).
If species 2 is sufficiently far
away from the hardcore limit, the system remains gapless.
However, in the vicinity
of the transition the scaling exponents of the liquids
are affected by the RG flow. As indicated,
the effective scaling exponent
of the hardcore bosons
is renormalized to a value that is smaller than 1, and therefore
we find both SF and CDW order, i.e. supersolid behavior.
The phase diagram of the dual mixture is of the following form:
the attractive and the repulsive regime are exchanged,
and in the gapped phase we again have the
mapping:
$CDW{\rightarrow} SDW_z$, $SDW_z{\rightarrow} CDW$, $SS{\rightarrow} SDW_{x,y}$, and $SDW_{x,y}{\rightarrow} SS$.
The gapless regime is unaffected.
The paired SF state discussed in [\onlinecite{kuklov}] corresponds
to the SS phase discussed here, whereas the dual SDW$_{x,y}$ phase
that appears for complementary filling corresponds to
the super-counter-fluid phase described therein.
Note that here these orders compete with either CDW or SDW$_z$ order,
and only appear as QLRO, not LRO, as in higher dimensions.
Both of these insights can only be gained by the using the LL description
and RG that is used in this paper.
\section{Bose-Fermi mixtures}\label{BFM}
For a BFM
we find that the order parameters $O_{CDW}$, $O_{SDW_z}$,
the polaron pairing operator
\begin{eqnarray}\label{O_PP}
O_{f-PP}&=&f_R f_L e^{-2i\lambda \Phi_b}
\end{eqnarray}
(see [\onlinecite{mathey,mathey2}]),
and $b$ can develop QLRO in the gapless regime.
In the gapped regime, the order parameters
\begin{eqnarray}
O_{PP}&\equiv& f_R b f_L b,\\
O_{PP'}&\equiv& f_R b^\dagger f_L b^\dagger,
\end{eqnarray}
in addition to $O_{CDW}$,
show
QLRO.
($O_{PP/PP'}$ are special cases of the polaron pairing operator
(\ref{O_PP}), extensively discussed
in [\onlinecite{mathey}] and [\onlinecite{mathey2}].)
In Fig. \ref{BFM1fig} we show
the phase diagram
of a BFM with hardcore bosons, and
in Fig. \ref{BFM2fig},
we vary the Luttinger parameter
of the bosons.
In both the gapless phase and the gapped phase, we find that
CDW and $f$-PP or PP, respectively, are mutually exclusive
and cover the entire phase diagram, cp. [\onlinecite{mathey,mathey2}].
The dual mapping
again maps attractive and repulsive regimes onto each other. Within
the
gapped phase we find the mapping
$CDW{\rightarrow} SDW_z$, $SDW_z{\rightarrow} CDW$, and $PP{\rightarrow} PP'$, the
gapless regime is unaffected.
%
\begin{figure}
\includegraphics[width=7cm]{fig4.eps}
\caption{\label{BFM2fig}
Phase diagram of a BFM,
in
terms of $U_{12}$, and
the Luttinger parameter of the second species ($K_2$), at
the fixed
velocity ratio $|v_1-v_2|/(v_1+v_2)=0.5$.
For large repulsive interaction the system undergoes phase separation (PS), for large attractive
interaction the system collapses (CL).
In the regime below the thick line the system opens
a gap, i.e. if species 2 is close to the hardcore limit.
However, for larger values of $K_2$,
the gapless phase is restored.
Close to the
transition, the properties of the fermions,
are still affected by the RG flow, leading
to CDW order.
}
\end{figure}
\section{Experimental detection}\label{detect}
The phase diagrams that have been derived and
shown in Figs. \ref{FFMfig}--\ref{BFM2fig}, are
given in terms of the parameters
that appear in the effective action.
With such a field theoretical approach
we can find the correct qualitative long-range
behavior, such as the functional form of the correlation functions.
However, it is also intrinsic to this
approach that the effective parameters
appearing can only be qualitatively related to the
underlying microscopic parameters.
Based on a phase diagram such as Fig. \ref{BBMfig}, for instance,
the following features for
a mixture of bosonic atoms with a short-range interaction can be
expected:
If one species is in the hardcore limit, and the other
is in between an intermediate interaction regime
and the hardcore limit, then for attractive
interaction between them, a gapped state can
be created, in which there is a competition
of SS pairing and CDW order. For repulsive interaction, and the
second species being very close to the hardcore regime, one can also
expect a gapped phase, in which we find SDW$_z$ order.
For the intermediate regime we expect a supersolid phase.
Before we conclude,
we discuss how the
predictions presented in this paper
could be measured experimentally.
Since the appearance of a gapped state has already been
demonstrated for the MI-SF transition in 1D [\onlinecite{stoeferle}],
and since it constitutes a
significant qualitative change in the system,
this would be
the first feature predicted in this paper to look for.
%
As demonstrated in [\onlinecite{RF}],
RF spectroscopy can be used
to determine
the presence and the size
of an energy gap.
To detect the rich structure of QLRO the following approaches
can be taken:
CDW order
will create additional peaks in TOF images, corresponding to
a wavevector $Q=2k_f$.
As demonstrated and pointed out in [\onlinecite{noise}],
the noise in TOF images
allows to identify
the different regimes
of both gapped and gapless phases.
As discussed in [\onlinecite{mathey,mathey2}],
a laser stirring experiment could determine
the onset of CDW order for fermions, or the supersolid regime
for bosons.
\section{Conclusion}\label{conclude}
In conclusion, we have studied mixtures of ultra-cold
atoms in 1D with commensurate filling.
We used a Luttinger liquid description which enables us to
study FFMs, BFMs, and BBMs in a single
approach.
We find that FFMs are generically gapped for both attractive and repulsive interactions,
whereas for BFMs and BBMs the bosons need to be close
to the hardcore limit.
We find a rich structure of quasi-phases in the vicinity
of these transitions, in particular a supersolid
phase for BBMs, that occurs
close to the hardcore limit.
Experimental methods to detect the predictions were also discussed.
We gratefully acknowledge important discussions with
D.-W. Wang, A.H. Castro Neto,
S. Sachdev, T. Giamarchi, E. Demler, and H.-H. Lin.
| {
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} | 6,213 |
{"url":"https:\/\/physics.stackexchange.com\/questions\/493736\/moment-of-inertia-for-a-random-polygon","text":"# Moment of inertia for a random polygon\n\nIs there an easy way to calculate the moment of inertia of an arbitrary 2-dimensional polygon, with respect to an axis perpendicular to the plane of the polygon (passing through the center of mass)?\n\nWhat I intended to do:\n\n1. Derive a formula for the moment of inertia of a triangle, with respect to an axis through a vertex of the triangle.\n2. Apply this formula on every triangle center of mass - vertex - next vertex in the polygon and add the results.\n\nWhat I achieved:\n\n\u2022 For a triangle with vertices $$(0,0)$$, $$(a,0)$$ and $$(b,c)$$, the moment of inertia with respect to the axis $$x=0\\wedge y=0$$ is:\n$$\\int_0^{b}\\int_0^{\\frac{c}{b}\\cdot x}x^2+y^2\\text{ }dy\\text{ }\\text{ }\\text{ }dx - \\int_a^b\\int_0^{\\frac{c}{b-a}\\cdot(x-a)}x^2+y^2\\text{ }dy\\text{ }\\text{ }\\text{ }dx$$ $$=\\int_0^{b}\\dfrac{\\left(c^3+3b^2c\\right)x^3}{3b^3}dx - \\int_a^b\\dfrac{c\\left(\\left(c^2+3b^2-6ab+3a^2\\right)x^3+\\left(-3ac^2-3ab^2+6a^2b-3a^3\\right)x^2+3a^2c^2x-a^3c^2\\right)}{3\\left(b^3-3ab^2+3a^2b-a^3\\right)}dx$$ $$=\\dfrac{b\\left(c^3+3b^2c\\right)}{12} - \\dfrac{\\left(b-a\\right)c\\left(c^2+3b^2+2ab+a^2\\right)}{12}$$ $$=\\frac{c^3a+ca^3+bca^2+b^2ca}{12}$$ $$=\\frac{ac}{12}\\cdot\\left(a^2+b^2+c^2+ab\\right)$$\n\u2022 Now we can calculate the moment of inertia of a triangle with vertices $$A(x_1,y_1)$$, $$B(x_2,y_2)$$ and $$C(x_3,y_3)$$ with respect to the axis through A as follows:\nThe Euclidean transformation that takes $$A$$ to $$(0,0)$$ and $$B$$ to $$(\\sqrt{x_2^2+y_2^2},0)$$, takes $$C$$ to $$(x_3',y_3')$$ (which can be calculated quite easily).\nAs a Euclidean transformation preserves the moment of inertia, we can apply the preceding formula.\n\u2022 Now we can apply this to every triangle center of mass - vertex - next vertex in the polygon and add the results to get the total moment of inertia.\n\nI haven't tested this yet (I'll do that later), but could someone please check my calculations and my method?\nFurthermore, if there is an easier way to calculate this, please let me know ;)\n\n\u2022 What if your object can't be broken into triangles like this? What if the center of mass isn't located in the polygon. Are you only considering regular polygons? \u2013\u00a0Aaron Stevens Jul 26 at 13:14\n\u2022 Please look also at this (Polygone) en.wikipedia.org\/wiki\/Second_moment_of_area \u2013\u00a0Eli Jul 26 at 14:54\n\u2022 @Aaron Stevens I'm only considering convex polygons. \u2013\u00a0Jonas De Schouwer Jul 26 at 15:30\n\u2022 Just as a side note, I think arbitrary polygon would be a better descriptor than random polygon. An arbitrary polygon is a polygon whose details have not been specified; a random polygon is one whose details are blindly picked out of a hat. In other words, randomness refers to a lack of predictability, which is not what you mean here. \u2013\u00a0J. Murray Jul 27 at 0:37\n\u2022 @J.Murray you're right. I didn't know there was a difference between both words. Fixed it now. \u2013\u00a0Jonas De Schouwer Jul 27 at 11:09\n\nTurning Eli's comment into an answer to my own post:\n\nApperently, there exists something called the second moment of area (see also: https:\/\/en.wikipedia.org\/wiki\/Second_moment_of_area).\nThis is the mathematical definition of the physical moment of inertia.\n\nThe second moment of area for an arbitrary shape $$R$$ with respect to an arbitrary axis $$a$$ is defined as: $$J_{a} = \\iint\\limits_{R}\\rho^2 dA$$\n\nWhen the axis is the $$x$$-axis, the second moment of area can be calculated as follows: $$J_x=\\iint\\limits_R y^2 dx dy$$ And when it's the $$y$$-axis: $$J_y=\\iint\\limits_R x^2 dy dx$$\n\nFor a polygon $$A_0A_1A_2\\cdots A_{n-1}$$ with $$A_i=(x_i,y_i)$$ and $$A_n=A_0$$, those are equal to: $$J_x = \\frac{1}{12}\\sum_{i=0}^{n-1}\\left(x_iy_{i+1}-x_{i+1}y_i\\right)\\left(y_i^2+y_iy_{i+1}+y_{i+1}^2\\right)$$ $$J_y = \\frac{1}{12}\\sum_{i=0}^{n-1}\\left(x_iy_{i+1}-x_{i+1}y_i\\right)\\left(x_i^2+x_ix_{i+1}+x_{i+1}^2\\right)$$\n\nNow, denote by $$z$$ the line perpendicular to the $$xy$$-plane that passes through the origin. Then: $$J_z = \\iint\\limits_R\\rho^2 dA = \\iint\\limits_Rx^2+y^2\\text{ }dA = \\iint\\limits_Ry^2\\text{ }dxdy + \\iint\\limits_Rx^2\\text{ }dydx = J_x+J_y$$ (this is the so-called perpendicular axis theorem)\nThanks to this, we can calculate $$J_z$$!\n\nI strongly believe that I was going to end up with this as well, due to the $$12$$ in the denominator and the fact that only subsequent $$x$$- and $$y$$-values are multiplied.\n\nWe can also derive the formula for the second moment of area w.r.t to the axis through the origin directly.\n\nLet $$A_0(=A_n), A_1\\ldots, A_{n-1}$$ be the vertices of your polygon $$P$$ where $$A_i = (x_i, y_i)$$.\n\nFor $$\\vec{r} = (x,y)$$ notice that $$\\operatorname{div}(r^2\\vec{r}) = 4r^2$$ so using the two-dimensional divergence theorem we get $$J_z = \\int_{P} r^2\\,dA = \\frac14\\int_P \\operatorname{div}(r^2\\vec{r})\\,dA = \\frac14\\int_{\\partial P}r^2\\,\\vec{r}\\cdot \\hat{n}\\,d\\gamma$$\n\nwhere we integrate over the boundary $$\\gamma = \\partial P$$ of the polygon with $$\\hat{n}$$ being the outward pointing normal.\n\nNotice that for $$i=0,\\ldots, n-1$$ you can parameterize the line segment $$[A_i, A_{i+1}]$$ with $$\\gamma_i : [-1,1] \\to \\mathbb{R}^2$$ given as $$\\gamma_i(t) = \\frac{A_{k+1}+A_k + t(A_{k-1}-A_k)}2, \\quad t\\in[-1,1]$$\n\nand the outward pointing normal is $$\\hat{n} = \\frac{(A_{k+1}-A_k)\\times \\hat{z}}{\\|A_{k+1}-A_k\\|}$$ where $$\\hat{z} = (0,0,1)$$. We have $$\\gamma_i'(t) = \\frac12(A_{k-1}-A_k)$$ so $$\\hat{n}\\,d\\gamma_i = \\frac12 (A_{k+1}-A_k)\\times \\hat{z}\\,dt$$. Therefore\n\n\\begin{align} J_z &= \\frac14\\int_{\\partial P}r^2\\,\\vec{r}\\cdot \\hat{n}\\,d\\gamma \\\\ &= \\frac14\\sum_{i=0}^{n-1}\\int_{\\gamma_i}r^2\\,\\vec{r}\\cdot \\hat{n}\\,d\\gamma_i \\\\ &= \\frac1{64}\\sum_{i=0}^{n-1}\\int_{-1}^1 \\big\\|A_{k+1}+A_k + t(A_{k-1}-A_k)\\big\\|^2\\,\\big(A_{k+1}+A_k + t(A_{k-1}-A_k)\\big)\\cdot (A_{k+1}-A_k)\\times \\hat{z}\\,dt \\end{align}\n\nUsing the triple product rule we get \\begin{align}\\big(A_{k+1}+A_k + t(A_{k-1}-A_k)\\big)\\cdot \\left((A_{k+1}-A_k)\\times \\hat{z}\\right) &= \\left(\\big(A_{k+1}+A_k + t(A_{k-1}-A_k)\\big)\\times (A_{k+1}-A_k)\\right)\\cdot \\hat{z}\\\\ &= 2(A_{k+1}\\times A_k)\\cdot \\hat{z}\\\\ &= 2(x_{k}y_{k+1}-x_{k+1}y_k) \\end{align} Hence by integrating it follows \\begin{align} J_z &= \\frac1{32}\\sum_{k=0}^{n-1}(x_{k}y_{k+1}-x_{k+1}y_k)\\int_{-1}^1 \\big\\|A_{k+1}+A_k + t(A_{k-1}-A_k)\\big\\|^2\\,dt\\\\ &= \\frac1{12}\\sum_{k=0}^{n-1}(x_{k}y_{k+1}-x_{k+1}y_k)\\left(x_{k+1}^2+x_{k+1}x_k+x_k^2 + y_{k+1}^2+y_{k+1}y_k+y_k^2\\right) \\end{align} which recovers your formula.\n\nInspired by this more general calculation.","date":"2019-09-19 04:55:32","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\": 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\": 56, \"wp-katex-eq\": 0, \"align\": 3, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9936809539794922, \"perplexity\": 329.23460828974106}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-39\/segments\/1568514573439.64\/warc\/CC-MAIN-20190919040032-20190919062032-00213.warc.gz\"}"} | null | null |
Q: JAVA JPA SPRING - IncorrectResultSizeDataAccessException: query did not return a unique result: 2 @Query(value = "SELECT i.productNumber FROM Product as i ORDER BY i.productNumber DESC")
public String getLastProductNumber(); // this is my query,
// this is my generator, I'd like to generate add by 1 when I published product everytime,
private String setProductNumber() {
try {
String value = productRepository.getLastProductNumber();
System.out.println("Get Last OrderNumber: " + value);
long currentValue = 0;
if (!(value == null || value.isEmpty())) {
currentValue = Long.parseLong(value);
}
String result = String.format("%08d", currentValue + 1);
return result;
} catch (EmptyResultDataAccessException e) {
String result = "00000001";
return result;
but it caused error after get 2 productNumber.
A: Hi please check this example
@Query(value = "SELECT i.productNumber FROM Product as i ORDER BY i.productNumber
DESC")
public List<String> getLastProductNumber();
| {
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La comunità montana del Vomano, Fino e Piomba (zona N) era stata istituita con la Legge regionale 7 settembre 1977, n. 59 della Regione Abruzzo, che ne aveva approvato lo statuto.
È stata soppressa dopo una riduzione delle comunità montane abruzzesi che sono passate da 19 ad 11 nel 2008.
Comprendeva tredici comuni della Provincia di Teramo. aveva sede nel comune di Cermignano.
Ne facevano parte:
Arsita
Atri
Basciano
Bisenti
Canzano
Castiglione Messer Raimondo
Castilenti
Castellalto
Cellino Attanasio
Cermignano
Montefino
Notaresco
Penna Sant'Andrea
Note
Voci correlate
Vomano
Fino (fiume)
Piomba
Vomano, Fino e Piomba | {
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Cyber Everywhere: Expand security capabilities with AI tools
Public and private sector enterprises need to consider expanding their use of AI-augmented cybersecurity tools to better defend their networks and assets.
As the range of cyberthreats continues to expand, and organizations remain hard-pressed to hire enough talent to keep up, cyber leaders recommend that executives explore AI tools to help assess and automate their security posture.
Security veteran Irfan Saif says that AI represents a range of different concepts such as intelligent automation, analytics and conversational AI as well as more sophisticated capabilities that start to approach what may be considered human intelligence, he says.
Saif, a principle and board member at Deloitte, also urges enterprise executives to think about AI in the context of machines helping humans, which he sees as "a much more viable, sustainable and scalable approach rather than thinking about AI in the context of human replacement," he explains.
Adding to the discussion, Deborah Golden, lead for Deloitte's U.S. Cyber Risk Services Practice says that this idea of partnership between people and AI-enabled technology will help organizations address the shortage of cybersecurity talent.
Golden and Saif share recommendations for executives leading public and private sector organizations on ways AI can combat new cyberthreats in the latest episode of the "Cyber Everywhere" podcast series produced by CyberScoop and underwritten by Deloitte:
Changes occurring in the cyberthreat landscape
"Bad actors — particularly those on the more sophisticated end of the spectrum — tend to adopt and adapt to changes in the technology landscape a bit faster than those that they are trying to attack," says Saif.
He cautions that AI is being used against enterprises, noting instances where AI has been used to mimic the activity of legitimate users and bypass various detection measures, he says.
Business case for AI-enabled tools
Golden says CIOs need to consider adopting AI-enabled tools to help available cyber talent achieve greater efficiencies at scale.
As the cyberthreat landscape continues to grow at exponentially, enterprises will need to keep investing in "structured and unstructured machine learning in a way that perhaps we've never looked at before," just to keep pace, she says.
Developing strategies and governance for AI
Saif says that the notion of "trustworthy AI" is gaining currency among security experts. The goal is to build a common language and framework to govern AI as a strategy and as a program "from the boardroom down to the server room."
"That is effectively taking critical principles of trust — whether that's ethics, whether that's explainability — all the sorts of things that people really want to understand when it comes to how to apply AI to business problems, how to manage and govern the data, and the inputs, the outputs and the use of that information," Saif says.
Irfan Saif currently co-leads Deloitte's U.S. artificial intelligence and cognitive advisory offering. He has more than 20 years of IT consulting experience, specializing in cybersecurity and risk management.
Deborah Golden has more than 25 years of IT experience spanning numerous industries, including government, life sciences, health care and financial services. She specializes in cybersecurity, technology transformation and privacy and governance initiatives.
Listen to the podcast for the full conversation on AI-augmented cybersecurity. You can hear more coverage of "Cyber Everywhere" on our CyberScoop radio channels on Apple Podcasts, Spotify, Google Play, Stitcher and TuneIn.
This podcast was produced by CyberScoop and underwritten by Deloitte.Deloitte is formally known as Deloitte & Touche LLP, a subsidiary of Deloitte LLP. For more details, see www.deloitte.com/us/about.
Cyber Everywhere: The growing threat of mis-, dis-, and malinformation
Deloitte cybersecurity leaders discuss the risks posed by MDM and share strategies to help combat these threats, often taking advantage of existing infrastructure and processes.
Threat intelligence increasingly depends on AI
Cyber Everywhere: Aligning the CISO role with the business strategy | {
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} | 4,181 |
Chris Weitz
TV Review: His Dark Materials
by May Perrin
The BBC and HBO team up to bring Philip Pullman's epic fantasy drama to life, in the form of His Dark Materials. Rising from the ashes of the disappointing 2007 The Golden Compass (dir. Chris Weitz), this series directed by Tom Hooper is the realisation of everything fans could hope...
Film Review – Cinderella
by Tom Watchorn
April 8, 2015 12:53
This enchanting remake of the animated 1950 classic has the very essence of Disney woven throughout. With a dynamic cast and stunning setting, this magical remake will capture your heart and make you fall in love with the fairy tale all over again. There is always a certain amount... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,603 |
{"url":"https:\/\/ctu-bern.github.io\/accrualPlot\/","text":"Accrual plots are an important tool when monitoring clinical trials. Some trials are terminated early due to low accrual, which is a waste of resources (including time). Assessing accrual rates can also be useful for planning analyses and estimating how long a trial needs to continue recruiting participants. accrualPlot provides tools for such plots\n\n## Installation\n\naccrualPlot can be installed from CRAN in the usual manner:\n\ninstall.packages('accrualPlot')\n\nThe development version of the package can be installed from the CTU Bern universe via\n\ninstall.packages('accrualPlot', repos = 'https:\/\/ctu-bern.r-universe.dev')\n\naccrualPlot can be installed directly from from github with:\n\n# install.packages(\"remotes\")\nremotes::install_github(\"CTU-Bern\/accrualPlot\")\n\nNote that remotes treats any warnings (e.g.\u00a0that a certain package was built under a different version of R) as errors. If you see such an error, run the following line and try again:\n\nSys.setenv(R_REMOTES_NO_ERRORS_FROM_WARNINGS = \"true\")\n\n## Overview\n\nThe first step to using accrualPlot is to create an accrual dataframe. This is simply a dataframe with a counts of participants included per day.\n\n# load package\nlibrary(accrualPlot)\n#>\n#> Attaching package: 'lubridate'\n#> The following objects are masked from 'package:base':\n#>\n#> date, intersect, setdiff, union\n\n# demonstration data\ndata(accrualdemo)\n\ndf <- accrual_create_df(accrualdemo\\$date)\n\nCumulative and absolute recruitment plots , as well as a method to predict the time point of study completion, are included.\n\npar(mfrow = c(1,3))\nplot(df, which = \"cum\")\nplot(df, which = \"abs\")\nplot(df, which = \"pred\", target = 300)\n\n### Acknowledgements\n\nThe package logo was created with ggplot2 and hexSticker with icons from Font Awesome (via the emojifont package).","date":"2023-02-09 13:08:01","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 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\": 0.19120819866657257, \"perplexity\": 8762.184259003037}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499966.43\/warc\/CC-MAIN-20230209112510-20230209142510-00691.warc.gz\"}"} | null | null |
\section{Introduction}\label{sec:intro}
Understanding the ability of gradient-based stochastic optimization algorithms to find good minima of non-convex objective functions has become an especially important problem due to the success of stochastic gradient descent (SGD) in learning deep neural networks. Although there exist non-convex objective functions and domains for which SGD will necessarily lead to sub-optimal local minima, it appears that for many problems of interest in deep learning, across domains as varied as natural language and images, these worst-case situations do not arise. Indeed, a number of recent works have developed provable guarantees for GD and SGD when used for objective functions defined in terms of neural networks over certain distributions, despite the non-convexity of the underlying optimization problem~\citep{brutzkus2018sgd,allenzhu.3layer,cao2019generalization,jitelgarsky20.polylog,frei2020singleneuron,frei2021provable}. To date, however, there has not been a framework which could unify the variegated approaches for guarantees in these settings.
In this work, we introduce the notion of \textit{proxy convexity}
and demonstrate that many existing provable guarantees for learning with neural networks trained by gradient-based optimization fall into a problem satisfying proxy convexity. First, let us define the \textit{learning problem} over a distribution $\calD$, where the goal is to minimize the expected loss
\begin{equation}\label{eq:opt.problem}
\min_{w\in \calW} F(w):= \E_{z\sim \calD} f(w; z),
\end{equation}
where $\calW$ is a parameter domain and $f(\cdot\ ;z)$ is a loss function. We are interested in guarantees using the online SGD algorithm, which uses a set of i.i.d. samples $z_t\iid \calD$ and has updates given by
\[ w_{t+1} = w_t - \eta \nabla f(w_t; z_t),\]
where $\eta>0$ is a fixed learning rate.
We now introduce the first notion of proxy convexity we will consider in the paper.
\begin{definition}[Proxy convexity]
We say that a function $f: \R^p \to \R$ satisfies $(g,h)$-\textit{proxy convexity} if there exist functions $g, h : \R^p \to \R$ such that for all $w,v\in \R^p$,
\[ \sip{\nabla f(w)}{w-v} \geq g(w) - h(v).\]
\end{definition}
Clearly, every convex function $f$ satisfies $(f, f)$-proxy convexity. We next introduce the analogy of proxy convexity for the Polyak--\L ojasiewicz (PL) inequality~\citep{karimi2016.linearconvpl}.
\begin{definition}[$g$-proxy, $\xi$-optimal PL inequality]
We say that a function $f: \R^p \to \R$ satisfies a $g$-proxy, $\xi$-optimal \textit{Polyak--\L ojasiewicz inequality} with parameters $\alpha\in (0,2]$ and $\mu>0$ (in short, $f$ satisfies the $(g,\xi, \alpha, \mu)$-\textit{PL inequality}) if there exists a function $g: \R^p \to \R$ and scalars $\xi\in \R$, $\mu>0$ such that for all $w\in \R^p$,
\[ \norm{\nabla f(w)}^\alpha \geq \f 1 2 \mu \l( g(w) - \xi\r) .\]
\end{definition}
As we shall see below, the proxy PL inequality is a natural extension of the standard PL inequality.
Our main contributions are as follows.
\begin{enumerate}[leftmargin = *]
\item When $f$ satisfies $(g,h)$-proxy convexity, and $f$ is either Lipschitz or satisfies a particular smoothness assumption,
then for any norm bound $R>0$, the online SGD algorithm run for polynomial (in $1/\eps$ and $R$) number of iterations satisfies the following in expectation over $z_1, \dots, z_T\sim \calD^T$,
\[ \min_{t<T} \E_{z\sim \calD} g(w_t; z) \leq \min_{\norm{w}\leq R} \E_{z\sim \calD} h(w; z) + \eps.\]
\item When $f$ satisfies a $(g, \xi, \alpha, \mu)$-proxy PL inequality and has Lipschitz gradients, SGD run for a polynomial (in $1/\eps)$ number of iterations satisfies the following in expectation over $z_1, \dots, z_T \sim \calD^T$,
\[ \min_{t<T} \E_{z\sim \calD} g(w_t; z) \leq \xi + \eps.\]
\item We demonstrate that many previous guarantees for neural networks trained by gradient descent can be unified in the framework of proxy convexity.
\end{enumerate}
As we will describe in more detail below, if a loss function $\ell$ is $(g,h)$-proxy convex or satisfies a $g$-proxy PL inequality, then the optimization problem is straightforward and the crux of the problem then becomes connecting guarantees for the proxy $g$ with approximate guarantees for $f$.
\paragraph{Notation.}
We use uppercase letters to refer to matrices, and lowercase letters will either refer to vectors or scalars depending on the context. For vectors $w$, we use $\norm w$ to refer to the Euclidean norm, and for matrices $W$ we use $\norm W$ to refer to the Frobenius norm. We use the standard $O(\cdot)$, $\Omega(\cdot)$ notations to hide universal constants, with $\tilde O(\cdot)$ and $\tilde \Omega(\cdot)$ additionally hiding logarithmic factors.
\section{Proxy Convexity in Comparison to Other Non-convex Optimization Frameworks}\label{sec:proxy.vs.other}
In this section, we describe how proxy convexity and proxy PL-inequalities relate to other notions in non-convex optimization. In Section~\ref{sec:additional.related}, we will discuss additional related work. First, recall that a function $f$ is $(g,h)$-proxy convex if there exist functions $g$ and $h$ such that for all $w,v$,
\[ \sip{\nabla f(w)}{w-v} \geq g(w) - h(v).\]
One notion from the non-convex optimization literature that is related to our notion of proxy convexity is that of \textit{invexity}~\citep{hanson1981invex}. A function $f$ is invex if it is differentiable and there exists a vector-valued function $k(w,v)$ such that for any $w,v$,
\[ \sip{\nabla f(w)}{k(w,v)} \geq f(w) - f(v).\]
It has been shown that a smooth function $f$ is invex if and only if every stationary point of $f$ is a global minimum~\citep{craven1985invex}. However, for many problems of interest involving neural networks, it is not the case that every stationary point will be a global optimum, which makes invexity a less appealing framework for understanding neural networks. Indeed, we shall see in Example \ref{example:single.neuron.proxy} below that if one considers the problem of learning a single ReLU neuron $x\mapsto \max(0,\sip wx)$ under the squared loss, it is not hard to see that there exist stationary points which are not global minima (e.g., $w=0$ assuming the convention $\sigma'(0)=0$). By contrast, we shall see that the single ReLU neuron does satisfy a form of proxy convexity that enables SGD to find approximately (but not globally) optimal minima. Thus even the simplest neural networks induce objective functions which are proxy convex and non-invex. We shall see in Example~\ref{example:deep.relu.proxy.pl} that proxy convexity appears in the objective functions induced by wide and deep neural networks as well.
To understand how the proxy PL inequality compares to other notions in the optimization literature, recall that an objective function $f$ satisfies the standard PL inequality~\citep{polyak1963,lojasiewicz1963} if there exists $\mu>0$ such that
\[ \norm{\nabla f(w)}^2 \geq \f \mu 2 \l[ f(w) - f^* \r],\]
where $f^* = \min_w f(w)$. Clearly, any stationary point of an objective satisfying the standard PL inequality is globally optimal. Thus, the presence of local minima among stationary points in neural network objectives makes the standard PL inequality suffer from the same drawbacks that invexity does for understanding neural networks trained by gradient descent. This further applies to any of the conditions which are known to imply the PL inequality, like weak strong convexity, the restricted secant inequality, and the error bound condition~(\citet{karimi2016.linearconvpl}).\footnote{\citet{karimi2016.linearconvpl} shows that these conditions imply the PL inequality under the assumption that the objective function has Lipschitz-continuous gradients.}
In comparison, the $(g, \xi, \alpha, \mu)$-proxy PL inequality is satisfied if there exists a function $g$ and constants $\xi>0$, $\alpha \in (0,2]$ and $\mu>0$ such that
\[ \norm{\nabla f(w)}^\alpha \geq \f \mu 2 \l[ g(w) - \xi \r].\]
It is clear that if a function $f$ satisfies the standard PL inequality, then it satisfies the $(f, f^*, 2, \mu)$ proxy PL inequality. Stationary points $w_0$ of objective functions satisfying the proxy PL inequality have $\norm{\nabla f(w_0)}=0$ which imply $g(w_0)\leq \xi$. In the case that $g = f$, the slack error term $\xi$ allows for the proxy PL inequality framework to accommodate the possibility that stationary points may not be globally optimal (i.e. have objective value $f^* = \min_w f(w)$), but could be approximately optimal by, for example, having objective value at most $\xi = C \cdot f^*$ or $\xi = C \cdot \sqrt{f^*}$ for some constant $C\geq 1$. When $g\neq f$, the proxy PL inequality allows for the possibility of analyzing a proxy loss function $g$ which is \textit{implicitly} minimized when using gradient-based optimization of the objective $f$.
At a high level, proxy convexity and the proxy PL inequality are well-suited to situations where stationary points may not be \textit{globally} optimal, but may be approximately optimal with respect to a related optimization objective. The proxy convexity framework allows for one to realize this through developing problem-specific analyses that connect the proxy objective $g$ to the original objective $f$. As we shall see below, rich function classes like neural networks are often more easily analyzed by considering a proxy objective function that naturally appears when one analyzes the gradient of the loss.
\section{Proxy PL Inequality Implies Proxy Objective Guarantees}\label{sec:proxy.pl}
In this section, we show that for loss functions satisfying a proxy PL inequality, SGD\footnote{We focus on online SGD in this paper for simplicity. Analogous optimization guarantees would hold for other variants of gradient descent that utilize samples in batches.}
efficiently minimizes the proxy. We leave the proofs for Section~\ref{sec:proofs}.
\begin{theorem}\label{thm:proxy.pl}
Suppose $F(w) = \E_{z\sim \calD} f(w; z)$ where $f(\cdot\ ; z)$ satisfies the $(g(\cdot\ ; z), \xi(z), \alpha, \mu)$-proxy PL inequality for some function $g(\cdot\ ; z):\R^p\to \R$ for each $z$. Denote by $G(w) := \E_{z\sim \calD} g(w; z)$. Assume that $f$ is non-negative and has $L_2$-Lipschitz gradients. Then for any $\eps>0$, provided $\eta < 1/L_2$,
online SGD with fixed step size $\eta$ and run for $T = 2\eta^{-1} (\mu \eps/2)^{-2/\alpha} f(w_0; z_0)$ iterations results in the following guarantee in expectation over $z_0, \dots, z_{T-1}\sim \calD^T$,
\begin{equation}\label{eq:proxy.pl.identity}
\min_{t<T} G(w_t) \leq \E_{z\sim \calD} \xi(z) + \eps.
\end{equation}
\end{theorem}
To get a feel for how a proxy PL inequality might be useful for learning neural networks, consider a classification problem with labels $y\in \{\pm 1\}$, and suppose that $N(W; x)$ is a neural network function. A standard approach for learning neural networks is to minimize the cross-entropy loss $\ell(y N(W; x)) = \log\big(1+\exp(-y N(W; x))\big)$ using gradient descent. Using the variational form of the norm, we have
\begin{align}\nonumber
\norm{\nabla \ell(y N(W; x))} &= \sup_{\norm{U}=1} \sip{\nabla \ell(y N(W; x))}{U} \\
&\geq -\ell'(y N(W; x)) \cdot y \sip{\nabla N(W; x)}{V},\label{eq:variational.norm.loss}
\end{align}
where $V$ is any matrix satisfying $\norm{V}=1$. Now, although the function $-\ell'$ is not an upper bound for $\ell$ (indeed, $-\ell' < \ell$), it \textit{is} an upper bound for a constant multiple of the zero-one loss, and can thus serve as a proxy for the classification error.\footnote{In fact, the function $z\mapsto [\ell'(z)]^2$ is also a proxy for the classification error; see~\cite[Appendix A]{frei2020halfspace}.} This is because for convex and decreasing losses $\ell$, the function $-\ell'$ is non-negative and decreasing, and hence by Markov's inequality,
\begin{align*}
\P(y \neq \sgn ( N(W; x) )) &= \P(y \cdot N(W; x) < 0) \\
&= \P( -\ell'(y N(w; x)) > -\ell'(0)) \\
&\leq \frac{1}{-\ell'(0)} \E[-\ell'(y N(W; x))].
\end{align*}
Thus, if one can bound the population risk under $-\ell'$, one has a bound for the classification error. Indeed, this property has been used in a number of recent works on neural networks~\citep{cao2019generalization,frei2019resnet,jitelgarsky20.polylog,frei2021provable}. This lets the $-\ell'$ term in \eqref{eq:variational.norm.loss} represent the desired proxy $g$ in the definition of the $(g, \xi, \alpha, \mu)$-proxy PL inequality. Thus, for neural network classification problems, the problem of showing the neural network has small classification error is reduced to constructing a matrix $V$ that allows for the quantity $y \sip{\nabla N(W; x)}{V}$ to be large and non-negative. The quantity $y\sip{\nabla N(W; x)}{V}$ can be thought of as a margin function that is large when the gradient of the neural network loss points in a good direction. Although we shall see below that in some instances one can derive a lower bound for $y \sip{\nabla N(W; x)}{V}$ that holds for \textit{all} $W$, $x$, and $y$, a more general approach would be to show that along the gradient descent trajectory $\Wt t$, a lower bound for $y \sip{\nabla N(\Wt t; x)}{V}$ holds.\footnote{Although our results as stated would not immediately apply in this setting, the proof would be the same up to trivial modifications.}
In the remainder of this section, we will show that a number of recent works on learning neural networks with gradient descent utilized proxy PL inequalities. In our first example, we consider recent work by~\citet{charles2018stability} that directly used a (standard) PL inequality.
\begin{example}[Standard PL inequality for single leaky ReLU neurons and deep linear networks]\label{example:single.neuron.standard.pl}
\citet{charles2018stability} showed that the standard PL inequality holds in two distinct settings. The first is that of a single leaky ReLU neuron $x\mapsto \sigma(\sip{w}{x})$, where $\sigma(z)=\max(c_\sigma z, z)$ for $c_\sigma \neq 0$. They showed that if $s_{\mathrm{min}}(X)$ is the smallest singular value of the matrix $X \in \R^{n\times d}$ of $n$ samples, then for a $\lambda$-strongly convex loss $\ell$, the loss $f(w) = \ell(\sigma(\sip{w}{x}))$ satisfies the standard $\mu$-PL inequality, i.e., the $(f, f^*,2, \mu)$-proxy PL inequality for $\mu = \lambda s_{\mathrm{min}}(X)^2 c_\sigma^2$~(\citet[Theorem 4.1]{charles2018stability}).
The same authors also showed that under certain conditions the standard PL inequality holds when the neural network takes the form $N(W; x) = W_L \cdots W_1x$ and the loss is the squared loss, $f(W) = \nicefrac 12 (y - N(W; x))^2$. In particular, they showed that if $s_{\mathrm{min}}(W_i)\geq \tau >0$ throughout the gradient descent trajectory, then $f$ satisfies the standard $\mu$-PL inequality for $\mu = L \tau^{2L-2} / \pnorm{(XX^\top)^{-1} X}F^2$~(\citet[Theorem 4.5]{charles2018stability}).
The standard PL inequality has been used by a number of other authors in the deep learning theory literature, see e.g. \citet[Theorem 1]{xie2017diverse},~\citet[Eq. 2.3]{hardt2018identity},~\citet[Theorem 1]{zhou2017characterization},~\citet[Theorem 3]{shamir2019exponential.linear}.
\end{example}
In our next example, we show that a proxy PL inequality holds for deep neural networks in the neural tangent kernel (NTK) regime.
\begin{example}[Proxy PL inequality for deep neural networks in NTK regime]\label{example:deep.relu.proxy.pl}
Consider the class of deep, $L$-hidden-layer ReLU networks, either with or without residual connections:
\begin{align*}
N_1(W; x) &= \sigma(\Wt 1 x),\quad N_l(W; x) = s_l N_{l-1}(W; x) + \sigma(\Wt l N_{l-1}(W; x)),\, l=2,\ldots,L,\\
N(W; x) &= \summ j m a_j [N_L(W; x)]_j,
\end{align*}
where $s_l =0$ for fully-connected networks and $s_l=1$ for residual networks. \citet[Theorem 4.2]{cao2019generalization}, ~\citet[Lemma 4.3]{frei2019resnet}, and~\citet[Lemma B.5]{zou2019gradient} have shown that under certain distributional assumptions and provided the iterates of gradient descent stay close to their intialization, one can guarantee that for the cross-entropy loss $f(W; (x,y)) = \ell(y N(W; x))$,
\begin{equation}
\norm{\nabla f(W; (x,y))} \geq C_1 \cdot -\ell'(y N(W; x)).
\end{equation}
By defining $g(W; z) = -\ell'(y N(W; x))$, the loss $f$ satisfies the $(g, 0, 1, 2C_1 )$-proxy PL inequality. Since the ReLU is not smooth, the loss $f$ will not have Lipschitz gradients, and thus a direct application of Theorem \ref{thm:proxy.pl} is not possible. Instead, the authors show that in the NTK regime, the loss obeys a type of semi-smoothness that still allows for an analysis simliar to that of Theorem \ref{thm:proxy.pl}.
\end{example}
\begin{example}[Proxy PL inequality for one-hidden-layer networks outside NTK regime]
Consider a one-hidden-layer network with activation function $\sigma$,
\begin{equation} \label{eq:onehiddenlayer.nn}
N(W; (x,y)) = \summ j m a_j \sigma(\sip{w_j}{x}),
\end{equation}
where the second layer weights $\{a_j\}_{j=1}^m$ are randomly initialized and fixed at initialization, but the $\{w_j\}_{j=1}^m$ are trained. Assume $\sigma$ satisfies $\sigma'(z)\geq c_\sigma >0$ for all $z$ (e.g., the leaky ReLU activation). Frei, Cao, and Gu have shown~\cite[Lemma 3.1]{frei2021provable} that there exists a matrix $V\in \R^{m\times d}$ with $\pnorm{V}F=1$ such that for distributions satisfying anti-concentration, for any $x,y$ and $W$,
\[ y \sip{\nabla N(W; x)}{V} \geq C_1[ c_\sigma - \xi(x,y)],\]
where $\E[\xi(x,y)] = O(\sqrt{\opt})$ where $\opt$ is the best classification error achieved by a halfspace over $\calD$. Thus, when $f(W; (x,y)) = \ell(y N(W; (x,y))$ is the cross-entropy loss,
\begin{align*}
\norm{\nabla f(W; (x,y))} &= \sup_{\norm{Z}_F=1} \sip{\nabla f(W; (x,y))}{Z} \\
&\geq -\ell'(y N(W; x)) \cdot y \sip{ \nabla N(W; x)}{V} \\
&\geq C_1 c_\sigma \cdot [-\ell'(y N(W; x)) - c_\sigma^{-1} \xi(x,y)].
\end{align*}
As in Example \ref{example:deep.relu.proxy.pl}, by defining $g(W; z) = -\ell'(y N(W; x))$, the loss $f$ satisfies the $(g, c_\sigma^{-1} \xi(x,y), 1, 2C_1 c_\sigma )$-proxy PL inequality. Thus, provided we can show that $f(\cdot ; z)$ has $L_2$-Lipschitz gradients for some constant $L_2>0$, Theorem \ref{thm:proxy.pl} shows that
\begin{align*}
\min_{t<T} \P_{(x,y)} (y \neq \sgn( N(\Wt t; x))) &\leq \min_{t<T} \f {\E_{(x,y)\sim \calD} [-\ell'(y N(\Wt t; x)]} {|\ell'(0)|} \\
&\leq 2 C_2 c_\sigma^{-1} \E_{(x,y)\sim \calD} \xi(x,y) + \eps \\
&= O(\sqrt{\opt}) + \eps.
\end{align*}
Provided $\sigma$ is such that $\sigma'$ is continuous and differentiable, then $f$ has $L_2$-Lipschitz gradients and thus the guarantees will follow. In particular, this analysis follows if $\sigma$ is any smoothed approximation to the leaky ReLU which satisfies $\sigma'(z)\geq c_\sigma>0$.
\end{example}
\section{Proxy Convexity Implies Proxy Objective Guarantees}\label{sec:proxy.convex}
In this section, we show that if $f$ satisfies $(g,h)$-proxy convexity, we can guarantee that by minimizing $f$ with gradient descent, we find a hypothesis for which $g(w)$ is at least as small as the best norm-bounded predictor as measured by the loss $h$. We present two versions of our result: one that relies upon fewer assumptions on the loss $f$ but needs a small step size, and another that requires a proxy smoothness assumption on $f$ but allows for a constant step size. The proofs for the theorem are given in Section \ref{sec:proofs}.
\begin{theorem}\label{thm:proxy.convexity}
Suppose that $F(w) := \E_{z\sim \calD} f(w; z)$ and $f(\cdot; z)$ is $(g(\cdot; z),h(\cdot;z))$-proxy convex for each $z$. Denote $H(w) := \E_{z\sim \calD} h(w; z)$ and $G(w) := \E_{z\sim \calD} g(w; z)$.
(a) Assume there exists $L_1>0$ such that for all $w$, $\E_{z\sim \calD} [\norm{\nabla f(w; z)}^2]\leq L_1^2$. Then for any $v\in \R^p$ and any $\eps>0$, performing online SGD on $F(w)$ from an arbitrary initialization $w_0$ with fixed step size $\eta \leq \eps L_1^{-2}$ for $T = \eta^{-1} \eps^{-1} \norm{w_0-v}^2$ iterations implies that, in expectation over $(z_0, \dots, z_{T-1})\sim \calD^T$,
\[ \min_{t<T} G(w_t) \leq H(v) + \eps.\]
(b) Assume there exists $L_2>0$ such that for all $w$, $\E_{z\sim \calD}[ \norm{\nabla f(w; z)}^2] \leq 2 L_2 \E_{z\sim \calD} g(w; z)$. Then for any $v\in \R^p$ and any $\eps>0$, performing online SGD on $F(w)$ from an arbitrary initialization with fixed step size $\eta \leq L_2^{-1}/2$ for $T = \eta^{-1} \eps^{-1} \norm{w_0-v}^2$ implies that, in expectation over $(z_0, \dots, z_{T-1})\sim \calD^T$,
\[ \min_{t<T} G(w_t) \leq (1 + 2\eta L_2) H(v) + \eps.\]
\end{theorem}
In order for $(g,h)$-proxy convexity to be useful, there must be a way to relate guarantees for $g$ into guarantees for the desired objective function $f$. In the remainder of this section, we will discuss two neural network learning problems which are non-convex and yet satisfy proxy convexity which leads to generalization guarantees. Our first example is the problem of learning a neural network with a single nonlinear unit.
\begin{example}[Single neuron satisfies proxy convexity]\label{example:single.neuron.proxy}
Consider the problem of learning a single neuron $x\mapsto \sigma(\sip{w}{x})$ under the squared loss, where $\sigma$ is the ReLU activation. The objective function of interest is
\[ F(w) = \E_{(x,y)\sim \calD} \nicefrac 12 (\sigma(\sip{w}{x}) - y)^2.\]
Denote
\[ F^* := \min_{\norm{w}\leq 1} F(w).\]
It is known that $F$ is non-convex~\citep{yehudai20}. Under the assumption that learning sparse parities with noise is computationally hard, it is known that no polynomial time algorithm can achieve risk $F^*$ exactly; moreover, it is known that (unconditionally) the standard gradient descent algorithm cannot achieve risk $F^*$~\citep{goel2019relugaussian}.\footnote{This stands in contrast to learning a single \textit{leaky} ReLU neuron $x\mapsto \max(\alpha x, x)$ for $\alpha \neq 0$, which as we showed in Example \ref{example:single.neuron.standard.pl} can be solved using much simpler techniques.} However,~\citet{frei2020singleneuron} showed that although $F$ is non-convex and no algorithm can achieve risk $F^*$, $F$ does satisfy a form of proxy convexity that allows for gradient descent to achieve risk $O(\sqrt{F^*})$. They showed that the loss function
\[ f(w; (x,y)) = \nicefrac 12 (\sigma(\sip{w}{x}) - y)^2\]
satisfies $(g,h)$-proxy convexity, where
\begin{align*}
g(w; (x,y)) &= 2 \l[ \sigma(\sip{w}{x}) - \sigma(\sip{v^*}{x}) \r]^2 \sigma'(\sip{w}{x}),\\
h(v; (x,y)) &= |\sigma(\sip{v}{x}) - y| = \sqrt{ 2f(v; (x,y)},
\end{align*}
where $v^*$ is the population risk minimizer of $F(w)$ (see their Eq. (3.13)). Moreover, they showed (see their Eq. (3.9))
\[ \norm{\nabla f(w; z)}^2 \leq 8 g(w; z).\]
Thus Theorem \ref{thm:proxy.convexity}(b) implies that SGD with step size $\eta \leq \nicefrac 1{8}$ and $T = 2 \eta^{-1} \eps^{-1} \norm{w_0-v^*}^2$ iterations will find a point $w_t$ satisfying
\begin{align*}
G(w_t) &= 2\E_{(x,y)} \l[ \l( \sigma(\sip{w_t}{x}) - \sigma(\sip{v^*}{x})\r)^2 \sigma'(\sip{w_t}{x}) \r] \\
&\leq ( 1 + 8 \eta) H(v^*) + \eps \\
&\leq ( 1 + 8 \eta) \E | \sigma(\sip{v^*}{x}) -y| + \eps \\
&\leq (1 + 8 \eta) \sqrt{ \E[(\sigma(\sip{v^*}{x}) -y)^2]} + \eps \\
&= O(\sqrt{F^*}).
\end{align*}
The authors then show that under some distributional assumptions on $\calD$, $G(w_t) = O(\sqrt{F^*})$ implies $F(w_t) = O(\sqrt{F^*})$~\cite[Lemma 3.5]{frei2020singleneuron}. Thus, the optimization problem for $F$ induces a proxy convex optimization problem defined in terms of $G$ which yields guarantees for $G$ in terms of $H$, and this in turn leads to approximate optimality guarantees for the original objective $F$.
\end{example}
In our next example, we show that a number of works on learning one-hidden-layer ReLU networks in the neural tangent kernel regime~\citep{jacot2018ntk} can be cast as problems satisfying proxy convexity.
\begin{example}[Proxy convexity for one-hidden-layer ReLU neural networks in the NTK regime]\label{example:onelayer.relu.ntk.proxy}
Consider the class of one-hidden-layer ReLU networks consisting of $m$ neurons,
\begin{equation} \nonumber
N(W; (x,y)) = \summ j m a_j \sigma(\sip{w_j}{x}),
\end{equation}
where the $\{a_j\}_{j=1}^m$ are randomly initialized and fixed at initialization, but the $\{w_j\}_{j=1}^m$ are trained. Suppose we consider a binary classification problem, where $y\in \{\pm 1\}$ and we minimize the cross-entropy loss,
\[ F(W) = \E_{(x,y)\sim \calD} f(W; (x,y)),\quad f(W; (x,y)) = \ell\big(y N(W; (x,y))\big),\quad \ell (z) = \log(1+\exp(-z)).\]
\citet[Proof of Lemma 2.6]{jitelgarsky20.polylog} showed that there exists a function $\tilde h(a, W, V; (x,y))$ such that the iterates of gradient descent satisfy
\[ \sip{\nabla f(W; (x,y))}{w-v} \geq f(W; (x,y)) - \tilde h(a, W, V; (x,y)).\]
Under the assumption that the iterates of gradient descent stay close to the initialization (i.e., the neural tangent kernel regime), they show that $\tilde h(a, W, V; (x,y)) \leq \eps$ under distributional assumptions, and thus $F(w)$ will satisfy $(f, \tilde h \equiv \eps)$-proxy convexity. The cross entropy loss satisfies $[\ell'(z)]^2 \leq \ell(z)$, and thus we can apply Theorem \ref{thm:proxy.convexity}(b) to get guarantees of the form $\min_{t<T} F(w_t)\leq \eps$ for the cross-entropy loss $F(W)$ of SGD-trained neural networks in the NTK regime.
In another problem of learning one-hidden-layer networks,~\citet[Proof of Lemma B.4]{allenzhu.3layer} show that there exists a proxy loss function $g(a, W; (x,y))$ such that provided the neural network weights stay close to their initialized values, $f(a, W; (x,y))$ satisfies $(g, g + \eps)$ proxy convexity. Again using that the cross-entropy loss satisfies $[\ell'(z)]^2 \leq \ell(z)$, Theorem \ref{thm:proxy.convexity}(b) shows that SGD-trained neural networks in the NTK regime satisfy $\min_{t<T} G(W_t) \leq \min_V G(V) + \eps$. They further show that the proxy loss $g$ is close to the cross entropy loss, i.e. $|\E[g(a, W; (x,y))] - \E[f(a, W; (x,y))]| \leq \eps$, implying a bound of the form $\min_{t<T} F(W_t)\leq \min_V F(V) + \eps$.
\end{example}
\section{Proof of the Main Results}\label{sec:proofs}
In this section we provide the proofs of the theorems given in Sections \ref{sec:proxy.pl} and \ref{sec:proxy.convex}.
We first give the proof of Theorem \ref{thm:proxy.pl} which provides guarantees for learning with objectives satisfying proxy PL inequalities.
\begin{proof}[Proof of Theorem \ref{thm:proxy.pl}]
Since $f$ has $L_2$-Lipschitz gradients, we have for any $w,w'$ and fixed $z$,
\begin{equation} \nonumber
f(w;z) \leq f(w';z) + \sip{\nabla f(w';z)}{w-w'} + \f {L_2}2 \norm{w-w'}^2.
\end{equation}
Taking $w=w_{t+1}$, $w'=w_t$, and $z=z_t$,
\begin{align} \nonumber
f(w_{t+1}; z_t) &\leq f(w_t; z_t) - \eta \norm{\nabla f(w_t; z_t)}^2 + \f{ \eta^2 L_2}{2} \norm{\nabla f(w_t; z_t)}^2 \\
&= f(w_t; z_t) - \eta \l[ 1 - \eta L_2/2\r] \norm{\nabla f(w_t; z_t)}^2.
\end{align}
Since $\eta <1/L_2$, we have $(1-\eta L_2/2)^{-1} \leq 2$, and thus we can rearrange the above to get
\begin{align} \nonumber
\norm{\nabla f(w_t; z_t)}^2 &\leq \f 1 {\eta (1 - \eta L_2/2)} [f(w_t; z_t) - f(w_{t+1}; z_t)] \\
&\leq \f 2 \eta [f(w_t; z_t) - f(w_{t+1}; z_t)].
\end{align}
Summing the above from $t=0$ to $t=T-1$ and using that $f$ is non-negative, we get
\begin{equation} \nonumber
\frac 1 {T} \summm t 0 {T-1} \norm{\nabla f(w_t; z_t)}^2 \leq \f{2 f(w_0; z_0)}{\eta T}.
\end{equation}
Using the definition of proxy PL inequality, this implies
\begin{equation} \nonumber
\frac 1 {T} \summm t 0 {T-1} (\mu/2)^{2/\alpha} (g(w_t; z_t) - \xi(z_t))^{2/\alpha} \leq \f{2 f(w_0; z_0)}{\eta T}.
\end{equation}
Taking the minimum over $t<T$ and re-arranging terms, this means
\begin{equation} \nonumber
\min_{t<T} (g(w_t; z_t)-\xi(z_t))^{2/\alpha} \leq \f{ 2 f(w_0; z_0)}{\eta T (\mu/2)^{2/\alpha}}.
\end{equation}
Therefore, we have
\begin{equation} \nonumber
\min_{t<T} g(w_t; z_t) \leq \xi(z_t) + \f 2 \mu \cdot \l( \f{ 2f(w_0; z_0)}{\eta T} \r)^{\alpha/2}.
\end{equation}
Taking $T = 2\eta^{-1} f(w_0; z_0) (\mu \eps/2)^{-2/\alpha}$ and taking expectations over $z_0, \dots, z_{T-1}$, we get \eqref{eq:proxy.pl.identity}.
\end{proof}
We next prove guarantees for SGD when the objective satisfies proxy convexity.
\begin{proof}[Proof of Theorem \ref{thm:proxy.convexity}]
By the definition of proxy convexity,
\begin{align} \nonumber
\norm{w_t - v}^2 - \norm{w_{t+1}-v}^2 &= 2 \eta \sip{\nabla f(w_t; z_t)}{w_t-v} - \eta^2 \norm{\nabla f(w_t; z_t)}^2\\ \nonumber
&\geq 2 \eta [g(w_t; z_t) - h(v; z_t)] - \eta^2 \norm{\nabla f(w_t; z_t)}^2.
\end{align}
Conditional on the values of $z_0, \dots, z_{t-1}$ (and hence on the value of $w_t$), taking expectations of both sides with respect to $z_t\sim \calD$ results in
\begin{equation}
\norm{w_t - v}^2 - \E_{z_t\sim \calD} \norm{w_{t+1}-v}^2 \geq 2 \eta [ G(w_t) - H(v)] - \eta^2 \E_{z_t\sim \calD} \norm{\nabla f(w_t; z_t)}^2. \label{eq:proxy.convexity.key}
\end{equation}
For case (a), this results in
\[ \norm{w_t - v}^2 - \E_{z_t\sim \calD} \norm{w_{t+1}-v}^2 \geq 2 \eta [ G(w_t) - H(v) - \eta/2 L_1^2 ].\]
Dividing both sides by $2\eta T$ and summing from $t=0, \dots, T-1$, we get
\begin{align} \nonumber
\f 1 T \summm t 0 {T-1} G(w_t) \leq \f 1 T \summm t 0 {T-1} H(v) + \f{\eta L_1^2}2 + \f{\norm{w_0-v}^2 - \E_{z_{T-1}\sim \calD} \norm{w_T-v}^2}{2\eta T}.
\end{align}
Taking expectations over $z_{0:t} = (z_0, \dots, z_{t-1}) \sim \calD^{t}$, we get
\[ \min_{t < T} \E_{z_{0:t}\sim \calD^{t}} G(w_t) \leq \frac 1 T \summm t 0 {T-1} \E_{z_{0:t} \sim \calD^{t}} G(w_t) \leq H(v) + \f{\eta L_1^2}{2} + \f{ \norm{w_0-v}^2}{2\eta T}.\]
In particular, for $\eta \leq \eps L_1^{-2}$ and $T = \eta^{-1} \eps^{-1} \norm{w_0-v}^2$, we get
\[ \min_{t < T} \E_{z_{0:t}\sim \calD^{t}} G(w_t) \leq H(v) + \eps.\]
For case (b), \eqref{eq:proxy.convexity.key} becomes
\begin{align*}
\norm{w_t - v}^2 - \E_{z_t\sim \calD} \norm{w_{t+1}-v}^2 &\geq 2 \eta [ G(w_t) - H(v) - \eta L_2 G(w_t) ] \\
&= 2 \eta \l[ (1 - \eta L_2) G(w_t) - H(v)\r] .
\end{align*}
Dividing both sides by $2\eta T(1 - \eta L_2)$ and summing from $t=0, \dots, T-1$, we get
\begin{align} \nonumber
\f 1 T \summm t 0 {T-1} G(w_t) &\leq \f 1 {1-\eta L_2} H(v) + \f{\norm{w_0-v}^2 - \E_{z_{T-1}\sim \calD} \norm{w_T-v}^2}{2\eta T(1 - \eta L_2)} \\ \nonumber
&\leq (1 + 2\eta L_2) H(v) + \f{(1 + 2 \eta L_2)\norm{w_0-v}^2}{2\eta T},
\end{align}
where in the last line we have used that $\eta \leq L_2^{-1}/2$ and that $1/(1-x)\leq 1+2x$ on $[0,1/2]$. Taking expectations over $z_{0:t} = (z_0, \dots, z_{t-1}) \sim \calD^{t}$, we get
\begin{align*}
\min_{t < T} \E_{z_{0:t}\sim \calD^{t}} G(w_t) &\leq \frac 1 T \summm t 0 {T-1} \E_{z_{0:t} \sim \calD^{t}} G(w_t) \\
&\leq (1 + 2 \eta L_2) H(v) + \f{ 2 \norm{w_0-v}^2}{2\eta T}.\end{align*}
In particular, for $T = \eta^{-1} \eps^{-1} \norm{w_0-v}^2$, we get
\begin{align*} \min_{t < T} \E_{z_{0:t}\sim \calD^{t}} G(w_t) \leq (1 + 2 \eta L_2) H(v) + \eps.
\end{align*}
\end{proof}
We note that under additional assumptions on $\calD$ and the loss function, we could improve the results from holding in expectation over the draws of the sample to high probability guarantees by using standard concentration arguments. This is easily done when the objective function satisfies proxy convexity: we can make a slight modification to the proof of Theorem \ref{thm:proxy.convexity} to argue inductively that until we reach a point with $G(w_t) \leq H(v) + \eta L_1^2 + \eps$, we have that that $\norm{w_t-v}^2 - \norm{w_{t+1}-v}^2 \geq \eps$. This implies that the norm of the predictors remain bounded throughout the trajectory of gradient descent until we reach the desired point with $G(w_t) \leq H(v) + \eta L_1^2 + \eps$, which can then be used in Rademacher complexity-type arguments~\citep{bartlett2003rademacher}. This type of argument was previously used by e.g.,~\citet{frei2020singleneuron}.
\section{Additional Related Work}\label{sec:additional.related}
The Polyak--Lojasiewicz inequality can be dated back to the original works of~\citet{polyak1963} and~\citet{lojasiewicz1963}. Recent work by~\citet{karimi2016.linearconvpl} proved linear convergence under the PL condition and showed that the PL condition is one of the weakest assumptions under which linear convergence is possible. In particular, they showed that the error bound inequality~\citep{luo1993errorbound}, essential strong convexity~\citep{liu2015essentialstrongconvexity}, weak strong convexity~\citep{necoara2019weakstrongconvexity}, and the restricted secant inequality~\citep{zhang2013restrictedsecant} are all assumptions under which linear convergence is possible and that each of these assumptions implies the PL inequality.
As we described in Section~\ref{sec:proxy.vs.other}, the standard PL condition was shown to hold under certain assumptions for neural network objective functions~\citep{hardt2018identity,xie2017diverse,zhou2017characterization,charles2018stability}. In addition to those covered in this paper, there are a number of other provable guarantees for generalization of SGD-trained networks which rely on a variety of different techniques, such as tensor methods~\citep{li2020relubeyondntk} and utilizing connections with partial differential equations by way of mean field approximations~\citep{mei2018meanfieldview,chizat2018globalconv,mei2019mean,chen2020generalized}.
In the optimization literature, recent work has shown that SGD can efficiently find stationary points and can escape saddle points~\citep{ge2015escapesaddle,fong2019escapesaddle}. As the proxy PL inequality implies guarantees for the proxy objective function at stationary points of the original optimization objective, our framework can naturally be used for other optimization algorithms that are known to efficiently find stationary points, such as
SVRG~\citep{allen2016variance,reddi2016svrg},
Natasha2~\citep{allenzhu2018natasha2}, SARAH/SPIDER~\citep{nguyen2017stochastic,fang2018spider}, and SNVRG~\citep{zhou2018snvrg}.
\section{Conclusion}
In this paper we have introduced the notion of proxy convexity and proxy PL inequality and developed guarantees for learning with stochastic gradient descent under these conditions. We demonstrated that many recent works in the learning of neural networks with gradient descent can be framed in terms of optimization problems that satisfy either proxy convexity or a proxy PL inequality. While the proxy convexity framework cannot unify all existing analyses of learning neural networks, we hope that it can reveal some of the principles underlying the success of SGD-trained neural networks.
\bibliographystyle{ims}
| {
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Thomas Woodrow Eck (March 29, 1914 – June 21, 1988) was an American football player and coach. He served as the head coach at the University of Massachusetts Amherst—known as Massachusetts State College until 1947—in 1945 and from 1947 to 1951, compiling a record of 17–23–4. Eck was the head coach when the Redmen, not known as the Minutemen until 1972, transitioned from independent status to their first official football conference, the Yankee Conference, in 1947.
Eck played college football for three years at Colgate University, from which he graduated in 1938. After coaching high school football in Massachusetts, he served as a special projects officer in the United States Army Air Forces during World War II. From 1952 to 1955, he coached football at Thornton Academy in Saco, Maine, tallying a mark of 33–4–2 that featured a 24-game winning streak. His teams at Thornton won two Western Maine Conference titles and two State of Maine Class FFF titles.
Head coaching record
College
See also
List of college football head coaches with non-consecutive tenure
References
1914 births
1988 deaths
Colgate Raiders football players
UMass Minutemen football coaches
High school football coaches in Maine
United States Army Air Forces personnel of World War II
United States Army Air Forces officers | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,781 |
The purpose of this article is to document a huge coincidence which we were a part of on this hike, so it is by no means a complete description of the hike.
On 9/05/16 Shelley and I woke from our 8,500' camp on day 2 of our 4-day hike on the Bechler River Trail in southwest Yellowstone. Winds were strong, temperatures were cold, and there was a mixture of rain and snow falling. Based on the crappy weather we decided to forego breakfast for a few hours and hike to lower ground, where hopefully the weather would be more dining friendly. We packed the majority of our supplies while still inside our tent, hopped out to quickly pack our tent and the remainder of our gear in miserable weather, and got the hell out of there.
We hiked for several hours in near whiteout conditions, including a trek across a meadow with stretches of 8-10" deep water which we didn't even attempt to find a way around. At one point a short while later I completely lost my bearings, thinking we were headed east (back towards our campsite) when in reality we were headed southwest (correct). We had to have complete faith in our GPS readings which provided the only reassurance that we were headed in the right direction. Such disorientation had never happened to me before. We didn't even think about pausing when we passed campsite 9D4, our reserved campsite for the night.
The weather finally started to improve after several hours and 1000'+ feet of elevation loss. Arriving at campsite 9B8 the sun suddenly appeared in full force. With a deer grazing nearby, a small waterfall behind the camp, nice flat tent spots, and the appearance of the sun, Shelley dubbed the site "Shangri La". Admittedly it was a nice site, but I suspect that part of the reason Shelley was so infatuated with it was because of the crappy weather we'd been hiking in all day. With the site unoccupied, Shelley was all for making it our camp, but since it was only 3:00 in the afternoon I was in favor of logging a few more miles before calling it a night.
We took a very short stroll down the trail and studied the first of our three fords of the Bechler River which we would have to make on this hike. Although fairly long it didn't look terribly deep, so I used this fact to make my case that we should continue hiking a bit more and get one of the three fords out of the way. We were still quite wet from the precipitation earlier in the day so it only made sense to put a ford behind us.
We walked back to Shangri La and Shelley debated with herself for a few minutes, then decided it would be best to continue with our hike a little longer. The ford was not too bad, although I thought it turned out to be surprisingly deep and swift for so far up the river; it made me dread the thought of the two fords which still awaited us downstream.
After the ford we continued one more mile down the trail to campsite 9B7. Since the site was unoccupied, and Shelley and I would never think of continuing and being forced to make an illegal campsite (ahem), we called it a day and began setting up our camp. While setting up camp two hikers appeared heading up the trail, and we grimaced at the prospect that this might be their site (Yellowstone backcountry campsites are made on an advance reservation basis). We breathed a sigh of relief as they passed the junction without a pause and continued up the trail.
A few minutes later two more hikers appeared, but this time we were not so fortunate and we braced ourselves for the coming meeting. We already knew that if the campers wanted us gone we would pack up and leave, but the two gentlemen immediately welcomed us to use their site, and selected a tent spot some distance from ours. The two hikers were Scott from Portland and Dwight from Fort Worth.
We got to know each other a little over dinner and shared a few stories of our adventures. We mentioned that we had climbed Colter Peak during our hike through the Thoroughfare region of the park a number of years ago, and Scott and Dwight told us of their aborted attempt of the same peak.
I awoke the next morning to cold temperatures but clear skies. By the time I put on all my damp clothing and emerged from our tent it was nearly 6:40 am, a massive sleep-in for me, but what else can you do on a cold dark morning but savor the warmth of your sleeping bag? With the help of some fuel from our stove tank I was able to get a small campfire going. When we had started the hike several days before campfires were banned due to wildfire hazard, but with the precipitationof the last 24 hours surely this ban had been lifted by now? Frankly I didn't care - I was cold and still wet from the previous day.
After heating our morning coffee and breakfast I boiled a pot of water for Scott and Dwight, partial payment for allowing us to poach their campsite. They had still not emerged from their tent, but we figured they would be rising shortly.
After they had joined us at the campfire and had their breakfast, we swapped a few more stories. Our hike of Colter Peak came up again, and Shelley mentioned that I had lost my GPS on that hike. Scott asked when this hike took place, and I told him it was August of 2008. It turned out that Scott had found my GPS about 3 weeks later when they had attempted to climb Colter Peak. One has to pause for a minute and contemplate the size of Yellowstone, along with the fact that we were hiking off-trail 17+ miles from the nearest road, to realize how unlikely this was. Eight years down the road, to run into the hikers who made this extremely lucky find just boggles my mind.
I'd actually posted a photo of Colter Peak on facebook.com, which mentioned my lost GPS in the caption. | {
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Combined transcriptome and metabolome analysis identifies defence responses in spider mite-infested pepper (Capsicum annuum)
Yuanyuan Zhang, Harro J. Bouwmeester, Iris F. Kappers
Laboratory of Plant Physiology
Plants regulate responses towards herbivory through fine-tuning of defence-related hormone production, expression of defence genes, and production of secondary metabolites. Jasmonic acid (JA) plays a key role in plant–herbivorous arthropod interactions. To understand how pepper (Capsicum annuum) responds to herbivory, leaf transcriptomes and metabolomes of two genotypes different in their susceptibility to spider mites were studied. Mites induced both JA and salicylic acid (SA) signalling. However, mite infestation and exogenous JA resulted in distinct transcriptome profiles. Compared with JA, mites induced fewer differentially expressed genes involved in metabolic processes (except for genes involved in the phenylpropanoid pathway) and lipid metabolic processes. Furthermore, pathogen-related defence responses including WRKY transcription factors were more strongly induced upon mite infestation, probably as a result of induced SA signalling. Untargeted analysis of secondary metabolites confirmed that JA treatment induced larger changes in metabolism than spider mite infestation, resulting in higher terpenoid and flavonoid production. The more resistant genotype exhibited a larger increase in endogenous JA and volatile and non-volatile secondary metabolites upon infestation, which could explain its stronger defence. Reasoning that in JA–SA antagonizing crosstalk, SA defences are prioritized over JA defences, we hypothesize that lack of SA-mediated repression of JA-induced defences could result in gain of resistance towards spider mites in pepper.
Journal of Experimental Botany
https://doi.org/10.1093/jxb/erz422
Tetranychidae
jasmonic acid
Mite Infestations
secondary metabolites
Herbivory
Zhang, Y., Bouwmeester, H. J., & Kappers, I. F. (2020). Combined transcriptome and metabolome analysis identifies defence responses in spider mite-infested pepper (Capsicum annuum). Journal of Experimental Botany, 71(1), 330-343. https://doi.org/10.1093/jxb/erz422
Zhang, Yuanyuan ; Bouwmeester, Harro J. ; Kappers, Iris F. / Combined transcriptome and metabolome analysis identifies defence responses in spider mite-infested pepper (Capsicum annuum). In: Journal of Experimental Botany. 2020 ; Vol. 71, No. 1. pp. 330-343.
