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Q: Having problems making a SUMO simulation work for omnet++ Trying to make my sumo simulation work on OMNeT++, but when modifying .launchd.xml i couldn't give permission to running sumo.
I'm currently using Ubuntu 18.04.2 LTS, sumo 0.32.0, Veins 4.7.1 and OMNeT++ 5.3.
I've searched for means to make a sumo simulation different from the erlangen example work. Until this point what I have found was modifying the erlangen.launchd.xml file to make it run my simulation and running
sudo python sumo-launchd.py -vv -c /home/gustavo/Downloads/sumo-0.32.0/bin/
However, everytime I tried to run it a message saying it lost connection appeared, so I tried creating a poly.xml archive with nothing in it (because I didn't want any buildings or something like that in the simulation) and it didn't work. I looked into the linux terminal and saw a message saying there was no sumo.cfg archive in the sumo-0.32.0/bin folder (I don't understand why it should, there is no .sumo.cfg archive from the erlangen example on that folder too), so I copied all the archives for the simulation (.net .rou .sumo.cfg and .poly) into the folder and tried again. This problem was solved but another error showed up in the terminal:
Could not start SUMO (/home/gustavo/Downloads/sumo-0.32.0/bin/ -c simulation.sumo.cfg): [Errno 13] Permission denied
I tried running it the command with sudo but it didn't solved it. Does anyone know how to make it work or another way of making my own sumo simulation work in OMNeT++?
<?xml version="1.0"?>
<!-- debug config -->
<launch>
<copy file="simulation.net.xml" />
<copy file="simulation.rou.xml" />
<copy file="simulation.poly.xml" />
<copy file="simulation.sumo.cfg" type="config" />
</launch>
I hoped to make my sumo simulation work on OMNeT++ because any other website that I looked didn't show that problem.
A: The parameter -c for the sumo-launchd expects the full path to the executable so you need to include sumo at the end:
sudo python sumo-launchd.py -vv -c /home/gustavo/Downloads/sumo-0.32.0/bin/sumo
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,833 |
Q: how to retrieve facebook, twitter, and google profile images without error using paperclip in rails 3.2 i have setup devise to handle my authentication and am also using omniauth to do the third party authentication which also works well for me.
i can retrieve the name, nickname, by setting this method in my user model
def self.from_omniauth(auth)
where(auth.slice(:provider, :uid)).first_or_create do |user|
user.provider = auth.provider
user.uid = auth.uid
user.username = auth.info.nickname
user.username = auth.info.first_name
user.email = auth.info.email
user.photo = auth.info.image
end
end
and i use paperclip to handle my image processing
whenever i try to authenticate a user with
user.photo = auth.info.image
i get an error like this
Paperclip::AdapterRegistry::NoHandlerError in OmniauthCallbacksController#facebook
No handler found for "http://graph.facebook.com/100006033401739/picture?type=square"
anyway around this or is there something am not doing right?
my OmniauthCallbacksController controller is as thus:
class OmniauthCallbacksController < Devise::OmniauthCallbacksController
def all
user = User.from_omniauth(request.env["omniauth.auth"])
if user.persisted?
flash.notice = "Signed in!"
sign_in_and_redirect user
else
session["devise.user_attributes"] = user.attributes
redirect_to new_user_registration_url
end
end
alias_method :twitter, :all
alias_method :facebook, :all
alias_method :google_oauth2, :all
end
A: The problem is that paperclip requires an image file and you are just passing an image url.
I'd suggest having 2 different attributes: photo and remote_photo:
*
*If logging in with facebook, google and twitter, you would set the remote_photo attribute to the auth.info.image returned.
*If uploading a photo, you would use the photo attribute.
However, if you want to download a person's facebook, google and twitter picture anyway, you can do this:
user.photo = URI.parse(auth.info.image) if auth.info.image?
This feature is kinda new so be sure that the version of Paperclip you are using is > 3.1.3.
A: From Paperclip version 5.2, the gem no longer loads IO adapters by default. It means if you pass an URI/HTTP strings/etc. as an attachment, it won't upload it automatically, as it used to do.
If you need to do that - for example, upload an photo that comes from Facebook Graph (omniauth integration) - add one or more adapters to your config/initializers/paperclip.rb, for example:
Paperclip::HttpUrlProxyAdapter.register
See other adapters here: https://github.com/thoughtbot/paperclip#io-adapters.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,265 |
Q: backup script to copy files by its extension I have created script to copy .png files from one directory to another. Script is as follows:
i="`cat /usr/local/app1/default.conf | grep -i values | sed -e 's/\values=//g' -e 's/,/ /g'`"
for data in $i
do
cp -rvp /usr/local/dir1/$data.png /home/user1/dir1
done
When I run the above script, this copies files from dir1 to /home/user/dir1 but at the end giving me error such as:
cp: cannot stat '/usr/local/dir1/wasw\r.png': No such file or directory
contents in default.conf:
values= wads sead seda okaw wasw
The command cp searches for these values 'wads', 'sead' etc but at the end I am getting the above error..
A: Actually I didn't get your use case, but all I understood is you want to find .png files from a directory and want to copy those files to another directory.If my understanding about your use case is correct then the below command helps,
find /home/your/source/path/ -name \*.png -exec cp {} /home/your/destination/path/ \;
A: You probably edited that default.conf file on windows machine and copied it over. There seems to be an extra character at the end of that line before the newline. Try od -c default.conf. It probably gives you something like:
0000000 v a l u e s = w a d s s e a
0000020 d s e d a o k a w w a s w
0000040 022 \n
0000042
Edit the file to remove the 022 character (which is \r) and try again.
A: If you want to keep the same code as you currently have, I suggest using sed a little more to its capacity :
i="`sed -n 's/'$'\r''//;s/,/ /g;s/values=//gp' /usr/local/app1/default.conf`"
for data in $i; do
cp -rvp /usr/local/dir1/$data.png /home/user1/dir1
done
How it works :
*
*The -n flag to sed tells it to not print the lines automatically
*The s/$'\r'// statement removes the \r characters that is probably causing your problems
*The added p at the end of the s/values=//gp statement prints the buffer only if a replacement was made (only if values= was present on the line)
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,966 |
Ascochyta caricae är en svampart som beskrevs av Ludwig Rabenhorst 1851. Ascochyta caricae ingår i släktet Ascochyta, ordningen Pleosporales, klassen Dothideomycetes, divisionen sporsäcksvampar och riket svampar. Inga underarter finns listade i Catalogue of Life.
Källor
Sporsäcksvampar
caricae | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,183 |
Severní Vněkarpatské sníženiny (polsky Podkarpacie Północne) jsou geomorfologická oblast na severní Moravě, ve Slezsku a v jižním Polsku v geomorfologické subprovincii Vněkarpatské sníženiny v Západních Karpatech. Podle polského členění podle Jerzyho Kondrackého nejde o oblast (makroregion), ale o jednotku o úroveň vyšší, tedy podprovincii.
Člení se na následující makroregiony (v českém členění makroregionu obvykle odpovídá geomorfologická oblast, avšak Ostravská kotlina není oblast, nýbrž celek). U každého makroregionu je uvedeno jeho číslo podle polského členění:
512.1 Ostravská pánev [CZ/PL] (Kouty - 333 m n. m.)
512.2 Kotlina Oświęcimska [PL] (Osvětimská pánev)
512.3 Brama Krakowska [PL] (Krakovská brána)
512.4-5 Kotlina Sandomierska [PL/UA] (Sandoměřská pánev)
Související články
Geomorfologické členění Česka
Geomorfologické členění Polska
Literatura
Geografický místopisný slovník, Academia, Praha, 1993.
Externí odkazy
Vněkarpatské sníženiny
Geomorfologické oblasti v Česku
Geomorfologické jednotky v Polsku | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,745 |
Investors worried about other IPOs due to loss of Paytm listing
November 29, 2021 by newsaid
New Delhi: The loss of Paytm listing has raised concerns among investors and entrepreneurs. He fears it could derail a string of expected Indian floatations, which were supposed to cement the country's position as a major destination for tech start-ups after the US and China. This was stated by the Financial Times.
The debacle has focused the bookrunners on IPOs including Paytm, its shareholders SoftBank, Alibaba, Goldman Sachs, Morgan Stanley and Citigroup.
India's head of equity capital markets at a western bank said, the concern for all of us is whether this affects the technical sentiment of broader India.
Evaluation is going to be very difficult.
Indian fintech company MobiKwik has delayed its IPO originally scheduled for November, saying it will list in due course, the report said.
Ashneer Grover, co-founder of Fintech BharatPe, said that Paytm has spoiled the Indian market.
According to a Financial Times report, Sandeep Murthy, partner at investment group Lightbox in Mumbai, said there may be some period of cooling off in fintech listings by early next year, but argued that it was natural.
Indian tech companies have raised a record $5 billion through listings this year, which is almost 10 times more than last year.
Prashant Gokhale, co-founder of Hong Kong-based research group Althea Capital, said Paytm's core business is not to make money and the move to cut marketing spend indicates that it is a better form of publicity with SoftBank and Warren Buffett prior to listing. Trying to show the bottom line.
A person with direct knowledge of Paytm's IPO pricing said there were deals chasing a lot of liquidity, especially with actions in China making India more attractive as a destination.
"Investors are desperate to leave, as prices were raised without improving fundamentals," the report said. The one who didn't get a lot of money chasing that deal is probably happy now.
Paytm's large Chinese ownership also poses a regulatory and reputational risk after India imposed tough restrictions on Chinese investments following military tensions last year.
Alibaba and its financial arm Ant sold shares in the IPO, while they still own about a third of the company, reports the Financial Times.
The least encouraging thing to me was that the market was not in a state of irrational enthusiasm, Lightbox's Murthy said of the company's launch.
Disclaimer: This is a news published directly from IANS News Feed. With this, the News Nation team has not done any editing of any kind. In such a situation, any responsibility regarding the related news will be of the news agency itself.
Government health expenditure share in GDP increased from 1.15 percent to 1.35 percent
Kerala-based Made Tech Startup wins Startup India Grand Challenge 2021
ट्यूनीशिया कैंप में सात खिलाड़ी COVID-19 के लिए सकारात्मक परीक्षण | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,321 |
About Clive
Negotiation Training
1-on-1 Negotiation Training
Corporate Negotiation Training
Barcelona and Arsenal need to Step into each other's Shoes
I'm not sure that all this public positioning around the Fabregas transfer deal by Barcelona and Arsenal really helps to get a deal done. Wenger calls Barcelona disrespectful; Barcelona speak publicly of their desire to bring home a player who is under contract to someone else.
This kind of positioning stops each party seeing the other side's point of view, which is normally an essential component in negotiating a successful outcome. If the roles were reversed and Arsenal were chasing the signature of an established Barcelona player who was English and yearned to return to England, then Wenger would have no hesitation about pursuing the player in public to try and improve his team. Equally, Barcelona would be just as outraged as Wenger is now about the "disrespectful" public pursuit of one their own key players.
If they could put themselves in the other side's shoes, then it would be easier for the clubs to find a price which reflects Arsenal's need for respect, Barcelona's need to continue achieving ground-breaking success, and the player's need to feel that he is back where he belongs.
As it is the public posturing prevents this kind of constructive negotiating, sours the climate, and means the deal takes longer to do. None of this is anybody's interest – especially Arsenal, who are running out of time to negotiate for replacements before the season starts…
By Clive Rich|July 20th, 2011|Blog|Comments Off on Barcelona and Arsenal need to Step into each other's Shoes
About the Author: Clive Rich
Clive Rich is an international deal maker and trouble-shooter. He has played a crucial role in negotiating and brokering more than £10bn worth of global deals for or with a broad spectrum of multi-nationals, major organisations and super-brands, including Sony, Yahoo, Apple, Napster, BMG, San Disk, My Space and the BBC. He is the author of The Yes Book as well as founder of Close My Deal.
Just In! The NEW 7-Steps to Negotiating Anything You Want.
Get the Yes Book
The Chaos Stage in a Negotiation
Heavy Turbulence grounds British Airways Negotiations
No substitute for cash; why the Glazers goal is to keep possession at Man Utd
Archives Select Month September 2013 August 2013 July 2013 April 2013 March 2013 February 2013 January 2013 November 2012 October 2012 June 2012 April 2012 March 2012 January 2012 December 2011 November 2011 October 2011 September 2011 August 2011 July 2011 June 2011 May 2011 April 2011 March 2011 February 2011 January 2011 December 2010 November 2010 October 2010 September 2010 August 2010 July 2010 June 2010 May 2010
Close my Deal
Rich Futures
Copyright 2013 CliveRich | All Rights Reserved | Contact Clive | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 6,226 |
Bitcoin is a term shrouded in mystery: what is it, exactly? Is it legal? What can you do with it? How is its value determined? All of these questions and more were recently answered when High Country Properties' own Bob Struwe represented a buyer in Utah's first, and one of the nation's largest, Bitcoin real estate transactions.
When the buyer approached Struwe, it was clear that it could be a challenge to make a deal happen. "We were running into a real problem," Struwe said. "His ideal scenario would be if a seller would take Bitcoin, and I knew that was going to be pretty remote." Many people have never heard of Bitcoin, and still others often associate it with illegal behavior that made headlines years ago. Luckily for the buyer, Struwe was not deterred and set out to make a deal happen.
The $3.5 million dollar deal Struwe was able to negotiate for his client is one of the largest Bitcoin transactions to date, and Utah's first. The deal took just 7 days to close, while most cash transactions average three to four weeks. One of the reasons this deal went through so quickly is the nature of Bitcoin itself. It's in both the seller and buyer's best interests to proceed quickly before the value of the currency changes.
Since finalizing the transaction, Struwe has been contacted by buyers from around the country and the world who are interested in purchasing property. With Park City real estate at a premium, and with Bitcoin emerging as a new way to purchase property, Struwe and High Country Properties is at the forefront of a new age in real estate. | {
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} | 251 |
La Coulonche is een gemeente in het Franse departement Orne (regio Normandië) en telt 493 inwoners (2005). De plaats maakt deel uit van het arrondissement Argentan.
Geografie
De oppervlakte van La Coulonche bedraagt 14,6 km², de bevolkingsdichtheid is 33,8 inwoners per km².
Demografie
Onderstaande figuur toont het verloop van het inwonertal (bron: INSEE-tellingen).
Externe links
Gemeente in Orne | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,945 |
4 NATO troops killed in Afghanistan's south, east
Two Afghan army soldiers, who were wounded in an IED explosion, are carried into a U.S. Air Force HH-60G Pave Hawk helicopter to be evacuated to Kandahar Air Field in Afghanistan's Kandahar province on Sunday, Oct. 3, 2010. Air Force pararescuemen and helicopter pilots from the 46th and 26th Expeditionary Rescue Squadrons are supporting ongoing military operations in southern Afghanistan. (AP Photo/David Guttenfelder)
By ROBERT KENNEDY, Associated Press Writer
KABUL, Afghanistan —
An Afghan herdsman guides a flocks of sheep in a suburban highway in Kabul, Afghanistan, Monday Oct. 4, 2010. (AP Photo/ Gemunu Amarasinghe)
An Afghan herdsman walks among buffalos in a suburban livestock market in Kabul, Afghanistan, Monday Oct. 4, 2010. (AP Photo/ Gemunu Amarasinghe)
Three NATO service members were killed by bombings in southern Afghanistan on Monday and an insurgent attack killed another in the east, raising the coalition's death toll to 11 in the first four days of October.
It has been the deadliest year for international troops in the nine-year Afghan conflict, and the escalating toll has shaken the commitment of many NATO countries with rising calls to start drawing down troops quickly.
NATO did not specify where Monday's bombings and attack occurred, and didn't give the service members' nationalities.
It also announced a joint Afghan-coalition unit launched a night mission that killed a senior Taliban leader named Farman and two other militants in eastern Paktia province. Farman "terrorized the local population by participating in attacks, kidnappings, interrogations and executions of Afghan civilians," NATO said.
An insurgent with the Haqqani network responsible for attacking coalition and Afghan troops was captured in an operation Sunday in eastern Khost province, the alliance said. The Haqqani network is a Pakistan-based faction of the Taliban with close ties to al-Qaida.
The group was started by Jalaluddin Haqqani, a commander once supported by Pakistan and the U.S. during the 1980s war against the Soviet Union in Afghanistan. Haqqani has since turned against the U.S., and American military officials have said his organization - now effectively led by his son, Sirajuddin - presents one of the greatest threats to foreign forces in Afghanistan.
Taliban spokesman Qari Yousef Ahmadi accused NATO of engaging in a propaganda campaign to demoralize the insurgents' moral by inventing Taliban leaders and alleging they were killed or captured.
"Most of the commanders' names NATO are using don't even exist," Ahmadi told The Associated Press. "This is just a game from the American side, nothing else."
In western Nimroz province Monday, a police convoy was ambushed in Khash Rod district, said provincial police chief Gen. Abdul Jabar Pardeli. Five militants were killed, three others wounded and two captured during a gunbattle. Police suffered no casualties, he said.
In other violence, a former district chief, Habibullah Aghonzada, was gunned down by assailants as he prayed at a packed mosque in Kandahar city on Monday, the governor's office said in a statement.
On Sunday, three insurgents died in an Afghan and NATO operation Sunday in Kandahar province's Arghandab district, the statement said. The raid in Khisroo village also recovered explosive material and an anti-personnel mine that were destroyed.
Afghan security forces, meanwhile, were attacked by militants in Kandahar's Panjwai district on Sunday, the statement said. No casualties were sustained by either side after a firefight.
Control of Kandahar, the Taliban movement's birthplace, is seen as key to the Afghan conflict. Afghan and NATO forces are engaged in a major operation to push out militants from strongholds there.
Associated Press writers Rahim Faiez in Kabul and Mirwais Khan in Kandahar contributed to this report. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,662 |
const tmp = require('tmp');
const path = require('path');
const copy = require('recursive-copy');
describe('[Regression](GH-637)', function () {
it("Should let test file locate babel-runtime if it's not installed on global or test file node_modules lookup scope", function () {
tmp.setGracefulCleanup();
const tmpDir = tmp.dirSync().name;
const srcDir = path.join(__dirname, './data');
return copy(srcDir, tmpDir).then(function () {
const testFile = path.join(tmpDir, './testfile.js');
return runTests(testFile, 'Some test', { only: 'chrome' });
});
});
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,377 |
On Wednesday (June 27), friends, family and fans of XXXTentacion, who was shot and killed during an apparent robbery on June 18, attended an open-casket public viewing for the slain rapper. Among the guests at the event, which is currently taking place at Sunrise, Fla.'s BB&T Center (the Florida Panthers Stadium), were former members of the XXL Freshmen Class including Lil Uzi Vert, PnB Rock and Ugly God. Denzel Curry, Fat Nick, Kid Trunks, Ronny J and some of X's friends were also in attendance.
Other attendees include Rolling Loud co-founder, Tariq Cherif and John Cunningham, who executive-produced X's ? album. During his time at the event, Cunningham could be seen gathering people to help carry handicapped fans to the stage so they could see X in his casket.
Local Florida news outlets report that the memorial, which is taking place today until 6 p.m. EST, is supposedly going to be attended by as many as 20,000 people. According to USA Today, at least 1,000 people had arrived to attend the memorial by 12 p.m. EST. Even with his legion of loyal fans, many of whom are wearing X's signature Revenge hoodies and XXXTentacion shirts, Uzi, PnB and Ugly God stood out.
Uzi, who's just pledged to start a foundation to help support X's family, stood at X's casket with his mother, Cleopatra Bernard, as they peered down at the rapper, who was wearing a denim jacket. His hair was as blue as it was the day he passed. X's platinum-selling song "Changes," which features PnB, played as the two stood at the casket of the late rapper. PnB, who posted an emotional behind-the-scenes video of himself and X recording the song in tribute days after his death, watched Uzi and XXXTentacion's mother as the song played. Ugly God could be seen sobbing.
For an event of such size, X's memorial is remarkably well-organized and attendees are said to be following directions given to them. Fans could be seen making X signs with their arms or crying as they walk on and off the stage on which the rapper's casket rests. Some knelt before the casket and said prayers before stepping off the stage. The late rapper's music played on repeat, as did intermittent messages from X.
At another point of the viewing, "Look At Me!," which was X's breakout single released in 2015, began playing over the loud speakers and the crowd began chanting.
The set up for the stage includes a dozen bouquets of silver leaves and black roses. Two large screens hang on either side of the curtain hanging behind the casket. The stadium's scoreboard played an X tribute video, which was created by Justin Staple, Quinton Dominguez, and Tyler Benz.
In the video, archival footage of X in concert or servicing his community can be seen, while his tracks like his ? song, "Alone, Part 3," play in the background.
XXXTentacion passed just over a week ago, but the Broward Sheriff's Office has already made a key arrest. Last Wednesday (June 20), they arrested Dedrick Williams, who they believe may have been the getaway driver for those who shot and killed X. Williams has pled not guilty to first-degree murder. The police are searching for two more suspects who they believe actually shot the 20-year-old rapper. Authorities named 22-year-old Robert Allen as a person of interest earlier today (June 27).
See the X tribute video played at the rapper's memorial for yourself below. | {
"redpajama_set_name": "RedPajamaC4"
} | 5,910 |
require 'rails_helper'
RSpec.describe AssetSubtype, :type => :model do
let(:test_subtype) { create(:asset_subtype) }
describe 'associations' do
it 'has an asset type' do
expect(test_subtype).to belong_to(:asset_type)
end
end
describe 'validations' do
it 'must have an asset type' do
test_subtype.asset_type = nil
expect(test_subtype.valid?).to be false
end
end
it '.full_name' do
expect(test_subtype.full_name).to eq("#{test_subtype.name} - #{test_subtype.description}")
end
it '.to_s' do
expect(test_subtype.to_s).to eq(test_subtype.name)
end
it 'responds to api_json' do
expect(test_subtype).to respond_to(:api_json)
end
end
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,224 |
Carriera
Dal 2011 al 2012 è stato l'allenatore della Nazionale di pallacanestro femminile del Brasile, che ha guidato ai Giochi olimpici di Londra 2012.