@article{3d4eb815638046528ee111925705e2f5,
title = "Combined transcriptome and metabolome analysis identifies defence responses in spider mite-infested pepper (Capsicum annuum)",
abstract = "Plants regulate responses towards herbivory through fine-tuning of defence-related hormone production, expression of defence genes, and production of secondary metabolites. Jasmonic acid (JA) plays a key role in plant–herbivorous arthropod interactions. To understand how pepper (Capsicum annuum) responds to herbivory, leaf transcriptomes and metabolomes of two genotypes different in their susceptibility to spider mites were studied. Mites induced both JA and salicylic acid (SA) signalling. However, mite infestation and exogenous JA resulted in distinct transcriptome profiles. Compared with JA, mites induced fewer differentially expressed genes involved in metabolic processes (except for genes involved in the phenylpropanoid pathway) and lipid metabolic processes. Furthermore, pathogen-related defence responses including WRKY transcription factors were more strongly induced upon mite infestation, probably as a result of induced SA signalling. Untargeted analysis of secondary metabolites confirmed that JA treatment induced larger changes in metabolism than spider mite infestation, resulting in higher terpenoid and flavonoid production. The more resistant genotype exhibited a larger increase in endogenous JA and volatile and non-volatile secondary metabolites upon infestation, which could explain its stronger defence. Reasoning that in JA–SA antagonizing crosstalk, SA defences are prioritized over JA defences, we hypothesize that lack of SA-mediated repression of JA-induced defences could result in gain of resistance towards spider mites in pepper.",
author = "Yuanyuan Zhang and Bouwmeester, {Harro J.} and Kappers, {Iris F.}",
doi = "10.1093/jxb/erz422",
journal = "Journal of Experimental Botany",
Zhang, Y, Bouwmeester, HJ & Kappers, IF 2020, 'Combined transcriptome and metabolome analysis identifies defence responses in spider mite-infested pepper (Capsicum annuum)', Journal of Experimental Botany, vol. 71, no. 1, pp. 330-343. https://doi.org/10.1093/jxb/erz422
Combined transcriptome and metabolome analysis identifies defence responses in spider mite-infested pepper (Capsicum annuum). / Zhang, Yuanyuan; Bouwmeester, Harro J.; Kappers, Iris F.
In: Journal of Experimental Botany, Vol. 71, No. 1, 01.01.2020, p. 330-343.
T1 - Combined transcriptome and metabolome analysis identifies defence responses in spider mite-infested pepper (Capsicum annuum)
AU - Zhang, Yuanyuan
AU - Bouwmeester, Harro J.
AU - Kappers, Iris F.
N2 - Plants regulate responses towards herbivory through fine-tuning of defence-related hormone production, expression of defence genes, and production of secondary metabolites. Jasmonic acid (JA) plays a key role in plant–herbivorous arthropod interactions. To understand how pepper (Capsicum annuum) responds to herbivory, leaf transcriptomes and metabolomes of two genotypes different in their susceptibility to spider mites were studied. Mites induced both JA and salicylic acid (SA) signalling. However, mite infestation and exogenous JA resulted in distinct transcriptome profiles. Compared with JA, mites induced fewer differentially expressed genes involved in metabolic processes (except for genes involved in the phenylpropanoid pathway) and lipid metabolic processes. Furthermore, pathogen-related defence responses including WRKY transcription factors were more strongly induced upon mite infestation, probably as a result of induced SA signalling. Untargeted analysis of secondary metabolites confirmed that JA treatment induced larger changes in metabolism than spider mite infestation, resulting in higher terpenoid and flavonoid production. The more resistant genotype exhibited a larger increase in endogenous JA and volatile and non-volatile secondary metabolites upon infestation, which could explain its stronger defence. Reasoning that in JA–SA antagonizing crosstalk, SA defences are prioritized over JA defences, we hypothesize that lack of SA-mediated repression of JA-induced defences could result in gain of resistance towards spider mites in pepper.
AB - Plants regulate responses towards herbivory through fine-tuning of defence-related hormone production, expression of defence genes, and production of secondary metabolites. Jasmonic acid (JA) plays a key role in plant–herbivorous arthropod interactions. To understand how pepper (Capsicum annuum) responds to herbivory, leaf transcriptomes and metabolomes of two genotypes different in their susceptibility to spider mites were studied. Mites induced both JA and salicylic acid (SA) signalling. However, mite infestation and exogenous JA resulted in distinct transcriptome profiles. Compared with JA, mites induced fewer differentially expressed genes involved in metabolic processes (except for genes involved in the phenylpropanoid pathway) and lipid metabolic processes. Furthermore, pathogen-related defence responses including WRKY transcription factors were more strongly induced upon mite infestation, probably as a result of induced SA signalling. Untargeted analysis of secondary metabolites confirmed that JA treatment induced larger changes in metabolism than spider mite infestation, resulting in higher terpenoid and flavonoid production. The more resistant genotype exhibited a larger increase in endogenous JA and volatile and non-volatile secondary metabolites upon infestation, which could explain its stronger defence. Reasoning that in JA–SA antagonizing crosstalk, SA defences are prioritized over JA defences, we hypothesize that lack of SA-mediated repression of JA-induced defences could result in gain of resistance towards spider mites in pepper.
U2 - 10.1093/jxb/erz422
DO - 10.1093/jxb/erz422
JO - Journal of Experimental Botany
JF - Journal of Experimental Botany
Zhang Y, Bouwmeester HJ, Kappers IF. Combined transcriptome and metabolome analysis identifies defence responses in spider mite-infested pepper (Capsicum annuum). Journal of Experimental Botany. 2020 Jan 1;71(1):330-343. https://doi.org/10.1093/jxb/erz422
10.1093/jxb/erz422Licence: CC BY-NC
Data from: Combined transcriptome and metabolome analysis identifies defence responses in spider-mite infested pepper | {
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\section{Introduction}
In this paper, we study properties of the local time of a random walk and special sites among a random walk range or a Gaussian free field.
Major parts of this paper are survey and contains some minor extensions of existing results.
As one of the motivation of this study, we are interested in the relation between the local time of a random walk and a Gaussian free field.
As a well known result, the generalized second Ray-Knight theorem makes the local time of random walk and a Gaussian free field closely linked as follows.
To state this theorem, we consider a reversible continuous-time random walk $\{S_t\}_{t\ge0 }$ on a graph $G$,
the total conductance (or weight) $\{\lambda_x\}_{x\in G}$ (See the definition in \cite{Sz}),
the local time
$$\tilde{K}(t,x):=\lambda_x^{-1}\int_0^t 1_{\{S_l=x \}} ds,$$
and
$$\tilde{\tau}_t:=\inf\{s: \tilde{K}(s,0)>t\}.$$
Pick a certain point in $G$ as the origin.
Let $\{\phi(x)\}_{x \in G}$ be the Gaussian free field with mean zero and
Cov$(\phi(x),\phi(y))=E^x[\tilde{K}(T_0,y)]$,
where $T_0$ is stopping time to the origin.
\begin{thm}[\cite{Eisen} :
The generalized second Ray-Knight theorem]
\begin{align*}
&\text{The law of }\{\tilde{K}(\tilde{\tau}_t,x)+\frac{1}{2}\phi(x)^2 \}_{x\in G} \text{ under } P \times \mathbf{P}\\
&\text{is the same as that of }\{ \frac{1}{2}(\phi(x )+ \sqrt{2t})^2 \}_{x\in G} \text{ under }\mathbf{P},
\end{align*}
where $\mathbf{P}$ is the probability of the Gaussian free field and
$P$ is that of the random walk starting the origin.
\end{thm}
A simple sorting of above yields the following.
\begin{crl}[\cite{Eisen}]
\begin{align*}
\{\frac{\tilde{K}(\tilde{\tau}_t,x)-t}{\sqrt{2t}} \}_{x\in G}
\to \{\phi(x )\}_{x\in G} \text{ in law as } t\to \infty.
\end{align*}
\end{crl}
We consider that this corresponds to the central limit theorem for the local time $\tilde{K}(\tilde{\tau}_t,x)$, and hence,
the limit of $\tilde{K}(\tilde{\tau}_t,x)$ as the central limit theorem is the corresponding Gaussian free field.
Then, we have the question:
how far are the distribution of the local time from that of the Gaussian free field at each time?
Then, we are interested in two processes and
especially study the relation between special points of a discrete-time simple random walk in
$\mathbb{Z}^2$ or $\mathbb{Z}^2_n$(:=$\mathbb{Z}^2/n\mathbb{Z}^2 $)
where the local time is large or small
and special points in $\mathbb{Z}^2_n$ where the corresponding Gaussian free field takes large value.
To explain properties of such points of a simple random walk in $\mathbb{Z}^d$, we give some definitions of a simple random walk.
(In the sequel, we consider only discrete-time simple random walk in $\mathbb{Z}^d$ (or $\mathbb{Z}^2_n$). )
Let $\{S_k\}_{k=0}^{\infty}$ be a simple random walk on the $d$-dimensional square lattice.
Let $d(x,y)$ be the Euclidean distance for $x,y \in \mathbb{Z}^d$.
Let $D(x,r):=\{y\in \mathbb{Z}^d: d(x,y)\le r\}$ and for $G\subset \mathbb{Z}^d$,
$\partial G:=\{y\in G: d(x,y)=1\text{ for some } x\in G^c \}$.
Let $P^x$ denote the probability measure for the simple random walk starting at $x$.
We simply write $P$ for $P^0$.
Let $K(n,x)$ be the number of visits of the simple random walk to $x$ until time $n$,
that is, $K(n,x)=\sum_{i=0}^n1_{\{S_i=x\}}$.
For $D\subset \mathbb{Z}^d$, let
$T_D:=\inf \{m\ge1: S_m\in D\}$.
In particular, we write $T_{x_1,...,x_j}$ for $T_{\{x_1,...,x_j\}}$.
Let $\tau_n:=\inf \{m\ge 0: S_m\in \partial D(0,n)\}$.
In addition, $\lceil a \rceil$ denotes the smallest integer $n$ with $n \ge a$.
\section{Some estimates for favorite points in $\mathbb{Z}^d$ with $d\ge 2$}
We call the most frequently visited site among all of the random walk range (up to a specific time) favorite point.
About fifty years ago, Erd\H{o}s and Taylor \cite{er} proposed a problem concerning the simple random walk in $\mathbb{Z}^d$:
how many times does the random walk revisit the favorite point?
In fact, \cite{er} showed
that for the simple random walk in $d\ge3$
\begin{align*}
\lim_{n \to \infty} \frac{\max_{x\in {\mathbb Z}^d}K(n,x)}{\log n}= \frac{1}{-\log P(T_0<\infty)}\quad \text{ a.s.}
\end{align*}
In addition, for $d=2$, they obtained
\begin{align*}
\frac{1}{4\pi} \le
\liminf_{n \to \infty} \frac{\max_{x\in {\mathbb Z}^2}K(n,x)}{(\log n)^2}\le
\limsup_{n \to \infty} \frac{\max_{x\in {\mathbb Z}^2}K(n,x)}{(\log n)^2}\le \frac{1}{\pi} \quad \text{ a.s.},
\end{align*}
and conjectured that the limit exists and equals $1/\pi$ a.s.
Forty years later, Dembo et al. \cite{Dembo} verified this conjecture,
that is, they showed that for a simple random walk in ${\mathbb{Z}^2}$,
\begin{align*}
\lim_{n \to \infty} \frac{\max_{x\in {\mathbb Z}^2}K(\tau_n,x)}{(\log n)^2}
=\lim_{n \to \infty} \frac{4\max_{x\in {\mathbb Z}^2}K(n,x)}{(\log n)^2}
= \frac{4}{\pi} \quad a.s.
\end{align*}
In addition, for $0<\alpha<1$, \cite{Dembo, Dembo2} defined the set of $\alpha$-favorite points
in $\mathbb{Z}^2$ such that
\begin{align*}
\Psi_n(\alpha):=\bigg\{x: K(\tau_n,x)\ge \bigg\lceil \frac{4\alpha}{\pi} (\log n)^2 \bigg\rceil \bigg\}.
\end{align*}
It is known that the law of large numbers for the random walk range in $\mathbb{Z}^2$ holds by \cite{jain}.
\cite{Dembo} showed the law of large numbers for $ \Psi_n(\alpha)$.
After this, \cite{rosen} made another proof of results of \cite{Dembo}.
In addition, there are many works studying properties of the favorite points for any dimension such as the number of visit to it and the location of it.
For example, in \cite{okada1} we showed that the favorite points of the simple random walk in $\mathbb{Z}^d$ with $d\ge2$ does not appear in the boundary of the random walk range from some time on a.s.
Additional open problems concerning geometric structure of the favorite points are raised by Erd\H{o}s and R\'ev\'esz \cite{er2, er3} and Shi and T\'oth \cite{shi} but almost no definite solution to them is known for multi-dimensional walks.
Now, we introduce results of \cite{okada1}.
Set $R(n) := \{S_0,S_1, \ldots, S_n\}$ as the random walk range until time $n$.
Moreover, we set $$M(n):=\max_{x \in \partial R(n)}K(n,x).$$
The first theorem provides us with sharp asymptotic behavior of $M(n)$.
\begin{thm}[\cite{okada1}]\label{k1}
For $d\ge2$
\begin{align*}
\lim_{n \to \infty} \frac{M(n)}{\log n}=\beta_d \quad \text{ a.s.},
\end{align*}
where
\begin{align*}
\beta_d=\frac{1}{- \log P(T_0<T_b)}
\end{align*}
and $b$ is a neighbor of the origin.
\end{thm}
To compare to $|\Psi_n(\alpha)|$, we define $\Theta_n (\delta)$ for $n \in \mathbb{N}$ and $0<\delta<1$ as
\begin{align*}
\Theta_n(\delta):=| \{ x\in \partial R(n) :
\frac{K(n,x)}{\log n}\ge \beta_d\delta \}|.
\end{align*}
In fact, it is known that the law of large numbers for the boundary of the range of a transient random walk holds by \cite{okada}.
Then, we have a question : does the law of large numbers for $\Theta_n(\delta)$ hold?
The following corresponds to the answer.
\begin{thm}[\cite{okada1}]\label{k2}
For $d\ge2$ and $0<\delta<1$,
\begin{align*}
\lim_{n \to \infty} \frac{\log \Theta_n(\delta)}{\log n}=1-\delta \quad \text{ a.s.}
\end{align*}
\end{thm}
Next, we will extend these results to that of some general boundary.
Fix $M\in \mathbb{N}$.
Let ${\cal H}={\cal H}(M):=\{H\subset \mathbb{Z}^d: \{0\} \notin H, |H|\le M, P(T_x < T_H )>0 \quad \forall x \in H^c \}$.
For any $H\subset \mathbb{Z}^d$ and $G\subset \mathbb{Z}^d$, let
$$\partial_H G:= \{y\in G: y+H \subset G^c \} .$$
Note that if we let ${\cal N}(0)$ be the set of neighbors of the origin, $\cup_{b \in {\cal N}(0)} \partial_{\{b\}} R(n)=\partial R(n)$ holds.
In addition, we need the condition ``$P(T_x < T_H )>0$ $\forall x \in H^c$" in the proof of Lemma \ref{hh+} below.
This condition verifies that $H$ does't not surround the origin.
\begin{thm}\label{p0}
For $d\ge3$ and $H\in {\cal H}$
\begin{align*}
\lim_{n \to \infty} \frac{|\partial_H R(n)|}{n}=q \quad \text{ a.s.},
\end{align*}
where
\begin{align*}
q:=P((\{S'_m\}_{m=0}^\infty \cup \{S_m\}_{m=0}^\infty) \cap H=\emptyset \text{ and }0 \in \{S_m\}_{m=1}^\infty)
\end{align*}
and
$\{S'_m\}_{m=0}^\infty$ denotes an independent copy of $\{-S_m\}_{m=0}^\infty$.
\end{thm}
Compare this to Theorem $2.1$ in \cite{okada}.
It is trivial that almost the same argument as in the proof of Theorem $2.1$ in \cite{okada} yields this result.
\begin{rem}
Note that any choice of $H\in {\cal H}$ does not realize $\partial _H G= \partial G$ in general.
However, Theorem \ref{p0} yields Theorem $2.1$ in \cite{okada} easily.
Note that for $H_1$, $H_2\subset \mathbb{Z}^d$ and $G \subset \mathbb{Z}^d$
\begin{align*}
\partial_{H_1} G \cap \partial_{H_2} G
= \partial_{H_1 \cup H_2} G.
\end{align*}
Set ${\cal U}_k := \{ A \subset {\cal N} (0) : |A |= k \}$.
Then, the inclusion-exclusion identity yields
\begin{align*}
\sum_{k=1}^{2d}
\sum_{H \in {\cal U}_k }
(-1)^{k+1}
|\partial_H R (n) | = | \partial R (n) |.
\end{align*}
In addition,
\begin{align*}
&\sum_{k=1}^{2d}
\sum_{H \in {\cal U}_k }
(-1)^{k+1}
P((\{S'_m\}_{m=0}^\infty \cup \{S_m\}_{m=0}^\infty) \cap H_k=\emptyset \text{ and }0 \in \{S_m\}_{m=1}^\infty)\\
=&P((\{S'_m\}_{m=0}^\infty \cup \{S_m\}_{m=0}^\infty) \not\supset {\cal N}(0) \text{ and }0 \in \{S_m\}_{m=1}^\infty).
\end{align*}
Therefore, the desired result holds.
\end{rem}
For $H\in {\cal H}$, we set $$M_H(n):=\max_{x \in \partial_H R(n)}K(n,x).$$
The following corresponds to the extension of Theorem \ref{k1}.
\begin{thm}\label{p1}
For $d\ge3$ and $H\in {\cal H}$,
\begin{align*}
\lim_{n \to \infty} \frac{M_H(n)}{\log n}=\beta_d(H) \quad \text{ a.s.},
\end{align*}
where
\begin{align*}
\beta_d(H)=\frac{1}{- \log P(T_0<T_H)}.
\end{align*}
\end{thm}
To compare to $\Theta_n (\delta)$, we define $\Theta_n (\delta,H)$ for $n \in \mathbb{N}$ and $0<\delta<1$ as
\begin{align*}
\Theta_n(\delta,H):=|\{ x\in \partial_H R(n) :
\frac{K(n,x)}{\log n}\ge \beta_d(H)\delta \}|.
\end{align*}
In view of Theorem \ref{p1} together with Theorems \ref{k1} and \ref{k2}, we have a question :
does the law of large numbers for $\Theta_n(\delta,H)$ hold?
The following corresponds to the answer.
\begin{thm}\label{p2}
For $d\ge3$, $H\in {\cal H}$ and $0<\delta<1$,
\begin{align*}
\lim_{n \to \infty} \frac{\log \Theta_n(\delta,H)}{\log n}=1-\delta \quad \text{ a.s.}
\end{align*}
\end{thm}
\begin{rem}
We extend Theorems \ref{p1} and \ref{p2} to Theorems \ref{k1} and \ref{k2}
by using the same argument as in Remark after Theorem \ref{p0}.
\end{rem}
As we will see, the proof of Theorem \ref{p1} is a minor modification of that of Theorem \ref{k1}.
Note that Theorem \ref{k2} follows from assertions developed in the proof of Theorem \ref{k1}.
Analogously, Theorem \ref{p2} follows in the same manner once we proved Theorem \ref{p1}.
Thus we argue only the proof of Theorem \ref{p1}.
To show Theorem \ref{p1}, let $T_x^0=\inf\{j\ge0: S_j=x\}$ and for $p\ge1$,
\begin{align*}
T_x^p=\inf\{ j>T_x^{p-1}: S_j=x\}
\end{align*}
with the convention $\inf \emptyset =\infty$.
To show the upper bound, we provide the following:
\begin{lmm}\label{up}
For $\beta>0$ and $ H\in {\cal H}$ there exists $C>0$ such that for any $n\in \mathbb{N}$
\begin{align*}
E[\tilde{\Theta}_n(\beta,H)]\le Cn^{1-\frac{\beta}{\beta_d(H)}},
\end{align*}
where
\begin{align*}
\tilde{\Theta}_n(\beta,H)
:=|\{ x\in\partial_H R(T_x^{\lceil \beta\log (n/2) \rceil }) :K(n,x)\ge \lceil \beta \log \frac{n}{2}\rceil \}|.
\end{align*}
\end{lmm}
\begin{rem}
Note that Lemma \ref{up} corresponds to Lemma $3.1$ in \cite{okada1}.
Therefore, if we obtain Lemma \ref{up}, by the same argument as in the proof of Proposition $3.1$ in \cite{okada1},
the desired upper bound follows with minor modifications.
\end{rem}
\begin{proof}
First, we introduce the elementary property corresponding to $(3)$ in \cite{okada1}.
For any intervals $I_0$, $I_1$, $I_2\subset {\mathbb N}\cup \{0\}$ with $I_0 \subset I_1 \subset I_2$, it holds that
\begin{align*}
R(I_0)\cap \partial_H R (I_2) \subset \partial_H R (I_1).
\end{align*}
Then, if we change $\partial$ in the proof of Lemma $3.1$ in \cite{okada1} to $\partial_H$,
we obtain the result with minor modifications.
\end{proof}
Next, to show the lower bound, we will define the following.
For $k\in {\mathbb{N}}$ and $\beta<\beta_d(H)$, let
$h_k=\beta\log P(T_0<T_H\wedge k)+1$.
Let $u_n:=\lceil \exp(n^2)\rceil$ and
$$I_n:=[\frac{u_{n-1}}{n^2}, u_{n-1}-k \lceil \beta_d(H) n^2\rceil]\cap {\mathbb{N}}.$$
For any $l\in I_n$, we introduce the event $A_{l,n}$ defined by
\begin{align*}
E_{l,n}:=\{ T_{S_l}^{j}-T_{S_l}^{j-1}<k \text{ for any }1\le j\le \lceil \beta n^2\rceil \},
\end{align*}
and
\begin{align*}
A_{l,n}:=\{S_l \in R(l-1)^c \cap \partial_H R(u_{n}) \}\cap E_{l,n}.
\end{align*}
Then, we set
\begin{align*}
Q_n:=\sum_{l \in I_n }1_{A_{l,n}}.
\end{align*}
\begin{lmm} \label{hh+}
Let $\beta<\beta_d$ and take $k\in{\mathbb{N}}$.
Then, there exists $c>0$ such that for any $n\in{\mathbb{N}}$,
\begin{align*}
EQ_n \ge c\exp(h_kn^2-2n).
\end{align*}
\end{lmm}
\begin{lmm} \label{hh}
Let $\beta<\beta_d$ and take $k\in{\mathbb{N}}$.
Then, there exists $C>0$ such that for any $n\in{\mathbb{N}}$,
\begin{align*}
\mathrm{Var} (Q_n) \le C\exp(2h_kn^2-4n)\times \frac{1}{n^{10}}.
\end{align*}
\end{lmm}
\begin{rem}
Note that Lemmas \ref{hh+} and \ref{hh} correspond to Lemmas $4.2$ and $4.3$ in \cite{okada1}.
Therefore, if we change $\partial_b$ in the proof of Proposition $4.1$ in \cite{okada1} to $\partial_H$,
Lemmas \ref{hh+} and \ref{hh} yield the lower bound with minor modifications.
\end{rem}
\begin{proof}[Proof of Lemmas \ref{hh+} and \ref{hh}]
Again, note that Lemmas \ref{hh+} and \ref{hh} correspond to Lemmas $4.2$ and $4.3$ in \cite{okada1}.
Especially, we need the condition ``$P(T_x < T_H )>0$ $\forall x \in H^c$" in ${\cal H}$ in the proof of Lemma \ref{hh+}.
This condition yields $P(T_0 \wedge T_H=\infty),P( T_H=\infty)>0$.
Lemma $4.4$, Corollary $4.1$, Lemmas $4.5$ and $4.6$ and the simple summation over $I_n$ in \cite{okada1} yield Lemmas $4.2$ and $4.3$ in \cite{okada1}.
If we change $b$ in the proof of \cite{okada1} to $H$,
we obtain results corresponding to Lemma $4.4$, Corollary $4.1$, Lemmas $4.5$ and $4.6$ in \cite{okada1} with minor modifications.
Especially, the facts $P(T_0 \wedge T_H=\infty),P( T_H=\infty)>0$ are used to show the assertion corresponding to Lemma $4.5$.
Other arguments are same as that of \cite{okada1}.
Therefore, desired results follow.
\end{proof}
\begin{proof}[Proof of Theorem \ref{p1}]
If we change $b$ in the proof of \cite{okada1} to $H$,
we obtain results corresponding to Proposition $4.1$ in \cite{okada1} with minor modifications.
\end{proof}
Now, we provide two open problems for $d=2$.
\begin{open}
For $d=2$, is it true that results corresponding to Theorems \ref{p1} and \ref{p2} hold?
\end{open}
We believe that it is true and explain why it is difficult to solve it.
If we solve the lower bound of this by the same argument as \cite{okada1},
we need to estimate the following:
for any $n\in \mathbb{N}$,
\begin{align*}
P(T_H > \lceil \frac{u_n}{n} \rceil),\quad
P(T_0\wedge T_H > \lceil \frac{u_n}{n} \rceil)
\end{align*}
and for any $n\in \mathbb{N}$ and $x\in {\mathbb{Z}^{2}}$
with $0<|x|< n\sqrt{u_{n-1}}$,
\begin{align*}
P(T_H \wedge T_{x+H}> \lceil \frac{u_n}{n} \rceil).
\end{align*}
Note that these correspond to (52), (53) or Lemma $4.7$ in \cite{okada1}.
In Corollary 2 and (1.2) of \cite{kesten}, they showed that for $H\subset \mathbb{Z}^2$,
\begin{align*}
\sum_{i\in H} P^i (T_H\ge n) \sim \frac{\pi}{\log n},
\end{align*}
where $a_n \sim c_n$ means $a_n/c_n \to 1$ $(n\to \infty)$ for sequences $a_n$ and $c_n$.
In addition, \cite{okada} yields
\begin{align*}
P^i (T_0 \wedge T_b \ge n) \sim \frac{\pi}{2\log n},
\end{align*}
where $b$ is a neighbor of the origin.
However, it is difficult to estimate the individual term for a general $H\subset \mathbb{Z}^2$,
that is, $P^i (T_H\ge n)$ for $i\in H$.
In addition, we will consider a certain extension of Theorem \ref{k1} for $d=2$.
\begin{open}
Consider $\tilde{D}_n\in {\cal H}$ with $\tilde{D}_n\subset D(0, a_n)^c$ and $a_n \to \infty$ as $n \to \infty$.
Then, how is the asymptotic of $M_{\tilde{D}_n}(n)$ with probability one?
\end{open}
We are especially interested in the case
that $\tilde{D}_n\in {\cal H}$ with $\tilde{D}_n\subset D(0, a_n)^c \cap D(0, 2a_n)$ and $a_n=n^{\beta}$ for $0<\beta<1/2$.
In fact, \cite{Dembo1} showed the largest disc completely covered until $\tau_n$ by the simple random walk in $\mathbb{Z}^2$ is $n^{1/2+o(1)}$.
Then, we conjecture that this argument yields that if we set $a_n\ge n^{1/2+\epsilon}$ for some $\epsilon>0$, it holds that
\begin{align*}
M_{\tilde{D}_n}(n)=\max_{x\in \mathbb{Z}^2} K(\tau_n,x) \quad
\text{for all sufficiently large }n\in \mathbb{N}\quad \text{ a.s.}
\end{align*}
\section{Some estimates for favorite points and late points of simple random walks and high points of Gaussian free fields in two dimensions}
First, we explain the high points of the Gaussian free field in $\mathbb{Z}^2_n$.
Originally,
Bolthausen, Deuschel and Giacomin \cite{Bol} showed that in probability
\begin{align*}
\lim_{n \to \infty}\frac{ \max_{x \in \mathbb{Z}_n^2 }\phi_n(x)}{\log n}=2\sqrt{\frac{2}{\pi}},
\end{align*}
where $\{\phi_n(x)\}_{x \in \mathbb{Z}_n^2}$ is the Gaussian free field defined in \cite{Bol,ol}.
In what follows, for $0<\alpha<1$, we define the set of $\alpha$-high points of the Gaussian free field by
\begin{align*}
{\cal V}_n(\alpha):=\bigg\{ x\in {\mathbb{Z}^2_n} :
\frac{\phi_n(x)^2}{2} \ge \frac{4\alpha}{\pi} (\log n)^2 \bigg\}.
\end{align*}
Next, we explain late points of a simple random walk in $\mathbb{Z}^2_n$.
Originally,
Dembo, Peres, Rosen and Zeitouni \cite{Dembo3} showed that
for a simple random walk in ${\mathbb{Z}^2_n}$ in probability
\begin{align*}
\lim_{n \to \infty} \frac{\max_{x\in {\mathbb Z}^2_n}T_x }{(n\log n)^2}= \frac{4}{\pi}.
\end{align*}
(Now, we consider $T_x$ as the stopping time to $x$ of a simple random walk in $\mathbb{Z}^2_n$.)
In what follows, for $0<\alpha<1$, we define the set of $\alpha$-late points in $\mathbb{Z}^2_n$
such that
\begin{align*}
{\cal L}_n(\alpha):=\bigg\{ x\in {\mathbb{Z}^2_n} :
\frac{T_x}{(n\log n)^2}\ge \frac{4\alpha}{\pi} \bigg\}.
\end{align*}
For any $0< \alpha,\beta <1$ and $\{G_n\}_{n=1}^{\infty}$ with $G_n\subset \mathbb{Z}^2$ (or $\mathbb{Z}_n^2$), let
\begin{align*}
&Q_j=Q_j (G_n):=\lim_{n\to \infty}\frac{\log
|\{ (x_1,...,x_j)\in G_n^j:d(x_i,x_l)\le n^{\beta} \text{ for any }1\le i,l \le j \}|}{\log n}\\
&\hat{Q}_j=\hat{Q}_j (G_n):=\lim_{n\to \infty}\frac{\log
E[|\{ (x_1,...,x_j)\in G_n^j:d(x_i,x_l)\le n^{\beta} \text{ for any }1\le i,l \le j \}||]}{\log n}
\end{align*}
if the right hand sides exist.
For any $0< \alpha,\beta <1$, set
\begin{align*}
&\rho_2(\alpha, \beta):=
\begin{cases}
2+2\beta-\frac{4\alpha}{2-\beta}&(\beta\le 2(1-\sqrt{\alpha}))
\\
8(1-\sqrt{\alpha})-4(1-\sqrt{\alpha})^2/\beta&(\beta\ge 2(1-\sqrt{\alpha})),
\end{cases}
\\
&\hat{\rho}_2(\alpha, \beta):=
\begin{cases}
2+2\beta-\frac{4\alpha}{2-\beta}&(\beta\le 2-\sqrt{2\alpha})
\\
6-4\sqrt{2\alpha}&(\beta\ge 2-\sqrt{2\alpha}).
\end{cases}
\end{align*}
All the known results for $Q_2({\cal L}_n(\alpha))$, $Q_2({\cal V}_n(\alpha))$ and $Q_2(\Psi_n(\alpha))$ (or the corresponding ones for $\hat{Q}_2$) are summarized in Table $1$ below.
\begin{table}[h]
\caption{Known results for $Q_j$ and $\hat{Q}_j$}
\begin{tabular}{|l||c|c|c|c|} \hline
& $Q_2$ in prob. & $\hat{Q}_2$ & $Q_j$ in prob. $\forall j\in \mathbb{N}$ & $\hat{Q}_j$ $\forall j\in \mathbb{N}$ \\ \hline \hline
${\cal L}_n(\alpha)$ &$\rho_2(\alpha, \beta)$ : \cite{Dembo2} &$\hat{\rho}_2(\alpha, \beta)$ : \cite{ho} & Open prob. by \cite{Dembo*, Dembo**} & Unsolved \\ \hline
${\cal V}_n(\alpha)$ & $\rho_2(\alpha, \beta)$ : \cite{ol} &$\hat{\rho}_2(\alpha, \beta)$ : \cite{ol} &Unsolved &Unsolved \\ \hline
${\Psi}_n(\alpha)$ & Open prob. by \cite{Dembo2}& Unsolved &Unsolved &Unsolved \\ \hline
\end{tabular}
\end{table}
If we write $"A: [b]"$ in the above table, the value of $Q_2$ (or $\hat{Q}_2$) is identified with
$A$ and $[b]$ is the reference which solved this problem.
In \cite{okada2}, we solve the problems concerning $Q_2(\Psi_n(\alpha))$ or $\hat{Q}_2(\Psi_n(\alpha))$.
Our result shows that all three exponents in almost sure sense (that is, $Q_2$) coincide with one another.
We also estimate this value in average (that is, $\hat{Q}_2$) and obtain a similar coincidence.
In addition, \cite{okada3} solves the problems concerning $Q_j({\cal L}_n(\alpha))$ or $\hat{Q}_j({\cal L}_n(\alpha))$ for $j \in \mathbb{N}$.
As stated in Theorem $1.1$, \cite{Eisen} gave a powerful equivalence in law.
Ding, Lee, and Peres \cite{Ding1, Ding2} gave a strong connection between the expected maximum of the Gaussian free field and the expected cover time.
Then, we are interested in the stronger relation between $\Psi_n(\alpha)$ and $ {\cal V}_n(\alpha)$ (or ${\cal L}_n(\alpha)$) and
search for the difference and the similarity.
We suggest the following problem.
\begin{open}
Consider the slightly different Gaussian free field from that defined by \cite{Bol, ol},
which is defined on $D(0,n)$ and whose covariance is $E^x[K(\tau_n,y)]$ for $x,y\in D(0,n)$.
In addition, consider the corresponding ${\cal V}_n(\alpha)$.
Is there the sequence of the coupling of the Gaussian free field in $D(0,n)$ and simple random walk in $\mathbb{Z}^2$ such that
for any $\epsilon>0$, $\epsilon<\alpha<1$ and all sufficiently large $n\in \mathbb{N}$,
\begin{align}\label{g*}
&P(\Psi_n(\alpha)\subset {\cal V}_n(\alpha-\epsilon) )=1\\
\label{g**}
\text{or }\quad&P( {\cal V}_n(\alpha) \subset\Psi_n(\alpha-\epsilon) )=1?
\end{align}
\end{open}
\section*{Acknowledgments. }
We are grateful to Prof. Kazumasa Kuwada for interesting discussion.
In addition, we thank referees for careful reading our manuscript and for giving useful comments.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,375 |
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\section{Introduction}
\IEEEPARstart{H}{eart} rate (HR) is one of the most convenient measurements that reflects the performance of the cardiovascular system and the overall health condition \cite{palatini2011role}. The abnormal escalation of HR indicates a possibility of any failures in the physique, but can often be detected and diagnosed prior to the emergence of the symptoms traditionally by monitoring the HR using an electrocardiogram (ECG) \cite{palatini2011role, reil2011heart, parak2015evaluation, nystoriak2018cardiovascular}. Daily activities can be categorized into a number of states with different physical activity levels (PAL), in which the extracted physical information can be used to forecast the possibility of having cardiovascular disease (CVD) \cite{mukhopadhyay2015wearable}. For instance, a study screening for metabolic dysfunctions by Fernandes \textit{et al.} \cite{fernandes2013resting} demonstrated a correlation between HR during the resting period and the risk of having CVD. Dyslipidemia and high blood glucose were found to be correlated with high HR during passive activity in adolescents. Similarly, a low HR during recovery state, a duration after an intense exercise, has been shown to indicate a possibility of coronary artery disease \cite{ghaffari2011abnormal}. Recently, many successful diagnostic processes of cardiac arrhythmia such as atrial fibrillation have adapted smart wearable devices to aid HR monitoring and further health deterioration prevention \cite{dorr2019watch}.
With the advanced technology, the concept of wearable devices has increased the mobility of the users, expanding the utilization of the apparatus into the daily life application beyond the confinement of the clinical establishments \cite{tedesco2019accuracy, sawangjai2019consumer, lakhan2019consumer, autthasan2019single}.
\iffalse
Consumer grade devices are generally portable, affordable, and user-friendly, equipped with trackers and sensors for measuring the real-time health status and various physiological parameters including neural oscillations, body temperature, respiration rate, blood pressure, sweat rate, gait, and posture, as well as other physical motions \cite{tedesco2019accuracy, lakhan2019consumer, autthasan2019single, sawangjai2019consumer}.
\fi
Many wearable ECG devices provide HR estimation with validated accuracy, however, with a requirement of contact to the chest area using a chest strap which reduces mobility, causes inconvenience, and limits the application of the devices \cite{spierer2015validation}. First described in 1937 \cite{hertzman1937observations}, Photoplethysmography (PPG) was introduced to estimate HR (HR$_\text{PPG}$) with the benefit of mobility and the ease of integration to ubiquitous devices \cite{asada2003mobile}. There are two types of PPG sensors: transmission and reflection \cite{ram2011novel, joseph2014photoplethysmogram}. A transmission type PPG sensor measures the light intensity on the other side of the light source. The reflection PPG sensor generally contains at least one LED and one photo-diode on the same side of the device as a base to detect a relatively small change in reflected light affected by the blood flow. The PPG signal reflects only a small pulsatile portion; only 0.1\% of total signal amplitude\cite{ram2011novel}. Subsequently, various noise components, such as ambient light, the electromagnetic coupling from the other sensors, and motion artifacts (MA), generated by the changes of the distance in the gap between a device and the skin, often disturb the PPG signals \cite{joseph2014photoplethysmogram}. Devices with PPG were previously reported to underestimate the HR$_\text{PPG}$ during an intense activity as movements occurred, preventing the detection of the peak-to-peak interval used to count the heartbeat per minute \cite{bai2018comparative, wallen2016accuracy}. Due to the higher probability of MA while wearing a wearable device, HR estimation by PPG may not yield as accurate readings, and therefore, needs further improvement. These techniques, requiring movement-related information, include an adaptive algorithm for minimum noise generation \cite{yousefi2013motion}, sparse signal reconstruction \cite{zhang2014troika}, and multi-channel spectral matrix decomposition \cite{xiong2016spectral}.
The effort to ameliorate the frameworks for HR monitoring and MA removal and reduction has led to the collection of datasets stored in various databases \cite{biswas2019heart}. In 2015, the IEEE Signal Processing Cup (SPC) database initiated the assembly of the state-of-the-art HR recordings from PPG sensors during different sets of exercises. Several frameworks, consisting of combinations of HR$_\text{PPG}$ estimation techniques, are being evaluated. The TROIKA framework, using sparse signal reconstruction, was tested on a dataset recorded from 12 subjects and was shown to perform with high estimation accuracy \cite{zhang2014troika}. Other frameworks, e.g., WPFV \cite{temko2017accurate} using Wiener filter and phase vocoder, particle filter \cite{nathan2017particle}, and CorNET \cite{biswas2019cornet} using deep learning algorithms, were explored to alleviate the MA. Although the endeavor to correct HR using raw accelerometer and gyroscope has been challenged by previous studies \cite{lee2018motion,pamula2018system,albadawi2018heart,culbert2016motion}, none has exploited the motion sensing elements such as physical activity, step count, and motion pattern, as featured in the improving HR estimation model. These previous works have been anchoring and assuming the existence of possible practical information from the raw data reported by the PPG sensors as well as equipped inertial measurement unit (IMU) motion sensors. However, commercialized devices generally do not grant permission for the alteration of the HR estimation algorithm and the removal of the prominent MA constituents.
The aforementioned studies unquestionably confirmed the benefit of using the accelerometer and/or gyroscope to curtail the undesired MA and correct the PPG signals. However, the PPG signals, logged as a set of time-series data, can be relatively massive, leading to the prominent concern over the storage capacity of any wearable device \cite{winfree2017modeling, henriksen2018using}. Hence, most of the consumer grade wearable devices often record only the HR$_\text{PPG}$. Therefore, we presented a post-calibration method using the information obtained from these wrist-worn devices as well as additional personal information from the users. We proposed two regression models, with and without the rolling window, to improve the HR$_\text{PPG}$ using the built-in sensing constituents as the main features as an attempt to achieve the most accurate HR estimation. Instead of executing artifact removal on the PPG signal, a robust post-calibration method was developed in which it can be directly applied to the derived HR$_\text{PPG}$ with MA residue. Feature selection was performed on the selected devices by testing the linear correlation and mutual information of each feature to the HR$_\text{ECG}$. Without the rolling window, six machine learning (ML) algorithms were formulated and trained with tuned hyperparameters. The rolling regression model has been developed to improve the ultimate estimated HR, abbreviated as HR$_\text{R}$, from the devices using the recorded movement data, attained directly from the devices.
Our contributions of this study can be summarized as:
\iffalse
\textcolor{red}{The PPG sensors employed were wrist-worn in combination with an ECG device attached to the chest area. The protocols established hitherto in these studies involved low to high intensity activities as well as vigorous exercising. However, the HR estimation using PPG sensors has not been explored much during resting and \textcolor{red}{laying down} states. To this end, HR$_\text{PPG}$ is categorized in this study into three different states of the possible daily activity: Resting state (RS), \textcolor{red}{Laying down} state (LS), and Intense treadmill activity state (IS).}
\fi
\begin{enumerate}
\item In addition to the state-of-the-art datasets \cite{zhang2014troika} in which only the recorded HR$_\text{PPG}$ during the intense treadmill activity state were monitored, we demonstrate the validations of HR measurement in three states including resting, \textcolor{red}{laying down}, and intense treadmill activity states performed by the four wrist-worn devices: Fitbit Charge HR \cite{Fitbit}, Apple Watch Series 4 \cite{Applewatch}, TicWatch Pro
\cite{TicWatch}, and E4 \cite{EmpaticaE4}.
\item In order to estimate HR, a novel approach of feature selection from a list of candidates, containing the extracted HR$_\text{PPG}$ (the raw HR provided) from the selected wrist-worn device, PAL, step count, gender, Pittsburgh sleep quality index (PSQI), and Body Mass Index (BMI), was introduced. The input features were chosen using a univariate linear regression with a null hypothesis testing, while the non-linear relationship was processed using the mutual information derived from the ECG (HR$_\text{ECG}$). The physical activity levels (PALs), step count, and rolling windows were subsequently exploited as the main features of the testbed to improve the HR$_\text{PPG}$.
\item The proposed process of post-calibration included further evaluation of the temporal information as features from the rolling windows. The rolling regression was also verified to enhance our proposed methods of HR estimation (HR$_\text{R}$) tested on our dataset of 29 participants (130 mins/participant). All results from the selected models (HR$_\text{ML}$ from ML models, HR$_\text{SF}$ from sensor fusion, and HR$_\text{R}$ from the rolling regression) were compared to demonstrate the most suitable post-calibration methods for HR estimation.
\end{enumerate}
The remainder of this study is organized into seven sections. The background of different candidate features and the ML algorithms are described in Section \ref{Background}. Our set of experiments in Section \ref{Methods} were divided into two main parts: physical data collection and data processing. The results of all the analyses are reported in Sections \ref{result:validation} and \ref{result:calibration}. The importance of the accuracy assessment and the improvement of HR estimation during the post-calibration process is further elaborated in Section \ref{Discussion}, followed by Section \ref{Conclusion} to conclude the proposed post-calibration methods of HR estimation and its accomplishments.
\section{Background}
\label{Background}
The accuracy of HR measurement is required for assessing the heart and the overall health status. Due to the uniqueness of our physical differences, we first determined a list of measurable candidate features exhibiting the potential to improve HR for each individual. A description of each possible feature is provided in this section, followed by the ML algorithms for regression problem: support vector regression (SVR), random forest (RF), Gaussian process (GP), artificial neural network (ANN), Logistic regression (LR) and k-Nearest Neighbors regression (kNN).
\begin{table*}[]
\centering
\caption{Comparison between technical specification of each wearable devices. \small{(* Raw data not available, ** No application in this study.)}}
\label{table:device-compare}
\begin{tabular}{lcccccc}
\hline
\hline
Device & Sensor & LED & Photodiode & 3-axis Accelerometer & Gyroscope & Price [\$]\\
\hline
Apple Watch Series 4 \cite{Applewatch} & ECG* + PPG & 4 (Green) 2 (Infrared) & 8 & Yes & Yes & 399.00\\
Empatica E4 \cite{EmpaticaE4} & PPG & 2 (Green) 2 (Red) & 2 & Yes & No & 1,690.00\\
Fitbit Charge HR \cite{Fitbit} & PPG & 2 (Green) & 1 & Yes** & No & 150.00\\
TicWatch Pro \cite{TicWatch} & PPG & 2 (Green) & 1 & Yes & Yes & 249.99\\
Polar H10 \cite{PolarH10} & ECG & - & - & Yes & No & 89.95\\
Biosignalsplux kit \cite{Biosignalsplux} & ECG & - & - & Yes & No & 1,349.93
\\ \hline \hline
\end{tabular}
\vspace{-5mm}
\end{table*}
\subsection{Wearable devices}
Three consumer grade wrist-worn wearable devices (Fitbit Charge HR \cite{Fitbit}, Apple Watch Series 4 \cite{TicWatch}, and TicWatch Pro \cite{TicWatch}), and one medical grade device class 2a (E4) \cite{EmpaticaE4} wristband available in the market were systemically validated against each other using our proposed protocols for the first time. The technical information for each device is exhibited in Table \ref{table:device-compare}. Fitbit (Fitbit Inc., San Francisco, CA, USA) has been listed as the best seller of the consumer grade wrist-worn wearable products and has been used in validation studies twice as many times as the other brands and 10 times more often than the other 131 brands used in clinical trials \cite{winfree2017modeling, henriksen2018using}. Apple (Apple Inc., Cupertino, CA, USA) dominates over a majority share in the technology sector. The PPG sensor in Apple Watch Series 4 contains more LEDs compared to the other products used to maximize the detection of a pulse wave which may contribute to the higher HR estimation accuracy. The HR$_\text{PPG}$ from TicWatch Pro (Mobvoi Information Technology Company Limited, Beijing, China) can be acquired by their software. TicWatch Pro operates with Wear OS from Google Inc., in which it supports Google service. The device has an energy saving mode, allowing the battery to last for up to 30 days within a single charge. E4 wristband from Empatica (Empatica Inc., Milano, Italy) publicly discloses one of the two algorithms that are parts of the HR calculation. The device also provides Electrodermal Activity (EDA) as well as Infrared to measure the skin temperature. Additionally, two ECG devices were used as the standard measurement of HR stemmed from the ECG (HR$_\text{ECG})$: Biosignalsplux kit (PLUX Wireless Biosignals S.A., Lisbon, Portugal) \cite{Biosignalsplux} and Polar H10 (Polar Electro, Kempele, Finland) \cite{PolarH10}.
\subsection{Candidate features}
\label{bg:features}
\subsubsection{Physical activity level (PAL)}
\label{section:PAL}
PAL is an estimation of the required physical activity in a day, which can be calculated by dividing the total energy expenditure (TEE) by basal metabolic rate (BMR) \cite{joint2004human}. BMR refers to the energy expenditure at a standard condition of resting. PALs from wrist-worn devices are assessed by the equipped accelerometer \cite{migueles2017accelerometer}. Its filtered and processed signals of the movement are used to compute the activity count per minute (cpm). The accelerometer-based PAL is computed from the change of the body movement in each axis of accelerometer and its interval \cite{hildebrand2014age}.
The current wearable technology enhances the fitness tracking and the measurement of PAL, allowing the classification and monitoring of the fitness intensity level and daily expenditure energy level in users of various ages and environmental conditions \cite{miller2010estimating, winfree2017modeling, awais2018physical}. Fitbit Charge HR, Apple Watch Series 4, and TicWatch Pro promptly report PALs, calculated from the signals detected by the furnished accelerometers using their restricted algorithms, whereas E4 only grants raw accelerometer count, in which an external method is required to compute PAL from the provided data. There are numerous PAL estimation methods depending on the locations of the devices in the physical activity research field \cite{migueles2017accelerometer}. The methods used in this study were based on the implementation of the four methods proposed by Freedson \textit{et al.} \cite{pamty2005calibration}, Troiano \textit{et al.} \cite{troiano2008physical}, and Crouter \textit{et al.} \cite{crouter2015estimating}, which are four of the most adopted PAL estimation methods included as a comparative experiment of physical activity \cite{awais2018physical}. The methods from Freedson \textit{et al.} \cite{pamty2005calibration} and Troiano \textit{et al.} \cite{troiano2008physical} were designed for hip placement. However, recent work by Knaier \textit{et al.} \cite{knaier2019validation} adopted the method from Troiano \textit{et al.} to perform PAL estimation with the accelerometer on the wrist in comparison to the hip, demonstrating that the algorithms can be adapted for both hip- and wrist-worn. Distinctively, only the two methods from Crouter \textit{et al.} \cite{crouter2015estimating}, using both vector magnitude (VM) and the vertical axis (VA), were originally designed for the PAL measuring on the wrist. These four PAL estimation methods differ in the threshold or the cut-point of the PAL. The cut-points for the four intensities of PAL (sedentary (SED), light physical activity (LPA), moderate physical activity (MPA), and vigorous physical activity (VPA)) were used following the original paper employing ActiGraph accelerometer by Freedson \textit{et al.} \cite{pamty2005calibration}. The four PALs are summarized in Table \ref{table:cutpoint}.
\begin{table}[t]\setlength\tabcolsep{2pt}
\centering
\caption{Physical activity level (PAL) cut-points in count per minute (cpm).}
\label{table:cutpoint}
\begin{tabular}{cccccc}
\hline
\hline
PAL threshold & Vector & SED & LPA & MPA & VPA\\
\hline
Crouter \textit{et al.} & VA &$\leq35$ & $36-360$ & $361-1129$ & $\geq1130$\\
Crouter \textit{et al.} & VM &$\leq100$ & $101-609$ & $610-1809$ & $\geq1810$\\
Freedson \textit{et al.} & VA &$\leq99$ & $100-759$ & $760-5724$ & $5725-9498$\\
Troiano \textit{et al.} & VA &$\leq100$ & $101-2019$ & $2020-5998$ & $\geq5998$
\\ \hline \hline
\end{tabular}
\vspace{-5mm}
\end{table}
\subsubsection{Step count}
Walking and running are two of the most common physical activities in daily life that consume energy \cite{wilkin2012energy}. In 2009, a study reported a positive relationship between the HR recovery level and the number of steps \cite{lubans2009relationship}. Almost all wrist-worn wearable devices are incorporated with the step counting feature as the main measurement to evaluate the daily activity performance of the user, especially in the fitness trackers. All devices that were used in this study, with the exception of the ECG sensor from Biosignalsplux kit, are equipped with the pedometer to record the steps along with either HR$_\text{ECG}$ or HR$_\text{PPG}$ continuously and simultaneously. Serving a different function from the previously mentioned cpm which can be determined from PAL, the steps per minute were calculated at each HR estimation data point from the aggregated step counting.
\subsubsection{Personal Health information}
Previous studies have reported the correlations between the level of cardiorespiratory fitness and gender \cite{lubans2009relationship, silvetti2001heart}. The measurements of HR in male individuals were found to be significantly higher in female individuals \cite{silvetti2001heart}. In addition to the previous features that contained the body movement information aforementioned in this study, gender was included as background information of each participant along with Body Mass Index (BMI) and Pittsburgh sleep quality index (PSQI) in the candidate feature list. BMI, one of the most measured personal physical information, was reported to be associated with pNN50, a portion of beat-to-beat difference that lasts longer than 50 milliseconds (ms), and the root mean square of successive differences (RMSSD) components of HR variability (HRV) \cite{koenig2014body}. Moreover, there are reports of the correlation between HRV and sleep quality \cite{wei2011subjective,fernandes2013resting} by applying PSQI \cite{buysse1989pittsburgh}. PSQI is a measurement of the sleep quality based on the self-reported sleep evaluation in the form of a questionnaire with the score ranges from 0 to 21. A PSQI that is lower than 5 indicates adequate sleep, whereas a larger score indicates a poor sleep quality. Age was excluded from the candidate feature list as no significant correlation between age and HR has been reported \cite{ryan1994gender}. Furthermore, the age range among the participants was not sufficient in this study.
\subsection{Machine learning algorithms}
\label{sec:ML}
\subsubsection{Support vector regression (SVR)}
\label{sec:R}
As an extension of the support vector machine (SVM), which was introduced to solve a classification problem \cite{cortes1995support}, SVR was derived for a regression problem \cite{drucker1997support}. The SVR approaches the generalized model by minimizing the generalized error bound while building a hyperdimensional function that deviates to the most $\epsilon$ from the training points. To maintain the balance of the generalization and the complexity of the model, a capacity constant ($C$) was introduced as a hyperparameter that needs to be carefully tuned, together with the $\epsilon$. The mathematical detail of this algorithm can be found in a library for SVM \cite{chang2007library} for simplicity in this Section.
\begin{equation}
\label{eq:3}
\begin{array}{l}
K_{\text{Poly}}(x_i,x_j) = (\gamma x_i x_j^\top + 1)^d\\
K_{\text{RBF}}(x_i,x_j) = exp(\frac{-\gamma||x_i-x_j ||^2}{2\sigma^2}) = exp(-\gamma||x_i-x_j ||^2)
\end{array}
\end{equation}
In the regression problem, the label $y_i$ can be predicted from the evidence, $x_i,x_j$, which includes the HR$_\text{PPG}$ and features from the wearable devices. The kernel ($K(x_i)$ ) acts as a feature transformation, mapping data points ($x_i,x_j$) to separable spaces in a higher feature space. Polynomial kernel ($K_{\text{Poly}}$) and radial basis function ($K_{\text{RBF}}$) were adopted as in \eqref{eq:3}. The kernels differ from each other in time complexity and parameters that require tuning. The $K_{\text{Poly}}$ has a tunable parameter $d$ that increases the complexity of the kernel, whereas $K_{\text{RBF}}$ has $\sigma$ that controls the generalization of the training data distribution. In all kernels, $\gamma$ is a core parameter that controls the influence of each training instance to the overall model. $\sigma$ in $K_{\text{RBF}}$ is a constant parameter which can be tuned together in the form of $\gamma$.
\subsubsection{Random forest (RF)}
\label{sec:RF}
RF is an ensemble learning method for decision tree (DT) \cite{liaw2002classification}. DT has a set of nodes, articulating hierarchically. One specific feature of DT includes its threshold, making a decision in which a region of a certain sample should be in. Each node creates two divided cuboid subspaces for its input feature. Therefore, the averaged target labels of the training data in each subspace, dividing by the deepest node of each branch, are considered a predicted value. In RF, multiple DT are built, each with a subset of features and training data randomly chosen with a replacement. Subsequently, the average of the results from all DT in RF is reported as a final regression value.
\subsubsection{Gaussian Process (GP)}
\label{sec:GP}
GP is a nonparametric learning method that uses a Bayesian approach to solve a regression problem \cite{rasmussen2006gaussian}. The model is defined by the mean function, often defined as zero, and a covariance kernel ($K$), based on a prior assumption of data. The GP is robust to noise since it is a distribution of multiple random functions. $K$ is a multivariate Gaussian distribution with a zero mean. At inference, GP uses all training data to provide a confidential interval of predicted value, based on the prior belief. The radial basis function kernel ($K_{\text{RBF}}$), as in \eqref{eq:3}, is used as the covariance kernel of GP. In GP, $\sigma$ in $K_{\text{RBF}}$ is considered as a length scale which scales each feature and can be learned from the data.
\subsubsection{Artificial neural network (ANN)}
\label{sec:ANN}
In recent years, ANN has become a well-known machine learning model in both classification and regression tasks of various domains \cite{acharya2003classification}. It outperforms the state-of-the-art traditional ML algorithms using a deep ANN, given that sufficient training data are provided. In this study, the depth of ANN was limited to a shallow network (3-5 hidden layers) to avoid the overfitting problem and generalized the model to work on cross-subject prediction. The fully connected layer architecture, which has hidden layers connecting from the input layer to the output layer with a variety of node's sizes in each layer, was applied.
\subsubsection{Logistic Regression (LR)}
\label{sec:LR}
LR is a statistical model that applies Sigmoid function to the linear combination of the candidate features and sets a threshold for dividing the data into a specific region, working well in binary classification \cite{hosmerdw}. When utilized with a regression task, the algorithm uses the output from Sigmoid function to predict a value without a threshold.
\subsubsection{k-Nearest neighbor regression (kNN)}
\label{sec:kNN}
Commonly known for the classification task, kNN is an approximate algorithm of the Bayesian classifier \cite{altman1992introduction}. The algorithm estimates the unconditional density function of the training data using the kernel method making calculation on k-nearest samples, prior to the prediction of the sample class. Regarding the regression task, the prediction is made by averaging the target labels of k-nearest training samples.
\begin{figure}[b]
\includegraphics[width=85mm]{stage.PNG}
\caption{Timeline of the experimental protocol. HR$_\text{ECG}$, HR$_\text{PPG}$, PAL, accelerometer data, and step count were collected during the three activity states (resting for 30 minutes, \textcolor{red}{laying down} for 60-90 minutes, and intense treadmill activities for 40 minutes).}
\label{fig:experimental}
\vspace{-6mm}
\end{figure}
\section{Methods}
\label{Methods}
The experimental protocol, as shown in Figure \ref{fig:experimental}, and the data recording are first introduced, followed by the descriptive information of the participants and the statistical analysis.
\subsection{Data collection}
\label{sec:StandardMeasurement}
The physical activity assessed in this study consists of three states: resting, \textcolor{red}{laying down}, and intense treadmill activity states. Two ECG monitoring devices were employed to collect HR$_\text{ECG}$ as a standard measurement. During the states with low energy expenditure, i.e., resting and \textcolor{red}{laying down} states, a single-lead ECG research grade device from the Biosignalsplux wearable platform is assembled. The ECG signals were wirelessly acquired through OpenSignals software. Although the Biosignalsplux kit has not been validated for clinical purposes, to our best knowledge, BITalino, which is a previously released platform from the same company, had been validated \cite{guerreiro2014performance} and has been used as a standard measurement \cite{tassignon2018continuous}. Despite supported with a high resolution of 16-bit, which can accommodate the sampling frequency up to 3,000 Hz, the Biosignalsplux kit is perhaps not designed for dynamic motion. Therefore, Polar H10, which provides interference-free ECG measurement with a chest strap HR sensor, was used as an alternative during the intense treadmill activity state, involving walking and running in a fixed amount of time interval, and the HR$_\text{ECG}$ obtained was applied as a standard measurement. Polar H7, the predecessor of Polar H10 in the same product line, had been validated previously \cite{plews2017comparison, gillinov2017variable} and was used as a standard measurement in this study \cite{eather2019efficacy}.
\subsection{Experimental Protocol}
The experimental design was \textcolor{red}{investigated} to imitate daily activities. The activities in resting, \textcolor{red}{laying down}, and intense treadmill activity states were devised accordingly, as shown in Figure \ref{fig:experimental}. Before beginning the experiments, skin preparation was performed to improve the quality of the signal acquisition \cite{kligfield2007recommendations}. Skin preparation gel and conductive gel were applied locally on the single-lead ECG placement. Informed consent was received from all participants following the Helsinki Declaration of 1975 (as revised in 2000), which was approved by the internal review board of Rayong Hospital, Thailand.
The recording in the resting state commenced with the participants relaxing in a living room environment while watching a collection of animated videos with a content rating of no violence or nudity present and simple language as the means to stabilize the mental activity that might affect the HR. This follows the reports on the direct relationship between a lower arousal rate and the media content with the recommended age in Parental Guidance (PG) \cite{fleming2001effects}. ``Peppa pig", a children's series which is recommended for PG 3+, was selected to simulate the resting state. The task in the resting state was continued for 30 minutes while the data were simultaneously collected as follows: ECG signal from Biosignalsplux, accelerometer signals from E4, PAL from the three consumer grade wearable devices, and HR$_\text{PPG}$ from Fitbit Charge HR, Apple Watch Series 4, TicWatch Pro, and E4.
After the resting state, the participants were asked to \textcolor{red}{lay down} in a \textcolor{red}{bedroom-like} environment. \textcolor{red}{
The recording began 5 minutes after the participants started to relax and lay down. The time interval of the experiment were arranged for 60 to 90 minutes.} The data were recorded with the same devices as in the resting state.
A set of physical activities in the intense treadmill activity state was then proceeded after the \textcolor{red}{laying down} state. The participants were instructed to rest for 10 minutes and thereafter stroll on the treadmill at three different speeds (walking: 2 kilometers per hour (kmph), brisk walking: 5 kmph, and jogging: 8 kmph). The duration of the physical cooling down period (walking: 2 kmph) after the main activities lasted for 5 minutes. The speed was slightly increased until reaching the specific speed of each portion. At the end of the set, the participants were instructed to rest for 10 minutes to allow for HR recovery. HR$_\text{ECG}$, HR$_\text{PPG}$, PAL, accelerometer data, and step count were recorded throughout the entire set. Biosignalsplux kit was exchanged with Polar H10 prior to the recording for HR$_\text{ECG}$ as Polar H10 is designed for usage during movement while the Biosignalsplux kit is designed for static activity. The recording protocol in other devices remained the same as in resting and \textcolor{red}{laying down} states.
\subsection{Participant} Healthy 29 individuals, age ranging from 15-33 years old, were recruited (Males: 17, Females: 12) to participate in this study. All participants completed the experimental protocol. The descriptive characteristics of the participants are summarized in Table \ref{table_characteristic}. \textcolor{red}{The participants were required to complete a demographic information form, PSQI, and written informed consent. The annual health checkup was checked to ensure that the participants have no CVD in his/her medical history.}
\begin{table}[bt]
\caption{Descriptive characteristics of 29 participants.}
\label{table_characteristic}
\begin{center}
\begin{tabular}{lr}
\hline\hline
& Mean (SE)\\
\hline
Age (year) & 24.62 (3.92)\\
Height (cm) & 162.25 (19.01)\\
Weight (kg) & 61.17 (13.96)\\
PSQI Score & 5.79 (2.92)\\
\hline\hline
\end{tabular}
\end{center}
\vspace{-5mm}
\end{table}
\subsection{Statistical analysis}
The HR$_\text{PPG}$ in all three states were collected using three consumer grade devices and one medical grade device. \textcolor{red}{Some data sampling rate was dropped in some devices due to the inconsistency of HR estimation. To establish a proper analysis, we considered only the data from the participants with a consistent sampling rate in all four devices.} The standard measurement from the Biosignalsplux kit was recorded \textcolor{red}{as} raw ECG signaling. A bandpass filter of 15-20 Hz was applied to the ECG signal to remove the unrelated components in the signal. The only remaining R-peak of the QRS complex is the strongest component in the ECG that indicates the heartbeat\cite{parak2011ecg}. With MNE package \cite{gramfort2013meg}, R-R or inter-beat interval was located and transformed to HR$_\text{ECG}$. Instead of smoothing noisy HR$_\text{ECG}$, occurring from the false detection of R-R interval with a moving average, a low-pass filter of 0.05 was applied to remove a sudden change (high frequency) in the HR$_\text{ECG}$. \textcolor{red}{Regarding discrepancy in HR estimation algorithms, the HR$_\text{ECG}$ was delayed for 10 seconds to match the lagged signal between HR$_\text{ECG}$ and HR$_\text{PPG}$ in all devices. On the other hand, Polar H10 promptly provides HR$_\text{ECG}$.}
The HR$_\text{PPG}$ from all wrist-worn wearable devices were compared with the baseline standard measurements (HR$_\text{ECG}$) from both Biosignalsplux kit and Polar H10. The mean absolute error (MAE) with a standard error of mean (SE) was individually reported for each device in every state. The value was then compared with the HR$_\text{PPG}$ from the other wrist-worn devices. The comparison between each wrist-worn device was reported, entailing the repeated measures of ANOVA, which required the same group of participants. A mean difference from MAE of each pair was computed with the indication of the significant p-value (p$<$0.05). \textcolor{red}{Only a device with the most significant error was used in the next experiment to improve HR in this study.}
\section{Result and Analysis: Heart rate validation} \label{result:validation}
The results of the performed validation of HR$_\text{PPG}$ provided by each wrist-worn device were presented in this section. \textcolor{red}{The MAEs of the baseline measurement HR$_\text{ECG}$ and the four wrist-worn devices were compared, as exhibited in \textit{Validating HR estimation from the wrist-worn device} of Figure \ref{fig:feature_analysis}, to select the device for post-calibration analyses.}
\begin{figure*}[bt]
\centering
\includegraphics[width=0.96\textwidth]{feature_analysis.PNG}
\caption{Overview diagram of our results and analyses.}
\label{fig:feature_analysis}
\vspace{-0mm}
\end{figure*}
\subsection{Data points collection}
The collected dataset contains 218,356 data points of HR$_\text{PPG}$ from all 29 participants recorded with four wrist-worn wearable devices in resting, \textcolor{red}{laying down}, and intense treadmill activity state. As mentioned previously, only the data from the participants with consistency in the sampling of HR$_\text{PPG}$ measured by all devices were used. The characteristics of the data are shown in Table \ref{table:datasets}. Each device operates with a different sampling rate, depending on the device specification, which varies from 1 Hz to 0.07 Hz. \textbf{Note:} E4 has a built-in algorithm that restrains the output of HR$_\text{PPG}$ if the quality of the observed R-R interval is not detectable.
The data were extracted by using Application Programmable Interface (API) from each hardware device, with the exception of Apple Watch Series 4 as a third-party software entitled "Cardiogram" was employed.
\begin{table}[tb]\setlength\tabcolsep{2pt}
\centering
\caption{Total numbers of recorded HR data point from each device.}
\label{table:datasets}
\begin{tabular}{c|cccc}
\hline
\hline
State & Fitbit (n) & E4 (n) & TicWatch (n) & Apple Watch 4 (n) \\ \hline\hline
RS & 6,063 (23) & 10,718 (24) & 16,149 (15) & 5,812 (18)\\
LS & 17,747 (26) & 36,207 (26) & 49,405 (16) & 17,897 (19)\\
IS & 11,994 (26) & 6,258 (27) & 28,217 (22) & 11,889 (28)\\
All & 35,804 (29) & 53,183 (29) & 93,771 (29) & 35,598 (29)\\ \hline
\hline
\end{tabular}
\caption*{ \footnotesize ``n" denotes the numbers of participant that displayed a consistent sampling rate of HR.}
\vspace{-2mm}
\end{table}
\subsection{Validation of HR estimation from wrist-worn wearable devices}
\label{sec:exp1}
\textcolor{red}{The HR$_\text{PPG}$ from each device was validated with HR$_\text{ECG}$ by using MAE.} The descriptive statistical errors in HR$_\text{PPG}$ from each device at different states are shown in Table \ref{table:DeviceErrorANOVA}. The lowest MAE in each state was emphasis with bold text. Pairwise comparison of MAE with repeated measure test is shown in Table \ref{table:pairwisecomparison}. Apple Watch Series 4 performed with significantly (p$<$0.05) lower MAE ($F(1.860,27.906) = 12.004$ in resting state, $F(1.284,19.254) = 9.448$ in \textcolor{red}{laying down} state, and $F(1.450,33.346) = 16.05$ in intense treadmill activity state) in comparison to the other devices, whereas Fitbit Charge HR yielded the highest MAE in the intense treadmill activity state. \textcolor{red}{Hence, Fitbit Charge HR, showing the highest MAE among all devices, was selected as the device of interest to investigate the improvement of HR$_\text{PPG}$ in the intense treadmill activity state}.
\begin{table}[tb]\setlength\tabcolsep{2pt}
\centering
\caption{Descriptive statistics of \textcolor{red}{the} device's HR errors with repeated measure test ANOVA.}
\label{table:DeviceErrorANOVA}
\begin{tabular}{c|cccc}
\hline\hline
\multicolumn{1}{l|}{\multirow{2}{*}{State (n)}}& \multicolumn{4}{c}{MAE $\pm$ SE (bpm)}\\
& Fitbit & E4 & TicWatch & Apple Watch \\ \hline\hline
\multicolumn{1}{l|}{RS (13)} & 2.95 $\pm$ 0.40 & 4.05 $\pm$ 0.34 & 2.97 $\pm$ 0.27 & \textbf{1.33 $\pm$ 0.08} \\
\multicolumn{1}{l|}{LS (13)} & 2.15 $\pm$ 0.32 & 3.55 $\pm$ 0.34 & 2.49 $\pm$ 0.28 & \textbf{1.10 $\pm$ 0.10} \\
\multicolumn{1}{l|}{IS (21)} & 10.35 $\pm$ 1.70 & 5.45 $\pm$ 0.48 & 4.45 $\pm$ 0.45 &\textbf{2.00 $\pm$ 0.46 }\\
\multicolumn{1}{l|}{All (11)} & 4.22 $\pm$ 1.17 & 3.90 $\pm$ 0.31 & 2.85 $\pm$ 0.22 & \textbf{1.65 $\pm$ 0.15} \\ \hline\hline
\end{tabular}
\caption*{ \footnotesize ``n" denotes the numbers of participant that displayed a consistent sampling rate of HR.}
\vspace{-1mm}
\end{table}
\begin{table}[!tb]\setlength\tabcolsep{2pt}
\centering
\caption{MAE Pairwise comparisons with repeated measure test ANOVA.}
\label{table:pairwisecomparison}
\begin{tabular}{ll|ccc}
\hline\hline
\multicolumn{2}{c|}{\multirow{2}{*}{Pair of devices}}& \multicolumn{3}{c}{MAE Mean Difference $\pm$ SE (bpm)}\\
& & RS & LS & IS \\ \hline\hline
Fitbit & E4 & -1.02 $\pm$ 0.37& -1.24 $\pm$ 0.21 & \textbf{4.49* $\pm$ 1.46}\\
& TicWatch & 0.56 $\pm$ 0.60 & 0.39 $\pm$ 0.72 & \textbf{5.56* $\pm$ 1.44}\\
& Apple & 1.75* $\pm$ 0.37 & 1.37* $\pm$ 0.43 & \textbf{7.57* $\pm$ 1.55} \\ \hline
E4 & Fitbit & 1.02 $\pm$ 0.37 & 1.24* $\pm$ 0.21 & -4.49$^\dagger$ $\pm$ 1.46 \\
& TicWatch & 1.58 $\pm$ 0.63 & 1.64* $\pm$ 0.66 & 1.07 $\pm$ 0.53 \\
& Apple & 2.77* $\pm$ 0.39 & 2.62 $\pm$ 0.40 & 3.07* $\pm$ 0.57 \\ \hline
TicWatch& Fitbit & -0.56 $\pm$ 0.60 & -0.39 $\pm$ 0.72 & -5.56$^\dagger$ $\pm$ 1.44 \\
& E4 & -1.58 $\pm$ 0.63 & -1.64$^\dagger$ $\pm$ 0.66 & -1.07 $\pm$ 0.53\\
& Apple & 1.19 $\pm$ 0.40 & 0.98 $\pm$ 0.38 & 2.01* $\pm$ 0.67 \\ \hline
Apple & Fitbit & -1.75$^\dagger$ $\pm$ 0.37 & -1.37$^\dagger$ $\pm$ 0.43 & \textbf{-7.57$^\dagger$ $\pm$ 1.55} \\
& E4 & -2.77$^\dagger$ $\pm$ 0.39 & -2.62 $\pm$ 0.40 & \textbf{-3.07$^\dagger$ $\pm$ 0.57} \\
& TicWatch & -1.19 $\pm$ 0.40 & 0.98 $\pm$ 0.38 & \textbf{-2.01$^\dagger$ $\pm$ 0.67}
\\ \hline\hline
\end{tabular}
\caption*{\footnotesize $*$ and $^\dagger$ indicate a significantly (at the 0.05 level) higher MAE and a significantly lower MAE, respectively.}
\vspace{-6mm}
\end{table}
\section{Result and Analysis: Heart rate post-calibration} \label{result:calibration}
The three analyses of the post-calibration of HR$_\text{PPG}$ using the ML models to improve the HR$_\text{Fitibit}$ detected by Fitbit Charge HR, the device of interest, are described and illustrated in \textit{Heart rate Post-calibration} of Figure \ref{fig:feature_analysis}: the ultimate HR$_\text{R}$ computed using selected features and ML model, model with external calculated PALs feature, and rolling regression.