Collegamenti esterni | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,688 |
var topiarist = require('topiarist');
var using = require('typester').using;
var deepEqual = require('deep-equal');
var AbstractPostableRef = require('./AbstractPostableRef');
function FakeWindowRef(fakePostableWorker, origin) {
AbstractPostableRef.call(this);
this.fakePostableWorker = fakePostableWorker;
this.origin = origin;
}
topiarist.extend(FakeWindowRef, AbstractPostableRef);
FakeWindowRef.prototype.postMessage = function(message, targetOrigin, transferList) {
using(arguments)
.verify('message').object()
.verify('targetOrigin').nonEmptyString()
.verify('transferList').optionally.isA(Array);
var targetWindowRefs = (targetOrigin == '*') ? this.fakePostableWorker.windowRefs :
[this.fakePostableWorker.windowRef(targetOrigin)];
var messageData = messageEvent(message, this.origin);
for(var windowRefIndex in targetWindowRefs) {
var targetWindowRef = targetWindowRefs[windowRefIndex];
for(var listenerIndex in targetWindowRef.listeners) {
var listener = targetWindowRef.listeners[listenerIndex];
listener(messageData);
}
}
};
function messageEvent(data, origin) {
var serializedData = JSON.parse(JSON.stringify(data));
if(!deepEqual(data, serializedData)) throw new Error('postMessage() invoked with unserializable data');
return {
data: serializedData,
origin: origin
};
}
module.exports = FakeWindowRef;
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,048 |
{"url":"https:\/\/www.physicsforums.com\/threads\/quasars-with-high-redshift-in-nearby-galaxies.505277\/","text":"# Quasars with high redshift in nearby galaxies\n\n## Main Question or Discussion Point\n\nhttp:\/\/www.sciencedaily.com\/releases\/2005\/01\/050111115201.htm\n\nPlease see this article. I've been trying to find the thread with 41 questions posted aces days ago. One of the questions was relating to high redshift objects like quasars appearing in near by galaxies. Someonevwanted a reference for such occurrences.\n\nYesterday I came across the abovevarticle, so I thought I'd post it here to get some possible explanations for the cosmological phenomenon.\n\nAny ideas? Or is this false reporting?\n\nphyzguy\nGeoffrey Burbidge had this opinion (that quasar redshifts are not cosmological) for decades, and steadfastly refused to abandon it despite mountains of evidence to the contrary. Note that this article is from 2005, and that Geoffrey Burbidge has since passed away and so will no longer be joining the debate. The obvious explanation is that the arrowed quasar in the picture is behind the nearby galaxy and is in fact as far away as its redshift indicates. There are currently very, very few astronomers who continue to believe that quasars are nearby.\n\nJonathan Scott\nGold Member\nGeoffrey Burbidge had this opinion (that quasar redshifts are not cosmological) for decades, and steadfastly refused to abandon it despite mountains of evidence to the contrary. Note that this article is from 2005, and that Geoffrey Burbidge has since passed away and so will no longer be joining the debate. The obvious explanation is that the arrowed quasar in the picture is behind the nearby galaxy and is in fact as far away as its redshift indicates. There are currently very, very few astronomers who continue to believe that quasars are nearby.\nThat explanation doesn't fit well with the quasar spectrum showing very little evidence of intervening material of the expected density along the line of sight at the appropriate redshift, although there could of course be a coincidental \"thin patch\" in the foreground galaxy along that line. It's true that a galaxy is nothing like as \"solid\" as it appears, but from what I've read about it (even from sceptics in this case) a line of sight that near to the core of a galaxy would normally be expected to show signs of much more intervening gas and dust, as has been observed in other cases where objects are seen through a foreground galaxy.\n\nIt appears that Morley B Bell in Canada is one astronomer who is still looking actively at the evidence; he has produced or co-authored several papers (search for M B Bell) analyzing quasar statistics and related observations which show that if you assume quasars redshifts are cosmological, the resulting explanations are complex and leave many unsolved puzzles, but if you assume the redshifts of young quasars have a large intrinsic component which decreases towards zero with time (as Arp claims) then all of these puzzles vanish and you suddenly only have one key thing to explain, the intrinsic redshift, for which there is admittedly thought to be no known explanation compatible with the standard interpretation of General Relativity.\n\nGiven that GR is already being adjusted heavily on galactic and cosmological scales by ideas involving dark matter and dark energy, I think that the chances are that quasars will demonstrate yet another case where we will eventually have to supplement or modify GR.\n\nbcrowell\nStaff Emeritus\nGold Member\nGiven that GR is already being adjusted heavily on galactic and cosmological scales by ideas involving dark matter and dark energy, I think that the chances are that quasars will demonstrate yet another case where we will eventually have to supplement or modify GR.\nThis is incorrect. Dark matter and dark energy don't require any modifications to GR. The GR used in modern cosmological models is the same GR Einstein published in 1915.\n\nbcrowell\nStaff Emeritus\nGold Member\nIt appears that Morley B Bell in Canada is one astronomer who is still looking actively at the evidence; he has produced or co-authored several papers (search for M B Bell) analyzing quasar statistics and related observations which show that if you assume quasars redshifts are cosmological, the resulting explanations are complex and leave many unsolved puzzles, but if you assume the redshifts of young quasars have a large intrinsic component which decreases towards zero with time (as Arp claims) then all of these puzzles vanish and you suddenly only have one key thing to explain, the intrinsic redshift, for which there is admittedly thought to be no known explanation compatible with the standard interpretation of General Relativity.\nA search for his name on arxiv, http:\/\/arxiv.org\/find\/astro-ph\/1\/au:+Bell_M\/0\/1\/0\/all\/0\/1 , gives 31 results. Does one of these papers contain the claim that \"if you assume quasars redshifts are cosmological, the resulting explanations are complex and leave many unsolved puzzles?\"\n\nphyzguy\n... if you assume quasars redshifts are cosmological, the resulting explanations are complex and leave many unsolved puzzles ...\nPlease list at least one of the 'unsolved puzzles'.\n\nPlease list at least one of the 'unsolved puzzles'.\nIsn't the accelerating expansion of the universe one of the unsolved puzzles which is linked to dark energy but not fully comprehended?\n\nbcrowell\nStaff Emeritus\nGold Member\nIsn't the accelerating expansion of the universe one of the unsolved puzzles which is linked to dark energy but not fully comprehended?\nJonathan Scott's statement was that the explanations were \"complex and leave many unsolved puzzles.\" There is nothing complex about a nonzero value for the cosmological constant. It's just a term in the Einstein field equations. It's also to be expected based on quantum field theory that the value is nonzero -- we just don't understand why there are so many cancellations that make it as small as it actually is.\n\nJonathan Scott also said that the issues arise \"if you assume quasars redshifts are cosmological.\" The main evidence for the nonzero cosmological constant comes from supernovae, not quasars.\n\nJonathan Scott\nGold Member\nThis is incorrect. Dark matter and dark energy don't require any modifications to GR. The GR used in modern cosmological models is the same GR Einstein published in 1915.\nAs you surely know, the concepts of dark matter and dark energy have arisen purely as an explanation for gravitational effects which are not explained by GR. This doesn't necessarily mean that GR is \"wrong\" but that it is not the whole story.\n\nJonathan Scott\nGold Member\nA search for his name on arxiv, http:\/\/arxiv.org\/find\/astro-ph\/1\/au:+Bell_M\/0\/1\/0\/all\/0\/1 , gives 31 results. Does one of these papers contain the claim that \"if you assume quasars redshifts are cosmological, the resulting explanations are complex and leave many unsolved puzzles?\"\nIf you ignore the hits on \"M.E.Bell\" and \"M.R.Bell\" and just stick to \"M.B.Bell\" I think you'll find that just about every entry in that list relates to such ideas, although that was not a literal quote.\n\nIn one I've recently looked at, even the title supports it: arXiv:0812.3130v1 \"The Peculiar Shape of the $\\beta_{ app} \u2212 z$ Distribution Seen in Radio Loud AGN Jets Is Explained Simply and Naturally In the Local Quasar Model\". This paper considers why \"superluminal\" blobs (conventionally attributed to beaming effects) cut off in a strange way for quasars with smaller redshift.\n\nThere are also questions of why the density of lines at various redshifts in quasar redshifts is so weakly related to the overall redshift. This is just one of the characteristics that is conventionally partly explained by the \"evolutionary\" concept that quasars of different ages have different characteristics.\n\nThere are also the obvious questions of quasar distribution in space; there is a surprisingly spherical hole around us if we assume cosmological distances. Again, this is conventionally covered by an evolutionary model which says that quasars suddenly stopped existing everywhere at some relatively recent time.\n\nAnother aspect is the Arp's original observation that many quasars appear to be have been emitted, often in pairs, by large galaxies of much lower redshift, including most of the brighest quasars. As we only have one sky, it's difficult to evaluate the probability of this occurring by chance, but it certainly looks interesting.\n\nI frequently hear statements that the idea of local quasars has been ruled out by the fact that some of them are surrounded by galaxies at the same redshift. However, this has always been consistent with Arp's original observation, which is that quasars closest to the host galaxy have high intrinsic redshifts and a point-like appearance, or unresolved nebulosity, but quasars further away have much smaller intrinsic redshifts and appear more like galaxies with active nuclei. Also, even if the surrounding material did appear to be a galaxy at the same redshift, this could still be just a matter of some unexplained intrinsic redshift of both the quasar and its surrounding galaxy; if it is possible to have intrinsic redshift in violation of what we expect from GR, we cannot use GR to rule it out at the galactic scale.\n\nThere are many other more technical complexities, for example relating to the \"metallicity\" shown in the spectrum failing to correlate with the evolutionary time scale, and \"fingers of God\" effects in the supposed spatial distribution.\n\nAnother very controversial observation is that the relative redshifts between quasars related to a given host galaxy seems to follow a specific approximate pattern. This might be explained for example by the idea that ejection speeds are relatively low (otherwise Doppler effects would hide this pattern) but ejections occur at regular intervals and the intrinsic redshifts decay in a curiously regular way. The official view is that this must be some form of selection effect in the way the observations are collected or processed, but such effects seem to be surprisingly common.\n\nExplaining intrinsic redshift is very difficult, and current attempts are far too speculative to discuss here. However, on the other side, the cosmological distance view of quasars has been presenting a whole series of weird effects ever since they were discovered, and although we have come up with some sort of possible explanation for each one the problems keep coming.\n\nChalnoth\nJonathan Scott also said that the issues arise \"if you assume quasars redshifts are cosmological.\" The main evidence for the nonzero cosmological constant comes from supernovae, not quasars.\nWell, the main evidence comes from multiple, independent pieces of data all converging on the same cosmology. Quasars don't feature prominently, because they aren't terribly good for estimating cosmological distances. But we do have supernovae, the cosmic microwave background, baryon acoustic oscillations, and measurements of the nearby expansion rate.\n\nJonathan Scott\nGold Member\nWell, the main evidence comes from multiple, independent pieces of data all converging on the same cosmology. Quasars don't feature prominently, because they aren't terribly good for estimating cosmological distances. But we do have supernovae, the cosmic microwave background, baryon acoustic oscillations, and measurements of the nearby expansion rate.\nI don't think the quasar questions relate directly to any of those things. Redshift of normal galaxies would appear to be a reliable indicator of cosmological distance.\n\nThere just seems to be a special problem with many quasars apparently being related to much nearer objects, which would require them to have a significant (but theoretically \"impossible\") intrinsic redshift, and there are also similar cases for apparent satellite objects which look more like small galaxies than quasars, where the intrinsic redshift is however less extreme.\n\nChalnoth\nI don't think the quasar questions relate directly to any of those things. Redshift of normal galaxies would appear to be a reliable indicator of cosmological distance.\n\nThere just seems to be a special problem with many quasars apparently being related to much nearer objects, which would require them to have a significant (but theoretically \"impossible\") intrinsic redshift, and there are also similar cases for apparent satellite objects which look more like small galaxies than quasars, where the intrinsic redshift is however less extreme.\nThere is no such coincidence problem when looked at properly. There is also no physical mechanism by which quasars can possibly have an intrinsic redshift while still being nearby.\n\nJonathan Scott\nGold Member\nThere is no such coincidence problem when looked at properly.\nAs Arp has previously pointed out many times, even if you just start from the brightest quasars in the sky (to avoid selection bias), there appears to be an obvious immediate relationship with major galaxies. As there's only one sky, it could just be a big coincidence, but it looks very plausible.\n\nThere is also no physical mechanism by which quasars can possibly have an intrinsic redshift while still being nearby.\nI agree there is no KNOWN physical mechanism, but it's not a huge leap to imagine the possibility of one, and it doesn't require violating any major principles (unlike some of Arp's suggestions).\n\nFor example, if GR turns out to be sufficiently inaccurate in the very strong field regime that gravitational collapse does not occur, that would allow the existence of quasi-stellar objects with significant intrinsic gravitational redshifts (and such objects could also hold together when spinning at relativistic speeds). That in itself would not also allow surrounding material to have the same intrinsic redshift, but if the surrounding material is actually being illuminated or stimulated by intense radiation from the central object, that could well give an illusion of a matching redshift.\n\nChalnoth\nAs Arp has previously pointed out many times, even if you just start from the brightest quasars in the sky (to avoid selection bias), there appears to be an obvious immediate relationship with major galaxies. As there's only one sky, it could just be a big coincidence, but it looks very plausible.\nThis is just flat-out not true. There is no coincidence whatsoever. Go ahead, try to find evidence that this is the case. I dare you.\n\nOh, and let me also point out that we have now observed the host galaxies associated with many quasars, and the expected jets of matter that such quasars would produce if they were due to supermassive black holes at the centers of galaxies.\n\nI agree there is no KNOWN physical mechanism, but it's not a huge leap to imagine the possibility of one, and it doesn't require violating any major principles (unlike some of Arp's suggestions).\nYes, it is a huge leap. A tremendous leap. An intrinsic redshift not related to velocity\/gravity requires basically violating all of relativity and quantum mechanics.\n\nTo find out how near or far the quasars are, is it not possible to do some calculations based on gravitational lending, using objects directly behind the objects of view. I've seen some rare pictures of quasars gravitational lending galaxies behind them. The other way round is far more common.\n\nWith regards to redshift being intrinsic in some objects. All you would need to find is another case of why wavelengths can stretch. Can the spin, rotation, radiation emissions, movement, somehow show this is possible? Maybe some sort of simulation program can show this occurring.\n\nphyzguy\nOf course it's possible to do gravitational lensing studies of quasars, and this has been done many times. There are many examples of multiply imaged quasars which are lensed by intervening galaxies. Not only are the images of the quasars consistent with gravitational lensing, but the time delays between the different images are fully consistent with GR as well. Here is an example: http:\/\/arxiv.org\/abs\/astro-ph\/0607513 . How anyone could look at these observations and conclude that quasars are not at cosmological distances is beyond me.\n\nChalnoth\nTo find out how near or far the quasars are, is it not possible to do some calculations based on gravitational lending, using objects directly behind the objects of view. I've seen some rare pictures of quasars gravitational lending galaxies behind them. The other way round is far more common.\nLensing doesn't give a very good estimate of the distance to the background (lensed) object, other than to show that the lensed object is significantly further away than the object that does the lensing. It does, however, provide information about the mass and mass distribution of the lensing object.\n\nWith regards to redshift being intrinsic in some objects. All you would need to find is another case of why wavelengths can stretch. Can the spin, rotation, radiation emissions, movement, somehow show this is possible? Maybe some sort of simulation program can show this occurring.\nSpin and rotation are basically the same thing and tend to broaden spectral lines. There is very little overall redshift.\n\nThe idea of radiation emissions doing anything here makes no sense whatsoever because we're looking at radiation emissions.\n\nFor movement to be the cause, you'd have to believe that every quasar is thrown away from us at relativistic speeds from the nearby galaxy it's supposed to be associated with. That's just nonsensical.\n\nThere are only two ways to cause a redshift of spectral lines: relative velocity and gravitational redshift. That is it.\n\nThanks for that information. I was just throwing a few ideas out there so that you could answer them and clarify as you did.\n\nIve not seen all the conflicting data, but it appears from what you've said that those quasars must be behind the galaxies. There being so many of these formations in space that the possibility of them appearing as they do is not so improbable.\n\nChalnoth\nThanks for that information. I was just throwing a few ideas out there so that you could answer them and clarify as you did.\n\nIve not seen all the conflicting data, but it appears from what you've said that those quasars must be behind the galaxies. There being so many of these formations in space that the possibility of them appearing as they do is not so improbable.\nThere's also the point to be made that gravitational lensing magnifies background objects, so that a quasar that just happens to be located near a foreground galaxy is likely to be made brighter by the lensing due to the foreground galaxy. The effect of this lensing needs to be taken into account if you're going to try to seriously ask the question of whether foreground galaxies and quasars are actually related.\n\nJonathan Scott\nGold Member\nThis is just flat-out not true. There is no coincidence whatsoever. Go ahead, try to find evidence that this is the case. I dare you.\nWhat do you mean \"no coincidence\"? By \"coincidence\" I would mean that the apparent patterns visible in the distribution of quasars were purely chance, so I would expect your position to be that it WAS purely coincidence.\n\nHave you looked at Arp's charts? Most of the lines and pairings seem quite obvious, and he includes all other similar quasars where relevant to show that he's not just picking out ones that show the pattern. I remember from some astronomy magazine many years ago being astonished to see Arp pointing out that even 3C273 seems to have been ejected from M49, with signs of a faint connection between the two.\n\nIt is admittedly difficult to evaluate the probabilities behind such patterns, given that it's not easy to define objectively what one would consider an \"unlikely\" grouping, but Arp makes some plausible attempts which suggest that the observed connections are really quite extraordinarily unlikely if all the bright quasars are background objects.\n\nOh, and let me also point out that we have now observed the host galaxies associated with many quasars, ...\nAs I previously mentioned, according to Arp's conclusions from his observations, host galaxies with similar redshifts are expected around mature quasars, and such quasars are expected to be at or near their cosmological redshift distances, for example as seen in gravitational lensing cases.\n\nHowever, young quasars would have an intrinsic redshift that would conventionally place them many times further away, so the apparent size of any nebulosity would be assumed to be much larger and it is not clear whether this might look like a galaxy even when unresolved. A mid-life quasar is very bright so that even if there is any nebulosity around it, it is difficult to obtain an independent spectrum, and even if that proves possible, it seems quite likely that the spectrum could be distorted by the radiation from the quasar itself.\n\n... and the expected jets of matter that such quasars would produce if they were due to supermassive black holes at the centers of galaxies.\nJets of matter are actually much easier to explain if the supermassive objects are NOT black holes, because they could then easily have magnetic fields which are many orders of magnitude greater than that expected for a black hole (by the \"no hair\" theorem) or even an accretion disk. There are theories that a black hole could hold a \"fossilized\" magnetic field which might be sufficient for the purpose, but that's quite a stretch.\n\nYes, it is a huge leap. A tremendous leap. An intrinsic redshift not related to velocity\/gravity requires basically violating all of relativity and quantum mechanics.\nI didn't say not related to velocity\/gravity. I pointed out that it could be related to gravity if collapse doesn't occur, which only requires a change to GR in a most extreme regime.\n\nWhat is approximately the threshold for a distance to be considered \"cosmological\"?\n\nJonathan Scott\nGold Member\nWhat is approximately the threshold for a distance to be considered \"cosmological\"?\nIt's not a threshold. The question is whether the redshift of quasars is taken to be purely due to their distance (following the usual cosmological redshift distance law) or whether a significant part of the redshift is intrinsic to the quasar.\n\nIt's not a threshold. The question is whether the redshift of quasars is taken to be purely due to their distance (following the usual cosmological redshift distance law) or whether a significant part of the redshift is intrinsic to the quasar.\nI know, I was thinking in terms of how near a quasar could be according to the concordance model.\n\nBTW, have you heard of the Wolf shift as a possible mechanism of intrinsic redshift?\n\nJonathan Scott\nGold Member\nBTW, have you heard of the Wolf shift as a possible mechanism of intrinsic redshift?\nIt doesn't seem very plausible to me. You can use it to shift one spectral line of coherent light in the lab, but I don't think it can be used to shift a whole spectrum, and bright quasars typically show multiple identifiable lines.","date":"2019-12-14 10:02:52","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\": 0.7090198993682861, \"perplexity\": 859.4590944344278}, \"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-2019-51\/segments\/1575540586560.45\/warc\/CC-MAIN-20191214094407-20191214122407-00180.warc.gz\"}"} | null | null |
The RBAPS project was a three and a half year project in Ireland and Spain working with farmers and stakeholders developing ways to reward farmers for delivering biodiversity on their lands. The key element of results-based method of delivering payments is that the amount of money paid to the farmer, reflects the quality of wildlife (biodiversity) that is delivered on their farmed land.
Result-based agri-environment payment schemes (RBAPS) award payments to farmers on the basis of the quality of the desired environmental outcome that is delivered. This contrasts with the standard 'prescription-based' model, where payments are awarded for complying with certain conditions, whether prohibitions or mandatory actions. For example, in a prescription-based agri-environment scheme (AES), a species-rich grassland option might specify certain grazing &/or mowing dates, livestock pressure, fertiliser and herbicide use, with the same payment made irrespective of the subsequent ecological quality of the grassland.
With result-based schemes, the habitat condition is scored (e.g. on a scale of 1-10), with the highest payment awarded to the best quality habitat. Assessments are based on objective assessment criteria (indicators), which are chosen to reflect the overall biodiversity and ecological integrity of the habitat while also responding to agricultural management practices. This method is based on the Burren model developed with farmers, farming representatives and ecologists (Parr et al. 2010).
Result-based schemes may involve payments awarded solely on results achieved or may be a blended model with payments for 'non-productive investments' which support the delivery of biodiversity (e.g. removal of scrub encroaching on species-rich grassland; or creating a chick feeding area on important wading bird habitat); and can be complemented by some prescriptive elements where necessary.
By linking payments to assessment criteria (which indicate the quality of the biodiversity) RBAPS make it financially beneficial for participating farmers to gain an understanding of the conditions needed for delivery of the biodiversity. This creates a new market for biodiversity; where those farmers who better deliver market requirements can be better rewarded.
Parr et al. (2010) Farming for Conservation in the Burren. Conservation Land Management; Autumn 2010. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,271 |
Q: How do I change the text property of a Textblock in WPF? I have a Form in WPF and a DispatcherTimer, every time the tick event fires I want to change the value of OrderLbl.Text from "Today's orders" to "this week's orders", and from "this week's orders to "This Month's Orders".
However, when I attempt to change the value of OrderLbl.text from the _timer_Tick event, it throws an exception saying an object reference is required, however when I reference this inside the tick event it will not change the value of OrderLbl.Text
Code is below;
public void Start()
{
System.Windows.Threading.DispatcherTimer DTimer = new System.Windows.Threading.DispatcherTimer();
DTimer.Tick += new EventHandler(_timer_Tick);
DTimer.Interval = new TimeSpan(0, 0, 5);
DTimer.Start();
}
private static void _timer_Tick(Object sender, EventArgs e)
{
if (OrderLbl.Text == "Today's Orders")
{
OrderLbl.Text = "This Week's Orders";
}
else if (OrderLbl.Text == "This Week's Orders")
{
OrderLbl.Text = "This Month's Orders";
}
//else
//{
// mw.orderlbl.text= "today's orders";
// go
//}
}
A: Remove the static keyword from the Tick handler:
private void _timer_Tick(Object sender, EventArgs e)
{
...
}
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,791 |
We have now been in business for just over a year and because of the people who trusted us with their insurance we have been able to donate over $5,000 to organizations that are helping the orphans and the poor in Liberia.
Above is a picture of Uniting Distant Stars, an organization right here in Minnesota that is helping students get trained in trades such as plumbing, electrician, baking, and many others. We don't just want to give money for work to be done, we want people to learn the skills so they can earn an income and bring up the development of the country.
A huge thanks to all everyone that has given us an opportunity to help them with insurance, we look forward to doing so much more! | {
"redpajama_set_name": "RedPajamaC4"
} | 8,953 |
<?php
/**
* The template for displaying all pages.
*
* This is the template that displays all pages by default.
* Please note that this is the WordPress construct of pages
* and that other 'pages' on your WordPress site will use a
* different template.
*
* @package volta
*Template Name: Full Width Template
*/
get_header(); ?>
<div id="primary" class="content-area-full">
<main id="main" class="site-main" role="main">
<?php while ( have_posts() ) : the_post(); ?>
<?php get_template_part( 'content', 'page' ); ?>
<?php
// If comments are open or we have at least one comment, load up the comment template
if ( comments_open() || '0' != get_comments_number() ) :
comments_template();
endif;
?>
<?php endwhile; // end of the loop. ?>
</main><!-- #main -->
</div><!-- #primary -->
<?php get_footer(); ?>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,638 |
// Copyright (c) Brock Allen & Dominick Baier. All rights reserved.
// Licensed under the Apache License, Version 2.0. See LICENSE in the project root for license information.
using System.Collections.Generic;
using IdentityServer4.Models;
namespace IdentityServer.IntegrationTests.Endpoints.Introspection.Setup
{
internal class Clients
{
public static IEnumerable<Client> Get()
{
return new List<Client>
{
new Client
{
ClientId = "client1",
ClientSecrets = new List<Secret>
{
new Secret("secret".Sha256())
},
AllowedGrantTypes = GrantTypes.ClientCredentials,
AllowedScopes = { "api1", "api2", "api3-a", "api3-b" },
AccessTokenType = AccessTokenType.Reference
},
new Client
{
ClientId = "client2",
ClientSecrets = new List<Secret>
{
new Secret("secret".Sha256())
},
AllowedGrantTypes = GrantTypes.ClientCredentials,
AllowedScopes = { "api1", "api2", "api3-a", "api3-b" },
AccessTokenType = AccessTokenType.Reference
},
new Client
{
ClientId = "client3",
ClientSecrets = new List<Secret>
{
new Secret("secret".Sha256())
},
AllowedGrantTypes = GrantTypes.ClientCredentials,
AllowedScopes = { "api1", "api2", "api3-a", "api3-b" },
AccessTokenType = AccessTokenType.Reference
},
new Client
{
ClientId = "ro.client",
ClientSecrets = new List<Secret>
{
new Secret("secret".Sha256())
},
AllowedGrantTypes = GrantTypes.ResourceOwnerPassword,
AllowedScopes = { "api1", "api2", "api3-a", "api3-b" },
AccessTokenType = AccessTokenType.Reference
}
};
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 782 |
\section{Introduction}
Inflation generates an adiabatic density perturbation,
which is generally thought to be responsible
for large scale structure and the cosmic
microwave background
(cmb) anisotropy, and it also generates gravitational waves which might
give a significant contribution to the latter
\cite{abook,kt,LL2}.
The spectrum of the density perturbation is conveniently
specified by a quantity $\delta_H^2$
and the spectral index $n$ is defined by
$\delta_H^2\propto k^{n-1}$ where $k$ is the comoving wavenumber.
At present there is no detectable scale dependence, and
observational limits on $n$
are only mildly constraining for inflationary
models, the most notable result being that `extended' inflation
\cite{extended}
is ruled out, except for rather contrived versions \cite{green}.
But in the forseeable
future one can expect a good measurement of $n$
and it is reasonable to ask
what it will tell us.
At a purely phenomenological level the answer to the question
is simple and well known.
The spectral index $n$ is given by \cite{LL1,davis,salopek}
\be
n-1\simeq -3\Mpl^2(V'/V)^2 + 2\Mpl^2 (V''/V)
\label{n}
\ee
where $\Mpl=(8\pi G)^{-1/2}=2.4\times 10^{18}\GeV$ is the reduced Planck
mass. The right hand side of this expression is a combination of the
potential and its derivatives,
evaluated at the epoch when
cosmological scales leave the horizon.
If $n$ is distinguishable from 1 the
combination will be measured, otherwise it will be constrained to near
zero. Moreover, the relative contribution of
gravitational waves to the mean-square low multipoles of the
cmb anisotropy is given by \cite{rubakov,starob,d13,andrew}
$r\simeq 5\Mpl^2(V'/V)^2$. Their eventual detection or
non-detection will determine or constrain the first term, so that measuring
$n$ determines or constrains $V''/V$. Finally,
with more data one might measure the effective $n$ over a range of scales
and also justify the use of more accurate formulas \cite{stly}, to obtain
limited additional information on the shape of $V$
while cosmological scales are leaving the horizon \cite{recon}.
This `reconstruction' approach is interesting, but it relies to a large
extent on detecting gravitational waves and will in any case always be
limited by the narrow range of scales accessible to cosmology.
Meanwhile,
since the advent of hybrid inflation \cite{l90,LIN2SC},
inflation model-building is beginning
to come back into the fold of particle physics. Like the earliest models
of inflation, but unlike those proposed in the intervening years,
hybrid inflation models work in the regime where all relevant fields
are small on the Planck scale \cite{CLLSW}.
If one accepts the usual view that
field theory beyond the standard model
involves supergravity \cite{susy}, this represents a crucial simplification
because in the context of supergravity the potential is expected to
have an infinite power series
expansion in the fields. For field values much bigger than $\Mpl$
one has no idea what form the potential will take, expect perhaps in the
direction of the moduli fields of superstring theory
\cite{bingall,paul}.
But in the small-field regime one is entitled to make the usual
assumption, that the expansion is dominated by a very few low-order
terms. As a result, inflation model-building has become a more varied,
yet better controlled activity.
Given this situation it is reasonable to go beyond the purely
phenomenological level, and ask how well a measurement of the
spectral index will discriminate between different models of inflation.
That
question is the main focus of the present paper.
There are of course many other aspects of inflation model-building.
The prediction Eq.~(\ref{delh})
of the normalization of $\delta_H$ provides another constraint
on the parameters of the inflationary potential, which can crucially
effect the viability of a given potential. It may be regarded
as fixing the magnitude of $V$ as opposed to its derivatives,
a quantity whose theoretical significance is not yet clear
(for some recent discussions
see Refs.~\cite{CLLSW,mutated,paul,lisa,graham}).
Another theoretical issue is the
difficulty of implementing slow-roll inflation in the context of
supergravity \cite{CLLSW,ewansgrav}.
These two things will be considered
briefly in the present paper, but other
aspects of inflation model building, in particular reheating
and `preheating' (see for instance Ref.~\cite{kls})
will not be discussed at all.
In most of the discussion
we assume that the slowly-rolling inflaton field is essentially
unique. This is the case in most of the models so far proposed,
and is necessary for the validity of the above formula for $n$.
But in Section VII we discuss the case
where there is a family of inflationary trajectories lying
in the space of two or more fields; in other words, where there is a
multi-component inflaton. The quantum fluctuation kicks the
field from one trajectory to another, and if
the trajectories are physically
inequivalent this gives a contribution
to the spectrum of the density perturbation which has to be added to the
usual one, coming from the fluctuation back and forth along the same
trajectory. In all cases, the best method of
calculating the spectrum of the adiabatic density perturbation
starts with the assumption that,
after smoothing on a super-horizon scale, the
evolution of the universe along each comoving worldline
will be the same
as in an unperturbed universe with the same initial inflaton field.
We also discuss briefly the
case of an isocurvature density perturbation,
giving a simple derivation of the fact that the
low multipoles of the cmb anisotropy are six times as big as for
an adiabatic density perturbation.
We assume that the scale dependence of
$\delta_H$ can be represented by a spectral index, $\delta_H^2\propto
k^{n-1}$, at least while cosmological scales are leaving the horizon.
This is the case if slow-roll inflation is continuously
maintained, unless there is a cancellation between the terms of
Eq.~(\ref{n}) for $n-1$. A feature, such as a shoulder in the spectrum,
can easily be generated if slow-roll inflation is briefly interrupted
at the epoch when the relevant scale is leaving
the horizon \cite{interrupt,c3,d7,d9,d10,d13}.
However to get any kind of feature during the few $e$-folds
corresponding to cosmological scales requires a delicate balance of
parameters. Observation does suggest a shoulder in the spectrum of the
{\em present} density perturbation, in the
tens of $\Mpc$ range, though at present the evidence is also consistent
with a smooth rise corresponding to $n\simeq 0.8$.
But the spectrum $\delta_H$ refers to the density perturbation in
the early universe, and the most natural explanation for a shoulder
in the spectrum of the observed density perturbation would be
that it is in the transfer function relating the present density
perturbation to the early one.
Indeed, any departure from the pure cold dark
matter hypothesis leads to a shoulder on just the required scales.
The conclusion is that there is little motivation to
depart from the slow-roll inflation paradigm, and its attendant
scale dependence $\delta_H^2\propto k^{n-1}$.
The discussion is confined to models of inflation leading to a spatially
flat universe on observable scales. This is the generic case,
though at the expense of moderate fine-tuning the
universe can be open (negatively curved) if we live inside a bubble formed at
the end of a preliminary era of
inflation \cite{open}.
Throughout the paper, $a$ denotes the scale factor of the universe
normalized to 1 at present and $H=\dot a/a$ is the Hubble parameter,
with present value $H_0=100h\km\sunit^{-1}\Mpc^{-1}$. The corresponding
Hubble distance is $cH_0^{-1}=3000h^{-1}\Mpc$.
From now on the units are $c=\hbar=1$.
The comoving wavenumber in a Fourier series expansion is denoted
by $k/a$, so that $k$ is the present wavenumber and $k^{-1}$
conveniently specifies the corresponding comoving scale.
The scale is said to be outside the horizon when $aH/k$
is bigger than 1 and inside it otherwise. A given scale of interest
leaves the horizon during inflation and re-enters it long afterwards
during radiation or matter domination.
The plan of the paper is as follows. In Section II the eventual accuracy
of the measurement of $n$ is discussed. In Section III some general
issues relating to inflation model-building are discussed, and
specific models are discussed in Section IV and V. In Section VI
various ways of implementing slow-roll inflation in the context of
supergravity are briefly discussed, focussing on their implications for the
spectral index. Section VII discusses inflation with a multi-component
inflaton, and the calculation of the spectrum of the density
perturbation in both this and the single-component case. A
summary is provided in Section VIII.
\section{How accurately will the spectral index be determined?}
Before discussing the theoretical predictions we need an idea of the
eventual accuracy to expect for the determination of the
spectral index of the adiabatic density perturbation. First let us define
this perturbation.
\subsection{The spectrum of the adiabatic density perturbation}
The evolution of small density perturbations in the universe is
governed by linear equations, with each Fourier mode
evolving independently. On scales well outside the horizon
the evolution is governed purely by gravity, and in each comoving
region its evolution with respect to the local proper time
is is the same as it would be in an unperturbed universe.
For the moment we are not interested in the
evolution during inflation and in the very early universe. Rather,
we focus on the relatively late era when
the non-baryonic dark matter has its present form, and the
radiation-dominated era immediately preceding the present
matter-dominated era has begun.
During this era, consider the perturbations $\delta\rho_i$ in the energy
densities of the various particle species. They are said to be adiabatic
if each of the densities $\rho_i$ is a unique function
of the total density $\rho$. The perturbations all vanish if we slice
the universe into
hypersurfaces of constant energy density. With any other slicing,
all species of matter have a common density contrast and so do all
species of radiation, with $\delta\rho_m/\rho_m=\frac34\delta\rho_r
/\rho_r$.
A different type of density perturbation would be an isocurvature one,
for which the total density perturbation vanishes but the separate ones
do not. The vacuum fluctuation during inflation will generate an adiabatic
perturbation, which is the subject of the present paper. At the end of
Section VII we consider briefly the isocurvature perturbation that might
also be produced.
The
adiabatic density perturbation is determined by the total density
perturbation $\delta\rho/\rho$. This in turn is determined
by a
quantity $\cal R$,
which defines the curvature of hypersurface orthogonal to
comoving worldlines (`comoving hypersurfaces').\footnote
{For each Fourier mode, the
curvature scalar $R^{(3)}$ is given by
${\cal R}= \frac14 (a/k)^2
R^{(3)}$ (it is a perturbation since the unperturbed universe is
supposed to be flat).
There is unfortunately no standard notation for $\cal R$.
It was first defined in Ref.~\cite{bardeen80}
where it was called $\phi_m$. It was called ${\cal R}_m$ in
Ref.~\cite{kodamasasaki}, and is a factor $\frac32k^{-2}$
times the quantity $\delta K$ of Ref.~\cite{lyth85}.
On the scales far outside the horizon where it is constant
(the only regime where it is of interest)
it coincides with the quantity $\xi/3$ of Ref.~\cite{bst}
and the quantity $\xi$ of Ref.~\cite{c3}.}
At least during the relatively late era that we are focussing on at the
moment, $\cal R$ is time-independent, and
\be
\frac{\delta\rho}{\rho}=\frac49 \left(\frac{k}{aH}\right)^2 {\cal R}
\ee
(As always we focus on a single Fourier component, well before horizon
entry.)
After horizon entry the evolution becomes more complicated, and
for radiation as opposed to matter it
involves the full phase-space densities. The main observational quantities
are
the present matter density perturbation,
which we can get a handle by observing the distribution and motion of
galaxies, and the cmb anisotropy. There is also some
information on the matter density
perturbation at earlier times.
Inflation typically predicts
that $\cal R$ is a a Gaussian random field, which is usually assumed and
is consistent with observation. Then all stochastic properties
of the perturbations
are determined by the spectrum ${\cal P}_{\cal R}$ of $\cal R$.\footnote
{The spectrum ${\cal P}_f(k)$
of a quantity $f$ is the modulus-squared of its Fourier
coefficient, times a factor which we choose so that
$f$ has mean-square value $\bar{f^2}=\int^\infty_0 {\cal P}_f
dk/k$.}
We are supposing that any scale dependence of the spectrum
has the form ${\cal P}_{\cal R}\propto k^{n-1}$ where $n$ is the
spectral index.
Instead of ${\cal P}_{\cal R}$ one usually considers
\be
\delta_H^2\equiv (4/25) {\cal P}_{\cal R}
\label{delhdef}
\ee
Then, on scales
$k^{-1}\gsim 10\Mpc$ where its evolution is still linear, the spectrum
of the present density perturbation $\delta\equiv\delta\rho/\rho$
is given by
\be
{\cal P}_\delta = \delta_H^2 (k/H_0)^4 T^2(k)
\ee
The transfer function $T(k)$ represents non-gravitational effects,
and is equal to 1 on the large scales
$k^{-1}\gg 100\Mpc$ where such effects are absent.
Information on ${\cal P}_\delta$
on scales of order $10$ to $100\Mpc$ is provided
by a variety of observations on the abundance, distribution and
motion of galaxies, and if the transfer function is known
one can deduce $\delta_H$.
The $l$th multipole of the cmb anisotropy
probes the scale
$k^{-1}\simeq 2/(H_0 l)= 6000h^{-1}l^{-1}\Mpc$, corresponding to the
distance subtended by angle $1/l$ at the surface of last scattering.
Since this `surface' is of order $10\Mpc$ thick the anisotropy will be
wiped out on smaller scales, corresponding to $l\gsim 10^4$.
It thus lies entirely within the linear regime.
The
mean-square multipoles seen by a randomly placed observer
can be calculated in terms of
$\delta_H^2(k)$ at $k^{-1}\simeq 2/(H_0 l)$, given another
transfer function encoding the effect of non-gravitational
interactions. At present
observations exist only for $l\lsim 100$, corresponding to
$k^{-1}\gsim 100\Mpc$.
The COBE measurement corresponds to
multipoles $l\sim 30$, and gives an accurate determination of
$\delta_H$ on the corresponding scales because the evolution is
purely gravitational (dominated by the Sachs-Wolfe effect).
On the
scale $k\simeq 5H_0$ corresponding to the
$l\simeq 10$ multipoles which define the centre of the measured
range one finds
\cite{delh,bunnwhite}
\be
\delta_H= 1.94\times 10^{-5}
\label{cobe}
\ee
with a $2\sigma$ uncertainty of $15\%$.
This assumes that gravitational waves give a negligible contribution.
Otherwise, the result is multiplied by a factor
$(1+r)^{-1/2}$ which can be calculated in a given model of
inflation.
In order to arrive at an observational value for
$n$
we need measurements of $\delta_H$ also on smaller scales. Here
transfer functions are needed and they
depend strongly on
the cosmology.
Since we are taking particle physics
seriously it is reasonable to assume that the cosmological constant
vanishes and (in the interests of simplicity) that the density is
critical. Then
the main uncertainties
are the value of the Hubble constant
$H_0=100h\km\sunit^{-1}\Mpc^{-1}$,
the value of the
baryon density $\Omega_B$, and the
nature of the dark matter.
The simplest assumption is pure cold dark matter. This case is viable at
present \cite{constraint2} with $h\simeq 0.5$, $\Omega_B\simeq 0.12$ and
\be
0.7 \lsim n \lsim 0.9
\ee
This assumption will be invalidated if a shoulder is observed
in the spectrum, or if
$h$ or $\Omega_B$ differ from the above values by more than $10\%$
or so.
A more robust
assumption is that there is an admixture
of hot dark matter in the form of a single neutrino species
\cite{mdm}, and taking
$\Omega_B<.15$ and $h>0.4$
a result at something like 95\% confidence-level is
\cite{constraint1,constraint2}
\be
0.7< n < 1.3
\label{nbound}
\ee
The uncertainty in $n$ comes mostly from the uncertainties in
$H_0$, $\Omega_B$ and the nature of the dark matter.
The quoted lower bound comes from the cmb anisotropy in the region of the
first Doppler peak and depends only on $\Omega_B$, though other
data
point to a lower bound that is not much weaker.
The upper bound depends strongly on $h$ and the nature of the dark
matter. With the same hypothesis about the dark matter one finds
$n<1.04$ if $h>0.5$ and $n<0.9$ if $h>0.6$, a very dramatic
improvement. The bounds on $n$ under other hypotheses about the dark
matter have not been explored in detail but one does not expect wildly
different results since the effect of modifying pure cold dark
matter is always to reduce the density perturbation on small scales.
The bounds on $n$ become tighter if there are significant gravitational
waves, but that is not the case in small-field models, nor is it the
case in most other models that have been proposed \cite{mygwave}.
In models where $r\simeq 5(1-n)$
(corresponding to an exponential potential, and approximately to
a power-law potential) the lower bound becomes $n>0.8$
\cite{constraint1}, which
rules
out extended inflation \cite{extended} except for rather
contrived versions \cite{green}.
To get an idea of the eventual accuracy to be expected for $n$, suppose that
$\delta_H$ has been measured at just
two wavenumbers $k$ and $q$, with $k>q$.
If the values of $\delta_H$ have fractional uncertainties
$\Delta_k$ and $\Delta_q$, the uncertainty in $n$ is \cite{knox}
\be
\Delta n = (\Delta_k^2 +\Delta_q^2)^{1/2} / \ln(k/q)
\ee
Other things being equal one will use the longest possible
`lever arm' $k/q$, so the idealization of using only two wavenumbers
is not too far from reality.
The longest useful lever arm at present
is provided by $k= (10h^{-1}\Mpc)^{-1}$, the smallest scale that
is linear now, and $q= 5H_0= (600h^{-1}\Mpc)^{-1}$,
the scale effectively probed by COBE.
If each piece of data has
a 10\% uncertainty this gives $\Delta n =.034$.
At present the uncertainties are $\Delta_q= 15\%$ as quoted above, and
{\em excluding the uncertainty in the transfer function}
$\Delta_k=30\%$ \cite{viana}. Thus we would already be not too far
from the situation of $10\%$ errors if only the cosmological parameters
and the nature of the dark matter were understood.
It seems quite reasonable to suppose
that good progress in this direction will have been made in a few years,
giving an accuracy of order 10\% on both pieces of data and an accuracy
$\Delta n\simeq .04$ for the spectral index.
Looking for much higher accuracy, one cannot use
data corresponding to multipoles of order $10$. The reason is that
the value of $\delta_H$ adduced even from perfect data is subject
to cosmic variance, arising from the fact that the multipoles are
measured only at our position. Using multipoles of order $l$
(over unit interval of $\ln l$) the uncertainty due to cosmic variance
is of order $1/l$. To achieve something like 1\% accuracy, one needs
data centered on $l= 100$, corresponding
to
$q=(60h^{-1}\Mpc)^{-1}$.
With 1\% uncertainty also at $k=(10h^{-1}\Mpc)^{-1}$,
one gets $\Delta n
=.008$. We see that improving the accuracy by a factor
10 only increases the accuracy on $\Delta n$ by a factor 4, because
of the necessarily shorter lever arm.
In the future one will be able to go to smaller scales, with
the best determination of the spectral index coming from
the cmb anisotropy with good angular resolution, but
an accuracy better than $\Delta n\sim .01$
will be difficult to achieve if only because of the uncertainty in the
cosmological parameters \cite{knox,jungman}.
Fortunately, this accuracy turns out to be all that is needed
to discriminate sharply between different models.
\subsection{Scale-dependence of the spectral index?}
If the power-law parametrization $\delta_H^2\propto k^{n-1}$ is
imperfect one can still define an effective spectral index by
$n-1\equiv d\ln\delta_H/d\ln k$ and it then makes sense to
ask whether the variation of the spectral index will ever be
observable.
In simple models
$n-1$ is proportional to $1/N$ if it varies at all,
where $N$ is the number of $e$-folds of
inflation remaining when the relevant scale leaves the horizon,
and we consider only that case.
Although the smallest scale on which $\delta_H(k)$ can be measured is
at present about $10\Mpc$, this limit will go down an order of magnitude
or two in the future, making the total
accessible range perhaps four decades ranging from
$6000h^{-1}\Mpc$ to say $0.6h^{-1}\Mpc$. This would
correspond to $\Delta \ln k =4.5$ (measured from the centre of the
range) and therefore to $\Delta N=4.5$.
As we shall discuss in the next section, the number of $e$-folds
after (say) the scale in the centre of the this range leaves the horizon
depends on the history of the universe after inflation, as well as
the energy scale at which inflation ends. For a
high (ordinary) inflationary
energy scale $N\simeq 50$ is reasonable, but a low scale combined with
an
era of thermal inflation \cite{thermal}
will give $N\simeq 25$. One or more
additional bouts of thermal inflation are not unlikely
from a theoretical viewpoint, but since each one reduces $N$ by
$\simeq 10$ there had better be at most one or two.
Assume first at most one bout of thermal inflation. Then the
fractional
changes in $n$ and $N$ are small so that
\be
\frac{\Delta n}{|1-n|} = \frac{\Delta N}{N}
\ee
Let us suppose that the range is split in half to yield two separate
determinations of $n$, corresponding to $\Delta N=4.5$.
Then one finds that
\be
\frac{\Delta n}{.01} =\frac{|1-n|}{0.11}\frac{N}{50}
\ee
From what has been said above, the scale dependence
$\Delta n$ will have to be bigger
than $.01$ if it is to be detectable in the forseeable future.
As a result its variation will be detectable only if it is quite far
away from 1.
Now suppose that there are two bouts of thermal
inflation so that $N$ varies from say 10 to 20 while cosmological scales
leave the horizon. Then $n-1$ will double over the observable range
so that its variation might be detectable even if $|n-1|$ is of order
$.01$. Three bouts would of course have a dramatic effect, with
four probably forbidden because cosmological scales would re-enter the
horizon during thermal inflation which would lose the prediction for the
density perturbation.
If the spectral index is strongly scale-dependent it is more useful
to consider the spectrum itself.
The case that we are considering, $n-1=q/N$, corresponds to
$\delta_H^2\propto N^q$. As discussed in Section VIII, the
strongest scale dependence obtained from a reasonably motivated
potential corresponds to $q=-4$, coming from a cubic potential.
In order to produce enough early
structure, $\delta_H$ should fall by no more than a factor 5 or so as
one goes down through the four orders of magnitude available to
observation \cite{constraint1}, which with the cubic potential
corresponds to requiring $N\gsim 11$ in the center of the range.
Thus the cubic potential
permits at most one additional bout of thermal inflation,
at least with a low scale for ordinary inflation.
Even if $N$ is not low enough to make it dramatic, the scale-dependence
of $n$ in this model might still be enough to improve the
viability of a pure cold dark matter model \cite{grahamnew}.
\subsection{Observational constraints on very small scales}
The above considerations apply only on cosmological scales. On smaller scales
there are only upper limits on $\delta_H$. The most useful
limit, from the viewpoint of constraining models of inflation,
is the one on the smallest relevant scale which is the one leaving the
horizon just before the end of inflation. It has been considered
in Refs.~\cite{jim} and \cite{lisa}, and according to the latter reference it
corresponds to $\delta_H\lsim 0.1$.
In models where $n$ is constant this corresponds
(using Eq.~(\ref{cobe}))
to $n-1<0.33(50/N)$, which is no stronger thant the constraint
coming from cosmological scales.
It represent a powerful additional constraint in models where
$n$ increases on small scales, as discussed in an extreme case in
Ref.~\cite{lisa,glw}.
\section{Slow-roll inflation}
Before discussing particular models
it will be useful to deal with some generalities.
Practically all models are of the
slow-roll variety \cite{kt,abook,LL2}. Such models are the
simplest, and at least while cosmological scales are leaving the horizon
they are motivated by the observed fact that the spectrum has mild scale
dependence, which is consistent with a power law.
Until Section VII we focus
on models in which the slow-rolling
inflaton field $\phi$ is essentially unique.
During inflation the
potential $V(\phi)$ is supposed to
satisfy the flatness conditions $\epsilon\ll 1$ and $|\eta|\ll1$,
where
\bea
\epsilon&\equiv &\frac12\Mpl^2(V'/V)^2\\
\eta&\equiv &\Mpl^2 V''/V
\eea
Given these conditions, the evolution
\be
\ddot\phi+3H\dot\phi=-V'
\label{phiddot}
\ee
typically settles down to the slow-roll evolution
\be
3H\dot\phi=-V'
\label{slowroll}
\ee
Slow roll and the flatness condition $\epsilon\ll 1$
ensure that the energy density
$\rho=V+\frac12\dot\phi^2$ is close to $V$ and is slowly varying.
As a result $H$ is slowly varying,
which implies that one can write
$a\propto e^{Ht}$ at least over a Hubble time or so.\footnote
{Here and in what follows,
slow variation of a function
$f$ of time means $|d\ln f/d\ln a|
\ll 1$, and slow variation of a function $g$ of wavenumber means
that $|d\ln g/d\ln k|\ll 1$.}
The
flatness condition $|\eta|\ll 1$ then ensures that
$\dot\phi$ and $\epsilon$ are slowly
varying.
A crucial role is played by the number of Hubble times
$N(\phi)$ of inflation, still remaining when $\phi$ has a given value.