\subsection{Analysis I - HR$_\text{ML}$: Improving HR estimation from the selected wrist-worn wearable device with the ML models}
Following the validation of the HR detection of the four wearable devices, the determination of the device of interest was considered depending on the highest error in HR$_\text{PPG}$. One of the four wearable devices was chosen due to its low performance. \textcolor{red}{The features from the selected device were extracted and used as the testbed in feature selection. This execution demonstrated the performance of HR estimation with the ML models, designated as HR$_\text{ML}$, using the information reported from the selected device. MAE was analyzed in all three states.} In the following set of experiments, the data from Fitbit Charge HR was chosen to be implemented with the improvement of HR$_\text{Fitbit}$ using selected features and several ML algorithms.
\label{section:exp2}
\subsubsection{Feature selection}
\label{section:feature_selection}
The features that were examined consisted of HR$_\text{PPG}$, personal health information, step count, and PAL. However, there has been no evidence that these features would benefit the model. In order to verify that each of the candidate physical features is informative and discriminative, in which the prediction of the HR$_\text{ML}$ based in these input features by our ML models essentially depends on, two basic feature selection methods were performed to test the significance of each feature to the HR$_\text{ECG}$, served as the baseline for most accurate measurement of the actual HR, in all four states (resting, \textcolor{red}{laying down}, intense treadmill activity, and all states combined). The linear relation between the feature and HR$_{\text{ECG}}$ was measured using a univariate linear regression by testing a null hypothesis that all of the regression coefficients are equal to zero. In other words, the model, which has no predictive capability using a certain feature, was tested. If the p-value of the linear relation test is smaller than 0.05, the null hypothesis can be rejected which indicates that the feature is informative and benefits the model to predict the HR$_\text{ECG}$. If the p-value is larger than 0.05, the information within the certain feature is not informative to the univariate linear regression model, which is the simplest discrimination model. In addition to the linear test, the mutual information for regression was carried out to measure the non-linear relationship, which cannot be determined by the correlation between each of the features. This relationship function is called entropy. An entropy estimated from k-nearest neighbor distances was computed following a previous study \cite{ross2014mutual} to measure the dependency between the two variables. A higher dependency level indicates a higher dependency of a certain feature with the HR$_\text{ECG}$. Battiti \textit{et al.} \cite{battiti1994using} previously investigated the dependency threshold in the mutual information test which demonstrated that the range of 0.2-0.4 was the best for trade-off point. A feature would be included in the model if it passed at least one of the two dependency tests. Means of the p-value and dependency level with SE for each state were computed across all leave-one-out cross-validation sets.
\begin{table}[t]
\centering
\caption{Univariate linear regression test and mutual information test of candidate features}
\label{table:feature_result}
\begin{tabular}{l|c@{\hspace{2em}}c@{\hspace{2em}}c@{\hspace{2em}}c@{\hspace{1em}}|c@{\hspace{2em}}c@{\hspace{2em}}c@{\hspace{2em}}c}
\hline
\hline
\multirow{2}{*}{Feature} & \multicolumn{4}{c|}{Univariate linear regression} & \multicolumn{4}{c}{Mutual information}\\
& RS & LS & IS & All& RS & LS & IS & All\\
\hline\hline
HR$_\text{\textbf{Fitbit}}$ & $*$ & $*$ & $*$ & $*$ & \checkmark & \checkmark & \checkmark & \checkmark \\
PAL & & $*$ & $*$ & $*$ & & & \checkmark & \checkmark \\
Step count & $*$ & $*$ & $*$ & $*$ & & & \checkmark & \checkmark \\
Gender & $*$ & $*$ & $*$ & $*$ & \checkmark & \checkmark & & \checkmark \\
PSQI & $*$ & $*$ & $*$ & $*$ & \checkmark & \checkmark & & \checkmark \\
BMI& $*$ & $*$ & $*$ & $*$ & \checkmark & \checkmark & \checkmark & \checkmark \\
\hline\hline
\end{tabular}
\caption*{\footnotesize$*$ indicates that p-value of univariate linear test is lower than 0.05. \\\checkmark indicates that the dependency level is higher than 0.3. }
\vspace{-2mm}
\end{table}
\label{result:features_selection}
A set of features for each state was selected using univariate linear regression and mutual information tests, as reported in Table \ref{table:feature_result}. A feature was considered as an important feature if a linear relationship was found (p$<$0.05) or the dependency level between the feature and HR$_{\text{ECG}}$ is larger than 0.3. HR$_\text{\text{Fitbit}}$ and BMI displayed a dependency of the relationship with HR$_{\text{ECG}}$ in all states whereas the other features showed a distribution between the pair in at least one of the two tests. Only PAL in the resting state failed both tests and was not included in the models in the mentioned state.
\subsubsection{ML Model selection}
\label{section:model_selection}
Six ML models (SVR, RF, GP, ANN, LR, and kNN) were trained to predict the HR$_\text{ML}$ using HR$_\text{PPG}$ from the device of interest, features from the feature selection method. Each ML model contains a kernel and different hyperparameters that required a fine-tuning to perform at its highest quality with a certain set of features. All possible sets of the hyperparameter values were compared, as a grid-search, with leave-one-out cross-validation folds. In each fold, one participant was held out as a testing set, one as a validation set, and the rest as a training set. All numerical data were normalized using standardization, based on the training set. Categorical data were encoded with zero and one before the training. The values for each hyperparameter that affect the model are displayed in Table \ref{table:hyperparameter}.
\begin{table}[t]\setlength\tabcolsep{2pt}
\centering
\caption{ A list of tuned hyperparameters \textcolor{red}{used in all models.}}
\label{table:hyperparameter}
\begin{tabular}{c|ll}
\hline
\hline
Model & Parameter/Kernel & Values\\
\hline\hline
SVR & $C$ / All & $0.001, 0.01, 0.1, 1, 10, 100$\\
& $\epsilon$ / All & $0.001, 0.01, 0.1, 1, 10, 100$\\
& $\gamma$ / All & $0.001, 0.01, 0.1, 1, 10, 100$\\
& $d$ / Poly & $2,3,4,5$\\
\hline
RF &Max features & 1, 2, 3\\
& Number of estimator & 200 - 2000 with a step of 4\\
&Max depth& 10 - 49 with a step of 3\\
&Min samples split&2 - 14 with a step of 3\\
&Min samples leaf&2 - 14 with a step of 3\\
\hline
GP & $\alpha$ &1e-10, 1e-7, 1e-5, 1e-3, 1e-1, 1\\
\hline
ANN & Number of hidden layer & 3, 4, 5\\
& Hidden unit in each layer
& 3 layers: (16,8,2), (16,8,4), (8,4,2)\\
& &4 layers: (16,8,4,2), (8,4,4,2)\\
& &5 layers: (16,8,4,4,2), (32,16,8,4,2)\\
& Learning rate& 0.01, 0.001, 0.0001\\
\hline
LR & $C$ & 0.001, 0.01, 0.1, 1, 10, 100\\
& Power & l1, l2, elasticnet\\
\hline
kNN & N Neighbors & 10, 20, 30, 40, 50, 100, 150, 200, \\
&&500, 1000\\
& Power & 1=Manhattan, 2=Euclidean, 3=Minkowski\\
\hline
\hline
\end{tabular}
\vspace{-4mm}
\end{table}
SVR, RF, GP, LR, and kNN models were constructed using Scikit-learn package \cite{scikit-learn}, while Keras package\cite{chollet2015keras} was used to construct a shallow ANN model.
In SVR, hyperparameters were tuned separately for each kernel. Both RBF kernel and polynomial kernel comprised $\gamma$, $C$, and $\epsilon$, in which they were tuned as core hyperparameters in SVR, where $d$ in polynomial kernel was tuned to increase the complexity of the model. All values used in the grid-search fine-tuning are shown in Table \ref{table:hyperparameter}.
Since RF has no individual kernel to be tuned, only the number of trees and early stop conditions, including the maximum number of features, the number of estimators, maximum depth, minimum sample split, and minimum sample leaf, were tuned as shown in Table \ref{table:hyperparameter}. In LR, $C$ was tuned similarly to SVR while the power function was included, whereas kNN displayed the k-nearest neighbor data points and the power hyperparameter metric that required tuning.
To construct an ANN, fully connected layer (2 and 3) with various numbers of hidden nodes were explored. The ANN was optimized using Adam optimizer with a learning ability tuned from 0.0001 to 0.1. Between each hidden layer, ReLU activation function was applied to add a non-linear transformation to the model with the exception of the last layer, which predicted the continuous value as an improved HR$_\text{ML}$.
A repeated measure test was used to test the performance of the six ML models and the hyperparameters. The MAEs of the validation sets were compared, and one ML model was adopted to be used in the subsequent experiments. The computed corrected sets of HR (HR$_\text{ML}$) from the chosen ML model were then compared with HR$_\text{ECG}$.
\label{result:model_selection}
For each of the activity state, the best performance of each ML model was tuned using the hyperparameters and validated with leave-one-out cross-validation. A set of tunable hyperparameters was selected based on the validation MAE. The results of all models are shown in Table \ref{table:tuning_result}. The result showed that SVR with RBF kernel outperformed all other models in every state with the lowest MAE.
\begin{table}[tb]
\centering
\caption{\textcolor{red}{Mean Absolute Error of each HR estimation model evaluated in the validation set with a list of tuned hyperparameters.}}
\label{table:tuning_result}
\begin{tabular}{c|ccc}
\hline
\hline
State & Model & Hyperparameter & MAE$\pm$ SE (bpm)\\
\hline
\hline
RS & \textbf{SVR(RBF) }& \textbf{(0.001,10,1) }& $\boldsymbol{2.53\pm
0.02}$ \\
&SVR(Poly) & (3,100,0.01,0.1)&$4.89
\pm0.05$ \\
& RF &(2,800,10,2,15)&$2.90\pm0.05$ \\
& GP(RBF) & (1e-7) & $2.64\pm0.18$\\
& ANN & (3,(16,8,4))& $6.28\pm2.65$ \\
& LR & (l1,0.1,1)& $5.06\pm0.59$ \\
& kNN & (500,2)& $4.23\pm0.527$ \\
\hline
LS& \textbf{SVR(RBF) }& \textbf{(0.001,10,0.001) }&$\boldsymbol{ 2.14
\pm0.02}$\\
&SVR(Poly) & (3,100,0.1,0.01)&$3.49\pm
0.06$\\
& RF &(2,800,10,2,15)& $2.53\pm0.02$\\
& GP(RBF) & (1e-7) & $3.20\pm0.10$\\
& ANN & (3,(16,8,4))& $3.63 \pm 0.93$\\
& LR & (l1,0.1,1)& $4.84\pm0.53$ \\
& kNN & (100,1) & $3.35\pm0.27$ \\
\hline
IS & \textbf{SVR(RBF) }& \textbf{(0.01,10,0.1)} & $\boldsymbol{7.79\pm0.02}$\\
&SVR(Poly) & (3,0.0001,1,0.001)&$10.80\pm
0.05$\\
& RF &(2,800,10,2,15)& $8.75\pm 0.05$\\
& GP(RBF) & (1e-7) & $19.6\pm 0.80$ \\
& ANN & (3,(16,8,4))& $11.19\pm3.01$\\
& LR & (l1,1.0)& $12.66\pm0.69$ \\
& kNN & (50,3)& $9.34 \pm0.57$ \\
\hline
All & \textbf{SVR(RBF) }& \textbf{(0.01,10,0.001)} & $\boldsymbol{4.10\pm
0.03}$\\
&SVR(Poly) & (2, 0.01,10,1)&$11.4\pm
0.07$\\
& RF &(2,800,10,2,15)&$4.57\pm0.02$ \\
& GP(RBF) & (1e-7) &$16.68\pm0.64$ \\
& ANN & (3,(16,8,4)) &$5.32\pm0.86$ \\
& LR & (1.0,l2)& $9.42\pm0.63$ \\
& kNN & (30,1) & $5.93\pm0.57$ \\
\hline
\hline
\end{tabular}
\caption*{ \footnotesize Note: Formats of hyperparameters for each ML model are as follows: ($\gamma,c,\epsilon$) for SVR (RBF), ($d,\gamma,c,\epsilon$) for SVR (Poly), (Max features, Number of estimator, Max depth, Min samples split, Min samples leaf) for RF, ($\alpha$) for GP, (number of hidden layer, (hidden unit in each layer) for ANN, ($C$, Power) for LR, and (N Neighbors and Power) for kNN.}
\vspace{-3mm}
\end{table}
The HR$_\text{ML}$ of the testbed encompassing the PAL and the step count extracted from Fitbit Charge HR were calculated. In comparison amongst the PAL related information, the results did not provide any significant difference. Moreover, when compared to the HR$_\text{Fitbit}$ with the MAE of $3.26 \pm 0.34$ bpm in the resting state, $2.33 \pm 0.23$ bpm in the \textcolor{red}{laying down} state, $9.53 \pm 1.47$ bpm in the intense treadmill activity state, and $5.02 \pm 0.64$ bpm in all states combines. Significant improvement was present in each activity with the exception of all states combined (Table \ref{table:sensor_fusion_result}).
\subsection{Analysis II - HR$_\text{SF}$: Improving HR estimation from selected wrist-worn wearable device using sensor fusion}
\label{section:experiment2_result}
\begin{table}[b]
\centering
\caption{Univariate linear regression test and mutual information test of PAL features from fusion device.}
\label{table:fusion_feature_result}
\begin{tabular}{l|c@{\hspace{2em}}c@{\hspace{2em}}c@{\hspace{2em}}c@{\hspace{1em}}|c@{\hspace{2em}}c@{\hspace{2em}}c@{\hspace{2em}}c}
\hline
\hline
\multirow{2}{*}{Feature} & \multicolumn{4}{c|}{Univariate linear regression} & \multicolumn{4}{c}{Mutual information}\\
& RS & LS & IS & All& RS & LS & IS & All\\
\hline\hline
Crouter$_{\text{VA}}$ & $*$ & $*$ & $*$ & $*$ & & & & \\
Crouter$_{\text{VM}}$ & $*$ & $*$ & $*$ & $*$ & & & & \\
Troiano$_{\text{VA}}$ & $*$ & $*$ & $*$ & $*$ & & & & \checkmark \\
Freedson$_{\text{VA}}$ & $*$ & $*$ & $*$ & $*$ & & & \checkmark & \checkmark \\
\hline\hline
\end{tabular}
\caption*{\footnotesize$*$ indicates that the p-value of univariate linear test is lower than 0.05. \\\checkmark indicates that dependency level is higher than 0.3.\\ VA and VM indicate that the PAL was computed based on vertical axis and vector magnitude of the accelerometer, respectively.}
\vspace{-3mm}
\end{table}
Bringing in an external sensor might be beneficial to the PALs evaluation. The PAL reported by Fitbit Charge HR was compared against the PALs calculated from the raw data from three-axis accelerometer equipped on E4, as E4 does not provide PAL of the user explicitly. Four PAL cut-points proposed by Crouter \textit{et al.} (VA and VM), Freedson \textit{et al.} (VA), and Troiano \textit{et al.} (VM) were applied to transform the raw data into PALs. VA used only y-axis in calculation whereas VM used vector magnitude. The PALs were tested with the feature selection method as described in Section \ref{section:feature_selection}. The features that passed the criteria were trained separately in each state, with the ML model and hyperparameters that yielded the lowest MAE from the previous experiment. Ultimately, a model called sensor fusion was performed to estimate HR$_\text{SF}$ with the testbed, the features from the selected device, and PAL in each state were compared among all ML models. The comparison between the PAL from the wearable device of interest and the calculated PAL from E4 was regarded as the baseline in the HR estimation process.
\label{section:result_exp3}
\textcolor{red}{The sensor fusion features for improving HR estimation models were validated in this experiment.} We used SVR with RBF kernel with a set of hyperparameters that achieved the lowest MAE from the validation set. The performance of SVR with RBF kernel on the testing set is reported in Table \ref{table:sensor_fusion_result} in comparison with other methods. The features from the testbed, i.e., PAL and step count, were enhanced by one of the four PALs from various thresholds (Crouter$_{\text{VA}}$, Crouter$_{\text{VM}}$, Troiano$_{\text{VA}}$,
wand Freedson$_{\text{VA}}$), which were introduced in Section \ref{section:PAL}. The feature selection, as in the previous experiment, was performed to test the linear relationship and the dependency between each feature and the HR$_{\text{ECG}}$. The results shown in Table \ref{table:fusion_feature_result} suggest that all PALs may possess a linear relationship with HR$_{\text{ECG}}$, although the dependencies were found only with Troino$_{\text{VA}}$ and Freedson$_{\text{VA}}$ in all states combined while only Freedson$_{\text{VA}}$ displayed dependency in intense treadmill activity state. The MAEs of the HR$_\text{SF}$ derived from the PALs of E4 using the method of sensor fusion are reported in Table \ref{table:sensor_fusion_result}.
\begin{figure}[tb]
\centering
\begin{subfigure}{.40\textwidth}
\includegraphics[width=0.85\textwidth]{time-Resting.png}\vspace{2mm}
\caption{Resting Stage}
\label{fig:timeseries-resting}
\end{subfigure}
\begin{subfigure}{.40\textwidth}
\includegraphics[width=0.88\textwidth]{time-Sleeping.png}\vspace{2mm}
\caption{\textcolor{red}{Laying down} Stage}
\label{fig:timeseries-sleeping}
\end{subfigure}
\begin{subfigure}{.40\textwidth}
\includegraphics[width=0.88\textwidth]{time-Intensity.png}\vspace{2mm}
\caption{Intense Treadmill Activity Stage }\vspace{2mm}
\label{fig:timeseries-intensity}
\end{subfigure}
\caption{Estimation results from Participant 11. A comparison between the ground truth of HR (HR$_\text{ECG}$) and HR computed by Fitbit Charge HR (HR$_\text{Fitbit}$) according to the three activity states; resting, \textcolor{red}{laying down}, and intense treadmill activity states. The window indexes are from HR$_\text{Fitbit}$ every 15 seconds.}
\label{fig:timeseries_plot}
\vspace{-4mm}
\end{figure}
\subsection{Analysis III - HR$_\text{R}$: Improving HR estimation from selected wrist-worn wearable device using rolling regression}
\label{section:exp4}
Improving the HR$_\text{ML}$ using the ML model with only PAL and personal health information does not fully utilize the ML algorithms and data as only a few features (low dimensionality) were trained to make a prediction in the previous experiment. Furthermore, it does not exploit the temporal information from the input, which is a time-series. To address these problems, a rolling regression was adopted, extended from the previous experiment, by considering a few data points in the time domain instead of only a single data point. A window of features was built using device-estimated HR$_\text{PPG}$, PAL, and step count within a window size, recorded every 15 seconds, controlling the amount of sequential data points to be utilized (5, 10, and 15 data points). Although the drawback of the rolling regression, in which a first few data points at the beginning in the time domain could not be predicted, is unavoidable, it was used as features to predict the subsequent HR$_\text{R}$. To compensate for the drawback, the chosen amounts of window size were optimized to be the smallest sizes that would not be considerably underperformed by the larger window size. For instance, a window size of 20 data points would affect the protocol as it would require up to 5 minutes of the data points. Therefore, the window sizes of 5, 10, and 15 data points were nominated for the comparison to determine the best possible window size. Similar to the previous experiment, the selected features, including gender, PSQI, and BMI, were used as the personal health information features without the rolling window. The performance of the selected rolling window model of the rolling regression was compared to all previous experiments using MAE: HR$_\text{ML}$, HR$_\text{SF}$, and HR$_\text{R}$.
\label{section:result_exp4}
Using the method of rolling regression, the ML algorithms exploited the features over the temporal domains to create a prediction. In this study, the temporal features were obtained from Fitbit Charge HR, including the HR$_\text{Fitbit}$, PAL, and step count. The sizes of the rolling window were tuned to match 5, 10, and 15 data points. The results of the three window sizes are reported in Table \ref{table:sensor_fusion_result} (Analysis III). The results between the HR$_\text{PPG}$ obtained from the devices, the HR$_\text{R}$, and the baseline (HR$_\text{ECG}$) were compared in Figure \ref{fig:timeseries_plot} in time-series.
\begin{figure}[]
\includegraphics[width=72mm]{blandaltman_before.png}\vspace{2mm}
\caption{Bland-Altman plot displaying the difference in HR between the standard measurement (HR$_\text{ECG}$) and Fitbit Charge HR (HR$_\text{Fitbit}$) before post-calibration with the rolling regression model (HR$_\text{R}$). The yellow line indicates an acceptable interval.}
\label{fig:blandaltman-before-correction}
\vspace{-5mm}
\end{figure}
\begin{figure}[]
\includegraphics[width=72mm]{blandaltman_after.png}\vspace{2mm}
\caption{Bland-Altman plot displaying the difference HR between the standard measurement (HR$_\text{ECG}$) and Fitbit Charge HR (HR$_\text{Fitbit}$) after post-calibration with the rolling regression model (HR$_\text{R}$). The yellow line indicates an acceptable interval.}
\label{fig:blandaltman-after-correction}
\vspace{-3mm}
\end{figure}
Bland-Altmans plots were used to compare the HR$_\text{ECG}$ and the HR$_\text{Fitbit}$ reported by Fitbit Charge HR. The confidence interval was set at 95\%. Before the correction, Figure \ref{fig:blandaltman-before-correction} shows 1,052 points out of the confidence interval range while Figure \ref{fig:blandaltman-after-correction} shows 406 points after correction. The two-tailed t-test between HR$_\text{Fitbit}$ and the correction of HR$_\text{R}$ using the rolling window regression with the window size of 10 indicated the error reduction of 33.44\% ($t(5323)=16.21$, p$<$0.05) in the resting state, 15.88\% ($t(15013)=9.82$, p$<$0.05) in the \textcolor{red}{laying down} state, 9.55\% ($t(8477)=7.65$, p$<$0.05) in the intensity state, and 18.73\% ($t(19313)=19.71$, p$<$0.05) in all states combined.
\begin{table*}[bt]\setlength\tabcolsep{2pt}
\centering
\caption{MAE of the HR estimation using the post-calibration approach with PAL related information from Fitbit Charge HR, sensor fusion PAL, and rolling regression of information from Fitbit Charge HR as main features on the testing set.}
\label{table:sensor_fusion_result}
\begin{tabular}{l>{\centering}p{3.5cm}>{\centering\arraybackslash}p{3.5cm}>{\centering\arraybackslash}p{3.5cm}>{\centering\arraybackslash}p{3.5cm}}
\hline
\hline
\multirow{2}{*}{HR estimation} & \multicolumn{4}{c}{MAE$\pm$ SE (bpm)} \\
& RS & LS & IS & All \\
\hline\hline\\[-1.5ex]
\multicolumn{5}{l}{(Heart Rate Validation) \textit{HR$_\text{Fitbit}$ from Fitbit Charge HR.}}\\
\hspace{1em}Fitbit & $3.26 \pm 0.34$ & $2.33 \pm 0.23$ & $9.53 \pm 1.47$ & $5.02 \pm 0.64$ \\ \hline \hline
\multicolumn{5}{l}{Heart Rate Post-Calibration}\\
\multicolumn{5}{l}{\textit{(Analysis I - HR$_\text{ML}$) Using improving HR$_\text{Fitbit}$ estimation model with PAL related information from Fitbit Charge HR. }}\\
\hspace{1em}PAL & *$2.90 \pm 0.35$ & *$2.15 \pm 0.20$ & *$9.25 \pm 1.32$ & $5.10 \pm 0.65$ \\
\hspace{1em}Step count & *$2.92 \pm 0.34$ & *$2.16 \pm 0.20$ & *$9.24 \pm 1.40$ & $4.92 \pm 0.54$ \\
\hspace{1em}PAL and step count& \multirow{1}{*}{*$2.92 \pm 0.34$} & \multirow{1}{*}{*$2.16 \pm 0.20$} & \multirow{1}{*}{*$9.15 \pm 1.31$} & \multirow{1}{*}{$4.88 \pm 0.54$} \\
\hline\\[-1.5ex]
\multicolumn{5}{l}{\textit{(Analysis II - HR$_\text{SF}$) Using improving HR$_\text{Fitbit}$ estimation model with sensor fusion PALs as the main feature. }}\\
\hspace{1em}Crouter$_{\text{VA}}$ & *$2.86 \pm 0.34$ & *$2.20 \pm 0.19$ & *$9.47 \pm 1.41$ & $5.02 \pm 0.69$ \\
\hspace{1em}Crouter$_{\text{VM}}$ & *$2.87 \pm 0.33$ & *$2.17 \pm 0.18$ & *$9.41 \pm 1.42$ & $5.02 \pm 0.69$ \\
\hspace{1em}Troiano$_{\text{VA}}$ & *$2.86 \pm 0.03$ & *$2.24 \pm 0.20 $ & $ 9.60 \pm 1.43$ & $5.12 \pm 0.72$ \\
\hspace{1em}Freedson$_{\text{VA}}$ & *$2.87 \pm 0.34$ & *$2.23 \pm 0.20$ & *$9.68 \pm 1.44$ & $5.14 \pm 0.72$ \\[1px]
\hline\\[-1.5ex]
\multicolumn{5}{l}{\textit{ (Analysis III - HR$_\text{R}$) Using improving HR$_\text{Fitbit}$ estimation model with a rolling window and PAL related information from Fitbit Charge HR as features. }}\\
\hspace{1em}5 points & *$2.55 \pm 0.40$ & *$1.99 \pm 0.22$ & *$8.65 \pm 1.05$ & *$4.15 \pm 0.34$\\[1px]
\hspace{1em}10 points & *$2.17 \pm 0.18$ & *$1.96 \pm 0.22$ & *$8.62 \pm 1.04$ & *$4.08 \pm 0.32$\\[1px]
\hspace{1em}15 points & *$2.16 \pm 0.18$ & *$1.97 \pm 0.23$ & *$8.60 \pm 1.03$ & *$4.03 \pm 0.30$
\\[0px]
\hline
\hline
\end{tabular}
\caption*{\footnotesize*An error reduction is significant when compared with HR$_\text{Fitbit}$ (p$<$0.05).}
\vspace{-5mm}
\end{table*}
\section{Discussion}
\label{Discussion}
The accuracy of the PAL assessment performed by the wrist-worn fitness trackers is imperative to the monitoring of the fitness of physical conditions and health. Numerous, yet validation voids, wrist-worn portable devices have been attractively targeting health-concerned purchasers in the rising wearable product market. Here in this study, we seek to evaluate the validity of the four chosen devices including Fitbit Charge HR, Apple Watch Series 4, TicWatch Pro, and E4 as well as the methods to improve the HR estimation during post-calibration using the data provided by these devices. One of the primary aims is to establish a universal experimental protocol including the three states of physical activity which mimic the daily activities in real life: resting, \textcolor{red}{laying down}, and intense treadmill activity states. Several studies have exploited different sets of study protocols with similar protocols involving active exercising on a treadmill and a cycle ergometer \cite{wallen2016accuracy, bai2018comparative, tedesco2019accuracy}. Although the studies differentiated between the activities at rest (sitting and lying), low intensity activities (household chores), and high intensity activities (walking and running), to our best knowledge, the protocol used in the present study entails different levels of active intensity as well as the \textcolor{red}{laying down} state. This may give rise to further insights on the HR measurement in the daily-like environment.
The results from the validation of the four chosen devices demonstrated and confirmed that the MAEs of the HR$_{\text{PPG}}$ from all the devices measured in the intense treadmill activity state exhibited higher than the other two states. The high MAE was in line with our initial hypothesis that MA might have caused an inaccuracy in the measurement by the PPG sensors. The comparison manifested that Fitbit Charge HR performed with a significantly higher MAE than TicWatch Pro and Apple Watch Series 4, and higher, but no significance, than E4 during the intense treadmill activity state. Interestingly, E4, the only medical grade class 2a wearable device from Empatica, performed with the second-highest MAE in all states, followed by TicWatch Pro. Despite the interference by MA, Apple Watch Series 4 achieved with the lowest MAE in all states compared to the other three devices. During the data collection in intense treadmill activity state, HR$_{\text{PPG}}$ from E4 could not be collected at a proper rate due to the HR estimation algorithm on the apparatus that disregards the HR$_{\text{PPG}}$ if peak-to-peak in the PPG signal is not obviously detected. To comprehensively develop the model for improving the HR estimation during the post-calibration process, only the HR$_{\text{Fitbit}}$ was considered as an input of the model, based on the pairwise comparison with repeated measure test of the MAE achieved by each pair of the devices. Fitbit Charge HR \textcolor{red}{generated} significantly higher MAE in the intense treadmill activity state which severely affected by MA.
It could be speculated that due to the lower amount of hardware components equipped on Fitbit Charge HR, which comprises one PPG sensor, two green LEDs, 1 Photodiode, three-axis accelerometer, and without a gyroscope, the accuracy of the HR$_{\text{Fitbit}}$ may perhaps be affected. However, the accuracy produced by each device may depend on the in-house algorithm. As displayed in Table \ref{table:device-compare}, although TicWatch Pro contains similar components to Fitbit Charge HR with an additional gyroscope, the device performed with a lower error rate than the medical grade E4. Thus, this suggests the importance of MA removal methods.
One appealing aspect of Fitbit Charge HR encompasses its functions of PAL computation, step count, and HR estimation simultaneously, in which we presumed that using these recorded data as the model features can improve the HR estimation during the post-calibration process, particularly in the intense treadmill activity state where the MA is likely to occur. From the MAE of HR$_\text{ML}$ in Section \ref{section:experiment2_result}, SVR with RBF kernel outperformed the other ML algorithms, i.e., SVR with polynomial kernel, RF, GP, ANN, LR, and kNN, in all states. The error in HR$_\text{Fitbit}$ on Fitbit Charge HR, which was severely affected by MA, could be lower using the ML model with the features that are regularly recorded by the device. The PAL and step count from Fitbit Charge HR were also compared. The models using PAL with step count as features slightly performed better in the intense treadmill activity states and all states, note that the models with separated features (Table \ref{table:sensor_fusion_result}) exhibited no difference. However, it is still not conclusive whether the PAL and step count solely would \textcolor{red}{mainly} benefit the HR estimation.
Due to the raw three-axis accelerometer data reported by E4, the four most adopted PAL estimation methods proposed by Freedson \textit{et al.} \cite{pamty2005calibration}, Troiano \textit{et al.} \cite{troiano2008physical}, and Crouter \textit{et al.} \cite{crouter2015estimating} were used to calculate the accelerometer-based PALs as a feature, as displayed in Table \ref{table:sensor_fusion_result}. The MAEs on Crouter$_{\text{VA}}$, Crouter$_{\text{VM}}$, and Freedson$_{\text{VA}}$ were significantly reduced in each activity except in all states whereas Troiano$_{\text{VA}}$ only displayed significant reduction in resting and \textcolor{red}{laying down} states. Furthermore, while the data from the three-axis accelerometer in Fitbit Charge HR could not be applied in this study, E4 contains additional axes of both VA and VM. However, the testbed containing the PAL and the step count acquired from Fitbit Charge HR was found to be similar to the four PALs calculated from E4 with no significant value exhibited. Thus, this method did not demonstrate that the PAL feature obtained from the external three-axis accelerometer could noticeably improve the calibration model when compared to the testbed.
The model using the PAL, step count, and rolling window sizes of 5, 10, and 15 points revealed a significant error reduction from HR$_\text{Fitbit}$ to HR$_\text{R}$ (p$<$0.05) in all activity states, especially in all states in which the error reduction was shown in the previous analysis using sensor fusion (Analysis II) as non-significant (p$>$0.05). In comparison between the three window sizes, no significant difference was achieved. Nonetheless, the rolling regression exhibited further significant improvement beyond the other methods on the testing set in the previous analyses. Although all sizes of the rolling window also displayed an improving trend of HR estimation, the rolling window size of 15 showed the most significance \textcolor{red}{compared to the other window sizes of 5 and 10 regarding the testbed.} It is reasonable to select the rolling window size of 10 as the finest performance of all the methods analyzed. Firstly, it performed with lower errors when compared to 5 points. Secondly, although 15 points might have performed with less errors when compared to 10 points in some states, it could compensate for the drawback of disregarding several more data points while achieving the significant improvement to reduce the error in which the window size of 15 would have. Interestingly, \textcolor{red}{the HR improvement was found in the laying down state, which was expected to create a lower production of MA errors as the human body would generally not create a fair amount of} movement during sedentary activity \cite{ram2011novel}. Furthermore, a significant difference was detected in the intense treadmill activity state, which is the state of interest to reduce the MA and improve the HR estimation using the post-calibration approach. This suggests a further improvement of the proposed methods for future investigation.
The public dataset produced by TROIKA is widely used and \textcolor{red}{has} been tested using different estimation models such as TROIKA, WPFV, and CorNET. However, these models did not engage in the post-calibration process. The works were applied with raw PPG signals to remove MA. Therefore, we deem that these frameworks are not comparable to our proposed method.
The post-calibration approach has \textcolor{red}{the} potential to be established as a standard practice to further improve the accuracy of the HR estimation both in the consumer grade and medical grade wearable products. However, many users have expressed concerns over the security mechanisms within the wearable devices as the personal and medical information \textcolor{red}{is} stored and collected. Recently, Zheng and colleagues have analyzed different cryptographic primitive techniques to facilitate and distribute the data from the sensors of the portable medical ECG devices in order to protect the wireless devices securely \cite{zheng2018critical}. Furthermore, the security within the wireless body sensor networks (WBSNs) has been explored to improve the heartbeats and the \textcolor{red}{signal} processing time, including noise removal, encoding technique, and feature extraction \cite{pirbhulal2018heartbeats}. Thus, future studies are encouraged to investigate the security system during the data collection for the post-calibration approach while draw a definitive framework to improve the efficacy of the post-calibration methods in various wearable devices.
\vspace{-2.2mm}\section{Conclusion}
\label{Conclusion}
This study presents the investigation on the accuracy of HR estimation in the four popular portable wrist-worn wearable devices: Fitbit Charge HR, Apple Watch Series 4, TicWatch Pro, and E4. Although these devices are favorable due to its convenience, the PPG signals affected by MA are not accurate. The devices are valid and reliable for measuring HR in sedentary and low PAL, whereas a higher error rate is detected during moderate to vigorous PAL. The HR correction should be performed before the report, especially in Fitbit Charge HR. A few post-calibration approaches to improve the HR estimation from PPG sensors are proposed in this study. Ultimately, the model with the rolling window was verified to yield the corrected HR with the least error.
Furthermore, the benefits of this validation would undoubtedly encourage the manufacturers of wearable devices to adopt these approaches towards consumers. The post-calibration would be propitious to the non-real-time analysis of \textcolor{red}{the} HR dataset, collected from the wearable devices, which may consequentially aid the calculation of energy expenditure and the diagnosis of CVD.
\ifCLASSOPTIONcaptionsoff
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| {
"redpajama_set_name": "RedPajamaArXiv"
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\section{Introduction}
In a search to discover semiconductor materials and metallic
interconnect for new generation miniaturized electronic devices,
nanostructures have been a focus of attention. Electronic devices,
such as transistors based on carbon nanotubes\cite{review},
attracted interest in nanowires. Rodlike Si nanowires have been
fabricated\cite{sinw1} with a diameter 1.3-7 nm.\cite{sinw2} It
has been shown that such Si nanowires can display metallic,
semiconducting and half-metallic properties depending on their
functionalization.\cite{sinw3} Being an alternative to silicon
based microelectronics GaAs is one of the most important materials
used in semiconductor physics. Due to the high electron mobility,
GaAs always carried a potential of being used in high speed
electronic devices. GaAs/AlGaAs
heterostructures\cite{esaki} have served as a media for the two
dimensional electron gas studies.
Similar to bulk crystals researcher have envisioned GaAs naowires
to be a potential alternative for Si nanowires. Recent advances in
fabrication technology made it possible to grow GaAs nanowires.
They are grown by metal catalysts in vapor-liquid-solid (VLS)
mechanism.\cite{harmand} Generally, GaAs nanowires are grown along
[111] direction in zincblende ($zb$) structure, whereas nanowires
with wurtzite ($wz$) structure with diameter as small as 10 nm are
also observed.\cite{mariager} Several models were developed to
predict the transition radius from wurtzite to zinc blende
structure.\cite{glas, dubrovskii} Actually, there is no sharp
transition but instead there are many different stacking
configurations with very similar energies and more sophisticated
models are needed to predict the ground state configuration.
Together with the crystal structure, surface facet structure is
also an important parameter affecting the structural and
electronic properties. Nanowires reported so far are either grown
in $zb$ structure with [$11\bar{2}$] or [$1\bar{1}0$] facet
orientation or in $wz$ structure with [$1\bar{1}00$] or [$11\bar{2}0$]
facet orientation.\cite{leitsmann, hiruma, ohlsson} One of the
unusual structures is so called A-wire, which has been grown in
defect free triangular shape on (111)A surface of
GaAs.\cite{wacaser} More complex structure of nanowires grown on
GaAs(111)B surface have been also reported.\cite{lisa}
There are many experimental and theoretical works on growth and
structure of GaAs nanowires, but not much work is done to predict
the electronic properties of these wires. Theoretical studies so
far have focused on the electronic properties of superlattices
composed of GaAs nanowires. For example, the electronic structure
of InAs/GaAs nanowire superlattices with radius R=10 nm was
examined using a semiempirical $sp^3d^5s^*$ tight-binding
model.\cite{niquet} Another atomistic tight-binding calculation
was carried out to reveal the electronic structure of freestanding
GaAs/Al$_{0.3}$Ga$_{0.7}$As nanowire superlattices oriented along
the [100] crystallographic direction.\cite{persson} Also a
first-principles investigation has been performed on the
hexagon-shaped, [111]/[0001]-oriented III-V semiconductor
nanowires, which was concentrated on the surface effects on the
structure and stability of these nanowires.\cite{leitsmann}
The purpose of this paper is to provide a detailed analysis of
GaAs nanowires, which is necessary for further experimental and
theoretical studies. To this end we present a systematic,
first-principles investigation on structural and electronic
properties of GaAs nanowires grown along [111] direction. Six
different types of GaAs nanowires are distinguished depending on
the shape of their cross sections and the crystallographic
orientation of their side surfaces. Their optimized atomic
structure and cohesive energies are calculated revealing
interesting trends between atomic structure and cohesive energy.
Based on the calculations of electronic structure and isosurface
charge density of specific states we analyzed the character of
states at the band edges and variation of band gap with diameter.
The effects of hydrogen saturation of dangling bonds of surface
atoms on the atomic and electronic structure are examined. We
found that most of bare GaAs nanowires are semiconducting and
remain semiconducting even before the passivation of surface
dangling bonds. Only one type is metallic due to the states
localized at the surface.
\section{Methods}
We have performed first-principles plane wave
calculations\cite{payne, vasp} within Density Functional Theory
(DFT)\cite{kohn} using ultrasoft pseudopotentials.\cite{vasp,
vander} The pseudopotentials having three electrons for Ga (4$s^2$ 4$p^1$),
five electrons for As (4$s^2$ 4$p^3$) and one electron for H (1$s^1$) were used. A plane-wave basis set with kinetic energy up to 250 eV was used. Cutoff energies used were at least 30 $\%$ higher than maximum values suggested.\cite{vasp} The exchange correlation potential is approximated by generalized gradient approximation (GGA) using PW91
functional.\cite{gga} For partial occupancies we use the
Methfessel-Paxton smearing method.\cite{methfessel} The adopted
smearing width is 0.1 eV for the atomic relaxation and 0.01 for
the accurate band structure analysis and density of state
calculations. All structures have been treated within a supercell
geometry using the periodic boundary conditions. Vacuum spacing was arranged so that the minimum distance between two atoms in adjacent unit cells were larger than 10 $\AA$, provided that atoms have negligible interaction at that far distances. We chose bare $wz3-96$ as a test structure and increased the vacuum spacings to 16 $\AA$. This resulted in an energy
difference around 0.2 meV/atom. In the self-consistent
potential and total energy calculations the
Brillouin zone (BZ) is sampled in the \textbf{k}-space within
Monkhorst-Pack scheme\cite{monk} by (1x1x9) mesh points for $wz$
and (1x1x7) mesh points $zb$ nanowires. Increasing the \textbf{k}-space sampling for bare $wz3-96$ structure from (1x1x9) to (1x1x15) resulted in a total energy difference around 1 meV. All atomic positions and lattice constant are optimized by using the conjugate gradient
method where total energy and atomic forces are minimized. The
criterion of convergence for energy is chosen as 10$^{-5}$ eV
between two ionic steps, and the maximum force allowed on each
atom is 0.05 eV/$\AA$. We have reduced the maximum force criterion
down to 0.0025 eV/$\AA$ in our test structure, bare $wz3-96$.
This had no considerable effect, since the change in energy and band gap was around 0.3 meV/atom and 0.006 eV, respectively. Clearly, the criterion for the maximum allowed force 0.05 eV/$\AA$ is appropriate for systems including large number of atoms.
\begin{figure}
\includegraphics[width=8cm]{fig1.jpg}
\caption{(Color online) Ideal and relaxed atomic structures of
bare GaAs nanowires considered in this paper. Numerals given in
parenthesis indicate the crystallographic directions perpendicular
to the surfaces. Numerals given to the bottom left of the
structures stand for the number of atom pairs per unit cell $N$.
$wz$ and $zb$ stand for structures having wurtzite and zincblende
stackings.}\label{fig:fig1}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{fig2.jpg}
\caption{(Color online) Cohesive energy per Ga-As atom pair versus
number of Ga-As atom pairs in the unit cell of different type of
relaxed nanowires. Horizontal axes presented inside the figure
are derived by fitting the diameter versus number of atom data of $wz$ and $zb$
nanowires to quadratic polynomials. Since $wz$ and $zb$ structures
have different number of atomic planes in the unit cell the fitting was done separately.}
\label{fig:fig2}
\end{figure}
\section{Structures and Cohesive Energies}
GaAs nanowires studied here are cut from ideal bulk structure
along [111] direction. Nanowires having wurtzite ($wz$) and zinc
blende ($zb$) stacking have four and six atomic layers in the unit
cell, respectively. Except A-wire, they have hexagonal cross
section. A-wires, by themselves have $zb$ stacking and display a
triangular cross-section with three ($11\bar{2}$) planar side
surfaces. Atomic structures of cross-section of all nanowires are
shown before and after relaxation (structure optimization) in
Fig.~\ref{fig:fig1}. Here we consider the largest members of all
types of GaAs nanowires. Upon relaxation the surfaces of ideal wires
undergo a reconstruction, while inner parts preserve the bulk
configuration. In spite of the fact that the indices of their
planar side surface are the same, $wz1$ and $wz3$ are still
different. $wz3$ structure have six identical surfaces, whereas
$wz1$ structure have three surfaces same as $wz3$ and three
surfaces with hanging Ga-As atom pairs before relaxation.
At the surfaces of some ideal structures in Fig.~\ref{fig:fig1}
the atoms can have the coordination number smaller than four.
Upon relaxation the coordination numbers may
undergo a change. Two adjacent surface atoms having low
coordination number can form new bonds, whereby these atoms
increase their coordination number and the nanowire, in turn,
lowers its energy (i.e it becomes more energetic). It turns out
that, the coordination number of surface atoms is crucial for the
value of the cohesive energy per atom pair. For example, $wz1$
nanowire with $N=109$ has three surfaces each having an atom pair
with coordination number two, while other three surfaces have
surface atoms with coordination number three. $wz2$ nanowires
having $N=$25, 60 and 85 atom pairs in the unit cell have surface
atoms with coordination number two only at the corners of the
hexagonal cross-section. On the other hand, $wz2$ nanowires having
$N=$42 and 114 atom pairs in the unit cell and all members of
$wz3$ nanowires have surface atoms with coordination number three.
In light of these arguments one expects $wz3$ to have larger
cohesive energy per atom pair than the rest of $wz2$ nanowires,
which, in turn, should be larger than that of $wz1$ nanowires.
These arguments are actually confirmed in Fig.~\ref{fig:fig2},
where we present the trends of cohesive energies per atom pair for
all nanowires considered here.
Ideal A-wires have surface atoms with coordination number of two
on each surface, but apart from that, it has a triangular
cross-section, which makes the surface to volume ratio even higher
compared to that of other types. Consequently, A-wires have the
lowest cohesive energy per atom pair as seen in Fig.~\ref{fig:fig2}.
\begin{table*}
\caption{Calculated cohesive energy per Ga-As atom pair $E_{c}$,
band gap $E_{g}$, lattice constant along the wire axis $c$ and
diameter $D$ values of relaxed nanowires are given. $D$ is defined
as the largest distance between two atoms in the same
cross-sectional plane. Here $N$ stands for the number of Ga-As
atom pairs in the unit cell. $N_S$ and $N_D$ stand for the number
of surface atoms and the total number of dangling bonds,
respectively. Surface atoms are defined as atoms making less than
four bonds, while the protruding bonds are defined as dangling
bonds.} \label{tab:table1}
\begin{tabular}{c|ccc|ccccc|ccc|cccc|ccc|ccc}
\hline\hline
Type&\multicolumn{3}{c|}{$wz1$}&\multicolumn{5}{c|}{$wz2$}&\multicolumn{3}{c|}{$wz3$}&\multicolumn{4}{c|}{A}&\multicolumn{3}{c|}{$zb1$}&\multicolumn{3}{c}{$zb2$}\\
\hline
$N$&31&64&109&25&42&60&85&144&24&54&96&28&43&61&82&31&73&133&19&37&61\\
$E_{c}$(eV)&7.71&7.89&8.01&7.78&7.99&7.94&8.05&8.11&7.94&8.05&8.11&7.55&7.69&7.77&7.83&7.78&7.89&8.02&7.48&7.75&7.88\\
$E_{g}$(eV)&1.06&1.02&0.92&0.81&0.96&0.95&0.92&0.88&1.45&1.10&0.90&0.81&0.77&0.85&0.70&1.07&0.14&0.58&M&M&M\\
$c($\AA$)$&6.64&6.63&6.63&6.58&6.59&6.61&6.61&6.63&6.60&6.61&6.63&9.94&9.89&9.88&9.87&9.97&9.95&9.85&9.97&9.87&9.84\\
$D($\AA$)$&15.9&23.6&31.6&14.5&18.1&23.1&28.4&32.0&12.5&20.5&28.6&12.6&16.3&19.8&24.5&12.7&20.9&29.0&9.2&14.5&18.3\\
$N_S$&24&36&48&24&36&36&48&60&24&36&48&29&36&45&54&30&48&66&24&36&48\\
$N_D$&30&36&54&27&36&43&51&60&24&36&48&38&42&54&66&30&54&66&30&42&54\\
\hline\hline
\end{tabular}
\end{table*}
Interestingly, $wz$ structures have relatively larger cohesive
energies than $zb$ structures, with $wz3$ structure being the
largest. The bulk $zb$ structure however, is energetically more
favorable than that of $wz$ structure by nearly 20 meV per Ga-As
pair, also confirmed by our calculations. As radii increase, the
cohesive energy per Ga-As atom pair values should converge to the
bulk value. It is energetically easier to form a $wz$ surface than
to form a $zb$ surface. In other words surface energy of $zb$
structure is larger. That is why, $wz$ structures become
energetically more favorable as the surface to volume ration
increases, namely as the radius of the nanowire decreases. So
there should be some point where cohesive energy of $wz$ and $zb$
structures cross each other. The radius at this point can be
interpreted as the critical radius for transition from $wz$ to
$zb$ structure or vice versa. Here it should be noted that,
energetically favorable does not mean that these structures will
start to grow in experiments. One should also include effects of
formation path, like nucleation growth. Actually, when we are
around the critical radius we can see some hybrid stacking
patterns like ABAC, which is defined as 4H
structure.\cite{dubrovskii}
\begin{figure}
\includegraphics[width=8cm]{fig3.jpg}
\caption{Band structure and charge density isosurface plots of
GaAs nanowires having $wz1$ structure. Energy band gap between the
valance and conduction band is shaded. Numerals given on top of
the bands stand for the number of GaAs atom pairs $N$ in the unit
cell. Charge density isosurfaces of specific states at
$\Gamma$-point are shown on the right hand side of the bands they
belong to. Isosurface charge densities correspond to three valence
and three conduction band edge states, ordered in the same manner
as bands themselves are. Here we also give the band structure of
infinite slab of bulk wurtzite structure consisting 11 atomic
layers with the same planar $(10\bar{1}0)$ surfaces as $wz1$
nanowires does. Zero of energy is set at the Fermi level
$E_F$.}\label{fig:fig3}
\end{figure}
Table \ref{tab:table1} gives the calculated values for the
structure and cohesive energies of nanowires after the relaxation.
Cohesive energies per Ga-As atom pair increase with increasing
diameter, approaching the bulk values, but we don't see the
critical radius because it is expected to be an order of magnitude
larger than that of our nanowires.\cite{dubrovskii} Surprisingly,
the lattice constant decrease as the diameter of nanowires in $zb$
structure increase, while for $wz$ structures the reverse
situation occurs. The ratio of number of surface atoms to the
total number of atom pairs give a measure of surface to volume
ratio, which is decreasing with increasing diameter. Note that the number
of surface atoms having coordination number of two is
$(N_{D}-N_{S})$.
\begin{figure}
\includegraphics[width=8cm]{fig4.jpg}
\caption{Same as in Fig.3 but for $wz2$ structure. Isosurface
charge densities correspond to three valence and three conduction
band edge states, ordered in the same manner as bands themselves
are.}\label{fig:fig4}
\end{figure}
\section{Electronic Structure}
Most of the relaxed GaAs nanowires presented in
Fig.~\ref{fig:fig1} are semiconducting even without hydrogen
saturation. As we will see, in some cases these bare GaAs
nanowires don't even have the surface states at the band edges.
This situation is in contrast with Si
nanowires.\cite{sinw1,sinw2,sinw3,sinw4} Si nanowires as cut
from the bulk crystal and subsequently relaxed are found to be
metallic. Their metallicity occurs due to the partial filling of
the dangling bonds surface states. Upon passivation of the
dangling bonds with hydrogen atoms the surface states are
discarded from the band gap and eventually Si nanowire becomes
semiconductor. In what follows, we will examine the electronic
structure of bare GaAs nanowires and reveal the effects of geometry
and the passivation of dangling bonds with hydrogen.\cite{bandgap}
\begin{figure}
\includegraphics[width=8cm]{fig5.jpg}
\caption{Band structure and charge density isosurface plots of
GaAs nanowires having $wz3$ structure. Energy band gap between the
valance and conduction band of bare nanowire is (yellow)
light-shaded. Numerals given on top of the bands stand for the
number of GaAs atom pairs $N$ in the unit cell. Charge density
isosurfaces of specific states at $\Gamma$-point are shown on the
right side of the bands they belong to. Isosurface charge
densities correspond to three valence and three conduction band
edge states, ordered in the same manner as bands themselves are.
Here we also give the band structure of infinite slab of bulk
wurtzite structure consisting 11 atomic layers with the same
planar $(10\bar{1}0)$ surfaces as $wz3$ nanowires does. Zero of
energy is set at the Fermi level $E_F$. The widening of the band
gap upon the termination of dangling bonds by hydrogen is shown by
(green) dark-shaded regions delineated by black curves at the
valance and conduction band edges.} \label{fig:fig5}
\end{figure}
\begin{figure}
\includegraphics[width=8cm]{fig6.jpg}
\caption{Interatomic distance distribution of the core and shell
part of bare and hydrogen saturated $wz3-96$ structures. The ball
and stick model illustrates the structure of $wz3-96$ nanowire,
while the shaded regions defines the core and shell parts of the
nanowires. (a) Interatomic distance distribution of interior atoms of bare
nanowire. (b) Interatomic distance distribution of exterior atoms of bare
nanowires. (c) Interatomic distance distribution of hydrogen passivated
exterior atoms. (d) Local density of states (LDOS) on surface atoms
(green/light) and on the remaining atoms (blue/dark) of bare nanowire.
(e) Same as (d) after passivation of surface atoms with hydrogen.} \label{fig:fig6}
\end{figure}
Figure~\ref{fig:fig3} presents results of the band structure and
charge density analysis performed for $wz1$ structure. Ideal
structure of these wires have Ga-As atom pairs hanging on three
surfaces, while other three surfaces have the same profile as
$wz3$ type. Upon relaxation these hanging pairs tend to bend towards
each other and lower energy by making unusual Ga-Ga and As-As
bonds. For example, $wz1$ structure having 64 atom pairs per unit
cell ($wz1-64$) have four hanging pairs before relaxation. After
relaxation first two and last two of them bend to each other and
form a stable structure. $wz1-31$ and $wz1-109$ nanowires have one
hanging pair after relaxation. As a result, all surface atoms of
$wz1-64$ structure have coordination number of three, while
$wz1-31$ and $wz1-109$ have six surface atoms with
coordination number of two (See Table~\ref{tab:table1}).
Isosurface charge densities show that three states at
the top of the valence band edge of $wz1-64$ and $wz1-109$
structures have bulklike character, while conduction band edge
states concentrate at the surface. It is hard to define the nature
of states in $wz1-31$ structure because it has low diameter.
To calculate the band structure of the relevant surface
we cut a slab from the bulk wurtzite structure parallel
to $(01\bar{1}0)$ surfaces so that the resulting
structure have 11 atomic layers. This slab has two
dimensional periodicity in the surface and a vacuum region between
adjacent surfaces, so that they don't interact. The band structure
of this slab plotted along \textbf{k}, parallel to [111]-direction, provide
us with information about the band structure of nanowires if they
were grown thick enough to have a reasonable bulk region in order
to reduce the corner effects. One expects the band gap of the
infinite slab structure to be lower than that of the nanowires.
Comparison of the bands of $wz1-109$ with those of $(01\bar{1}0)$
surface confirm their similarity and expected size effect.
Figure~\ref{fig:fig4} shows the band structure and charge density
analysis for $wz2$ structures. Here all surface atoms of $wz2-42$
and $wz2-114$ structures have coordination number of three, while
the rest of considered $wz2$ structures have surface atoms with
coordination number of two at the corners. Note that the valence
band edge of $wz2-42$ and $wz2-114$ structures mimic that of
infinite slab structure, while other structures fail to do that.
We see that the band gap of $wz2-42$ is larger than that of
$wz2-114$ structure in agreement with the quantum confinement
effect. On the other hand, however, $wz2-25$ structure have the
lowest band gap. This is caused by large surface to volume
ratio, which make surface effects pronounced. Isosurface charge
densities show that the valence band edge have bulklike character,
while conduction band edge states concentrate on the surface.
Band structure and charge density analysis of $wz3$ structure is
illustrated in Fig.~\ref{fig:fig5}. This structure has the largest
cohesive energy per atom value compared to other types of
nanowires in Fig.~\ref{fig:fig1}. In this nanowire all surface
atoms make three bonds and surface states derived therefrom do not
occur in the band gap. Consequently, the band structure of the
related infinite slab is very similar to that of $wz3-96$
nanowire. Charge density of the states at both band edges are
spread throughout the nanowire cross-section showing the bulk
character. Since band gap is not diminished by surface states
bands, one can clearly see the quantum confinement effect in these
nanowires.
\begin{figure}
\includegraphics[width=8cm]{fig7.jpg}
\caption{Band structure and charge density isosurface plots for
A-wires.}\label{fig:fig7}
\end{figure}
Passivation of dangling bonds of semiconducting nanowires by
hydrogen atoms, generally, results in significant changes in the
electronic structure. These changes depend on whether the passivation
is done before or after the relaxation of bare nanowires. We find
the latter case more suitable for the simulation of the experimental procedure.\cite{sinw3} Figure~\ref{fig:fig5} includes information about the effect
of hydrogen passivation of all surface dangling bonds on the band
structure of $wz3$ nanowires. In contrast to Si nanowires (where
surface states of bare structure were carried out from forbidden
region to the band continua upon passivation with
hydrogen)\cite{sinw3,ref1,ref2}, the band edge states of $wz3$ GaAs
nanowires remain in their place after the hydrogen saturation.
This conclusion is corroborated by the analysis of isosurface
charge density of states at the center of BZ located
at both edges of band gap. We found that the character and charge
distribution of these states do not undergo a change after
passivation of surface dangling bonds with hydrogen atoms. Also
the similarity in the profile of band edges before and after
hydrogen saturation is found to be striking. Hydrogen atoms mostly affect the surface states, which, in $wz3$ structure, are found in the valence band continua. That is why, the effect of hydrogen saturation is not reflected on the band edges in the way it was in silicon nanowires having surface states at the band edges. Here the increase of the band gap occur not because the edge states are cleared out, but because the atomic structure of the nanowires become more bulklike. This effect is illustrated in Fig.~\ref{fig:fig6}, where we analyzed the interatomic distance distribution of $wz3-96$ structure before and after the hydrogen saturation. Plots given here are done by making a histogram of interatomic distances and then smearing it out. The first plot indicates the interatomic distance distribution of the core region of bare nanowire. We get the same result also in the core region of hydrogen saturated
structure and the peaks match the first, second, third and fourth
nearest neighbor distances of bulk GaAs in $wz$ structure. The
crucial difference between bare and hydrogen saturated structures
is reflected in the shell part of the nanowires. One can clearly
see that hydrogen saturated structure have more bulklike
character.
Furthermore we performed atom projected density of states analysis
by calculating the localized density of states on the surface
atoms, as well as on the core atoms before and after passivation
with hydrogen atoms. As seen in Fig.~\ref{fig:fig6} (d) and (e),
surface as well as core atoms have comparable contributions to the
state distribution at both edges of band gap. These results further
corroborate the fact that GaAs nanowires like $wz3-96$ having
surface atoms with coordination number of three do not have
dangling bond surface states in the band gap. For nanowires with
large $D$ the passivation with hydrogen have negligible effects on
the band gap. Note that the increase in the band gap with decreasing
diameter holds also for hydrogen saturated structures.
\begin{figure}
\includegraphics[width=8cm]{fig8.jpg}
\caption{(Color online)Band structure and charge density
isosurface plots of GaAs nanowires having $zb1$ structure. Energy
band gap between the valence and conduction band of bare nanowire
is (yellow) light-shaded. Numerals given on top of the bands stand
for the number of GaAs atom pairs $N$ in the unit cell. Charge
density isosurfaces of specific states at $\Gamma$-point are shown
on the right hand side of the bands they belong to. Isosurface
charge densities correspond to three valence and three conduction
band edge states, ordered in the same manner as bands themselves
are. Here we also give the band structure of infinite slab of bulk
wurtzite structure with the same planar $(11\bar{2})$ surfaces as
$zb1$ nanowires does. Zero of energy is set at the Fermi level
$E_F$. The widening of the band gap upon the termination of
dangling bonds by hydrogen is shown by (green) dark-shaded regions
for $N=31$, 73 and surface.}\label{fig:fig8}
\end{figure}
In Fig.~\ref{fig:fig7} we present the band structures and
isosurface charge densities of selected states at $\Gamma$-point
of A-wires. Note that, we cannot have an infinite slab
corresponding to the large A-wires because these nanowires have
triangular cross-section. All relaxed A-wires have surface atoms
with coordination number of two. The band gap have no obvious
trend with varying diameter and is determined by surface states
especially falling in the conduction band edge. For small $D$ the
edges of band gap are determined by surface states with low
dispersion. As the diameter increases the valence band edge starts
to show bulk character, while states at the conduction band edge
remain to display surface character. Even for $A-82$ nanowire the
flat band states at the edge of the conduction band are located on
one of the planar side surface.
The effect of surface states on the band gap is dramatic for the
bare GaAs nanowires having $zb$ stacking. The values of band gap
as large as 1.5 eV occurring in $wz$-type nanowires reduce to the
values as small as $\sim$0.2 eV in $zb1$-type nanowires. The
calculated band gap can even be closed in $zb2$-type nanowires. It
appears that many of the dangling bond surface states, in
particular those associated with the surface atoms having
coordination number of two, are placed in the band gap. Therefore
one expect dramatic changes in the electronic properties after the
passivation of surface dangling bonds with hydrogen atoms. The
electronic structures of bare $zb1$-type nanowires are illustrated
in Fig.~\ref{fig:fig8}. Here all surface atoms of $zb1-31$ and
$zb1-113$ nanowires have coordination number of three after
relaxation, but certain surface atoms of $zb1-73$ structure cannot
find a pair to form a bond and stay with coordination number of
two. As a result the band gap of $zb1-73$ does not have a value
between that of $zb1-31$ and $zb1-133$, as one would expect. To
get more information about the nature of the states at both edges
of the band gap we have saturated all dangling bonds of $zb1-31$
and $zb1-73$ structures by hydrogen. After the hydrogen saturation
the band edge profile does not stay the same, as it was in the
case of $wz3$ structures. This means that the band gap of bare
nanowire has increased upon hydrogen passivation of dangling
bonds, since the band gap is cleared from the surface states.
Apparently, the band edge states of the bare $zb1$ nanowire
originate from the surface, while in $wz3$ structure they have
bulklike behavior. The different behavior of triply coordinated surface atoms in $zb1$ and $wz3$ structure is attributed to their structural orientation. Surface atoms of $zb1$ structure make atomic rings perpendicular to the wire axis. Dangling bonds of these atomic rings form several minibands (with low dispersion along $\Gamma$Z) in the band gap, with charge density localized at the surface. Surface atoms of $wz3$ structure, however, form atomic chains along the wire axis. The dangling bonds of these atomic chains form surface states with high dispersion, which fall in the band continua. The effect of hydrogen saturation is investigated in the case of a slab in Fig.~\ref{fig:fig8}. Here the first and third valence band edge states of bare slab structure are labeled as A and B, respectively. These dispersive states have bulklike character.
Upon the passivation band A rises to A' forming the conduction band edge, and band B rises to B' forming the valence band edge of the passivated structure. Surface states having low dispersion are lowered to valence band continua.
\begin{figure}
\includegraphics[width=8cm]{fig9.jpg}
\caption{Same as in Fig.~\ref{fig:fig8} but for $zb2$ structure.
The shaded region in the band structure of $zb2-37$ corresponds to
the band gap opened after the passivation of dangling bonds with
hydrogen. $zb2-19$ and $zb2-61$ nanowires are not passivated with hydrogen}\label{fig:fig9}
\end{figure}
The effects of the dangling bond surface states are even more
dramatic in $zb2$-type bare GaAs nanowires in Fig.~\ref{fig:fig9},
where the band gaps are closed and the nanowires become metallic.
Therefore, $zb2$-type is the only type we have considered to have
metallic character. It is interesting that the corresponding
infinite slab structure have a finite band gap. $zb2$ structures
would have a band gap which is larger than that of the slab
structure if they would had no partly filled corner states
crossing the Fermi level. Isosurface charge density plots support
the idea of band edge states being originated from the dangling
bonds at the corners of hexagonal cross-section. We further tested
these arguments by passivating the dangling bonds of $zb2-37$ with
hydrogen. Upon hydrogen passivation all flat surface state bands
in the range of energy from -0.5 eV to 0.5 eV disappeared and a band
gap of 0.9 eV opened. The resulting band profiles near the edges
of conduction and valence bands are similar to those of the slab
$(\bar{1}10)$ slab surface given at the right hand side of
Fig.~\ref{fig:fig9}.
\section{Discussions and Conclusions}
We have performed first-principles DFT calculations to reveal the
atomic and electronic structure of six different types of bare and
hydrogen saturated GaAs nanowires. Nanowires considered have a diameter less than 3 nm,
and at this sizes $wz3$-type have the highest cohesive energy per
atom pair, while A-wires have the lowest. In general, $wz$
nanowires have higher cohesive energy than $zb$ nanowires, but the
difference in cohesive energies decrease with increasing diameter.
We found that all bare GaAs nanowires are semiconducting, except $zb2$-type. In the latter structure, dangling bond states of atoms having coordination number of two cross the Fermi level and hence the structure becomes metallic. For bare GaAs nanowires in $wz$ structure with surface atoms all having coordination number of three, the dangling bond states associated with these surface atoms do not appear in the band gap, but rather in the band continua. Therefore, quantum (size) confinement effect is apparent in their band gap variation with radius. For these $wz$-type nanowires the band gap may increase upon the passivation of dangling bonds with hydrogen, since bonding of surface atoms become more bulklike.
Surface states of GaAs nanowires in $zb$ structure fall in the band gap, even if all surface atoms have coordination number of three. In $wz$ structure surface atoms form atomic chains along the wire axis, while in $zb$ structure they form non-interacting atomic rings perpendicular to the wire axis. Hydrogen saturation dramatically increases the band gap of $zb$ structures by clearing the surface states from the forbidden region. Generally, increasing diameter results in more bulk like valence band edge, however conduction band edge behave more surface like.
For reasons explained in detail, the band gap variation of GaAs nanowires is rather complex and
depends on their type and geometry, diameter, relaxation and also
whether the dangling bonds of surface atoms are passivated with
hydrogen. We believe that present results are valuable for further
research on GaAs and other III-V compound nanowires dealing with
their doping, forming heterostructure and multiple quantum well
structure and their fuctionalization to get new electronic and
magnetic properties.\cite{gudiksen,bjork}
\begin{acknowledgments}
Part of the computations have been carried out by using UYBHM at
Istanbul Technical University through a grant (2-024-2007).
\end{acknowledgments}
| {
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} | 9,489 |
Editor's Notes: James Corbett's excellent report The Last Word on Terrorism defines "terrorism" concisely and brilliantly exposes doublespeak used by the media and such people as Henry Kissinger. This piece ties in with the recent events in the Middle East region, and the US policy that will pick its partners, or launch an attack. The dollar's muscle comes from the ultimate form of suppression and violence, yet the people in the streets are challenging the brutal rule of law and corruption of partners of an oil oligarchy and the concentration and corruption of wealth. The whole issue is getting closer to home as corporations are ruling our political system and driving deals for the rich and putting the squeeze on everyone else.
In breaking news, too, following the Rolling Stone article, "Another Runaway General as Army Deploys Psy-Ops on U.S. Senators", there are links to one the greatest military psychological operations in the 20th century, regarding the claims that the Apollo missions landed and launched off from the moon from 1969 to 1972. I realize this is difficult to believe since many of us were convinced that this had happened throughout our lives. While many know that 9/11 was an inside job with military-grade explosives bringing down 3 World Trade Center buildings in controlled demolitions, using airliners and patsy hijackers for a false flag operation, many still believe in the visitations of the moon and will ridicule others for thinking otherwise. FN has had to look into this issue to understand the military pursuit of lies at all cost following assassinations, coup d'etat in the US and other governments, and control of the media and distortion of science. FN challenges you to check out the films we are screening this month.
explode Hydrogen and Atomic bombs in the Van Allen belts.
Do the research – don't be mind controlled!
FN also recently posted an article that was published in Scientific American, "Can Geoengineering Save the World from Global Warming?" Thus, we updated our resource page on Military Weather – Chem/Nukes in Space. I wish I was making all this stuff up, but awakening to our reality is much safer than remaining ignorant. I appreciate those open to this information, and especially those who are helping others to awaken. If you want to know a good first step in encouraging others to awaken to the grand deceptions of our day, share the FN website and I also recommend, which was agreed to by unanimous consensus by 9'11 truth leaders and representatives of 9/11 truth groups – to endorse and promote Chris Pratt's film, DECEPTIONS to help bring 9/11 truth into context of other grand deceptions, while maintaining a focus in demanding a new and thorough investigation into the events of September 11, 2001.
of the Middle East are transforming the region themselves.
"9/11: Science and Society" Now on Youtube!
David Ray Griffin exposing the truth of 9/11.
Audio is in archives or go to this MP3 link! | {
"redpajama_set_name": "RedPajamaC4"
} | 330 |
Diuridinae – podplemię roślin z rodziny storczykowatych (Orchidaceae). Obejmuje 2 rodzaje i około 100 gatunków występujących w Azji Południowo-Wschodniej, Australii i Oceanii.
Systematyka
Podplemię sklasyfikowane do plemienia Diurideae z podrodziny storczykowych (Orchidoideae) w rodzinie storczykowatych (Orchidaceae), rząd szparagowce (Asparagales) w obrębie roślin jednoliściennych.
Wykaz rodzajów
Diuris Sm.
Orthoceras R. Br.
Przypisy
Storczykowe | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,465 |
"In this ingenious mess of a novel, with all the bullshit paranormal characters that a superhero-habituated modern audience requires, Jarett Kobek clearly and calmly explains our genocidally idiotic mess of a culture as it plunges enthusiastically into a genuine, non-fictional damnation that Batman isn't going to rescue it from. Brilliantly funny and sociologically terrifying, _Only Americans Burn in Hell_ is the best satire of our contemporary nightmare that you will ever see, and very possibly the last. Read it while you're still neurologically capable." **Alan Moore**
"This time Kobek has called all of his own craziest bluffs and rocketed straight over the ionosphere, into sheer blue sky and beyond—this book breathes in outer space. One wishes the phrase 'takes no prisoners' had been saved for when we'd need it. If you don't find yourself busting a gut laughing, then you're probably still in denial of how deeply you feel implicated." **Jonathan Lethem**
"Jarett Kobek's books are an attempt to explode what the novel could still be, as radical as Samuel Richardson or Laurence Sterne's attempts to define what it was in the first place. _Only Americans Burn in Hell_ is a fantasy work about mythic Amazons time-travelling to modern America of the type currently clogging multiplexes—but one infected by anxieties about sexual politics, the ethics of the digital world and the horrorshow of the Trump administration. Kobek makes you laugh and think at the same time, engaging both the head and the gut." **Stewart Lee**
" _Only Americans Burn in Hell_ is a smoking hot and hilarious dissection of why the world is in such a mess right now. While you watch Jarett Kobek pour gasoline on everything—international politics, Internet culture, the book business, American presidents, Christianity, capitalism, the fantasy genre—you will be so mesmerised and laugh so much that your faith in humanity will be restored by the time he lights the match. Jarett Kobek is one of our most groundbreaking writers." **Dorthe Nors**
"There's a chance that when the dust settles on the cultural and political insanity of the early twenty-first century, only one writer will remain relevant: Jarett Kobek. With scathing wit, shocking insight and brutal honesty Kobek demolishes social media and the publishing industry, introduces us to a Saudi Prince hopped up on DMT, and conjures perhaps the most important and hilarious fairy story ever written." **Ivy Pochoda**
"To think of Jarett Kobek as merely ('merely!') an American Houellebecq would be sorely to miss the point. His energy, intellect, wit, sensibility, erudition, tenderness, and—yes—obnoxiousness add up to something wholly original, and absolutely necessary. _Only Americans Burn in Hell_ extends the vibrant, reckless critiques offered by _I Hate the Internet_ into our present moment, and perhaps a little bit beyond: one reads it with a sense of elation, gratitude and relief that someone is saying these things out loud. So far as that goes, Kobek may be the only contemporary American novelist who matters." **Matthew Specktor**
# **Only Americans
Burn in Hell**
ALSO BY JARETT KOBEK
_ATTA_
_I Hate the Internet_
_The Future Won't Be Long_
_Do Every Thing Wrong! XXXTentacion Against the World_
First published in Great Britain in 2019 by Serpent's Tail, an imprint of Profile Books Ltd
3 Holford Yard
Bevin Way
London
WC1X 9HD
_www.serpentstail.com_
Copyright © 2019 by Jarett Kobek
Cover Design: Peter Dyer
Cover Photograph: iStock
The moral right of the author has been asserted.
All rights reserved. Without limiting the rights under copyright reserved above, no part of this publication may be reproduced, stored or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise), without the prior written permission of both the copyright owner and the publisher of this book.
A CIP record for this book can be obtained from the British Library.