By definition $dN=-H\,dt$, and
the
slow-roll condition together with the flatness condition
$\epsilon\ll 1$ lead to
\be
\frac{dN}{d\phi}=-\frac{H}{\dot\phi}=\frac{V}{\Mpl^2V'}
\label{Nrelation}
\ee
This leads to
\be
N=\left| \int^\phi_{\phi_{\rm end}} \Mpl^{-2}\frac V{V'} d\phi
\right|
\label{nint}
\ee
The slow-roll paradigm is motivated, while cosmological
scales are leaving the horizon, by the observed fact that
$\delta_H$ does not vary much on such scales. In typical models
slow-roll persists until almost the end of inflation, but it can
happen that it fails much earlier. The loop-correction model of
Ref.~\cite{ewanloop} is the only one so far exhibited with this property,
and as discussed there the only effect of the failure of slow roll is
that one needs to replace the above expression for $N(\phi)$ with
something more accurate, when it is used to work out the value of
$\phi$ corresponding to cosmological scales.\footnote
{In principle one might be
concerned that the quantum fluctuation during non-slow-roll inflation
might become big and produce unwanted black holes. But in the
slow-roll regime the spectrum $\delta_H$ becomes {\em smaller }
as the speed of rolling is increased (ie., as $\epsilon$ is increased)
and it seems reasonable to suppose that this trend persists beyond the
slow-roll regime.}
\subsection{The predictions}
The quantum fluctuation of the inflaton field
gives rise to an adiabatic density
perturbation, whose spectrum is
\be
\delta_H^2(k) = \frac1{75\pi^2\Mpl^6}\frac{V^3}{V'^2}
=
\frac1{150\pi^2\Mpl^4}\frac{V}{\epsilon}
\label{delh}
\ee
In this expression, the
potential and its derivative are evaluated at the epoch of
horizon exit for the scale $k$, which is defined by $k=aH$.
This result was in essence derived at about the same time in
Refs.~\cite{hawking,starob82,guthpi,bst}.
To be precise, these authors give results which lead more or less
directly to the desired one after the
spectrum has been defined, though that last step is not
explicitly made and except for the last work only a
particular potential is discussed.
(The last three works give results equivalent
to the one we quote, and
the first gives a result which is approximately the same.)
Soon afterwards it was derived again, this time with an explicitly
defined spectrum \cite{lyth85}.\footnote
{Strictly
speaking none of these five derivations is completely satisfactory.
The first three make simplifying assumptions,
and all except the third assume
something equivalent to the constancy of $\cal R$
without adequate proof. But as discussed in
Section VII the constancy of $\cal R$, and therefore the validity of the result,
is easy to establish
for a single-component inflaton.
Also, none of them properly considers the effect of the inflaton field
perturbation on the metric, but again this is easily justified to
first order in the slow-roll approximation \cite{LL2}.
A little later a formalism was given that takes the metric perturbation into
account \cite{mukhanov,sasaki}, which formed the basis for the more
accurate calculation of \cite{stly}.}
Inflation also generates
gravitational waves, whose relative contribution to the mean-squared
low multipoles of the cmb anisotropy is
\be
r = 5\Mpl^2(V'/V)^2
\ee
The contribution of the gravitational waves was estimated
roughly in Ref.~\cite{rubakov}, and in
Ref.~\cite{starob} a result equivalent to $r=6.2\Mpl^2(V'/V)^2$
was obtained under certain approximations. The more correct factor
$\simeq 5$, for the multipoles most relevant for the determination of
the COBE normalization of $\delta_H$, was given in Refs.~\cite{d13,andrew}.
Returning to the spectrum of the density perturbation, comparison
of the prediction
(\ref{delh}) with the value deduced from the COBE observation of the cmb
anisotropy gives (\ref{cobe}), ignoring the gravitational waves\footnote
{In this context
there is no point in
including the effect of gravitational waves on the cmb anisotropy,
since
the prediction for $\delta_H$ that is being used has an error of at least
the same order. If necessary one could include the effect
of the gravitational waves using the more accurate formula for
$\delta_H$ \cite{stly,recon}.}
\be
\Mpl^{-3} V^{3/2}/V' = 5.3\times 10^{-4}
\label{cobecons}
\ee
This relation provides a useful constraint
on the parameters of the potential. It can be written in the equivalent
form
\be
V^{1/4}/\epsilon^{1/4}=.027\Mpl=6.7\times 10^{16}\GeV
\label{vbound}
\ee
Since $\epsilon$ is much less than 1, the
inflationary energy scale $V^{1/4}$
is at least a couple of orders of magnitude
below the Planck scale \cite{lyth85}.
Our main focus is on the
spectral index of the density perturbation. It may be defined by
$n-1\equiv 2d\ln \delta_H/d\ln k$.
Since the right hand side of
Eq.~(\ref{delh}) is evaluated when $k=aH$, and since
the rate of change of
$H$ is negligible compared with that of $a$ we have
$d\ln k=H dt$. From the slow-roll condition
$dt=d\phi/\dot\phi=-(3H/V')d\phi$
and therefore
\be
\frac{d}{d\ln k}
=-\Mpl^2 \frac{V'}{V} \frac{d}{d\phi}
\label{nexpression}
\ee
This gives
\cite{LL1,davis,salopek} the formula
advertized in the introduction,
\be
n-1=-6\epsilon+2\eta
\label{n2}
\ee
More accurate versions of the formulas for $\delta_H$ and $n$ have
been derived \cite{stly} which provide estimates of
the errors in the usual formulas. They are
\bea
\frac{\Delta\delta_H}{\delta_H}&=&O(\epsilon,\eta)\\
\Delta n &=& O(\epsilon^2,\eta^2,\gamma^2)
\label{deltan2}
\eea
where $\gamma$ is a third flatness parameter defined by
$\gamma^2\equiv \Mpl^4 V'''V'/V^2$.
We see that the usual formula $n-1=-6\epsilon+2\eta$ is valid
(barring cancellations) if and only if
there is a third flatness condition
(in addition to $\epsilon\ll 1$ and $|\eta|\ll 1$) which is
\be
\gamma^2\ll\max\{\epsilon,\eta\}
\label{flat3}
\ee
In typical models it
is an automatically consequence of the other two,
but it is in principle an additional requirement.
In most models, $\epsilon$ and $\gamma$ are
both negligible compared with $\eta$. Then
$n=1+2\eta$, and
\be
\Delta n\sim |1-n|^2
\ee
This theoretical error
will be comparable with the eventual observational uncertainty
(say $\Delta n\sim .01$) if
it turns out that $|1-n|\gsim 0.1$, in which case the use of the
more accurate formula would be needed to fully interpret the
observation. However, as will become clear an accuracy
$\Delta n$ of order $.01$ is (in the models proposed so far)
needed only if $|1-n|\lsim 0.1$. Thus the
more accurate formula for the spectral index is not required for the
purpose of distinguishing between models.
Differentiating the usual expression one finds \cite{running}
\be
\frac{d\ln (n-1)}{d\ln k}= [-16\epsilon\eta
+24\epsilon^2 + 2\gamma^2]/(n-1)
\label{running}
\ee
Barring cancellations, we see that
$n-1$ varies little over an interval
$\Delta \ln k\sim 1$ if and only if all three flatness conditions are
satisfied.
Cosmological scales correspond to only $\Delta \ln k\sim 10$,
and in most models $n-1$ varies little over this interval too. Then
one can write $\delta_H^2\propto k^{n-1}$, the usual
definition of $n$. (A model where the two terms of $n-1$ cancel,
allowing it to vary significantly, is mentioned in Section V(D).)
\subsection{A simpler formula for the spectral index}
In almost all models proposed so far,
$\epsilon$ is negligible so that one can write
$n-1=2\eta$.
Consider first the potential
$V\simeq V_0(1\pm\mu\phi^p)$,
with the constant term dominating. We shall encounter potentials of this
form frequently, and since the constant term dominates they give
\be
\frac{\epsilon}{\eta}=\frac{p}{p-1}\frac\mu2 \phi^p\ll \frac p{p-1}
\ee
Except in the case $p\simeq 1$, we see that $\epsilon$
is indeed negligible in these models.
This result applies whether or not $\phi$ is small on the Planck scale.
If it is small, one
can argue that $\epsilon$ is negligible irrespectively of the form of
the potential.
To see this, use
the estimate $\Delta \ln k\simeq 9$ for the range of cosmological
scales. It corresponds to $9$ $e$-folds of
inflation.
In slow-roll inflation the quantity $V'/V$ has negligible variation over one
$e$-fold and in typical models it has only small variation over the
9 $e$-folds. Taking that to be the case, one learns from
(\ref{Nrelation})
that in small-field models the contribution of $(V'/V)$ to the spectral
index is $3\Mpl^2 (V'/V)^2\ll (1/9)^2=.04$,
which is indeed negligible.
This upper bound on $(V'/V)$ means \cite{mygwave}
that the effect of gravitational
waves on the cmb anisotropy will be too small ever to detect,
because owing to cosmic variance the relative
contribution
$r\simeq 5\Mpl^2(V'/V)^2$
has to be $\gsim 0.1$ if it is to be detectable
\cite{turner}.
The effect is actually undetectable
in most large-field models as well \cite{mygwave}.
\subsection{The number of $e$-folds}
A model of inflation will give us an inflationary potential $V(\phi)$,
and a prescription for the value $\phi_{\rm end}$ of the field at
the end of inflation. What we need to obtain the predictions
is the value of $\phi$ when cosmological scales
leave the horizon, which is given in terms of $N$ by
Eq.~(\ref{nint}).
This quantity can in turn be
determined if the history of the universe after inflation
is known.
Consider first the epoch when the
scale
$k^{-1}=H_0 ^{-1}= 3000h^{-1}\Mpc$ leaves the horizon, which can be
taken to mark the
beginning of cosmological inflation.\footnote
{This is roughly the biggest cosmologically interesting
scale. The absolute limit of direct observation is $2H_0^{-1}$,
the distance to the particle horizon. Since
the prediction is made for a randomly placed observer
in a much bigger patch,
bigger scales in principle contribute to it,
but sensitivity rapidly decreases outside our horizon.
Only if the spectrum increases sharply on very large scales
\cite{gz} might there be a significant effect.}
Using a subscript 1 to denote this epoch, $N_1= \ln(a_{\rm end}/a_1)$.
Since this scale is the one entering the horizon now,
$a_1H_1=a_0H_0$ where the subscript 0 indicates the present
epoch and
\be
N_1=
\ln\left(\frac{a_{\rm end}H_{\rm end}}{a_0H_0}\right)
-\ln\left(\frac{H_{\rm end}}{H_1}\right)
\ee
The second term will be given by the model of inflation and is usually
$\lsim 1$; for simplicity let us
ignore it. The first term is known if we
know the evolution of the scale factor between the end of cosmological
inflation and the present.
Assume first that the end of inflation gives way promptly to
matter domination, which is followed by a radiation dominated era
lasting until the present matter dominated era begins,
one has \cite{LL2}
\be
N_1=62-\ln(10^{16}\GeV/V_{\rm end}^{1/4})
-\frac13\ln(V_{\rm end}^{1/4}/\rho_{\rm reh}^{1/4})
\ee
($\rho_{\rm reh}$ is the `reheat' temperature, when radiation domination
begins.)
With $V^{1/4}\sim 10^{16}\GeV$ and instant reheating this gives
$N_1\simeq 62$, the biggest possible value. However,
$\rho_{\rm reh}$ should probably be no bigger than
$10^{10}\GeV$ to avoid too many gravitinos, and using that value
gives $N_1=56$, perhaps the biggest reasonable value.
With $V^{1/4}=10^{10}\GeV$, the
lowest scale usually considered, one finds $N_1=47$ with instant reheating,
and $N_1=39$ if reheating is delayed to just before nucleosynthesis.
If there is in
addition an era of thermal inflation
\cite{thermal} lasting about 10 $e$-folds this figure is reduced to
$N_1\simeq 29$. Subsequent eras of the thermal inflation are
not particularly unlikely from a theoretical viewpoint,
but since each era would subtract $\simeq 10$ from $N_1$
this would have dramatic observational consequences with
more than two eras practically forbidden.
The smallest scale that will be directly probed in the forseeable
future is perhaps
four orders of of magnitude lower than $H_0^{-1}$, which corresponds to
replacing $N_1$ by $N_1-9$.
A central value, appropriate for use when calculating
the spectral index, is therefore $N\simeq N_1-4.5$,
and assuming at most one bout of thermal inflation
it probably lies in the range
\be
24\lsim N \lsim 51
\ee
In the models that have been proposed, the predicted value of $n-1$
is either independent of $N$, or is proportional to $1/N$.
In the latter case, taking $N=25$ instead of the usual $N=50$
or so doubles the predicted value of $n-1$.
\subsection{Hybrid and non-hybrid models of inflation}
In non-hybrid models of inflation, the potential $V(\phi)$
is dominated by the inflaton field $\phi$. The potential
has a minimum corresponding to the vacuum value (vev) of $\phi$,
at which $V$ vanishes (or anyhow is much
smaller than during inflation).
Inflation ends when the minimum is approached, leading to a
failure of one of the flatness conditions, and
$\phi$ then oscillates about its vev.
In hybrid
models, the potential $V(\phi,\psi)$
is dominated by some other field $\psi$,
which is held in place by its interaction with $\phi$.
In ordinary hybrid models $\psi$ is fixed, whereas in
`mutated' models it is slowly varying as it adjusts to minimize the
potential at fixed $\phi$, but either way we again
end up with an effective
potential $V(\phi)$ during inflation. However,
inflation now typically ends
when the other field is destabilized, as
$\phi$ passes through some critical value $\phi_c$, rather than by
failure of the flatness conditions. After inflation ends
both $\phi$ and the other field oscillate about their vevs,
though they might be very inhomogeneous depending on the
model.
\subsection{The power-series expansion}
In the context of supergravity one expects the potential $V$ to have
an infinite power series expansion in the fields \cite{susy}.
It vanishes
at its minimum, which corresponds to the vacuum expectation values
(vevs) of the fields.
Assuming for simplicity
that odd terms are forbidden by some symmetry, the expansion for a
single field with all others fixed at their vev
will be of the form (with
coefficients of either sign)
\be
V(\phi)
=V_0 +\frac12 m^2\phi^2 +\frac14\lambda\phi^4+\lambda' \Mpl^{-2}\phi^6
+\cdots
\label{power}
\ee
The minimum, at which $V$ vanishes, corresponds to the vev of
$\phi$.
In the usual applications of particle theory,
to such things as the standard model, neutrino masses, Peccei-Quinn
symmetry or a GUT, one is interested only in field values
$\phi\ll \Mpl$. In this regime it is reasonable to expect
the series to be dominated by one, two or at most three terms.
To be precise, one expects the quadratic and/or the quartic
terms to be important,
plus at most one higher term. As a result there is good control
over the form of the potential, and plenty of predictive power.
It is attractive to suppose that inflation model-building lies in the
same regime, and indeed one may even hope that the fields
occurring in the inflation model are ones that already occur in some
particle physics application.\footnote
{For a recent proposal for identifying the inflaton field with a flat
direction of a supersymmetric extension of the standard model,
see Ref.~\cite{lisa}; note though that the
non-inflaton field is a distance of order $\Mpl$ from its vacuum value
in the models considered there.}
As one considers bigger field values the justification for keeping
only a few terms becomes weaker. For values of order $\Mpl$ one
may still argue for keeping only a few terms, giving for instance a vev
of order $\Mpl$. For field values one or more orders of magnitude bigger
there is no theoretical argument for such an assumption.
Equally there is none against it (except in the case of
the moduli fields of superstring theory).
Returning to the small-field case, not
all terms need be significant. In particular the
$\phi^4$ term can be absent, or at least suppressed by factors like
$(100\GeV/\Mpl)^2$ which make it completely negligible. If that happens,
$\phi$ is said to correspond to a flat direction in field space
(see for example Ref.~\cite{thermal} and references cited there).
As we shall note later a `flat' direction in this sense is a good
candidate for ordinary hybrid inflation. (Note though
that this use of the word `flat' is not directly related to
the `flatness conditions'
$\epsilon\ll1$ and $|\eta|\ll 1$ that are needed
for slow-roll inflation {\em per se}.)
The above form of the expansion is appropriate if $\phi$ is the modulus
of a complex field charged under a $U(1)$ (or higher) symmetry,
the vevs of other fields charged under the symmetry being
negligible. In general odd
terms will appear, though the linear term
is still forbidden if the origin is the fixed point of some lower
symmetry, such as a subgroup $Z_n$ of a $U(1)$.
In non-hybrid models, where only the inflaton field is relevant,
the above expansion is what we need, with $\phi$
the inflaton field. In hybrid inflation, where the dominant contribution
to the potential comes from a different field $\psi$, we need a similar
expansion in both fields simultaneously.
If two or more fields are significant we have to worry about the fact
that in the context of supergravity one does not expect the fields to be
canonically normalized. Rather, the kinetic term will be of the form
\be
{\cal L}_{\rm kin} = \frac12\sum_{a,b} h_{ab} \partial_\mu \phi_a
\partial^\mu \phi_b
\label{hmetric}
\ee
If the metric $h_{ab}$ has vanishing curvature then one can redefine the
fields to get the canonical normalization $h_{ab}=\delta_{ab}$.
In supergravity the curvature does not vanish, but it will be negligible
in the small-field regime. For field values $\gsim\Mpl$ it will become
significant and in general must be taken into account.
But even in a hybrid inflation model, the non-inflaton field is fixed
(or nearly so) during inflation, and one can always
impose canonical normalization on a single-component inflaton field
even though that might not be the most natural choice in the context of
supergravity.
Unless otherwise stated, the inflationary potential $V(\phi)$ is
supposed to refer to a canonically normalized inflaton field
in what follows.
In the above discussion we have in mind the case of Einstein gravity,
corresponding to ${\cal L}=-\frac12\Mpl^2 R + \tilde{\cal L}$. The
second term is the Lagrangian defining the particle theory, which we
have in mind is some gauge theory formulated in the usual context of
essentially flat spacetime.
This is a reasonable assumption because the energy scale when
cosmological scales leave the horizon during inflation
is at least two orders of magnitude below
the Planck scale. Modifications of Einstein gravity are sometimes
considered though, such as replacing $\Mpl$ by a field or adding
an $R^2$ term. In these cases, and many others, one can recover
Einstein gravity by redefining the fields and
the spacetime metric, at the expense of making the non-gravitational
part of the Lagrangian more complicated. In this way one typically arrives
at a model of inflation with large fields and a non-polynomial
potential, such as the one in Eq.~(\ref{another}). Models of this kind are
in practice constructed without reference to supergravity,
and ignore our lack of understanding of the large-field behaviour
of the potential just as much as large-field models invoking say a power-law
potential.
\subsection{The difficulty of keeping $\eta$ small in supergravity}
In small-field models of inflation, where the theory is under control,
it is reasonable to work in the context of
supergravity. This is a relatively recent activity because although
small-field models were the first to be proposed \cite{new,singlet}
they were soon abandoned in favour of models with fields first of order
the Planck scale \cite{primordial} and then much bigger \cite{CHAOTIC}.
Activity began again after hybrid inflation was proposed
\cite{l90,LIN2SC}, with the realization
\cite{CLLSW} that the model is again of the small-field type.
In Ref.~\cite{CLLSW} supersymmetric implementations of
hybrid inflation were considered, in the context of both global
supersymmetry and of supergravity.
According to supergravity, the potential can be written as
a `$D$ term' plus an `$F$ term', and it is usually supposed that
the $D$ term vanishes during inflation.
All models proposed so far are of this kind, except for
hybrid inflation models with the $D$ term dominating
\cite{ewansgrav,bindvali,halyo}; such models are
quite special and are expected to give a distinctive
$n=0.96$ to $0.98$ as briefly mentioned in
Section V.
For models where the $D$ term vanishes,
the flatness parameter $\eta=\Mpl^2 V''/V$ generically receives
various contributions of order $\pm 1$. This
crucial point was first emphasized in Ref.~\cite{CLLSW}, though it
is essentially
a special case of the more general result,
noted much earlier
\cite{dinefisch,coughlan}, that
there are contributions of order $\pm H^2$ to the mass-squared of every
scalar field. Indeed, in a small-field
model the troublesome contributions to $\eta$ may
be regarded as contributions
to the coefficient $m^2$ in the expansion (\ref{power}) of the inflaton
potential.
Since slow-roll inflation requires
$|\eta|\ll 1$, either there is an accidental
cancellation, or the form of the $F$ term is non-generic.
An accident is reasonable only if $\eta$ is not too small,
corresponding to $n=1+2\eta$ not too close to 1.
Pending a full discussion in Section VII, the $\eta$ problem
will be mentioned as appropriate when discussing the various models.
\subsection{Before cosmological inflation}
The only
era of inflation
that is directly relevant for observation is the one beginning when
the presently observable universe leaves the horizon.
This era of `cosmological inflation'
will undoubtedly be preceded by
more inflation, but all memory of earlier inflation is lost apart from
the starting values of the fields at the beginning of cosmological
inflation. Nevertheless, one ought to try to understand the earlier era
if only to check that the assumed starting values are not ridiculous.
A complete history of the universe will
presumably start when the energy density is at the Planck scale.
(Recall that $V^{1/4}$ is at least two orders of magnitude
lower during cosmological inflation.) The
generally accepted hypothesis is that the scalar fields
at that epoch take on chaotically varying values as one moves around
the universe, inflation occurring where they have suitable values
\cite{CHAOTIC,abook}.
It is indeed desirable to start descending from the Planck scale
with an era of inflation
for at least two reasons. One is
to avoid the universe either collapsing or becoming practically empty,
and the other is to have an event horizon so that the
homogeneous patch within which we are supposed to live is
not eaten up by its inhomogeneous
surroundings. However, there is no reason to suppose
that this initial era of inflation is of the slow-roll variety.
The motivation for slow-roll comes from the observed
fact that $\delta_H$ is almost scale-independent, which applies only
during the relatively brief era when cosmological scales are
leaving the
horizon. In the context of supergravity,
where achieving slow-roll inflation requires rather delicate
conditions, it might be quite attractive to suppose that
non-slow-roll inflation takes the universe down from the Planck scale with
slow-roll setting in only much later. A well known potential
that can give non-slow-roll inflation is
$V\propto \exp(\sqrt{2/p}\phi/\Mpl)$, which gives
$a\propto t^p$ and corresponds to non-slow-roll inflation in
the regime where $p$ is bigger than 1 but not much bigger.
If slow-roll is established well before cosmological scales leave the
horizon it is possible to have an era of `eternal inflation'
during which
the motion of the inflaton field is dominated by the
quantum fluctuation.{For eternal inflation taking place at
large field values see
Ref.~\cite{eternal}. The corresponding phenomenon for inflation near a
maximum was noted earlier
by a number of authors.}
The condition for this to occur is
that the predicted spectrum $\delta_H$ be
formally bigger than 1 \cite{stochastic}.
Coming to the beginning of cosmological inflation, the
possibilities are actually so varied
that one can contemplate a wide range of initial field values.
Going back in time, one might find a smooth
inflationary trajectory going all the way back to an era when
$V$ is at the Planck scale
(or at any rate much bigger than its value during cosmological
inflation). In that case
the inflaton field will probably be decreasing during inflation.
Another natural possibility is for the inflaton to
find itself near a maximum of the potential
before cosmological inflation
starts.
Then there may be eternal inflation followed by
slow-roll
inflation.
If the maximum is a fixed point of the symmetries it is quite natural for
the field to have been driven there by its interaction with other
fields. Otherwise it
could arrive there by accident, though this
is perhaps only reasonable if
the distance from the maximum to the minimum
is $\gsim \Mpl$
(see for instance Ref.~\cite{natural3} for an example).
In this latter case, the fact that eternal inflation occurs near the
maximum may help to enhance the probability of inflation starting
there.\footnote
{The idea that
inflation could start at a maximum of the potential through
non-thermal effects was first promoted in Ref.~\cite{nontherm}.
If the maximum is a fixed point the field might alternatively
be placed there through thermal corrections to the potential
\cite{new}, but
this `new inflation'
hypothesis is difficult to realize and has fallen out of favour now that
alternatives have been recognized.}
\section{Non-hybrid models}
Non-hybrid models of inflation proposed so far
fall into two broad classes.
Either the inflaton field is of order $10\Mpl$ when cosmological scales
leave the horizon, moving towards the origin
under a power-law potential $V\propto \phi^p$, or it is $\lsim \Mpl$
and moving away from the origin under a potential
$V\simeq V_0(1-\mu\phi^p)$. The first case is often called
`chaotic' inflation because it provides a way of
descending from the Planck scale with chaotic initial field conditions.
The second case is often called `new' inflation because that was the
name given to the first example of it.
\subsection{Power-law and exponential potentials}
The simplest potential giving inflation is
\be
V= \frac12 m^2\phi^2
\ee
and obvious generalizations are $V=\frac14\lambda\phi^4$,
and $V=\lambda M_{\rm Pl}^{4-p}\phi^p$ with $p/2\geq 3$.
Potentials of this form were proposed as the simplest realizations
of chaotic initial conditions at the Planck scale
\cite{CHAOTIC}.
Inflation ends
at $\phi_{\rm end}\simeq p\Mpl$, after which $\phi$
starts to oscillate about its vev $\phi=0$.
When cosmological scales leave the horizon
$\phi= \sqrt{2Np}\Mpl$. Since the inflaton field is
of order 1 to $10\Mpl$, there is no
particle physics motivation for a power-law potential
though equally
there is no argument against it.
The model gives $n-1=-(2+p)/(2N)$, and gravitational waves
are big enough to be
eventually observable with $r=2.5p/N=5(1-n)-2.5/N$.
The COBE normalization
corresponds to $m=1.8\times 10^{13}\GeV$ for the quadratic case.
For $p=4,6,8$ it gives respectively
$\lambda=2\times 10^{-14}$, $\lambda=8\times 10^{-17}$,
$\lambda=6\times 10^{-20}$ and so on.
The energy scale is $V^{1/4}\sim 10^{16}\GeV$ (corresponding to
$\epsilon$ not very small)
but as we are not dealing with a broken symmetry there is no
question of relating this to a GUT scale.
The same prediction is obtained for a more complicated potential,
provided that it is proportional to $\phi^p$ during cosmological
inflation,
and in particular $\phi$ could have a nonzero vev $\ll\Mpl$
\cite{shafi1,kls}. The more general case, where more than one power of
$\phi$ is important even while cosmological scales are leaving the
horizon, has also been considered \cite{hodges}.
It can give rise to
a variety of predictions \cite{blumhodges}
for the scale-dependence of $\delta_H$
but of course requires a delicate balance of coefficients.
The limit of a high power is an exponential potential,
of the form $V=\exp(\sqrt{2/q}\phi)$. This gives
$\epsilon=\eta/2=1/q$
which lead to
$n-1=-2/q$ and $r=10/q$.
This is the case of
`extended inflation',
where the basic interaction involves non-Einstein gravity but the
exponential potential occurs after transforming to Einstein gravity
\cite{extended}.
However, simple versions of this proposal are ruled out by observation,
because the end of inflation corresponds to a first order phase
transition, and in order for the bubbles not to spoil the cmb
isotropy one requires $n\lsim 0.75$ in contradiction with
observation \cite{LL1,green}.
\subsection{The inverted quadratic potential}
Another simple potential leading to inflation is
\cite{nontherm,bingall,natural3,paul,natural,ovrut,natural2,%
natural4,kinney,kumekawa,izawa}
\be
V=V_0-\frac12m^2\phi^2 +\cdots
\label{natural}
\ee
We shall call this the `inverted' quadratic potential, to distinguish it
from the same potential with the plus sign which comes from the simplest
version of hybrid inflation.
The dots indicate the effect of higher powers, that are supposed
to come in after cosmological scales leave the horizon. As long as they
are negligible, $V_0$ has to dominate for inflation to
occur.
This potential gives $1-n=2\Mpl^2m^2/V_0$. If $m$ and $V_0$ are regarded
as free parameters, the region of parameter space permitting slow-roll
inflation corresponds to $1-n\ll 1$. Thus $n$ is indistinguishable from
1 except on the edge of parameter space. However, in the context of
supergravity there are two reasons why the edge might be regarded as
favoured. One is the fact that
$\eta$ generically receives contributions of order
1. Since slow-roll inflation
requires $|\eta|\ll 1$, either $\eta$ is somewhat reduced from its
natural value by accident, or it is suppressed because the theory has a
non-generic form. In the first case one does not expect $\eta$ to be
tiny, and in the present context this means that $1-n=2\eta$ will not be
tiny either.
The other reason for expecting $n$ to be significantly below 1, which is
specific to this potential, has to do with the magnitude of the vev
$\phi_m$. If the inverted quadratic form
for the potential holds until $V_0$ ceases to dominate,
one expects
\be
\phi_m\sim \frac{V_0^{1/2}}{m}
=\left(\frac{2}{1-n} \right)^{1/2} \Mpl
\ee
(This is also an estimate of $\phi_{\rm end}$ in that case.)
To understand the potential within the context of particle
theory the vev should not be more than a few times
$\Mpl$, which requires $n$ to be well below 1.
We see that to have $n$ indistinguishable from 1, the
potential should steepen drastically after cosmological scales leave the
horizon. Even given that condition, one also requires that the smallness
of $\eta$ is not an accident in the context of supergravity.
So far no model has been proposed that satisfies both of these
conditions, though a steepening of the potential with an accidental
suppression of $\eta$ has recently been considered \cite{izawa}.
In the next section we shall also consider the inverted quadratic
potential in the context of hybrid inflation, where some other field is
responsible for $V_0$. In that case inflation ends at
some $\phi_{\rm end}\ll \Mpl$, and one can be dealing with a version of
supergravity that keeps $\eta$ tiny in which case
$n$ will be indistinguishable from 1.
When cosmological scales leave the horizon
$\phi\sim \phi_{\rm end} e^{-x}$ where $x\equiv N(1-n)/2$.
The COBE normalization gives
\be
\frac{V_0^{1/2}}{\Mpl^2}= 5.3\times 10^{-4}
N^{-1} x e^{-x}\frac{\phi_{\rm end}}{\Mpl}
\ee
For $\phi_{\rm end}\sim \Mpl$ this gives $V_0^{1/4}> 5\times 10^{14}\GeV$
(using $n>0.7$). But in a hybrid inflation model $\phi_{\rm end}$
and $V_0^{1/4}$ could be much lower.
Staying with non-hybrid models, there have been several
proposals for realizing the inverted quadratic potential
without steepening. Since there is no steepening they
will all give $n$ significantly below 1, and are not expected to
occupy the regime $0.9\lsim n<1$ populated by models to be discussed
later. Let us list them briefly.
\subsubsection{Modular inflation}
If $\phi$ is the real or imaginary part of
one of the superstring moduli, its potential is generally expected
\cite{bingall,paul} to be of the form
$V=\Lambda^4 f(\phi/\Mpl)$, with $f(x) $ and its derivatives roughly
of order 1
in the regime $|x|\lsim 1$.
In that region one expects the flatness parameters
$\eta\equiv\Mpl^2V''/V $
and $\epsilon \equiv \Mpl^2(V'/V)^2/2$ to be both roughly of order 1,
and one might hope to find within it a region where they both happen to
be significantly
below 1 so that slow-roll inflation can occur.
In the regime $|x|\gg 1$ one expects
either a potential that rises too steeply to support inflation,
or one that becomes periodic.
It is usually supposed that inflation takes place near a maximum.
(As the maximum is not usually supposed to be a fixed point
of a symmetry there is no reason why the interaction of the modulus
should drive it there, but the existence of `eternal'
inflation may still be held to favour it as a starting point.)
One then expects
the potential to be of the inverted quadratic form, with
no significant steepening after cosmological scales leave the
horizon.
So far, investigations using specific models \cite{bingall,natural2,macorra}
have actually concluded that viable inflation does not occur.\footnote
{Ref.~\cite{bento} claims to have been successful, but
an analytic calculation of that model reported in Ref.~\cite{ournew}
finds that it is not viable.}
Many complex fields other than the modulus are expected
in the context of supersymmetry, and one could take
$\phi$ to be the real or imaginary part of one of them.
Although the potential for such fields is not understood for
$\phi\gg\Mpl$ it is not unreasonable to suppose that there is a
vev at $\phi\sim \Mpl$, and if so one can again expect an inverted
quadratic form for the potential.
\subsubsection{`Natural' inflation}
The prediction for $n$ depends only on the form of the potential while
cosmological scales are leaving the horizon. A potential
reducing to the inverted quadratic one in this regime is
\be
V= V_0 \cos^2(\phi/M)
\label{gbpot}
\ee
This potential typically arises if $\phi$ is the pseudo-Goldstone boson
of
a $U(1)$ symmetry broken by non-perturbative effects.
Inflation using it
was first discussed
in Ref.~\cite{bingall}, and then in Ref.~\cite{natural} where it was called
`natural' inflation, and has subsequently been considered by several
authors \cite{natural3,ovrut,natural2,natural4,glw,ewanprep}.
Suggestions as to the identity of $\phi$ have been that
it is the imaginary part of a modulus
such as the dilaton \cite{bingall,natural2},
the angular part
of an ordinary complex field
\cite{natural,natural2,natural3,natural4,glw,ewanprep}
or something else \cite{ovrut}. For any of them to be reasonable
$M$ should be not too far above the Planck scale, corresponding
to $\eta$ not too small and $n$ significantly below 1.
Even when $\eta$ is not tiny, one still feels more comfortable with a
definite mechanism for keeping it well below 1 in the context of
supergravity, and the $U(1)$ symmetry is such a mechanism
provided that it is broken only by the superpotential
(Section VII).
\subsubsection{A loop-corrected potential}
It was pointed out in Ref.~\cite{ewanloop} that a
different mechanism for keeping $\eta$ small
might be provided by a loop correction of the
form $A\phi^2\ln(\phi/B)$ \cite{ewanloop}.
Since the loop correction has to be large one needs to allow a renormalization
group modification of it. The most reasonable model to
emerge has an inverted quadratic potential without steepening,
leading again to $n$ well below 1.
\subsection{Cubic and higher potentials}
If the quadratic term is heavily suppressed or absent, one will have
\be
V\simeq V_0 ( 1-\mu \phi^p +\cdots)
\label{higher}
\ee
with $p\geq 3$. For this potential one expects that the integral
(\ref{nint}) for $N$ is dominated by the limit
$\phi$ leading to \cite{kinney}
\be
\phi^{p-2}=[p(p-2)\mu N \Mpl^2]^{-1}
\label{phiend}
\ee
and
\be
n\simeq1-2\left(\frac{p-1}{p-2}\right)\frac1 N
\ee
It is easy to see that the integral is typically dominated by the $\phi$ limit,
if higher terms in the potential (\ref{higher}) become significant only when
$V_0$ ceases to dominate at
$\phi^p\sim \mu^{-1}$. Then, in
the regime where $V_0$ dominates,
$\eta=[(p(p-1)\Mpl^2/\phi^2]\mu\phi^p$, and if this expression becomes of order
1 in that regime inflation presumably ends soon after.
Otherwise inflation ends when $V_0$ ceases to
dominate. At the end of inflation one therefore has
$\Mpl^2\mu\phi_{\rm end}^{p-2}\sim 1$ if $\phi_{\rm end}\ll
\Mpl$, otherwise one has $\mu\phi_{\rm end}^p\sim 1$.
(We are supposing for simplicity that $p$ is not enormous, and dropping
it in these rough estimates.)
The integral
(\ref{nint}) is dominated by the limit
$\phi$ provided that
\be
N\Mpl^2\mu\phi_{\rm end}^{p-2}\gg 1
\label{criterion1}
\ee
This is always
satisfied in the first case, and is satisfied in the second case
provided that $\phi_{\rm end}\ll \sqrt N \Mpl$ which we shall assume.
If higher order terms come in more quickly than we have supposed, or if
inflation ends through a hybrid inflation mechanism then
$\phi_{\rm end}$ will be smaller than these estimates, and
one will have to see whether the criterion (\ref{criterion1})
is satisfied.
Continuing with the assumption that it is satisfied,
the COBE normalization is \cite{kinney}
\be
5.3\times 10^{-4}= (p\mu\Mpl^p)^{\frac1{p-2}} [N(p-2)]^{\frac{p-1}{p-2}}
V_0^{\frac12} \Mpl^{-2}
\ee
In particular, for $p=4$ the dimensionless coupling is
$\lambda\equiv 4V_0\mu=2.8\times 10
^{-13}(50/N)^3$. At least in the context of non-hybrid models
such a tiny value may be problematical, as
was recognized in the first models of inflation
\cite{new,singlet} using the essentially equivalent non-supersymmetric
Coleman-Weinberg loop correction $A\phi^4(\phi/B)$
(see however Ref.~\cite{langbein} for a dissenting view about this
latter case).
As we noted earlier, a strong suppression of the quadratic term
may well occur in a supergravity theory.
An explicit example has been given recently
\cite{grahamnew}. In this proposal, one finds a potential
\be
V=V_0(1+\beta\phi^2\psi - \gamma\phi^3 +\cdots)
\ee
where $\psi$ is another field. Then, with $\beta$ and $\gamma$ of order 1 in
Planck units, and initial values
$\psi\sim \Mpl$ and $\phi\simeq 0$ one can check that
the quadratic term is driven to a negligible value
before cosmological inflation
begins.
This proposal gives $\phi$ a vev of order $\Mpl$.
Some particle-physics motivation for
small-field models with $p\geq 3$ is given in
Refs.~\cite{natural4,kinney}, though not in the context of
supergravity.
One could contemplate models in which
more than one power of $\phi$ is significant while
cosmological scales leave the horizon, but this requires a delicate
balance of coefficients. Models of this kind were also discussed a long
time ago \cite{primordial,olive}, again with a vev of order $\Mpl$,
but their motivation was in the context of setting the initial value of
$\phi$ through thermal equilibrium and has disappeared with the
realization that this `new inflation' mechanism is not needed.
They could give a range of predictions for $n$.
\subsection{Another potential}
Potentials have been proposed
that are of the form
\be
V\simeq V_0(1- e^{-q\phi/\Mpl})
\label{another}
\ee
with $q$ of order 1.
This form is supposed to
apply in the regime where $V_0$ dominates, which is $\phi\gsim \Mpl$.
Inflation ends at $\phi_{\rm end}\sim \Mpl$, and when
cosmological scales leave the horizon one has
\bea
\phi&=& \frac1q \ln(q^2 N) \Mpl\\
n-1&=&-2\eta= -2/N
\label{r2pot}
\eea
Gravitational waves are negligible.
The most reasonable-looking derivation of a potential of this form
in the context of particle theory
\cite{ewansgrav} starts with a highly non-minimal kinetic term.
In that case $q$ can have different values such as
$1$ or $\sqrt2$.
Another derivation \cite{r2,burt} modifies Einstein gravity
by adding a large $R^2$ term to the usual $R$ term, but with a
huge
coefficient,
and a third \cite{vplanck} uses a variable Planck mass. In both cases,
after transforming back to Einstein gravity one obtains the above form
with $q=\sqrt{2/3}$.
This potential is mimicked by $V=V_0(1-\mu \phi^{-p})$ with
$p\to\infty$ (Table 1).
\section{Hybrid inflation models}
In hybrid inflation models \cite{l90,LIN2SC}, the
slowly rolling inflaton field $\phi$ is
not the one responsible for most of the energy density.\footnote
{We take this to be the definition of the term `hybrid', which is
coming to be the standard usage though somewhat narrower than
in Refs.~\cite{LIN2SC,LIN2SC2} where it was first introduced.}
That role is
played by another field $\psi$, which is held in place
by its interaction with $\phi$ until the latter reaches a critical
value $\phi_c$. When that happens $\psi$ is destabilized
and inflation ends. As was pointed out in Ref.~\cite{CLLSW},
$\phi_c$ is typically small in Planck units which means that hybrid
inflation models are typically small-field models.
The end of inflation can correspond to either a first \cite{l90}
or a second order \cite{LIN2SC}
phase transition, but the second-order case (corresponding to the absence of a
potential barrier) is simpler and is the only one that will be
considered here.
It has been considered by many authors
\cite{LL2,LIN2SC2,CLLSW,ewansgrav,lisa,glw,ewanloop,ournew,%
mutated,bindvali,halyo,ewanprep,%
silvia,wang,dave,qaisar,dvaliloop,lazpan,qaisarlatest}.
(The first-order case amounts to an
Einstein gravity version of extended inflation \cite{extended,green},
and has been considered in Refs.~\cite{AdamsFreese,CLLSW}.
For a non-Einstein version of second-order hybrid inflation see
Ref.~\cite{bertolami}.)
\subsection{Ordinary hybrid inflation}
The potential for the original model of (second-order) hybrid inflation
is \cite{LIN2SC}
\bea
V&=&\frac14\lambda(M^2-\psi^2)^2 +\frac12m^2\phi^2
+\frac12\lambda'\psi^2\phi^2 \\
&=&V_0-\frac12 m_\psi^2 \psi^2 + \frac14\lambda\psi^4
+ \frac12 m^2 \phi^2 +\frac12\lambda'\psi^2\phi^2
\label{fullpot}
\eea
The field $\psi$ is fixed at the origin if $\phi>\phi_c$, where
\be
\phi_c^2=m_\psi^2/\lambda'=\lambda M^2/\lambda'
\ee
In this regime slow-roll inflation can take place, with
the quadratic potential
\be
V=V_0+ \frac12 m^2\phi^2
\label{vord}
\ee
In the context of particle theory one expects the
couplings $\lambda$ and $\lambda'$ to be $\lesssim 1$.
In order to have inflation at small $\phi$
the first term must dominate after cosmological
scales leave the horizon, because at the other extreme one arrives at
$V\propto \phi^p$ which requires $\phi\sim 10$ to $20\Mpl$.
To fully justify the low-order polynomial form of the potential
one would also like to have $M\ll\Mpl$, which also
turns out to be a requirement
for inflation to end promptly at $\phi_c$ \cite{CLLSW}.
It is reasonable to also consider $M\sim \Mpl$, which has been done
in Refs.~\cite{lisa,glw}.
The same inflationary potential is obtained with other forms for the
last term of the potential (\ref{fullpot}), an example being
\cite{lisa}
$\lambda'\Mpl^{-2} \psi^2\phi^4$, leading to
\be
\phi_c^2=\Mpl m_\psi/\lambda'=\Mpl M \lambda^{1/2}/\lambda'
\label{phic2}
\ee
Other prescriptions for
$\phi_c$ are provided by mutated hybrid inflation as described
later.
When cosmological scales leave the horizon
\be
\frac{\phi}{\phi_c}=e^{\frac{n-1}{2}N}
\label{phih}
\ee
At least with the above two prescriptions
for $\phi_c$
this implies that we are dealing with a model
in which the inflaton field is small \cite{CLLSW}.
The quadratic inflationary potential gives \cite{LL2}
$n=1+2\Mpl^2 m^2/V_0$. From
the flatness condition $\eta\ll 1$, the
allowed region of parameter space corresponds to $n-1\ll 1$, so
one expects $n$ to be indistinguishable from 1 unless there is some
reason for the parameters to be near the edge of the allowed region.
Two reasons might be cited for wanting to be near the edge, but neither
is very compelling. One would be the fact that in
$\eta\equiv \Mpl^2 V''/V$ is of order 1 in
a {\em generic} supergravity theory.
If the smallness of $\eta$ is due to an accident it should not be too small.
However, in contrast with the inverted quadratic potential discussed
earlier, we are now dealing with a model in which the inflaton field is
small. As discussed in Section VII this makes it more attractive
to suppose that the smallness of $\eta$ is ensured from the start,
in which case there is no reason why it should not be tiny.
A more subtle consideration
is the proposal of Ref.~\cite{lisa} that
$m\sim m_\psi \sim 100\GeV$ and $M\sim \Mpl$ are favoured values, which
taken literally
does indeed give $\eta\sim 1$.
However the COBE normalization, to be discussed
in a moment, would then require a precise
choice of $\phi_c$, and in particular with either of the
prescriptions one would need a precise value of $\lambda'$ which is
{\em not} of order 1. Thus, while the suggested orders of magnitude
may be reasonable as rough estimates,
it cannot be said that the extreme edge of
the allowed region is really favoured, and indeed of the six examples
displayed in the Figures of Ref.~\cite{lisa} all except one
have $n$ indistinguishable from 1.\footnote
{Since the actual models used in Ref.~\cite{lisa}
are formulated in the context of global supersymmetry the smallness of
$\eta$ should presumably be viewed as an accident for them.
But one can imagine that a different
implementation of the proposal might keep $\eta$ small automatically
and we have proceeded with the discussion on this assumption.
If the smallness of $\eta$ is really accidental, very small values
are disfavoured and one should reject the other five examples
in favour of the last one.
This point seems to have been overlooked in Ref.~\cite{lisa}.}
The conclusion is that in the ordinary hybrid inflation model
one expects $n$ to be
indistinguishable from 1, though a value significantly
above 1 is not out of the question.
The COBE normalization is
\be
5.3\times 10^{-4}=\Mpl^{-3}\frac{V_0^{3/2}}{m^2\phi}
\label{cobenormhyb}
\ee
where $\phi$ is given by Eq.~(\ref{phih}).
With $\phi_c$ given
by either of the above prescriptions this imposes \cite{CLLSW}
a limit
$n\lesssim 1.3$ (assuming that
$V_0$ dominates the potential and that $M\lsim \Mpl$).
What about higher powers of $\phi$ in the inflationary potential?
They can be ignored if their contribution is
insignificant when $\phi$ has the value given by (\ref{cobenormhyb}).
For a quartic term $\frac14\lambda\phi^4$
this requires $\lambda\ll 10^{-7}(1-n)^3$ which seems very small.
However, the quartic term can be
eliminated altogether,
by identifying $\phi$ with one of the `flat' directions of
particle theory.
If a higher power $\phi^p$ does dominate
we have
\be
n=1 + 2\left(\frac{p-1}{p-2}\right)\frac1 N
\ee
For $p=4$ the COBE normalization requires
\cite{dave}
a very small dimensionless coupling
$\lambda\equiv4\mu V_0\lsim 10^{-12}$. This case is therefore less
attractive than the $p=2$ case. With higher powers,
inflation can
be interrupted and the requirement that the interruption is
over before cosmological scales leave the horizon
places an additional restriction
on the parameter space \cite{dave}.
The case where quadratic, cubic and quartic terms may all be important
has also been discussed \cite{wang}, though the possibility of inflation
being interrupted was ignored. As with the example discussed at the end
of the next subsection (and the large-field model of
Ref.~\cite{hodges}) there are enough parameters to allow a variety of
possibilities for the scale dependence of $\delta_H(k)$, though a
delicate balance between the parameters is required to achieve this.
A linear term
$p=1$ is forbidden if the origin is a fixed point of appropriate
symmetries, and even if a linear term is present the case that it
dominates is somewhat artificial (see the example at the end of
the next subsection). For the record, it gives a
spectral index indistinguishable from 1 in a
small-field model, because $V''=0$.
To summarize, ordinary hybrid inflation leading to a a quadratic inflationary
potential $V(\phi)$ is a very attractive model. The full potential
$V(\phi,\psi)$ need contain only quadratic and
quartic terms, and need have no small couplings after the quartic term
in $\phi$ has been eliminated by identifying $\phi$ with a `flat' direction.
Of course, whether the model can be successfully embedded in a more
complete theory is a bigger question, which is not the subject of
the present work, and is still a long way from being answered.
\subsection{Inverted hybrid inflation}
Instead of $\phi$ rolling towards the origin it might roll away from it.
The simplest way of achieving this `inverted' hybrid inflation
\cite{ournew} is to have\footnote
{As was pointed out a long time ago \cite{weinberg}, a potential of this kind
with the fields in thermal equilibrium
leads to high temperature symmetry restoration, the inverse of the usual
case. The same thing is happening in our non-equilibrium situation.}
\be
V = V_0 - \frac{1}{2} m_\phi^2 \phi^2 + \frac{1}{2} m_\psi^2 \psi^2
- \frac{1}{2} \lambda \phi^2 \psi^2 + \frac14\lambda_\phi\phi^4
+\frac14\lambda_\psi\psi^4
\label{first}
\ee
There is supposed to be a minimum at nonzero $\phi$ and $\psi$,
at which $V$ vanishes, but at fixed $\phi$ there is a minimum
at $\psi=0$ provided that
\be
\phi < \phi_{\rm c} = \frac{m_\psi}{\sqrt{\lambda}}
\ee
In this regime one can have inflation with the inverted quadratic potential
\be
V=V_0-\frac12m^2\phi^2
\ee
This is the same potential that we discussed already, with
the hybrid inflation mechanism now ending inflation while the field is still
small, instead of the steepening of the potential that has to be
postulated in a non-hybrid model. The hybrid mechanism looks more
natural, because it involves only
quadratic and quartic terms, with no need for the dimensionless
couplings of quartic terms to be small.
One
could replace the quadratic term $-\frac12m^2\phi^2$ by
a higher order term, but the hybrid inflation mechanism would not then
offer any simplification compared with the corresponding
single-field models.
If two or more terms are comparable while cosmological scales leave
the horizon, there are more complicated
possibilities including a significant
variation of the spectral index.
For example, one could have
\be
V=V_0-\frac12m^2\phi^2+\frac14\lambda\phi^4
\ee
This has a minimum at $\phi_m=m/\sqrt\lambda$ and we suppose that
$V_0$ still dominates there. Then, if
$\phi_c\sim \phi_m$ the spectral index could flip between the values
$n=1\pm2\Mpl^2m^2/V_0$ on cosmological scales.