ISBN 9781788162203
eISBN 9781782835349
## Contents
Introduction: Thank You for Your Honesty
Chapter One: Certain Facts about Celia, the Queen of Fairy Land
Chapter Two: Some Facts about Fern
Chapter Three: How Fairy Land Escaped the Clutches of Global Capitalism
Chapter Four: Child, Be Strange
Chapter Five: Wonder Women
Chapter Six: Willkommen im Dschungel
Chapter Seven: The House on the Hill
Chapter Eight: Gentlemen Prefer Blood
Chapter Nine: Cleaning up the Mess
Chapter Ten: On the Streets of Los Angeles, There the Wild Beast Slumbers
Chapter Eleven: Let Slip the Dogs of War
Chapter Twelve: hello from sex drenched hollywood
Chapter Thirteen: Routine Humiliations
Chapter Fourteen: When Y Meets X
Chapter Fifteen: Until the Wheels Fall Off and Burn
Chapter Sixteen: Drink of Me, Eat of Me
Chapter Seventeen: How It All Went Down
Chapter Eighteen: Bleak House
Chapter Nineteen: Exeunt Rusticano
Chapter Twenty: What Rusticano Didn't Say
Chapter Twenty-One: καταδυσόμεθ᾽ εἰς Ἀΐδαο δόμους
Chapter Twenty-Two: Literary Fiction
Chapter Twenty-Three: The Full Throat of Christian Virtue
Chapter Twenty-Four: The Man Who Said Bo! to a Goose
# **Only Americans
Burn in Hell**
**JARETT KOBEK**
## Introduction
## Thank You for Your Honesty
The last time anyone thanked me for my honesty was in an email sent by the Office of Development and Alumni Relations at New York University, an institution of higher learning centered in New York City's Greenwich Village.
NYU has three distinguishing characteristics.
The first is that it's my alma mater.
I graduated in 2002 AD, after giving the university an absurd amount of money for an undergraduate degree.
This is why the school begs me for money.
It's like a junkie who can't stop.
NYU's second distinction is its inhuman cost.
In 2017 AD, the tuition was $46,170 for a year. Throw in campus housing and administrative fees, and the total was $63,472.
To put this in context: as of 2016 AD, the American median income was $57,617 per person.
You can't charge $63,472 and expect much more than a mixture of the rich and the gullible.
The gullible emerge from NYU in a state of financial ruin, indebted for a substandard education that they could've received for about 1/8th of the price at a state-run university.
Welcome to adulthood!
Time to pay back $253,888!
With compounding interest!
The rich kids come out fine.
The rich kids are always fine.
The third thing that distinguishes NYU is its Abu Dhabi campus, which opened in 2014 AD.
The idea behind the Abu Dhabi campus was to construct a mirror-world NYU that bestowed the same substandard education, and thus conferred the same substandard degree, as the Greenwich Village campus.
The only difference was that the mirror-world campus would be located on Happiness Island in the United Arab Emirates, an absolute monarchy funded by the world's seventh-largest oil reserve.
Nothing says academic freedom like petrol feudalism!
Before the Happiness Island campus had its grand opening, an article appeared in the _New York Times_ which detailed the nature of NYU's new venture.
The school's administration had arranged a deal with the government of the United Arab Emirates, in which the oil monarchy would cover the whole expense, and construction, of the mirror-world campus.
Picture this: a repressive regime renowned for its human rights abuses makes a deal with a bunch of very naïve and very greedy American bureaucrats.
What could possibly go wrong?
The oil monarchy sent labor recruiters around the Indian subcontinent.
The recruiters told people that they could make big money if they came to Abu Dhabi and helped build the mirror-world campus on Happiness Island.
When people of the Indian subcontinent arrived in Abu Dhabi, happiness proved elusive.
The workers were stuck in subhuman housing and paid dirt-poor wages.
When they tried to strike for the money they were promised, they had the shit beat out of them by the police.
And the workers couldn't leave Happiness Island.
Their passports had been confiscated.
They were slaves.
And although putting people into human bondage and making them build college campuses was a time-honored tradition, it'd been a very long while since any American institution of higher learning had involved itself in this sort of disgrace.
On August 30th, 2017 AD, I received an email from NYU's Office of Development and Alumni Relations.
The email was from a Senior Annual Giving Officer named Corey, and it informed me that Corey was coming to Los Angeles.
I live in Los Angeles.
Corey wanted to have lunch or get coffee.
For years, I'd received emails from NYU. All of the emails begged for money.
But none of them had extended a personal invitation of food or caffeine.
Corey's email made me wonder if NYU employed a clipping service to search for media mentions of prominent alumni.
In the sixteen months prior to Corey's email, I'd lived as a minor literary sensation off the strength of my novel _I Hate the Internet_. Some of the news stories about my book's unlikely success had mentioned that I was an alumnus.
Whenever someone thanks you for your honesty, what they mean is _shut the fuck up._
Being thanked for your honesty is like someone tattooing the word SEXY on their upper arm.
If it has to be said aloud, its opposite is sure to be true.
"Your face is very stupid!"
"Thank you for your honesty."
"Madame, everyone in this room knows that your wife is a living grotesque!"
"Thank you for your honesty."
"Never invade Russia in the winter!"
"Thank you for your honesty."
"Your prolonged substance abuse is destroying your body, your employment prospects, and the mental health of your family members!"
"Thank you for your honesty."
I wrote back to Corey.
This is what I wrote:
**Thu, Aug 31, 2017 at 1:26 AM**
**From: Jarett Kobek**
**To: Corey**
**Subject: RE: Meeting with NYU in LA?**
Dear Corey,
Thanks for the offer, but I've long disconnected myself from NYU.
It's impossible to imagine supporting an institution that allowed slave labor to build an entire campus in Abu Dhabi and has failed, completely, to redress the situation in any meaningful fashion.
Thanks,
Jarett
On the surface, my email would appear to be motivated by a principled stance.
A principled stance is the euphemism that people like myself, who are hopelessly mired in the Looney Left, use to describe those moments when they say or do something that ruins a party by taking exception to a harmless comment or action.
I suppose that it was a principled stance.
It's appalling that I attended an institution which placed a fig leaf atop global evil.
And it's repulsive that the fig leaf was built by slaves.
And, really, I'm sorry how mixed I made that fig-leaf metaphor.
But only a mixed metaphor can contain the existential horror of NYU.
Also: I'm a terrible writer.
But, really, this was why I sent my note to Corey: I just wanted NYU to stop asking for money.
You can't imagine how much email starts coming in after you've been a minor literary sensation.
In the same month that Corey extended his invitation for food or caffeine, a major American publisher issued my follow-up to _I Hate the Internet_. It was a novel that ended up with the title _The Future Won't Be Long._
It was a massive commercial failure.
Less than 300 copies sold in its first six months!
_I Hate the Internet_ sold 300 copies in its first two weeks!
Reader, this was shocking.
If for no other reason than the simple fact that _The Future Won't Be Long_ was published by Penguin Random House.
Penguin Random House is the biggest publishing conglomerate in the world. It's a multibillion-dollar multinational corporation owned by another multibillion-dollar multinational corporation called Bertelsmann, which spent much of World War Two producing Nazi propaganda and using Jewish slaves to work in its factories.
My book was backed by Nazi money!
And it still failed!
So what happened?
For decades, everyone who had any pretense to High Culture wasted fathomless hours talking about theorists like Michel Foucault and Jean Baudrillard.
These people with pretenses to High Culture had advanced the idea that reading incomprehensible French books gave them special insight into the way the world works.
Sometimes they expressed this pretense in unreadable texts called master's theses and doctoral dissertations.
One of Baudrillard's ideas was very popular. He'd theorized that there would be a moment when reality collapsed into fiction, at which point it would then be impossible to distinguish the fake from the actual.
He called this the Hyperreal.
But what neither Baudrillard nor his readers could ever locate was the exact moment when the Hyperreal would replace the real.
It was a mystery, floating-point arithmetic without any definitive beginning.
But then it happened.
On November 8/9th, 2016 AD, while I was asleep in London's Little Venice, passed out in someone's former childhood bedroom above Blomfield Road, the real became Hyperreal.
Donald J. Trump, the world's best approximation of living fiction, whose body appears to be constituted of media coverage stitched together with plastic surgery, was elected to the Presidency of the United States of America.
When this happened at around 6AM Greenwich Mean Time, a film crew was on Blomfield Road. They were shooting footage for a film called _Paddington 2_.
The film was about a very fussy bear with a posh accent, its cartoon body generated by computers. The bear goes to prison and makes friends with inmates whose bodies were generated by loveless sexual reproduction.
My smartphone started vibrating.
People were sending me text messages of shock and awe.
They were freaked the fuck out.
_What just happened?_ they asked.
It turned out that the people who were the least prepared for the Hyperreal were the same people who'd spent decades talking about the Hyperreal.
They had no special insight into anything!
A fog descended upon them.
Trust me, I know what I'm talking about.
These people are my friends.
And, holy shit, these people did not see this thing coming.
And, double holy shit, did it ever make them annoying.
Only two people have ever thanked Donald J. Trump for his honesty. The Ugandan dictator Yoweri Museveni and David Duke, a former Grand Wizard of the Ku Klux Klan.
Great company!
No one else has ever thanked Donald J. Trump for his honesty.
And with good reason.
The President could not be honest.
This was not because the President went out of his way to exist in a state of perpetual falsehood.
The President could not be honest because he existed in a moral universe where there was no truth and there are no lies.
He was hopelessly insane.
He lived in the Hyperreal.
Ideas floated into his head, ideas floated out.
And the whole world jumped at their utterance.
If the country was bombarded, every day, by a morass of awful noise that displayed at best a partial relationship to the truth, and if the citizens of that country were expected to run around like chickens with their heads cut off in response to this awful noise, then why not empower someone to make a different kind of noise?
Why not get someone who would make noise in a different direction?
To steal a joke from the comedian Stewart Lee: it was like being given a room in a fleabag motel, and, in protest at its unsatisfactory conditions, shitting in the room's bed before realizing that you had nowhere else to sleep.
But people did it anyway.
They shit the bed.
They voted for Donald J. Trump.
A fictitious being with, at best, a tenuous connection to reality ended up at the head of the world's most powerful military and the world's biggest economy.
He was from the fourth branch of American governance: the Celebrity.
And he had taken over the first branch: the Executive.
Reality collapsed into fiction.
And you would think, reader, that the best time to be backed by Nazi money was after a living caricature had inaugurated the Hyperreal.
But you'd be wrong.
Here was the implicit sales pitch behind my book: _The country is collapsing, reality has gone mad, and a White Supremacist just murdered a woman by driving his Dodge Challenger into a crowd of protestors. So please buy my book about drug parties in the 1980s AD!_
No one wanted to read that shit.
And all the Nazi loot in the world couldn't make it otherwise.
This book that you are reading was going to be a cracked attempt at the sorry bullshit that people in the Hyperreal actually want to read, which are mindless tales about supranatural creatures.
I had come up with what I thought was a funny contrast between narrative voice and subject matter.
I was going to write a fantasy novel in the imagined voice of an alcoholic from southeastern New England. It was going to be the _The Hobbit_ as told by a gin-room rummy from Fall River, Massachusetts.
It was going to go something like this:
Fuckin' Bilbo the little midget over here, he crawls into the prickers and what does he see but some fuckin' trolls sittin' at a fuckin' fire.
"Wow, I says, wow. You tell me, guy, what the fuck am I gonna do with some trolls?" says Bilbo the Dildo. "Who am I, a fuckin' Terminator? I ain't gettin' myself eaten just cause some big shot Poindexter thinks he's a wizard."
But life kept interrupting.
Things went screwy.
It's possible that I had a nervous breakdown.
Somehow I ended up writing a novel that is not only about whimsical undying characters who live on a magical island called Fairy Land, but is also a book that functions as an accidental allegory for a social media hashtag.
This state of affairs seems like a perfect statement about the present moment.
Corey wrote back about a week after I'd sent my email.
This is what he wrote:
**Thu, Sep 7, 2017 at 11:15 AM**
**From: Corey**
**To: Jarett Kobek**
**Subject: RE: Meeting with NYU in LA?**
Dear Mr. Kobek,
Thank you for your response, your honesty, and your candor.
Best wishes,
Corey
## Chapter One
## Certain Facts about Celia, the Queen of Fairy Land
Here are some things that you should know.
The first of these things: Celia was an immortal and undying being, possessed of supranatural powers.
The second: Celia lived on Fairy Land, which was an island in the sea past the sun.
The third: Celia was Fairy Land's Regnant Queen.
The fourth: Celia had become Queen of Fairy Land when she and the other undying women on the island in the sea past the sun had decided to expel or murder all of the men in Fairy Land.
The fifth: when all of the men were dead or expelled, the women of Fairy Land gathered together and called upon Celia to reign as their Queen.
The sixth: Celia accepted the women's call and wore the crown of Fairy Land.
The seventh: Celia liked to fuck.
Here's another thing that you should know: everything is going to be okay.
It isn't easy living in a world where every device of mass communication has been designed to tell you that you're horrible.
It's no picnic being taunted by a Greek chorus when your only economic future is feudalism.
Take a deep breath.
Make sure your exhale is longer than your inhale.
You're on Planet Earth until you're dead.
Everything between now and then is survival.
And survive is what you'll do until you don't.
Calm down.
For the length of time that it takes you to read this book, everything will be fine.
Despite her status as an immortal and undying being from Fairy Land, Celia was pretty uptight in her understanding of gender and sexual norms.
If Celia had not been so rigid in her embrace of gender and sexual norms, Celia could have just fucked some of the other immortal and undying women on the island past the sun.
It could have been very _Wonder Woman_.
But Celia was a hardliner.
Which meant that Celia was into fucking men.
And Celia lived on an island where all of the men had been expelled or killed.
The construction of Celia's monarchy had screwed up Celia's sex life.
So it wasn't very _Wonder Woman._
It was very _Game of Thrones_.
_Wonder Woman_ and _Game of Thrones_ were both literary intellectual properties that had been developed past their humble origins into huge media spectacles.
_Wonder Woman_ was about an undying woman named Diana who lived on an island in the sea. Diana left her island of lesbians to kill a bunch of Germans.
_Game of Thrones_ was about unpleasant people in a fantasy medieval world.
The unpleasant people in _Game of Thrones_ killed and fucked each other while reinforcing a worldwide hegemony that replicated, for no particular reason, the racial, sexual, and cultural prejudices of the British colonial era.
Both media spectacles were pornography about war.
This pornography was very popular with people in the United States of America.
The United States of America was a warrior nation that liked to fuck up the shit of weaker countries through unending battles, through the dropping of bombs, through the wholesale slaughter of the poor.
The huge media spectacle of _Wonder Woman_ was released in 2017 AD, by which point the United States of America had been at war with the country of Afghanistan for sixteen years.
It was the longest war in the history of the United States of America.
It was sixteen years of turning illiterate Muslim peasants into bloody red streaks of chalk.
Almost everyone in the United States of America pretended that it wasn't happening.
But they loved _Wonder Woman_.
And they loved _Game of Thrones._
Celia knew a thing or two about being transformed into a media property.
Back in 1599 AD, a guy from England had written a short book called _The Most Pleasant History of Tom a Lincoln_.
Some of the book was true, in that it recounted events that had happened to Celia.
Most of it was bullshit.
Amongst other nonsense, _Tom a Lincoln_ was about how Celia had met Tom a Lincoln, who was also called the Red-Rose Knight. He was King Arthur's bastard son.
_Tom a Lincoln_ was about how Celia had allowed the Red-Rose Knight to enter Fairy Land after his boat had washed up on the island in the sea past the sun.
_Tom a Lincoln_ was about how Celia had fucked the Red-Rose Knight, and how as a result of that fucking, Celia had birthed a son named the Fairy Knight.
About two years after _Tom a Lincoln_ was first published, Celia was given a copy of the book. She discovered something that happens to anyone who becomes the subject of media coverage.
Celia discovered the people who create media coverage are depraved beasts that will say anything for money.
The author of _Tom a Lincoln_ was a guy named Richard Johnson. He is described thusly in the _Oxford Dictionary of National Biography_ :
Richard Johnson was in every sense a derivative writer: his romances synthesize a mass of traditional materials along with some more sophisticated modern texts, _The Faerie Queene_ among them; he retails familiar ballads, songs, and jests under a light disguise of novelty; and his secondhand pamphlets are aimed at the prides and prejudices of a readership of London citizens and their families. His career is a paradigm of popular commercial writing for the press in his time...
In other words, a total fucking hack.
May they be with us always!
Richard Johnson wrote that Celia had killed herself after King Arthur's bastard son left Fairy Land and didn't return.
This lie was presented in a very dramatic fashion, with the Red-Rose Knight trying to return to Fairy Land but facing ill winds which kept his ship from reaching the island.
Richard Johnson had written out Celia's suicide note, which he said was inked in her own blood.
It was very sad.
It dripped with pathos.
It was stuffed with treacle.
It didn't sound anything like Celia.
No one in Fairy Land had any idea how Richard Johnson had learned about the Red-Rose Knight and his visit to Fairy Land.
So many of the details were wrong. Especially the part about the Red-Rose Knight's valiant resistance to Celia's sexual advances.
Especially the suicide.
How could Celia, an undying being, kill herself?
And why would she do it for a mortal man?
Richard Johnson had written a bit about Celia's son, the Fairy Knight. In Part II of _Tom a Lincoln_ , the Fairy Knight performs all manner of great deeds and wins the world.
This was sort of true.
In their sexual congress, Celia and the Red-Rose Knight had indeed created the Fairy Knight. But the Fairy Knight hadn't performed all manner of great deeds or won the esteem of the world.
All that happened to the Fairy Knight was that he lived out his early life in Fairy Land until he was banished in his sixteenth year.
No one from Fairy Land ever saw him again.
Richard Johnson omitted that Celia and the Red-Rose Knight had a second child.
When the Red-Rose Knight arrived on the shores of Fairy Land, he and his shipmates had entered the kingdom and never left.
Why would they?
They were surrounded by supranatural women. Many of these women were like Celia. They were hardliners when it came to gender and sexual norms.
And they hadn't seen any men in a very long time.
The Red-Rose Knight's men lived like princes, fucking their tiny brains out in grottos where the flowers sang songs in time with the sexual thrusting while the trees swept their branches along the rutting lovers' flesh.
While they fucked out their tiny brains, the Red-Rose Knight and his men were enacting the general bullshit con on women that is heterosexuality.
The rules of the game go like this: for every thousand remarkable women, the really beautiful ones, the really smart ones, the really smartly beautiful ones and the really beautiful smart ones, there's about one semi-okay man.
Heterosexuality is a giant joke played on the women of the world.
Here's the punchline: if you're a woman, and you want to experience the biological imperative of sex with a man, you pretty much have to bed down with a sack of worthless crap.
And don't forget: this was back in the early medieval period, so the Red-Rose Knight and his men were no beauties.
They really were sacks of worthless crap.
But they were the only dicks in town.
Richard Johnson wrote that the Red-Rose Knight's men knocked up most of the women of Fairy Land.
This isn't true.
The only person who got knocked up was Celia.
She gave birth to the Fairy Knight.
Then, about a year later, she had her daughter.
Celia named her daughter Fernstuff Wirethorne, Keeper of the Sacred Flame, Fiery Green Horsetender, and Mistress Magical of Fairy Land and Its Environs.
No one on Fairy Land could be bothered saying Celia's daughter's full name.
Everyone called Celia's daughter Fern.
Not long after the birth of this daughter, the Red-Rose Knight keeled over and died of an ailment that had yet to be named.
He was murdered by typhoid fever, which meant that he'd been killed by water filled with human shit.
Unbeknownst to everyone, one of the Red-Rose Knight's men was an asymptomatic carrier of typhoid fever.
The carrier's name was Orson.
Orson's hobbies included skipping rocks across ponds, speaking in a high-pitched voice while imitating his mother's folk wisdom, and pretending that he was a beautiful princess waiting to be rescued by a dashing knight.
Being an asymptomatic carrier of typhoid fever meant that although Orson never showed any signs of having the disease, his body was in a state of constant typhoid production.
Orson was like a factory worker under crony capitalism: he was making something, but he didn't share in the gains of that production.
One night, Orson was corralled into helping prepare a feast. This had happened because one of the usual preparers of food was busy fucking out his brains in a shadowy elm grove by the Ancient Rocks of Forever.
And so Orson helped prepare the feast for the Red-Rose Knight, the Red-Rose Knight's men, and all the women of Fairy Land.
Orson placed the Red-Rose Knight's drinking vessel on the communal table.
Because it was the early medieval period, Orson hadn't washed his hands after using the latrine.
He got his left index finger in the Red-Rose Knight's water.
The women of Fairy Land were immune to typhoid fever.
When the Red-Rose Knight died from drinking too much water filled with Orson's shit, he resolved an ethical dilemma.
The dilemma was this: despite liking to fuck, Celia also believed in and embodied the organizational principles of Fairy Land, and the preeminent organizational principle of the Realm was that all men had to be killed or banished.
Celia had stretched this rule for a very long time. For years, she'd let the Red-Rose Knight and his men stay on Fairy Land.
Her citizenry had started to complain.
At the very moment when the Red-Rose Knight died from consuming too much of Orson's shit, Celia had been trying to figure out how to tell the father of her children that he was to be banished from her island.
After the Red-Rose Knight died, the women of Fairy Land killed all of the Red-Rose Knight's men who'd survived their encounter with Orson's shit.
With one exception.
Rusticano was allowed to live.
Orson was the first to lose his head.
Unlike in his fantasies of being a princess, no one saved him from death.
His hands were filthy.
## Chapter Two
## Some Facts about Fern
A magical bullshit thing had happened when Fern was born. Maybe it was because she was the Queen's daughter, maybe it was because she was King Arthur's granddaughter. Maybe it was because she was the first and only woman born in Fairy Land after the expulsion and murder of its men.
Whatever the cause, the effect of this magical bullshit was that the health of the Realm of Fairy Land was tied, directly, to Fern.
She was its living avatar.
In the times when Fern was happy, Fairy Land was a paradise, full of joy and pleasure. The harvests were incredible, the livestock flourished, and the lesbianism was euphoric and multi-orgasmic.
When Fern was angry, Fairy Land was miserable. The harvests were pathetic, the animals all perished, and the lesbianism drifted into a mythical bed death.
In the times when Fern experienced feelings of vulgar existentialism, wondering about the purpose of her or any other life, the whole of Fairy Land entered a state of paralysis, of grinding malaise without discernible beginning or end.
In the five days before Fern ovulated, Fairy Land was hell on Earth.
One of the ways by which Fern dealt with being the living avatar of Fairy Land was to go on vacation.
Most women on Fairy Land never left the island. There was no rule against travel, but enormous social pressure kept the citizenry from venturing into the wider world.
Some people went on trips, sometimes, but it was always awkward.
No one left as much as Fern.
Fern's first departure from Fairy Land was in the Year of the Silken Cutthroat, which roughly corresponded to 1349 AD, 749 AH, and 5109 AM.
Fern set off on a little boat, with no crew, and sailed across the sea to France.
Once she had landed, she made her way by horse to Paris.
Are you wondering how Fern managed to do all of this?
Don't forget: Fern's mother was Celia, the Queen of Fairy Land. Fern was the daughter of an undying being possessed of supranatural abilities, and Fern herself was the living avatar of a magic realm.
Money and horses and boats and all of that?
Fern waved her hand.
Fern performed magic.
And if you're like the Los Angeles-based artist William E. Jones, when you read about Fern performing her magic, you thought to yourself: "Every time that the supernatural enters fiction, it's a cheap shortcut around the craft of storytelling."
And you're right.
Fern's bullshit magic really was a cheap shortcut around the craft of storytelling.
But take a deep breath.
Calm down.
Everything is going to be fine.
Just remember: this level of unprofessionalism has been positively reviewed by the _New York Times._
When Fern got to Paris, the French capital was not what she had imagined.
It was 1349 AD.
The Black Death had arrived.
Tumors were sprouting from people's skin and then bursting open into fireworks of wretched fluid.
The Black Death was rotting people's flesh with gangrene until the people died.
They were living beings and they had lives and loves and hates and cares and worries and now they were lifeless matter.
Food for worms.
Trash scattered around the streets of Saint-Germain-des-Prés.
Stinking to high heaven.
It was a pretty shitty vacation.
When she traveled around Paris in 1349 AD, stepping over the bodies of the tortured urban poor, Fern collected news and information about the outside world.
It'd been about a century since the people of Fairy Land had learned much about what men were doing to the planet.
The news was not good.
It never was.
Fern was nothing if not resolute.
When she returned to Fairy Land, she decided to rest, but also decided that she would go again into the wider world.
And so she went on more vacations.
It was the late medieval and early modern periods. Fern saw unfathomable amounts of human suffering, but other than the Lisbon earthquake of 1755 AD, nothing was ever as bad as Paris in 1349 AD.
Fern became a seasoned traveler.
She learned to put up with a lot of crap, as long as she got the manic contact high and psychic relief that comes with being far from home.
And as she moved around the world, she collected more news.
Fern brought back this news in physical formats.
At first it was books, which eventually turned into other forms of media. Newsbooks, broadsides, wire recordings, shellac and vinyl records, audio cassettes, magazines, newspapers, reel-to-reel recordings, LaserDiscs, CDs, VHS tapes, DVDs, HD DVDs, Blu-ray Discs.
This is how Celia learned about Richard Johnson and _Tom a Lincoln_.
Unlike the sojourns of the island's other women, Fern's trips abroad were embraced by the residents of Fairy Land.
There was a simple reason.
As soon as Fern left the island, her mood stopped influencing their lives.
During Fern's seventh trip abroad, which was supposed to be for six months, but lasted about two years, the residents of Fairy Land revised their previous opinion on Fern's forays away from the island.
They had noticed a material change in the quality of their life.
It wasn't anything that anyone could explain.
It wasn't anything that had a definite beginning or end.
But there was a difference in the air, in the very luster of the trees, in the smell of things, in the crispness of life.
It was as if the island had entered into a long, flat period of depression.
People went through the motions, people did what they always did, but something was off. There was a pointlessness that made a mockery of the simplest actions.
Without Fern, the women of Fairy Land had been stripped of magical charm.
They were seeing life as it was.
They were witnessing existence with a dead honest clarity.
And life was brutal.
When Fern returned to the island, the depression lifted.
The magical charm returned.
Here then was Fern's version of the bitter twist in the faery stories and folk tales that mortals used to tell each other before the world anesthetized itself with prescription opioids, anal gangbang pornography, and the illusion of individual freedom in the pyramid of global order.
Without Fern, the taste of the Queen's honey was neither sweet nor bitter.
The sacred oak groves went unkempt.
The birdsong rang hollow.
The lamps burned less bright.
The lesbianism evoked orgasms that offered all the dull-eyed joy of being frigged off inside a stripmall swingers' club.
A deal was struck.
Fern would still leave Fairy Land, but for no period longer than it took for Fairy Land to be stripped of its magical charm.
Which was roughly a year.
All of which brings us to the Year of the Froward Worm, which roughly corresponded to 2017 AD, 1438 AH, and 5777 AM.
Fern had left Fairy Land about eighteen months earlier, during the Year of the Misplaced Butter.
She'd told everyone that she was going to Los Angeles, which was a city on the west coast of the United States of America, the warrior nation that had made a cottage industry of transforming illiterate Muslim peasants into char and bone.
Los Angeles was responsible for a disproportionate amount of the media produced in the United States of America.
The women of Fairy Land were well versed in this media.
They had magicked up an Internet connection and used it to pirate television shows and films produced in Los Angeles, which they then watched on a television they'd magicked up out of some old twigs and a bit of wool.
Fern had visited Los Angeles on several occasions. None of the other women from the island had visited Los Angeles.
In the Year of the Misplaced Butter, Fern announced that she was returning to the city.
It'd been about five years since her last visit.
"You will leave us for the full year?" asked Celia.
"Yes," said Fern. "But worry not, Mother, I shall return as ever."
Fern did not return. She was gone well into the Year of the Froward Worm, which roughly corresponded to 2017 AD, 1438 AH, and 5777 AM.
Flatness settled on Fairy Land.
Celia looked out at her kingdom. All she saw was the citizenry's empty faces and the graying of the flora and fauna. The lesbianism was mega-fallow.
"How long has my daughter been gone?" Celia asked her court advisors.
"By our counts," said the Chieftess of Celia's High Council, "One year, seven months, and six days."
"Have efforts been made to contact her?" asked Celia. "There has been no response, my queen," said the Chieftess. "We must go and find her."
## Chapter Three
## How Fairy Land Escaped the Clutches of Global Capitalism
The Twenty-First Century AD was full of people who had filthy hands.
In some places, like rural Bangladesh, the filthy-handed people were no different than Orson, the imaginative man who'd used early medieval hygiene to assassinate the Red-Rose Knight.
Their hands were covered with shit.
The exploitive global hierarchy of capitalism had denied them the basic mechanisms of modern life.
They had no plumbing.
The people who exploited the global hierarchy also had filthy hands.
But their hands weren't covered with shit.
Their hands were stained with the blood of the poor, which, like climate change and Islamic-themed terrorism, was a semi-accidental byproduct of exploiting the global hierarchy.
There were a lot of explanations as to why capitalists liked exploiting the global hierarchy.
Some of these explanations were purely psychological.
Some of the explanations were entirely about money.
Some explanations attributed an innate evil to the global capitalists.
But the most logical explanation, really, was that people became global capitalists only after they'd entered a secret contest to see who could own the ugliest house.
Reader, look into your heart.
Pore through your memories.
When was the last time you went to a really rich person's house and found it anything but hideous?
Another reason why global capitalists kept rural Bangladeshis covered in shit is that keeping rural Bangladeshis covered in shit ensured an unequal distribution of the world's wealth and resources, with a disproportionate amount of that wealth going to the global capitalists.
And it was an open secret that the acquisition of vast wealth was the quickest way for a human to become a supranatural being.
It was a documented scientific fact that, after an individual had accumulated vast wealth, then they reached what was called the Cash Horizon.
Beyond the Cash Horizon, the wealth-accumulating individual was transformed into a supranatural being.
In other words: _the rich were not human_.
If you're wondering why the rich felt the need to become supranatural creatures, then good for you!
It's the obvious question.
And here's the answer: there was a sense that by becoming supranatural creatures, the rich could conquer death and thus avoid their certain destination of Hell.
But even with the Cash Horizon, the rich still died.
Death remained unconquered.
And Hell was filled with the rich.
So don't say that this book lacks a happy ending.
In addition to owning unspeakably ugly homes and being able to withstand mephedrone psychosis while attending black-tie galas, those who passed the Cash Horizon were granted the ability to hear the rare Lou Reed outtake "Doin' the Dookie."
Written for the Velvet Underground in 1965 AD but not recorded until sessions for Reed's 1973 AD masterpiece _Berlin_ , "Doin' the Dookie" had been sequenced to appear on that album's A-side, but was swapped out at the last minute in favor of "Oh Jim."
The lyrics of "Doin' the Dookie" were what anyone'd expect, Dylanesque nonsense about hip gender-bending junkies punctuated, loosely, by exclamations:
_Oooohhh, fleet-foot Sam had his can of jam_ ,
_Bertha Mason was feeling kinda kooky, huh_ ,
_And her girl Will, yeah_ ,
_He was looking pretty spooky_ ,
_And they was all Doin' the Dookie_ ,
_Oh whoa, they was Doin' the Dookie_ ,
_For you and me._
_Doin' the Dookie!_
_Oh wee._
A rogue engineer, stoned beyond belief on Moroccan hash, had misplaced the master tape of "Doin' the Dookie" inside an aquarium.
Ten years later, when the tape was fished out, it was discovered that the aquarium's chemical-soaked water had enacted an alchemical corruption, transforming Reed's recording into high-frequency sound beyond the range of normal human hearing.
The music was still there.
The lyrics were still there.
They simply could not be heard by human beings.
But if a person's net worth had passed the Cash Horizon, then their enhanced senses allowed them to hear "Doin' the Dookie."
And if you're wondering, reader, what it's like to be a superhuman being whose money has pushed you well past the Cash Horizon, you'd do worse than to consider a character who'll show up in later chapters of this book.
This character is named His Royal Highness Mamduh bin Fatih bin Muhammad bin Abdulaziz Al Saud.
HRH was a son of the House of Saud, which was the monarchy that ruled much of the Arabian Peninsula and owned the world's second-largest oil reserve.
HRH was from serious money.
HRH never even had a chance to be human.
HRH came screaming into this world and the money made him into a supranatural creature.
"Doin' the Dookie" was a lot like Fairy Land.
It was both there and not there, invisible to 99.9999 per cent of the world's population.
But Fairy Land hadn't gone invisible by being lost in an aquarium.
Fairy Land had become invisible when the women of Celia's realm used magic to align the island with an unconquered principle of everyday deception.
The principle worked like this: the physical appearance of any given object, be it animal or mineral, arrived with a series of common expectations.
As long as the appearance of that object was maintained, the vast majority of human beings would never notice any deviation from common expectations, and, in fact, people would go out of their way to ignore those deviations.
The most obvious place where this principle operated was within the publishing industry of the United States of America.
Despite decades of effort, and thousands of Internet thinkpieces about the inclusion of marginalized voices, publishing was a dirty business that had done nothing to alleviate a system of ghettoizing its authors based on their physical appearances and socio-economic points of origins.
The books of the publishing industry rested on a cheap shorthand, with each of its marketing demographics defined by the implicit prejudices of the American upper middle class.
And if you think that's an exaggeration, ask yourself this: how many well-received books of Literary Fiction published over the last thirty years do you remember being written by a poor person?
In the unlikely event that a person was allowed to publish a book which spoke beyond the simple facts of their socio-economic origins, then the message of that book was ignored.
Consider _The Women of Brewster Place_ by Gloria Naylor, a novel about several African-American women who all live in the same urban development.
The text explicitly states that the titular Brewster Place, the urban development itself, is a machine that manufactures the lives of its women.
The book is an exploration of the way by which the machine crafts, structures, and demolishes its product.
It's a dark, mechanistic text about the nature of urban living, about the secret lines of power, and about the way that Twentieth-Century AD architecture created new perversions and desires.
Remember when J.G. Ballard, a white English colonial, wrote the exact same shit?
You thought it was genius!
You gave him his own adjective!
When Naylor wrote the same thing, no one even noticed.
But _The Women of Brewster Place_ was authored by someone whose points of origin fulfilled the paltry expectations of America's upper middle class, a group of people who wanted little more from Black women writers than triumph over individual adversity, folksy homespun wisdom, sexual suffering, and horrible deaths.
And before anyone suggests that this is revisionist thinking about Naylor, cramming some weird bullshit into her work, go and read _1996_.
Read _1996_ and then try convincing yourself that Naylor wasn't a writer obsessed with the world of secret persuaders.
And, hey!
Speaking of publishing, let's talk turkey!
It's inevitable that this book will draw comparisons to writings by the late Kurt Vonnegut, who was an American novelist from the Twentieth Century AD.
I couldn't escape the comparisons with _I Hate the Internet_!
At least one question from the audience at every book event!
And I won't escape them with this book!
Total theft from _Breakfast of Champions_!
Even down to Fairy Land!
Most of the comparisons between this book and the writings of the late Kurt Vonnegut will occur in cheap little reviews on Goodreads.com and Amazon.com, which are Internet websites owned by a guy named Jeff Bezos.
These websites are where the American readership makes sure that American authors know their fucking place, and further ensures American authors know that their place is the equivalent to that of a moon-faced kid being shoved into some mud by a bully.
"How do you like that mud, you little shit?" asks the American readership. "This is what happens when you try to do anything! Fucking eat it, you pig!"
"Mgjhasdhashfs fdasmmmppfkjjsad," reply American authors, their pie-holes crushed into a mélange of star-rankings, facile two-sentence comparisons, and moronic assumptions about authorial motivation.
Quick!
Here's how to murder a culture: create a system in which every fucking thing, no matter how small or tedious, is smothered in bullshit instant commentary and hot takes by the stupidest people on the planet.
Good luck.
You're gonna need it.
But who the fuck are the dinosaurs reviewing books on websites?
Losers!
Who reads books?
Nobody!
Who uses a website?
Nobody!
It's all smartphones now.
Here's a text message that a well-known Hollywood screenwriter sent me, unbidden, on December 25th, 2017 AD, while I was trying to watch the 1981 AD film _Christiane F. – Wir Kinder vom Bahnhof Zoo_ :
That shit is disgusting.
But it's also brilliant, an entirely new kind of writing that's unfathomable in its complexity and immediacy.
The screenwriter didn't write it.
It comes from nowhere. It's a chaintext that people were sending each other in the days before Christmas. This was how the world talked to itself.
And by any measurable standard, it's much more interesting than reading a book.
Despite the notions thrown about whenever a prestigious novelist gives birth to another tedious narrative bound in paper, the actual function of novels in American society was very different than anyone liked to admit.
Yes, reader, you could shit in some high cotton and talk to your friends about how reading ennobled the human spirit, and how literature connected people to one another, and how the whole enterprise promoted a humanistic understanding of Life in Our Time.
But then, of course, you would be no different from the Xanax-addled Brooklynites who earn small amounts of money by writing crap articles critiquing the implicit racial and gender politics of television dramas about werewolves and vampires.
And, reader, you are many things.
Some good, some bad.
But you're better than the children who pretend, for money, that they're upset about the latest episode of _Supernatural._
You're not that kind of liar.
I can think of one reason why I can't escape comparisons to Kurt Vonnegut.
And it ain't because my work is so indebted to his own.
It's because Vonnegut was the same as me: another con artist ripping off the French writer Louis-Ferdinand Céline.
A bunch of people have talked shit about Céline.
I don't blame them!
Besides being one of the best writers of the Twentieth Century AD, he was also a rabid anti-Semite who collaborated with the Nazis.
But I can't judge!
I too have collaborated with Nazis!
I was published by Penguin Random House!
But the real reason why I can't escape the Vonnegut comparisons is not because our books are rip-offs of the same anti-Semite, but rather that the entire conception of the Serious Novel is a hideous stew of baked-in prejudices.
These prejudices are so omnipresent that they're invisible.
Whenever someone writes a work of incandescent prose about privileged people whose artistic, cultural, and familial foibles result in a plot-and-character-driven catharsis, no one goes on Goodreads.com and accuses them of ripping off Henry James.
But they should!
All of that crap, all of the good writing, the well-structured paragraphs, the emphasis on plot, the unexpected quirks of prose, the pretend lives of pretend people which resolve into a reflection of Our Time and Our Selves!
It's all technique!
Henry James was doing that shit before your parents were fertilized zygotes!
It's older than old hat.
Ancient technology!
And that's how we've defined the Serious Novel.
By pretending that technique from the Nineteenth Century AD can encompass the horror of the Twenty-First Century AD.
And because of that definition, most Serious Novels are so fucking boring that they have zero hope of competing with smartphones. Imagine a very cranky human being who, while riding public transit, gets upset when they witness other people using smartphones.
"No one reads anymore," laments the very cranky human being. "Look at all these kids using smartphones!"
And you nod your head in agreement, don't you, reader?
You think it's ever such a shame that the public is no longer willing to engage with long tedious narratives bound in paper. How terrible you find it that smartphones have killed literacy!
You agree with that crank!
But the problem isn't the smartphone!
It isn't the people using their smartphones!
It's that books got defined down!
There's one working standard for judging quality!
_Is this tedious narrative bound in paper less boring than watching peoples' slack faces as they ride a crosstown bus?_
I don't blame anyone for using a smartphone to alleviate boredom while riding public transit. I know that pictographic messages about sexual encounters with Santa Claus are slightly less boring than reading novels about Life in Our Time.
So, no, reader, I'm not like that crank.
I don't blame anyone for getting addicted to their smartphones.
I only blame people for their terrible attempts at reviewing my work.
Vonnegut, Vonnegut, Vonnegut!
He invented the short sentence!
He invented the short paragraph!
He invented jokes!
## Chapter Four
## **Child, Be Strange**
Before going to Los Angeles, Celia had left Fairy Land on one previous occasion.
This was when she went to the city of London on the island of Great Britain.
She traveled in the Year of the Sulky Octopus, which roughly corresponded to 1608 AD, 1017 AH, and 5369 AM.
Celia had arrived in the middle of the Little Ice Age, which was a long period of freezing winters and terrible cold.
Celia went to London a few days after Christmas, which was a holiday that celebrated the birth of an itinerant preacher from Galilee who'd promulgated an ideology of love, non-violence, and forgiveness.
Somehow this ideology of love and forgiveness, which was called Christianity, had been transformed into a religion responsible for tens of millions of deaths.
History is so fucking weird.
More Vonnegut!
He invented Jesus!
Prior to Celia's first departure, Fern had returned with news from a peregrination abroad: _Tom a Lincoln_ , the book by Richard Johnson, had been adapted into a play.
"A play?" asked Celia. "Whatever is a play?"
"Some people are chosen to embody roles around a theme. The chosen people speak words as if they themselves were their embodied roles."
"You say that they have made a play of my life?" asked Celia.
"Yes," said Fern.
"Someone will speak as me?" asked Celia.
"Yes," said Fern.
"I must attend," said Celia.
Fern could not go to England with her mother.
She'd been away from Fairy Land for about a year.
Whenever Fern returned from a vacation, she'd stay on Fairy Land for at least two years, which was long enough to chase away even the slightest hint of the island's collective depression.
One of Fairy Land's more aggressive women was drafted into service as Celia's escort.
Her name was Rose Byrne.
When the women of Fairy Land had banished or murdered all of the island's men, Rose had been one of the more violent and vocal agitators.
Rose had argued against banishment. She wanted to kill all the men.
She hadn't killed all the men, but she had murdered more men than anyone else on the island.
She'd cut off their heads.
She'd hung them from gibbets.
She'd boiled them in oil.
She'd drowned them in ale.
She'd crushed them with rocks.
She'd buried them in sand, covered their heads with honey, and let their skulls be picked clean by ants.
About two centuries before Fern first left the island, Rose began taking her own trips away from Fairy Land.
Rose's trips abroad were very short affairs.
She only left long enough to sail a skiff to a distant land, get blotto stinking drunk, and then brutalize unsuspecting men in dirty taverns.
But the violence tourism had taught Rose how to travel, which made her useful as Celia's companion.
Celia was the Regnant Queen.
She wasn't traveling by boat.
She did some faery bullshit and opened a magic window to London.
The magic window opened in Southwark, on the south side of the river, between the bear-baiting ring and St. Saviour's church.
A bunch of awful London people stood around, gaping at Celia and Rose Byrne.
The awful London people had seen a lot of things in their miserable London lives, but they'd never witnessed the spontaneous materialization of a fairy queen and her disagreeable companion.
One of the awful Londoners was a drunken scoundrel.
He only had one eye.
The scoundrel began dancing like a chicken, in the hopes that Celia or Rose would give him coin for alcohol.
"Let us anon, lady," said Rose. "Before I rip this one's arms from his shoulders and beat him about the head with his own appendages."
"Come on, missus," said the scoundrel. "Come on, I'm a righteous chicken and I'm a-dancing for you!"
Before Fern left England, she'd put a faery glamor on the location where the play of Celia's life would be performed, which was the Hall at Gray's Inn.
Gray's Inn was one of the four Inns of Court, which were places where upper-class families sent their sons to train as barristers.
A barrister was a fancy lawyer.
The inmates of Gray's Inn were learning to exploit England's ad hoc legal system.
This training helped the inmates' families abuse the poor and retain an iron hold over the country's unjust social structure.
It was good work if you could get it.
Which you couldn't.
Because you were poor.
Celia cast a spell.
The spell created a long thin tendril of magical light, like a ropey strand of saliva, that led from the faery glamor on Gray's Inn to Celia's location in Southwark. The tendril snaked through the streets of London, creating the most effective route to Gray's Inn.
It was a little like getting directions from a smartphone, but without supplying every stupid fucking detail of your sad little life to the sociopaths who operate megalithic American corporations.
Celia and Rose left the Londoners and followed the tendril.
"Come on, missus," cried the one-eyed scoundrel after Celia and Rose. "Come on, don't you want to pluck me old feathers? Don't you want to tug on the old beak? I've got some nice meat on me old chicken bones!"
The tendril led Celia and Rose over London Bridge.
There were human heads on spikes attached to the bridge's southern gate.
Celia and Rose passed through the gate, taking no notice of the human heads, which were in various stages of decomposition. It was nighttime, so the heads weren't very visible, and, anyway, a bunch of men's heads on spikes was nothing new to the women of Fairy Land.
London Bridge was lined with buildings and shops on either side, and the passage was narrow, and as Celia and Rose followed the tendril, they often found themselves in darkness illuminated only by the tendril's light.
The tendril brought Celia and Rose into Holborn, which was mostly countryside in the greeny northwest of the city.
The tendril brought Celia and Rose through the Holborn gate of Gray's Inn.
There was a crowd of people, all headed in the same direction as Celia and Rose.
"What a great number have come to see this play of my life," said Celia.
"Why would they not?" asked Rose. "What else would the dogs do? Bark at sparrows, chase cats up trees, and, by the smell of them, shit themselves every other Tuesday."
Just past the gate, there was a little bookshop under the sign of a white bear. It was tended by a man named Henry Thomes.
Henry Thomes stood in front of his shop, crying out at passersby.
"Books, books, books," he shouted. "Books of the Red-Rose Knight. Parts one and two. Books of the Red-Rose Knight. Read about the Red-Rose Knight in _Tom a Lincoln_!"
Celia stopped.
"The book has two parts?" she asked.
"The writer published the second but last year."
"I will have this second part," said Celia.
"For you, the cost is but four pence."
"Pence?" asked Celia.
"Pennies," said Henry Thomes.
"The swine asks for money," said Rose. "We have spoken of money, lady. Do you remember?"
"Money," said Celia. "I have no money."
"No money, no book," said Thomes.
"Would you take some ham?" asked Celia. "I believe Rose is carrying cured ham on her person. We could share it with you."
"What am I to do with your old hog?" asked Thomes. "What I need is coin."
Celia and Rose followed the crowd into the Hall at Gray's Inn. They entered into a temporary autonomous zone called the Kingdom of Purpoole.
Almost every Christmas season, the young men of bleeding privilege who studied at Gray's Inn would throw a huge party, creating a pseudo-monarchy of Purpoole, in which one of their number would be made Prince.
The Prince would rule for the season with his own courts, ministers, and government.
He was expected to put on masks, revels, plays, and dances. The current Kingdom had been established on the 12th of December.
A pupil named Thomas Rudde, of Higham Ferrers near Northampton, was made the Kingdom's prince.
As the two women entered the Hall, the subjects of the Kingdom of Purpoole were escorting guests to their seats.
Prince Thomas was watching over his court.
Prince Thomas was drunk as a skunk. He'd been drunk for sixteen days.
He saw Celia.
Some of the tendril's magic light had rubbed off on Celia. She glowed with the power of Fairy Land.
"How now," Prince Thomas cried from his throne. "Who is this that comes amongst us? See how her face and bosom glow with light of the waxing crescent! Why, I shall avail myself of her company."
The Prince leapt from his throne and took Celia's arm in his own.
Prince Thomas was too drunk to notice that Rose Byrne had taken out her sword and was about to murder him.
Celia raised her hand, staying Rose's assault.
"Sweetest creature," said Prince Thomas. "Who art thou with thy fiery raiment?"
"I am Celia, Regnant Queen of Fairy Land."
Prince Thomas laughed and laughed and laughed.
"What a jape!" he cried. "Which man of Gray's Inn has architected such a jest?"
"Why are you laughing at my lady?" asked Rose.
"Many jibes arise throughout a Christmas Revel, but I know not of any previous happenstance when a character from imagination has come to life and presented herself at our court."
"My lady is no product of imagination," said Rose. "She is the Regnant Queen of Fairy Land. She has come to see the play."
"Tell me," said Celia. "What is your name?"
"I am Prince Thomas of this, the Kingdom of Purpoole."
"I thought us in the Kingdom of England," said Celia.
"In these walls, I am the true prince. All that happens within is for my benefit and by my leave."
"As we are both monarchs," said Celia, "shall we not watch the play together?"
"Excellent," said Prince Thomas. "I have no throne for a queen, but my minions will find you some grand chair upon which to rest your bones and flesh."
"Who whispered to you that my flesh wanted rest?"
Prince Thomas roared with laughter.
The play was presented by the Queen Anne's Men.
The audience sat around three sides of the great hall. The fourth side was kept behind a curtain, which was used for scene changes during the play.
Celia watched.
Celia watched.
Celia watched.
At first the play was pretty fucking boring, some old shit about whether or not King Arthur could fuck the Red-Rose Knight's mother. Then the living incarnation of Time came out and showed a bunch of other shit that happened, none of which was that interesting, and then an abbess put King Arthur's bastard son, who was a baby called Tom a Lincoln, into the hands of a shepherd. Then Time came out again and Tom a Lincoln was much older and he and his fellow shepherds took up weapons and abandoned their sheep. Tom's friends crowned him with a laurel of roses, thereby making him the Red-Rose Knight, and then all of the former shepherds camped out on a heath and robbed people, and then they ended up dragged to the court of King Arthur.
King Arthur and the Red-Rose Knight fought each other until their sublimated incestuous homoeroticism convinced King Arthur to accept the Red-Rose Knight as his son, and then the Red-Rose Knight and King Arthur kicked the shit out of the French, and then the Red-Rose Knight took some of Arthur's men on boats and they went sailing around the world. Time came back on stage and said some shit. And then finally, the Red-Rose Knight and his men turned up on Fairy Land.
And Celia was there, watching herself, watching a man dressed up as Celia, watching as the man dressed up as Celia spoke words that Celia had never said and acted out deeds that Celia had never done.
The sexual morality of Fairy Land wasn't prudish, but it was an out-of-body experience to watch a fictional iteration of yourself bed down with a makeshift knight.
In its many lies, Richard Johnson's _Tom a Lincoln_ had contained no mention of Rusticano.
But in the play at Gray's Inn, Rusticano was about 30 per cent of the action.
A musical intermission occurred after the Red-Rose Knight left Fairy Land. There was a great amount of social mingling, with young rakes talking to women, and an outrageous amount of drinking.
"You are far more fair than the one who acts out your story," said Prince Thomas.
"I am not a man," said Celia. "Of course I am more fair."
"You would be surprised," said Prince Thomas. "Many of the boys who play as ladies are very comely, and it is said that most are paid catamites. I promise you, my queen, that the Celia of our drama shall find himself enveloped by one of Gray's brutes before the night is through."
"The lust of men can be overpowering. It was not the case with the true Red-Rose Knight. He mewled like a kitten."
"Some men, often those who are princes, are known to roar like lions."
"A sound that I am certain could shake my bones," said Celia.
Celia didn't pay attention to the rest of the play, which was claptrap about the Red-Rose Knight leaving Fairy Land and getting another girl pregnant and then Celia killing herself by jumping off a rock.
It wasn't much different from _Tom a Lincoln_.
After the applause died down, Prince Thomas turned to Celia and asked, "How then, my fair elf queen, did you like the play of your own life?"
"It was very strange," she said. "But was it a good play? We have no such entertainments in Fairy Land."
"It was passable," said Prince Thomas. "I have seen better, I have seen worse. But look at you, still your dusky skin is illuminated by the light of moon. My word, lady, what kind of woman are you?"
"I have told you," said Celia. "I am the Queen of Fairy Land."
"A queen of Clerkenwell, more like, a sister of Luce," said Prince Thomas. "What a jest! Dressed as a queen! Did they send you here to inquire of me, my girl, as you inquired of the Red-Rose Knight? Are you this prince's tribute? Is it my bed that next you target?"
"Where do you sleep?" asked Celia.
"I keep a chamber in the south court. Beyond this door and a small walk."
"Is it fit for a queen to consort with a prince?"
"Our two kingdoms, my queen, are not as of other kingdoms," said Prince Thomas. "So why should our congress be ruled by their practices? My lady, you arise in me the sacred demon of ungovernableness!"
Celia followed Prince Thomas to his chamber in the south court.
It was tepid British sex with the chinless scion of an upper-class family.
But it'd been almost a thousand years since Celia had fucked.
She took what she could get.
When Celia emerged from Prince Thomas's chambers, she found Rose Byrne standing outside of the building.
"My lady," said Rose Byrne. "Have you finished with your antics?"
"I believe so," said Celia.
"Let us anon. Fairy Land is waiting."
"A word," said Celia.
"Yes, my lady?" asked Rose Byrne.
"You saw the false Rusticano."
"Who could miss the spectacle?"
"When we return to Fairy Land," said Celia, "you are free to speak of the play in any fashion that you might wish. My one request is that you not inform anyone of the false Rusticano."
"I do as you command," said Rose Byrne.
"I would not have him know of the insult," said Celia. "For the peace of us all."
As they walked towards the Holborn gate, Rose handed Celia a small book.
"Part II," said Rose, "of _Tom a Lincoln_."
"How did you come by this?" asked Celia.
"I had some time while you were at your frolic," said Rose. "I convinced the little man that he wanted my cured ham."
"Was there any violence?"
"Only a bit," said Rose.
## Chapter Five
## **Wonder Women**
Then about four hundred years happened.
The industrial revolution poisoned the Earth's atmosphere, the United States of America was founded on the dual principles of genocide and human slavery, and soccer became very popular.
And Fern lost herself in Los Angeles.
Which meant that Celia had, once more, to leave Fairy Land. She took Rose Byrne with her.
Those four hundred years, by the way, were some of the most monumental in the planetary existence of homo sapiens.
Fern had warned Celia about the changes, back before her disappearance, and Celia had caught some glimpses on Fairy Land's woolen television.
If you asked people living in Los Angeles during the Year of the Froward Worm about the last four hundred years, they'd almost certainly talk about things like the Internet, smartphones, and air travel.
But the women of Fairy Land were immortal and undying beings, and they viewed the previous four hundred years in a very different light than the people living in Los Angeles.
The women of Fairy Land knew that most of the technological developments of the previous four hundred years were about as impressive as an old dog learning a new trick, only to discover that the dog's new trick was something useless like shelling pumpkin seeds, translating the _Apocalypse of the Pseudo-Methodius_ out of Syriac, or building a career in the American recording industry by performing parodies of popular songs.
Smartphones, the Internet, and air travel were only refinements of a principle that had governed human behavior from its very beginnings.
All the technology really did was create new ways for a person to be annoyed by the neighbors.
Fern and Celia knew where the real change had been.
They knew what the real difference was between Los Angeles in the Year of the Froward Worm and, say, the early medieval period or the Ancient Hellenic era.
Fern and Celia knew that the real change had come with the development of indoor plumbing and, specifically, the management of sewage.
Celia and Fern were more sensitive than usual to the problem of human waste and its effective management.
After all, they'd both watched the Red-Rose Knight be assassinated by Orson's shit.
The effective management of human sewage had been developed about one hundred years prior to the Year of the Froward Worm.
Homo sapiens had been on Earth for about two hundred thousand years, which means that it took the planet's dominant species roughly one hundred and ninety-nine thousand nine hundred years before someone realized that people shouldn't do a poo on the living-room floor.
So don't get your hopes up.
When Celia and Rose Byrne went to Los Angeles, they had difficulty in figuring out where they should arrive.
It wasn't like London in the Seventeenth Century AD.
Fern hadn't left any magical beacons hanging around to guide her mother through the landscape.
Los Angeles County was four thousand square miles.
When Celia cast her spell that opened a magic window onto Los Angeles, she had to do a little faery fudging, asking that the window open on the place which would be the most hospitable to their arrival.
She didn't specify the exact nature of this hospitality.
The magical window opened in the lobby of the Vista Theater, which was a movie house in the neighborhood of Los Feliz.
The Vista, which was a giant single-screen theater, had been built in 1923 AD.
The exterior façade of the building was Spanish Colonial Revival, but its interior décor was early Twentieth-Century AD Egyptian kitsch, which meant that the theater was filled with Pharaonic heads and hieroglyphics.
Celia and Rose Byrne arrived on the evening of Thursday, June 2nd, 2017 AD.
This evening hosted the Vista's first screening of _Wonder Woman_ , the huge media spectacle in which a lesbian named Diana left her magical island with the intention of beating the shit out of some Germans.
For decades, the Vista had been managed by a man named Victor Martinez.
A curious feature of Victor's tenure was his delight in dressing up as characters from the films that showed at the Vista.
Victor's appearances in these outfits were always more enjoyable than the films themselves.
When the Vista had shown _Iron Man_ , which was about a war profiteer who learned that war profiteering could be more profitable if the war profiteer built a suit of armor and personally killed Muslims with his own mechanical hands, Victor Martinez wore a version of the war profiteer's suit of armor.
When the Vista had shown _The Hobbit: An Unexpected Journey_ , which was about a fussy midget drawn into an unlikely adventure by a slightly pompous wizard, Victor Martinez dressed as the wizard.
When the Vista had shown _Pirates of the Caribbean 3_ , which was a film about a pirate rapist with a charming accent, Victor Martinez dressed as the pirate rapist.
Because _Wonder Woman_ was about a female character, Victor Martinez did not dress as the film's lead role on June 2nd, 2017 AD.
Instead, he dressed as the film's male sidekick, an indistinct American soldier during World War One.
Another of the Vista's employees, who was a woman, dressed as the lead character of _Wonder Woman_.
They stood outside the theater, greeting attendees.
Victor Martinez and his fellow employee were not the only people dressed in costumes on June 2nd, 2017 AD.
A curious feature of early Twenty-First-Century AD life was that fans of media spectacles liked to dress up as characters which appeared within those media spectacles.
In the case of the Vista's premiere screening of _Wonder Woman_ , this was really weird.
No one had seen the film!
It could have been a total piece of shit!
Unlike Victor and his fellow employee, who had a vested economic interest in the film's success, the people who dressed in costume at _Wonder Woman_ had no stake in the property.
_Wonder Woman_ had arrived at the Vista anointed in a sold-out madness emblematic of the United States of America in the Twenty-First Century.
This madness was long-brewing and the result of multiple historical occurrences and tendencies.
Some of these historical occurrences and tendencies had been running for decades.
Some had been running for centuries.
The culmination of these historical occurrences and tendencies was the recent election of Donald J. Trump to the Presidency of the United States of America.
The Presidency was the highest office in the country, to which individuals were elected every four years through an arcane process that had been designed, originally, to make sure the United States was cool with enslaving people from Africa.
Enslaving people from Africa was great business, and it was the economic bedrock of the fledgling nation, and it involved owning human beings who would be forced into labor and receive no benefits from that labor.
About seventy years after its founding, the country held a big debate as to whether or not it was cool to enslave people from Africa.
After this debate had killed about 716,000 poor White people fighting for the economic masters, and 36,000 Black people fighting for their freedom, everyone decided that enslaving people from Africa probably wasn't too cool.
Because it was no longer too cool to enslave people from Africa, which was the country's explicit purpose, the United States entered a malaise.
It had lost its demon.
The purpose of the Presidency shifted.
If its original function no longer existed, then surely some new purpose could be found.
It turned out that the Presidency was really good at making war.
After all, it had overseen about seventy years of war on Africa.
So new wars were made.
Decades and decades and decades of war.
By the time that Donald J. Trump was elected to the Presidency, the elections which chose the President had transformed from referendums about who would best administer the international slave trade into contests about who'd get the chance to reduce illiterate Muslims into pulpy masses of intestines.
Even by the dubious standards of candidates for the United States Presidency, Donald J. Trump was a wretched specimen.
He was the most famous person who had ever lived.
He was the most famous person who would ever live.
He was orange, he wore a stupid wig, and he was a pawn of multinational corporations.
He was hella racist.
By any honest account, he was into sexually assaulting women.
It was rumored that he was a speed freak, which would explain the difference between his public appearances as President and his public appearances in the 1980s AD and 1990s AD, when he'd been a fixture of New York City's tabloid culture.
In the early days, the President had been, if not especially bright, then at the very least coherent.
By the time that he won the right to turn Muslims into shattered masses of agony, the President could barely speak.
Amphetamine abuse has a terrible effect on the brain.
For decades, the political liberals of the Celebrity branch of American governance had profited off Donald J. Trump's crass public persona.
They'd given him deals for books that he hadn't written and stuck him on television whenever they thought it'd turn a buck.
Trump, who pretended on television that he was a billionaire, was big entertainment dollars.
His media persona was this: he was a total fucking jerk!
And he was rich!
He was great entertainment in a country that fostered a delusion in its poor that they too, someday, would be rich enough to treat other poor people like shit.
Donald J. Trump ran for the Presidency, and won, by embracing political viewpoints in direct opposition to the very people who had created him.
The liberals in the Celebrity branch of American governance had made a beast which they could not control.
It was like Mary Shelley's _Frankenstein_ , _or, The Modern Prometheus_ , a novel about a scientist who creates a monster out of spare human body parts that he's dug up from graves. The monster gets angry. Things go badly.
There were some differences.
The monster in _Frankenstein_ , made of rotten human remains, had a body that was slightly less disturbing than the body of the President, which was made of media coverage stitched together with plastic surgery.
The monster in _Frankenstein_ didn't have a speed habit.
And the monster in _Frankenstein_ had a more honest relationship to literacy.
The monster in _Frankenstein_ was into reading Milton, Plutarch, and Goethe.
By contrast, the monster who was the President just put his name on books that other people had written and then took money from political liberals in the publishing industry.
_What's the harm?_ asked the publishing industry.
_It's all just business_ , said the publishing industry.
199,900 years of shitting in the living room.
Anyway, the election of Donald J. Trump made America go nuts.
To be fair, the country had always been pretty crazy.
War, genocide, and slavery aren't good for anyone's mental health.
But after Trump assumed the Presidency, the madness got worse.
The people who'd voted for Trump went nuts because they'd won and had no idea what to do with their impossible victory.
The country's political liberals went nuts because Trump put them in the position of facing an undeniable and yet unpalatable truth.
This was the truth that the political liberals could not deny and could not face: beyond making English Comp courses at community colleges very annoying, forty years of rhetorical progress had achieved little, and it turned out that feeling good about gay marriage did not alleviate the taint of being warmongers whose taxes had killed more Muslims than the Black Death.
You can't make evil disappear by being a reasonably nice person who mouths platitudes at dinner parties. Social media confessions do not alleviate suffering. You can't talk the world into being a decent place while sacrificing nothing.
The socialists didn't go nuts.
They were the people who'd thought about the complex problems facing the nation and decided that an honest solution to these problems could be achieved with applied Leftism.
But don't get your hopes up.
Despite being correct in their thinking, the socialists were the most annoying people in America. When they spoke, it was like bamboo slivers shoved under a fingernail.
I don't know why.
It was the single biggest American tragedy of the last one hundred years.
By the Year of the Froward Worm, too much warmongering had splintered the national psyche into a series of tribes.
The most obvious schism was between the public voices of the liberal warmongers and the public voices of the tribe that had helped Donald J. Trump win his impossible victory.
For the sake of clarity, let's call this second tribe the Fucking Assholes.
The noise from the public voices of the liberal warmongers had become the dominant voice of the haute bourgeoisie. This contingent was represented by a mixture of high-grade celebrities, op-ed writers, Democratic party apparatchiks, and the mentally ill. A great number of these public voices had passed the Cash Horizon.
For varied reasons, the public voices of the liberal warmongers had devised an idea that was extraordinarily profitable for the arch-capitalist class: that the Celebrity branch of American governance, and its products, could be read as a proxy for the struggles and strife of the great American unwashed.
The public voices of the Fucking Assholes were represented by a mixture of low-rent celebrities, op-ed writers, Republican party apparatchiks, and the mentally ill. A great number of people in these public voices had passed the Cash Horizon.
The public voices of the Fucking Assholes agreed with the public voices of the liberal warmongers: the Celebrity branch of American governance, and its products, could be read as a proxy for the struggles and strife of the great American unwashed.
The only difference of opinion was about the interpretation of this proxy.
Both sides accepted the unchallenged underlying thesis.
The argument proved to be very profitable for the arch-capitalist class who actually owned the Celebrity branch of American governance.
Everything was an advertisement.
And if you're wondering about the opinions of the non-public voices, then go and fuck off back to the Dark Ages.
You're revealing a thinking that's very Twentieth Century AD, with atavistic tendencies towards logic and dreams of a populace that hasn't been preyed upon by the mind-altering substances of the pharmaceutical industry. That shit is ancient news.
You either agreed with the country's priestly castes, and their apparatuses of sycophants, novitiate aspirants and true believers, or you found yourself on the receiving end of a barrage of hatred and death threats.
Here was the difference between the priestly castes, many of whom had opinions on deadline for money, and everyone else: sane people shut the fuck up, nodded their heads, and did what they needed to survive in a toxic political landscape.
In an era when public discourse was the bought-and-paid property of roughly twenty companies, and the airing of an opinion could subject a person to unfathomable amounts of abuse and recrimination, the only reasonable option was to be quiet.
So when you next fawn over someone's brave public thoughts, repeat the following: _The contours of discourse are so horrendous that one thing has become certain. Any individual offering up a public opinion necessarily must be either hopelessly stupid or insane. I am engaging with a product of madness and idiocy._
Regarding the public opinions offered up in this book, they are the products of both idiocy and bad craziness.
But at least I have some justification for engaging with the stupidity and insanity of this book.
I wrote the thing.
Reader, what's your excuse?
Here was one thing that all the priestly castes agreed upon in the run-up to the election in the Year of the Misplaced Butter: Donald J. Trump could not, should not, and would not be President.
It was impossible.
But Donald J. Trump won anyway.
A creature created by the Celebrity branch of American governance had taken over the Executive branch, the conflation of entertainment into political life was complete, and it had happened without the blessing of the high clergy, and it shut out the vast majority of people who were from the Celebrity branch of American governance.
By the way, all of this is why one's political tools should probably be comprised of effective organization, decent arguments, an understanding of the actual political landscape, as opposed to an imaginary map built as a reflection of one's own virtue.
If the only tool in your political arsenal is shame, don't be surprised what happens when you meet a shameless man.
Enter _Wonder Woman_ in 2017 AD.
There'd been about fifteen years of films about superheroes, which were intellectual properties about supranatural beings like Celia.
These films were all the same: a supranatural being reenacted American foreign policy by responding to an existential threat through exaggerated violence, generally after another supranatural being reenacted 9/11, which was when some Muslims blew up two ugly buildings in New York and facefucked reality into a cartoon.
What differentiated _Wonder Woman_ from the rest of the super-hero films was that its lead character was female.
Because the country was run by a monster created by liberals in the Celebrity branch of American governance, and because liberals were totally disconnected from the political structure of their country, and because the film mapped to easy marketing demographics, _Wonder Woman_ was freighted with a swollen ideology.
It arrived as a place where the unexamined ideologies of American life could belong to women as easily as men.
If you think this is an exaggeration, please read the following quotes from "Want to Take Political Action This Weekend? Go to the Movies", an article written by Melissa Goodman for the website of the Southern California branch of the American Civil Liberties Union:
Political action doesn't always have to take the form of marching, holding a house party or calling your local representative. You can make a bold and necessary political statement just by buying a movie ticket.
Go see _Wonder Woman_...*
That was politics at the mid-point of 2017 AD.
It arrived in an article on the website of an organization dedicated to civil liberties which suggested that an alternative to applied Leftist action was to patronize media produced by a massive multinational corporation owned by the same old shits who'd been ruining the world for centuries.
This was the madness of the moment.
People had lost the ability to tell the difference between the Celebrity and the other three branches of American governance.
Because the world has gone stupid and elected a rogue member of the Celebrity branch of American governance into the Executive, allow me to point out the difference: representation in the traditional three branches of government really does matter, because the people who end up in the government are the people who make policy and laws.
In other words, these are the people who determine whether or not you will be able to make a living wage.
These are the people who shape your lives.
People who end up in the Celebrity branch of American governance are the people who make movies and television and huge profits for the same old shits who rule the world.
In other words, these are the people who are taking your money.
I know of what I speak.
I'm one of them.
I've duped you into buying my turgid work.
Unless you've pirated this book.
If you have, then good for you!
Do me a favor. Steal _The Future Won't Be Long_!
And, yes, reader, I know the arguments about why it's important to see diverse faces in television and in films.
And, yes, I realize that no one agrees with me on this topic.
But I'm sorry, arguing about the shadow theater of the entertainment industry is not politics.
What did everyone at the Vista Theater see when they made a bold political statement by giving money to the people who'd ruined the world?
_Wonder Woman_ was a film made by people baptized in the primordial ooze of unconscious American life.
The attendees saw a story about the unexamined glory of American foreign policy, of the meaningfulness of war and violence, and a story about how a woman could be like a man in her ability to simulate genocide.
A woman named Diana lives on an island full of lesbians. Her mother is the Queen of the island. Everyone lives in paradise, doing what everyone who's ever met a lesbian knows that all the world's lesbians do, which is train for perpetual war. This goes on for millennia until one day an American in an airplane crashes on the island. Diana rescues the American, only to find that the reason he crashed is because a bunch of Germans were firing materiel at the plane. The Germans invade the lesbian paradise. The lesbians murder all the Germans. The Germans murder some of the lesbians. The American gets naked and feels insecure about the size of his penis on an island full of lesbians and then confesses that he's working as a spy against the Germans, who have developed biological weaponry. Some nonsense happens where Diana gets convinced that Ares, the Greek god of war, is responsible for the chaos. Diana and the American go out into the world with the intention of murdering a bunch of Germans and stopping Ares from developing biological weapons. Then Diana goes to London where, as Celia once discovered, English shit is widely acknowledged as Europe's most toxic. Then she goes to France with a motley crew of drunkards, and for some reason only the dark-skinned drunkards are capable of belief in the supra-natural. Then Diana kicks the shit out of some Germans for about forty minutes, performing ritualistic genocide that saves the fictional world while adhering to an unspoken embrace of American foreign policy. Somewhere in here, weirder members of the audience cheer and cry because they've imbibed enough primordial ooze that they believe the appropriate solution to the horror of men is to adopt the tactics of men. In other words, the committing of genocide has become so ingrained and unexamined in the American psyche that there is no longer any purpose in questioning whether or not one should commit genocide. The real question is who gets to kill. And for some reason it's important that women have opportunities to butcher their fellow living beings. Just ask the ACLU. Then Diana kills Ares, who turns out to be an Englishman in a bowler hat, which is probably the only realistic thing in the entire film, and then the war ends and everyone is happy because Diana has committed genocide against the right people at the right time and there's no way that the roman numeral at the end of World War One could possibly predicate a sequel.
At least the genocide simulator of _Wonder Woman_ gave some people at the Vista an opportunity to dress in goofy costumes.
And it was those costumes that brought Celia and Rose Byrne to the premiere.
The magic bullshit window had chosen well.
Celia and Rose Byrne were clothed in Fairy Land's haute couture, which over the last season had moved into animal pelts.
Had they arrived anywhere else in Los Angeles, their outfits would have drawn a lot of attention.
At _Wonder Woman_ , they were just making a political statement.
They arrived through the magic bullshit faery window, popping dead center into the lobby of the Vista, right in front of the concessions counter.
They saw a lot of people going into the twin double doors of the theater.
They both remembered _Tom a Lincoln_ at Gray's Inn.
They knew what it looked like when people went to a show.
Celia and Rose followed the crowd inside.
They found two seats to the back right of the theater.
They watched the movie.
* <https://www.aclusocal.org/en/news/want-take-political-action-weekend-go-movies>
## Chapter Six
## **Willkommen im Dschungel**
Around the time when I started writing Chapter Twelve of this book, right between two paragraphs in which I insult George R.R. Martin and _Game of Thrones_ , I underwent an unexpected religious experience.
To make sense of this: at the beginning of June 2017 AD, I decided that I should go see the band Guns N' Roses perform live at the Staples Center.
What can you say about Guns N' Roses?
Back in the 1980s AD, they were total Hollywood scumbags, the dregs of the dregs, homeless trash who became the most famous people in the world.
It's the greatest faery story ever told.
The band carried on for about five years before flaming out.
Lead vocalist Axl Rose was left in control of the name, but all of the other original members quit or were fired. A period of twelve years followed.
This period included the album _Chinese Democracy_ , mocked because it took forever to be released, but which is actually pretty good.
Anyway, they were a great band, and their iconography haunted my childhood and is about 70 per cent of the reason why I live in Los Angeles.
In 2016 AD, three of the original members reformed the band and ventured out on a reunion tour.
I saw their August 19th, 2016 AD show at Dodger Stadium.
Because 2016 AD was a year in which I had made a significant, but not substantial, amount of money, I bought a General Admission ticket to the pit.
It cost about $280.
I was way in front.
I was next to the stage.
The whole thing was filmed by a professional camera crew.
If there's ever a live DVD, you'll see me. I'm the guy with no hair looking very uncomfortable as he stands next to a group of models who are younger than the songs being performed.
When a second American leg of the tour was announced for 2017 AD, with the Los Angeles dates in late November, I decided that I should buy another ticket.
Because 2017 AD was a year in which I earned an even more significant, bordering on substantial, amount of money, I bought a General Admission ticket to the pit.
It cost about $550.
Which is manifestly insane.
But I have a lot of disposable income.
This is because I don't spend any money.
In the twenty-two months following the release of my novel _I Hate the Internet_ , I made just under $200,000, net, pre-tax, pre-agents' commissions, and the only things I bought were a cemetery plot and two tickets to see Guns N' Roses.
On June 30th, 2017 AD, I purchased a General Admission pit ticket to see the Guns N' Roses show at the Staples Center.
The show was scheduled to occur on November 24th, 2017 AD.
Because I like useless ephemera, I paid an extra $5 to have a physical ticket.
The ticket arrived about a week later.
It was sent via postal mail.
Then, in August of 2017 AD, right around when my novel _The Future Won't Be Long_ was published by Penguin Random House, ensuring that I made significantly less money in 2018 AD than I did in 2017 AD, I found a surprise in my postal mail.
I'd been sent a second ticket.
I compared the two tickets.
Except for the barcodes, they were identical.
Barcodes are bits of black ink and numbers printed on every ticket. Whenever you try to enter an event, someone's there with a device that scans the barcode and ascertains the ticket's validity.
The tickets had different barcodes.
There were two options: (1) believe that the second ticket supplanted the first or (2) believe that both tickets were valid.
I opted for a soft belief in the second option.
I now had two tickets to see Guns N' Roses at the Staples Center.
Which meant that I had to find someone to come with me.
I called Arafat Kazi.
Arafat Kazi is my best friend.
He used to be the fattest man in Bangladesh.
Now he's an American citizen and had recent gastric bypass surgery. Hundreds of pounds of fat have melted off his body, but their absence has draped him in a suit of empty skin.
He's also a drummer.
We met in 2001 AD, when he was an undergrad at Boston University.
One of the very first things that we talked about was his taste in music, which in those days was almost entirely Heavy Metal.
He was into Iron Maiden and Judas Priest.
The worst bands of all time!
One of our few overlaps in taste was Guns N' Roses.
We built a friendship talking about the band.
"Dude," I said into the telephone. "I have this extra ticket to see Guns N' Roses that was mailed to me by mistake. You've got to come to Los Angeles for Thanksgiving."
"Okay, dude," he said. "I'll do it. Can you pay my plane fare?"
Fast forward to November 23rd, 2017 AD.
Thanksgiving Day! Celebration of genocide with disgusting food!
Around 9PM, I picked Arafat up at the airport and brought him to my apartment.
I suggested that he sleep on the pull-out, but he insisted on taking the floor.
He passed out around midnight.
About twenty hours before Arafat's arrival, an astonishing thing happened: somehow _The Future Won't Be Long_ was shortlisted for the _Literary Review_ 's Bad Sex in Fiction Award.
The _Literary Review_ was a London magazine for, quote, People Who Devour Books, unquote.
The Bad Sex in Fiction Award was an award that, quote, honoured an author who has produced an outstandingly bad scene of sexual description in an otherwise good novel. The purpose of the prize is to draw attention to poorly written, perfunctory or redundant passages of sexual description in modern fiction, unquote.
The shortlisting of _The Future Won't Be Long_ generated more emails than any other thing that had happened in my life.
When I woke up that morning and examined my inbox, it was flooded.
Draw your own lesson, reader.
Here was mine: people remain unbelievably primitive.
The emails had a 50/50 split.
Half of the people felt bad for me and wanted to make sure that I was okay.
The other half understood the shortlisting for what it was: absolutely fucking awesome, even if it did produce a moral compromise.
The moral compromise emerged from the fact that I am a hopeless case.
I loathe human attempts at establishing status.
I object to the general idea of awards and literary awards in specific.
But.
The Bad Sex in Fiction Award?
For the first time in my life, there was something that I wanted to win.
I knew that I wouldn't.
The shortlisted passage wasn't a sex scene.
It was an absurd, pretentious character describing her reaction to sex in a manner that was absurd and pretentious.
There was no way that it fit the bill.
My novel does, in fact, contain an actual sex scene.
It's two pages long. It's disgusting. It's redundant. It's perfunctory. It's so pretentious that at the moment of climax, it mentions James Boswell, a writer from the Eighteenth Century AD.
So why wasn't it nominated?
Here's my theory: the actual sex scene in _The Future Won't Be Long_ is the description of a down-and-dirty homosexual encounter.
Major league assfucking!
And all of the passages shortlisted for the Bad Sex in Fiction Award were exclusively heterosexual.
Clearly, the pretentious passage for which I had been nominated was a stand-in for the pretentious passage which contained a description of actual sex.
I was weighing this in my mind while I waited for Arafat Kazi to get off his plane.
Everyone else in the airport terminal waited with anticipation for the arrival of their friends, lovers, and family.
And I was there too, and I was trying to decide if the liberal intelligentsia believed homosexuals are incapable of having bad sex.
With Arafat crashed out, I fell asleep around 3AM after beginning to write Chapter Twelve.
When I woke up at 10AM, he wasn't in the apartment.
I checked my email and found the following:
**Fri, Nov 24, 2017 at 9:18 AM**
**From: Arafat Kazi**
**To: Jarett Kobek**
**Subject:**
Hey dude, I couldn't sleep from friction of excess skin on floor, so I got a hotel. About to go to sleep for a couple of hours. It's 9:18 am.
Sent from Arafat's iPhone.
We met for a late lunch at Musso & Frank, which is the oldest restaurant in Hollywood, and also the setting for a short story that I wrote about the film director Wes Anderson using a urinal. The story is titled "Wes Anderson Uses a Urinal." You can find it in a recently published anthology called _Mixed Up: Cocktail Recipes (and Flash Fiction) for the Discerning Drinker (and Reader)._
Before I left for lunch, I checked my ticket on the website from which I'd ordered it, and discovered something that I hadn't noticed before.
Despite tickets to the pit being General Admission, my purchase had been assigned a seat.
The reason I'd adopted a soft belief in both tickets' validity was on the basis of General Admission. Why would any General Admission ticket be assigned a seat number?
Ipso facto, one ticket couldn't replace the other.
But now my belief was shattered. It was clear that there was only one seat.
Ergo, one ticket.
A sensation of dread crashed on me.
I'd made Arafat Kazi fly out to Los Angeles and bought his plane ticket and we'd been talking about this stupid concert for months and now he was staying in a hotel because the empty skin which draped his body had made it impossible to sleep on my apartment floor.
And he wasn't getting inside.
Before I left, I made a vow to the universe: if Arafat Kazi got into the pit to see Guns N' Roses at the Staples Center, then I would stop worrying about the outcome of my life.
I would take it as a sign that everything would be fine, even if my last novel had commanded a high advance and turned out to be a commercial failure.
I'm not sure why I made this vow.
It happened while I was urinating.
Shades of Wes Anderson.
I went to Musso & Frank. I ordered a hot turkey sandwich. Arafat got a French Dip sandwich. Then we ordered dessert.
I had a piece of key lime pie.
Arafat had something called the Diplomat Pudding.
I had, and have, no idea what's in a Diplomat Pudding.
It looked disgusting.
We left the restaurant and walked for a few blocks.
Arafat used his smartphone to hail an Uber, which was a private car operated by a company that's single-handedly set back the American labor movement by about seventy years.
The car brought us to his hotel. We sat around his room for an hour and a half, talking about Muslims in America.
Arafat's a Muslim.
I'm half a Muslim.
Break out the misspelled placards!
"Dude, I know people, you know," he said, "who have jobs as bank managers, who are nice when you see them, and then you go back home and see that ten minutes after you parted, they've posted about Sharia law on Facebook."
"I read about this poll a few months ago," I said. "They asked people of every possible demographic how they felt about people from every other demographic. And, dude, Muslims polled worse than anyone else in America. With every single demographic. When they asked Muslims about other Muslims, dude, they still polled worse than everyone else."
"Well, dude," he said, "I think you've got to realize that even though people express a public opposition to the rhetoric, when that rhetoric comes from the top, it still seeps in."
Then I convinced him to change his clothes.
He'd packed an outfit that he wanted to wear to the concert, but earlier that afternoon, he'd decided against it.
We argued, but I won the day with the following thought: "If you're dressed like a circus performer, there's a better chance of them letting you inside."
This was the outfit: hot pink pants and a striped multi-colored psychedelic shirt.
Arafat also had a cap which matched the shirt.
He changed his clothes.
It was incredible.
He really did look like a circus performer.
We took another Uber to the Staples Center, which is a circular-shaped venue where the Los Angeles Lakers play basketball and imbue the city's cocaine addicts with a sense of regional superiority.
The driver parked across the street from the venue.
We got out of the Uber.
We walked over to the Staples Center and discovered that there was a special line for people with General Admission tickets. It was much shorter than the normal line, which was full of sane people who hadn't paid $550 to see middle-aged men perform thirty-year-old songs.
I gave Arafat one of my tickets.
"Let's see how it goes," I said.
At the front of the line, a pleasant woman tried to scan the barcode on my ticket.
It didn't work.
"What about his?" I asked.
She scanned Arafat's ticket.
It worked.
We tried to convince her that she should let us both in. She said that she couldn't. We'd have to talk with the box office manager.
We walked away and then she called us back.
Because she'd scanned Arafat's ticket, and it worked, one of us would have to go inside.
I took Arafat's ticket.
I went inside.
He said he'd go talk to the manager.
When I got inside, there was a table for General Admission tickets, and the young woman working at the table was checking barcodes and names against a list of people authorized to be in the pit. If your barcode matched an entry on the list, then she'd put a purple leopard-spotted paper bracelet on your left wrist.
The bracelet was your pass into the pit, and even with that, you still had to get through about four more security checks.
I got down to the pit, which was about five feet from the stage.
_Arafat would have loved this_ , I thought.
_It's awful that he isn't here_ , I thought.
_Everything's ruined_ , I thought.
There were fifteen other people in the pit. They were pressed up against a railing that separated the pit from the stage.
I was the only person standing in the middle, not pressed up against anything.
It felt awkward.
I went back to the round concourse of the Staples Center.
And then I did what all pathetic writers do.
I found the bar.
With a bloodstream full of overpriced vodka, I texted Arafat.
I wrote that he shouldn't worry, that he should just get a scalped ticket on his smartphone, and I'd pay him back.
At the very least, I thought, he could get a ticket in the cheapseats. It'd be a shared memory even if we were apart.
But he didn't respond.
Showtime was at 7:30PM.
Around 7:20PM, I decided that I should go back to the pit.
I again went through the phalanx of security.
When I got into the arena, I saw only one thing.
Arafat Kazi, standing in the pit, his circus performer costume as bright as the sun.
He'd talked his way in!
I was so happy that I insisted we pose for a picture where I was kissing his greasy fucking head.
The show was amazing. Guns N' Roses was the best band I'd ever seen.
They were so good that they were even better than when I saw them at Dodger Stadium, where they'd been brilliant. They were good in the way that people are good only when they hate the alternative so much they'll do anything to avoid it.
And in the case of Guns N' Roses, this was the alternative: go home and lead a normal life.
The next day, Arafat Kazi woke up and took a train to San Diego.
He sent me a series of text messages:
My head is still spinning
Nothing makes sense
I think it was a capstone moment in our friendship
That's what the final scene in the movie about us would be
This is as formative as anything we've shared
There are two options here.
You can believe that Arafat Kazi getting into the pit to see Guns N' Roses at the Staples Center was the byproduct of a random universe acting out in its mechanistic complexity.
But to believe this, you have to accept a chain of events so unlikely as to be incalculable in their probability.
You have to accept a universe so random in its possibilities that it was able to produce the unlikelihood that Arafat Kazi, the only person alive who could talk his way into a $550 ticket, would have a best friend who would be mailed, by accident, two tickets to the same concert after stumbling into the impossibility of making a bunch of money from writing a novel, and that this best friend would see the second ticket and know exactly how it should be used.
And you would have to accept that all of this would happen while someone was dressed like a circus performer.
The other option is to do what I've done.
You can accept that the universe, for whatever reason, wanted Arafat Kazi and myself to be in the pit to see Guns N' Roses at the Staples Center. It wanted us to have that formative experience. It wanted to write that last scene in the movie about our lives.
You can accept that a divine hand was involved in the whole process, easing our path, guiding the journey.
You can accept that I saw the face of God.
And you're going to have to forgive me, because the worst possible time to see the face of God is in the middle of writing a novel.
It's going to make a mess of everything.
The last few chapters of this book are going to dissolve into a hectoring lecture about Jesus Christ.
Sorry about that.
Don't say you weren't warned.
Anyway, here I am, the author, Jarett Kobek, and I say to you, reader, that I was in the Staples Center, I was in the pit, I was at Guns N' Roses, I was with Arafat Kazi, I was shortlisted for the Bad Sex in Fiction Award, and I saw the face of God.
And it looked like this:
And this:
## Chapter Seven
## **The House on the Hill**
And while Celia and Rose Byrne were seeing _Wonder Woman_ at the Vista, another attendee at the same screening was a man named Francis Fuller. He'd been a director of films and television for about thirty years between 1950 AD and 1980 AD.
Fuller began his filmmaking career as a young native Angelino who was queerer than a three-dollar bill and made short experimental films on 8MM and 16MM.
These films were more expressionistic than narrative, featured aggressive editing, and were shown at makeshift cinemas for audiences of people who smoked too much marijuana and had too much sex with strangers.
Embarrassed as he later would be by his works of youth, Fuller admitted that they'd helped earn him admission to the film school at USC, where he'd gained a fundamental understanding of the craft.
After graduation, he'd bummed around Hollywood until 1963 AD, when his life had changed through a meeting at a cultural salon hosted by the former actor Samson de Brier.
It was a night when everyone'd been smoking too much tea, and too many people'd been talking about Thelonious Monk. Everyone was crammed into a little house in the backyard of de Brier's property on Barton Avenue.
Fuller was bored. He didn't know fuck all about jazz.
He looked around de Brier's tiny cottage and saw an exceedingly corpulent man pressed up against a Venetian mural.
Fuller went over and said hello to the corpulent man.
The man turned out to be a lush named Aram Menechian, who'd come to Hollywood with the intention of laundering some of his brother's ill-gotten money.
Fuller said that he had a screenplay. Fuller said that he'd gone to USC. Fuller mentioned that _Time_ magazine had sneered at his short films.
Fuller walked out of de Brier's salon with an offer from Menechian to produce the screenplay.
The screenplay was entitled _Handspun Roses_ and for two years, it'd been sitting in Fuller's bedroom at his parents' house in Riverside County.
_Handspun Roses_ was a loose adaptation of Elizabeth Gaskell's "The Poor Clare." The action was transposed to the San Fernando Valley.
Despite its reliance on narrative, the finished film exhibited the same qualities as Fuller's experimental work, this time exercised in service of the horror genre.
_Handspun Roses_ caught the attention of Roger Corman, who gave Fuller work directing several more feature-length films, including a black-and-white psycho-biddy starring Myrna Loy.
As the 1960s AD wore on and became the 1970s AD, Fuller found himself working in television. He directed bonecheap made-for-TV films and countless episodes of sitcoms and evening soap operas.
He missed the old days of handheld 16MM cameras, when you could tell ultra-butch straight boys that you were making a movie and watch as they put themselves into homoerotic situations for the sake of maybe kinda getting famous.
But the TV money was good.
And Fuller retained a certain silverback-daddy sex appeal.
And he'd bought his own home on Glendower Avenue in Los Feliz, which was an upper-middle-class neighborhood north of the Vista Theater.
Fuller grew old.
Work dried up, but he'd managed his investments, and he drew a pension, and thanks to Proposition 13, the taxes on his property were almost non-existent. He'd never reached the Cash Horizon, but he'd gotten pretty close.
Fuller lived on, a lonely geriatric in the pink house where once he'd thrown lavish parties full of rent boys and rough trade.
Some fans wrote to him, and there'd been one last non-union effort with a crowdfunded adaptation of "Young Goodman Brown" by Nathaniel Hawthorne, and there were always emails to be answered.
By 2017 AD, Francis Fuller knew that he was nearing the end and that very little excitement would come again.
He'd returned to the primary activity of his youth, when the world seemed full of promise: he went to the movies.
The films had changed.
The glamor and the glitz were gone. Most movies were parables about American foreign policy and had an intended audience of bloodthirsty men.
That's why he was at _Wonder Woman_.
He saw everything that played the Vista.
When _Wonder Woman_ finished simulating genocide, Francis Fuller went to the lobby and thought about using the bathroom.
There was a line of young men who needed to urinate. Fuller was too old to be pushed about in the queue. He decided to wait until the bathroom was empty.
And it was while he waited that he saw the two most astonishing women.
They were dressed a bit like Diana, the hero of _Wonder Woman_ , but instead of wearing bondage-themed body armor, they were wearing animal pelts. Real fur!
And they were so muscular.
But not at all.
And so femme.
And yet not.
He couldn't determine their ages. Were they very old?
Or were they very young?
Francis Fuller couldn't help himself.
He had to talk to them.
The conversation turned into Francis Fuller giving Celia and Rose Byrne a ride in his vintage Jaguar. He drove them to his house on Glendower Avenue.
Celia and Rose Byrne ended up in Francis Fuller's living room, where, because of effective sewage management, only one person had ever voided their bowels.
The house was high enough on the hill that Celia and Rose could look through Fuller's picture window and see the whole of the city.
It was infinite lines of car headlights, the north–south avenues intersecting with the east–west boulevards, a fathomless grid of industrial pollution and greenhouse gases.
"We are in a new world," said Celia to Rose Byrne.
"It is much worse than on our television," said Rose Byrne.
"One does not expect much," said Celia. "But one maintains hope. The mortals I have known in my life have been pleasant enough. How can they have created such a nightmare?"
"The human condition, my dears," said Francis Fuller as he came from his kitchen, holding a tray with three cups of black Darjeeling tea.
There were many things that Francis Fuller couldn't imagine.
He'd spent most of his professional life making films about supranatural entities and now he had brought supranatural entities into his own home.
And he had no idea.
Fuller couldn't imagine the level of danger implicit in the women's presence.
If _Wonder Woman_ was a genocide simulator, then Rose Byrne was genocide.
Other than individuals in the military arm of the United States of America and former Presidents of the United States of America, she'd killed more human beings than anyone on Earth.
They sat in Fuller's living room, drinking his tea.
Celia looked at the décor.
It was shabby old furniture surrounded by vintage framed movie posters, all of which were advertisements for 1940s AD films produced at RKO by Val Lewton.
_Cat People, I Walked With a Zombie_ , _The Leopard Man_ , _The Seventh Victim_ , and _Isle of the Dead_.
"You have a wonderful eye," Francis Fuller said to Celia. "Not many people pay attention."
"My queen is a rare being," said Rose Byrne.
"I've known some rare queens," said Francis Fuller. "They're all dead now. Except Ken Anger. I heard he was still down on Hollywood Boulevard, you know, screaming at anyone who'll pretend he's interesting. The last time I saw him was at Curtis Harrington's funeral. Poor Curtis, he and Ken had a thing back in the '40s. The funeral was ghastly. Ken was even worse than usual and spent the whole time heckling anyone stupid enough to speak from the podium. He made a whole show over Curtis's body, kissing the corpse. But that's Hollywood. It's always been like this."
Like most readers of this book, Celia and Rose Byrne had absolutely no idea what Francis Fuller was talking about.
"There's something I have wondered," asked Celia. "How do people hear the stories that they put into films and plays?"
"Hear them?" asked Francis Fuller.
"Yes," said Celia. "How did the actors in _Wonder Woman_ hear about Diana and her island and her journey into the world and her queen mother?"
"Honey," said Francis Fuller, "that answer is too long. We live in the era of the mega-franchise."
"But where did the story come from?" asked Rose Byrne.
"Comic books," said Francis Fuller. "These days, all of the movies come from the funny papers."
Celia and Rose Byrne had never seen comic books, which were cheap little periodicals that contained American power fantasies.
But Fern had brought home many a newspaper and the women of Fairy Island had pored over them, paying especial attention to the comic strips that arrived printed in full color in the Sunday editions.
"You mean that the story of Diana came from _Krazy Kat_?" asked Celia. "Or _Blondie_?"
_Krazy Kat_ was an old newspaper comic strip about a cat struck with love for a mouse that liked throwing bricks at the cat's head. The cat was named Krazy. The mouse was named Ignatz.
_Blondie_ was an old newspaper comic strip about a Jazz Age flapper who married a man with an insatiable appetite for sandwiches. The flapper was named Blondie. The husband was named Dagwood.
Celia had seen both strips in the early 1940s, when Fern had brought home copies of the _New York Journal-American_.
"Something like that," said Francis Fuller. "Recycled old pap. That's what the flickers are these days. When I was in the business, things were different."
"You made films?" asked Rose Byrne.
The doorbell rang.
Francis Fuller jumped up. His octogenarian bones buckled under the sudden thrust of his mass.
Fuller answered the front door, which was in a foyer off the living room.
Standing on his doorstep was Adam Leroux.
Leroux was Fuller's makeshift assistant.
He was twenty-eight years old.
Leroux had first shown up in Fuller's life after Leroux sent an email asking about _Handspun Roses_. A correspondence ensued, wherein many topics about old Hollywood were discussed. This led to Fuller's discovery that Leroux lived in Los Feliz. An invitation was extended for Leroux to visit Fuller's home.
When Leroux arrived for the first time, Fuller was delighted.
The young man was so handsome and butch.
Fuller was fascinated by the short story of Leroux's life, which had included a few military years in Iraq, where Leroux, who was poor, had shot Muslims at the behest of rich people.
For his part, Leroux was drunk on proximity to someone who'd directed films and known people like Anaïs Nin, James Whale, Susan Sontag, Dorothy Dean, and Orson Welles.
It wasn't long before Leroux was coming over every day and helping Fuller with his memoir.
They were a Hollywood odd couple of the Twenty-First Century AD.
The old man, decaying in his earth-tone suits, and his young assistant, body covered in tattoos, head pierced with metal, dressed in black T-shirts and jeans.
"Adam," said Fuller. "You'll never believe who's here."
Fuller brought Leroux into the living room.
Leroux had a sixth sense.
He'd killed enough Muslims, and had enough Muslims try to kill him, that he knew when he was in danger. One look at Rose Byrne, broadsword dangling against her exposed thigh, and he understood that he was staring at death.
Leroux had a seventh sense.
He was a Hollywood assistant with no future who'd attached himself like a barnacle to an old ship. Fuller was his one shot.
Leroux had an almost preternatural sensitivity to moments when his hold on the old man was threatened.
There'd been other guests who pinged off Leroux's seventh sense.
The ones who wouldn't leave.
The ones who'd stolen memorabilia.
The ones who'd take advantage of Fuller for the sake of social media.
Leroux knew that Fuller was a man who collected stray dogs.
But none of the others had carried a sharpened sword forged in the fires of Fairy Land.
Fuller made introductions between his guests and Leroux.
"They were just asking me the most wonderful question," said Fuller. "They asked me how the people who made _Wonder Woman_ had heard the story of Princess Diana and the island of the Amazons."
"I haven't seen it," said Leroux.
"We just came from a screening at the Vista," said Fuller.
"How was it?" asked Leroux.
"Moronic," said Fuller. "But you know, at my age, and in this town, I don't expect much."
There was more small talk, with Fuller and Leroux explaining the magic of moviemaking to Celia and Rose Byrne.
These efforts failed, as both men relied on the expectation that the women were conversant with Hollywood's shared cultural history, which was an American religious mythos that had penetrated every recess of the globe except Fairy Land and some remote tribes in South America.
Fuller's bladder, which had dogged him for several years, again demanded voiding. He excused himself and went to the bathroom at the back end of the house.
"Francis is older than he looks," said Adam. "He gets tired very easily. People don't realize how much these conversations take out of him."
"It is said that aging past usefulness is the worst thing that can befall a person," said Celia.
"He's still useful," said Adam Leroux. "He's working on his memoirs."
"What is a memoir?" asked Rose Byrne.
"His personal history," said Adam Leroux. "He's known some very interesting people. There's a whole chapter about Joanna Cassidy."
The women of Fairy Land didn't respond.
"Francis is too kind to say it himself," said Adam Leroux, "but you should probably get on your way. It's close to his bedtime."
"We have nowhere to go," said Celia. "We are newly arrived in Los Angeles."
"I don't see how that's Francis's problem," said Adam Leroux.
"Are you telling us that my queen should leave?" asked Rose Byrne.
"That's exactly what I'm suggesting."
Rose Byrne stood up from Francis Fuller's shabby couch, took out her sword, and chopped off Adam Leroux's head.
He tried to defend himself but he was a mortal and Rose Byrne was an old hand.
All his military training and killing of Muslims were for naught.
He didn't stand a chance.
His head rolled around the living room.
His body twitched out its last bioelectric moments of life.
Rose Byrne stormed to the bathroom, where Fuller was sitting on the toilet with his penis tucked between his legs, struggling against age to void his bladder.
She drove her sword into his chest.
"Oh," said Francis Fuller.
## Chapter Eight
## **Gentlemen Prefer Blood**
On the very same day that Rose Byrne chopped off the head of Adam Leroux, HRH Mamduh bin Fatih bin Muhammad bin Abdulaziz Al Saud was guest speaker at the Lunch Series put on by Harvard University's Program for Constitutional Government.
The title of the talk was this: "Teaching Foundational Classics to the Mid-East: What It Means and Why It Matters."
It was held in room K354 of the CGIS Knafel Building on Harvard's campus in Cambridge, Massachusetts, which was a satellite city across the Charles River from Boston.
Harvard University was a hedge fund that masqueraded as an institution of higher learning. It was one of the places where the world's upper classes enjoyed grade inflation as they became economic war-lords of the technocratic elite who mouthed platitudes about equality while crushing the global poor.
The political philosopher Harvey Mansfield introduced HRH.
Mansfield explained that HRH was an alumnus of Harvard, having received a Master's in Public Policy at the Kennedy School before earning his Doctorate of Philosophy from the London School of Economics.
Mansfield explained an initiative funded by HRH's non-profit wing.
It was a multi-disciplinary program that brought promising students from the Middle East and funded their undergraduate education at Harvard, with a focus on a broad liberal arts education and exposure to the foundational influences of Western thought.
After Harvey Mansfield finished speaking, HRH addressed the room.
HRH talked about education being the cornerstone of liberal democracy.
HRH talked about the paucity of books in Arabic translation.
HRH said that while a great many students from the Middle East were receiving educations in America, their focus was on STEM, and that this had left them disconnected from ideas underpinning the basic political philosophies of the Twentieth Century AD and Twenty-First Century AD.
HRH talked about how it was impossible to expect events like the Arab Spring to resolve productively if people in the Middle East weren't exposed, in advance, to a diversity of ideas about governance.
HRH finished with this: "I am not an expert like some of the people in this room, but I am resolute in my belief that if human rights are to emerge, we must first educate humans, and then teach them what is right."
The audience applauded.
Harvey Mansfield opened the event to questions from the audience.
The first question was familiar.
The questioner told HRH that she had Googled him and found his interviews refreshing and unexpected. Then she asked: "I was wondering if you could speak about the reaction of the Saudi government to your more provocative statements?"
HRH smiled.
His bridgework was fucking fantastic.
"Madame," said HRH, "I was raised in the hotels of Europe and America. I hold citizenship in Malta. I do not speak as a member of my family. I speak as an inhabitant of the world."
The next question was also familiar.
It was being asked on every American campus by people who were terrified of college students.
"I don't know if you've followed any stories," said a man in a suit. "There's been a thing happening where the students at our universities have been asking for safe spaces. If you're not familiar with the term, and I wasn't until a few months ago, a safe space is a place where the students can be coddled away from hearing ideas that they don't like, and it's disguised under the idea of oppression. You're from a region beset by conflict. I tell my students that there are no safe spaces in Aleppo. Do you have an opinion on this phenomenon?"
HRH smiled.
His bridgework was fucking fantastic.
"I always err on the side of generosity. If people require safe spaces, then I see nothing wrong with providing them, as long as the institution tempers their presence with a robust environment of educational rigor."
When the questions were over, pleasantries were exchanged.
HRH texted his manservant Dmitri Huda.
"HEY NONNY HEY, ARE THINGS IN ORDER?????" asked HRH.
"Yes, Dennis," texted Dmitri Huda. "I'm downstairs."
HRH's father Fatih bin Muhammad bin Abdulaziz Al Saud was the second-richest man in the Middle East. He built a fortune after being exiled from the Kingdom.
This exile followed the parking-lot execution of Misha'al bint Fahd bin Muhammad bin Abdulaziz Al Saud.
Fatih bin Muhammad was a convenient scapegoat for the assassination.
It was said that he encouraged delusions of romance in Misha'al.
He was given the riyal equivalent of $200,000.
He was kicked the fuck out.
He traded off the family name, got into construction and concrete, and used that money to diversify his holdings. When he had established his fortune, he decided to do what all people do when they want to legitimate their place in the hierarchy of global evil.
He wrote a book.
First published in French as _Le chemin du conquérant arabe: les leçons d'un prince saoudien_ , an English translation appeared in 1999 AD under the title _The Conqueror's Path: Business Lessons from a Saudi Prince_.
It was a CEO-style autobiography married, awkwardly, with Fifteen Lessons that Fatih bin Muhammad had learned through the ups and downs of doing business on an international scale. Each lesson was expanded with historical parallel and floating anecdote.
_Il Principe_ meets _Trump: The Art of the Deal_.
It sold in small numbers until references began appearing in the songs of well-known hip-hop artists, who adopted the book's maxims of worldly success into anthems of global capitalism.
Sales exploded.
Fatih bin Muhammad became The Conqueror.
One of The Conqueror's Fifteen Lessons, present in _Le chemin du conquérant arabe_ , was the idea that a successful businessman, particularly if he comes from a place unfamiliar to his potential financial partners, must take up stratagems to evoke comfort in others.
Following this advice, HRH had adopted many names in different languages.
In Chinese, HRH was called 野生花卉, which meant Wild Flower.
In Spanish, HRH was called _el Diablo árabe_ , which meant The Arabic Devil.
In Turkish, HRH was called _Küçükkutsaldağ_ , which meant The Little Holy Mountain.
In German, HRH was called _Der Meister der Weltschmerzes_ , which meant Master of the World's Sorrows.
In English, HRH was called Dennis, which meant Dennis.
Dmitri Huda had commandeered a surface parking spot on Cambridge Street.
HRH came out of the Knafel Building.
HRH walked towards the car.
Dmitri Huda jumped out of the driver's seat and rushed to the rear passenger door of the gun-metal 2016 AD Bentley Mulsanne.
"Dmitri! Play not the dogsbody!" cried HRH. "What do you take me for? Have I too lost the ability to walk? Must I next crawl?"
Dmitri Huda returned to the driver's side door.
"Do you behold this complex, Dmitri?"
HRH pointed to a series of drab buildings on the other side of Cambridge Street.
"This august institution is the Cambridge Rindge and Latin School."
"I see," said Dmitri.
"It is notable for its alumni," said HRH. "Most prominent are the actors Matt Damon and Ben Affleck. Followed only by Dzhokhar Tsarnaev and his brother Tamerlan, who together orchestrated the bombing of the Boston Marathon. When news of the blasts reached my ears, it evoked salad days misspent in Cambridge. I sensed in my inner heart that the perpetrators would be revealed as local yokels. Only the trite provincialism of a Bostonian would suggest the Marathon as a target. Dmitri, if you wish to further your spiritual development, you should consider the occult principles of this complex. It always produces its monstrosities in pairs."
HRH climbed into the back seat.
Dmitri Huda returned to the driver's seat and started the engine.
"You know the location?" asked HRH.
"It's in satnav."
HRH opened the refrigerated bar.
Inside there was a vaporizer and a bag of marijuana.
"Is this indica or sativa?" asked HRH. "I will not suffer the mellow vibes of indica. Not tonight. I must invigorate with the lush and vibrant caress of sativa."
"It's sativa," said Dmitri.
HRH vaped sativa.
HRH pressed a button, which deployed a bespoke Android tablet embedded in the reverse of the front passenger seat.
HRH engaged with the bespoke Android tablet.
HRH opened the YouTube app.
HRH streamed "Dark Avenger" by the American heavy metal band Manowar.
"Dark Avenger" played through the Naim audio system.
"Drive on, Dmitri," shouted HRH over the 1,100 watts of pulsing metal power. "Bring me to my destiny!"
HRH's destiny was an old factory in Waltham that had been gentrified into offices and loft apartments.
For a solid century, the building had manufactured watches. Now it crafted the aspirant lives of the haute bourgeoisie.
Dmitri Huda navigated the Bentley from Cambridge to West Cambridge to Watertown and through the other suburbs. It was that New England experience: the transition between multiple disparate landscapes in less than forty minutes of travel. Dense urbanity giving way to small-town life to post-industrial decay.
When they arrived at the old factory, Dmitri Huda idled in the parking lot.
"Remain here," said HRH. "I am sure to stride forth, triumphant in my victory."
HRH emerged from the 2016 AD Bentley Mulsanne with a rattlesnake suitcase under his arm.
Here's something that Harvey Mansfield didn't explain in the CGIS Knafel Building: HRH had been hipped to the possibility of a Doctorate in Philosophy at the LSE by Saif al-Islam Gaddafi.
Saif al-Islam Gaddafi, famous for being the son of a lunatic dictator who blew up a passenger plane over Scotland and was beaten to death after hiding in a drainage pipe, had demonstrated how this possibility worked.
The vampire of the LSE sucked blood money.
Its conscience was soothed with paid holidays for the administrative staff and faculty, all the better to generate white papers and editorials in the _Telegraph._
In terms of education, the metropolitan area was lousy with debauched Eton boys who would handle your coursework and dissertation.
They only asked what anyone asked.
Lucre, filthy lucre.
One needn't spend much time in the Old Smoke, but it did help to make the occasional appearance. Besides, as Dr. Johnson had told Mr. Boswell, when a man tires of London, he tires of life.
And if, during his salad days, the stout erections of HRH's penis had carried any information, it was that his corporeal form had yet to tire of life.
HRH managed his way through the old factory until he came to the fourth-floor apartment.
HRH knocked on the door.
A sex worker, who held a lease on the apartment, opened the door. "You must be Dennis."
"Madame," said HRH. "You have identified me with utter precision and laser focus."
The sex worker moved from the doorway.
HRH passed into the apartment.
The sex worker led HRH down a small staircase to the apartment's lower level, which housed a bedroom, a kitchen, and a living/dining space.
"You will please to remind me," said HRH. "Did my assistant forward the funds through Venmo? Or must I be discreet in my placement of the requisite white envelope on your granite countertop?"
"We got the money," said a male voice from the living/dining space.
It was the sex worker's bodyguard.
He was a large man. He was wearing a blue-and-silver sports jersey advertising his avowed fandom of the New England Patriots and the team's star quarterback Tom Brady.
The bodyguard was sitting in front of a television. The bodyguard was watching the television with its speaker muted.
He was reading the closed-captioned subtitles, which conveyed the story of an attack on a casino resort in Manila. Thirty-seven people had been shot and killed.
"I was unaware that another soul would be present," said HRH.
"Is that a problem?" asked the bodyguard.
"My dear fellow," said HRH. "I flourish on company. What a stout, robust lad you seem! Shall you too join us in our deluge of flesh and avarice? I should like very much to see and feel that frame of yours in its bounding action. What thighs you have, my liege!"
"I'll pass," said the bodyguard.
Above the bodyguard's head, there were three clusters of helium-inflated balloons, tied together in a haphazard fashion to create letters from the Roman alphabet.
The balloons said:
HRH walked over to the exterior wall of the apartment, placing his hand on its exposed brick. Its windows looked out over the Charles River.
"Fear death by water," said HRH. "As my manservant drove me towards this monolithic structure, it occurred to me that perhaps my father had some hand in its conversion. The Conqueror is consumed with a smothering love for Boston and its environs. The redevelopment of Boylston in the Fenway was his own initiative."
"Let's get going?" asked the sex worker.
"Wunderbar, my dear lady!" cried HRH. "To the stables!"
In the bedroom, two other sex workers were waiting.
HRH and the original sex worker entered.
Each of the sex workers had been picked by Dmitri Huda via an arcane process that began with The Erotic Review, which was the Internet's top community of escorts, hobbyists, and service providers.
The Erotic Review's vast userbase was comprised of people who fucked sex workers and then went on The Erotic Review and reviewed the performance, looks, and personalities of recently fucked sex workers.
The Erotic Review offered its reviewers the option to confirm whether or not a recently fucked sex worker provided specific sexual activities during the recent fucking. These included: (1) cum in mouth (2) touch pussy (3) lick pussy (4) two-girl action (5) more than one guy at a time (6) multiple pops allowed.
After Harvard University invited HRH to be a guest speaker, Dmitri Huda had contacted a sex worker whom he'd procured several years earlier using The Erotic Review.
The sex worker wrote back. She wasn't available. She was working in Dallas.
She recommended a friend, who got Dmitri Huda in touch with an agency that sometimes did cross-over work with people from FetLife.
The agency said that it could satisfy HRH's demands: three girls, athletic, Ivy League educated, very bi, 420-friendly, unafraid of BDSM, and willing to go anal.
Dmitri asked the agency to procure helium-inflated balloons.
The balloons were HRH's way of making sure that his requests had been fulfilled to the utmost. Past experience had demonstrated that if the balloons were not present, then other requests would also be ignored.
HRH had learned this trick by reading about Van Halen's tour rider.
HRH put his rattlesnake suitcase on the bed.
"Good day, ladies," HRH said. "We meet now in this temporality but I believe that we have known each other always."
HRH opened the rattlesnake suitcase and extracted a vaporizer and a small, clear plastic bag that contained an off-white powder.
"First, mes chères amies," said HRH, "You shall watch as I consume dimethyltryptamine. Fear nothing, my sweets, for the effects are not long lasting. This ease of use has earned the substance a wonderful soubriquet. They call it The Businessman's Trip."
HRH sat in a plush chair purchased from IKEA in Stoughton.
HRH vaped DMT.
HRH's eyes went blank.
HRH's breathing became labored.
One of the sex workers got up from the bed and waved her hand in front of HRH's face.
HRH didn't respond.
"He's out," said one sex worker to the other sex workers. "Don't worry. These guys are the easy ones. We give them what they can't get in Dubai."
"What's that?"
"The kissing and the cuddling."
HRH went on an inner trip.
There was a psychedelic tunnel.
HRH went through the psychedelic tunnel.
Everything looked like a Mandelbrot set transformed into quivering nerves.
HRH turned back and saw himself in the IKEA chair, surrounded by sex workers.
HRH continued through the psychedelic tunnel.
HRH came through on the other side.
HRH found himself in a mystical land, surrounded by elfin creatures, with fractal trees sprouting forth from the earth. The elfin creatures spoke a strange language that sounded more like buzzing than words.
HRH tried to talk but his words came out as shattered glass.
HRH didn't know it, but the dimethyltryptamine had sent an astral projection of his soul to Fairy Land.
This happened to every user of dimethyltryptamine, leading to endless reports on Erowid.org and Reddit.com. And some very bad writing by Terence McKenna and Tao Lin.
Terence McKenna, Tao Lin, and the users of Erowid.org and Reddit.com thought that they had traveled in fourth-dimensional space and held forth with cybernetic elves.
But really, they were just in Fairy Land, and the astral projection was creating a perceptual filter that prevented full comprehension of the experience.
The women of Fairy Land could see the spiritual projections of dimethyltryptamine users.
The souls appeared like flickering lights.
The women of Fairy Land thought that these lights were ghosts of the People Who Came Before.
They didn't know that the flickering lights were just some old assholes on drugs.
The trip wore off.
HRH came back into consciousness, back to the watch factory.
HRH jumped out of the IKEA plush chair.
"Another entheogenic experience!" said HRH. "Further communion with the divine! I seek knowledge! Soon I shall have the answer!"
"That wasn't very long," said one of the sex workers.
"As I said, madame," said HRH. "It is the trip of a businessman."
"You must inform me," said HRH to the sex worker who leased the apartment. "What is your WiFi network and its password?"
"The network is arcticmonkeys," said the sex worker. "The password is doiwannaknow. All lower case, no spaces."
HRH opened his rattlesnake suitcase and removed an Amazon Echo Dot.
It was the shape and size of a hockey puck.
HRH put his hands into his pantaloons.
HRH fished out his smartphone.
HRH engaged with his smartphone.
HRH opened the Amazon Alexa app.
HRH plugged in the Amazon Echo Dot.
HRH used the Amazon Alexa app to get the Amazon Echo Dot on the sex worker's WiFi network.
Perhaps you are wondering about the exact nature of the Amazon Echo Dot.
Reader, its nature was two-fold.
The Amazon Echo Dot was a device that connected to the Internet and responded to voice command. Its users could ask the Amazon Echo Dot to play music, which would emerge from its onboard speaker. If the Amazon Echo Dot was networked with a television, it could be used to play films and television. It could be used to order products through Amazon.com, which was a website dedicated to the destruction of the publishing industry. And the Amazon Echo Dot could be used to relay information.
To achieve these tasks, users would say the word, "Alexa," which was the magic phrase that alerted the Amazon Echo Dot that an instruction was forthcoming. Then the user would say an instruction, which would be something like, "Play _Jersey Shore_ " or "I want to shop for cat food."
The Amazon Echo Dot would then respond with the synthesized voice of a woman and attempt to follow the user's command. This synthesized voice was the personality of the device. It was the ghost in the machine. Its name was Alexa.
The Amazon Echo Dot was one in a line of Echo products offered by Amazon.com, each offering some variation in shape and size, but retaining the same core functionality.
Reader, this was the surface nature of the Amazon Echo Dot.
Its true nature was this: it represented concrete evidence of the disconnect between issues that journalists believed were of public importance and the swells of indifference that these issues produced in the public.
Following the election of Donald J. Trump to the Presidency, there was a clamor about the manner in which his campaign may have misappropriated the private information of millions of Americans.
This was all of the media coverage distilled: the users of social media had provided their private information with no intention of it being deployed for anything other than their banal self-expression on platforms owned by megalithic corporations. Its unauthorized use in a political campaign represented a grievous breach of ethics and corporate governance.
If you took this media coverage at face value, reader, you would believe that most people were outraged about turning their private information over to megalithic corporations.
But listen to someone who became a minor literary sensation on the basis of a book that critiqued turning over one's personal information to megalithic American corporations.
Enthusiastic journalists wrote twenty-seven thousand articles about _I Hate the Internet._
It gave the impression of a book that was all dominant, all powerful, all consuming.
But I've read the only writing about _I Hate the Internet_ that matters.
Royalty statements.
And so I speak with the authority of someone who managed to get an obscene amount of press coverage for what was, ultimately, an obscure book: most people do not give a fuck or a tuppence about what happens to their private information.
A hockey puck that was always listening.
It was indistinguishable from espionage devices. It sent the inner workings of private homes to a corporation with one of the largest market caps in the world.
There was no illusion about its purpose.
Its nature was both its virtue and its advertising hook.
And at a moment when journalists were producing hundreds of thousands of words about privacy on social media networks, millions of people were buying the Amazon Echo Dot.
By the end of the Year of the Froward Worm, it was a necessary accoutrement of every middle-class American home.
HRH disrobed.
"Please, please, my sweets, you too must remove your store-bought modesty."
The sex workers disrobed.
"I have requested you because of your academic pedigrees. For this evening of transcendence, I want no Masshole curs trapped in the amber of ignorance. Is it true that each of you has received an Ivy League education?"
"Yes," said the sex worker with the lease on the apartment.
"Yes," said the second sex worker, who was lying. She had an undergraduate degree from Babson.
"Yes," said the third sex worker.
"I believe you to possess what is necessary," said HRH. "Tonight I demand that you address me only as Enver Hoxha, the former Albanian head of state. Cast back your imaginations to that glorious moment when Hoxha rejected the reforms of Khrushchev as revisionist Leninist–Marxism and took up the cause with Red China. Imagine it! An isolated European country, surrounded by its enemies and the sea, aligning itself with Maoist principles!"
HRH opened up his rattlesnake suitcase.
"As I do not doubt that you learned while earning your Ivy League degrees," said HRH, "a key difference between Stalinist Marxist–Leninism and Maoism is the Maoist belief in reeducation. The Stalinists would excommunicate the unwanted, while the Maoists enacted programs of reeducation. Why else did the Symbionese Liberation Army bring Patty Hearst into the fold? They demonstrated her to be a fascist insect preying upon the life of the people and through class consciousness reformed her into a revolutionary."
HRH removed a device from the suitcase.
The sex workers couldn't see what HRH was holding.
"Tonight," said HRH, "we shall query Alexa and discover what she knows about the People's Republic of Albania. When Alexa fails in her knowledge, then your acres of skin will be reeducated. Tonight, the fleshzone is a labor camp and you are its prisoners. Arbeit macht frei, meine Mädchen."
One of the sex workers caught a glimpse of what HRH was holding.
It was a rhino-hide chicotte, restored and recovered from the Congo Free State.
"Before we embark upon our merriment," said HRH, "I suggest that we test the ability of Alexa to provide us with information."
HRH stood over the Amazon Echo Dot.
"Alexa," said HRH. "Why does the caged bird sing?"
"The caged bird sings," said the Amazon Echo Dot, "because its heart is still free and using song is an efficient way for birds to communicate over distance."
"That's actually kind of cool," said one of the sex workers.
"Schnell! Schnell!" said HRH. "It is time to make a great leap forward."
"Alexa," said HRH, "who was Enver Hoxha?"
HRH pronounced _Enver Hoxha_ properly: En-ver Ho-dja.
"Hmm," said the Amazon Echo Dot. "I don't know that one."
"Alexa," said HRH, "who was Enver Hoxha?"
HRH pronounced _Enver Hoxha_ in phonetic English: En-ver Hox-ha.
"Here's something I found on Wikipedia," said the Amazon Echo Dot. "The Rwandan Genocide also known as the genocide against the Tutsi was a genocide of mass slaughter of Tutsi in Rwanda by members of the Hutu majority government."
"Alexa," said HRH, "who was Enver Hoxha?"
HRH again pronounced _Enver Hoxha_ in phonetic English: En-ver Hox-ha.
"I'm not quite sure how to help you with that," said the Amazon Echo Dot.
"It should be rather clear that we have long hours of Maoism ahead," said HRH.
There was a moment when the labor camp screaming grew so loud that the bodyguard burst into the room.
He found two of the sex workers on the bed, crying, bleeding.
The chicotte had offered bitter instruction.
The third was being forced to hold the Amazon Echo Dot over her head for as long as her arms would allow. She'd been instructed to address the Amazon Echo Dot only as Aten, after the Egyptian Sun disc.
"What the fuck is going on in here?" asked the bodyguard. "What the fuck is this shit?"
"My darling patriot," said HRH. "You have joined us at last. I have waited for your flesh all these long hours. Would you care to act the Gomorrhean? I am your willing receptacle, and if you like, I can ease the path towards priapism with a surfeit of cocaine. Snow is general all over Ireland. It is a dead certainty that a man with your thighs must ache with a clutched need to relieve the vital center. Let rain down your frothing spittle like Agent Orange upon the Vietnamese peasantry!"
The orgasm occurred.
The Amazon Echo Dot was playing "Blood of My Enemies."
"Blood of My Enemies" was a song by Manowar.
HRH threw back his head and cried out, "All of my foes shall perish before me! To Asgard the Valkyries fly! 诉苦!"
The fleshzone decommenced.
HRH unplugged the Amazon Echo Dot.
HRH repacked his rattlesnake suitcase.
HRH left a white envelope on the kitchen granite countertop.
The white envelope contained a very generous tip.
Dmitri Huda was waiting in the Bentley.
HRH climbed into the rear passenger seat.
"Was it everything you'd hoped?" asked Dmitri Huda.
"I met a charming fellow named Steve," said HRH. "He informs me that he was raised in Lowell."
"That's what everyone loves about you, Dennis," said Dmitri Huda. "You always make friends in new places."
HRH vaped sativa.
"What is my agenda for the morrow, Dmitri?" asked HRH.
"You're doing a TEDx at Brandeis," said Dmitri Huda. "Have you forgotten?"
"I never forget," said HRH. "I remember everything."
## Chapter Nine
## **Cleaning up the Mess**
So there was Francis Fuller's house on Glendower Avenue, with its low property taxes and its grand view of Los Angeles. It was full of blood and bodies.
Celia examined the headless corpse of Adam Leroux and wondered about the wisdom of bringing Rose Byrne to Los Angeles.
A psychotic sidekick made sense amongst the lawless stupidity of Jacobean London, but in a world dominated by a professionalized police force, it could prove problematic to be accompanied by the supranatural embodiment of genocide.
"You might have waited," said Celia to Rose Byrne. "I am certain I would have persuaded him with my charms."
"He was a warrior, lady," said Rose Byrne. "I could see it in his eyes."
"I have not bedded with a man in four centuries."
"We have concerns beyond the bowers of pleasure," said Rose Byrne.
"As you say."
Celia walked to the bathroom.
Francis Fuller's body, impaled on Rose Byrne's sword, sat on the toilet.
Blood was everywhere.
Because Celia had engaged with the woolen television of Fairy Land, a sense of _déjà vu_ washed over her.
She remembered, vaguely, a scene from the television adaptation of _Game of Thrones_. It was from the end of Season Four, when the mad dwarf Tyrion Lannister assassinated his own father while the latter sat above a latrine.
Celia's _déjà vu_ was a common feeling. The world was saturated with media. The memory of unreal things had imposed themselves upon the real. The President was a creation of television. The appearances of things were more important than the things themselves.
Celia returned to the living room and stood above Adam Leroux's unliving body. She stared out at the forever infinity headlights of Los Angeles.
She cast a spell.
It was a 1970s AD neutron bomb sort of magic, erasing all traces of both Francis Fuller and Adam Leroux while leaving Fuller's personal property intact.
Celia had no idea how long she would be in Los Angeles.
She needed a place to crash.
Why not keep the house on the hill?
It was the darkest of faery magic, the ancient stuff where children would walk the ferny path and never be seen again, lingering only as memories, leaving behind crying peasant mothers who talked about lost daughters wandering over green hills with the seely folk, until the mother herself died and the missing girl became nothing but a legend, just a name sung in a ballad that had been corrupted by endless performances over decades and then centuries.
It was the total effacement of humanity.
Goodbye, Francis Fuller.
You lasted for one of this book's longer chapters.
Goodbye, Adam Leroux.
You managed about a thousand words.
An entire segment of obscure film history was rewritten. Fuller's early experimental efforts disappeared. _Handspun Roses_ never happened. The films produced by Roger Corman evaporated. Myrna Loy's filmography lost one of its stronger late entries.
The television stuff didn't change much, because television was the result of an industrialized process in which the people behind the camera were interchangeable. Francis Fuller's name was struck from the collective credits of _Charlie's Angels_ and _Dynasty_ , but the episodes themselves were unaltered and lingered in the unpopular consciousness.
Almost all of Fuller's friends and family were dead, so there were hardly any gaps in individual memories.
Paragraphs disappeared from a few books. Alterations occurred in a handful of sad men's underwhelming master's theses. Some very old webpages evaporated. A few torrents stopped being listed on Cinemaggedon and Karagarga.
If he were alive, Francis Fuller would have been astonished at how small his life had been, at how easily the hole was patched.
He was like everyone else.
He thought that he was more important than he actually was.
But no one was any more or any less important than anyone else.
You can beg the Earth to stop turning, but it never listens.
And, please, reader, don't get amped up on this statement of your relative position of egalitarian non-importance.
You're still not qualified to review this book on Amazon.com.
The same thing happened with Adam Leroux.
His memory went out.
His family forgot him.
His friends forgot him.
He was struck from the computerized databases of surveillance and corporate marketing that dominated modern life.
Someone else got his car.
Someone else got his apartment.
Someone else got his French bulldog.
Someone else got his vintage 45 Grave T-shirt.
All of the Muslims that Adam Leroux had killed were like the episodes of television directed by Francis Fuller.
Their corpses were the end result of an industrialized process. The person pulling the trigger wasn't a big deal.
The Muslims were still dead.
The one place where the faery magic didn't have any effect was Adam Leroux's Instagram account.
Instagram was a social media platform that existed on telephones and computers. Its users shared pictures of their squalid lives, which fostered the illusion of a human connection while generating revenue for Facebook, which was a publicly traded company headquartered near San Francisco.
Instagram was also history's single most successful terrorist attack on the self-esteem of women.
Adam Leroux had managed to avoid most of social media.
Facebook, the company that owned Instagram, had another social media platform which was also called Facebook.
The company was named for the platform, which had started out as a student project at Harvard University.
Harvard was where HRH had received his Master's in Public Policy.
The Harvard version of Facebook, the ur-Facebook, had been designed to rate whether or not the hedge fund's female students were sexually attractive.
The ur-Facebook evolved into actual Facebook, spreading beyond the hedge fund's campus, and conquered the world.
Adam Leroux only logged into his Facebook account about once every three years, which gave him a slightly unique perspective when he checked it in the year 2016 AD.
He'd last been a heavy user of Facebook in 2008 AD, when the most annoying thing on the social media platform was people insisting that they were so happy and so in love with their latest semi-monogamous partner.
Things had changed.
By 2016 AD, no one was boasting about how their latest semi-monogamous partner made them so much happier than their previous semi-monogamous partners.
Now Adam Leroux's friends were bombarding each other with images of murdered bodies and shrieking about the corruptible nature of human beings while they apologized for social privilege which derived from their relative position in the global hierarchy.
"Fuck this shit," said Adam Leroux, logging out of Facebook for what would be the last time.
Another social media platform called Twitter held even less appeal.
Twitter was a place where people practiced bumper-sticker morality while other people threatened to rape and murder each other for expressing simple sentiments about banal objects.
"I like cats," a user typed into Twitter.
_I will fucking rip your ugly fucking shit face off you fucking jew cuck jew_ , replied Twitter.
"Crayons are good," a user typed into Twitter.
_Your soul will be mine in hell as you suck molten fire from my demonic warted prick_ , replied Twitter.
"My grandma wears a knitted hat," a user typed into Twitter.
_I am coming to kill and rape you until you are dead and raped you assfucked pussy_ , replied Twitter.
Twitter was also where Donald J. Trump ruled over America.
Donald J. Trump on Twitter was the ultimate tool of distraction.
Each day of Donald J. Trump's Presidency, his administration dismantled some aspect of the federal government, terraforming America into a dystopian misery, but no one talked about it and very few media outlets gave it any coverage.
All anyone paid attention to was Donald J. Trump's activity on Twitter, where he issued mean-spirited and stupid opinions about nonsense.
Concerned about Donald J. Trump stacking the federal bench with crypto-conservatives who believe that dinosaurs were made of chocolate pudding?
Shut the fuck up!
The President is upset about professional sports!
On Twitter!
Worried about nuclear war?
Who fucking cares?
The President called an actress ugly!
On Twitter!
Adam Leroux stayed away from Twitter.
But the multi-tentacled hivemind of global capitalism was nothing if not adaptable.
It had become necessary to enchain every human being with some form of social media. New platforms were being developed every minute of every day, attempting to unlock each individual mind.
In Adam Leroux's case, it turned out that Instagram was the key.
And I could easily write some very long and possibly pithy descriptions of Instagram's terrorist attack on female self-esteem, explaining how it had become the #1 destination on the Internet for plastic surgery disasters, for a plethora of fake asses, fake tits, hair removal, skin lightening, lip enhancements and Botox, and how female celebrities with certain physical features used their Instagram followers to advertise products that they'd been paid to hawk, and how the products were inevitably chemical warfare on the natural beauty of women, and how all of this was a sustained spiritual attack and how I myself know a handful of amazing people who'd gone haywire with plastic surgery inspired by Instagram.
But why bother with that?
Here is the simplest way to describe how awful Instagram was for women: it had weaponized yoga.
Instagram had created an environment where ridiculously blonde women from the ridiculous upper classes could flaunt their ridiculous lifestyles comprised of samosas and endless Caribbean vacations and could, somehow, wrap this excess of capitalism in a blanket of spirituality, photographs of Downward Dogs and Warrior Poses, the language of body-positive affirmation, and cloying truisms about the ability of anyone to achieve their dreams if they put enough effort and faith into the achievement of those dreams.
Yoga was one of the many weapons of mass destruction employed in Instagram's terrorist war on women's self-esteem.
A tool to bludgeon people with the things that they couldn't have.
Impossible bodies, impossible wealth, impossible life.
If anything could have resisted, it was yoga.
Yoga was as old as the hills.
It was ancient technology. It was almost as old as Fairy Land. And it too had fallen.
It was like everything else on Instagram.
Just another weapon in a long war.
So don't even ask about the fucking Kardashians.
Because heterosexuality is a bullshit con on women, the accidental byproduct of Instagram's remorseless terrorist war was the even more remorseless arousal of Adam Leroux's sexual desire.
His particular demesne was Instagram accounts belonging to women who were strippers in the city of Philadelphia.
Adam Leroux liked their fake asses, he liked their fake tits, he liked their fake lips, he liked their fake hair.
Say what you will about the strippers of Philadelphia, but they had a leg up when it came to Instagram. They'd done something nearly impossible.
They'd monetized their participation in Instagram's terrorist war on women's self-esteem.
Their primary motivation for using Instagram was to advertise to potential customers.
They posted pictures of themselves and alerted the world about which nights they'd be working the clubs.
Adam Leroux's attention was an accidental byproduct of this monetization.
Adam Leroux had discovered these women in 2015 AD.
Using his own Instagram account, he had spent almost two years commenting on their photos.
Here are some of the choicer comments that Adam Leroux had posted to Instagram:
_(1) bae i wanna crawl up in that a$$ like a small wood land animal and die_
_(2) would lick that pussy until u exploded just one taste its all im asking_
_(3) beautiful face bootiful body y wont u let me touch_
_(4) girl u got wot i need and wot i need is a$$ lol_
_(5) wont u let me show u a good time my hand to god above ill come to philly and teach u bout brotherly love and u can buy whatever u like_
Adam Leroux had left thousands of these comments.
For some inexplicable reason, the dark magic of Fairly Land had left them unaffected.
The comments remained long after Leroux's death.
He'd spent the last year of his life imagining that his literary output would be as the co-writer of Fuller's memoir.
But the old man's life and memory was gone.
This was Adam Leroux's legacy.
Comments on Instagram that expressed his infinite and endless thirst for the surgically inflated buttocks of Philadelphia's strippers.
Welcome to the future.
## Chapter Ten
## **On the Streets of Los Angeles, There the Wild Beast Slumbers**
Being a serial killer, Rose Byrne was in her post-murder cool-down phase.
She was sleeping in the master bedroom.
Celia watched television.
The content that she saw was different than what had played on the woolen television of Fairy Land, where all of the programs had been pre-selected and pirated by the island's more knowledgeable women.
The television on Fairy Land had focused on what the American liberal intelligentsia suggested was worth watching: shows from Netflix, from HBO, a select peppering of BBC, the Amazon.com adaptation of Chris Kraus's _I Love Dick_ , and some basic cable like _Mad Men_ or _Breaking Bad_.
By contrast, sitting in the living room of the former Francis Fuller, there was no pre-selection. There was only what aired on television in the middle of an average day.
It was what Los Angeles produced for the 99.5 per cent of Americans who weren't part of the country's liberal intelligentsia.
Celia saw an episode of _Judge Judy_ , in which a multimillionaire fake judge ritually abused the poor while adjudicating their small claims court cases.
She saw an episode of _Dr. Phil_ , in which a multimillionaire fake therapist ritually abused the poor while oozing a synthetic variant of empathy.
She saw an episode of _Family Feud_ , in which a multimillionaire comedian asked the poor to produce sexual innuendo in exchange for the promise of money.
She saw an episode of _Laura Luke's Paternity Court_ , in which a multimillionaire fake judge humiliated poor African-American women for engaging in the biological imperative of sex.
She saw an episode of _Divorce Court_ , in which a multimillionaire fake judge convinced poor African-Americans that they should embrace the global hegemony by creating two consumer households where there had originally been one.
She saw an episode of _Dr. Oz_ , in which a multimillionaire Turkish-American doctor hawked pseudoscience to the poor while embarrassing the fuck out of the five other Turkish people who lived in America.
She saw an episode of _The Real_ , in which a group of multi-millionaire women from marginalized backgrounds pretended that their money hadn't taken them past the Cash Horizon.
She saw an episode of _TMZ Live_ , in which a multimillionaire lawyer/feudal lord encouraged his cow-eyed millennial vassals to explain the sexual dysfunction of Twitter celebrities.
She saw an episode of _Keeping Up with the Kardashians_ , in which a family of multimillionaires proved that the biggest existential threat to the African-American male was not the Ku Klux Klan or the organized brutality of law enforcement or the school-to-prison pipeline but, in fact, the family themselves.
She saw an episode of _The Ellen DeGeneres Show_ , in which a multimillionaire comedian excreted a synthetic variant of sisterhood.
She saw an episode of _My 600-lb Life_ , in which a multimillionaire doctor ritualistically abused poor people who'd destroyed their bodies with a toxic diet of repressed homosexuality, junk food, and prescription painkillers.
She watched CNN, MSNBC, and Fox News, which were 24-hour news channels dedicated to obsessive, and non-stop, coverage of Donald J. Trump.
These television networks were watched by the elderly and the insane.
These networks served a valuable social function.
They were voluntary euthanasia through informational poison.
Celia shut off the television.
She wanted to go home.
The next day, Celia stood in the living room of the house on the hill.
She looked out over the infinite vastness of Los Angeles.
She cast a spell.
It was some bullshit magic that was intended to solve an intra-narrative problem while moving forward the storytelling.
The spell was supposed to create a direct line of smartphone navigation to Fern. It was supposed to be another bullshit tendril of ropey saliva.
But Celia's spell did nothing.
It fizzled.
Here is why Celia's spell fizzled: Fern was nobody's fool.
Fern knew that her mother would try to find her.
Months before Celia took possession of the house on the hill, Fern had cast her own spell, which blocked any attempts to establish a ropey strand of smartphone navigation.
As Celia's spell fizzled, Rose Byrne watched from the alpine-blue couch. She looked like a teenager who's been told by her parents that the whole family is going on a sea cruise themed around an intellectual property geared towards children.
Celia tried to recast her magical bullshit spell.
It fizzled for a second time.
The two women from Fairy Land conferenced as to what was wrong.
Neither of them suspected Fern of blocking Celia's spells.
Rose Byrne said that perhaps Fern was no longer in Los Angeles, but it was pointed out that this wouldn't block the ropey smartphone navigation.
Besides, Celia could sense Fern's presence in Los Angeles. It was one of those fucked-up faery things, just a green feeling that her daughter was present in the same rough geographical locale.
Rose Byrne suggested that as they were in the United States, they could emulate the practices of the American security apparatus.
She proposed that they track where Fern had spent her money and then triangulate her location based on clusters of purchases in a localized region.
Celia cast a spell.
It did nothing.
Fern was from Fairy Land.
She was using an older, weirder form of magical bullshit than money.
Rose Byrne suggested summoning Rusticano.
But no one wanted that.
The women of Fairy Land were stumped.
Then Celia remembered something Maeveen Licksweet had told her.
There'd been a period, back in the Nineteenth Century AD, when Maeveen Licksweet had spent a great deal of time away from Fairy Land. She'd traveled around the world for reasons that she never shared with anyone.
But she did talk about something that she'd noticed in Udine, where she'd spent three weeks.
Maeveen's landlady in Udine was a widow who'd convinced herself that whenever she slept, she went on a spiritual journey into barren fields where she did battle with witches.
In her dreams, the widow would beat the witches with bundles of fennel and the witches would beat the widow with stalks of sorghum.
One day, after Maeveen returned to her lodgings, the widow asked if Maeveen's room had been painted.
_Of course not_ , said Maeveen. _Why would I paint a room? And what is paint, really?_
_Then why is the room the color of wolves?_ asked the widow.
Maeveen thought this was more witch nonsense, but she followed the landlady into the room.
At first, Maeveen couldn't see what the widow was talking about. But then she caught it out of the corner of her eye. A faint glow permeated everything.
If Maeveen acknowledged the glow, the widow would chatter on for ages about the color of wolves.
Maeveen cast a spell that messed up her landlady's mind.
The widow shut the fuck up.
The rest of Maeveen's time in Udine was quiet.
As Maeveen traveled throughout the Italian peninsula, she kept looking out of the corner of her eye. In each of her quarters, in each new city, the glow appeared after she'd been in residence for roughly a week.
Maeveen spent some time thinking about the glow's cause.
She realized that it was herself, in her magical puissance, having an effect on her lodgings.
It was a byproduct of being a citizen of Fairy Land in the mortal world.
After Maeveen reported this story to the women of Fairy Land, the few who did leave the island noticed that they too had the same effect on their lodgings.
Celia recalled Maeveen's story and realized that although she was unable to find Fern, she could seek out the radiation traces of her daughter's puissance.
Celia cast a spell, with as broad a mandate as possible, to look for sources of preternatural power in Los Angeles.
But Los Angeles was as bad as Fairy Land.
It was full of magical bullshit.
It had been built on magical bullshit.
It was nothing but magical bullshit.
About fifty ropes of smartphone navigation saliva emerged from the living room of the house on the hill and stretched out into Los Angeles County.
"We have little choice," said Celia. "We shall follow each until we find the one that brings us to Fern."
Two practical matters arose.
Celia pointed out that their clothes, the haute couture of Fairy Land, were going to attract attention.
She cast a spell.
Celia wasn't well versed enough in contemporary American fashion to pick clothes, so she let the magic do the work of a personal stylist.
The magic made the women look like recent transplants to Echo Park, which was a traditionally Latino neighborhood that had gentrified into a fashionable enclave of upscale dining and high-level annoyance.
The women's fur-clad haute couture transformed into designer denim, vintage metal T-shirts, Balenciaga sneakers, and Marni handbags.
Rose Byrne's T-shirt said: EMPEROR.
Celia's T-shirt said: SAVATAGE.
Neither of the women knew it, but the magic had failed in its job as a personal stylist.
Vintage metal T-shirts were the hot look of the previous summer.
The other practical matter was one of transportation.
Los Angeles was too big for the women to walk, and the smart-phone saliva didn't interface with magic windows, so teleportation was prevented.
Celia remembered the former Francis Fuller's vintage black Jaguar XJ-S, which was parked in the driveway.
The women went outside and looked at the car.
Neither of them knew how to drive.
Celia suggested that she cast a bullshit spell of knowledge which would teach Rose Byrne how to drive.
For the first time in her life, Rose Byrne was about to find a natural place for her ingrained psychosis. She had become a driver in the hellscape of Los Angeles, just another murderous freak steering several thousand pounds of death machine.
Celia got in on the passenger's side.
Rose Byrne got behind the wheel.
Her bullshit magical training took over. Her psychosis flowed into the machine and then back into her own body. She was ready.
She backed out of the driveway.
She followed one of the ropey strands of smartphone navigation.
Celia fiddled with the Jaguar's radio until sound came through the car's paltry speakers. The radio was tuned to 89.9FM, KCRW, one of Los Angeles County's several stations affiliated with National Public Radio.
National Public Radio was, in part, radio sponsored by the American state. It was a relic of another era, which is to say the mid-1960s AD, when there was still currency in the idea that civic institutions could serve, and enrich, the lives of the citizenry.
What a jest!
What a jape!
KCRW was broadcasting the afternoon NPR news show, which was called _All Things Considered_.
As Rose Byrne followed her saliva-based smartphone navigation, the women heard the stories of the day.
The lead story was of some interest to both Celia and Rose Byrne, as it was about a recent Islamic-themed terror attack on London Bridge.
Like all terror attacks, the London Bridge incident had evoked a general aura of stupidity, and like all terror attacks in London, it had produced a plethora of people with silly accents waiting to give interviews to the vultures called reporters.
"Oi, guv, I tell you what, terrorism is bad stuff, innit, hey, guvvy?" said KCRW.
NPR dedicated thirty seconds to the importance of Donald J. Trump's tweet about the terror attack. He'd insulted the Mayor of London.
What an asshole.
Rose Byrne drove the Jaguar into Hollywood.
## Chapter Eleven
## **Let Slip the Dogs of War**
The only good advice that anyone ever gave me about writing came from the author Stephen Prothero.
He said something like this: "If there's an obvious comment about your book, don't run from it. Just include the comment in the book itself and make it part of the text. Get there first."
In the spirit of those words, let's address the big fat elephant in the room.
Let's talk about how you can't write a novel about an island of women who banish and murder all of their male co-citizens and not have everyone think that you're writing an allegory about #MeToo.
#MeToo was a hashtag.
Hashtags were a method for a bunch of people on social media to comment on the same topic, roughly at the same time.
You took an alphanumeric phrase and put the # symbol in front of that phrase and appended the phrase to a comment on social media.
#FuckTrump was a popular hashtag.
So was #NotMyPresident.
This book is not an allegory.
It was begun in August of 2017 AD.
#MeToo didn't start until October of 2017 AD.
The first 12,000 words of this book were written before October of 2017 AD.
#MeToo kicked off with an article in the _New York Times_ and a follow-up in the _New Yorker_. Both articles were about a film producer named Harvey Weinstein.
He had produced nearly every middlebrow American film of the last twenty years, he was a bully, he was a braggart, he was physically repulsive, and he was in deep with the Democratic Party.
And he was also a serial sexual abuser of women and a rapist.
With every news story there is a visible layer, the one that plays out in media coverage, and then there is an unconscious layer, the story serving as a medium through which unspoken social undercurrents are made manifest.
And the unconscious layer of the Harvey Weinstein story was all about Donald J. Trump.
They were both disgusting fat slobs from New York City, they were both from the Celebrity branch of American governance, they were both deep into politics, and it was a barely kept secret that both of them were pigs with women.
Had Donald J. Trump not won the election, #MeToo would not have happened.
The psyche of the haute bourgeoisie would not have bruised.
There would have been no waves of outrage.
And no one would have scrutinized Harvey Weinstein, who had decades of extraordinary access to Donald J. Trump's opponent.
He would have been on the winning side.
And everyone always falls in line behind a winner.
The election of 2016 AD produced a problem: Donald J. Trump had both won and lost.
He was a beneficiary of the Electoral College, which was a system of proportional representation designed by America's founders to ensure that no one would ever outlaw owning slaves from Africa.
The Electoral College didn't stop America from outlawing slavery, but it did seriously screw up the Twenty-First Century AD.
Here's how the Electoral College worked: the general election, in which the will of the people was expressed, meant nothing.
A candidate could win a majority of votes and still lose the election.
This is exactly what happened in 2016 AD.
Donald J. Trump lost the popular vote and won the Electoral College.
Millions more people voted for Donald J. Trump's opponent than voted for Donald J. Trump. Way more Americans had decided that his opponent was the appropriate person to turn Muslims into garam masala.