The number of $e$-folds taken to flip is $\Delta N\sim
\Mpl^{-2}(V/V')\phi_m \sim (\Delta n)^{-1}$ where
$\Delta n$ is the magnitude of $1-n$ before and after the
flip. For the flip to occur with $\Delta N<10$ (the range of
cosmological scales) requires $\Delta n\gtrsim 0.1$.
For $\Delta n\lsim .1$,
the whole of cosmological inflation could
take place with $\phi$ midway between the maximum and minimum. In
particular it could be near the point of inflexion so that the potential
is practically linear. By choosing the same point as the origin we
recover the case $p=1$ discussed earlier. We see that at least in this
particular realization that case is either unnatural because the
cosmological epoch has to occupy a special place on the trajectory,
or is trivial because the entire trajectory (not just its linear part)
gives $n$ indistinguishable from 1.
\subsection{Mutated hybrid inflation}
In both ordinary and inverted hybrid inflation, the other field $\psi$ is
precisely fixed during inflation. If it varies,
an effective potential $V(\phi)$ can be generated even if the
original potential contains no piece that depends only on $\phi$.
This mechanism was first proposed in Ref.~\cite{mutated}, where it was called
mutated hybrid inflation. The potential considered was
\be
V = V_0 - A\psi + B\psi^2 \phi^2 + C\psi^2
\ee
The last term serves only to give $V$ a minimum at which it vanishes,
and is negligible during inflation. All of the other terms are significant,
with $V_0$ dominating.
For suitable choices of the parameters inflation takes place
with $\psi$ held at the instantaneous minimum, leading to a potential
\be
V= V_0 (1-\mu \phi^{-2})
\ee
Shortly afterwards the mechanism was rediscovered \cite{lazpan} and called
`smooth' hybrid
inflation emphasizing that any topological defects associated with
$\psi$ will never be produced
(in contrast with the case of ordinary and inverted hybrid inflation).
The potential considered there was $V=V_0-A\psi^4+B\psi^6\phi^2
+C\psi^8$, where again the last term is negligible during inflation
and $V_0$ dominates the remainder. It leads to
$V=V_0(1-\mu \phi^{-4})$.
Retaining the original name, the most general mutated hybrid inflation
model with only two significant terms is \cite{ournew}
\be
V = V_0 - \frac{\sigma}{p} \Mpl^{4-p}\psi^p + \frac{\lambda}{q}
\Mpl^{4-q-r}\psi^q \phi^r
+\dots
\ee
In a suitable regime of parameter space,
$\psi$ adjusts itself to minimize $V$ at fixed
$\phi$, and
$\psi\ll\phi$ so that the slight curvature of the
inflaton trajectory does not affect the field dynamics. Then,
provided that $V_0$ dominates the energy density, the effective potential
during inflation is
\be
V=V_0(1-\mu \phi^{-\alpha})
\label{vmut}
\ee
where
\bea
\mu&=&\Mpl^{4+\alpha} \left(\frac{q-p}{pq}\right)
\frac{\sigma^{\frac q{q-p}}\lambda^{-\frac{p}{q-p}}}
{V_0} >0\\
\alpha&=&\frac{pr}{q-p}
\eea
For $q>p$, the exponent $\alpha$ is positive as in the examples already
mentioned, but for $p>q$ it is negative with $\alpha<-1$.
In both cases it can be non-integral, though integer values are the most
common for low choices of the integers $p$ and $q$.
The situation in
the regime $-2\lsim\alpha<-1$ is similar to the one that we
discussed already for the case $\alpha=-2$;
the prediction for $n$ covers a continuous range below 1 because it
depends on the parameters,
but to have a small-field model the potential has to be steepened
after cosmological scales leave the horizon. An example of such
steepening is provided in the next subsection.
For choices of $\alpha$ outside the range $-2\lsim\alpha<0$,
the integral (\ref{nint}) is dominated by the limit $\phi$
provided that we are dealing with a small-field model, corresponding to
\be
\phi_{\rm end}\simeq \mu^{1/\alpha}\ll \Mpl
\label{phiend2}
\ee
Then one has \cite{ournew}
\be
n = 1 - 2 \left(\frac{\alpha+1}{\alpha+2}
\right)\frac{1}{N}
\ee
This prediction is listed in the Table for some integer values of
$\alpha$, along
with the limiting cases $\alpha=\pm\infty$ and $\alpha=0$.
Of the various possibilities regarding $\alpha$, some
are preferred over others in the context of supersymmetry.
One would prefer \cite{ournew} $q$ and $r$ to be even if
$\alpha>0$ (corresponding to $q>p$)
and $p$ to be even if $\alpha<0$. Applying this criterion with $p=1
$ or $2$ and $q$ and $r$ as low as possible leads \cite{ournew}
to the original mutated hybrid model, along with the cases
$\alpha=-2$ and $\alpha=-4$ that we discussed earlier
in the context of inverted hybrid and single-field models.
In the original model (at least)
the order of magnitude of the inflationary energy scale
$V_0^{1/4}$ can be understood if supersymmetry is broken
by gaugino condensation in a hidden sector.
A different example of a mutated hybrid inflation potential is given in
Ref.~\cite{glw}, where $\psi$ is a pseudo-Golstone boson
with the potential (\ref{gbpot}). Depending on the parameter values
it might reduce in practice to a potential of the form discussed above,
to one of the kind discussed in the next subsection or to something
different.
\subsection{Mutated hybrid inflation with explicit $\phi$ dependence}
So far we have assumed that the original potential has no piece that
depends only on $\phi$. If there is such a piece it has to be added
to the inflationary potential (\ref{vmut}). If it
dominates while cosmological scales leave the horizon,
the only effect that the $\psi$ variation has
on the inflationary prediction is to
determine $\phi_c$ through Eq.~(\ref{phiend2}).
\subsection{Hybrid inflation with a loop-corrected potential}
In the
context of supersymmetry one expects the correction to be
of the form $A\phi^2\ln(\phi/B)$ or $A\ln(\phi/B)$,
depending on the mechanism of supersymmetry breaking during inflation.
We discussed an application of the first case earlier, and now consider
the second.
The prediction for $n$ with a loop correction of the form
$A\ln(\phi/B)$ is the same
as for a potential $V_0(1-\mu\phi^{-\alpha})$ with $\alpha\simeq 0$,
and as in the Table it is between .96 and .98.
It was first proposed \cite{qaisar,dvaliloop,qaisarlatest}
to give a slope to the flat classical potential coming out of a
globally supersymmetric model that had been written down
in Ref.~\cite{CLLSW}. The superpotential is
\be
W=\sigma (\Phi_1\Phi_2 + \Lambda^2) \Phi_3
\label{wglobal}
\ee
where $\sigma$ is a dimensionless coupling, $\Lambda$ is a mass scale
and the
$\Phi_n$ are complex fields. This gives the classical potential
\be
V=\sum_n\left|\frac{\partial W}{\partial \Phi_n}\right|^2
\label{vglobal}
\ee
With $\Phi_1=\Phi_2=0$, $V$
has the constant value $\sigma^2\Lambda^4$, and the loop correction
gives it a small slope making $|\Phi_3|$ the inflaton of a hybrid
inflation model.
As had been pointed
out in the earlier reference, one
expects that in this model supergravity corrections will give
$\eta\equiv\Mpl^2|V''/V|\sim 1$, preventing inflation.\footnote
{Sufficient accidental suppression of $\eta$ is unlikely because the
loop correction gives $\eta=.01$ to $.02$, and $W$ is not of a form that would
guarantee the smallness of $\eta$ for a reasonable Kahler potential.}
However, it has
recently been noted \cite{bindvali,halyo}
that a loop correction of the same form is likely to
be responsible for the slope of a hybrid inflation model dominated by
the $D$ term, of the type that was proposed at tree level
in Ref.~\cite{ewansgrav}. As pointed out in that reference, such a
$D$ term model
has the nice feature that there is no problem
about keeping
$\eta$ small.
\section{The difficulty of inflation model-building in supergravity}
In this section the problem of keeping $\eta$ small in the context of
supergravity is explained, and various proposed solutions to it are listed.
Most of them have been encountered already.
\subsection{The problem}
The
kinetic terms of the complex
scalar fields
$\Phi_n$ are given in terms of the Kahler potential $K$
(a real function of $\Phi_n$ and its complex conjugate
$\bar\Phi_n$)
by
\be
{\cal L}_{\rm kin} = (\partial_\mu\bar\Phi_n)
K_{\bar n m}
(\partial^\mu\Phi_m)
\ee
Here and in the following expressions, a subscript $n$
denotes the derivative with respect to the $\Phi_n$
($\bar n$ the derivative with respect to $\bar\Phi_n$)
and a summation over repeated indices is implied.
Following Ref.~\cite{ewansgrav},
focus on a given point on the inflationary trajectory,
and choose it as the origin $\Phi_n=0$.
For the analysis leading to the usual slow-roll predictions to be valid
we need the relevant fields to be canonically normalized
in a small region around this point, which is ensured by choosing
the fields so that\footnote
{Strictly speaking this choice ensures
only that equations involving at most second derivatives of the fields
are valid. It ensures the validity of the prediction for
$\delta_H$ but not, strictly speaking, of the prediction for $n$.
The correction analogous to $\gamma$ in
Eqs.~(\ref{deltan2}) and (\ref{running}) will
involve both $V'''$ and the higher order contributions
to $K_{\bar n m}$. It could be worked out in a particular model
from the general formalism of
Ref.~\cite{nak}, specialized to a single field, and we are working on
the assumption that it is negligible.}
\be
K_{\bar n m}=\delta_{nm}+O(\Phi_n^2)
\label{kinetic}
\ee
This is analogous to using a locally inertial frame in general
relativity.
The `curvature scale', beyond which higher order terms become
significant,
is expected to be of order the Planck
scale $|\Phi_n|\sim \Mpl$.
The potential $V$ consists
of a `$D$' term and an `$F$' term, and
the problem arises when the $F$ term dominates, which
is usually taken to be the case.
The $F$ term involves the superpotential $W$ (a holomorphic function
of the complex fields $\Phi_n$) and is
\be
V=e^{K/\Mpl^2}\tilde V
\label{vexp}
\ee
where
\be
\tilde V=(W_n+\Mpl^{-2}WK_n)K^{n \bar m} (\bar W_{\bar m} +
\Mpl^{-2}\bar W K_{\bar m})
-3\Mpl^{-2}|W|^2
\ee
The matrix $K^{n \bar m}$ is the inverse of $K_{\bar n m}$.
This expression depends on $K$ and $W$ only through the combination
$G\equiv K+\ln|W|^2$, so it is invariant under the transformation
$K\to K-F-\bar F$, $W\to e^F W$ where $F$ is any holomorphic function.
As a result one can choose
$K$ and $W$ so that about the given point on the trajectory
\be
K=\sum_n|\Phi_n|^2+\cdots
\label{kahlerexp}
\ee
The inflaton field $\phi$ can be chosen to be $2^{-\frac12}$ times the
real part of one of the $\Phi_n$, and one then finds
\be
\eta\equiv\Mpl^2V''/V=1+\Mpl^2\tilde V''/\tilde V
\ee
The first term, which alone would give $\eta=1$,
comes from the factor $e^{K/\Mpl^2}$. To
have $|\eta|\ll1$ it must be cancelled by the
second term.
Examination of $\tilde V$ shows that there are
indeed contributions
of order 1, which may be positive or negative,
but generically their total will not be $-1$ to high
accuracy. They come from the quadratic term
in the expansion (\ref{kahlerexp}) of $K$, and \cite{ewanpers}
from the quartic $|\Phi|^4$ term through
$K^{\bar n m}$. Thus
the minimal supergravity
approximation, of keeping only the quadratic term,
cannot be used in this context.
\subsection{Solving the problem}
How severe the problem is depends on the magnitude of $\eta$.
If $\eta$ is
not too small then its smallness could be due to accidental
cancellations. Having $\eta$ not too small requires
that $n-1=-6\epsilon+2\eta$
be not too small (unless $3\epsilon\simeq\eta$ which is not the case
in models that have been proposed)
so the observational bound
$|n-1|<.3$ is already beginning to make an accident look unlikely.
An accidental cancellation is
being assumed by proponents of
modular inflation as discussed earlier.
A toy model for accidental cancellation would be the superpotential
$W=A(\Phi-\Phi_0)^2$. The inflaton is supposed to be the real part of
$\Phi$ and with minimal supergravity its mass-squared vanishes
provided that \cite{holman,graham}
$\Phi_0=\Mpl$. Including the quartic
term in $K$ the same thing will occur for some other value $\Phi_0=B$,
where $B$ depends on the coefficient of the quartic term but is still of
order $\Mpl$. So $\eta$ will be accidentally
suppressed if $|\Phi_0-B|\ll \Mpl$.
Barring an accidental cancellation, the smallness of $\eta$
requires a non-generic form for the supergravity potential.
With minimal supergravity, a simple choice that works is
$W=V_0^{1/2}\Phi$, which leads to the global supersymmetry result,
corresponding to an exactly flat potential
\cite{CLLSW}, $V=V_0$.
Taking $W=V_0^{1/2}\Phi(1-\frac12\mu\Phi^p)$ could then give \cite{kumekawa}
a non-hybrid inflation
potential $V\simeq V_0(1-\mu\phi^p)$. Alternatively,
taking $W$ of the form (\ref{wglobal})
could give \cite{CLLSW} hybrid
inflation with $\Phi_1=\Phi_2=0$, leading again to
$W=V_0^{1/2}\Phi$ and an absolutely flat potential
whose slope might come from a loop
correction \cite{qaisar,dvaliloop,qaisarlatest}.
But the quartic contribution to $K$ is expected to spoil minimal
supergravity, in which case these choices will not work.\footnote
{A possible intermediate
strategy \cite{izawa} might be to use one of these forms to
eliminate the contribution to $\eta$ of the minimal (quadratic) term,
and then suppose that the quartic term is accidentally
suppressed. Such an accidental {\em suppression} is perhaps
less objectionable than the accidental {\em cancellation} which is
otherwise required.
However, in the model of Ref.~\cite{izawa} the suppression
needs to be considerable, and the other parameters also have to be
chosen delicately to get a viable model. If one were to apply the idea
to the model of Refs.~\cite{CLLSW,qaisar,dvaliloop},
with the slope coming from
a loop correction, the suppression would need to be severe since
$\eta\sim .01$.} In the context of nonminimal supergravity
there are the following proposals, mostly involving things that have
been mentioned earlier.
\begin{enumerate}
\item
The potential might be dominated by the
$D$ term rather than by the $F$ term \cite{ewansgrav} in which case its
variation will probably be dominated by a loop correction
\cite{bindvali,halyo} of the form
$A\ln(\phi/B)$, giving
$n=.96$ to $.98$.
\item
One can impose restrictions on the form of the $F$ term
\cite{ewansgrav}, which are of a type that might emerge
from superstring theory provided that it is in a perturbative regime.
This scheme is sufficiently flexible that it can accomodate
practically
all versions of hybrid inflation \cite{ewansgrav,mutated,ournew}, as
well as the large kinetic term model of Ref.~\cite{ewansgrav}.
Thus it can accomodate more or less any measured value of $n$.
\item
It has been known for a long time that supergravity models
of the `no-scale'
type possess a `Heisenberg symmetry' that can eliminate the
mass-squared of order $H^2$ for generic scalar fields. The problem has
been to stabilize the `Polonyi' or (in the modern setting) modulus
field occurring in such theories. It was known from the beginning that
an ad hoc prescription can fix the modulus to give viable models
of inflation, though the focus was not on small-field models
(for reviews see Refs.~\cite{olive,abook} and for a different ad hoc
prescription see Ref.~\cite{hitoshi1}).
Recently it has been
noted \cite{gaillard}
that a loop correction might provide the stabilization
automatically. As yet no model of inflation based on this latter scheme
has
been proposed. Since the modulus will adjust itself to
the current minimum of the potential during inflation
\cite{hitpers}
(instead of being absolutely fixed as in the early ad hoc schemes)
one may be looking at something resembling
a mutated hybrid inflation model.
\item
The tree-level contribution to $\eta$ might be cancelled, within a
limited interval of $\phi$, by a loop correction
\cite{ewanloop},
leading to an inverted quadratic potential with
$n$ considerably below 1.
\item
Identifying the inflaton field with a pseudo-Goldstone boson
keeps its potential absolutely flat in the limit of unbroken symmetry.
Explicitly breaking the symmetry
\cite{natural,natural2,natural4,kinney,glw,ewanprep} then gives a
nonzero $\eta$, which can be small if
the symmetry is broken only by $W$
and not by $K$, or if most of the potential comes from another field
\cite{ewanprep}.
This solution to the problem would
probably give $n$ significantly below 1 in the former case, but
indistinguishable from 1 in the latter.
\item
The form of the potential $V(\phi)$
might depend on some other field, which is driven
to a value corresponding to negligible $\eta$ before cosmological
inflation starts. This is the proposal of Ref.~\cite{grahamnew}
mentioned in Section IV,
leading to
$n=.84$ to $.92$. For the proposal to work
one still needs a mechanism
for keeping the potential flat in the direction of the other field
(a global $U(1)$ in the above model) so
it might become the inflaton
in a different regime of parameter space.
\end{enumerate}
\subsection{Inflation as a probe of supergravity}
From these considerations we see that inflation is a very powerful probe
of supergravity. In most models, the $F$ term
dominates the potential, and $\eta$ generically receives
various contributions of
order $1$ which must cancel. To keep it small requires
either an accidental cancellation (reasonable only if $\eta$ is not
too small) or non-generic
forms for $K$ and $W$. In constructing suitable forms,
it is not permissible to make the `minimal supergravity'
approximation of
ignoring the higher order terms in the expansion
(\ref{kahlerexp}) of $K$.
Having ensured the smallness of $\eta$, one is not in general left with
the global supergravity result $V=\sum|W_n|^2$. It
does indeed hold for solution 2 of the problem,
but both $W$ and $K$ need to have very special forms.
Finally, if
the $D$ term dominates there is no problem about keeping $\eta$ small,
but to require this is itself a strong constraint
on the model.
\section{Inflation with a multi-component inflaton}
So far we have assumed that the slow-rolling inflaton field is
essentially unique. What does `essentially' mean in this context?
A strictly unique inflaton trajectory would be one lying
in a steep-sided valley in field space. This is not very likely in a
realistic model. Rather there will be a whole
family of possible inflaton trajectories, lying in the space of two or
more real fields $\phi_1$, $\phi_2, \cdots$.
In this case, it quite useful to
think of the inflaton field as a multi-component object with
components $\phi_a$.
But even for a multi-component inflaton field it
may well be the case that different choices
of trajectory lead to the same universe, and if that is the case
we still have an `essentially' unique inflaton field.
A familiar example of the essentially unique case
is if the
inflaton field
is the modulus of a complex field charged under a global $U(1)$
symmetry.
Then the possible inflaton trajectories are the radial
lines and the $U(1)$ symmetry ensures that all trajectories correspond
to identical universes during inflation. Moreover, the transition
to a universe composed of matter plus radiation
will be the same for all of them. Thus the inflaton field
is essentially unique.
More generally, the inflaton field
will be essentially unique if the
inflaton trajectories are practically straight in the space of
canonically normalized fields, and if also the transition from
inflation to a matter/radiation universe is the same for each
trajectory.
The essentially non-unique case is when different possible
trajectories correspond to differently evolving universes.
Then trajectories near the classical one
cannot be ignored,
because the quantum fluctuation
kicks the inflaton field onto them in a random way.
From now on, `multi-component' will refer to this case.
Even in the single-component case there is a quantum fluctuation,
which kicks the inflaton back
and forth along the classical trajectory. This causes an adiabatic
density perturbation, and in general the
effect of the orthogonal fluctuation onto nearby trajectories is to
cause an additional adiabatic density perturbation.
(Exceptionally it might cause an isocurvature perturbation as
mentioned at the end of this section.)
Multi-component inflaton models generally have just two components,
and are called double inflation models because the trajectory can
lie first in the direction of one field, then in the direction of the
other. They were first proposed in
the context of non-Einstein gravity
\cite{starob85,d1,d2,d4,d5,d11,noncanon,davidjuan}.
By redefining the fields
and the spacetime metric one can recover Einstein gravity,
with fields that are not small on the Planck scale and
in general non-canonical kinetic terms and a non-polynomial
potential. Then models
with canonical kinetic terms were proposed
\cite{c3,d7,d9,d10,d13,c1,c2,d3,d6,d8,isocurv,salopek95},
with potentials such as
$V=\lambda_1\phi_1^p+\lambda_2\phi_2^q$.
These potentials too inflate in the large-field regime
where theory provides no guidance about the form of the potential.
However there
seems to be no bar to having a small-field multi-component model,
and one may yet emerge in
a well-motivated particle theory setting.
In that case a hybrid model might emerge, though
the models proposed so far
are all of the
non-hybrid type (ie., the multi-component inflaton is entirely
responsible for the potential).
In this brief survey we have focussed on the era when cosmological
scales leave the horizon. In the hybrid inflation
model of Ref.~\cite{lisa,glw},
the `other' field is responsible for the last several $e$-folds
of inflation, so one is really dealing with a
two-component inflaton (in a non-hybrid model).
The scales corresponding to the last
few $e$-folds are many orders of magnitude shorter than
the cosmological scales, but it turns out that the perturbation
on them is big so that black holes can be produced.
This phenomenon was investigated in Refs.~\cite{lisa,glw}.
The second reference also investigated the possible production of
topological defects, when the first field is destabilized.
\subsection{Calculating the spectrum of the curvature perturbation}
We noted at the beginning of Section II that the adiabatic density
perturbation on scales well outside the horizon
is specified by a quantity $\cal R$, which
defines the curvature of comoving hypersurfaces.
Going back in time from the epoch of horizon entry, it soon achieves a
constant value, which is maintained
at least until the beginning of the
radiation-dominated era preceding the present matter-dominated
one, and unless otherwise stated $\cal R$ denotes this constant value.
On the assumption that it is a gaussian random field, which is
generally the case if it is generated by a vacuum fluctuation of the
inflaton field, its stochastic properties are completely determined
by its spectrum
${\cal P}_{\cal R}$.
In this section we will see how to calculate the spectrum,
first for a single-component inflaton and then for a multi-component
one. In both cases we use an approach that has only recently been
developed \cite{salopek95,ewanmisao}, though its starting point
can already be seen in the first derivations of the spectrum
\cite{hawking,starob82,guthpi}.
This is the assumption that
after smoothing on a scale well outside the
horizon (Hubble distance) the evolution of the universe
along each comoving worldline will be practically
the same as in a Robertson-Walker
universe.\footnote
{`Smoothing' on a scale $R$ means that one replaces (say) $\rho({\bf x})$
by $\int d^3x' W(|{\bf x}'-{\bf x}|) \rho({\bf x}')$ with
$W(y)\simeq 1$ for $y\lsim R$ and
$W\simeq 0$ for $y\gsim R$. A simple choice is to take
$W=1$ for $y<R$ and $W=0$ for $y>R$
(top-hat smoothing).}
One always makes such an assumption when doing
cosmological perturbation theory. When a quantity is split into an
average plus a perturbation, the average is
identified with the `background' quantity that is supposed to correspond
to a Robertson-Walker universe (ie., to an absolutely homogeneous and
isotropic one).
And it is accepted without question that
the criterion for the size of the averaging region is that it be much
bigger than the horizon.
The averaging scale will of course be chosen to be a comoving one.
When splitting a quantity into an average and a perturbation it is
usually taken to be much bigger than the one corresponding to the
whole presently observable universe. But
in the early universe it makes equal sense
to make it much smaller.
This is indeed often done implicitly (sometimes explicitly
\cite{myaxion}) for a {\em single} small region, namely the one around
us. The crucial idea behind the present approach
\cite{salopek95,ewanmisao} is to recognize that the comparison of
{\em different} regions provides a simple and powerful technique for
calculating the density perturbation.
As we discuss later, it is quite different from the
usual one of writing down, and then solving, a closed set of equations
for the perturbations in the relevant degrees of freedom
(for instance the components of the inflaton field during inflation).
Roughly speaking the present approach replaces the sequence `perturb then solve'
by the far simpler sequence `solve then perturb', though it is actually
more general than the other approach.
For the case of a single-component inflaton it gives
a very simple, and completely general, proof of
the constancy of $\cal R$ on scales well outside the horizon.
For the multi-component case it allows one to follow the evolution of
$\cal R$, knowing only the evolution of the {\em unperturbed}
universe corresponding to a given value of the initial inflaton field.
So far
it has been applied to three
multi-component models \cite{salopek95,davidjuan,glw}.
\subsection{The case of a single-component inflaton}
We begin with a derivation of the usual result for the single-component
case. The assumption about the evolution along each comoving worldline
is invoked only at the very end,
when it is used to establish the constancy of $\cal R$
which up till now has only been demonstrated for special cases.
Otherwise the proof is the standard one \cite{LL2}, but it provides a
useful starting point for the multi-component case.
A few Hubble times
after horizon exit during inflation, when $\cal R$
can first be regarded as a classical quantity, its spectrum
can be calculated using the relation
\cite{kodamasasaki,LL2}\footnote
{In \cite{LL2} there is an incorrect minus sign on the right hand side.}
\be
{\cal R}({\bf x})=H\Delta\tau({\bf x})
\label{r1}
\ee
where $\Delta\tau$ is the separation of the comoving hypersurface
(with curvature $\cal R$)
from a spatially flat one coinciding with it on average.
The relation is generally true, but we apply it at an epoch
a few Hubble times after horizon exit during inflation.
On a comoving hypersurface the inflaton field $\phi$
is uniform, because the momentum density $\dot\phi {\bf \nabla}\phi$
vanishes. It follows that
\be
\Delta\tau({\bf x})=-\delta\phi({\bf x})/\dot\phi
\ee
where now $\delta\phi$ is defined on the flat hypersurface.
Note that the comoving hypersurfaces become singular (infinitely
distorted) in the slow-roll limit $\dot\phi\to 0$, so that
to first order in slow-roll any non-singular choice of hypersurface
could actually be used to define $\delta\phi$.
The spectrum of $\delta\phi$
is calculated by assuming that well before horizon entry
(when the particle concept makes sense) $\delta\phi$ is a
practically free field in the vacuum state. Using the flatness and
slow-roll conditions one finds, a few Hubble times after horizon
exit, the famous result \cite{LL2}
${\cal P}_\phi=(H/2\pi)^2$, which leads to the
usual formula (\ref{delh}) for the spectrum.
However, this result refers to $\cal R$ a few Hubble times after horizon
exit, and we need to check that $\cal R$ remains constant until
the radiation dominated era where we need it. To calculate the rate of
change of $\cal R$ we proceed as follows \cite{lythmuk,LL2}.
In addition to the energy density $\rho$ and
the pressure $P$, we consider a locally defined
Hubble parameter
$H=\frac13D_\mu u^\mu$ where $u^\mu$ is the four-velocity of
comoving
worldlines and $D_\mu$ is the covariant derivative.
Our $3H$ is often called $\theta$ in the literature.
The universe is sliced into comoving hypersurfaces, and each
quantity is split into an average (`background') plus a
perturbation, $\rho({\bf x},t)=\rho+\delta\rho({\bf x},t)$
and so on. (We use the same symbol for the local and the background
quantity since there is no confusion in practice.)
As usual, $\bf x$ is the Cartesian position-vector of a
comoving worldline and $t$ is the time. As we are working to first order
in the perturbations they `live' in unperturbed spacetime.
The locally defined quantities satisfy
\cite{hawkingellis,lyth85,lythmuk,LL2}
\be
H^2=\Mpl^{-2}\rho/3-\frac23\frac{k^2}{a^2}\cal R
\label{localfr}
\ee
(The equation is valid for each Fourier mode, and also for the
full quantities with the identification
$(k/a)^2=-\nabla^2$. From now on that is taken for granted.)
This is the Friedmann equation except that
$K\equiv(2/3)k^2\cal R$ need not be constant.
The evolution along each worldline is
\bea
\frac{d\rho}{d\tau}&=&-3H(\rho+P)
\label{cont}\\
\frac{d H}{d\tau} &=& -
H^2 - \frac12 \Mpl^{-2} (\rho+3P)
+\frac13\frac{(k/a)^2\delta P}{\rho+P}
\label{raych}
\eea
Except for the last term these are the same as in an unperturbed
universe. If it vanishes $\cal R$ is constant, but otherwise one finds
\be
\dot{\cal R}=-H\delta P/(\rho +P)
\label{rdot}
\ee
In this equation we have in mind that $\rho$ and $P$ are background quantities,
though as we are working to first order in the perturbations it would
make no difference if they were the locally defined quantities.
The equation shows that $\cal R$ will be constant if $\delta P$ is negligible.
We now show that this is so, by first demonstrating that
$\delta\rho$ is negligible, and then using the new viewpoint
to see that $P$ will be a practically
unique function of $\rho$ making $\delta P$ also negligible.
Extracting the perturbations from
Eq.~(\ref{localfr}) gives
\be
2\frac{\delta H}{H}=\frac{\delta\rho}{\rho } -\frac23\left(
\frac{k}{aH}\right)^2 \cal R
\label{perturbfre}
\ee
This allows one to calculate the evolution of $\delta \rho$ from
Eq.~(\ref{cont}), but we have
to remember that the proper-time separation of the hypersurfaces
is position-dependent. Writing
$\tau({\bf x},t)=t+\delta\tau({\bf x},t)$ we have
\cite{kodamasasaki,lythmuk,lyst}
\be
\delta(\dot\tau)= -\delta P/(\rho+P)
\label{deltataudot}
\ee
Writing $\delta\rho/\rho\equiv (k/aH)^2 Z$
one finds \cite{lythmuk}
\be
(fZ)'=f (1+w) \cal R
\label{zexpr}
\ee
Here a prime denotes $d/d(\ln a)$ and
$f'/f\equiv(5+3w)/2$ where $w\equiv P/\rho$.
With $w$ and $\cal R$ constant, and dropping a decaying mode, this gives
\be
Z=\frac{2+2w}{5+3w} \cal R
\ee
More generally, integrating Eq.~(\ref{zexpr})
will give $|Z|\sim |\cal R|$ for any reasonable variation of
$w$ and $\cal R$. Even for a bizarre variation there is no scale
dependence in either $w$ (obviously) or in $\cal R$ (because
Eq.~(\ref{penult}) gives it in terms of $\delta P$, and we will see that
if $\delta P$ is significant it is scale-independent).
In all cases
$\delta\rho/\rho$ becomes negligible on
scales sufficiently far outside the horizon.
The discussion so far applies to each Fourier mode separately, on
the assumption that the corresponding perturbation is small.
To make the final step, of showing that $\delta P$ is also negligible,
we need to consider the full quantities $\rho({\bf x},t)$
and so on. But we still want to consider only scales that are well
outside the horizon, so we suppose that all quantities are smoothed
on a comoving scale somewhat shorter than the one of interest.
The smoothing removes Fourier modes on scales shorter than the
smoothing scale, but has practically no effect on the scale
of interest.
Having done this, we invoke the assumption that the evolution of the
universe along each worldline is practically the same as in an unperturbed
universe. In the context of slow-roll inflation, this means that
the evolution is determined by the inflaton field
at the `initial' epoch a few Hubble times after horizon exit.
To high accuracy, $\rho$ and $P$ are well
defined functions of the initial inflaton field and
{\em if it has only one component}
this means that they are well defined
functions of each other. Therefore $\delta P$ will be very small
on comoving hypersurfaces because $\delta \rho$ is.\footnote
{If $k/a$ is the smoothing scale, the
assumption that the evolution is the same as in an unperturbed
universe with the same initial inflaton field
has in general errors of order $(k/aH)^2$.
In the single-component case, where $\delta P$ is also of this order,
we cannot use the assumption to actually
calculate it, but neither is it of any interest.}
Finally, we note for future reference that $\delta H$ is also
negligible because of Eq.~(\ref{perturbfre}).
\subsection{The multi-component case}
It is assumed that while cosmological scales are leaving the horizon
all components of the inflaton have the slow-roll behaviour
\be
3H\dot \phi_a = - V_{,a}
\ee
(The subscript $,a$ denotes the derivative with respect to $\phi_a$.)
Differentiating this
and comparing it with the exact expression $\ddot\phi_a
+3H\dot \phi_a +V_{,a}=0$ gives consistency provided that
\bea
\Mpl^2 (V_{,a}/V)^2 &\ll& 1\\
\Mpl^2 |V_{,ab}/V| &\ll& 1
\label{flat2}
\eea
(The second condition could actually be replaced by a weaker one but let
us retain it for simplicity.)
One expects slow-roll to hold if these flatness conditions are
satisfied. Slow-roll plus the first flatness condition
imply
that $H$ (and therefore $\rho$) is slowly varying, giving
quasi-exponential inflation. The second flatness condition
ensures that $\dot\phi_a$ is slowly varying.
It is not necessary to assume that
all of the fields continue to slow-roll after cosmological scales
leave the horizon. For
instance, one or more of the fields might start to oscillate, while the others
continue to support quasi-exponential inflation, which ends only
when slow-roll fails for all of them.
Alternatively, the oscillation of
some field might briefly interrupt inflation, which resumes when its
amplitude becomes small enough. (Of course these things might happen
while cosmological scales leave the horizon too, but that case
will not be considered.)
The expression (\ref{r1}) for
$\cal R$ still holds in the multi-component case. Also,
one still has
$\Delta\tau=-\delta\phi/\dot\phi$ if $\delta\phi$
denotes the component of the vector $\delta\phi_a$
parallel to the trajectory. (The momentum density
seen by an observer orthogonal to an arbitrary hypersurface is
$\dot\phi_a {\bf \nabla} \phi_a$.) A few Hubble times after horizon exit
the spectrum of
every inflaton field component, in particular the parallel one,
is still $(H/2\pi)^2$. If $\cal R$ had no subsequent variation this
would lead to the usual prediction, but we are considering the case
where the variation is significant.
It is given in terms of $\delta P$ by
Eq.~(\ref{rdot}), and when $\delta P$ is significant it can be calculated
from the assumption that the evolution along each
worldline is the same as for an unperturbed universe with the same
initial inflaton field. This will give
\be
\delta P=P_{,a}\delta\phi_a
\ee
where $\delta\phi_a$ is evaluated at the initial epoch and the
function $P(\phi_1,\phi_2,\cdots,t)$ represents the evolution of
$P$ in an unperturbed universe. Choosing the basis so that one of
the components is the parallel one, and remembering that all components
have spectrum $(H/2\pi)^2$, one can calculate
the final spectrum of $\cal R$. The only input is the evolution of
$P$ in the unperturbed universe corresponding to a generic initial
inflaton field (close to the classical initial field).
In this discussion we started with Eq.~(\ref{r1}) for the initial
$\cal R$, and then invoked Eq.~(\ref{rdot}) to evolve it.
The equations
can actually be combined to give
\be
{\cal R}=\delta N
\ee
where $N=\int Hd\tau$ is the number of Hubble times
between the initial flat hypersurface and the final comoving one on
which $\cal R$ is evaluated. This remarkable expression was given
in Ref.~\cite{starob85} and proved in Refs.~\cite{salopek95,ewanmisao}.
The approach we are using is close to the one in the
last reference.
The proof that
Eqs.~(\ref{r1}) and (\ref{rdot}) lead to ${\cal R}=\delta N$
is very simple. First
combine them to give
\be
{\cal R}({\bf x},t)=H_1\Delta\tau_1({\bf x})-
\int_{t_1}^{t} H(t) \frac{\delta P}{\rho+P}
\label{penult}
\ee
where $t_1$ is a few Hubble times after horizon exit.
Then use Eq.~(\ref{deltataudot}) to give
\be
{\cal R}({\bf x},t)=H_1\Delta\tau_1({\bf x})+
\int_{t_1}^{t} H(t) \delta \dot\tau({\bf x},t)dt
\label{penult1}
\ee
Next note that because $\delta H$ is negligible
this can be written
\be
{\cal R}({\bf x},t)=H_1\Delta\tau_1({\bf x})+
\delta \int_{t_1}^{t} H({\bf x},t) \dot\tau({\bf x},t)dt
\label{ult}
\ee
Finally redefine $\tau({\bf x},t)$ so that it
vanishes on the
initial {\em flat} hypersurface, which gives the desired
relation ${\cal R} = \delta N$.
In Ref.~\cite{ewanmisao} this relation is derived using an
arbitrary smooth interpolation of hypersurfaces
between the initial and final one, rather than by making the sudden jump
to a comoving one. Then $H$ is replaced by the corresponding quantity
$\tilde H$
for worldlines orthogonal to
the interpolation (incidentally making $\delta \tilde H$ non-negligible).
One then finds $\cal R=\delta\tilde N$. One also finds that the right
hand side is independent of the choice of the interpolation, as it must
be for consistency. If the interpolating hypersurfaces are chosen to be
comoving except very near the initial one, $\tilde N\simeq N$ which
gives the desired formula ${\cal R}=\delta N$.\footnote
{The last step is not spelled out in Ref.~\cite{ewanmisao}.
The statement that $\tilde N$ is independent of the interpolation
is true only on scales well outside the
horizon, and its physical interpretation is unclear though it drops out
very simply in the explicit calculation.}
\subsection{Calculating the spectrum and the spectral index}
Now we derive explicit formulas for the spectrum and the spectral index,
following \cite{ewanmisao}.
Since the evolution of
$H$ along a
comoving worldline will be the same as for a homogeneous universe
with the same initial inflaton field, $N$ is a function only of this
field and we have
\be
{\cal R}=N_{,a} \delta\phi_a
\ee
(Repeated indices are summed over
and the subscript $,a$ denotes differentiation with respect to $\phi_a$.)
The perturbations $\delta\phi_a$ are Gaussian random fields generated by the
vacuum fluctuation, and have a common
spectrum $(H/2\pi)^2$. The
spectrum $\delta_H^2\equiv (4/25){\cal P}_{\cal R}$
is therefore
\be
\delta_H^2= \frac{V}{75\pi^2\Mpl^2}N_{,a}N_{,a}
\ee
In the single-component case, $N'=\Mpl^{-2} V/V'$
and we recover the usual expression. In the multi-component case
we can always choose the basis
fields so that while cosmological scales are leaving the horizon
one of them points along the inflaton trajectory, and then its
contribution gives the standard result with the orthogonal
directions giving an additional contribution.
Since the spectrum of gravitational waves is independent of the number
components (being equal to a numerical constant times $V$)
the relative contribution $r$ of gravitational waves to the cmb
is always {\em smaller} in the multi-component case.
The contribution from the orthogonal directions
depends on the whole inflationary potential after the relevant scale
leaves the horizon, and maybe even on the evolution of the energy
density after inflation as well. This is in contrast to the contribution
from the parallel direction which depends
only on $V$ and $V'$ evaluated
when the relevant scale leaves the horizon.
The contribution from the orthogonal directions will be at most of order
the one from the parallel direction provided that all $N_{,a}$ are at
most of order $\Mpl^{-2} V/V'$.
We shall see later that this is a
reasonable expectation at least if $\cal R$ stops varying after the end
of slow-roll inflation.
To calculate the spectral index we need the analogue of
Eqs.~(\ref{nexpression}) and (\ref{Nrelation}). Using
the chain rule and $dN=-Hdt$ one finds
\bea
\frac{d}{d\ln k}
&=&-\frac{\Mpl^2}{V}V_{,a} \frac{\partial}{\partial\phi_a}\\
N_{,a} V_{,a} &= & \Mpl^{-2} V
\label{nv}
\eea
Differentiating the second expression gives
\be
V_{,a} N_{,ab}+ N_{,a} V_{,ab} = \Mpl^{-2}V_{,b}
\ee
Using these results one finds
\be
n-1 = -\frac{\Mpl^2V_{,a}V_{,a}}{V^2}
-\frac2{\Mpl^2N_{,a}N_{,a}}
+2\frac{\Mpl^2N_{,a} N_{,b} V_{,ab} }
{VN_{,d}N_{,d}}
\label{multin}
\ee
Again, we recover the single field case using $N'=\Mpl^{-2} V/V'$.
Differentiating this expression and setting $\Mpl=1$ for clarity
gives
\bea
\frac{dn}{d\ln k}&=&
-\frac {2}{V^3}V_{,a} V_{,b} V_{,ab}
+\frac2{V^4} (V_{,a}V_{,a})^2
+\frac 4 V \frac{(V-N_{,a}N_{,b} V_{,ab} )^2}{(N_{,d}N_{,d})^2}\nonumber\\
&+&\frac2 V \frac{N_{,a} N_{,b} V_{,c} V_{,abc} }{N_{,d}N_{,d}}
+\frac4 V \frac {( V_{,c} -N_{,a} V_{,ac}) N_{,b} V_{,bc} }
{N_{,d}N_{,d}}
\eea
A correction
to the formula for $n-1$ has also been worked out \cite{nak}.
Analogously with the single-component case, both this correction
and the variation of $n-1$ involve the first, second and third
derivatives of $V$.
Provided that the derivatives of $N$ in the orthogonal directions are
not particularly big, and barring cancellations,
a third flatness
condition
$V_{,abc}V_{,c}/V^2\ll \max\{\sum_a(V_{,a})^2,\sum_{ab}|V_{,ab}|\}$
ensures that both the correction
and the variation of $n-1$ in a Hubble time are small.
(One could find a weaker condition that would do the same job.)
These formulas give the spectrum and spectral index of the
density perturbation, if one knows the evolution
of the homogeneous universe corresponding both to the
classical inflaton trajectory and to nearby trajectories.
An important difference in principle from the single-component
case, is that the classical trajectory is not uniquely specified by
the potential, but rather has to be given as a separate piece of
information. However, if there are only two components the classical
trajectory can be determined from the COBE normalization of the
spectrum, and then there is still a prediction for the spectral index.
This treatment
can be generalized straightforwardly \cite{ewanmisao}
to the case of
non-canonical kinetic terms of the form (\ref{hmetric}), that is
expected in
supergravity.
However, in the small-field regime one expects the
curvature associated
with the `metric' $h_{ab}$ to be negligible, and then one can recover
the canonical normalization
$h_{ab}=\delta_{ab}$ by redefining the
fields.
\subsection{When will $\cal R$ become constant?}
We need to evaluate $N$ up to the epoch where ${\cal R}=\delta N$
has no further time dependence. When will that be?
As long as all fields are slow-rolling, $\cal R$ is constant if and only
if the inflaton trajectory is straight.
If it turns through a small angle $\theta$,
and the trajectories have not converged appreciably since horizon exit,
the fractional change in $\cal R$ is in fact $2\theta$.\footnote
{Thinking in two
dimensions and taking the trajectory to be an arc of a circle,
a displacement $\delta\phi$
towards the center decreases
the length of the trajectory by an amount $\theta\delta\phi$,
to be compared with the decrease $\delta\phi$ for the same
displacement along the trajectory. (The rms displacements will indeed
be the same if the trajectories have not converged.) Since
The
speed along the new trajectory is faster in inverse proportion to the
length since it is proportional to $V'$
and $V$ is fixed at the
initial and final points on the trajectory. Thus the perpendicular
displacement increases $N$ by $2\theta$ times the effect of a
parallel displacement, for $\theta\ll 1$.}
Since slow-roll
requires that the change in the vector $\dot\phi_a$ during
one Hubble time is negligible, the total angle turned is $\ll N$.
Hence the relative contribution of the orthogonal directions cannot
be orders of magnitude bigger than the one from the parallel
direction, if it is generated during slow-roll inflation.
(In two dimensions the angle turned cannot exceed $2\pi$ of course, but
there could be say a corkscrew motion in more dimensions.)
Later slow-roll may fail for one or more of the fields,
with or without interrupting inflation, and things become more
complicated, but in general there is no reason why $\cal R$ should stop varying
before the end of inflation.
Now let us ask what happens after the end of inflation
(or to be more precise,
after significant particle production has spoiled the above analysis,
which may happen a little before the end).
The simplest case is if
the relevant trajectories have practically converged to a
single trajectory $\phi_a(\tau)$, as in Ref.~\cite{glw}.
Then $\cal R$ will not vary any more (even after inflation is over)
as soon as the trajectory has
been reached. Indeed, setting $\tau=0$ at the end of
inflation, this unique trajectory corresponds to a post-inflationary universe
depending only on $\tau$. The fluctuation in the initial field values
causes a fluctuation $\Delta\tau$
in the arrival time at the end of inflation, leading to a
time-independent ${\cal R}=\delta N
=H_{\rm end}\Delta\tau$.
What if the trajectory is not unique at the end of inflation?
After the completion of the transition from inflation to a universe of
radiation and matter, Eq.~(\ref{rdot}) tells us that
$\cal R$ will be constant provided that $\delta P$ and $\delta \rho$
are related in a definite way. This is the case
during matter domination
($\delta P=0$) or radiation domination
($\delta P=\frac13\delta\rho$).\footnote
{If the `matter' consists of a nearly homogeneous oscillating
scalar field
one actually has $\delta P=\delta\rho$ (=$\frac12
\delta (d\phi/d\tau)^2$), since on comoving hypersurfaces
$\phi$ and therefore $V$ is constant. But
this still makes $\delta P$ negligible.}
Immediately following inflation there might be
a quite complicated situation, with `preheating' \cite{kls}
or else the quantum fluctuation of the `other' field in hybrid models
\cite{CLLSW} converting most of the inflationary potential energy
into marginally relativistic particles in much less than a Hubble time.
But after at most a few Hubble times
one expects to
arrive at a matter-dominated era so that $\cal R$ is constant.
Subsequent events will not cause $\cal R$ to vary provided that they
occur at definite values of the energy density, since again $P$ will
have a definite relation with $\rho$.
This is indeed the case for the usually-considered events, such as
the decay of
matter into radiation and thermal
phase transitions
(including thermal inflation).
The conclusion is that it is
reasonable to suppose that $\cal R$ achieves a constant
value at most a few Hubble times after inflation.
On the other hand one cannot exclude the possibility that one of the
orthogonal components of the inflaton provides a significant additional
degree of freedom, allowing $\cal R$ to have additional variation before
we finally arrive at the radiation-dominated era preceding the present
matter-dominated era.
This leaves the transition, lasting probably at most a few Hubble times,
from inflation to the first epoch of matter domination.
It is conjecture in Ref.~\cite{nak} that
the variation of $\cal R$ during the transition
may still be negligible
if slow-roll holds almost until the end of inflation.
(This is in the present context of the first-order slow-roll
calculation, not the second-order one that is the main focus
of Ref.~\cite{nak}.)
The question requires detailed study however.
\subsection{Working out the perturbation generated by slow-roll
inflation}
If $\cal R$ stops varying by the end of inflation, the
final hypersurface can be located
just before the end (not necessarily at the very end
because that might not correspond to a hypersurface of constant energy
density). Then, knowing the potential
and the hypersurface in field space that corresponds to the end of
inflation, one can work out $N(\phi_1,\phi_2,\cdots)$
using the equations of motion for the fields, and the expression
\be
3\Mpl^2 H = \rho =V+\frac12\frac{d\phi_a}{d\tau}\frac{d\phi_a}{d\tau}
\ee
To perform such a calculation it is not necessary that
all of the fields continue to slow-roll after cosmological scales leave
the horizon. In particular, the oscillation of
some field might briefly interrupt inflation, which resumes when its
amplitude becomes small enough.
If that happens it may be necessary to take
into account `preheating' during the interruption.
In general all this is quite complicated, but there is one
case that may be extremely simple, at least
in a limited regime of parameter space.
This is the case
\be
V=V_1(\phi_1) + V_2(\phi_2) + \cdots
\ee
with each $V_a$ proportional to a power of $\phi_a$.
For a single-component inflaton this gives inflation ending at
$\phi_{\rm end} \simeq \Mpl$, with cosmological scales leaving
the horizon at $\phi\gg\phi_{\rm end}$.
If the potentials $V_a$ are identical we recover that case.
If they are different, slow-roll
may fail in sequence for the different components,
but in some regime of parameter space
the result for
$N$ (at least) might be
the same as if it failed simultaneously for all components.
If that is the case one can derive simple formulas \cite{d7,salopek95},
provided that cosmological scales leave the horizon
at $\phi_a\gg \phi_a^{\rm end}$ for all components.
One has
\be
Hdt=-\Mpl^{-2}\frac{V}{V_1'}d\phi_1=
-\Mpl^{-2}\sum_a \frac{V_a}{V_a'} d\phi_a
\ee
It follows that
\be
N=\Mpl^{-2}\sum_a \int_{\phi_a^{\rm end}}^{\phi_a}\frac{V_a}{V_a'} d\phi_a
\ee
Since each integral is dominated by the endpoint $\phi_a$, we have
$N_{,a} = \Mpl^{-2}V_a/ V_a'$ and
\be
\delta_H^2 =\frac{V}{75\pi^2\Mpl^6} \sum_a \left(\frac{V_a}{V_a'}
\right)^2
\ee
The spectral index is given by
Eq.~(\ref{multin}),
which simplifies slightly
because $V_{,ab}=\delta_{ab}V_a''$.