Which made sense.
Under the previous President, Donald J. Trump's opponent had been the Secretary of State, which meant that she'd been intimately, and professionally, involved with the obliteration of Muslims.
And say what you will of Donald J. Trump, but for all of the endless accusations hurled in his direction during the Year of the Misplaced Butter, no one ever suggested that he'd killed a Muslim.
Experience matters!
It wasn't as if Donald J. Trump's victory was unprecedented. Recent history had contained another split between the Electoral College and the popular vote.
2000 AD!
Everyone forgot.
But not me.
Here are three emails between me and a woman who shall remain nameless:
**Sat, Nov 5, 2016 at 11:26 AM**
**From: XXXXX (xxxx@xxxx.com)**
**To: Jarett Kobek**
**Subject: quick**
election prediction in 1, 2, go!
**Sat, Nov 5, 2016 at 12:42 PM**
**From: Jarett Kobek**
**To: XXXXX (xxxx@xxxx.com)**
**Subject: Re: quick**
TRUMP
possible popular/electoral split
**Sat, Nov 5, 2016 at 1:26 PM**
**From: XXXXX (xxxx@xxxx.com)**
**To: Jarett Kobek**
**Subject: Re: quick**
Omg not again. Not again. I cannot take another popular/electoral split. I will lose my goddamned mind.
She didn't lose her goddamned mind.
But everyone else did.
The Weinstein story exploded and went metastatic in a way that stories don't go in an era of media fragmentation and a politically divided citizenry.
It was all-consuming, a black hole at the center of a depraved galaxy.
It opened two floodgates.
The first floodgate had held back a torrent of stories about men who worked in the Celebrity branch of American governance and their proclivities towards sexual assault.
The second floodgate was ancient magic.
It'd been there for a very long time, holding back all of women's awful experiences with men from the dawn of civilization.
And now it was open.
There was an organic outpouring of stories.
These appeared on social media under the hashtag of #MeToo.
Women wrote about being sexually harassed, about being raped, about being treated like idiots. It amounted to a profound discomfort with the way that sexual politics worked in the post-industrial civilized world.
And let's be clear.
Whatever the merit of any individual statement, the general intent of #MeToo was undeniable. It was people saying that a society built around the whims of men is a recipe for a disaster.
And if you disagree with that, go and look out the fucking window.
Or inside your smartphone.
And, reader, don't mistake me for one of your dopey male acquaintances who, after #MeToo broke, went and posted statements on Facebook about how they were learning to be better people, when all they were really saying was this: _Please don't get me fired because I tried to fuck you when I was drunk at the office holiday party._
I wrote an entire book about the horror of a society built around the whims of men, and I did it long before there was any obvious reward for performing this particular piety.
It's called _I Hate the Internet_.
It made me famous in Serbia.
Serbia!
Despite its obvious virtues, #MeToo demonstrated why the Twenty-First Century AD may preclude the possibility of meaningful political protest.
In August of 2017 AD, Donald J. Trump returned to Trump Tower, which was a giant golden skyscraper that he'd built over Manhattan's Fifth Avenue.
This was where Donald J. Trump had lived before he moved into the White House.
This was where he had staged his bid for the Presidency.
He hadn't been back since he'd become President and earned the right to bomb the living fuck shit out of Muslim peasants in the name of American freedom.
A few days before his return, there'd been a White Supremacist rally in Virginia where a young woman was killed when a Neo-Nazi drove his car into a crowd of counter-protestors.
On the very same day as Donald J. Trump's return, I happened to be staying on the eleventh floor of the Warwick Hotel, which is about two blocks south and one block west from Trump Tower.
From my hotel room, I could hear the protests outside of Donald J. Trump's former home.
I walked over to Trump Tower, where the NYPD had blocked off Fifth Avenue.
Donald J. Trump still hadn't arrived.
Like the novel _The Life and Opinions of Tristram Shandy_ , Trump Tower was empty of its eponymous hero.
About two thousand people were barricaded on both the west and east sidewalks.
People were holding signs.
People were wearing T-shirts relevant to their political protests.
People were using their cellphones to record video of the protests.
People were chanting.
They were screaming: THIS IS WHAT DEMOCRACY LOOKS LIKE!
When they screamed THIS IS WHAT DEMOCRACY LOOKS LIKE, I think what they meant was this: _Donald J. Trump, here is the face of the American public, and we oppose you in all of your manifoldperversions. We repudiate you in your evil. A change is gonna come, Bubba_.
The scene was straight out of Tolkien.
A few thousand people, restrained by the Orcish Host of the NYPD, had been corralled into pre-approved places from where they shouted impotent chants at an impregnable empty golden tower.
The protestors were right.
It really was what democracy looked like.
In an era when significant amounts of social protest occurs on the Internet, it necessarily means that all of that social protest is monetized.
And not by the protestors.
#MeToo generated unbelievable amounts of web traffic.
For months, it was an international spectator sport.
Almost every time that someone interacted with #MeToo, they were generating income for Facebook or Google or Twitter, which were the three companies that dominated advertising and political expression on the Internet.
Here's a list of the major institutional holders of Facebook, circa September 2017 AD: The Vanguard Group, BlackRock, Fidelity Investments, State Street Corporation, T. Rowe Price Associates, Capital World Investors (a subsidiary of Capital Group), Northern Trust, Morgan Stanley, Invesco, Geode Capital Management.
Together, these ten companies owned just over 31 per cent of Facebook.
Here's a list of the major institutional holders of Google, circa September 2017 AD: The Vanguard Group, BlackRock, Fidelity Investments, State Street Corporation, T. Rowe Price Associates, Capital Research Global Investors (a subsidiary of Capital Group), Capital World Investors (a subsidiary of Capital Group), Northern Trust, BNY Mellon, Wellington Management.
Together, these ten companies owned just over 31 per cent of Google.
Here's a list of the major institutional holders of Twitter, circa September 2017 AD: The Vanguard Group, ClearBridge Investments, BlackRock, Morgan Stanley, Slate Path Capital, State Street Corporation, OppenheimerFunds, Coatue Management, First Trust, Northern Trust.
Together, these ten companies owned just over 27 per cent of Twitter.
With one exception, none of these institutional holders was operated in any meaningful sense by anyone other than some old white guys in suits.
And the job of these white guys in suits was to make money for the people who owned everything.
In the case of the one institutional holder that was run by a woman, the woman in question had inherited the company from her father.
This literally was the Patriarchy.
And #MeToo had made them, and their clients, a huge amount of money.
The general consensus of opinion was that Twitter, more than any other company headquartered in and around the San Francisco Bay Area, had destroyed America.
It had turned everyone into kindergarteners, it had murdered journalism, and it had almost certainly helped Donald J. Trump get elected.
In the seven years following its initial public offering in 2011 AD, Twitter had never made a dime. It lost money for twenty-seven straight quarters.
Yet when it posted its results for the fourth quarter of 2017 AD, which was the time period encompassing the Weinstein revelations and the subsequent social fallout, Twitter revealed that in the final three months of the year, the company had made $91,000,000.
It was a #MeToo miracle!
And it couldn't have happened to a nicer group of men!
In the early days, it felt as if the organic uprising of women was going to be the main story. It was one of those rare moments of social openness where the rules are up for grabs.
Anything could happen.
But this was America.
#MeToo became the same story as every story in America: a nexus of how power and money played out amongst the Celebrity branch of American governance.
The revised story fixated on the three industries that were the locus of Donald J. Trump's power: the entertainment industry, journalism, and politics.
The organic outcry was lost amidst stories of the appalling behavior of certain men with professional careers in the public sphere. These stories tended to run the gamut: they went from unfortunate comments to groping to flat-out rape.
A handful of the stories weren't even about sexual harassment.
They were about consensual relationships with deeply unsavory people, which had been recontextualized after the #MeToo moment.
Literally every woman alive who'd engaged in the biological imperative of sex with men had undergone the routine humiliation of consensual sex with at least one deeply unsavory person.
This was the bullshit con of heterosexuality.
But most of those women, who were poor and didn't work in media, weren't given the opportunity to write opinion pieces for _Variety_ about their shitty ex-boyfriends and old lovers.
Their shitty ex-boyfriends and old lovers weren't members of the Celebrity branch of American governance.
The unspoken social undercurrent of the revised story revealed itself.
#MeToo became about the way in which encounters with men had stymied the ambitions of women who had wanted to achieve upward social mobility in the industries that were the nexus of Donald J. Trump's power.
Which, look, by itself this was no small problem.
But it's a very far cry from what kicked the whole thing off, which was a story about a serial rapist who actively worked to destroy people after he raped them.
All of which creates an atmosphere that makes it very fucking hard to write a book about an island of women and not have everyone think you're allegorizing a hashtag.
The whole thing's ruined before it even started!
And, reader, trust me, I can imagine the responses to this chapter before they're typed by dullards into social media, and they all boil down to something like this: "Who the fuck does this guy think that he is?"
To which I reply in advance: on the topic of #MeToo, I have more innate moral authority than most people in America.
And this isn't because of inborn privilege.
There's a simple explanation as to why I have innate moral authority on the topic.
I'm almost certainly the only person alive who was sexually harassed in front of a crowd of 280 people by a woman who pens _New York Times_ opinion pieces about sexual harassment, and I'm absolutely certain that I'm the only person alive who experienced this sexual harassment several years after winning a $1.2-million judgment in a lawsuit against an Internet stalker who libeled me as a rapist.
## Chapter Twelve
## hello from sex drenched hollywood
Smartphone saliva brought the Jaguar XJ-S to Hollywood, a neighborhood that was being victimized by the international capitalist class's money laundering.
The money laundering took the form of cruddy new apartment buildings and ugly hotels.
Hollywood was also a neighborhood that had become a hotspot of nightclubs, places where people went to dance, get high, and challenge the received sexual wisdom of the upper middle class.
Several blocks before their arrival, the women of Fairy Land knew their final destination.
They knew this because the navigation rope had wrapped itself around its target, which was the thirteen-story Fontenoy Apartments on Whitley Avenue.
From a distance of several blocks, the women of Fairy Land could see the building glowing.
The Fontenoy was an early Los Angeles folly, from back in the 1920s AD, dressed up in nouveau-riche ornamentation and a French-Norman roof.
When they arrived on Whitley Avenue, Rose Byrne parked the Jaguar in the Fontenoy's underground parking structure, right after Rose Byrne used magic to blast open the structure's automatic gate.
She took a parking spot that was reserved for someone on the tenth floor.
Celia cast a spell on the car, creating a glamor that caused human beings' eyes to malfunction.
When human beings looked at the Jaguar, they didn't see a vintage car designed by the British.
They saw a series of orange construction cones and were surprised by neither the appearance of the cones nor the implication that a parking spot, an inert piece of concrete demarcated by lines of paint, was out of order.
_Oh_ , they thought. _Here's something else that's broken._
Everyone in America possessed an unconscious, and sometimes conscious, acknowledgement that their empire was in decline.
But gone were the halcyon days when one could expect the whole thing to end through an invasion of the Mongols or the Ottomans or the Huns.
Gone were the sweet moments when barbarian hordes would pull down the walls of your capital city and murder all of your cousins.
Now an empire died of a thousand tiny wounds.
Postal carriers stopped delivering mail.
Air travel became a horror.
Infrastructure went to shit.
Trains crashed.
And parking spots went out of order.
Because the women of Fairy Land were traditionalists, they didn't ride the elevator from the basement, but rather walked out of the underground parking structure.
They emerged back on Whitley Avenue.
Several years earlier, the faceless entity that owned the Fontenoy had installed a security gate. Rose Byrne blasted it open with magic and then did the same thing with the front door, which was also locked.
In the lobby, they passed through a small room that looked like a bordello, and walked to the elevators opposite the front entrance.
Celia pressed the call button.
The doors opened.
They got into the elevator.
Ropey strands of salvia could bring Fairy Land's women to a generalized magical destination, but it could not indicate why that destination was magical or what they should do when they got there.
Given that this chapter occurs at this book's rough three-fifths mark, it's pretty obvious that Fern isn't in the Fontenoy.
Neither of the women know that.
Which is shameful ignorance and demonstrates the limits of their preternatural powers.
If the women of Fairy Land are really supranatural beings, unbound by the laws of nature and capable of casting spells that alleviate issues of plotting and characterization, you'd think that they'd have the resources to check the page number.
Anyway.
They're in the elevator and they're looking at the buttons which lead to the Fontenoy's other twelve floors. If Fern is in the building, they have no idea what floor she's occupying.
The women of Fairy Land don't have any choice.
They're going to have to explore each apartment in the building, one by one, until they can determine whether or not Fern is present within the structure.
Which she obviously isn't, if for no other reason than the fact that this chapter, like almost every chapter in this book, isn't really about anyone finding Fern. This chapter is a poorly fleshed-out fictional pretense to write about something that isn't fictitious.
This is, after all, a novel written in an era when the entire purpose of fiction has been outmoded and destroyed by vast social changes.
Another thing that the women of Fairy Land don't know is that they're in the most magical place in Los Angeles.
The Fontenoy is where the American Twenty-First Century AD was invented.
They started on the second floor, bursting into the apartment nearest the elevator.
No one was home, but Rose Byrne did have an interesting conversation with a yellow parakeet.
They burst into the next apartment, where three young men were smoking marijuana and watching television.
In anticipation of the Season Seven premiere of _Game of Thrones_ , the three young men had entered into a covenant.
After the June 26th, 2016 AD finale of Season Six, each of the young men had gone to the source material and read every published volume of George R.R. Martin's magnum opus.
1,736,054 words of pure shit!
But reading the books had not slaked their thirst, and in anticipation of the approaching Season Seven premiere on July 16th, 2017 AD, the young men had agreed to spend the summer rewatching every extant episode of the televised adaptation.
As the women of Fairy Land burst into the apartment, the young men were watching the eighth episode of Season Three.
The television was displaying a scene in which the mad dwarf Tyrion Lannister is in a boudoir with his unwilling wife Sansa Stark. They've just been married and Tyrion's father has ordered Tyrion to break his bride's maidenhead.
The dwarf, in anticipation of this horror, has gotten ridiculously drunk.
With great reluctance, his bride sheds her clothing.
He stops her. If she does not want to sleep with him, he shall never force her.
Then the dwarf passes out.
This scene is of some interest because both the televised adaptation, and its source material, feature a character who's drunk himself silly and refuses to sleep with someone on moral grounds, rather than the obvious explanation of too much alcohol rendering him unfit for the congruous act.
This scene, in both book and television formats, points to the place where George R.R. Martin's _Game of Thrones_ is a divergent universe from the one in which we live.
It ain't the elves.
It ain't the fucking dragons.
It ain't the kid who can see the future.
It ain't the snow zombies who function as an obvious insult to the people of Scotland.
What makes _Game of Thrones_ diverge from our universe is one very special magical rule.
This is the magical rule which creates the divergent universe: no male character in _Games of Thrones_ ever experiences erectile dysfunction.
The three young men were stoned enough that they all imagined someone had left the front door unlocked. They thought that the women of Fairy Land had the wrong apartment.
One of the young men chatted up Celia while Rose Byrne demanded to know about Fern.
Another of the young men made a joke about the absurdity of inquiring about ferns when clearly another green plant was the apartment's dominant spirit animal.
The women of Fairy Land didn't get the joke.
And this wasn't because the language of the joke was slightly mixed in its metaphors.
The joke was like all jokes about marijuana.
Terminally unfunny.
On it went, apartment by apartment, floor by floor.
They burst into apartment #403 and found a woman named Ashley Lopez sitting on her living-room floor.
She was practicing Transcendental Meditation, a technique in which the practitioner repeated a mantra, in the silence of their own mind, after having blown about $1,000 on a seven-day course to learn an easy trick that any old asshole can Google in about five minutes.
With her mindfulness practice disrupted, Ashley opened her eyes and saw the women standing over her.
She didn't question their presence.
It was one of those faery things, a biochemical process. The supranatural entities were emitting pheromones that calmed the human psyche.
"Can I help you?" asked Ashley Lopez.
"We are looking for my daughter," said Celia. "Have you seen her?"
"What's her name?" asked Ashley Lopez.
"Her name is Fern," said Celia.
Rose Byrne looked at the decorations on Ashley Lopez's living-room walls.
It was some witchy nonsense: a reproduction of The Tower from the Thoth tarot, the hieroglyphic monad of John Dee, a banker's cheque endorsed by Austin Osman Spare, a Stele of Revealing, a mural of Tiamat, a painting by Steffi Grant, the logo of the Builders of Adytum, and other bullshit.
Ashley Lopez was locked into a ceremonial magick groove.
Ashley still believed in things like gods and primal magic and art nouveau and the manifestations of expression that dominated human consciousness before the psychic cataclysm of World War One.
What can you do?
Everyone's got something.
Ashley Lopez was confronted by the women of Fairy Land, who were actual magic.
All of her ceremonial magick was of no use.
On those lonely evenings when Ashley Lopez crossed the Abyss and went on the Dark Pilgrimage to Chorazin, the whole thing was about psychological insight into her own self and the limits of identity.
Which was a real change from the old days.
In the old days, magick used to be goofy shit like necromancy, which was the art of raising the dead, and demonology, which was the art of making the Spirits of Hell do your bidding.
The defining aspect of demonology was the bathetic juxtaposition of its methods and its aims.
The Spirits of Hell, who were supranatural beings capable of unimaginable feats, were summoned by the demonologist and asked to perform silly little tasks like facilitating intercontinental travel, or making another person have sex with the demonologist, or causing the reputation of a demonologist's enemy to suffer grievous ruin.
By the Year of the Froward Worm, no one needed the Spirits of Hell to help them travel to Asia or get fucked or ruin an enemy.
Now people just owned smartphones.
Ashley Lopez's tenancy in the Fontenoy was foreordained by a lifetime of practicing ceremonial magick.
In addition to challenging the limits of her identity, the ceremonies had blasted open her seven chakras and made her susceptible to the unseen but very real magical currents running throughout Los Angeles.
When she signed her lease, it was like a magnet being drawn to metal.
The Fontenoy was the most magical place in Los Angeles.
Way back in 1989 AD, a young man had moved onto the ninth floor of the building.
He was, just, like you know, this guy.
His name was Matt Drudge.
He'd been raised around Washington DC, which was the capital city of the United States of America, and that proximity gave him a fixation on the currents of power.
He bummed around Hollywood for about half a decade. And this was the old Hollywood, the Hollywood of the Yucca Corridor, the Hollywood that existed prior to the infestation of the international capital class's money laundering.
It was gang territory. It was full of drug dealing. It was full of prostitution.
In 1994 AD, Drudge's father paid him a visit.
He was appalled by his son's life.
At the time, Drudge was selling T-shirts at CBS Studios in Century City, which was on the other side of the hills that hold the HOLLYWOOD sign.
The old man bestowed a gift upon his son from Circuit City on Sunset Boulevard: an IBM PC compatible computer.
This was before the release of Microsoft's Windows 95 destroyed the American West Coast, another psychic cataclysm, and oddly, one that's never been written about in any meaningful detail.
Drudge's computer had a modem, which was a stupid little device that connected to telephone lines and allowed his computer to call up other computers.
Using his modem, Matt Drudge discovered the Internet. And this was the old Internet, the Internet of Usenet and #hack on EFnet, the Internet that existed prior to the infestation of the international capital class's money laundering.
Drudge's first utterance on the Internet, ever, was three days after Christmas 1994 AD at 1:48PM.
It said:
**hello from sex drenched hollywood**
Drudge replied to himself at 3:31PM. His response said:
**we are so sex drenched here in hollywood. 65% of us city dwellers have herpes**
And so, on a cloudy Wednesday afternoon, on the ninth floor of the Fontenoy, the Twenty-First Century AD was born.
Ashley Lopez had lived in the Fontenoy for five years, performing ceremonial magick and using all kinds of magickal phrases, and she'd never said anything with as much power as the one phrase which had baptized a century.
She'd never said anything as important, or as ominous, as hello from sex drenched hollywood.
No one could have known that Matt Drudge was the only authentic genius of the Twenty-First Century AD.
He was the only person in the world who understood how the Internet really worked.
And he had found his demon.
Not long after he'd written about 65 per cent of people in Hollywood having herpes, Drudge founded an email newsletter obsessed with the currents of power in American life.
The newsletter was about the entertainment industry and politics, which, by virtue of the Celebrity branch of American governance, were the same thing.
The newsletter was called the _Drudge Report_.
It offered its readers a very gossip-inflected take on the issues of the day.
Everything broke in 1998 AD.
_Newsweek_ , which was a magazine that offered milquetoast political and cultural reporting, decided not to run a story about an alleged affair between the sitting President, William Jefferson Clinton, and a twenty-two-year-old White House intern named Monica Lewinsky.
Drudge learned about the spiked story and sent word to his mailing list.
He didn't know it, but he'd murdered the gentleman's agreement between news journalists and politicians, which was more or less a tacit acknowledgement that politicians could fuck around in private as long as Washington bureau chiefs were invited to dinner parties in Georgetown.
And Drudge had, accidentally, trashed the American idea of good governance, fostering an environment in which the Republicans would go on to impeach William Jefferson Clinton, and learn that the way to power was through publicity stunts and using the Legislative branch not to govern but rather to obstruct.
After the Lewinsky thing, Drudge's fame went nuclear, went global.
He got a short-lived TV show. He got a radio show.
His newsletter evolved into a webpage that collated links to articles on other websites, and, on occasion, featured some of Drudge's own reporting and, in times of emergency, an animated siren GIF.
The links to other websites were written by Drudge himself in an ultra-minimalist headline style. hello from sex drenched hollywood.
The webpage was three columns of black text on a white background.
There was no flash and no glut.
The design never changed.
Not once in two decades.
It was perfect in the way that Steve Jobs, a psychopath who enslaved Chinese children and made them build electronic devices which allowed American liberals to write treatises on human rights, had envisioned perfection: the absolute and seamless melding of form and function.
By the Year of the Froward Worm, Drudge's website received ten billion visits per year.
In the late 1990s AD, there was an unbelievable amount of bullshit about how the Internet was going to offer new platforms of expression that leveled the playing field, and how computers would produce an enormous flowering of creativity and new opportunities.
What no one admitted, or perhaps even realized, was that while the Internet would indeed create a million opportunities for people to express their ignorant-ass opinions on topics about which they knew nothing, those opinions would not offer any real benefit to the ignorant-ass people who offered them.
The ignorant-ass opinions would only enrich the people who owned the platforms of expression.
And the people who owned the platforms of expression were the same old shits who ruled the world.
Here was the genius of Drudge laid bare: he understood, before anyone else, that the way to make money on the Internet was by monetizing other people's content.
After Drudge shattered journalism, the international capitalist class gathered up the fragments and ground them into dust.
The noble profession transformed from attempts at a first draft of history into a quest for eyeballs on websites.
In the process, seasoned professionals lost their jobs and were replaced with cocaine-addled children from Brooklyn who worked for spare change.
The international capitalist class didn't care.
Journalism had always been a pain in their ass.
What they wanted was traffic on the websites that they'd funded.
And Drudge drove that traffic.
Even though Drudge's website consisted almost entirely of links to other websites, it provided a coherent and linear worldview. The links were like a jigsaw puzzle. If you read Drudge for a week, you could piece together who he was and what he thought.
He made sense of an era in which the world had become incomprehensible, and when the traditional arbiters of American life had given up any hope of explaining the global situation.
His website was the Internet's unmoved mover, just about the most read news site in English, and his millions of daily readers would deluge any site that he linked.
And even more importantly, he was read by absolutely everyone who was anyone in media. He drove entire cycles with headlines that were no more than fifteen words in length.
He was literally the most powerful voice in America.
And if you think that's an exaggeration, consider this: for all of the explanations floated as to why Donald J. Trump won the Presidency with his impossible victory, no one has ever suggested the most obvious.
Which is that Donald J. Trump won the Presidency because Matt Drudge decided that Donald J. Trump should win the Presidency, and did everything he could to cast the best possible light on Trump's many missteps.
Donald J. Trump's impossible victory had come via a very small margin: 77,744 votes cast in the three states had determined the Electoral College.
0.02 per cent of the US population.
By November 6th, 2016 AD, Drudge's website received that many visitors every two and a half minutes.
If you want to know about the American Twenty-First Century AD, I recommend watching two videos.
One is available on the website of C-SPAN, which is a non-profit organization that hosts an archive of media related to the governance and affairs of public life in the United States.
The other video is on YouTube, which is an expensive attempt by Google to make copyright law irrelevant.
The first video is Matt Drudge's appearance on November 11th, 1997 AD at the Annenberg School for Communication, which was a division of the University of Southern California, an institution of higher learning that used things like a School of Communication to cloak its relationship with the military–industrial complex.
The second video is Matt Drudge's incredibly weird October 6th, 2015 AD appearance on _The Alex Jones Show_ , which was a radio program hosted by the eponymous Alex Jones, a disgraceful little man who believed that poisoned water turned frogs into homosexuals, that 9/11 was an inside job, and that clouds were made of Muslims.
The USC appearance occurred several months before _Newsweek_ and Lewinsky, which makes it a valuable document of Drudge before he broke the story that would define his life. It features Drudge on a panel with several high priests of journalism.
The first high priest is Michael Kinsley, who'd been on TV and written for the _New Republic_ , and who was the editor of Slate.com, which was a news website funded by Microsoft with money that they'd made from ruining the West Coast.
The second high priest is Todd S. Purdum, then the Los Angeles bureau chief for the _New York Times_ , which is the definitive American organ of sober judgment, good taste, and quality reporting.
By contrast, Matt Drudge was a guy with an email account.
He got his email from a company called L.A. Internet Inc.
He paid for his own Internet access.
He worked out of the ninth floor of the Fontenoy.
Everyone on the stage can't imagine that Lewinsky is coming. Both Purdum and Kinsley think that Drudge has already issued the story that will define his life.
Back on August 10th, 1997 AD, Drudge sent a report to his newsletter.
The report quoted an anonymous GOP operative who said that a Clinton aide named Sidney Blumenthal had beaten his wife.
The story was untrue.
Drudge issued a retraction.
Blumenthal sued Drudge for $30,000,000.
Prior to this incident, media coverage of Drudge had been geewhiz! articles about what he was doing, about how the Internet was really strange, and about how strange it was that Drudge was a weird person doing something strange on the Internet.
The minute after the Blumenthal thing, the knives were out.
You can see it in the video of the USC panel.
Kinsley and Purdum suggest that Drudge's methods are abhorrent, they tell him that he's a flash in the pan, they say that he's irresponsible, they repeatedly insult him to his face.
The smugness is unbearable.
It's actually shocking.
Drudge, meanwhile, defends himself to the best of his abilities and talks about his ideas of what the Internet is going to do to journalism, which is create a nation of citizens who operate the news, unfiltered and without editorial interference, and unrestrained by the social mores of the upper middle class.
When he speaks, he sounds slightly naïve and a little self-righteous. But think about this: he's a guy who makes about $3,000 a month and he's being sued for $30,000,000 by a Presidential aide. And he's on a stage where he is, by any conventional metric, seriously outclassed by his fellow panelists.
When Drudge speaks, it's clear that he's attempting to be understood.
He's a person asking to be taken seriously.
His exchanges with his fellow panelists are, effectively, Patient Zero diagnosing his own disease, and its symptoms, to aging doctors who don't read the new research.
And they hate him.
The loathing is palpable.
During the last ten minutes of the video, there's an audience Q&A.
The only question is asked by a future psychotic named Andrew Breitbart.
Breitbart would go on to be Matt Drudge's assistant, handling the afternoon shift of the _Drudge Report_.
In the Q&A, Breitbart asks why the mainstream media gave Hunter S. Thompson free reign to lie and distort the truth while not allowing Drudge any latitude in his own reporting. Breitbart suggests that this lack of latitude derives from Drudge's conservative-leaning politics.
One doesn't like to praise the devil, but this isn't the stupidest path of inquiry.
But here's the real significance: Breitbart is the only person, throughout the entire event, who doesn't insult Drudge or treat him like a child who's been caught stealing cookies.
Breitbart went on to found the Breitbart News Network, a website which by the Year of the Froward Worm had become the dominant voice of the Far Right in America.
When Breitbart died in 2012 AD, presumably from a toxic mix of being both a drug freak and a huge fucking asshole, a guy named Steve Bannon ended up in control of the Breitbart News Network.
In August of 2016 AD, he became Chief Executive Officer of Donald J. Trump's Presidential campaign.
When Trump assumed the Presidency, Bannon went to the White House.
When Blumenthal sued Drudge, Drudge didn't have any resources to mount a legal defense. He was on the wrong side of the Democrats. He was on the wrong side of the White House.
And this was before Lewinsky!
The only people who helped him, and assumed the cost of his legal liabilities, were people on the Far Right.
They did his case mostly pro bono with occasional donations from supporters.
The video of Drudge on _The Alex Jones Show_ is something else.
Before Google made a gesture towards political theater by declaring Alex Jones to be persona non grata, he filmed every episode of his radio show and put the videos on YouTube. The Drudge was no different.
Because of this, as the episode is being recorded, Drudge refuses to emerge from the shadows. He lets Jones interview him, but the image remains fixed on Jones.
Matt Drudge, the only genius of the new century, has hijacked another forum.
For the first, and only, time in the history of _The Alex Jones Show_ , Alex Jones shuts the fuck up.
Drudge talks about many of the same ideas that he expresses in the USC video, but now he's less nervous, and now he's embittered.
If, back in 1997 AD, he was Matt Drudge, who was just, like you know, this guy, now he's MATT DRUDGE, GOD OF ALL NEW MEDIA.
He's still talking about citizen reporting, but he's dispirited by the rise of the corporate groupthink and the way that it's influenced the homogeneity of the news. In a moment of sounding uncomfortably like the present author, he denounces social media.
He boasts of his independence from everyone.
And then it gets depressing.
Drudge sings the praises of Alex Jones. He sees the radio host as a lonely man who wages war against that corporate homogeneity, which is true from a certain perspective, but which ignores the true insanity of Alex Jones, a person who believes that the late singer-songwriter Jeff Buckley was a robot built by Muslims.
At first, it feels like maybe Drudge is being polite.
But then he starts throwing out his own crazy ideas.
He suggests there's a cover-up of Hillary Clinton's lovers, with the implication being that there's scores of women who've had the former Secretary of State's tongue in their birth canals.
He says that Clinton is old and sick and that there's a cover-up about her impending death.
He claims there are 80 million illegal immigrants living in the US.
Things are different than back in 1997 AD.
The coherent worldview has changed and encompassed some very dubious thoughts.
There's an edge in this interview that's nowhere to be seen in the early days.
This is a person who knows that he'll never be understood.
While Michael Kinsley sneered at Drudge for an hour in 1997 AD, he was wrapped in a delusion about the nature of his job. He thought that he was a person who offered the world a valuable service, but actually, all he did was lure people into looking at advertisements.
In the video, he can't imagine that, within about twelve years, it'll turn out that the Internet is better at advertising than newspapers, and that his colleagues in journalism, all the hallowed practitioners of the art, are going to be chasing Patient Zero's vision of the future, reducing institutions of sober judgment into op-ed factories that, try as hard as they might, will never be able to compete with the sheer entertainment psychosis of a seventeen-year-old denouncing Jews on YouTube.
Another thing that he can't imagine: by the Year of the Froward Worm, anonymous and unsupported allegations on the Internet will be the backbone of his entire industry.
And the last thing that Kinsley can't imagine is that he's insulting the one person who could have helped.
Drudge was, and is, the only person who understands the Internet.
And he was insulted so badly that he sought refuge with the scum of the world, and he took all of that genius and all of its attendant power, and he befriended the people who were nice to him.
That's how history works.
That's how politics work.
You figure out how to get along with people you find unpalatable. You figure out how to make a decent argument that convinces people who don't agree with you. You don't throw away people because you think they're powerless and worthless.
Or you end up like Michael Kinsley.
Totally forgotten and left behind.
Just a smug asshole no one remembers in a video that no one watches.
Here's a pro-tip for the Democrats.
If you want to win Presidential elections, there's a very simple thing that you can do.
It's too late to harness Matt Drudge's unbelievable influence over the national dialogue.
He's an autodidact and you insulted him.
You can't make friends now.
But you could always kill him.
Call out the Clinton death squads!
## Chapter Thirteen
## **Routine Humiliations**
To understand how I ended up being sexually harassed in front of 280 people by a woman who pens _New York Times_ opinion pieces on the topic of sexual harassment, you have to understand what my career was like before the success of my novel _I Hate the Internet_.
It was non-existent.
I'd published a novella called _ATTA_ , which was a psychedelic biography of the lead 9/11 hijacker.
It had moved a surprising amount of copies for a short work published on an independent press, and generated a great deal of secondary academic writing, but for a variety of reasons, no one noticed that any of this had happened.
After _ATTA_ came out, I worked on another book, which would eventually turn into _The Future Won't Be Long._ I wasted two years trying to get the thing published.
None of it came to anything.
When I wrote _ATTA_ , I was living in Los Angeles.
By the time that it was published in 2011 AD, I had moved to San Francisco.
In 2014 AD, I moved from San Francisco and ended up back in Los Angeles.
While I lived in San Francisco, the only positive thing that had happened, career wise, was that I ended up doing a writer's residency in rural Denmark.
This was in the summer of 2013 AD.
While I was at the residency, I met the Danish writer Dorthe Nors.
In addition to being a truly lovely person, Dorthe also happens to be one of the best writers in the world. Her books _Minna Needs Rehearsal Space_ and _Mirror, Shoulder, Signal_ are fucking intellectual masterpieces.
But she's a woman, which means that while she's become very successful, her work is always reviewed in a specific way: no one pays attention to the intellect and everyone looks for the moral instruction.
Dorthe and I became friends.
She was on the cusp of becoming a literary superstar.
In 2017 AD, she was nominated for a Man Booker International.
No one deserved the award more.
In the unique case of Dorthe, I suspend my disdain for awards.
Dorthe doesn't just deserve the Man Booker International.
She deserves every award.
She should win the Nobel Prize in Literature.
She should win Motor Trend's Car of the Year.
Bad Sex in Fiction!
As Dorthe was transforming into a superstar, she helped me out in whatever ways that she could. This is how I ended up getting an email in the summer of 2014 AD from a guy named Adrian Todd Zuniga.
Adrian Todd Zuniga is the founder and the host of a thing called Literary Death Match.
He'd met Dorthe somewhere in Europe, at one of the ten billion literary festivals that extend invitations to Dorthe.
She told him that he should have me participate in Literary Death Match.
So he reached out.
I said yes.
Saying yes to Literary Death Match was a moral compromise of the highest order.
To understand why, I need to explain the thing.
Literary Death Match works like this: four writers are given the opportunity to read their work.
Unlike normal readings, Literary Death Match happens in two rounds.
In each round, two writers perform their work, and then their work is critiqued by three judges. These judges are often celebrities.
The judges choose one writer as the victor of each round, and then the two victors face off against one another in a final round which involves a humiliating game.
Whoever demonstrates the greatest capacity for making a fool of themselves is the winner of Literary Death Match.
This is awful shit. It's the clusterfuck of debasement that has overtaken writing.
Everyone pretends that they're on the same side, everyone pretends that they're friends, and everyone makes awful pronouncements about the seriousness of their work while maintaining their aw shucks relatability, and sometimes writers are rewarded for their pomposity with badly rendered line drawings of their faces on bookshop walls.
And sometimes, if the writer is a good little boy, people will reward his pomposity with the gift of a tote bag.
Most of these tote bags have an aphorism or a logo printed on their sides.
The aphorisms and logos are always very positive about publishing.
I've never bought a tote bag in my life.
But I've still got about twenty hanging in my kitchen.
One of them says BOOKS.
I knew what Literary Death Match was.
I abhorred it.
And I still said yes.
That's how desperate I was.
Summer of 2014 AD was particularly bad.
I'd finished writing the manuscript for _I Hate the Internet_ and two things had become apparent: (1) it was the most significant piece of work that I'd done and (2) absolutely no one would publish it.
When I was offered Literary Death Match, these two things had left me beyond debased.
I was thinking, honestly, that if I won the thing, it'd at least give me another meaningless credential to put in query emails to agents who would refuse to represent my manuscript.
The iteration of Literary Death Match to which I'd been invited occurred on July 10th, 2014 AD, and it was held at Largo at Coronet on La Cienaga Boulevard.
Largo is one of those venues that people who aren't from Los Angeles can't possibly understand. It's where the Celebrity branch of American governance entertains itself in a 280-seat venue.
If your response to the existential horror of Donald J. Trump is a desire to have your liberal pieties reinforced with a joke about _Star Trek_ , then you should fly to Los Angeles and go to Largo.
The comedian Patton Oswalt will be waiting with your chuckles.
The other writers who were performing at Literary Death Match were Aimee Bender, Jay Martel, and Annabelle Gurwitch.
Jay Martel was the producer of _Key & Peele_, which was a popular sketch comedy show in which two African-American actors who'd grown up as members of the middle classes performed skits based around the hilarity inherent in the accents of poor African-Americans.
Annabelle Gurwitch was an actress who'd found some success as a writer of books about her sex drive as she approached the age of fifty.
Aimee Bender was a literary writer. She taught creative writing at the University of Southern California, and was director of that university's Creative Writing PhD program.
I've never read her work, but my friend Dean Smith was in the audience at Largo with his boyfriend Mike Kitchell, and Dean Smith said that he'd read Aimee Bender's book _The Particular Sadness of Lemon Cake_.
The judges at Literary Death Match were Amber Tamblyn, Jody Hill, and Dana Gould.
Amber Tamblyn was an actor and a poet with several volumes of published poetry. She'd done a lot of good in the world, having convinced people to give money to the poet Diane di Prima when Diane di Prima had serious healthcare issues and needed help with the costs.
At the time of Literary Death Match, Amber Tamblyn was just coming off a starring role in Season Eleven of the sitcom _Two and a Half Men_ , which was the highest-rated television show in America.
Jody Hill was a director and writer of films and television. Through the terrible magic of Los Angeles, we'd met about eight years earlier, but neither of us could remember where.
Dana Gould was a stand-up comedian and a former writer for _The Simpsons_.
To state the bleedingly obvious: I was the freak.
Everyone else at Literary Death Match had significant amounts of money and significant amounts of success, and with the exception of Aimee Bender, all of them were representatives from the Celebrity branch of American governance.
I was poor and I wrote psychedelic biographies of Islamic-themed terrorists.
Thanks, Dorthe!
The first round of Literary Death Match was Aimee Bender versus Jay Martel.
I was in Largo's green room with Annabelle Gurwitch.
She was charming.
When Aimee Bender and Jay Martel stopped reading, the judges chimed in and offered opinions on their work. The judges ended up going with Aimee Bender.
I should say that I had never been to a Literary Death Match.
So I had no idea what the judges' critiques would be like.
I certainly wasn't expecting what I saw during the first round, which was a rah-rah all-in-together-now malice masked by a layer of bonhomie.
If you want to imagine an analogue, think about Celebrity Roasts, which are spectacles where a celebrity will attend an event that honors the celebrity by having other celebrities say cruel things about the honored celebrity.
Literary Death Match wasn't anywhere as cruel as a Celebrity Roast.
But it was the same atmosphere.
Somewhere in the middle of this, when Amber Tamblyn was talking, she mentioned that she was drunk.
The next round happened.
Annabelle Gurwitch and I had decided in the green room that she'd read first.
She did.
And then I read.
My appearance at Literary Death Match occurred after years of countless San Francisco literary readings. If I'd learned anything, it was how to work an audience.
I fucking killed.
And then it was time to hear from the judges.
Annabelle Gurwitch and I sat in chairs. Stage right.
The judges were seated stage left.
We watched as our performances were dissected with the jokey malice of a Celebrity Roast. In front of an audience of 280 people.
You'll forgive me, but I can't remember a word of what anyone said about Annabelle Gurwitch.
And you'll forgive me when I say that I can barely remember what Jody Hill and Dana Gould said about me, although I do remember that one of them talked about how innovative it was that I read my piece off an iPad.
It wasn't an iPad.
It was an Android tablet.
The last judge to comment on my piece was Amber Tamblyn.
She'd been taking notes throughout the event, and she began by reading one of her notes. This is what her note said, give or take: "This guy is wearing white pants. That's hot."
She was a person who was infinitely more successful than me, with infinitely more money. She was on America's highest rated television show. She was published by serious New York presses. And she was in a position of actual, literal judgment on my merit as a writer, and that judgment, if positive, could affect the success of my work and my future.
And she was drunk and making sexualized comments.
In front of an audience of 280 laughing people.
And all I could do was sit there, take it, and pretend to laugh.
While my friends watched.
By any conceivable metric used during #MeToo, this was sexual harassment.
But, seriously, who fucking cares?
Amber Tamblyn wasn't even the worst.
One time, I was assaulted by a rabid fan outside of the Echo Park Film Center, and another time, I received unsolicited emails from a beloved elder statesman of the literary scene fantasizing about sucking my cock.
He remains a friend.
And I know that as a society we've descended into revenge narratives in which a lesser figure remembers some stray incident from the past and uses it to attack someone who's significantly more famous.
Speak bitterness!
This is our entertainment.
And I realize how uncomfortably close this chapter reads to those narratives.
But that's not what this is.
Because here is an everlasting truth: if you get Diane di Prima money, you should be allowed to sexually harass the living shit out of everyone in the world.
If I were to give advice to anyone who wants to enter the public sphere, this is what I would say: don't.
If this theoretical person insisted on entering the public sphere, I would say: recognize the binary presentation inherent in mass media. A public figure can either be good or evil. There are no shades of gray.
So recognize this binary and do yourself a favor: do not cloak yourself in virtue.
Cloak yourself in vice.
Being cloaked in virtue creates an impossible situation: the presentation of self as infallible.
And you will fail.
And when you do, the mass media will be waiting, and the public will feast on your corpse. Nothing tastes better than false virtue.
But cloaking yourself in vice?
There's nowhere to go but up.
The early days will be difficult, but if you can last four years, you will be an unshakable fixture.
Who knows?
Do this long enough and you might become President of the United States of America!
If the Queen of England trips over a dog, it's a national scandal.
When Liam Gallagher kicks an old man down the stairs, no one even blinks.
That's Liam being Liam.
But when our kid hugs a jaundiced paraplegic?
All of that said, it remains a very peculiar experience to be sexually harassed by someone who pens opinion pieces for the _New York Times_ on the social scourge of sexual harassment.
If there's one aspect of every opinion piece on the social scourge of sexual harassment, it's that they all contain an implicit core: that there are ways to make the world a better place.
Which, of course, there are.
But when the tools used to make a better world are owned by the Patriarchy, the best outcome you're going to end up with is a discussion about the social mores in the workplaces of the haute bourgeoisie.
And, remember, that's the best case.
Here's one much worse: that, in the end, everyone's life is still dominated by the whims of the very rich and the social mores of the slightly rich. And that this new reality is exploited by the people who understand that appearances are more important than reality.
All of which is to say that by fixating on sex, the discussion around sexual harassment misses the key element.
Which is the harassment.
The people who end up in positions of power end up in those positions because they are very, very good at humiliation.
That's their skill.
That's how they end up as CEOs.
Everyone who has ever had a job has been humiliated by their boss.
This is the nature of the thing.
And, yes, it sucks that the men who end up in power are so fucking crude that the only way they can imagine humiliating women is with sex.
But every single boss who's humiliating his women underlings is also humiliating his male underlings.
This is who we, as a society, put into power.
Remind me: how many obsequious movies and books and articles have been written about Steve Jobs?
In the end, having a job, even a job like writing, is about interfacing with money, and the biggest lie of our society is that the individual currencies of money are units that measure value.
Money doesn't measure value.
Money is the measure of humiliation.
What would you do for a dollar?
What would you do for ten dollars?
What would you do for a million dollars?
What would you do for a billion dollars?
So of course Amber Tamblyn would sexually harass me at Literary Death Match.
Why wouldn't she?
She'd been put into a position of power at an event predicated on the perpetual humiliation of writers.
## Chapter Fourteen
## **When Y Meets X**
After the Fontenoy was a bust, Celia and Rose Byrne spent weeks and weeks exploring magical strands of smartphone navigation, which gave the women a decent internal map of Los Angeles and its surrounding environs.
One ropey strand of salvia took them to the Self-Realization Fellowship Lake Shrine, where they wandered around a lake decorated with religious kitsch.
Another strand took them to the site of Jack Parson's hermitage in Pasadena, where L. Ron Hubbard learned about ceremonial magick and imbued himself with the ideological basis of what would become Scientology.
Another strand, and by far the longest, took them out of Los Angeles and all the way to 274 Coast Boulevard in La Jolla, where, during World War Two, Anna Kavan had spent several months hard drinking and going ga-ga for an architect while looking at ridiculous California coastal splendor.
Another took them to 6026 Barton Ave, the address at which Samson de Brier held his cultural salons, where the former Francis Fuller had made the deal for _Handspun Roses_ several decades before Celia wiped out any memory of the film or its director.
Another took them to a lecture at the Philosophical Research Society on Los Feliz Boulevard, which had the virtue of being very close to the house on the hill.
Another took them to the Bellagio gate of Bel Air, where buses from the San Fernando Valley dropped off the permanent servant class of Latino Americans to perform domestic duties in the homes of the Celebrity branch of American governance.
Another took them to a one-room structure behind 7508 Sunset Boulevard, where the members of Guns N' Roses had lived in depravity.
And there were other places, arbitrarily chosen by authorial whims: the former St. Francis Hotel, and the shack on North Genovese where Marjorie Cameron spent the final years of her life, and the site of the former Motel Hell on Hollywood Boulevard, and the former Security Pacific National Bank Building on Hollywood Boulevard, where in the early 1980s AD a tribe of street freaks called the Night People took up residence while the bank still operated out of the bottom floor.
And there were others too.
Countless others.
But nowhere did they find Fern.
Meanwhile, the women of Fairy Land spent their evenings in bars.
They sampled places like Frank N Hanks and the HMS Bounty before settling on Tenants of Trees as their regular haunt.
Tenants of Trees was a Silverlake bar that was home to a fairly pleasant outdoor patio.
It was a human meat market filled with the sexual desperation of people who'd made the mistake of following their dreams and moving to Los Angeles.
Celia used the meat market to engage in reckless sex with some of the city's more pathetic men.
One night, Celia and Rose Byrne were sitting in an open-air room off the patio.
"I have seen too much of this mortal world," said Rose Byrne.
Rose Byrne was wearing a T-shirt that said: CRIMSON GLORY.
Celia was wearing a T-shirt that said: KING DIAMOND.
"Another drink, I think," said Celia.
Celia had cast a spell on Tenants of Trees which gave them an open and bottomless tab.
Celia made her way to the bar, passing a man and woman involved in a meat-market transaction. The transaction was comprised of monosyllables.
"That's, you know, so dumb," said the woman.
"Shit, isn't it," said the man.
"Right, don't you think?" asked the woman.
"Fuck," said the man.
"What you, like, do, I've done," said the woman.
Celia sat on a stool at the center of the bar.
The bartender, a young woman with full-sleeve tattoos, was serving other customers. She didn't see Celia.
A man on the stool to Celia's left turned his body in her direction.
"Whenever I espy a woman in licensed tour apparel, I am stricken with a fevered and paralyzing round of myxomatosis," said the man.
"I beg your pardon?" asked Celia.
"King Diamond, madame," said the man. "Your shirt. Is this not a reproduction of the _Abigail_ artwork?"
"I suppose," said Celia.
Celia had no idea about King Diamond, the eponymous vocalist of the heavy metal band King Diamond, or the band itself, or the band's 1987 AD concept album _Abigail_.
As with every other day, magic had chosen her outfit.
Through the coincidental power endemic to fiction, the man was also not wearing an outfit of his choice.
He was wearing a pair of banana-yellow shorts with a fringe trim.
And, like Celia, he was also wearing a T-shirt.
Unlike Celia, his T-shirt did not advertise a heavy metal band from the 1980s AD.
His T-shirt said this:
The man's T-shirt was very long.
That morning, with his body smarting from the previous night's Abu Ghraib-themed BDSM/taqiyya session, HRH had done a Skype interview.
The journalist was from Portland, Oregon. The interview subject was the Klaus Mann Center, a homeless shelter in Portland that HRH had opened in 2007 AD. The shelter had a specific focus on LGBTQIA+ youth.
"I believe," said HRH into a laptop that displayed the computerized face of the interviewer, "that it is our duty to protect the least fortunate of society."
"It's very unusual, though, isn't it?" asked the interview.
"I should hope that this belief is universally held," said HRH.
"You're a Saudi prince," said the interviewer.
"The royal flesh is my own," said HRH. "Yet do not forget, I am a citizen of St. Kitts and Nevis."
"I was only curious if things like the Klaus Mann Center made family reunions awkward," said the journalist.
"Whenever is a reunion of family not a-drip with awkwardness?" asked HRH.
"One last question. Why name a shelter in America after a German writer?"
"I had wished to christen the enterprise after Annemarie Schwarzen bach," said HRH. "An advisor warned me against both the length of her name and its linguistic closeness to that of film star and former California governor Arnold Schwarzenegger. In her stead, I opted for that friend of her bosom, Klaus Mann, a man whom I rate as a personal hero. He was tortured by his father. When the Nazis willed themselves to power, Klaus fled into exile, and beyond the snug confines of the Weimar Republic, he found that his fey lust for the bodies of other men caused great pain. He committed suicide. Yet I consider his life a triumph. Through the torrents of suffering, he authored several brilliant books and one unvarnished masterpiece. He inspires us all."
After the interview, HRH went to a board meeting at the Venice Beach offices of Snapchat.
Snapchat was a smartphone app that had achieved a long-standing dream of corporate America: cornering the ever elusive market of child pornography.
Following a tip received at an orgy full of unattractive men and female sex workers, all of whom were in the thrall of MDMA, HRH had gotten in on the Series A funding of Snapchat.
Snapchat was a late-period capitalist innovation: a corporation either worth nothing or everything, and one with such a complex relationship to money that it was impossible to judge the company's failure or success.
The Series A funding had earned HRH a seat on the board.
HRH arrived wearing a suit that'd been tailored in London by Gieves & Hawkes.
By the end of the board meeting, the suit was so stained that HRH had to borrow clothes from an employee of Snapchat.
"There is a curious lacuna in _Abigail_ , and one that is never revealed through the stylized vocals of King Diamond," said HRH to Celia. "Speak not of the ludicrous sequel. We are not barbarians, madame. We consider texts unburdened by _a priori_ knowledge. As King Diamond sings, we meet the ghost of Count de LaFey, and also his unfaithful wife, and their descendant Jonathan and his wife Miriam. One almost need not even mention Abigail herself. The stillborn child of de LaFey's wife, conceived in the sullen pits of adultery. Although the main thrust of the album concerns itself with Abigail's attempts to possess Miriam, represented as the symbolic transition from eighteen to nine, I remain struck by our ignorance of Abigail's father. Her sire is the one player never identified. I wonder, madame, have you any theories as to the identity of this unfortunate progenitor?"
"I beg your pardon?" asked Celia.
"Is this swine bothering you?" asked Rose Byrne.
From her bench in the open-air room, Rose Byrne had been keeping an eye on Celia.
At first, she wasn't concerned when she saw Celia talking with HRH.
She'd seen Celia speaking with a legion of meat-market men.
But then she noticed HRH's face pushing too close to Celia.
Celia was inching backwards on her stool.
Rose Byrne decided to intervene.
Her broadsword was in a scabbard.
The scabbard was hanging from her battle belt.
Throughout their journeys across Los Angeles, the broadsword had occasioned enough comment that Celia had cast a spell making the weapon invisible to mortals.
But it was always there.
HRH turned to Rose Byrne.
HRH looked Rose Byrne up and down.
"If there is any one thing that I am able to recognize within an instant, it is a servant," HRH said to Celia.
"Her name is Rose Byrne," said Celia.
"Wonderful!" cried HRH. "A dwarf with a broadsword! Straight from the pages of John Ronald Reuel! Madame, you offer no end of surprises! Where did you find such a creature? I must have her! Another blinkered specimen for my menagerie of the damned! How much must I offer to purchase this beauty?"
"We have no use for money," said Celia.
"I am not for sale," said Rose Byrne.
"Sir, I know not how it is that you see the broadsword," said Celia. "I suggest that you leave us in peace. Rose Byrne is disagreeable and her weapon was sharpened this very morn."
"You issue threats, madame?" asked HRH.
"A statement of reality," said Celia.
"Are you as unpleasant as she claims?" HRH asked Rose Byrne. "Your apple face betrays no wrath. I see only dwarven mirth. Sing me a song of the misty mountains cold!"
"I am a whirlwind," said Rose Byrne.
"If I fluster your companion any further, you will use this sword on my person?" asked HRH. "You will murder my body in Tenants of Trees?"
"Without a doubt," said Rose Byrne. "Your head will roll on the tiles."
"Wonderful!" cried HRH. "Wonderful!"
HRH jumped off his bar stool.
HRH kneeled on the ground before Rose Byrne.
HRH bent his head.
"Come now, you broken creature of Khazad-dûm! Here is my neck! Make swift with your cut. Pretend that I am the bastard offspring of Charles the First and the Great God Pan! I will be the martyr of the people! Chop, chop, cut, cut, make your haste!"
Rose Byrne's previous murders in Tenant of Trees had required a great deal of magic.
Many lives had been erased.
Celia saw no need for the bother.
She cast a spell to transport HRH out of Tenant of Trees.
But the spell fizzled.
Rose Byrne stood over HRH.
Rose Byrne ached with the ideated reality of a serial killer.
She moved her hand to her broadsword.
But she could not remove it from its scabbard.
HRH rose from the ground.
"As I imagined," HRH said to Rose Byrne. "One more disappointment in the litany that is life."
HRH sat to Celia's left.
"Sir," said Celia. "Who are you that you stayed her hand?"
"I am the alpha and the omega," said HRH.
"I beg your pardon?" asked Celia.
"I will use a metaphor that I hope gives clarity," said HRH. "Think of yourself as a being of rare luck. You sit in the presence of a superhero. Throughout the livelong day I am a mild-mannered financial wizard and neoliberal philanthropist. By night, I venture forth and make the world sane. I am like nothing you have ever met. Wait until you read my press coverage. As with the faint hopes of a penile inadequate, it is measured not in length but thickness!"
Celia stood.
"Come, Rose," said Celia. "Enough of this."
They walked out of Tenant of Trees.
HRH turned to the woman sitting at his left.
She twenty-five years old.
She was an aspiring actress.
She was from Kissimmee, Florida.
She had moved to Los Angeles to follow her dreams.
Her body was filled with the following psychoactive agents: Paxil, Lexapro, and a microdose of LSD.
"My dear," said HRH to the aspiring actress, "I wonder if you have ever perused the speeches of Cesar Chavez?"
## Chapter Fifteen
## **Until the Wheels Fall Off and Burn**
By the way, all of the women in Fairy Land, and the Fairy Knight too, had Afro-textured hair and skin loaded with eumelanin in the stratum basale of their epidermis.
When the women of Fairy Land wandered around Los Angeles in their vintage metal T-shirts, this is what people thought: _Hey, there are some Black girls_.
This was followed by another thought: _Wait, Black girls like Megadeth? How is that possible?_
And this wasn't because the people of Los Angeles were essentializing, which was a crude mental process by which inherent characteristics were attributed to an arbitrary and socially constructed grouping of humans.
The people of Los Angeles weren't having this thought because they were racist and believed it unlikely that Black girls would enjoy the sounds of Megadeth.
The people of Los Angeles were having this thought because they were shocked at the bad taste of anyone in a Megadeth T-shirt.
Megadeth were awful.
No one's accusing you, reader, of having read this book with a mental image of lily-white faeries as its main characters.
But let's be honest.
All of this book's other readers have done exactly that.
It's a cruel narrative trick that relies on ingrained cultural assumptions about mythological beings, character names with Celtic origin, and the underlying biases of fantasy literature.
But it's not as if there weren't a few clues back in Chapter Four.
Prince Thomas of the Kingdom of Purpoole clearly refers to Celia's skin as "dusky."
Also, Prince Thomas suggests that Celia's a queen of Clerkenwell and a sister of Luce.
And, as everyone knows, Black Luce or Negro Lucy was a woman of Sub-Saharan African descent who ran a brothel in Clerkenwell during the last decade of the Sixteenth Century AD and the first decade of the Seventeenth Century AD.
It's not you, reader.
It's everyone else.
But that's racial prejudice, isn't it?
And yes, reader, I understand the peril into which I've thrust myself by suggesting that Richard Johnson's made-up characters possess imaginary physical characteristics which group them into an arbitrary social construct.
Nothing could be more controversial.
Someone might get upset!
On the Internet!
Where important things happen!
No one likes to talk about it, but we live in a world where a significant proportion of the population believes that Batman is real.
Batman is a comic-book character.
Here is his origin story: he was born super rich, and his rich parents were murdered in an alley while Batman watched, and then when Batman's trust fund matured, he used the money to enact a systemized campaign of violence against the poor.
Batman goes out every night and makes the world sane.
Most of Batman's true believers don't believe in the physical reality of Batman.
It isn't that kind of belief.
It's religious.
But then again, there are always the ones who think they can talk with gods.
In 2014 AD, there was a news story about a pair of twelve-year-old girls who stabbed another twelve-year-old girl. When the girls were apprehended, they were asked why they had tried to murder their BFF. The girls told the cops that they were killing for Slender Man.
Slender Man was an imaginary supranatural character that had been created by someone on the Internet.
Slender Man wore a bad suit and he hung out with children and he inspired tedious academic papers by bottomfeeders.
When the girls were asked why Slender Man wanted them to kill, they said that Slender Man would reward their human sacrifice with a resplendent palace in Hell, where they would rule for eternity amongst the damned.
The bottomfeeders who wrote academic papers about Slender Man weren't that different from the girls who stabbed their BFF.
They were looking for tenure at state-funded universities, which meant that they too were seeking a resplendent palace in Hell, where they too would rule for eternity amongst the damned.
The media played the stabbing for its obvious shock.
Given the character's origins, which were heavily documented and easily verifiable, how could anyone think that Slender Man was real?
Psychological examinations revealed that one of the assailants was in regular telepathic communication with Mr. Spock from _Star Trek_ , all four of the Teenage Mutant Ninja Turtles, and Lord Voldemort from the _Harry Potter_ books.
The _Harry Potter_ books were a series of fantasy novels about an English boarding school, wherein the most fantastical thing that happened was the complete absence of buggery and same-sex handjobs.
Mr. Spock from _Star Trek_?
Why not?
Lord Voldemort?
All right.
But the _Teenage Mutant Ninja Turtles_?
Unless she'd somehow encountered self-published black-and-white comic books from the 1980s AD, the twelve-year-old was presumably receiving communications from the most commonly known versions of these intellectual properties.
And the commonly known versions were characterized by nothing more than their irrepressible hunger for pizza and their use of an American dialect of English that sounded like the media stereotype of California surfers.
They said shit like: "Cowabunga, dude and dudettes! I can't wait to gnosh on some gnarly pizza and get, like, weirded out! Mondo nutsiness! Time to boogie!"
Imagine that horror beamed into your fucking head.
The right question wasn't why someone would believe in the reality of Slender Man.
This was the right question: _Why wouldn't they?_
America was full of millions of people who posted to the Internet, daily, about the importance of Batman, and insisted on interpreting prevailing social trends through the prism of Batman.
These people believed in Batman, they knew that Batman was real, and they invested Batman with religious faith.
Batman was a new god.
Batman had risen from the rankest nether regions of pop culture, nurtured on the Internet after September 11th, 2001 AD, which was when a bunch of Muslims facefucked the collective psyche of mankind and transformed reality into a shitty disaster movie from the mid-1990s AD.
Life became a cartoon.
A new pantheon was required.
And there was Batman.
And there was Mr. Spock.
And there were the Teenage Mutant Ninja Turtles.
And there was Harry fucking Potter, still unbuggered, still longing for the strong and nurturing caress of a same-sex handjob.
All of these intellectual properties were no different than Slender Man.
They were just some crap that someone had made up.
And they all had definite, and well-documented, points of origin.
And this is why writers run into terrible peril when they write about supranatural characters that directly, or accidentally, touch upon hot-button issues like race or gender.
The problem is never race or gender.
That's only the smokescreen.
The problem is the supranatural creatures.
The writer risks profaning a new religion.
Like all religious people, the new religion's adherents are completely insane.
But they're not so insane that they're willing to make a direct argument about their religion.
You can't say that Batman is real.
Not in public.
Not yet.
So they grasp at the obvious.
And like any zealots, they demand obsequious gestures as retribution for the profane.
One obsequious gesture that emerged around the Year of the Froward Worm was the employment of what were termed sensitivity readers.
Authors hired sensitivity readers, who were apparently of marginalized backgrounds, to read through the authors' manuscripts and identify issues of bias or grotesque cultural misrepresentation.
Basically, it was a writer hiring someone from the Internet to tell the writer why they were wrong before other Internet people could tell the writer why they were wrong.
Imaginary narratives about fantasy worlds were being fact-checked!
By people who were about ten minutes away from making a sacrifice to Slender Man!
Like most efforts of the liberal intelligentsia to maintain plausible deniability about one's culpability in the global order of exploitation, the concept of the sensitivity reader dripped with unexamined racism.
It essentialized to an extreme degree, suggesting that there were inalienable qualities specific to arbitrary social constructs, and furthermore, that any one individual could comprehend, and identify, biases against millions of people based on nothing more than the accident of their birth.
Even the name was insane: it suggested that people from arbitrary social constructs had an innate sensitivity that differentiated them from other human beings, and that this sensitivity was based in a unique moral superiority.
And it is this thought—that the arbitrary circumstances of birth give the ability to comment on a slim range of human suffering—which has animated a central motif of the book that you are reading.
The motif in question is the idea that the purpose of the Presidency of the United States of America is the transformation of Muslims into aching piles of ash and steaming puddles of blood.
As the towelheaded son of a dirty fucking immigrant camelfucker, I've focused on the most personally applicable aspect of the American War Machine and transformed it into a reccurring joke.
Yet, reader, does not this approach suffer from the sin of narrowness?
It's not as if the American War Machine has limited itself to the execution of Muslims in the Middle East and North African region.
It's not as if the American War Machine only fucks up the relatives of people who self-identify on the Internet as #MENA.
Ever since 9/11, the American War Machine has unleashed total chaos upon the world.
By the Year of the Froward Worm, seventy-two sovereign states were involved in its conflicts.
That's 39 per cent of the world's countries.
By the Year of the Froward Worm, about 13,486,400 refugees came from five countries: Syria, Afghanistan, South Sudan, Myanmar, and Somalia.
All five of these places had been touched by the American War Machine.
Five had been fucked with by the Central Intelligence Agency, the major intelligence agency of the American War Machine.
Four had hosted members of the American War Machine's military.
Three had been bombed by the American War Machine.
Two had hosted major American War Machine military operations.
One had hosted the longest war in the history of the American War Machine.
As I write this, America wages a secret war in Sub-Saharan Africa.
According to the best available information, this secret war is taking place in the following twenty countries: Mauritania, Senegal, Mali, Liberia, Burkina Faso, Ghana, Nigeria, Chad, Cameroon, the Central African Republic, Gabon, the Democratic Republic of Congo, Burundi, Tanzania, Uganda, Kenya, Somalia, Ethiopia, Djibouti, and Botswana.
The secret war is conducted under an American combatant command named AFRICOM.
Much like the multinational conglomerate that owns Penguin Random House, AFRICOM is headquartered in Germany.
If I had to guess, I'd suggest that about 0.5 per cent of the American population knows that AFRICOM exists.
Even that estimate is wild in its optimism, as it would mean that around 1.6 million people in the United States know their country is waging a secret war against Sub-Saharan Africa.
And based on the evidence, I find this to be impossible.
Here is that evidence: if fifty people freak out on Twitter about issues of racial misrepresentation in a cultural product about supra-natural creatures, it generates coverage in the house organs of the American liberal intelligentsia.
Oh, the articles they'll write!
There is fun to be done!
There are points to be scored!
There are games to be won!
Fifty people is nothing.
Which means that the threshold for generating media interest is very low.
So if 1.6 million people know about the secret war in Sub-Saharan Africa, wouldn't this topic receive endless media coverage?
Insert your own joke here.
## Chapter Sixteen
## **Drink of Me, Eat of Me**
At the end of September in the Year of the Froward Worm, the women of Fairy Land left the house on the hill and followed the last strand.
Saliva-based navigation was responsive to changes in traffic patterns, and while the shortest route to their destination would have been to take Los Feliz Boulevard to the 5 onto the 10 and then come in through 4th Street, a traffic accident had made the 5 a complete horror show.
The saliva-based navigation directed the women on to Vermont to the 101 to the 110 and had them come in through 6th Street.
Because their destination was in Skid Row, and because they were on 6th Street, Rose Byrne drove the Jaguar XJ-S through the most abject scene of American cruelty.
What you have to realize about America is that America was a mug's game, it was a bullshit con, and nothing proved how fucked the country was more than Los Angeles's homeless population.
Official estimates in the Year of the Froward Worm, based on nothing, were 58,000 people.
Unofficial?
More like 100,000.
More people than had won Donald J. Trump his Electoral College victory!
And even that number might be low.
It was impossible to say.
There'd always been homeless people in Los Angeles, but the first decade of the Twenty-First Century AD hosted two events which pushed the situation into overdrive.
The first event was when nineteen Muslims attacked the United States with airplanes on September 11th, 2001 AD.
This provided the pretext for a series of unending wars in the Middle East.
Lots of Americans, like the former Adam Leroux, went over to foreign countries and killed a huge number of Muslims, and had the process fuck up their heads and bodies.
And unlike depictions of PTSD in cultural products made by Hollywood professionals, the consequences of this damage were more severe than flashback montages after someone mistook a traffic sign for a Shi'ite.
The other event was in 2007/8 AD, when predatory banking practices had collapsed the economy and obliterated the homes and wealth of a disproportionate number of African-American and Latino peoples.
So there were a bunch of former soldiers, who'd been given a glimpse of humanity at its worst, and as a result had been rendered unfit for society.
And there were a bunch of people without any money or homes.
And don't forget: the weather in Los Angeles was tolerable in every month of the year, which was untrue of almost every other place in the country.
And also don't forget: the Twentieth Century AD was about the ruthless exploitation of peoples' natural weaknesses for mind-altering chemicals, and this exploitation had been legitimized by every rung of society.
Despite best efforts by the money laundering of the international capitalist class, the neighborhood of Skid Row had not changed that much, and because the Jaguar XJ-S was on 6th Street, the women of Fairy Land had a perfect view.
The streets were lined with canvas tents and human bodies.
There were about fifty tents on each side of every block.
The women passed SROs, which were single-room-occupancy hotels that catered to the homeless. The women passed missions, which were Christian charities that attempted to feed and clothe the homeless. The women passed the Skid Row building of the Los Angeles Police Department, which was a state-funded apparatus that, amongst other tasks, kicked the shit out of homeless people.
What none of this conveys, really, is the squalor.
And, hey, it's mildly fucked up to write about the most destitute people in the country and say that they were living in filth.
It's not as if they don't know.
But at the same time, it's a fact.
The filth was off the charts.
There was trash and piss and shit and it was everywhere and it had been there for so long that it had changed the color of the sidewalks and the streets.
There's a way that you, reader, can measure the ways in which the American government had failed its most vulnerable citizens.
Google!
Google was a company that'd made more money off advertisements than any other company in the history of the world, but it had been founded by people who were embarrassed by a business model dependent upon advertising lawn chairs, car insurance, and Viagra.
To deflect the embarrassment, the company cloaked itself in an aura of innovation and some old bullshit about the expansion of human knowledge.
Google maintained this façade by providing web and mobile services to the masses.
The most beloved of these services was the near daily alteration of the company's logo as it appeared on the company's website.
Almost every day, the Google logo transformed into cutesy, diminutive cartoons of people who'd done something with their lives other than sell advertisements.
These cartoons were called Google Doodles.
They encompassed the whole spectrum of achievement, with a special focus on scientific achievement and the lives of minorities. In its own way, this was a perfect distillation of politics in the San Francisco Bay Area.
Whenever they appeared, the Google Doodles were beloved and celebrated in meaningless little articles on meaningless little websites.
They were not met with the obvious emotion, which would be total fucking outrage at a massive multinational corporation co-opting a wide range of human experience into an advertisement for that very same corporation.
Here was the perversity of Twenty-First-Century AD life: Native-American women had a statistically better chance of being caricatured in a Google Doodle than they did of being hired into a leadership position at Google.
And no one cared.
People were delighted!
They were being honored!
By a corporation!
But look, reader, before you assume your bien-pensant righteousness about the tech industry, let me point out that it's not as if publishing is any better.
Of the _New Yorker_ 's forty-seven issues published in the Year of the Froward Worm, ten featured a cutesy illustration of a Black woman on the cover.
By my count, and this may be low because it's impossible to verify everyone's identity, the _New Yorker_ published fourteen pieces by Black women.
If you assume an average contributor count of thirteen people per issue, then that's 611 contributors across the year, which comes out to exactly 2.29 per cent of the magazine's 2017 AD contributions being authored by Black women.
And ten out of forty-seven is 21.28 per cent.
Anyway.
In 2007 AD, Google introduced Google Street View.
Google Street View was a massive invasion of privacy.
It worked like this: Google bought cameras that could take a full 360-degree image.
Google strapped these cameras atop cars, and then hired people to drive these cars around America, while the camera took photographs every five feet. Then, using GPS geolocation, Google matched the images taken by the cameras to virtual locations on Google Maps.
You could put an address into Google Maps and see that location's real-world appearance at the exact moment when Google committed a privacy violation.
In 2014 AD, a timeline feature was introduced, which allowed the user to view the full history of Google's privacy violations.
In some places, this didn't mean anything, because Google had only sent a car out once.
In major cities, like Los Angeles, you could use the Street View timeline to look at a dense archive of imagery.
Reader, here is a game that you can play.
Go to Google Maps and search for "5th Street & Crocker Los Angeles."
Go to Street View.
Google will display its most recent invasion of privacy.
If you're savvy, you'll be able to figure out how to use the timeline.
If you aren't, ask a friend.
Go to the earliest image on the timeline, which should be from 2007 AD.
What you will see is an intersection in Skid Row.
While not in the best shape, it is not overrun with human misery.
Now move forward through the timeline.
Watch as the years pass by and watch as the human misery accumulates. Watch as the tents rise up. Watch as the suffering mounts. Watch as the bullshit con of America fails its most vulnerable citizens. Watch as liberal democracy dies.
And, yes, reader, it is sad.
And, yes, it is a shame.
But here we are.
You and me.
Or as they say in Turkish: _sen ve ben bebek._
And we're still doing nothing.
Worse than HRH!
But doing nothing is better than Google, a corporation which has decided that, facing a social cataclysm, the appropriate course of action is to violate the privacy of the homeless and then post the evidence on the Internet.
The ropey smartphone navigation directed Rose Byrne to turn left from 6th onto Stanford Avenue.
The women saw their destination before they arrived.
The strand of magical saliva was wrapped around a two-story building surrounded by single-story warehouses.
The single-story warehouses were full of companies involved in the importing, exporting, and wholesaling of seafood.
But the women didn't need navigational saliva to tell them where they were going.
There was a line of disheveled people coming out of the two-story building.
Rose Byrne found a parking spot in front of the TUNA EXPRESS CO.
The women climbed out of the Jaguar and walked over to the building wrapped in smartphone navigation.
Celia's body was resonating with a green feeling.
Fern was in the building.
And if this were a book written by someone who still had the ability to build suspense or cared about meaningful plot resolution, there'd be about three-to-four thousand words about how Celia went in the building and found Fern and discovered what Fern was doing in Los Angeles.
And it would be so dramatic.
Your heart would be in my hands.
But this book isn't being written by that kind of someone.
I'm burnt out.
Donald J. Trump was elected to the Presidency of the United States!
So there's really no point.
Stop hoping that books will save you.
Stop pretending.
Everyone else has.
You aren't getting your three-to-four thousand words.
You're getting about four hundred and fifty.
The women of Fairy Land went into the building and found Fern on the top floor.
She was bringing homeless people into a backroom.
There was a tense reunion.
Celia demanded that Fern come home.
Fern refused.
Celia asked Fern what was so important about staying in Los Angeles.
Fern brought Celia into the backroom.
Fern showed Celia what the homeless people were doing in the backroom.
They were drinking the blood of the Fairy Knight, who was sitting in a plastic chair and had a tube coming out of a vein in his left arm.
The homeless guzzled his blood from the tube.
Fern said that she had found her brother nine months earlier.
He was hopelessly insane and haunting the boardwalk at Venice Beach.
Fern used magic to bring the Fairy Knight out of his insanity.
He awoke into sanity and said that he had been wandering the world for centuries.
The Fairy Knight said that while he was insane, he had converted to Christianity.
It'd happened in Avignon.
But then he'd gone so mad that he'd forgotten about everything.
Now that he was sane, he wanted to emulate one of the most basic Christian ideas, which was to give of himself to the poor.
As a magical being, his blood could serve as endless succor and would flow without end.
He wanted Fern to serve his magical blood to the homeless.
Fern cast some spells.
Fern found the building on Stanford Avenue.
The Fairy Knight opened shop.
The Fairy Knight gave succor to the most rejected people in America.
His blood nourished the poor and healed the sick.
Fern wanted to be with the Fairy Knight.
She wasn't going home.
She didn't care if the women of Fairy Land had to live without any charm in their lives.
Everyone else in the world lived without charmed lives.
If the worst thing that happened to the women of Fairy Land was a loss of charm in their lives, then they were doing better than the rest of the planet.
She too had converted to Christianity.
It had happened long before she rescued the Fairy Knight.
And now her brother's blood had given her life meaning.
## Chapter Seventeen
## **How It All Went Down**
Celia sent Rose Byrne back to Fairy Land.
There was much protestation, but the Queen is the Queen.
That's it.
Rose Byrne's gone from the book.
Celia spent the next few months in Los Angeles.
She cast a spell that taught her how to drive, and because she had a decent internal map from her forays into saliva navigation, she found her way around the city.
Sometimes she went to Hollywood Boulevard and strolled atop the Walk of Fame, dodging the tourists, and doing a supra-natural trick where she saw the whole history, the layers of time superimposed on one another, going back to the beginning, to the Hadean.
And other times she went to Stanford Avenue and talked with her children as the homeless drank the blood of her son.
Her children proselytized to their mother.
They told her about Jesus Christ and his redemptive powers that would give mortals life after death and wash away their sins.
Celia's children kept talking about Heaven and the crucifixion and eternal life and the Epistles of Paul.
They wanted Celia to convert to Christianity.
Celia couldn't get on that trip.
Celia could smell the bullshit.
One Sunday morning, Celia went for a walk.
She took the precarious route down Glendower Ave, which had been built for the rich and thus didn't have usable sidewalks, and went to Vermont Ave.
She walked past gigantic Moreton Bay figs.
The trees reminded her of Fairy Land.
She traveled west on Los Feliz Boulevard and then south for several blocks on Edgemont, passing into a significantly poorer area with the crossing of every east–west boulevard.
At the corner of Fountain, she heard singing.
The voices were coming from a white building on the northwest corner.
The sound reminded Celia of the Ceremony of the Grunting Skyrock, a recent addition to Fairy Land's festival calendar.
The Ceremony of the Grunting Skyrock had been instituted in the Year of the Pleasurable Caravan, which roughly corresponded to 1000 AD, 390 AH, and 4760 AM.
In the Year of the Unmemorable Salt, which directly preceded the Year of the Pleasurable Caravan, a rock had fallen from the sky and smashed into Fairy Land.
Somehow the magic of Fairy Land had prevented any property damage, but the meteorite did leave one hell of a hole.
The women of Fairy Land kept the meteorite in its hole until someone realized that a giant rock from the sky was as good excuse as any to throw a party.
The Ceremony of the Grunting Skyrock involved a lot of choral singing.
For reasons that were always mysterious, the songs that the women sang were filthy tavern ballads about sex and human beings who couldn't stop pissing their own pants.
One of the songs, which was old Turkic-Roma magic, went like this:
_Bu kimin donu_
_Kaynanamin donu_
_Ben yikamam onu_
_Bok kokuyor donu_
Despite their lyrical subjects, the songs sounded beautiful. When a hundred voices rise up as one, all individual imperfections disappear into a flawless unity.
And that's what Celia heard coming out of the white building.
Celia went into the white building. It turned out to be the HOPE International Bible Fellowship, housed in what had once been the Fountain Avenue Baptist Church.
The building wasn't much changed from when it opened in 1929 AD. It was the same shape and it still had people sitting in its pews and they were still listening to bullshit about how to worship Jesus Christ.
Celia took a seat in a back pew, next to a small woman.
The Queen of Fairy Land watched as the humans went through the motions, none of which made any sense, and she sat through the sermon, which she couldn't quite understand.
This wasn't because Celia didn't have a firm theological basis in Bible study.
Celia couldn't follow the sermon because it relied on two conflicting cultural shorthands that were presented as if they were in harmony.
Christian sermons in American life were always more about America than Christianity, and America was the ideological enemy of Christianity.
When the service was over and the Christians had stopped singing and shaking hands, Celia wondered what the hell she'd just seen.
The woman sitting beside Celia noticed the Fairy Queen's confusion.
"You are new here?" asked the woman.
"Is this a church?" asked Celia. "I have read about churches but I have never been inside a church."
"Yes, this is a church," said the woman.
"My children have become Christians," said Celia. "My son and my daughter."
"My children," said the woman, "they are not so good about church. You are lucky."
"Am I?" asked Celia.
"Yes," said the woman. "I wish my children they were thinking about Jesus."
"My children will not stop," said Celia.
"Beautiful," said the woman.
"I am not certain," said Celia. "It has been very painful."
The woman reached into her purse. She pulled out a book. She put the book in Celia's hands.
"Read this," said the woman. "You will make sense of your children."
Celia looked down at the book.
On its black cover, there were gold foil letters that said:
The pre-Internet library of Fairy Land had never included a copy of the Bible.
Not in any of its forms or translations.
This was an oversight, particularly as the Bible was one of the three most influential literary works ever published. The other two were القرآن and the seven volumes of J.K. Rowling's _Harry Potter_.
The _King James Version_ of the Bible was a 1611 AD English translation of the Christian Bible, which was originally put together in the Fourth Century AD, and was comprised of two parts, the Old Testament and the New Testament.
The Old Testament was a collection of documents that'd emerged from the Jewish faith.
There existed another version of the texts, used by Jewish people in a very different way than Christians, called the תַּנ"ַּךְ.
There were a lot of stories in the Old Testament, but the underlying Christian interpretation suggested that it was a book about YHWH, a divinity who had created the world and all of the living beings on the planet, and then spent thousands of years torturing his creations.
The New Testament was primarily about the life of Jesus Christ, his disciples, and the implications of his message as it carried through the world.
Despite having never read the Christian Bible, Celia was the one person on Earth who had an innate critical apparatus for comprehending the disjunction between the Old and New Testaments.
She'd spent about four hundred years reading and thinking about _Tom a Lincoln_ , which was another book split into two parts.
Of this structure, Richard S.M. Hirsch writes: "[Richard Johnson]... had early on decided... to organize his matter in two parts, in this case showing heroic exploits in Part I, and the moral retribution for them in Part II."
In other words, Part I of _Tom a Lincoln_ was about a father who did some weird shit, and Part II was about the father's son paying the price for that weird shit.
Which was the Christian Bible in a nutshell.
Celia brought the _King James_ back to the house on the hill.
She read.
It took several weeks.
The _King James_ wasn't _Game of Thrones_ long, but it was pretty close.
The Old Testament was ancient, and other than the _Song of Solomon_ , which induced at least one meat-market visit to Tenants of Trees, reading it felt like being back in Fairy Land, like inhabiting a universe of unclear moral rules, where the brutality of magic could break your spirit on nothing more than a whim.
The New Testament was different.
Celia couldn't grasp the epistles.
_The Revelation of St. John the Divine_ was a bore.
Even _Acts of the Apostles_ was difficult.
But she understood the gospels, which were four narratives about Jesus Christ and his life.
And because Celia'd developed that critical faculty, she could weed out an author's made-up bullshit from the reality upon which he'd built his narrative.
For centuries, she'd been doing this with _Tom a Lincoln_ , seeing where the fictional account of herself differed from the reality, and comparing the Red-Rose Knight's pillow talk about his childhood with Richard Johnson's early chapters.
Here was Celia's conclusion: Jesus was weird as fuck.
This was the actual Jesus, Jesus without the Christ, not the totemic icon used as justification for a thousand awful wars and for millions of deaths.
This was not the Jesus of Fern and the Fairy Knight.
This was not Jesus of the Church or the churches.
This was not the Jesus of the mean-spirited American, the smiling face that blessed slavery and indigenous genocide, the impetus behind KILL A QUEER FOR CHRIST.
This was the real Jesus.
Until his advent, the ancient world had never placed any intellectual premium on kindness or mercy.
Even good people like Diogenes of Sinope or Epicurus had the habit of talking about virtue as a thing that could be developed by the self for the self. If people wrote or thought about sacrifice, it was in the service of the state or the dominant group.
Never in service of the powerless.
And then Jesus arrived from Bumfuck Shitsville, which, despite its proximity to Sepphoris, is what Nazareth was, and he spoke Aramaic with a hick Galilean accent, and he was a carpenter's son, and he hung around with the filth of society. Sex workers, illiterate fisherman, lepers, literary agents, and tax collectors.
He was nobody.
And he talked funny.
And what he said, the core message, delivered in that hick accent, was this: _stop being a total fucking dick_.
If someone hits your face, offer them the other cheek.
Forgive those who trespass against you.
Serve others before serving yourself. The poor shall inherit the Earth.
Throw away your possessions.
Mercy is the greatest good.
Don't cast the first stone.
Worry not over money.
Embrace the sick.
These ideas have been repeated so many times that they've become platitudes, bumper-sticker morality for the users of Twitter and depressed women of Instagram.
They're like anything in an era of mass production.
Reduced into meaninglessness, transformed into marketable product.
T-shirts.
Words divorced from ideas.
Sharp edges smoothed down.
Yet the intent remains. Jesus had asked his followers to follow a moral code that violated every known precept of human nature.
Consider, by contrast, the trilogy of plays by Aeschylus known as the _Oresteia_.
The _Oresteia_ goes like this: Agamemnon, from the House of Atreus and King of Mycenae, returns home after being away for over a decade. He's been in Troy, where he practiced the art of ethnic cleansing.
Before Agamemnon left for war, the goddess Artemis ordered him to sacrifice his daughter.
So he did.
He killed his daughter and sailed off to be a hero.
While Agamemnon spent ten years practicing ethnic cleansing across the Aegean Sea, his wife Clytemnestra stewed over the murder of her daughter. She took a lover named Aegisthus.
When her husband returns to Mycenae, Clytemnestra and Aegis-thus murder Agamemnon and then assume the crown.
Years later, the son of Clytemnestra and Agamemnon, a guy named Orestes, comes to Mycenae. On orders from the god Apollo, he murders his mother and Aegisthus.
Unfortunately, there are these mythological things called the Erinyes.
The Erinyes, or the Furies, are the living embodiment of vengeance. They torment anyone who breaks the basic rules of society.
One of these rules: don't fuck with hospitality customs.
Another: don't kill your mother.
The Furies chase Orestes all over Greece, until one night, they fall asleep and Apollo spirits Orestes away to Athens, where the matricide begs help from the goddess Athena.
Athena puts Orestes on trial in Athens. He gets prosecuted, he has to defend himself.
The trial ends with a split jury. Athena casts the final vote in favor of Orestes, which frees him, and which also pisses the Furies off.
They scream and shout about the sorrow they're going to wreak upon the world as revenge for the insult. They spit and they foam.
Athena, meanwhile, is the face of reason and calm. She soothes the Furies, slowly, suggesting a better function for them in the world. Why rage when you can help mankind and be worshipped? Who wants all that grief when life can be easy?
The Furies agree and undergo a metaphysical transformation.
They become the Kindly Ones.
They've been tamed by Athena, the personification of Wisdom.
The _Oresteia_ is an allegorical representation of a major event in human history. It's a stand-in for the establishment of civil justice. It's about how societies maintain order in the face of outrageous crimes.
The theme is so universal that all you have to do is engage with any website for about five minutes before you find yourself in the middle of the same debate.
The _Oresteia_ offers a comprehensible vision that works on shared assumptions of how human beings operate.
You might not be able to claim blood for blood, but the court system still allows you a claim of retribution. Wrongs are made right and the world is put into order.
There will be justice.
But not vengeance.
If Jesus had been advising Orestes, this is what he would have said: _Forgive your mother for killing your father. Ask her to kill you next. If she refuses, bring her into your home and feed and clothe her. Love her. And expect no reward for doing as I command you. There is nothing you stand to gain by this mercy other than mercy itself._
One must have as much sympathy for the perpetrator of a crime as for the crime's victim.
This is an inhuman standard.
Even Celia, who wasn't human, couldn't wrap her head around it.
Taken to its furthest logical extreme, the implication is that people don't have to follow the scripts of their lives.
You are more than your base urges.
You don't have to be as terrible as everyone else.
You don't have to burn with pointless judgment.
There is another way.
And it is guided by absolute mercy and radical compassion.
This crazy hick showed up in sophisticated ol' Jerusalem, where everyone posted on social media about the decline of society.
And he spoke of love and forgiveness and mercy and brotherhood.
And he told the people of Jerusalem that they didn't have to follow the scripts of their lives.
So they killed him.
199,900 years of shitting in the living room.
He was crucified, given the lowest of all deaths.
"Ow, that really hurts," said Jesus when the Roman legionnaire Casca Longinus thrust his spear into Jesus' side.
"Give a fuck, me," said Casca Longinus. "Haddaway and shite, you poof."
Then Jesus died.
And maybe he came back to life.
Who fucking knows?
Anything's possible in a world so supranatural that Donald J. Trump ends up in control of 6,800 nuclear warheads.
## Chapter Eighteen
## **Bleak House**
To understand how I ended up winning a $1.2-million judgment in a lawsuit against an Internet stalker who libeled me as a rapist, we have to go back to the early dim days of when I first decided to be a writer.
This was back around 2007 AD, when I was newly arrived in the city of Los Angeles.
I went west after the collapse of a romantic relationship that had lasted seven years, and I had moved to Los Angeles with the unconscious desire to be one of the people who come to California to die.
Much to my surprise, it turned out that moving to Los Angeles wouldn't kill you.
So I had to do something.
Being a writer seemed as bad a fate as any.
In the first decade of the Twenty-First Century AD, there was a vogue called blogging.
Blogging happened when people operated websites and used those websites to publish their own inane commentary on the issues of the day.
There was a sense, then, that one could somehow launch blogging into a career as a writer.
Don't ask me to explain this.
I did the same thing as every other pathetic would-be writer in the first decade of the Twenty-First Century AD.
I started my own blog.
I offered inane commentary on pointless bullshit.
My blog attracted a small but dedicated readership. I'm sure that the daily writing probably helped in some way, but fuck me if I can tell you how.
One member of that small but dedicated readership would end up becoming a huge problem.
At the time, I didn't know their real name, but they'd left several comments on my blog, and they'd always use the same pseudonym: "Oyster the Clown."
The comments were about me being a big ol' homo.
In June of 2008 AD, the same person had sent me an email.
It made no sense.
This was the full extent of the communication between me and Oyster the Clown.
By 2009 AD, I'd stopped writing on the blog.
The website was still there, with its senseless opinions getting no younger, but I couldn't be bothered.
I was doing a million other things, including figuring out how to get books published.
If my career as a writer felt non-existent when I was sexually harassed by Amber Tamblyn, then in 2009 AD I was something below that.
My career wasn't even a career.
It was a stupid little idea on which I'd wasted too much time when I could have been doing things that actually made money.
Literally no one knew me as a writer.
There was nothing to know.
I should also mention that this happened before I lived in San Francisco.
I had yet to be exposed to the mendacity of the people who make money off the Internet.
My faith in humanity was not yet murdered.
I was much softer.
Over Thanksgiving of 2009 AD, while I was celebrating the genocide of the indigenous peoples of the Americas, someone went on the website of _Vice_ and left comments on about fifty articles.
_Vice_ was a media platform that specialized in gross-out journalism and videos in which a sneering idiot from Brooklyn would visit a war-torn locale and contextualize the havoc in terms that could be understood by American children.
The comments said two things.
"Jarett Kobek is a rapist" or "Jarett Kobek is a pedophile."
_Hello_ , said I to myself, _you're neither a rapist nor a pedophile! Why, these comments on the Internet are simply not true!_
Because I am good with computers, I was able to figure out that these comments had been posted from Woodland Hills, California, which was about twenty miles from my apartment in Los Angeles.
I was also able to figure out that they had been posted by Oyster the Clown.
This was not a happy moment.
It's difficult to be libeled as a rapist and a pedophile on the Internet and not feel as if the sky is collapsing on your head.
It is an awful thing to experience.
_Someone is out to get you_ , said I to myself.
At the time, reader, I didn't know it but I was encountering the very strange and new experience of someone writing Jarett Kobek fanfiction.
Generally speaking, fanfiction is written whenever someone decides that they want to tell a story about an intellectual property to which they have no legal rights.
A good example would be when a Batman true believer wants to offer up a prayer and types a little story about Batman kicking the shit out of The Joker.
Or snogging The Riddler.
Or whatever.
These stories tend to go into the Internet.
Alas, many of them, like the Jarett Kobek fanfiction, are about pedophilia and rape!
And, reader, as we've read about someone else's Jarett Kobek fanfiction, I shall write a bit of my own.
I'll tell you a story about the failure of _The Future Won't Be Long_.
You'll have to pardon me, as this fanfiction will be short on both pedophilia and rape.
But it will employ the grotesque language of business.
Which is almost as bad.
If you believe in brands, then you must also believe that the success of any brand derives from its ability to reflect and be defined by its core values.
If, following the self-published US release of _I Hate the Internet_ , you can conceive of a Jarett Kobek brand, then you must also conceive that its core value was this: _fuck you_.
Self-publishing meant that _I Hate the Internet_ had erupted into the world with no permission, no rules, and disconnected from the social and class strictures dominating American writing.
And the novel's text had done something nearly impossible: it had shit on the rich not from a sense of envy but rather one of superiority.
The brand said this: _I denounce thee._
_I denounce thee, publishing._
_I denounce thee, civility._
_I denounce thee, you masters of reality._
_Fuck you._
After _I Hate the Internet_ was released and succeeded beyond his wildest ambitions, Jarett Kobek couldn't imagine any direction other than going to one of the five major publishers.
At the moment of his triumph, Jarett Kobek suffered a failure of imagination.
He flung himself at Penguin Random House with all the vigor of a dog returning to its own vomit.
He allowed himself to be published in the trade dress of a literary writer.
He revealed himself as a class pretender, as someone who believed that he could operate on the level of Jonathan Franzen, as the kind of fraud who'd take that misbegotten Treblinka money and run run run.
It was the smart decision.
But the smart decision was what it always is.
The anti-life equation.
The death of fuck you.
And, boy, did Jarett Kobek ever pay the price.
In the end, his ultimate _fuck you_ was to himself.
Anyway.
_Vice_ deleted the comments.
I spent the new few weeks Googling my own name, obsessively, wondering when Oyster the Clown would strike again.
But nothing happened.
Silence.
In early April of 2010 AD, I visited San Francisco, where I delivered a paper on the underground comix artist Rory Hayes at a comic-book convention.
During my visit, I received an email informing me that I'd been subscribed to the mailing list of Biggayfrathouse.com, a website dedicated to a Big Gay Frat House in San Francisco's Castro District.
The email carried the IP address of the person who subscribed me.
An IP address is the individual marker of any point of access to the Internet.
The IP address in the email resolved to a Comcast Cable account in Washington DC.
In about 1,000 words, this will be an important detail.
I Googled for my own name and discovered that a few minutes after I'd been subscribed to the mailing list of Big Gay Frat House, someone had gone to the website of CNN and posted two comments on an article about the screenwriter Diablo Cody's pregnancy.
The first comment was from someone calling themselves, "oyster."
The first comment read: "Abort it now!"
Just below, "Jarett Kobek" had commented: "I do enjoy a good fetus rape."
Things again fell silent.
On May 3rd, 2010 AD, an article that I'd written was published both in print and online.
It detailed a visit that I'd made in 2009 AD to northern Iraq, where I'd spent a small amount of time at Lalish, the central religious shrine of the Yezidi, who are a persecuted religious culture from Syria and Iraq.
Getting the article published was a total pain in the ass.
This was well before the Yezidi were genocided by the Islamic State in 2014 AD, which meant that the Yezidi were not yet a story that appealed to the editorial class.
And the ultimate thrust of what I wanted to write was an unpopular message on the verge of America's supposed withdrawal of military troops from Iraq.
The thrust was this: _We've made a huge mess and these people will pay the price._
It took a year, but I ended up publishing with the _NYU Alumni Magazine_ on the advice of my friend Rich Byrne, who said that glossy alumni magazines tended to pay serious money.
He wasn't kidding.
I got $1,800 for a 1,500-word piece.
The editorial process was tortured, and the article was a disaster, and somehow the whole thing ended up as a holiday in other people's misery.
It functioned in the exact same way as videos on _Vice._
Someone shows up in a crisis zone and leaves anointed with a superficial knowledge of other people's pain.
On the night of the article's publication, the situation with Oyster the Clown exploded.
Hundreds of comments were left across a wide spectrum of websites.
These were the usual: gay/rapist/pedophile.
The really dangerous stuff was the accounts opened in my name.
Facebook accounts.
Accounts on one of Google's early attempts on social media.
And most insidious of all, an account on YouTube, which contained a surprising amount of personal information in the profile data.
The YouTube account had been used to leave endless comments on videos of children.
These comments were not savory.
There was other stuff too.
I'm not going to bother to recount it here.
This went on for about a month.
New comments, new accounts.
Meanwhile, I was finishing the manuscript of _ATTA_ , and I knew that it was the first significant writing that I'd done, and I further knew that its completion would necessitate getting in touch with professionals in the publishing industry.
I also knew that the first thing that professionals in the publishing industry would do, if they were considering the manuscript, was search for my name on Google.
The results would be the fake Jarett Kobek perving out on videos of children and hundreds of comments about my pedophilia.
And the job of any competent publishing-industry professional is finding an excuse to say no.
By this point, I'd wasted about two or three years on the bullshit of writing.
I didn't need anyone else's help fucking up my life.
It was too late to go back now.
Something had to be done.
At the time, I was poor as fuck.
But class in America is a weird thing.
Half of it is money.
Half of it is social access.
I had no money, but I did have social access.
I ended up talking to a friend of a friend, who was a lawyer at the Electronic Frontier Foundation.
They passed on the name of a law firm in San Francisco that routinely dealt with this shit.
"What would you do?" I asked the friend of a friend.
"Sue the fucker," said the friend of the friend.
So I did.
I sued the fucker.
My attorneys were Ridder, Costa & Johnstone of San Francisco.
And here's another way that I was poor but not poor.
I paid the attorneys with money that I got from my family.
If you ever want to sound like an insane person, cold call some attorneys and tell them that you're being impersonated on the Internet by someone who is libeling you as a rapist and a pedophile.
Imagine that conversation!
As I'd spent some time discussing the situation with friends, I knew the first question that the attorneys would ask.
"Do you have any idea who's doing this?"
"No," I said.
In that initial phone call, the lawyers said that in their experience almost all of these cases derived from romantic entanglement.
An ex-boyfriend, an ex-girlfriend, or the ex-boyfriend or ex-girlfriend's new partner.
When men were targeted, it was always the same: rapist, pedophile, homosexual.
With women, it was: slut, whore, skank.
Often accompanied by boudoir media taken in a haze of coercion or deluded innocence.
Remember: this happened back in 2010 AD.
A more innocent time!
What made the actions of my stalker so egregious were their relative rarity.
Hardly anyone was dealing with this shit.
By the Year of the Froward Worm, which roughly corresponded to 2017 AD, 1438 AH, and 5777 AM, about 40 per cent of online political and social discourse was indistinguishable from the treatment I'd received at the hands of Oyster the Clown.
Seven years after my misery, and everyone was being smeared as a rapist and a pedophile!
One of the common points in the literature available to victims of stalking is the idea that the victim will go through a period of self-recrimination. They will hunt down their every ill deed and wonder which one was the cause of their current misfortune.
The literature is uniform in its rejection of the victim bearing any responsibility for their misfortune.
_It's not your fault_ , says the literature.
But, in my case, I find this to be bullshit.
I went through my period of self-recrimination, and my only conclusion was that the whole thing was my fault.
I had put myself out there.
No one had asked me to write a blog.
No one had asked me to be a writer.
I had done this to myself.
This was another instance in which my situation had anticipated the political and social tactics of the Year of the Froward Worm.
People who made the mistake of putting themselves out there, with the delusion that they should have a voice in the public sphere, were sifted through a purity test in which every public utterance that they'd ever made was given ruthless scrutiny.
If you were delusional enough to be an artist or a writer, you had to anticipate that the only possible reaction your work could receive was unfathomable amounts of hatred.
Let's say that Donald J. Trump, the President, decides that he's going to ban all Muslims from entering America.
Let's say that he effects this ban by issuing an Executive Order, which was a way for the President to do whatever the fuck he wanted under the pretext of the law without having Congressional approval.
Now let's imagine some slightly clueless person with a Twitter account.
This person is enraged by Donald J. Trump's Muslim ban.
This isn't his America!
His America doesn't ban Muslims!
His America just murders them by dropping bombs on peasant villages!
This person decides that they want to criticize Donald J. Trump's Muslim ban.
The way by which this slightly clueless person enacts his criticism is with a stupid little cartoon.
He draws big fat Donald J. Trump riding a beleaguered elephant, which is the go-to caricatured symbol of the Republican party. The elephant is trampling an America flag.
A dialogue balloon comes out of big fat cartoon Donald Trump's mouth and it says, "I'm protecting America."
At the bottom of the cartoon, another dialogue balloon comes out of the elephant's mouth.
It says, "Plus he hates Ragheads. He's not crazy about Spics either."
Whose life will be ruined for at least several years?
Will it be the person who banned Muslims and ripped apart families and is literally killing people in the Middle East while ensuring that Palestinians live in misery?
Or will it be the person who attempted to criticize the person who bans Muslims, and in doing so used exaggerated rhetoric in an admittedly awkward attempt to strike at the truth?
Who will suffer?
Who will be haunted until their end of days?
The rich person or the poor person?
You'll never guess what happens.
When I wrote the first draft of this chapter, I decided that it was only sporting to give my readers an opportunity to contribute to my eventual destruction, which is now the unavoidable fate of anyone who has ever been a writer.
What I had put in this very spot, reader, were some cheap misdeeds from my past.
The unstated joke about these cheap misdeeds was that none were particularly damning.
In the end, I'm just a shy, bookish person.
Happily, between the writing of that first draft and the publication of this book, I've committed a far greater sin than anything I could have confessed from my past.
In that window of time, I wrote and published a short book in defense of XXXTentacion.
XXXTentacion was a young musician who was shot to death at the age of twenty.
He was murdered after he'd been arrested and accused of beating the living daylights out of his pregnant girlfriend.
And the shooting occurred after he'd bragged, in an interview, about a homophobia-inspired beating of a fellow inmate in Florida juvenile detention.
These two incidents had caused much morality written on deadline.
He was the person who unified everyone across the political spectrum in their disgust.
He was the new O.J. Simpson.
And I defended him.
Without ambiguity, without shame.
I had sympathy for the devil.
Without repentance, without prejudice.
So there's your raw material.
Go right ahead.
You can use almost any page of that XXXTentacion book to fuck up my career!
Rob me of the opportunity to contribute shitty opinion pieces to a dying news media!
Deprive me of the ability to be hired as faculty at a small liberal arts college where I can delude the stupid and the rich into thinking that they'll be writers!
The future is in your hands!
Slender Man commands you!
But, reader, I give you fair warning.
You might be able to fuck up my shit, but no matter how much you huff and puff, you'll never take away my tote bag that says BOOKS.
When my lawyers asked me who I thought might be responsible, I offered a crazy person's answer: I suggested that my blogging had worked as sorcery and I'd summoned up a demon that was haunting me for the crime of hubris.
The suggestion was politely ignored.
My attorneys filed suit in Los Angeles Superior Court.
They received power of subpoena.
Power of subpoena meant that they could send out demands to the Internet service providers in Woodland Hills and Washington DC.
And when the subpoenas came back, we'd have the name of my stalker.
There's a story in here that I can't tell you, because it would go back on a promise that my attorneys made to a third party, but we very quickly ended up with the name of the person responsible for all the bother.
I'd been harboring the delusion that, when the name was revealed, it would play out like Agatha Christie, and the unmasking would give me a sense of understanding and wisdom.
But that's not what happened.
It was someone that I didn't know.
At all.
Hadn't met.
Not once.
With no connection to anyone that I'd ever known.
There was no reason behind the stalking and libel.
It was random.
And I could give you my stalker's name, right now, immortalizing them in the annals of literature. There's nothing stopping me. I can't be punished for reporting on public records available in the case files of Los Angeles Superior Court.
But as I've been writing this chapter, I've reflected on how life plays out.
And how strange things have become.
Back when I was being stalked, there was no question that I was on a level playing field with my stalker.
I was no one.
He was no one.
But things have changed.
I'm an international bestseller, I've done countless radio appearances, I've been on television more times than is good for the spiritual health of any one person, I've been chased at the Frankfurt Book Fair by a swarm of book paparazzi, sat through about one hundred and fifty interviews, I've been hotboxed by Alan Moore, I've had Carl Bernstein talk to me about how his son plays guitar for Demi Lovato, I've informed Seymour Hersh about my cat's irritable bowels, and I've annoyed Zadie Smith for about forty minutes at a reception filled with billionaires, Salman Rushdie, and the Jordanian royal family.
I'm famous in Serbia.
I'm writer-famous in Germany and the United Kingdom.
The power differential has shifted.
And we must embrace mercy above all things.
When my attorneys gave me my stalker's name, I spent about a week putting together a picture of who'd been fucking with my shit.
There was a near vacuum of information, but he had a page on the Internet Movie Database, and I was able to figure out that he was a thirty-four-year-old man from Washington DC.
He was a failed screenwriter.
Unemployed and living with his parents.
And the namesake of his father.
He was a junior.
At the very moment when Junior was fucking up my shit from the family's million-dollar row house, Senior was in the United States Senate, working as an assistant to a long-serving Republican.
I was being fucked with by Republicans in Washington DC!
For the first time in my life, I felt like a true American artist.
The father had spent decades working his way up through the Republican hierarchy, until he ascended into a job with the Republican Finance Committee. At one point, ABC News had called him, "One of the Republican party's top officials."
It had all fallen apart in 1996 AD, when the RFC was hit with a sexual harassment lawsuit that specifically focused on Senior. It alleged that on a near daily basis, Senior expected to fondle his female subordinates.
The culmination came in October 1996 AD, when _20/20_ , a television show, ran a report on the lawsuit and on Senior in specific.
It contained footage of Senior at a Republican holiday party, dressed like Santa Claus, leering at younger woman.
It also contained footage of Senior being confronted at the Republican National Convention by the reporter Brian Ross, asking Senior to explain why he had sexually harassed women while dressed like Santa Claus.
This was the beginning of the end.
Senior kept getting demoted to lesser and lesser positions in the Republican hierarchy until he ended up working in the Senate as an assistant.
Reader, let me say this: I'm sorry that this book has included two sexualized mentions of Santa Claus.
None of this is what I wanted.
This is what life has done to me.
In May of 2000 AD, the _Washington Times_ ran a puff piece about the previous home of my stalker's parents. This was where they were living before they bought the row house.
It was the sort of rubbish that newspapers run whenever a rich person wants to sell their home. Stuffed with quaint, folksy detail.
My stalker's mother is quoted in the article, talking about how the home was haunted.
She tells a story about how the ghosts had fucked with a wheel of cheese.
I was being stalked by a person who'd grown up with a haunted wheel of cheese whose father had been exposed on national news for leering at women while dressed as Santa Claus.
That's life.
No one ever said it'd be easy.
It's very easy to laugh about these absurdities, but there was another way to look at the situation and quake with dread.
I had stuck my nose in a hornet's nest.
These were very rich people.
And they were very well connected.
They were consummate Republican insiders.
And I was fucking with them.
Reader, I could supply you with endless details about the intrigues of the case and how my stalker dodged being served with the lawsuit, and how his parents aided and abetted him in dodging service, and how he finally accepted service after I had my attorneys call his sister and leave a message on her work voicemail.
But I'm only going to give the basics.
Before my stalker accepted service, Senior ended up on the phone with my attorneys, and he tried to talk them out of the lawsuit.
He threatened and he blustered.
He finally claimed that Junior suffered from "nerve problems" which kept him from working, but that he'd try to get his son on board with the lawsuit.
He said that he couldn't understand why anyone would waste the resources suing his son, given that his son had no money.
I can't remember what my attorneys said in response.
Had I been asked, I would have said this: _I'm from Rhode Island_.
Rhode Island is the smallest of America's fifty states.
It has the longest official name: The State of Rhode Island and Providence Plantations.
I haven't lived in Rhode Island for nearly twenty years.
But I'm always from Rhode Island.
Whenever someone from Rhode Island is on reality television, they're always the worst of the worst of the worst. They're the people television producers cast because they know that the presence of a Rhode Islander is a shortcut to endless drama.
Rhode Island was founded in the Seventeenth Century AD by the most obnoxious people in the New World.
People like Roger Williams and Anne Hutchinson.
They had fled England because their bad personalities threw their neighbors into murderous rages. They sailed across the Atlantic and settled in bullshit Massachusetts hellholes like Salem and Boston.
And then their bad personalities promptly threw the new neighbors into murderous rages.
They were banished from Massachusetts.
And they had to go somewhere.
So they went and founded Rhode Island.
The crazy never left.
It's still there.
Think of it like this: unlike every other colony in New England, Rhode Island never had a witch trial. But we sure as fuck dug up some old corpses, cut out their hearts, and called them vampires.
If Senior had asked me why I was suing his son, this is what I would have said: _Your son made a terrible error in judgment. He thought that the limits of his own imagination were the boundaries of the universe. But he had no idea about the chaos of Rhode Island. There's a reason why it says INRI on the cross._
After the phone call with my attorneys, Senior convinced his son to accept service.
And then they stonewalled.
Months went by.
By stonewalling, what Senior did was this: he hung his only son out to dry.
I wasn't stopping.
I was from Rhode Island.
And my attorneys were fucking sharks.
If you're ever sued, there's good strategy and there's stupid strategy, and then there's the worst strategy.
Which is to do nothing.
And that's what my stalker did.
Despite all of my attorneys' phone calls, despite the constant forwarding of documents and filings related to the case.
He did nothing.
His parents would not help him.
We kept going, kept moving the case along, and we ended up filing a default motion.
When a defendant refuses to interact with the civil courts, the plaintiff enters a motion of default. When the motion is granted, the plaintiff then enters the documents for a default judgment.
If the court accepts the plaintiff's plea for judgment, it means that the plaintiff has won the case. There is an accepted absolute veracity in the plaintiff's filings.
The defendant agrees, tacitly, that all of the plaintiff's claims are true. The defendant agrees, tacitly, that they are not contesting their responsibility.
My attorneys filed the proposed default judgment.
On February 11th, 2011 AD, Justice Daniel J. Buckley of the Los Angeles Superior Court signed off on the full requested amount.
$1,235,144.75.
One million two hundred thirty-five thousand one hundred forty-four dollars and seventy-five cents.
My attorneys had derived this figure as the ultra-extreme of what the law allowed.
I haven't tried to collect on the judgment, which has compounding 10 per cent yearly interest.
As of this writing, my stalker owes me $2,406,947.70.
By the time this book is published, the amount will be more.
And that's the last contact I ever had with my Internet stalker.
All of the comments he left about me are gone.
All of the accounts he made in my name are gone.
It's like none of it ever happened.
But given that it did, I hope you'll forgive me when I express discomfort with an entire society, and its decaying journalistic apparatus, orienting itself around the destruction of individuals based on things posted to the Internet.
## Chapter Nineteen
## Exeunt Rusticano
A magical island devoid of its charm would be like feminism in a society where all the men had been expelled or murdered. Pretty fucking pointless.
Celia did the only thing that she could.
She called in the Big Dog.
She summoned Rusticano.
He'd been off Fairy Land for three hundred years.
There had been no word in a century.
But he had lived on Fairy Land for centuries and the background radiation of its magic had made him undying.
He was somewhere on Earth.
Celia cast a spell.
Rusticano was transported straight into the house on the hill.
He arrived midsentence, in a burst of light and untamed magic.
"... und deshalb empfehle ich immer einen Tampon," he said.
In the Gray's Inn adaptation of _Tom a Lincoln_ , Rusticano occupies the traditional role of an Elizabethan/Jacobean clown.
He's a lower-class brute in a fictional world where everyone else in resplendent in their finery and goes on about Fairy Queens and dragons.
Rusticano is the one who, in the middle of a speech about the redemptive blood of the Savior, can be relied upon to unleash the world's most unholy fart.
It's the simplest of things, but the device works.
The device always works.
Upper-class social mores, as constructed by the middle-class people who create cheap entertainments, are structured around a pretense that people with money and power believe themselves to be something more than dumb animals.
Enter Rusticano's fart.
What greater rebuke to the pretense of non-animal man than the trumpet-like sound of stinky methane being expelled from a clown's ass?
But that was just a play written to amuse rich kids.
There was a real Rusticano, and other than the name, he shared nothing in common with his fictional iteration.
Rusticano was a human oddity.
He was the one man who'd come to Fairy Land and escaped the hangman's noose.
After the Red-Rose Knight was killed by Orson's shit, the women of Fairy Land rounded up all of the Red-Rose Knight's men.
The men died screaming.
Every single one of the men had defended themselves with weaponry and brute force.
Except Rusticano.
In the massacre, Rose Byrne had been the chief executioner, but she didn't command the task force. Leadership fell to Celia, who stood on a chariot pulled by Fairy Land's meanest buckskin stallions.
Her warrior women followed behind like a bridal train.
When the chariot arrived at the Babbling Brook of Sorrow, where Rusticano spent his days, Rusticano did not flee. He did not pull out his sword.
He stood and faced the women.
"Hail Rusticano," cried Celia. "The time has come for your demise."
"What is my crime, lady?" asked Rusticano.
"No crime, sir," said Celia. "Only that you are a man in Fairy Land. No men shall remain on this island. Your leader is dead. Your friends are dead. You too shall join them in nevermore."
Rusticano kneeled down and picked up a rock.
The women of Fairy Land drew back their arrows and unsheathed their swords.
Rusticano tossed the rock into the Babbling Brook.
"Are you certain," he asked, "that I am a man?"
"What else would you be?" asked Celia.
"No one has ever told Rusticano what makes a man, so how can Rusticano judge for himself? Can you tell me?"
"A man is the opposite of a woman," said Celia.
"Do I not have the same arms and legs as you, do I not have the same head? Should not an opposite be in direct opposition to the thing it opposes?"
"Your body is different than ours," said Celia.
"Are your bodies so similar?" asked Rusticano. "Look at the skinny maid, holding her axe. Is her body the same as the fat one who cripples the horse? Are those differences so much less than the distance of my body from yours?"
"Too much prattle, fair Rusticano," said Celia. "Your talk will not save you."
"Lady," said Rusticano. "I am happy to die if you wish it. But I have shared your salt and lived as your guest, and I have given you my own gift, and by those sacred and ancient terms, I claim certain rights. I demand to know why I should die. Tell me, then, what makes a woman and what makes me not a woman."
"Do you bleed with the moon?" asked Celia.
"I do not," said Rusticano. "But is that what makes a woman? I vow to you here that I shall take the sword to my flesh with every full rise and let flow as much blood as you demand."
"Where are your breasts?" asked Celia.
"Again, lady," said Rusticano. "I have no breasts but I see at least three women here with chests flatter than mine. Are they not women?"
"What of that prick between your legs?" asked Celia.
"Are we so certain that it is a prick?" asked Rusticano.
"What else would it be?" asked Celia.
"I would caution against defining a thing by its appearance," said Rusticano. "How many chairs are there on this island? Each looks different from the others and yet we see them and know that they are chairs. This is because a chair is a thing for sitting upon, and as with a prick, it is defined not by its appearance but rather by its function. So tell me, lady, if this is a prick, then what is its function?"
"You piss through your prick," said Celia. "And you put it inside a woman."
"These are the two functions of a prick?" asked Rusticano. "These are what define a prick, and you say that having a prick is what defines me as a man?"
"Yes," said Celia. "I put this to you as the reason why you must die."
"And will you swear by this definition?" asked Rusticano. "Will you swear by it before the hospitality under whose banner I now march?"
"I swear it," said Celia.
"And you speak that oath with the full force of your reign?" asked Rusticano. "This definition is the law of Fairy Land?"
"It is," said Celia.
"If the two functions of the prick are to piss on the earth and make shame in a woman," said Rusticano, "then are not all women of Fairy Land halfways a man? For do not all of you piss the same as me? You may claim that your piss issues forth from a different place than my own, but Rusticano says that the piss is defined by itself in its own state of being and not its source. When someone speaks a word, do you concentrate on the teeth and the tongue? Nay, you heed the final issuance. When my piss is on the ground, does it demonstrate any difference from your own puddles? Nay, lady, I suggest that this cannot be a function that defines a prick."
"You are right," said Celia. "A prick is a prick because it goes into a woman."
"Then lady," said Rusticano, "I ask you to find a single woman here on Fairy Land in whom I've entered. I've pricked none of your island. I've pricked no woman anywhere in the world. How often did you laugh when the Red-Rose Knight made sore jests about my untried virginity? I submit to you that by your own definition what hangs here is no prick. I know not what I am, lady, but I further submit that Rusticano is no man."
Rusticano stayed on Fairy Land.
Centuries ticked off.
He was transformed into an immortal supranatural creature, but he never developed the ability to cast spells.
Rusticano was not accepted as a full member of Fairy Land's society.
He was just this person who lived in a cave near the Babbling Brook of Sorrow.
Rusticano developed a reputation.
He was someone who could talk his way through anything.
Sometimes the women of Fairy Land relied on Rusticano to solve problems.
Like when Freita Muscleback and Bianca Findlay both fell in love with Youna Shifa.
Love affairs on the island were especially fraught with peril.
If something went really wrong, everyone would be stuck dealing with the consequences for centuries.
This model of old lovers' inescapability was something that the mortal world would later duplicate in the form of Facebook.
The social media platform had doomed everyone in the mortal world to the worst possible fate: living in a small town where they never ever lost contact with the people whom they'd fucked in high school, and worse yet, seeing the people whom they'd fucked in high school post daily updates on the topic of White Supremacy.
White Supremacy was a rhetorical device that'd been developed to describe the unfathomable social advantages that allowed the dominant social group in America to experience hereditary social wealth, primarily at the expense of people descended from African slaves who, once upon a time, literally had been that wealth.
The very expression of the concept drove a lot of people crazy.
They denied that it existed.
They stamped their feet and put their fingers in their ears.
But c'mon.
Of course White Supremacy was real.
The author knows better than anyone.
He is nothing but the product of White Supremacy.
He was raised in a single-parent household!
Child of divorce!
Rhode Island!
Traumatized by an early violent death in the family!
And his father was a Muslim!
And an immigrant from the Middle East!
And a member of the proletariat!
And an alcoholic!
#MENA!
And even with these obvious deviations, White Supremacy still carried me to the promised land of a commercial failure published with Nazi money by Penguin Random House.
Meanwhile, Byron Crawford, who is the best writer that you're not reading, was self-publishing his own books.*
And Ernest Baker had to do the same thing with _Black American Psycho_ , one of the previous five years' most interesting novels.
Published through fucking Amazon.com!
Print on demand!
So, yes, fucking obviously.
White Supremacy is real.
But the fact of its existence in no way alleviates the tedium of Facebook updates on the topic. Particularly those written by your high-school sweetheart.
On the day when Freita Muscleback and Bianca Findlay realized that they were both in love with Youna Shifa, they approached Rusticano.
Rusticano was in his cave by the Babbling Brook of Sorrow.
"What is the nature of the problem?" asked Rusticano.
"We both love Youna," said Freita. "Has anyone ever told you about Boadicea Thrumpguts?"
"The bloody story," said Rusticano. "I know it well. I wonder if the nature of your problem is not love, but rather something else. You would agree that love is predicated on an object of love? If one loves, then one must necessarily love something?"
"That is right," said Freita.
"Would you also agree that if love is predicated on an object of that love, then one cannot love nothing?"
"Yes," said Bianca. "It is impossible to love nothing."
"If the lover must necessarily love something or someone, then does the lover love the beloved based on contingent actions, or does the lover love because she recognizes a quality in the beloved that is a mirror of a greater love?"
"I am not sure that I follow, Rusticano," said Bianca.
"Do you love your mother?" asked Rusticano.
"Yes, Rusticano," said Bianca.
"On what basis do you love her?" asked Rusticano. "Is it because of an action that she performed?"
"No," said Freita. "One is born with a love of her mother."
"If this love is not contingent," said Rusticano, "then may we say that it is based not on the actions of the beloved, but rather some quality that the lover can recognize even before she speaks words?"
"You are correct," said Freita.
"Then why are you worrying over whether the actions of the beloved affect the nature of the lover's love?"
"I don't understand," said Freita.
"When your beloved is occupied with something other than yourself, such as preparing a meal or hunting a deer, do you stop loving them?"
"No," said Bianca.
"If you agree that the beloved does not impede the lover's love with the performance of tasks unrelated to the lover's love, then why should the lover's love be affected by the beloved's love of another? Do you fault your beloved when she expresses love for her sister? Does the lover's love diminish because the beloved loves a family member?"
"No," said Bianca.
"If you say that a beloved's love for another does not detract from the lover's love, then would you admit that when the lover experiences jealousy, it cannot be from any diminishing of love that is reactive to the beloved's actions?"
"Yes," said Freita.
"Rusticano would suggest that this jealousy is not above love, but rather its opposite, which is the fear of loving nothing. When your beloved loves another, and your fears rise up, you must remember that your fear is not of a fear of your own diminished love, but rather fear that you are loving nothing. But as we have proven it is impossible to love nothing, then this fear is without base and is a meaningless thing."
"You are right there, Rusticano," said Freita.
This went on for hours.
By the end of it, Freita and Bianca had agreed to share the love of Youna Shifa. It was going to be the Fairy Land version of San Francisco polyamory. The only problem was that no one had bothered to ask Youna Shifa if she loved Freita or Bianca.
She didn't.
The centuries passed.
Rusticano grew bored with the island. He asked Celia to send him to the mortal world, where he would make his way.
She agreed.
The only condition was that once Rusticano left the island, he could never return.
He departed.
Word filtered back through the usual channels.
Rusticano was in Spain.
Rusticano was in Germany.
Rusticano had opened a business.
Rusticano's business sold luggage.
Rusticano's business was evolving into fashion.
After the psychic cataclysm of World War One, there were no more reports.
At the very moment that he disappeared in a flash of untamed magic, Rusticano was sitting in a Coffee Fellows at München Hauptbahnhof's northeast corner.
He was eating an egg bagel sandwich and talking to his friend Liv Lisa Fries.
And then, like that, he was standing face-to-face with Celia in the house on the hill.
"My lady," he said. "It has been too long."
"I have need of you, Rusticano," said Celia. "The debt comes due."
"Payment in full," he said. "I am yours to command."
Celia told Rusticano about her children and their conversion to Christianity.
"What would you have me do?" asked Rusticano.
"Talk them away from their folly," said Celia. "Get them out of that building. Convince Fern to come back to Fairy Land."
Celia tried to drive Rusticano to Stanford Avenue, but Rusticano insisted that before they departed, he be allowed to drive the Jaguar around the neighborhood.
He said that he was a fan of vintage British engineering.
"How will you find your way?" asked Celia.
"Rusticano keeps a smartphone upon his person," said Rusticano. "But I require that you cast a spell and turn on its international roaming."
Celia cast the spell.
Rusticano owned a Samsung Galaxy Note 8. He'd installed LineageOS 14.1, an open source fork of CyanogenMod, which was itself an open source fork of Google's Android OS.
When Rusticano returned from his neighborhood sojourn, he carried a black duffle bag.
He did not explain the bag.
He informed Celia that she could drive them to Stanford Avenue.
They headed downtown.
"In my experience, people with religious beliefs are the least open to reason," said Rusticano. "Yet Rusticano has his ways. My only request is that you do not interfere with my deeds."
"By my leave," said Celia, "I vow that I shall not interrupt you."
"No matter the action?"
"No matter the action," said Celia.
"The nature of the problem is that they are in this building and will not leave this building?" asked Rusticano.
"They believe they are doing the work of Jesus Christ," said Celia.
"They always do," said Rusticano.
"Have you read the Bible?" asked Celia.
"Once," said Rusticano. "Many years ago."
"Jesus was weird as fuck," said Celia.
"I imagine that had he come to Fairy Land, a crucifixion would have been the least of his worries."
Celia parked the car in front of the TUNA EXPRESS CO.
They followed the line of bodies inside the building.
They went up to the second floor.
They went into the backroom.
Fern was pumping out her brother's blood into the mouths of several homeless men.
The Fairy Knight was on the table, half dazed.
"You say that you believe in Jesus Christ and do his works?" asked Rusticano of Fern.
"We do," said Fern.
"Nothing can shake you in your faith?" asked Rusticano.
"Nothing," said Fern. "We have been baptized. We are his."
"What say you, Fairy Knight?" asked Rusticano. "Are you too unshakeable in your faith?"
"Totally, completely, utterly," groaned the Fairy Knight.
Rusticano paused.
Rusticano thought.
"You may not remember," said Rusticano, "but the Red-Rose Knight was my bosom friend. I knew the man when he was still Tom a Lincoln, and I was there when we crowned his head with a laurel of roses."
"We remember," said Fern.
"I wager that I knew the man better even than your mother understood him," said Rusticano. "I never shared his bed, but I spent more time with him than any other."
"Our father would not object to our service," said Fern.
"Oh, I have no doubt that he would not," said Rusticano. "The Red-Rose Knight would offer no dissent from this practice. But do you heed Rusticano when he says that the Red-Rose Knight was the stupidest man that he ever met? Your father was a jumped-up fool who believed that he was better than Lincoln, and for his delusion, all he earned was a few years between the thighs of a fairy queen before he drank a glass of water filled with filth. I grant you that this is more than most men, but it does not change anything. Children, your father was a jackanapes, and while the royal blood of England and the royal blood of Fairy Land flows through your veins, I fear that you have not inherited any of your mother's wisdom. Do you know that the woman asked me here to dissuade you from your perverse hobby? Rusticano agreed. Rusticano thought about his best method of attack, considering every possible avenue of criticism. Then Rusticano remembered. There is only one way to deal with religious people, and there is only one way to deal with the grandchildren of King Arthur, a man who I also met, and who, if I may say so, rivaled your father for his stupidity."
Rusticano had left Lincoln with the Red-Rose Knight because he'd hated the grotesque meanness of the English. When he returned to the mortal world, to mainland Europe, he found that not much was different.
It'd been 1,000 years and there was still so much evil.
The names had changed but shits were still ruling the world.
Everything was violence. Everything was war.
The poor were still starving.
The only difference now was that Rusticano himself could not die.
And he could not return to his cave by the Babbling Brook of Sorrow.
Rusticano decided to embrace the values of the mortal world.
He married a woman named Evette, a tanner's daughter, and with her started a family.
The early years were good.
Rusticano was in love.
But it wore off.
After a decade, nothing about Evette or their children could alleviate the hollow feeling.
He didn't want to see his wife die.
He didn't want to learn that his children were not immortal.
He didn't want to pretend that a domestic life had any meaning.
Food, clothes, shelter.
Every fucking day.
Without end.
Faking his own death, Rusticano abandoned his family and made his acquaintance with those who had passed the Cash Horizon. He went to interesting parties, he hoarded money, and he had reckless sex with people who were bored out of their minds.
These antics could amuse for a moment, maybe even for a week, but in the end, everything was empty. The parties and the people were dull as dishwater.
Alcohol helped.
Alcohol always helped.
Thinking that honest labor might give his life some purpose, Rusticano took up a trade.
Rusticano founded his own company.
He had great success with handcrafted luggage. The company expanded and Rusticano experienced even greater success with fashion.
But the rewards of his business were only iterations of the same boring things.
Food, shelter, clothing, sex.
The world moved on and entered the Twentieth Century AD.
Rusticano could see the score.
He watched as liberal democracies consumed themselves with internal divisions about their relative social order, and he watched as these liberal democracies placed the petty squabbles about these internal divisions upon a foundation of ghettos and the foreign dead.
People had stopped arguing about the divine right of kings and now screamed at each other about human rights, about how terrible it was that some inequality in the internal society had made a mockery of that society's values, and then retreated into their homes and feasted on the mass-murdered flesh of animals while their militaries dropped bombs on distant locales and the mechanisms of their societies destroyed the poor with unfair labor practices.
There was no place for Rusticano's wordplay and his reason.
Not in the mortal world.
The Twentieth Century AD had only one rule: might made right.
Rusticano was immortal.
He was stronger than everyone else.
He could do anything he wanted.
Rusticano opened his black duffel bag and took out a five-gallon plastic fuel can that he'd purchased from Rite Aid at the corner of Vermont Avenue and Hollywood Boulevard.
Rusticano unscrewed the cap on the fuel can.
Rusticano splashed out five gallons of gasoline that he'd purchased from the Shell station at the opposite corner of Vermont Avenue and Hollywood Boulevard.
He did it so fast that neither Celia nor Fern nor the Fairy Knight could stop him.
The homeless people were too tripped out on the Fairy Knight's blood to realize what was going on. And anyway, they were people who America had deemed useless, so they weren't particularly surprised to be abused by a stranger.
"You may now thank me for my honesty," said Rusticano.
Rusticano used a lighter that he'd bought at Rite Aid to ignite the gasoline.
Everything burned.
As their flesh caught fire, but they did not die, both the Fairy Knight and Fern said the Lord's Prayer.
It was for nothing.
They might as well have said a prayer with a proven track record.
They might as well have said this:
_Arafat Kazi got into the pit to see Guns N' Roses at the Staples Center_.
The building burned.
Celia burned.
Rusticano burned.
Fern burned.
The Fairy Knight burned.
The homeless burned.
Undying beings came out of the charred wreckage.
Rusticano looked at Celia.
Celia looked at Rusticano.
"Lady," he said. "Your children are no longer in the building. I would ask that you send me back."
Celia cast a spell and Rusticano disappeared.
One minute he was there.
Then: blink.
He was in a Coffee Fellows at the northeast corner in the München Hauptbahnhof.
Celia's magic had not accounted for the difference in time zones and the operating hours of Coffee Fellows.
It was midnight.
Rusticano was trapped behind locked doors.
He threw a table through a plate glass window and walked down Prielmayerstraße towards the former Bürgerbräukeller.
As with the Gray's Inn play, he had wrecked another narrative.
His actions had precipitated an unsatisfying end to this book, causing its story to dissolve into a hectoring lecture about Christianity divorced from any pretext of plot.
Exeunt Rusticano.
* Technically, Byron Crawford shares the title of the Best Writer That You're Not Reading with Fiona Helmsley.
## Chapter Twenty
## **What Rusticano Didn't Say**
Rusticano was nobody's fool.
He knew that the best arguments against Christianity were surprisingly terrible, and furthermore that these arguments relied on philological research, observations about historical injustice, scientific empiricism, and the issue of theodicy, which was the fancy way of asking _Why does God let bad things happen to good people?_
When people asked _Why does God let bad things happen to good people?_ what they really meant was this: "Why does God let bad things happen to me?"
It's a very Twenty-First-Century AD argument.
Cooked in a base of insipid narcissism.
The strongest of the bad arguments were the ones that relied on scientific empiricism, which pointed out the impossibility of God creating his son in human form.
And the impossibility of that human form rising from the dead.
But appeals to rationality and scientific truth, while making great sense on paper, faced a very significant problem.
The world kept getting weirder.
The everyday lives of everyday people were completely insane.
Roughly 100,000 people controlled the fates and destinies of 7,799,900,000 people, and the 7,799,900,000 let themselves be subject to the whimsies of the 100,000.
But let's be clear: the madness of everyday life was its own issue.
It didn't have any relationship to whether or not Christianity was bullshit.
Obviously, Christianity was total bullshit.
It was the most insane bullshit!
But it was impossible to make an argument against superstition and magical nonsense, and have it stick, when that argument was delivered from a society where every citizen was a magician.
And yes, reader, that includes you.
You too are a magician.
Your life is dominated by one of the oldest and most perverse forms of magic, one with less interior cohesion than the Christian faith, and you invest its empty symbolism with a level of belief that far outpaces that of any Christian.
Here are some strips of paper and bits of metal!
Watch as I transform these strips of paper and bits of metal into: (a) sex (b) food (c) clothing (d) shelter (e) transportation that allows me to acquire strips of paper and bits of money (f) intoxicants that distract me from my endless pursuit of strips of paper and bits of metal (g) leisure items that distract me from my endless pursuit of strips of paper and bits of metal (h) pointless vacations to exotic locales where I will replicate the brutish behavior that I display in my point of origin as a brief respite from my endless pursuit of strips of paper and bits of metal (i) unfair social advantages that allow my rotten children to undertake their own moronic pursuits of strips of paper and bits of metal.
Humiliate yourself for strips of paper.
Murder for the strips of paper.
Humiliate others for the strips of paper.
Worship the people who've accumulated such vast quantities of strips of paper that their strips of paper no longer have any physical existence and are now represented by binary notation.
Treat the vast accumulators like gods.
Free blowies for the moldering corpse of Steve Jobs!
Fawning profile pieces for Jay-Z!
The Presidency for billionaire socialite and real-estate developer Donald J. Trump!
Kill! Kill! Kill!
Work! Work! Work!
Die! Die! Die!
Go on.
Pretend this is not the most magical thing that has ever happened.
Historical arguments against Christianity tended to be delivered in tones of pearl-clutching horror, usually by subpar British intellectuals pimping their accent in America, a country where sounding like an Oxbridge twat conferred an unearned credibility.
Yes, the Crusades were horrible.
Yes, the Inquisition was awful.
Yes, they shouldn't have burned witches in Salem.
Yes, there is an unfathomable amount of sexually abused walking wounded.
Yes, every Christian country has oriented itself around the rich and done nothing but abuse the fuck out of its poor.
But it's not like the secular conversion of the industrialized world has alleviated any of the horror. Read the news.
Murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder, rape, murder.
Despair.
All secularism has done, really, is remove a yoke from the rich. They'd always been horrible, but at least when they still paid lip service to Christian virtues, they could be shamed into philanthropy.
Now they use market forces to slide the whole thing into feudalism.
New York University built a campus with slave labor!
In the Twenty-First Century ad!
And has suffered no rebuke!
Applications are at an all-time high!
The historical arguments against Christianity are as facile as reviews on Goodreads.com, and come down to this: _Why do you organize around bad people who tell you that a Skyman wants you to be good?_
To which the rejoinder is: yes, the clergy sucks, but who cares how normal people are delivered into goodness?
As for philological research.
C'mon, get real.
None of these arguments would've worked on Fern or the Fairy Knight.
How does one supranatural creature tell two other supranatural creatures that they shouldn't believe in a fourth supranatural creature?
Rusticano couldn't make that argument.
So he made a fire.
You can't talk people out of religion.
But you can violently assault them and burn down their churches. Or spraypaint MUHAMMAD PROPHET OF BUTCHERS on their masjid.
It won't work, you won't disabuse people of their belief.
But anything's better than subjecting the world to another lecture about atheism.
## Chapter Twenty-One
## καταδυσόμεθ᾽ εἰς Ἀΐδαο δόμους
The 2016 AD Aston Martin Vanquish made its way up Sunset Boulevard.
Dmitri Huda had the night off.
HRH was in the driver's seat.
HRH turned left onto Silver Lake Boulevard.
A sex worker was in the passenger seat.
She looked out of the passenger-side window at the Silversun liquor store.
HRH followed Silver Lake Boulevard to the 2016 AD Aston Martin Vanquish's final destination, which was HRH's mid-century Los Angeles home overlooking the Silver Lake Reservoir.
HRH's personal breaking point with Beverly Hills had come in the autumn of 2015 AD, when the media turned its attention to HRH's cousin HRH Majed bin Abdullah bin Abdulaziz Al Saud.
HRH Majed had, allegedly, thrown a party over three days.
The party had, allegedly, spanned from September 21, 2015 AD to September 23, 2015 AD.
On September 25th, 2015 AD, three women, who'd been employed by HRH Majed to provide housekeeping services during the party, filed a civil lawsuit with the Superior Court of Los Angeles.
In the complaint, the women were anonymous. They were listed as JANE DOES 1 through 3.
On October 22nd, 2015 AD, the three JANE DOES filed an amended complaint. And this second complaint caught media attention.
In the complaint, it was alleged that over the alleged three nights of alleged drug abuse, HRH Majed bin Abdullah bin Abdulaziz Al Saud had: (1) allegedly attempted to urinate on or around JANE DOES 1 through 3 while saying, allegedly, "I want to pee pee!" (2) allegedly threatened to kill JANE DOE 1, allegedly saying, "Tomorrow I will have a party with you and you will do everything I want, otherwise I will kill you" (3) allegedly jumped on top of JANE DOE 2 while she was seated and allegedly started rubbing his body against her body in a sexual manner and then, allegedly, shouted: "I am a prince and I do what I want! You are a nobody" followed by, allegedly, "You're not a woman! You're nobody! I'm a prince and I'll do what I want and nobody will do anything to me!" (4) allegedly grabbed JANE DOE 2's arm and allegedly kicked her on the knee while maintaining his grasp, allegedly leaving nail marks on her wrist and bruise marks on her thigh (5) allegedly instructed JANE DOE 3 as follows: "You're going to go upstairs. I'll be up there in two minutes and you'll do whatever I want. If not, then I'll kill you" (6) allegedly forced JANE DOE 1 and JANE DOE 2 to watch as his penis was stroked, allegedly, by a male employee who was allegedly on his knees before HRH Majed (7) allegedly forced JANE DOE 1 to watch as a different man, by request, allegedly farted in HRH Majed's face (8) allegedly told JANE DOE 1: "I will pay you to lick my entire body. If you make me feel good, you'll feel good too."
The complaint said very little of HRH Majed's subsequent arrest for alleged forced oral copulation, brought after a neighbor saw a woman smeared with blood, shouting for help, and trying to climb an eight-foot wall.
But the media reports said quite a bit.
To be fair to HRH Majed Majed bin Abdullah bin Abdulaziz Al Saud: he never faced a criminal trial stemming from his arrest at the hands of the LAPD.
The Los Angeles District Attorney's office declined to press charges.
And, through his representatives, HRH Majed denied everything stemming from both the arrest and the civil trial. A representative said: "I will not dignify these salacious allegations – which the District attorney found to be unsupported by evidence... The decision by the D.A.'s office not to file charges shows that the accuser's stories cannot be substantiated... The sheikh is very happy to put it behind him and move on with his life."
And, it must be said: the lawsuit was filed anonymously and never went to trial.
None of its claims were even proven in a court of law.
And on December 12, 2017 AD, all parties filed for dismissal.
"Can you conceive of how difficult it is for an Arabian prince to be arrested in Beverly Hills?" HRH had asked Dmitri Huda after the media descended. "Even with these false charges and obvious calumnies, how does one achieve such a miracle?"
"His father died in January," said Dmitri Huda. "Maybe he's distraught."
"I wish that my father would die in January! Or any other month! You would never find myself dismantling an equitable arrangement with law enforcement! If one cannot trust the LAPD, then one can trust nothing. The heat is on, Dmitri."
And so HRH escaped to Silver Lake.
The 2016 AD Aston Martin Vanquish pulled into HRH's driveway.
A small plastic device sent out a radio transmission that instructed a motor to open the garage door.
The 2016 AD Aston Martin Vanquish pulled into the garage.
HRH removed the crystal key from the ignition.
HRH and the sex worker walked up the stairs and into HRH's hallway.
"Is this you?" asked the sex worker, pointing to a photograph of a young HRH and Ronald Wilson Reagan, taken during the last year of the former actor's Presidency, when Alzheimer's disease had begun transforming the former actor's brain into useless mush.
"I am indeed the prepubescent so pictured," said HRH.
The sex worker walked past the photograph of HRH and Richard Milhous Nixon, taken in the early 1990s AD, just before the former President's death. In the photograph, an unfortunate wisp of a moustache was present on HRH's upper lip.
HRH walked past the photograph of himself and William Jefferson Clinton, taken in the late 1990s AD. In the photograph, HRH had become a young man.
The sex worker walked past the photograph of HRH and George Walker Bush.
HRH walked past the photograph of himself and George Herbert Walker Bush, taken only moments after HRH was photographed with George Walker Bush.
"The noblest soul ever to vomit upon the Prime Minister of Japan," said HRH.
The sex worker walked past the photograph of HRH and Donald J. Trump, taken around 2005 AD. The two men were in Bangkok. They were surrounded by pleasure girls.
"Fuck," said the sex worker. "You know him?"
"He is a friend of my father," said HRH.
"Who the fuck is your father?" asked the sex worker.
"He is called The Conqueror," said HRH.
HRH walked past the photograph of himself and James Earl Carter Jr., taken around 2006 AD, in the halcyon days when HRH laundered medical marijuana money through environmental NGOs.
The sex worker walked past the photograph of HRH and Barack Hussein Obama, taken in late 2009 AD.
First Lady Michelle Obama had invited HRH to a White House dinner after HRH funded an initiative to help celebrate Women's History Month.
HRH and the sex worker were 420-friendly.
The sex worker smoked from a waterpipe that she found in HRH's living room.
HRH vaped indica.
"Now is the time, O you budding sapling of May," said HRH. "Your clothes must take their absence from your flesh."
When the sex worker took off her clothes, HRH was surprised to see that her chest, torso, and left outer thigh were inked with a multicolored tattoo.
The tattoo depicted a monster of vaguely anthropoid outline, but with an octopus-like head whose face was a mass of tentacles, a scaly, rubbery-looking body, prodigious claws on hind and fore feet, and long, narrow wings behind.
In a textbook example of the caricature endemic to the modern tattoo artist, a few of the monster's tentacles wrapped around the sex worker's nipples, and another stretched down to her mons pubis, while a final tentacle went around the left buttock and appeared to terminate at the sex worker's rectum.
"Rare is the treasure who adorns her essential skin with the cosmic horror of H.P. Lovecraft."
"Thanks," said the sex worker.
"Tell me, which of Lovecraft's works do you rank as the finest?"
"I like 'The Thing on the Doorstep'," said the sex worker.
"O my darling, with each step you reveal new depths," said HRH. "Your perversity knows no bounds."
HRH vaped indica.
"The Thing on the Doorstep" was about a man who marries a woman only to discover that her mind has been replaced with the malevolent consciousness of the woman's father. Everything revolves around open concerns of homosexuality, incest, trans people, and flat-out bestiality with a fishwoman.
On several occasions in his dewy youth, HRH had masturbated while reading the story.
"My plan had been to roger you senseless," said HRH to the sex worker. "I was to leave you drooling and dazed like a donkey attacked by the silent killer of encephalomyelitis. Yet damn your eyes, you have revealed yourself as a beast who should not suffer the usual rounds of amorous pursuits. For you, I shall unleash the highest form of depredation. I will permit entry to my inner sanctum, to the chamber where the grandest perversity flourishes. Fear not. I possess no red room of pain. This is neither _Fifty Shades of Grey_ nor _The Amityville Horror_."
The sex worker followed HRH upstairs.
HRH led the sex worker into a bedroom.
There were two DXRacer chairs, a desk, and a giant LCD display attached to an Alienware Area-51 desktop computer.
HRH sat in one of the chairs.
"Position your meat in the other receptacle," said HRH. "Join me at this terminal to infinity."
"Are you trying to make me watch porn?" asked the sex worker. "I've seen porn. But it's your money."
HRH powered on the Alienware Area-51 desktop computer.
HRH opened the Google Chrome web browser.
HRH directed the Google Chrome web browser to <https://www.twitch.tv>.
<https://www.twitch.tv> was the URL of Twitch, a subsidiary of Amazon.com, which was a website dedicated to the destruction of the publishing industry.
Amazon.com was owned by Jeff Bezos, who also owned Goodreads.com, the Internet Movie Database, Blue Origin, and the _Washington Post_ , which was a newspaper with a slogan that said: "Democracy Dies in Darkness."
This motto implied that without a free press casting illumination upon the powerful, American democracy would devolve into a hollow shell.
This motto had been handpicked by Jeff Bezos.
Depending on fluctuations in the stock market, on some days Jeff Bezos was the richest man in the world.
Which meant that the motto was slightly disingenuous.
You don't get as rich as Jeff Bezos without knowing exactly how people beyond the Cash Horizon wreaked havoc upon democracy: they said what they were going to do, in public, and then they did it.
In 2003 AD, about twenty people in the United States government told the world that it was going to kill a metric fuckton of Iraqis.
The whole world cried out, in unison, and demanded that the United States government not kill a metric fuckton of Iraqis.
But the Iraqis still died by the metric fuckton.
Remind me: who gave a shit about darkness?
The second problem with the motto was that it was based on an ahistorical assumption, which was that Americans lived in a democracy.
America was never a democracy.
It was a Republic.
It had been designed as a Republic.
The country's founders were horrified by democracy.
American democracy couldn't die because American democracy had never existed.
out of .
The front page of Twitch's website displayed a live video stream from the Overwatch League.
Two regional teams battled each other in the game _Overwatch_ while a live audience watched.
The stream was narrated by two men who'd patterned their vocal style after sports commentators.
"Dallas is going to swap things up a bit. Getting aggressive here and you have to worry a bit," said one of the commentators.
"I mean that's huge for Dallas," said the other commentator. "Because now they're going to have a player advantage. It's a six versus five. That's a big hit on the way out."
"What the hell is this?" asked the sex worker.
"Are you not a millennial, madame?" asked HRH. "Is this not your natural domain?"
The vast majority of streams on Twitch were very different than Overwatch League.
A random person played video games in their home and broadcast this over the Internet. Twitch hosted the action, providing a central place for the meeting of broadcasters, who were called streamers, and their viewers.
The video game action occupied most of any individual stream. A small box, containing live video of the user, appeared in one of the stream's corners. On the right side of the screen, the stream's viewers commented in a scrolling livechat.
Depending on which streaming software was used, and depending on which plug-ins the streamer had configured, various graphics were displayed when viewers interfaced with the platform's monetization.
In other words, the people watching videos on Twitch could give money to the people broadcasting on Twitch and these donations would show up in the stream itself.
"Over many arduous months, I have cultivated a personal fandom of several Twitch channels," said HRH. "Permit a demonstration."
HRH navigated to the Twitch channel of an unremarkable young man.
The young man was playing _Fortnite: Battle Royale._
The sex worker watched as the young man navigated his video game avatar across an island landscape, destroying objects and simulating genocide against the other players connected to the same _Fortnite_ server.
HRH navigated to the Twitch channel of a pretty young woman who lived in Tokyo.
The woman was not playing a video game. She was interacting with the livechat. She was receiving donations whenever she impersonated a character from _Final Fantasy XV_.
HRH navigated to the Twitch channel of a young woman who was dressed as Diana from _Wonder Woman_. The woman was drinking AriZona Iced Tea and playing _South Park: The Fractured but Whole_.
HRH navigated to the Twitch channel of a man in his twenties, who was cursing wildly as he attempted to play a game called _Cuphead_.
" _Cuphead_ is a crowd-funded odyssey into an ersatz replica of animation from the Great Depression," said HRH. "I have never indulged, but I am informed that it is a work of manifold difficulty."
HRH navigated to the Twitch channel of a young woman who lived in Sidcup.
The Sidcup woman was playing _The Sims 4_ , a piece of software that simulated the appearance of a Twentieth-Century AD suburban life that been murdered by the international capitalist class.
The sex worker watched the Sidcup woman demonstrate the décor of a simulated house in _The Sims 4_.
The house in _The Sims 4_ was very moderne Danske.
It stood in contrast to the visible décor of the woman's Sidcup home.
"This is live?" asked the sex worker.
"Twitch is where the Western world's underclasses go to demonstrate their lack of utility in the face of increasing mechanization and globalized manufacturing," said HRH. "Education has failed them. These children produce nothing but hours of live video. Each day hosts an onslaught of countless banal gigabytes. Millions of other children hang upon these performers, watching their every gesture and nuance."
"It's people playing video games?" asked the sex worker.
"What you are witnessing is the death of traditional media. Do you think these children have the capacity to thrill to the slight characterization that you discovered in Lovecraft? Do you believe that after hours of this plotless false intimacy they will return to television? Here we encounter the terminal point for millennia of narrative. Goodbye the Ferrari, Tony Kushner."
"I feel fucking old," said the sex worker. "And I'm only twenty-seven."
"Worry not. All of the Shropshire lads who salivate over MILF pornography will seek to unlock your wisdom of the ages. Forget you not, madame, that blood is a rover."
"Is this what we're doing tonight?" asked the sex worker. "Are we going to fuck or what?"
"Such crassness!" cried HRH. "Delightful! Delightful! Did I not inform you that I would demonstrate the greatest perversity? Do not think that Twitch itself constitutes the horror. There remains another dimension."
HRH scrolled down on the webpage hosting the Sidcup woman's Twitch channel.
HRH clicked the donate button.
The donate button opened another browser tab in Google Chrome.
HRH switched to this tab.
HRH filled out the form on the donate page.
HRH clicked donate.
A notification appeared on the Sidcup woman's stream.
It informed the woman and her viewers that HRH had donated £2,000.
The woman pulled off her headphones and began to cry.
"One cannot donate to any Twitch channel which experiences true popularity," said HRH. "Fellows with an audience in the hundreds of thousands will not evidence the appropriate response when presented with a mere £2,000."
The Sidcup woman screamed into her computer: "No. Oh my God. Oh my fucking God. What? No. No. No. Oh my God. No. Oh my God. No. No. No. Fuck. Fucking Hell. Oh my God, no. No. No. Fuck. No. What? What? WHAT?"
"How much money do you have?" asked the sex worker.
"The zeroes pile up like the bloated corpses of dissident intellectuals at Dachau," said HRH. "Imagine the earnings from a weapons-for-hostages scheme with the Islamic Republic of Iran and multiply that figure by a billion."
HRH leaned back in his DXRacer chair.
HRH vaped indica.
"It strikes my mind that perhaps there is a way to raise the pleasure," said HRH. "Would you care to indulge?"
HRH taught the sex worker to navigate channels on Twitch.
The sex worker navigated channels on Twitch.
The sex worker found channels belonging to sadder members of the Twitch community. People with four viewers, people who were streaming games that no one liked, people who were talking to an audience of no one.
The sex worker donated $1,000 to a bald man with a goatee who was playing the VGA remake of _Quest for Glory II: Trial by Fire_.
The sex worker donated $3,000 to a man who was playing _World of Warcraft_.
The sex worker donated $5,000 to a woman in Seoul. The woman was not playing a video game. She was watering her plants and singing along to "Lip & Hip" by 현아.
All three streamers pulled off their headphones. One started crying. All started cursing. One talked about dreams coming true.
"You see?" asked HRH. "One can change a life with nothing more than a donation of $3,000. Streaming video is the intellectual sweatshop of the future."
HRH told the sex worker to take it up a notch.
She donated $20,000 to a young woman dressed in _Sailor Moon_ cosplay.
Her shriek was so piercing that both HRH and the sex worker had to cover their ears.
"Shall we go for the big score?" asked HRH. "Do you wish to inhale the sweet smell of success?"
"What?" asked the sex worker.
"$100,000," said HRH.
"You actually have this much?"
"The bodies of Dachau. Arms sales to the Islamic Republic of Iran," said HRH. "For one night only, my cherub, with the contours of your Cthulhoid membrane illuminated by a liquid crystal display, money is of no concern."
"Let's do it," said the sex worker.
"My one request is that I pick your victim," said HRH.
HRH navigated to the Twitch channel of a young woman who was dressed like a sexy unicorn.
The sexy unicorn wasn't playing a game. She was speaking to the people in her channel's livechat.
"Okay, SweetA, thanks for the sub," said the sexy unicorn.
"No, DuskDot, I don't own a gun," said the sexy unicorn.
"Here she is," said HRH. "I have watched this one for a great long while. Her popularity is minimal. Her desperation is great. With one click, you will change her life forever. Imagine the surprise!"
The sex worker clicked on the donate button.
The sex worker filled out the form.
The sex worker donated the money.
A notification rose up on the sexy unicorn's Twitch stream.
The sexy unicorn sat in stunned silence.
The sexy unicorn could not believe what she was seeing.
The sexy unicorn checked to see if the donation was real.
The sexy unicorn threw off her headphones.
The sexy unicorn screamed.
The sexy unicorn started dancing in her lower-middle-class bedroom.
HRH leaned back in his DXRacer chair.
HRH vaped indica.
HRH smiled.
HRH experienced the shudder of a tantric orgasm.
"Do you realize that we've just changed that girl's life?" asked the sex worker. "We totally fucking changed everything."
"I am aware," said HRH.
"I can't believe it," said the sex worker.
The sexy unicorn was still dancing.
The sexy unicorn started jumping on her bed.
"She's probably never seen that kind of money in her life," said the sex worker.
"I guarantee that it is a new experience," said HRH. "Here, madame, is the true perversity. This is from where the greatest pleasure derives. You sit there and you believe yourself enmeshed in generosity, in the glow of altruism, in the spirit of human giving, but tonight you have done nothing but practice a refined form of cruelty."
"What?" asked the sex worker.
"You have taken that child and thrust her into a higher tax bracket," said HRH. "Do you believe that a peasant can handle a sudden influx of filthy lucre? Like yourself, she too is ignorant of the difference between money and wealth. She will spend this sum on clothes, on a new car, on trinkets and baubles, and when she has drained the swamp, there awaits the taxman. She will have no hope of paying. She will travel on, haunted by ever increasing debt. Her best chance will be bankruptcy after seven years. She will murder her credit and she will have learned nothing and she will own nothing. All of this because of a random act of violence perpetuated by a stranger while she was dressed in a unicorn costume that emphasized her heaving bosom. It will be your fault. You did this to that child. You have destroyed her."
HRH vaped indica.
## Chapter Twenty-Two
## Literary Fiction
While Fern's body burned with gasoline, and shrieks of human agony assaulted her ears, she had a thought.
Here was her thought: _This is not how I imagined things would turn out._
To fathom Fern's disappointment, you'll have to cast your mind back to the Year of the Salted Earth, which roughly corresponded with 1997 AD, 1417 AH, and 5757 AM.
Fern spent that year out of Fairy Land.
Her existence was a minor cultural stereotype.
She was living as a pretend artist in a St. Mark's Place apartment between Avenue A & First Avenue on the island of Manhattan, which was a borough of New York City.
Fern had been in and out of New York City for almost a decade, starting in the Year of the Unquenched Longing, which roughly corresponded to 1989 AD, 1407 AH, and 5747 AM.
It that year, Fern met a young woman named Denise. They dated for a short time, but it was fleeting. Denise had to move to Boston.
Before she left New York, Denise introduced Fern to the demimondes of Manhattan's East Village and the Lower East Side, two overlapping ethnic ghettos that had transformed into cesspools of petty crime and cheap drugs and were gentrifying into cesspools of international money laundering and the expensive drugs required to fuel international money laundering.
It was in the East Village, where people were wearing terrible leather jackets and even worse denim jeans, that Fern met a boy named Anthony.
Anthony was from Long Island, which was an island next to Manhattan.
The western part of Long Island encompassed Queens and Brooklyn, two of New York City's boroughs.
At its eastern end, farthest from New York City, Long Island was full of property soon to be the exclusive domain of the ultra-wealthy, where the ruling class would throw parties that commingled the Celebrity branch of American politics with the people who really ruled the world, namely its international merchant bankers.
In the space between the boroughs and the money, there was a heaping mass of vast suburbs.
Anthony was from the middle. He'd grown up in the heaping mass.
Fern met Anthony in a bar on Second Avenue.
The bar was full of ersatz punk rockers and old drunks from the Ukraine.
Their attraction was so obvious, and so apparent, that it made an audible noise.
All of the bar's drunks heard the noise.
The ersatz punk rockers heard the noise.
Because the bar was full of brains pickled in alcohol, and because its patrons were sitting in a relative darkness designed to hide the shame of their existences, neither the drunks nor the punk rockers could identify the sound's origin.
The noise sounded like this:
"What the fuck?" asked one of the ersatz punk rockers.
"Hи хуя́ себе́" said one of the drunks.
Then she went back to her drink.
Fern met Anthony in the Year of the Baroque Promise, which roughly corresponded to 1990 AD, 1411 AH, and 5751 AM.
After the love connection made its audible sound, Anthony talked to Fern about the Krautrock band Amon Düül II.
He said stupid shit like: "I found _Yeti_ at Bleecker Bob's and I had no idea what it was. 'Archangels Thunderbird' was one of those moments, you know? It fucking changed my whole fucking life. My God, those drums, that guitar."
This was the surface babbling of a human being who knew, on the cellular level, that he stood before the firestorm which would consume years of his life.
As his mouth spoke, so too did his subatomic particulars cry out: _Fuck me fuck fuck me fuck me love me love me I am yours fuck me fuck me flesh of my flesh burn me burn me my soul is boring a hole this second hole is penetrating the hole of your face the skull of your bone look at me here I am yours and yours alone and you are mine touch me I am the one for whom you have been waiting please please please please please. Kiss me, my darling, for I too am like you, I am a kinder from Bahnhof Zoo._
Unlike the love connection, the crying out of Anthony's subatomic particulars happened on a level of quantum physics that was inaudible to human ears.
Not even people who had passed the Cash Horizon would have heard.
But in their case, the inaudibility was irrelevant: the rich are incapable of love or its recognition.
Fern was neither human nor past the Cash Horizon.
She was from Fairy Land.
She heard every word.
They talked, they hung around the East Village, they fell into bed, they wandered through the city, and because they'd both consumed endless amounts of media, they were imbued with the photogenic qualities of New York City, and these qualities freighted their wanderings with cultural weight.
Everything was ridiculously romantic.
On their third date, Anthony and Fern were walking in Washington Square Park. They were in the park because they were headed to Jones Street. Anthony had talked Fern into seeing some folk singers at Caffe Vivaldi.
The folk singers in question were absurd historical anachronisms. They were as bad as the people who wrote novels and poetry in the Twenty-First Century AD.
One of the folk singers was a woman named Bianca.
She was in Anthony's Philosophy program at the New School for Social Research, and she was doing a doctoral dissertation on Spinoza.
"Why don't you ever talk about your family?" Anthony asked Fern as they passed the statue of Giuseppe Garibaldi.
"What do you mean?" asked Fern. "You don't talk about your family," said Anthony. "Do you have any brothers or sisters?"
At Caffe Vivaldi, they sat through an assortment of folk singers who sang 1930s AD ballads about the coming wave of international socialism. Then Bianca got on at the microphone against the back wall.
"Hi," she said into the microphone. "My name's Bianca. I'm going to sing a few songs, but while I'm setting up I thought my friend James could do a song. James is a folk singer. He moved to New York last week. I don't think he wants to do this, but if you give him a round of applause, I'm sure he'll come up. Let's have a warm welcome for James."
Bianca handed James her guitar. He put his mouth too close to the microphone.
"Uhm, hello people," James said, popping his p, "I'm, uh, I'm pretty nervous. This is the, uh, the first time I've ever performed in New York. I've never been in the city before, not before Thursday. I'm from Columbus, Ohio. Don't judge. We all, uh, have to be from somewhere and Columbus is pretty much just as good as pretty much anywhere. Well, kinda. Uhm, you know, sometimes back in Columbus my stuff doesn't really go over. I thought I'd play a classic from 1935, maybe one you haven't heard before. Someone played it for me last night on reel-to-reel. So, uhm, can you please be gentle? Kindness never killed anyone."
Fern thought about the simplicity of music on Fairy Land. Music without filter, music as in ancient times, the voice and the instrument, a holy sound in supplication to the divine.
The purity of what humanity had lost in its era of machines and computers and cars and airplanes. The lost society, the fallen dream, the missing kindness.
James looked so innocent, begging for mercy.
_I know how he feels_ , thought Fern. _Oh, please, please, please, let him be good._
James cleared his throat. He checked the guitar's tuning.
He played his song. This is what he sang:
_Every time I fuck them men_
_I give 'em the doggone clap_
_Oh, baby, I give 'em the doggone clap_
_But that's the kind of pussy that they really like_
_You can fuck my cock_
_Suck my cock_
_Or leave my cock alone_
_Oh, baby, honey, I piss all night long_
_If you suck my pussy, baby_
_I'll suck your dick_
_I'll do it to ya, honey, till I make you shit_
_Oh, baby, honey, all night long_
Long before the Year of the Unspoken Promise, Fern'd concluded that there was nothing new to experience, that all her future years would feature repeats of previous days.
She was like a sexy vampire in a novel by Anne Rice. She was bored by eternal life.
And then, in a bar in the East Village, surrounded by Ukrainian drunks and terrible black leather jackets, she discovered something new.
Meeting Anthony was like being in San Francisco in 1965 AD prior to America's construction of received drug experiences and dosing with high-grade Owsley lysergic acid diethylamide.
Unexplored territory.
It was insane love, _l'amour fou_ , sex magick, the post-coital sparkle of two souls in unison wandering through a fluorescent-lit grocery store at 11:30PM, stoned, drunk, lunacy born of a shared experience, tongue in the mouth as guns fire overhead.
More Bad Sex in Fiction!
Nomination forthcoming!
The vast suburbs of Long Island were built with a specific and exact purpose: to isolate their residents from the perceived chaos of New York City, which was conceptualized as the presence of racial minorities.
In Anthony's youth, he'd sensed vibrations beyond the vast suburbs, and grasped on an intuitive level that the very experience of the suburbs, and their pretense of isolation, were the byproducts of an economic scheme over which he, and everyone he knew, had no control.
America was a prison for the young: a person either went runaway and threw themselves on the lusts of strangers, or they integrated into the sorting mechanisms of the haute bourgeoisie and hoped that a natural gift would carry them into one of the economic scheme's higher echelons.
Anthony chose the latter.
He smoked too much pot, he read too many books, he drank too much beer.
He dated a vegetarian girl who wore Malcolm X glasses, had a Siouxsie and the Banshees poster above her bed, and owned an ill-tempered ferret named Pumpkin.
He did well in high school.
After earning an undergraduate degree at the University of Chicago, Anthony ended up in New York City, on the island of Manhattan, doing a Philosophy PhD at the New School for Social Research.
Which is where he met Fern.
During those ridiculously romantic wanderings around New York City, Fern's thoughts were haunted.
She'd met Anthony at an inopportune time.
She had to return to Fairy Land.
For two years.
And she couldn't tell the truth.
Imagine the scene: Fern explains to Anthony, who is focusing on a proposed marriage between rational materialism and strict empiricism, that she is a supranatural creature from Fairy Land and that her father was the bastard son of King Arthur and that her mother is the Regnant Queen, and that, oh yeah, all of this has been the subject of Elizabethan pulp fiction and a Jacobean play, and double oh yeah, Fern could not die and was capable of supernatural feats of magic.
She cast two spells on Anthony.
The first drenched him in the radiation of primal magic, altering his brain so that Fern's periodic disappearances wouldn't register as significant events.
Whenever the biochemistry of Anthony's brain produced a thought like: _It's fucking weird as shit that I haven't seen Fern inseventeen months_, it was replaced by another thought: _Fern's gone to Bloomingdale's_.
The other spell drenched him with a second dose of primal magical radiation and created an energy field that rerouted social inquiries.
If someone asked Anthony why they hadn't seen his girlfriend, the energy field would mess up their minds. The inquisitor would forget that they hadn't seen Fern. They'd forget her entire existence until the next time they encountered her in the flesh, at which point their brains would be stuffed with false memories of seeing Fern's nonexistent paintings at hopeless group shows around SoHo.
The spells sat on, and in, Anthony's body.
They imbued him with the bitter puissance of Fairy Land.
Fern left New York City.
The affair came in dense clusters of contact and absence: one year on, two years off. It was the ultimate long-distance relationship, minus the benefits of then-contemporary modern communication.
There were no letters, no phone calls, no nothing.
Fern disappeared and reappeared.
And the magic deluded Anthony into thinking that she'd never left.
In the Year of the Mechanized Baptism, which roughly corresponded to 1993 AD, 1413 AH, 5753 AM, Fern was back in New York City.
One night, while Fern's presence was changing the color of the bedroom, Anthony got on the telephone with his mother.
His mother had been born on Long Island.
She still lived on Long Island.
She told Anthony about his uncle's various bodily ailments, which included dementia, fecal and urinary incontinence, spontaneous bleeding, a lack of mobility, a loss of skin elasticity, and kidney disease.
Then she suggested that it was only a matter of time before her brother would return home from the state-funded institution in which he convalesced.
"He's not coming back," Anthony said to his mother. "No one gets better when they're suffering full-body failure."
"You're talking crazy," said Anthony's mother. "He's still young!"
A few weeks earlier, Anthony had left Fern on Manhattan and returned to Long Island, where he'd visited his uncle in the state-funded institution.
Anthony walked past the recreation room and found his uncle's room, where his uncle's useless machine of a body had been positioned in a chair.
The useless machine could not get up from the chair. It needed a functioning machine, in the form of a social worker, to help it stand.
This caused its own problem, because every millimeter of the useless machine was wracked with pain. When it was touched, waves of agony ran through the useless machine.
The useless machine could not talk.
The useless machine had wires coming out of its arms and a wire running through its penis into its bladder.
The useless machine was wearing socks that were stained with an instance of the useless machine's uncontrollable diarrhea.
So when Anthony's mother said that her brother was still young, Anthony started screaming.
Fern came out of the bedroom and watched as her lover's face turned red and watched her lover's mouth emit violent sounds and inadvertent spittle.
"You don't understand anything!" cried Anthony into the telephone.
"The body isn't something you can just fuck around with!" cried Anthony into the telephone.
"You've never been sick, you have no idea what it's like!" cried Anthony into the telephone.
That night, when Fern and Anthony engaged in some bad fictional sex, Anthony sobbed like an infant.
In the Year of the Mechanized Baptism, New York City played host to one of its storied events: the Whitney Biennial.
The Biennial was a display of artworks. It occurred every two years at the Whitney Museum of American Art.
Generally speaking, artworks were human-made abstract representations of three-dimensional reality.
Anthony wanted to go see a film by the Los Angeles-based artist William E. Jones.
The film was called _Massillon_ and it was included in the 1993 AD Whitney Biennial. Amongst other things, the film was about Jones growing up mega-homosexual in post-industrial Ohio.
"Before we see the film," Anthony said to Fern, "we should check out the show. The whole thing costs six bucks."
Everyone who'd come into contact with the energy field residing in Anthony's body believed that Fern spent most of her time painting. Anthony himself believed this.
When Anthony extended the invitation, Fern couldn't say no.
The Whitney Biennial was a professional obligation.
They walked uptown to the Biennial, which was housed in the Whitney Museum at the corner of 75th & Madison.
On the way, Anthony and Fern found themselves trapped in an unpleasant discussion.
The topic of this unpleasant discussion was familiar.
It was a reliable source of discord.
This was the topic: Fern's unwillingness to discuss her past.
Anthony was deeply suspicious that Fern was hiding vital information.
Which, of course, she was.
But Anthony's body was awash in huge amounts of testosterone and primal magic.
He could not imagine the information that Fern was hiding.
No one could!
Anthony's body had funneled his suspicion into some serious masculine bullshit. He was fixated on Fern's sexual history prior to the advent of their rutting congress.
He was convinced that she had a long history of shameful encounters.
From a certain perspective, this was true: Fern had more than her fair share of Fairy Land relationships, and she'd been visiting the mortal world since the Fourteenth Century AD.
But Anthony's thoughts were more pedestrian.
He was consumed with fleeting images of suburban fingerbanging, semen-smeared threesomes, and an excess of New York City blowjobs.
Don't forget: he was from Long Island.
"I just want to know the truth!" he shouted. "I can handle it!"
Even in the best of times, the Biennial was notorious for producing a high level of annoyance.
Everyone who visited an iteration of the Biennial left the Whitney Museum and complained about how the abstract representations of three-dimensional reality in the Biennial were the wrong abstract representations of three-dimensional reality to be displayed in a space dedicated to abstract representations of three-dimensional reality.
Unlike previous Biennials, the 1993 AD iteration had overthrown the tyranny of certain kinds of abstract representations of three-dimensional reality and replaced them with different abstract representations of three-dimensional reality.
The 1993 AD show was conceived and executed to engage with voices marginalized from the mainstream of the art world and American culture. It included people descended from the indigenous tribes of the Americas, and people descended from people brought in chains to support America's original economic scheme, and people exploring the subjugation of women.
And because it was the Year of the Mechanized Baptism, the Biennial occurred before the American capitalist class realized the inherent profitability in men who had sex with other men.
So the Biennial also included mega-homosexuals like William E. Jones.
Back in those days, William E. Jones was an excluded voice.
Now he's a faithful viewer of Reality TV!
_RuPaul's Drag Race_ !
Things change!
Capitalism can eat anything!
The 1993 AD Biennial presented two particularly controversial representations of three-dimensional reality.
The first was a video of the Rodney King assault, which was shot by a plumber and depicted the Los Angeles Police Department beating the shit out of a motorist descended from people brought in chains to support America's original economic scheme.
The second were the admissions badges, which everyone had to wear if they didn't want to be kicked out of the Whitney Museum.
The badges were designed by the artist Daniel Joseph Martinez and spelled out various iterations of the following phrase: I CAN'T IMAGINE EVER WANTING TO BE WHITE.
Everyone freaked out.
They freaked out so hard that they created a physical, and conceptual, environment of malice and paranoia.
If you think this is an exaggeration, reader, then I recommend that you read contemporary reviews of the 1993 AD Whitney Biennial. If you can find anything more positive than qualified sneering and affronted guilt, then you are a much better researcher than me.
And remember: the qualified sneering and affronted guilt came from people who were sympathetic to the show.
Fern and Anthony paid their collective $12 and were given admissions badges.
Fern's admissions badge said: EVER WANTING.
Anthony's admissions badge said: IMAGINE.
They wandered through the rooms and galleries of the Whitney Museum, bombarded by abstract representations of three-dimensional reality.
When Fern and Anthony arrived at the Cindy Sherman photographs of mannequins and exaggerated plastic genitalia and reproductive organs and BDSM masks, Anthony looked into Cindy Sherman's abstract representations of three-dimensional reality.
And Cindy Sherman's abstract representations of three-dimensional reality looked into Anthony.
"Here we find the nature of knowledge," said Anthony.
"What?" asked Fern.
"We can't know anything that isn't first filtered through our senses. There is no knowledge beyond that which is observed. There is no first truth. Sherman is presenting us with a moral lesson on instructive epistemology. This is the real nature of sex. This is the desire of all men. It may arrive disguised, but this is what happened when you allowed yourself to be fucked by beasts. The constructed nature of sex, informed by the media, informed by society, informed by ten thousand years of patriarchal society. Cindy Sherman has stripped it away. Now you can see them how they saw you. This is the truth of letting yourself be fingerbanged by animals."
For centuries, people had been dragging Fern to exhibitions of abstract representation of three-dimensional reality.
She'd been stuffed full of the nonsense that people said in salons, in museums, and in galleries. She'd suffered through endless men talking about abstract representations of three-dimensional reality.
And Fern was nobody's fool.
She hadn't endured these exercises in tedium and learned nothing.
She had cottoned on to the underlying, and unjustifiable, delusion that animated every one of these discussions: the religious belief that art, rather than money, was the most influential thing imagined by human beings.
And here was the only person with whom she'd ever fallen in love and he was condescending to her with an even older bullshit than discussions about abstract depictions of three-dimensional reality, and he was disguising it as bullshit about abstract depictions of three-dimensional reality.
Fern cast another spell on Anthony.
Right there in the Whitney Museum.
Right there in the Biennial.
It was one of the weirdest spells cast by anyone from Fairy Land.
The underlying nature of art was the ability of human beings to perceive an implied whole from the presentation of its parts.
Imagine the human face abstracted to the furthest degree:
No human face has ever looked like this.
And yet your brain, reader, has interpreted it as a face.
Fern's spell was intended to scramble Anthony's ability to apprehend the whole from the presentation of its parts. When the spell took its effect, and Anthony looked at the above, he would see this:
The spell was designed to wear off when they left the Whitney.
Its assumed virtue was this: it would make Anthony stop condescending to Fern about epistemology when really he was telling her she was a slut for sucking so much dick back in the suburbs.
But something went wrong.
For two years, Anthony had been saturated with the radiation of primal magic, and those two spells had sat in and on his body. They had done peculiar things to his biology.
This wasn't like messing up the mind of a landlady in Udine.
This was magic without precedent.
Anthony's biology rejected the third spell.
When Anthony's biology rejected the spell, the feedback caused an invisible magical explosion.
This explosion created a magical avatar of the 1993 AD Biennial.
In its most abstract form.
Here was the abstraction: the 1993 AD Whitney Biennial, focusing on artworks touching on issues of identity and social discord and presenting a critique of how very rich people had constructed the world, existed entirely as the largesse of very rich people.
The Museum and its Biennial were gifts from beyond the Cash Horizon.
All of the bad reviews, all of the upset, all of the guilt, all of the empowerment, all the renewed focus on marginalized voices.
It had happened because some very rich people wanted bragging rights. Through the arcane processes of those who had passed the Cash Horizon, the social capital of these bragging rights would be transformed into an actual capital.
And if you think that's an exaggeration, reader, then you could always look at the patrons listed in the catalogue of the 1993 AD Whitney Biennial.
The list is this: Emily Fisher Landau, The Greenwall Foundation, Philip Morris Inc., Sony USA Inc., Henry and Elaine Kaufman, The Lauder Foundation, Mrs. William A. Marsteller, The Andrew W. Mellon Foundation, Mrs. Donald Petrie, Primerica Foundation, The Samuel and May Rudin Foundation Inc, The Simon Foundation, and Nancy Brown Wellin.
Andrew W. Mellon was a war profiteer. He made a killing during the Spanish-American War, a conflict that was precipitated by an imaginary attack on an American sea vessel.
Primerica Foundation was the philanthropic wing of Primerica, a multi-level marketing operation that targeted lower- and middle-income Americans and got them to buy term-life insurance, as opposed to whole-life insurance, and invest the difference in mutual funds operated by Primerica subsidiaries. Multi-level marketing, by the way, was almost indistinguishable from a pyramid scheme.
Philip Morris Inc. sold a very pleasurable form of suicide.
Nancy Brown Wellin was the daughter of George Brown, who co-founded Brown & Root, which ended up as a Halliburton subsidiary.
Brown & Root supplied almost all of the logistical support for the Vietnam War, a conflict that was precipitated by an imaginary attack on an American sea vessel.
And here's a funny anecdote, apropos of nothing.
Nancy Brown Wellin was at the Armstrong Ranch on February 11, 2006 AD.
This was the day when, and the place where, then Vice President of the United States Richard B. Cheney, architect of the First and Second American Wars against Iraq, was trying to murder innocent animals and accidently shot a lawyer in the heart.
It was like gladiators before a Roman emperor.
You fight, sure, because otherwise another gladiator would kill you, but ultimately your life and your death and your fighting were interchangeable.
It was all someone else's entertainment.
You were paying obeisance to the Cash Horizon.
Then Fern's spell did something funny: it took that abstract representation of the Whitney Biennial and shot it forward through time.
The abstraction landed on the Twenty-First Century AD.
And that abstract representation infected the Internet and all human culture.
Fern doomed everyone in the Twenty-First Century AD to the worst possible fate: rehashing the Cultural Wars of the 1980s AD and 1990s AD, with all of its direct and internecine fighting, and doing it purely for the amusement and enrichment of people who had moved past the Cash Horizon.
When her spell fizzled, Fern took a good look at Anthony.
It was one of those things: when you live with someone, it's harder to notice subtle changes in their appearance.
Fern had missed it.
But now she could see.
And something was dreadfully wrong.
Fern left New York City.
In the Year of the Speckled Band, which roughly corresponded to 1995 AD, 1415 AH, and 5755 AM, Fern returned to New York City and moved back into Anthony's apartment on St. Mark's Place.
Anthony was not well.
It could not be ignored.
Anthony himself seemed unaware of the change.
Anthony kept plugging away at his PhD.
Anthony contributed to a handful of minor academic papers trapped in an arduous process of backbiting peer review.
Anthony kept teaching classes at Eugene Lang College, the undergraduate division of the New School for Social Research.
But his every step was tormented.
Fern cast spells trying to remove the primal magic and its radiation, but these too were repulsed by Anthony's biological transformation.
Fern left New York City.
In the Year of the Salted Earth, which roughly corresponded with 1997 AD, 1417 AH, and 5757 AM, Fern came back to New York City.
Anthony's apartment was empty of Anthony.
His possessions were there.
Anthony was not.
There was an eviction notice taped to the apartment's front door.
Fern cast a spell that handled the pressing issue of outstanding and future rent.
Then she tried to find her boyfriend.
It took some high-grade magic, and a ride on the Long Island Rail Road with a transfer at Jamaica station, but Fern found Anthony in the same state-run institution where his uncle's useless machine had run out of fuel.
Anthony had his own room.
The useless machine of his body had sprouted wires that were attached to other machines that monitored, and influenced, his weakening vital signs.
Sometimes he was lucid. Sometimes his useless machine would stop processing data.
Fern touched his face.
Anthony woke up. His milky eyes focused on Fern.
"I wondered when you'd show up," he said. "How was Bloomingdale's?"
Anthony's mother was in and out of the room.
His siblings were in and out of the room.
Fern never left.
She cast a spell which made her invisible to Anthony's family and the state-funded institution's staff.
When Anthony slipped back into consciousness, he and Fern would speak.
"Oh God," said his mother. "Now he's talking to himself!"
Fern tried to remedy Anthony with magic, but his body repulsed the spells.
She stayed in the room and watched as her boyfriend died.
She knew that she was the one who had assassinated him.
A day before Anthony died, he told Fern that he'd managed to complete his PhD dissertation.
"A lot of Maimonides," he said. "More than I would have thought fucking possible."
"And it was accepted?"
"I'm a doctor now," said Anthony. "Not that it helps."
When the end came, it was gentle, except for a brief moment in which Anthony began speaking with the dead.
"I see her there," said Anthony, his useless machine arm lifting itself and pointing to the empty doorway. "Why are you here, Edith? Keep away! Keep away! You never understood. Everything you said was a lie. Every word. Keep away! Keep away!"
Anthony's family thought that Anthony was talking to himself.
Fern looked at the doorway with the eyes of Fairy Land.
And for a moment, a luminescent human form was present.
It was a woman dressed in costume from Eighteenth-Century AD America. She was carrying a bouquet of flowers in her right hand and a scythe in her left. A fake beard was plastered on her brow.
One minute Anthony was there.
The next he was gone.
His mother wept.
Fern couldn't figure out how this woman had given birth to Anthony.
She couldn't understand how any of his family shared his lineal biology.
The things that they'd argued over while he lay in his sick bed.
Money, property, romances.
He was a man who'd dedicated his life to escaping the suburban isolation of Long Island. He'd thrown himself into the world. He had not left his home in shame or fear, but with the spirit of a conqueror, with the thirst of someone who wanted to know everything.
_He is me_ , thought Fern. _I am him_.
She too was from an island.
She too had chafed at the isolationism.
She too had fled everything.
Fern returned to their apartment.
She found a xeroxed copy of Anthony's PhD dissertation.
She read it.
All 263 pages.
She had absolutely no idea what the hell it said.
She walked to Fifth Avenue and went into the Graduate Faculty building of the New School for Social Research.
Before its acquisition by the university, the building had housed a department store.
It still felt like a space dedicated to shopping.
Fern took an escalator to the second floor and wandered past an abstract representation of three-dimensional reality. The abstract representation was a painting that depicted the Bacchae.
Fern found Anthony's advisor.
He was in his office.
By human standards, he was on the threshold of being ancient.
By Fern's standards, he looked like a baby.
"Can I help you?" he asked.
"I am Anthony's girlfriend," she said.
"It's a shame," said Anthony's advisor.
"Yes," said Fern. "It is."
There was a long silence.
"I read his dissertation," said Fern.
"Did you?" asked the advisor.
"I did not understand a word," said Fern.
"I'm afraid that we don't write for the layman," said the advisor.
Fern left the New School for Social Research and went south down Fifth Avenue, and through Washington Square, and through the West Village until she found herself on Jones Street.
She stood before Caffe Vivaldi.
She went inside.
Folk singers were performing.
The historical anachronicity of Caffe Vivaldi had increased after seven years of globalization.
Fern sat down. She ordered a cappuccino.
She realized that one of the folk singers performing historical anachronisms had also performed historical anachronisms on the night of Fern's third date with Anthony.
In the Year of the Baroque Promise.
The same person.
Doing the same thing.
Singing the same songs.
James wasn't there.
He'd taken his filthy mouth back to Columbus.
The advisor'd said that Anthony's dissertation was one of the best that he'd ever read, that Anthony was a star pupil, that Anthony had conquered everything he'd set out to conquer, that if Anthony had lived he would have made an immeasurable mark on the field, and even if Fern couldn't understand Anthony's abstract depiction of three-dimensional reality, she should take pride in it. The advisor was working on posthumous publication. The advisor would write an introduction that served as an in memoriam.
One of the folk singers sang a song by the Carter Family. It was called "Can't Feel at Home."
Part of it went like this:
_Over in glory land there is no dying there_
_The saints are shouting victory, there's singing everywhere_
_I hear the voice of them that I have heard before_
_And I can't feel at home in this world anymore_
You could be like Anthony and go out into the wide world and chase the only thing that was worth chasing, which was neither money nor power, nor love or comfort, but knowledge.
Escape the suburbs, rise through the social ranks, read more philosophy than is good for anyone, achieve a practical application in 263 pages.
_How does the world work?_
The thought was trapped like methane in tar, rising up, until she heard the folksinger, until she'd spoken to the advisor, until she'd read the dissertation and understood nothing.
_Oh I have a loving mother over in glory land_
_I don't expect to stop until I shake her hand_
_And I can't feel at home in this world anymore_
You could figure out how the world worked.
Anthony had.
You could develop a working model of how everything fit together.
Anthony had.
And it would mean nothing.
Knowledge was not power.
One person learning how the world worked had zero impact on how the world worked.
The boy who escaped his island only to be poisoned by the girl who'd escaped hers.
Years of fleeing Long Island.
Centuries of visiting the mortal world.
And he died six miles from the house in which he'd been raised.
The folk singer did one more verse:
_Heaven's expecting me, that's one I know_
_I fixed it up with Jesus a long time ago_
_He will take me through though I am weak and poor_
_And I can't feel at home in this world anymore_
Fern left New York City.
## Chapter Twenty-Three
## The Full Throat of Christian Virtue
Back in the Twentieth Century AD, there was a genre of writing called Science Fiction.
The writers of Science Fiction speculated about many possible futures.
A great deal of these possible futures involved robots, which were machines that emulated the bodies and practices of humans.
In some books of Science Fiction, the robots were friendly.
In some books of Science Fiction, the robots were mean.
In some books of Science Fiction, the robots replaced the humans.
In some books of Science Fiction, the robots were designed for pleasure.
But the writers of Science Fiction got the robots wrong.
In the whole history of Science Fiction, across all those tedious narratives bound in paper, not a single writer predicted the actual world in which we live.
No one ever suggested that the robots would be total fucking jerks controlled by Russia, or that the robots would use social media platforms to inflame emotions around the hot-button issues facing the 1993 AD Whitney Biennial, and that this use of social media would be part of a campaign to ensure that liberal democracy ate itself from within, and that in using these social media platforms, the robots would enrich a transnational class of oligarchs.
And no one ever suggested that the robots' use of social media would be quoted in articles written by actual journalists.
And no one ever suggested that these robots, in their mean spirit, would be indistinguishable from a plurality of the actual humans who used social media.
And, yes, reader, I know what happens to any writer who makes the mistake of mentioning Russia: instant Twitter accusations of working for the Russian government!
Let me state for the record that I don't work for Vladimir Putin, who is the President of the Russian Federation, or the FSB, who are the state security agency of the Russian Federation.
But I would!
Do you think I want to write hack bullshit about Fairy Land?
Buy me, Vladimir Vladimirovich, buy me!
If you want to understand the Hell in which we live, I suggest taking a look at changes in the American publishing industry throughout the Twentieth Century AD and Twenty-First Century AD.
At the beginning of the Twentieth Century AD, publishing was a father-and-son business. People had a press, they published writers they liked, and hopefully it worked out.
In the 1960s AD, the industry faced existential challenges of distribution, cost, and the sudden realization that the people no longer had to read trash for their numbing dose of daily entertainment.
These challenges resulted in in a wave of consolidation and mergers.
Where there had been, say, a hundred publishers, there were now about thirty.
The mergers continued throughout the 1970s AD, decelerated for a little while, and then kicked off again during the 1980s AD.
The latter decade introduced a new element: the presence of multinational conglomerates.
After the Democratic President William Jefferson Clinton signed the Telecommunications Act of 1996 AD, which deregulated rules of ownership, there was a wave of mega-media mergers that extended well beyond publishing.
Long before this happened, most of the United States' major publishers had been bought up by mega-corporations. In the new mergers, publishing was an afterthought. It was garnish on the meal.
By the mid-2010s AD, this was the state of the publishing industry: there were five major publishers, all owned by mega-companies, with three of the five owned by corporations not based in the United States.
The Big Five were Penguin Random House, Hachette, HarperCollins, Simon & Schuster, and Macmillan.
Macmillan was owned by Holtzbrinck Publishing Group, which was based in Germany.
Penguin Random House was owned by Bertelsmann, which was based in Germany, and which I've insulted enough to ensure that I'll be banished from American publishing for the foreseeable future.
Hachette was owned by Lagardère, which was based in France and was, for most of its history, powered by the manufacture and sales of weapons.
Simon & Schuster was owned by CBS, which was based in the United States.
HarperCollins was owned by Rupert Murdoch, and about whom I will soon say enough bad things to ensure that I'm banished from American publishing for the rest of eternity.
Technically, _The Future Won't Be Long_ wasn't published by Penguin Random House.
Technically, it was published by Viking, which once upon a time had been the Viking Press before it was eaten by Penguin Books and became Viking Penguin, and before Viking Penguin was eaten by Penguin Putnam, and before Penguin Putnam was rebranded as the Penguin Group and was eaten by Penguin Random House.
Reader, if you follow this metaphor of consumption to its logical end, you may imagine my failed novel as the excrement that follows such a hearty meal.
If there's one media outlet that has dominated the tone and tenor of American life since some Muslims facefucked life into a shitty disaster movie, it's Fox News, which is a network found on cable television.
Generally speaking, Fox News offers news from a Far Right perspective, and is consumed by an ongoing advocacy that Muslims should be reduced to a heaping pile of agonized screaming ash.
If you're in any liberal American home, and you want to invoke a series of paleoconservative values while implying your moral superiority to the people who hold them, all you have to do is wave your arms around as if you've been stung by a bee and shout this: "FOX NEWS! FOX NEWS! FOX NEWS!"
Everyone will know what you mean.
They'll know that you mean this: "Republicans want to burn gay people alive and put Black people in cages and fuck up everything that I believe! But that stops here! After the digestifs!"
I shop in a grocery store designed for the haute bourgeoisie.
The prices are ridiculous.
Other than the organic produce, every product in my local grocery has, somewhere on its packaging, a goofy narrative about the company that manufactures the product.
In my neighborhood, it is impossible to go to the local grocery store and buy mustard without encountering a whimsical tale about rural people from Northern California and Oregon and how their quirky values are reflected in the ingredients of their products.
These quirky values are why it costs $3 for a vegan cookie.
The narratives go something like this:
Twenty years ago, my wife Betty and I were in our kitchen, talking about the taste of the mustard that our parents bought. All of the store brands weren't anything like what we remembered, and they were made with pre-processed ingredients and contained preservatives. These chemicals might have allowed for a longer shelf life, but they reduced flavor, and even worse, no one knew what they did to people's health. "I wish someone would go back to old-fashioned values," I said. "Why won't someone make a mustard that tastes great and is good for people?"
Then Betty asked a question that changed our lives.
"Why don't we do it?"
I have watched hundreds of people read these narratives.
And as I have watched people read these narratives, the thought has occurred to me that people are more conscientious about their mustard than they are about the media they consume.
Reader, I have written a narrative in the voice of the man who owns HarperCollins and Fox News.
To acclimate you to its message, I've written this narrative in the style of stories that one finds printed on jars of organic mustard:
**MEET RUPERT**
Hi, I'm Rupert Murdoch. I'm having a cuppa in my country home in Mayfair, part of a little town that the lads like to call London. You probably don't know much about my story, but ooh, crikey, I reckon it's a real ripper.
Over sixty years ago, when I inherited an Australian newspaper from my father, I knew that people didn't want a landscape chocka with media outlets producing a true spectrum of thought. The world was crying out for an oligarchical structure of media ownership, where a handful of companies controlled everything and created a false dichotomy of public opinion.
I took my father's little newspaper and used it to gain an iron control over Australia's media landscape, and I funneled the obscene profits into a slow campaign against other countries. My first target was the English. Those old bogans couldn't resist my crass strategy of big tits paired with disgusting opinions for the ill-educated masses.
I moved on to America and did the very same thing. It was ace. The molls in the American government were some real wantons, and they deregulated their media landscape so that me and a few other big'un blokes could consolidate control over almost every outlet in the country. Television, film, newspapers, and publishing. Those Americans were bang up for it. What a bunch of naughty slags.
Maybe you'll recognize one of my profitable divisions. It's called Fox News. It does a cracking job of getting the olds upset about global warming and Christmas.
I also own HarperCollins, and one of the things that Harper-Collins does is publish books by American liberals. Strewth, it's a great deal! I use Fox News to make money off rightward turns of public opinion, and then I make money off the reaction to those rightward turns of public opinion by publishing books which the ideological opponents of Fox News quote like gospel scripture.
When I'm chopping logs for my old wood stove in Mayfair, I like to ask myself whether the liberal writers on HarperCollins, who are enmeshed in the media and entertainment industries, are so stupid that they don't know they're taking money from me, or if they're so cynical and motivated by their own atomized interests that they don't care. I never do make up my mind. Who can decide with that lot of saddos?
Remember when the feminist Internet sheilas were deadset about that Paki comedian fella Aziz Ansari rooting a young moll? That was a real laugh. Aziz was in a few of my movies. _Epic_ , _Ice Age: Continental Drift_ , and _What's Your Number?_ I jolly well paid for his holiday in the sun. Remember the original article that told the world about Aziz's rooting? It was published on a website called Babe.net. Guess who's an investor?
Do you recall when the benders at the _Guardian_ unleashed a real corker and said that _Empire_ , a hip-hop-themed television drama, was 'audaciously honest on Black issues'? Crikey, do you care to hazard a guess who produced _Empire_? Want to guess who owned the network? Guess who made the real money off the advertisements and sales into foreign territories? That _Guardian_ article was a shock! They made me sound a bloody golly!
I'm getting on in years, but I think I've done pretty well. Maybe some sook dags say I play the larrikin, but I run a family businesses. Ooh, crikey, I hope I've stayed true to my values. I know that when my time comes and I go meet the Great Sky Cunt, I've raised a right crop of young'uns who'll steer my works in the right direction.
If people from the Right Wing want to gain moral instruction, they go watch Fox News, and Rupert Murdoch makes money off the advertisements that are aired on the network.
If people from the so-called American Left want to gain moral instruction, they go buy a book published by a Certified Liberal who is being published by HarperCollins, and Rupert Murdoch makes money off the sale.
The purpose of anyone expressing a public opinion in American life, or consuming one, is this: to make money for about 1,500 people.
And don't think I'm singling out Rupert Murdoch.
Other than the phone hacking, anything you could say about Murdoch was true of 1,499 other individuals.
For instance: the American cable network which served as the ideological counterweight to Fox News.
It was called MSNBC. It stole Fox News's playbook and changed the cheap conservative opinions into cheap liberal ones.
Millions of people watched it every night, convinced that they were being given the inside scoop on how the Trump Presidency would crumble.
Because MSNBC wasn't a jar of mustard, it didn't come with a short narrative about its values, so maybe you can't blame its viewers for being ignorant of who was manufacturing their opinion.
But still.
The letters N-B-C appear in MSNBC.
And as everyone remembers, NBC was the broadcast network that aired fourteen seasons of _The Apprentice_.
_The Apprentice_ starred Donald J. Trump, and it was on that show where he honed the skills of televised humiliation and abuse which he would use to win the Presidency.
His last episode aired on February 16th, 2015 AD.
He declared his candidacy for the Presidency on June 16th, 2015 AD.
Comcast Corporation, which owns NBC, made big money off Donald J. Trump before he won the Presidency.
And then they made money after.
And, look, I can't judge any writer who gets paid by Rupert Murdoch.
I took money from Penguin Random House, and if I hadn't had a huge commercial failure, I'd be no different than anyone else.
I'd still be there, just another haute bourgeoisie aspirant chasing my small piece of the global media landscape.
I'd be hoping to crawl through the window before they locked it from the inside.
And to put an even finer point on it: through media coverage which generated advertising revenues, _I Hate the Internet_ made money for Rupert Murdoch.
I didn't even sign a contract with the devil and I still work for him.
Now here I am, disgruntled, and I'm like those Science Fiction writers of the Twentieth Century AD.
I see the future.
If you look at the corporate history of publishing, it's been the reallocation of assets from smaller pools of capital into larger pools of capital.
Within twenty years, at least one major American publisher will be majority owned by a conglomerate from either China or the Middle East.
Probably Qatar.
Maybe Saudi Arabia.
And then your moral instruction will come from writers who are cashing cheques signed by repressive regimes with long histories of human rights abuses.
Your opinions will come from writers who will be no different than New York University.
They will be founts of knowledge and they will be economically powered by hegemonies built with slave labor.
And you'll still be more concerned about who made your mustard.
None of this would be of any consequence.
Regardless of what is printed on tote bags, in normal circumstances books have no impact on the governing of any society.
And neither does television.
Popular entertainment is meaningless.
In a sane world, I'd be using the example of publishing to illustrate the increasing consolidation of wealth and money in the hands of a transnational global oligarchy, and bitching about how this excludes freaks from achievement in the arts.
But something terrible happened in 2016 AD: the ghosts of one million dead Iraqis cried out for a just revenge against their killers.
And the world listened.
And so a rogue member of the Celebrity branch of American governance took over the Presidency.
And Penguin Random House publishes his books.
And so does Simon & Schuster.
And so does Macmillan.
And so does HarperCollins.
But not Hachette.
There's still hope!
Ignore the arms dealing of its corporate parent!
Except:
La Librairie Hachette craignait, à juste titre, que les résistants n'appliquent à la lettre le programme du Conseil National de la Résistance (CNR) et ne nationalisent cet exceptionnel outil que les nazis admiraient et dont ils avaient envisagé de faire la base d'une énorme entreprise européenne placée sous leur contrôle... Obligés de céder, ils firent tout pour maintenir leurs positions au plus haut niveau dans la reconfiguration du capital envisagée. À la Libération, pour être sûrs que nul ne songerait à les accabler, ils firent réécrire une partie de leurs archives, en ajoutant par exemple qu'au cours d'une entrevue, Laval s'était montré glacial alors que, dans les faits, il avait été d'un commerce agréable, ou d'autres remarques que l'historien éprouve les plus grandes difficultés à repérer quand il consulte aujourd'hui ces documents savamment élagués en 1945.*
Imagine a litter of three-month-old kittens. They are locked in a box. No light penetrates the box. There is a steady supply of oxygen. There is no food or water.
The kittens are kept in the box beyond the point of starvation and dehydration.
They shriek and they moan, and they rend each other with their claws.
They kill each other.
The dead are eaten by the living.
One kitten will survive the rest, nourished on the corpses of its siblings, but its suffering will be the longest and, in its final days, it will die the worst death, lacking even the analgesic numbness that comes with inflicting pain on another living being.
Because they are dumb animals trapped in the immediacy of a terrible situation, none of the kittens will ask the right question.
None of the kittens will ask: "Who locked me in this box?"
The defense mechanisms that you've been given as a member of a Western liberal democracy will not save you and they will not save your children.
It will take several decades, but your future, and theirs, is digitally inflected feudalism.
There's a slow train coming.
Everyone knows it.
Your life, and your body, will have only one purpose.
You will make money for monsters beyond the Cash Horizon.
You will be the slave of HRH.
And because you will not kill the rich or mandate a wave of socialism, the best idea that you'll have will be to exercise your franchise at the ballot box, where you will choose a candidate who'll sell you down the river at the first flash of cash.
And your second-best idea will be to go out in public and fight with another poor person while a third poor person captures the action on a smartphone that they will turn into a monetarily profitable video for Facebook, Twitter, and Google.
And your third-best idea will be to become a cynical asshole who lies for money and writes thinkpieces to manipulate the emotions of naïve morons on the Internet.
And your fourth-best idea will be to become one of the naïve morons, and you will make money for your global overlords by pretending into devices built by slaves that the worst thing in the world is whenever a honky gas station attendant insults someone from Honduras.
And your worst idea will be to keep your head down and try to make a reasonably decent life while buying more shit and imagining that you have a special relationship with sports teams, the Celebrity branch of American governance, and intellectual property in which you have no economic stake.
None of this will save you.
If there are still historians in the future, and that's a big if, and their histories are not sanitized at the behest of centralized organizations, my guess is that the Twenty-First Century AD will be seen as the time when all the reasonably decent ideas developed by the Left were co-opted and conquered by the Right.
Identity politics, performance art, fluidity with mass media, total freedom of speech, post-modernism.
The mistake was in thinking that these tactics were the specific province of one ideology.
But a gun doesn't care who it shoots.
When those historians of the future write about 9/11, which was when some Muslims facefucked reality into a shitty disaster movie, it will be seen as the beginning of a moment that ended with the election of Donald J. Trump and the inauguration of the Hyperreal.
9/11 was avoidable, but its psychic message wasn't: the destinies that Americans believed were their birthrights, and the birthrights of their children, were not inevitable.
Those destinies were the accidental byproduct of an unparalleled prosperity boom.
And that boom ended about fifty years ago.
And to fill the hole, the American people embraced the one sector that never dies, the one industry that never goes away, the one stain that spreads forever.
They went to war.
Against the world.
Against each other.
Donald J. Trump was the natural consequence of an entire society that adopted unending slaughter as its central function.
If you believe that this is simply an American issue, you're wrong.
A few months before Trump's election, America's biggest partner in the war against Iraq, the United Kingdom, also heard the crying out of a million dead Iraqis.
And a majority of its citizens attempted suicide by withdrawing from the European Union.
And because no one ever mentions it: most of the European countries that have seen a Far Right resurgence were part of the so-called Coalition of the Willing.
The Coalition of the Willing was a shitty euphemism that President George Walker Bush used to describe the countries that his administration had beguiled or bullied into America's second war against Iraq.
And I know, reader, that causality isn't causation.
But still.
I remember.
And I hear the wailing ghosts of a million dead Iraqis.
And those future historians will also say this: when confronted by the total co-option of their tactics, and facing their greatest existential threat, American liberals doubled down on the very tactics that had been co-opted by their ideological enemies.
They fought over a corpse.
As a member of the Hyperreal, one message has resounded since the day of your birth: the discrediting of Christianity from refined intellectual opinion.
Christianity has been discredited for a very specific reason.
At its core, it exists in opposition to money.
And refined intellectual opinion is just another commodity.
Try some, buy some.
It's all just filthy lucre for Penguin Random House.
On those long and empty nights when I hear the voices of a million dead Iraqis, sometimes the ghosts speak in full paragraphs.
Sometimes they say this: _Jarett_ , _our terrible vengeance has reconfigured your entire society into a shitty iteration of the Church._
_We have restored the concept of original sin and we have forced your liberal intelligentsia to genuflect before it. We have ensured that they will do this in the form of vacuous statements regarding White Privilege issued between tweets about consequence-free hallucinogenic use, dank memes, and bespoke doggie daycare._
_We have elevated a faux-Left who write books for Rupert Murdoch, and we have ensured that your society is doomed to rehash the battles of the 1993 AD Whitney Biennial with all the efficacy and power of the Byzantine scholars who debated how many angels could fit on the head of a pin while the Ottomans were sacking Constantinople in 1453 AD._
_And, yes, Jarett, we find it very helpful that you've pointed out the apocryphal nature of this story about your ancestors and Byzantine scholars._
_We thank you for both your honesty and your needless pedantry._
_If you hadn't noticed, we're using the heightened language of rhetoric to make a greater point than mere factual happenstance._
_Besides, your taxes paid for our blood._
_So give us a break._
_We have transformed your fellow citizens of the Hyperreal into a laity with no control over the debate or the direction of their society. We have ensured that all they can do is offer useless peasants' revolts into cyberspace while enriching their feudal overlords and strengthening theostracism of their fellows. We have created the world's greatest device of excommunication in the form of the Internet._
_And, yes, Jarett, we have heard your objection to the archaic use of the word "cyberspace."_
_The paucity of books in Arabic translation ensured that a bootleg edition of William Gibson's_ Neuromancer _did not appear on the streets of Baghdad until February 2003, just prior to your country's invasion of Iraq. Its cover artwork was a heavily pixelated-and-dithered Internet JPG of a painting by Rowena Morrill._
_As such, we apologize to you for our lack of a fresher reference point and for failing to use_ au courant _terminology to describe your vast and unending apparatus of state surveillance, social shame, and pointless judgment._
_We note that in your fathomless need to play the pedant, you have emailed William Gibson and asked him whether or not an Arabic translation of_ Neuromancer _appeared in Baghdad. We further note William Gibson's response, in which he wrote that to his best knowledge there is no Arabic translation of the book._
_To this end, we refute William Gibson and assert again that a bootleg version of_ Neuromancer _appeared in Baghdad in February of 2003, arriving on the same shelves as the novels of Saddam Hussein. We dare you, and William Gibson, to prove that this did not occur._
_We have ensured that your feudal overlords benefit from the Church. They are the ones who make money off inflections of dogma about the 1993 AD Whitney Biennial. They own the device of excommunication. The hierarchal social pyramid of the classroom textbook has returned._
_You are near its bottom, but because you made the curious life choice to be an entertainer, you are not quite a serf. You are a jongleur. The third-oldest profession. Keep singing your songs. See how much effect it has on the world._
_We will tell you about the delusion that animates your stories. You were born at the tail end of the only fifty years in history when life got noticeably better. You grew up in an historical anomaly and you have mistaken the contours of this anomaly for The Way Things Work._
_The other 4,950 years of recorded history were bitter slogs through the wretched lives of miserable people suffering beneath unfair systems of governance. The weight of those years is against you, Jarett. What kind of idiot would assume that five anomalous decades are a better predictor of the future than the other 4,950?_
_Let's not pretend. Only people in about twenty countries had better lives during those fifty years. If you haven't killed them yet, perhaps you should inquire with the people of Iraq about whether or not they experienced a significant increase in their quality of life while suffering beneath an oppressive system of oil feudalism propped up by British Petroleum._
_Our greatest vengeance, Jarett, is that we have recreated the Church and removed from it any hope of the Christian virtues. Your entire society has reconstituted itself around a cruel medieval structure and stripped away that structure's slim benefits._
_Your Twenty-First Century AD is about everything interesting from your Twentieth Century AD being transformed into a very shitty religion ruled over by a high clergy of the haute bourgeoisie. They pray to monsters. Their faint wish is to somehow avoid their feudal destinies. But they too will fall._
_Everything will be top and bottom._
_There will be no middle._
_Now you live in a world where there is no hope, no charity, and no fraternity._
_Please enjoy Batman._
_Please enjoy Harry Potter, even if he is an unfulfilled ghulat al-latah._
_Please enjoy the Presidency of Donald J. Trump._
_Please enjoy Brexit._
_Please enjoy the rise of the Far Right._
_Good luck with the future._
_You will most certainly need it._
_PS: We also apologize for the instance last spring when we expressed surprise that your given name isn't spelled with two Rs and one T._
_But you killed us, Jarret._
_You did it with drone warfare._
_You did it with a child's toy._
_You did it with a radio-controlled airplane._
_Get over yourself._
So what the fuck, reader, why not?
If for no reason other than the bloody-minded perversity of the damned, you might as well embrace the most discredited idea in Western life.
You might as well ride dirty with Jesus.
And his ultimate message.
It's not like anything else is working.
You are more than your base impulses.
You don't have to follow the script of your life.
Don't be a dick.
The only things that they can't monetize are individual acts of kindness.
It occurs to me that I never explained how Arafat Kazi talked his way into the pit.
He found the box office manager.
Arafat Kazi said that he'd bought a ticket.
But that the ticket wouldn't scan.
And then he apologized.
And apologized again.
And again.
And again.
Think about it from the perspective of the box office manager: presumably this was a person who'd spent a great deal of his life talking to people who wanted free admission.
Surely, he was hardened against grifters and schemers.
But none of those people were dressed like circus performers.
They were not holy fools clad in motley.
And none of them apologized for the bother.
And none of those people got a free ticket.
And that's why I'm a Christian.
* Mollier, Jean-Yves. "L'édition française dans la tourmente de la Seconde Guerre mondiale." _Vingtième Siècle. Revue d'histoire_ 2011/4 (n° 112).
## Chapter Twenty-Four
## **The Man Who Said Bo! to a Goose**
After Rusticano, there was nothing else for it.
Celia went back to Fairy Land.
Fern went with her.
The Fairy Knight was left in the mortal world, doomed to wander for an uncertain term, but with the promise that his mother and sister would keep in touch.
Fern's return to Fairy Land occasioned much joy.
Magical charm returned to the island.
The lesbianism was explosive.
What could Fern do?
The experiment had failed.
Life had not turned out how she wanted.
Everything that she'd hoped would carry her through had turned out hollow.
In the end, all that remained was where she'd grown up.
Welcome to true adulthood, Fern.
And, sometimes, Fairy Land was visited by the fractured shimmering astral projections of people tripping on dimethyltryptamine.
As always, the women of Fairy Land believed that these astral projections were remnants of The People Who Came Before.
The astral projections tried communicating with the women.
But their voices came out like Morse code sent over a telegraph wire.
Dot, dash, dot, dash, dash, dot, dot.
One spirit appeared with greater frequency than the others.
Its form had become better defined, more human.
With each of its appearances, the spirit inched closer to speech.
Its words had begun to sound like English.
Like this: _lrhsssrsssslrlrllrlrrlrllrrssssllsssslrssssrrssssrlrlrssssslrllssssrlrlssssrlrlsssrrrlll_.
One night, as Fern walked past the Warbling Yews of Nevermore, she came upon this spirit.
It looked like a man.
It spoke like a man.
"Madame," the spirit said to Fern, "I perhaps wonder if your elvish brain can be run through its robust Mandelbrotian paces."
Fern stared at the spirit.
She'd heard the story of The People Who Came Before.
Who hadn't?
They'd been the original inhabitants of Fairy Land.
And they'd grown so weary of life that they made a bargain with a creature calling itself Eru Ilúvatar.
In exchange for the blessing of eternal peace, The People Who Came Before had traded away their narcissistic senses of selves.
They'd lost all that my/me/mine bullshit.
They'd lost the curse of language.
And then they'd disappeared.
Fern was freaked the fuck out.
It wasn't that one of The People Who Came Before was speaking.
With magic, none of the rules are ever set in place.
Weird shit happens all the time.
Fern was freaked the fuck out because she couldn't imagine how, by any possible quirk of magic, one of The People Who Came Before would materialize in Fairy Land while wearing a T-shirt that said this:
"Oh most noble spirit," said Fern. "Do you speak now from the Great Beyond?"
"If, in your greeny estimation," said the fractured astral projection, "the Great Beyond is the ketamine-flecked restroom of the local hotspot and private events space known as KABIN, then, yes, this hearty voice shouts from the Great Beyond."
"The restroom of KABIN," said Fern. "What are you doing?"
"I indulge in brief respite. I am in attendance at a fundraiser hosted by 2020 Democratic Presidential hopeful Senator Kamala Harris," said HRH. "Former Attorney General! Top Cop! Straight from the sewer milieu of San Francisco single-party politics! Law and order for the chaos of Trump's America! The iron fist of the prison–industrial complex sheathed in a red velvet glove!"
"You are in America," said Fern. "What part of America?"
"The District of Columbia," said HRH. "Yet my time in your world is as fleeting as the sanity of an unprepared pop sensation thrust into the charnel house of post-industrial fame. Will you not answer my question, madame, in the quick, while still we share our brief moments? I have traveled across time and space. If nothing else, I am a seeker!"
"What is the question?" asked Fern.
"For fifteen years, I have pondered one thought," said the projection of HRH. "My brain is as tormented as a hardened platoon of Achaeans struck down by the arrows of Apollo."
"What is your question?" asked Fern.
"How can one resolve the idiocy of Varg Vikernes," asked HRH, "with his undeniable aesthetic success? When I listen to his recorded works, it causes a great grotesque feeling in the interior self. I am experiencing the horrors of racism. Yet I thrill to the music. I must resolve this dilemma! Can you cut the knot, madame?"
Through all of the coincidental nonsense of fiction, Fern knew what HRH was asking.
She knew all about Varg Vikernes.
This was because of Anthony.
Her dead boyfriend had co-authored a posthumously published academic paper on Norwegian Black Metal.
This paper had appeared in 22:4 of _Popular Music and Society_.
Anthony's co-authorship had been, mostly, a favor to a fellow doctoral student named Jacob.
Jacob had been browsing compact discs at Generation Records on Thompson Street when he'd stumbled across the Fierce Recordings reissue of Darkthrone's _Transilvanian Hunger_.
This was back in The Year of the Speckled Band, which roughly corresponded to 1995 AD, 1415 AH, and 5755 AM.
When he first held the compact disc and its jewel case, Jacob had no idea what the hell was in his hands.
But the ultracontrast black-and-white cover art convinced him into an impulse buy.
Jacob went home and listened to _Transilvanian Hunger_.
He used the Internet, then in its pre-Google days, to search on Darkthrone and _Transilvanian Hunger_.
Jacob used a search engine called AltaVista.
AltaVista helped Jacob find out that Darkthrone was one of the foremost bands in the second wave of a subgenre called Black Metal.
AltaVista also helped Jacob find out that the words printed on the album's back insert—Norsk Arisk Black Metal—translated to NORWEGIAN ARYAN BLACK METAL.
Jacob had that old familiar feeling.
Heavy Metal, of which Black Metal was a subgenre, was like all rock music in the Twentieth Century AD: totally indebted, and dependent upon, the influence of African-American blues and R&B.
But there had been a trend in Heavy Metal.
Its practitioners had gazed towards the structures and presumed virtuosity of Classical Music.
Heavy Metal was a genre that pulled away from the African-American influence and sought inspiration amongst received conceptions of European tradition.
Jacob saw Black Metal as the furthest possible extension of this trend.
_Transilvanian Hunger_ was an album defined by its abject rejection, ideologically and aesthetically, of the African-American influence.
By virtue of this approach, and its resulting sound, the album was something totally new in quasi-popular music.
That old familiar feeling arrived whenever Jacob stumbled into the cheap wordplay that animates minor academic papers.
In an instant, he came up with a title: "Why are Black People Absent from Black Metal?: National identity, artistic convention and racist ideology in a new subgenre of heavy metal music."
Jacob got to work.
And he involved Anthony.
Jacob returned to Generation Records in search of more Norwegian Black Metal.
The early Internet had recommended several bands.
One name in particular kept popping up: Burzum.
Burzum was a one-man outfit.
Burzum's one man was Varg Vikernes.
In 1995 AD, while Fern was in New York City and worried about the effects of her magic on Anthony's health, she had spent a great deal of time listening to Burzum.
Anthony called it research.
But the truth was that he really liked Burzum.
And so did Fern.
Years after Anthony's death, Fern had ended up at a waterfront penthouse party in the Karşıyaka district of İzmir, Turkey.
The party was full of soft Turkish college boys who hadn't yet done their mandatory military service.
Or had parents rich enough, and well connected enough, to buy their sons out of mandatory military service.
On the penthouse's television, a pirated download was playing.
The pirated download was an iterative copy of the 2008 AD documentary film _Until the Light Takes Us_.
_Until the Light Takes Us_ was about Norwegian Black Metal, and it had been named after the English translation of a Burzum album.
The Burzum album was called _Hvis lyset tar oss_.
It had been released in 1994 AD.
The party wasn't much more than soft Turkish boys giggling as they looked at pornographic videos on each other's smartphones.
So Fern had no real choice.
She watched _Until the Light Takes Us._
It all came back.
Anthony in 1995 AD.
New York City.
Black Metal.
And as she watched _Until the Light Takes Us_ , Fern saw the story of Varg Vikernes.
How he'd come from Bergen, how he'd gotten involved with the Oslo Black Metal scene based out of the record store Helvete, how he'd dedicated himself and his music to a racist doctrine of vaguely Satanic neo-paganism, how he'd started burning down old wooden churches as a protest of the Semitic Christian invasion of Norway, how he'd murdered Euronymous, who was the owner of Helvete and founding member of the band Mayhem, and how Euronymous himself was no piece of work, having stumbled upon the corpse of Mayhem's lead vocalist after Mayhem's lead vocalist had blown his head open with a shotgun, and how Euronymous photographed the body and later made necklaces from the body's skull fragments.
Varg Vikernes was the star of _Until the Light Takes Us_ , unfathom-ably pompous, unchallenged, serving out his sentence for the murder of Euronymous, spouting neo-Nazi Norsk ideology from within prison walls, adopting the same insufferable persona that he'd developed for the Norwegian press of the early 1990s AD.
As she watched, Fern remembered how Burzum had soothed Anthony in his creeping pain.
She remembered Anthony telling her that Varg Vikernes was the key to the whole Norwegian Black Metal scene: he had released his own albums as Burzum, but he'd also played bass on Mayhem's _De Mysteriis Dom Sathanas_ and written the lyrics to one side of _Transilvanian Hunger_.
And yet Varg Vikernes was a total fucking idiot.
He was just a crap Nazi with an Odin fetish.
And after being released from prison, he'd done the same thing as every crackpot with his glory days behind him: he'd started a YouTube channel.
But the albums that he'd recorded before prison?
Absolutely fucking brilliant.
So when the fractured astral projection of HRH confronted Fern with his inquiry about Varg Vikernes, Fern could answer his question.
Ever since that penthouse party in Karşıyaka, she'd been wondering how Varg Vikernes's brief period of aesthetic genius could emerge from the body of an idiot.
"It's my guess," said Fern to the astral projection of HRH, "that generating art, and experiencing it, has no connection to the possession of intelligence. There have been millennia of humans writing words and making music and printing posters that insult politicians. Nothing has changed. Still you wallow in your filth. Still you elevate pigs above you. Only a fool would seek intellect amongst human aestheticians. Better if you look for inspiration amongst your plumbers."
"Madame," said HRH, "when next I relieve the vital center, with your words alone shall I shake, rattle, and roll."
HRH felt his spirit begin to return to his body.
"Yet my short time is not up! As you have expressed some interest in America, may I repay your kindness? Do you wish to know the secret of this great depraved land?"
"Amuse me, mortal," said Fern.
"America is a land in which one can be employed by Boeing or Lockheed Martin or General Dynamics," said HRH, "and in that employment be hired and paid for only one purpose. The sales and development of weaponry dedicated to the eradication of Muslim flesh. The more Mohammedians that one kills, the greater the rewards. Endless wealth! A rise in society! Invitations to the best galas! Come and find me in the golden hour of cocktail reception at the Saudi embassy. The drinks are, alas, without alcohol, but I am waiting and watching for you."
The astral projection of HRH began to shimmer.
"Pray, madame, do not think that the blessed war industry would exclude you on the basis of your fairer sex. Four of the five major American defense contractors are headed by women CEOs resolute in their dedication to massacre. When America bathes in the splattered excreta of Musulmans, the country is nothing but liberté, égalité, and sororité. Yes, madame, you also could build those bombs and you too could work towards the complete obliteration of the Saracens. Not one eyebrow raised! The world will be yours! I myself have had the pleasure of a healthy conversation with Marillyn Hewson following her triumphant acceptance of an Edison Achievement Award. She was touched to the core when informed that both my father and myself are shareholders who follow her good works. Extinguish enough lives and they will reward you with a profile piece in the Style section of the _New York Times_. Beau Brummell for the Blackwater generation! The pantsuits and pumps that power the putrefaction parade!"
The astral projection of HRH started to disappear.
"I take my leave," said HRH. "I must make quickest haste! America will not scold or shame you for the mass manufacture of weapons with no possible function other than wholesale slaughter. Feel free to murder tens of thousands. Carte blanche! More filthy lucre for The Conqueror!"
HRH disappeared.
His voice came in a final paragraph:
"Yet if you wish to maintain your position as a resoundingly fêted killer of the distant peasantry, then there is one mistake that you must never make. Never consume Zolpidem and power up your smartphone. For if you do, madame, perhaps you will discover the truest meaning of _in vino veritas_. What if, in your drugged haze, you log on to Twitter and refer to your victims with an unfortunate slur? The social and corporate structure of America longs for tsunamis of Mohammedian blood. Yet the human resources department will have zero tolerance for the scourge of online Islamophobia. Kill them all, O my elfen dearest, but never call them Ragheads! In America, the entire society will scrape and bow before your bloody conquest. But no one will ever thank you for your honesty."
The author, age 7, with his father
**Jarett Kobek** is a Turkish-American writer living in California. His novella _ATTA_ , a psychedelic biography of the 9/11 hijacker Mohamed Atta, was an unexplained bestseller in parts of Canada. His novel _I Hate the Internet_ was a bestseller everywhere, doing especially well in Serbia. His follow-up novel, _The Future Won't Be Long_ , wasn't a bestseller anywhere but did receive a shortlisting for the _Literary Reveiw's_ 2017 Bad Sex in Fiction Award and was published in the United States by a company that printed propaganda for Nazi Germany. So there's always hope.
Thanks for reading!
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