The simplest case is
$V=\frac12m_1^2\phi_1^2+\frac12m_2^2\phi_2^2$.
Then $n$ is given by the following formula
\be
1-n= \frac1 N\left[
\frac{(1+r)(1+\mu^2 r)}{(1+\mu r)^2} + 1\right]
\ee
where $r=\phi_2^2/\phi_1^2$ and $\mu=m_2^2/m_1^2$.
If $\mu=1$ this reduces to the single-component formula
$1-n=2/N$.
Otherwise it
can be
much bigger, but note that our assumptions will be valid
if at all in a restricted region of the $r$-$\mu$ plane.
\subsection{Comparison with the usual approach}
The usual approach is to use cosmological perturbation theory to find
a closed system
of linear equations, for perturbations in the relevant degrees of
freedom. For each Fourier mode there is a set of coupled differential
equations, which can
be solved with enough effort. This approach
has been used to establish the constancy of $\cal R$ in special cases
for a single-component inflaton \cite{usual}, and
to calculate $\cal R$ at the end of inflation with a multi-component
inflaton \cite{c1,c2,c3,noncanon,davidjuan}.
When it can be formulated, the usual approach
will lead to the same result as the present one, since
on scales far outside the horizon all spatial gradients in the equations
will become negligible.
On the other hand it is not clear that the desired closed system of
equations will always exist.
It might for instance happen
that the small-scale quantum fluctuation of the `other' field
in a hybrid model
generates field gradients which play a crucial role in the transition
from inflation to a matter-dominated universe \cite{CLLSW}.
In that case the evolution of (say) $P$ on large scales is sensitive
to the small-scale behaviour, and one will not be able to develop a
closed system of equations for a given large-scale
Fourier component.
A striking demonstration of the greater
power of the present approach
is provided by
the issue of the constancy
of $\cal R$ for a single-component inflaton. In the usual approach
one has to make simplifying assumptions, and even then
the equations are
sufficiently complicated that it is possible to make mistakes and
arrive at the incorrect
conclusion that $\cal R$ is not constant \cite{grishchuk}.
The present approach makes redundant all proofs of the constancy of
$\cal R$ based on the usual approach.
\subsection{An isocurvature density perturbation?}
Following the astrophysics usage, we classify a density perturbation as
adiabatic or isocurvature with reference to its properties well before
horizon entry, but during the radiation-dominated era preceding the present
matter-dominated era.\footnote
{In some of the theoretical literature this
kind of classification is made also at earlier times, in particular
during inflation, and also during the present matter-dominated era.
From that viewpoint the density perturbation always starts out with
an isocurvature component in the multi-component case,
and it always ends up as adiabatic
by the present matter-dominated era.}
For an adiabatic density perturbation, the density of each particle
species is a unique function of the total energy density.
For an
isocurvature density perturbation the total
density perturbation vanishes, but those of the individual particle
species do not.
The most general density perturbation is the sum of an adiabatic and
an isocurvature perturbation, with $\cal R$ specifying the adiabatic density
perturbation only.
For an isocurvature perturbation to exist the universe
has to possess more than the single degree of freedom provided by the
total energy density.
If the inflaton trajectory is unique, or has become so by the end of
inflation, there is only the single degree of freedom corresponding to
the fluctuation back and forth along the trajectory and
there can be no isocurvature perturbation.
Otherwise one of the orthogonal fields
can provide the necessary degree of freedom.
The simplest way for this to happen is for the orthogonal field
to survive, and acquire a potential so that it
starts to oscillate and becomes matter.\footnote
{If the potential of the `orthogonal' field
already exists during inflation the inflaton
trajectory will have a tiny component in its direction, so that it is
not strictly orthogonal to the inflaton trajectory. This makes no
practical difference. In the axion case the potential is usually
supposed to be generated by QCD effects long after inflation.}
The start of the oscillation
will be determined by the total energy density, but its amplitude
will depend on the initial field value so there will be an isocurvature
perturbation in the axion density. It will be compensated,
for given energy density, by the perturbations in the other species of
matter and radiation which will continue to satisfy the adiabatic
condition $\delta\rho_m/\rho_m=\frac34\delta\rho_r/\rho_r$.
The classic example of this is the axion field
\cite{abook,kt,myaxion}, which is simple because
the fluctuation in the direction of the axion
field causes no adiabatic density
perturbation, at least in the models
proposed so far. The more general case, where one of the components of
the inflaton may cause both an adiabatic and an
isocurvature perturbation
has been looked at in for instance Ref.~\cite{isocurv}, though not in the
context of specific particle physics.
If an isocurvature perturbation in the non-baryonic dark matter density
is generated during inflation,
it must not conflict with observation and this imposes strong
constraints on, for instance, models of the axion
\cite{myaxion,LIN2SC}.
An isocurvature perturbation in the density of one a species of matter
may be defined by
the `entropy perturbation' \cite{kodamasasaki,lyst,LL2}
\be
S= \frac{\delta\rho_m}{\rho_m}-\frac34\frac{\delta\rho_r}{\rho_r}
\ee
where $\rho_m$ is the non-baryonic dark matter density.
Equivalently, $S=\delta y/y$, where $y=\rho_m/\rho_r^{3/4}$.
Since we are dealing with scales far outside the horizon,
$\rho_m$ and $\rho_r$ evolve as they would in an unperturbed
universe which means that $y$ is constant and so is $S$.
Provided that the field fluctuation is small $S$ will be proportional to
it, and so will be a Gaussian random field with a nearly flat spectrum
\cite{myaxion,LL2}.
For an isocurvature perturbation, $\cal R$ vanishes
during the radiation dominated era
preceding the present matter
dominated era. But on the
very large scales entering the horizon well after matter
domination, $S$ generates a nonzero $\cal R$ during
matter domination, namely
${\cal R}=\frac13 S$. A simple way of seeing this, which
has not been noted before, is through the relation
(\ref{rdot}).
Since $\delta\rho=0$, one has $S=-(\rho_m^{-1}+\frac34\rho_r^{-1})
\delta\rho_r$. Then, using $\delta P=\delta\rho_r/3$,
$\rho_r/\rho_m\propto a$ and $Hdt=da/a$ one finds
the quoted result by integrating Eq.~(\ref{rdot}).
As discussed for instance in Ref.~\cite{LL2}, the large-scale
cmb anisotropy coming from an isocurvature
perturbation is $\Delta T/T=-(\frac13+\frac 1{15})S$, where
$S$ is evaluated on the last-scattering surface. The
second
term is the Sachs-Wolfe effect coming from the curvature perturbation we
just calculated, and the first term is the anisotropy $\frac14
\delta\rho_r/\rho_r$ just after last scattering (on a comoving
hypersurface). By contrast the anisotropy from an adiabatic perturbation
comes only from the Sachs-Wolfe effect, so for a given
large-scale density
perturbation the
isocurvature perturbation gives an anisotropy six times bigger.
As a result
an isocurvature perturbation cannot be the dominant contribution to
the cmb, though one could contemplate a small contribution
\cite{iso}.
\section{Discussion and conclusion}
Let us summarize. Except in the last section we have focussed on models
involving a single-component inflaton field, since they are simple
and give a relatively clean prediction for $n$.
The models considered include
non-hybrid ones where the inflaton
field $\phi$ dominates the potential during inflation, and hybrid ones
where this role is played by a different field $\psi$.
In the latter case $\psi$ minimizes the potential during inflation,
so that one still ends up with an effective potential $V(\phi)$.
It is usually assumed that
$V(\phi)$ (non-hybrid) or $V(\phi,\psi)$ (hybrid)
consists of one or a few low-order terms in a power-series expansion,
but this
has motivation in the context of supergravity only if the fields are at
most of order $\Mpl$. In the hybrid case one can still obtain
a non-polynomial $V(\phi)$ during inflation since $\psi$ may be a
function of $\phi$ (the case of mutated hybrid inflation).
A non-polynomial potential can also emerge in a theory starting
out with non-canonical kinetic terms or non-Einstein gravity,
as well as from a quantum correction.
A mathematically simple potential giving inflation
is the one first proposed to implement chaotic initial conditions
at the Planck scale, $V\propto \phi^p$, but with this potential
cosmological scales leave the horizon when
$\phi\sim 10\Mpl$. It
gives $n-1=-(2+p)/(2N)$. It also gives a significant
gravitational wave contribution to the cmb anisotropy, being practically
the only viable
potential proposed so far that does. (Extended inflation, which
also gives a significant contribution,
is ruled out by observation
except in contrived versions.)
Virtually all other models so far proposed
give, during inflation, a potential effectively
of the form
$V=V_0(1\pm \mu\phi^p)$ with the constant term dominating.
With this potential the fields can be $\lsim \Mpl$, and in many
cases $\ll \Mpl$.
For the plus sign $p$ is a positive integer, but for the minus sign
it can be negative, or even non-integral.
In the last case, $p$ just below zero mimics the case of the potential
$V_0[1-A\ln(B/\phi)]$ and $p\to-\infty$ mimics
$V_0(1-e^{-q\phi})$, both of which
have good particle physics motivation (respectively from a
quantum correction or a theory starting out with non-canonical
kinetic terms).
Except for $0\lsim p\lsim 2$
the prediction for
$n$ depends only on the exponent, as is shown in
the Table for some integer values. The only important exceptions are the
quadratic potentials
$V=V_0\pm \frac12m^2\phi^2$ which give
$n=1\pm 2\Mpl^2 m^2/V_0$. The flatness conditions
require $\Mpl^2 m^2/V_0$ to be considerably less than 1.
Unless there is a reason why the parameters should be on the edge of
the allowed region, one therefore
expects $n$ to be indistinguishable from 1 in this case.
In the context of supergravity there may or may not be
such a reason, depending on the model.
Looking at the Table, the
most striking thing is that most of the potentials make
$n$ close to 1, but not very
close. In fact practically all of the listed potentials are ruled out
unless $n$ lies in one of the intervals $.84<n<.98$ or $1.04<n<1.16$.
Another interesting feature is the dividing line
between positive and negative values of $p$, which occurs
somewhere in the range
$n=.92$ and $n=.96$.
Although any selection is as yet tentative, there is no doubt that some
of these potentials are more favoured than others.
Values of $n$ significantly bigger than 1
seem unlikely; the potentials $V=V_0(1+\mu\phi^p)$ with $p>2$ are not
favoured by particle theory, whereas the quadratic potential
is likely to give $n$ indistinguishable from 1.
Values $n<.84$ are also unlikely, unless the
inverted quadratic potential emerges from one of the
non-hybrid settings discussed in Section IV.
This leaves the regime $.84<n\leq 1.00$, and within that are a few
potentials
that might be regarded as
favoured theoretically. A very subjective
selection corresponds
to the five rows marked of the Table marked by **.
The first and last cases are the quadratic and inverted quadratic
potentials $V_0\pm\frac12m^2\phi^2$, thought of as being
derived respectively
from
ordinary and inverted hybrid inflation; since there is no reason
for the parameters to be on the edge of the region allowed by the
flatness condition $\eta\ll 1$ one expects
$n$ to be
indistinguishable from 1 in these models. The second case is the
loop-corrected potential that might arise if the $D$ term
dominates, mimicked by the potential $V_0(1-\mu\phi^{-p})$
with $p\simeq 0$. The third case is the potential $V_0(1-\mu\phi^{-2})$
which comes out of the original mutated hybrid inflation model
\cite{mutated}.
The fourth is the cubic potential advocated in Ref.~\cite{grahamnew}.
With this last potential, for small $N$,
the scale dependence of the spectrum
(not well-represented by constant $n$) becomes
strong enough to give a useful lower bound on $N$. It was estimated
at the end of Section II as $N\gsim 11$.
To summarize the situation regarding $n$ in these models, it
is clear that a measurement of it will
give valuable discrimination between different potentials.
Values of $n$
significantly above
1 are disfavoured theoretically.
We have looked briefly at inflation model-building in the demanding
context of supergravity, focussing on the problem of keeping the
inflaton mass small enough in the face of generic contributions
of order $\pm H^2$.
We also considered models with a multi-component inflaton,
and we have looked in some detail at the calculation
of the spectrum in both this and the single-component case.
In both cases the most powerful calculational technique
\cite{salopek95,ewanmisao} starts with the observation
that after smoothing the
relevant quantities on scales far outside the
horizon, the evolution of the universe along each comoving worldline
will be the same as for an unperturbed universe with the same
initial inflaton field. In the single-component case this justifies
the usual assumption that $\cal R$ is constant. In the multi-component
case it leads to a simple formula for $n$, whose only input is the
unperturbed evolution. Finally, we looked at the case of an isocurvature
perturbation, giving a simple derivation of the previously obscure
fact that the low multipoles of the cmb anisotropy are six times as big
as for an adiabatic density perturbation.
\begin{table}
\centering
\caption[table]{Predictions for $n$ are displayed for
inflationary potentials of the
form $V=V_0(1\pm \mu\phi^p )$, with the first term dominating.
The sixth row represents the limiting case $p\to 0$ from below,
with the minus sign.
In most cases the prediction
is proportional to $1/N$,
where $N$ is the number of $e$-folds of inflation after cosmological
scales leave the horizon.
Results are given for $N=50$ and $N=25$, corresponds to different
cosmologies after inflation.
The predictions for $n$ are quoted to only two
decimal places, because a better observational accuracy
would be very hard to achieve. The symbols {**} mark
the rows corresponding to five potentials
that might be favoured on the basis of
theory. The first of them is expected to give $n$ indistinguishable from
1,
but the last may or may not depending on how the potential is derived.}
\begin{tabular}{cllllc}
&$V(\phi)/V_0$ & $(n-1)(N/50)$ & \multicolumn{2}{c}{$n$} & \\
&& & $N=50$ & $N=25$ &
\\ \hline
&$1+\mu\phi$ & \ \ $.00$ & $1.00$ & $1.00$ &\\
{**}&$1+\mu\phi^2$ & $1.00$\,? &
\multicolumn{2}{c}{$n=1+4\mu$}&{**}\\
&$1+\mu\phi^3$ & \ \ $.08$ & $1.08$ & $1.16$ &\\
&$1+\mu\phi^4$ & \ \ $.06$ & $1.06$ & $1.12$ &\\
&$1+\mu\phi^\infty$ & \ \ $.04$ & $1.04$ & $1.08$ & \\
\hline
{**}&$1-\mu\phi^{-0}$ & $-.02$ & \ $.98$ & \ $.96$ &{**}\\
{**}&$1-\mu\phi^{-2}$ & $-.03$ & \ $.97$ &\ $.94$ &{**} \\
&$1-\mu\phi^{-4}$ & $-.03$ & \ $.97$ &\ $.93$ &\\
&$1-\mu\phi^{\pm\infty}$ & $-.04
$ & \ $.96$ &\ $.92$ &\\
&$1-\mu\phi^{4}$ & $-.06$ & \ $.94$ &\ $.88$ & \\
{**}&$1-\mu\phi^{3} $& $-.03$ & \ $.92$ &\ $.84$ & {**}\\
{**}&$1-\mu\phi^2$ & $1.00$\,??? & \multicolumn{2}{c}{$n=1-4\mu$}& {**}\\
&$1-\mu\phi$ & \ \ $.00$ & $1.00$ & $1.00$ &\\
\end{tabular}
\end{table}
\newpage
\underline{Acknowledgements}:
I am grateful to CfPA and LBL, Berkeley, for the provision of financial
support and a stimulating working environment when
this work was started.
I am indebted to Ewan Stewart for many helpful
discussions about supergravity, and about multi-component inflaton
models of inflation. I have
also received valuable input from Andrew Liddle,
Andrei Linde, Hitoshi Murayama and Graham Ross and David Wands.
The work is partially supported by grants from PPARC and
from the European Commission
under the Human Capital and Mobility programme, contract
No.~CHRX-CT94-0423.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 795 |
Salto, anteriormente Salto Oriental, é uma cidade e capital do departamento de Salto, Uruguai, situada a 498 km de Montevidéu. Localiza-se na margem oriental do rio Uruguai, limitando-se com a cidade de Concordia, na Argentina. A Ponte Salto Grande, construída sobre a represa homônima, conecta as duas cidades.
O nome do local provém do fato de que o rio Uruguai possuía uma elevação (conhecida como Itú na língua guarani) derivada das rochas em seu leito. O centro da cidade situa-se a cerca de 18 km a sul da barragem e do reservatório. A Estrada Nacional 3 é a principal via de comunicação com o resto do departamento. A localidade é servida pelo Aeroporto de Salto.
História
Primeiros povos, e fundação de Salto
Segundo documentações arqueológicas, estima-se que já existiam povos indígenas há cerca de 10.000 anos na região de Salto. A fundação oficial da cidade ocorreu em 8 de novembro de 1756.
Durante a Guerra do Guarani, o governador do Rio da Prata e o Marquês de Valdelirios pediram ao governador de Montevidéu para se mudar para o norte com um exército de 400 homens para a conclusão do Tratado de Madrid. Em 1757, o governador de Montevidéu e o governador de Buenos Aires construíram o Forte de Santo Antônio, onde 100 homens se fixaram porque não conseguiam acompanhar a navegação no rio Uruguai. O forte foi uma instalação militar que iniciou a atual organização geográfica da cidade de Salto, e da cidade de Concórdia, na Argentina. Nessa época, passaram tropas espanholas que lutavam para acabar com um dos sítios de Portugal na Colônia do Sacramento. As instalações serviram por sete anos para abastecer o exército. O Forte foi abandonado em 1763, mas deixou residentes na região que hoje corresponde ao território de Salto.
Exílio dos jesuítas
Em 16 de junho de 1768, o forte foi ocupado pelo governador de Buenos Aires com 1.500 soldados para ratificar a expulsão dos jesuítas de todo o território colonial espanhol, como ordenado pelo Rei Carlos III. Começaram através da captura de uma das missões da Companhia de Jesus, denominada Reducción de Yapeyú. O Forte de Santo Antônio serviu como depósito de armas, e mais tarde como uma prisão para os sacerdotes. Enquanto os jesuítas eram capturados, o tenente Nicolás Garcia administrava o forte.
O forte e seus habitantes foram destruídos pela grande inundação do rio da prata. O complexo militar foi reconstruído, mas dessa vez no lado ocidental do rio, onde hoje se localiza a cidade argentina de Concórdia. No final do século XVIII, a cidade já tinha residentes permanentes.
População
Segundo o censo de 2011, a cidade tem uma população de 106.011 habitantes, dos quais 53,131 são homens, e 53,880 são mulheres. Isto torna a terceira maior cidade do Uruguai depois de Montevidéu e de Ciudad de la Costa. Com sua área metropolitana, a população chega aos 125.000 habitantes. A estimativa de 2017 é que a cidade tenha chegado aos 112 mil habitantes.
Economia
A economia da cidade é movida principalmente pelos setores do comércio, turismo, serviços e indústria, recendo destaque pela produção de frutas cítricas. A cidade é cercada por um cinturão de fazendas envolvidas na produção destes frutos, cuja maturação é favorecido por um clima mais quente que o do sul. A uva Harriague, base dos vinhos Tannat, é a única variedade reconhecida internacionalmente para o país.
Cidadãos notórios
Luis Alberto Suárez, futebolista.
Edinson Cavani, futebolista.
Pedro Rocha, futebolista.
José Leandro Andrade, futebolista.
Mauricio Moreira, ciclista. | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,773 |
Moore v. Kobach
As announced on June 19, 2018, the ACLU of Kansas filed a class-action lawsuit against Kansas Secretary of State Kris Kobach on behalf of three individual plaintiffs, who had their constitutional rights to privacy violated by Sec. Kobach's office.
In the case, Moore v. Kobach, lead plaintiff Scott Moore shared a name and birthdate with a different man from Naples, Florida, and Kobach's Crosscheck program proceeded to "match" them as the same person. Kobach then shuttled Moore's information to Florida officials via unencrypted emails, leaving Moore vulnerable to identity theft. Moore's personal information was exposed in 2013, but he only learned of the breach this year when he received a postcard and a one-year subscription to LifeLock, an identity theft protection company.
Kobach has implemented the "Crosscheck" data matching program, purported to identify people registered to vote in more than one state, in a careless way that breaks the law and puts registered voters in Kansas at risk of identity theft and fraud. Outside analyses by data management professionals have found that Crosscheck's matching criteria yields false positives more than 99% of the time - however, Sec. Kobach continues to use this faulty data in his illegal experiment to reduce citizen participation in Kansas elections and to perpetuate the false narrative of voter fraud that courts have held as illegal (see Fish v. Kobach). Crosscheck is exclusively funded by Kansas taxpayers and therefore free of charge to participating states. Citing both data inaccuracy and privacy concerns, eight states across the country have opted out of the Crosscheck program.
This case seeks remedy for the flagrant disregard with which Sec. Kobach has used Crosscheck to recklessly and routinely share personal, private information of thousands of Kansas voters in ways that are unsafe, unsecured, and cavalier. The action seeks to prohibit Koback from continuing to maintain, share, and release sensitive voter registration information to other states through the Crosscheck program. The action also seeks remedy for past disclosures, as Kobach's disclosure of the plaintiffs' private voter data is an unconstitutional violation of the Kansas Public Records Act, which prohibits government disclosure of social security numbers.
The ACLU of Kansas calls on the Court to bring justice for Mr. Moore and the other plaintiffs and to also bring the state of Kansas closer to a truthful accounting of the enormous costs Crosscheck has on the privacy rights of citizens and on the institutions of our democracy itself.
*UPDATE* On February 1, the Court denied the defense's motion to dismiss, concluding that precedent establishes that a constitutional right to privacy and that our clients' complaints indeed pose plausible claim for relief.
pdfMoore v. Kobach - complaint
pdfDef MTD.pdf
pdfMoore Court Order - Denying MTD.pdf
Colorado Withdraws from Crosscheck
Illinois drops out of controversial Crosscheck anti-voter fraud...
Illinois ends participation in multi-state voter database | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,335 |
Q: Efficient JPA query to find if at least one record exists with given condition Given following JPA queries to find if there is at least one record with given condition, which one is more efficient (performance)?
A)
boolean notFound = entityManager
.createQuery("SELECT f.id FROM Foo f WHERE f.active = true")
.setMaxResults(1)
.getResultList()
.isEmpty();
B)
boolean notFound = entityManager
.createQuery("SELECT COUNT(f.id) FROM Foo f WHERE f.active = true")
.getSingleResult() == 0
A: If you have more than a trivial amount of Foo records, your DB supports something like a LIMIT clause, and the JPA impl is clever enough to use it, and DB recognizes it as equivalent to a EXISTS query, then option 1 should outperform option 2.
Here is a nice explanation.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,528 |
{"url":"https:\/\/wsxq2.55555.io\/blog\/2018\/11\/19\/Python%E5%AD%A6%E4%B9%A0%E7%AC%94%E8%AE%B0\/","text":"# Python\u5b66\u4e60\u7b14\u8bb0\n\nPosted by wsxq2 on 2018-11-19\nTAGS:\u2002 python2.7TODO\n\n## 0 Coding Style\n\nThe most important points:\n\n\u2022 Use 4-space indentation, and no tabs.\n\u2022 Wrap lines so that they don\u2019t exceed 79 characters.\n\u2022 Use blank lines to separate functions and classes, and larger blocks of code inside functions.\n\u2022 When possible, put comments(#) on a line of their own.\n\u2022 Use docstrings.\n\u2022 Use spaces around operators and after commas, but not directly inside bracketing constructs: a = f(1, 2) + g(3, 4).\n\u2022 Name your classes and functions consistently; the convention is to use CamelCase for classes and lower_case_with_underscores for functions and methods. Always use self as the name for the first method argument (see A First Look at Classes for more on classes and methods).\n\u2022 Don\u2019t use fancy encodings if your code is meant to be used in international environments. Plain ASCII works best in any case.\n\nMore detailed information is in PEP 8\n\n## 1 Python Interpreter\n\n### 1.1 Invoking the Interpreter\n\nPython\u2019s default path is \/usr\/local\/bin\/python(Linux) or C:\\python27(Windows)\n\n\u2022 start:\n1. python\n2. python [-i] <source file>\n3. python -c command [arg] ...\n4. python -m module [arg] ...\n\u2022 Argument Passing:\n\nWhen known to the interpreter, the script name and additional arguments thereafter are turned into a list of strings and assigned to the argv variable in the sys module. You can access this list by executing import sys. The length of the list is at least one; when no script and no arguments are given, sys.argv[0] is an empty string. When the script name is given as '-' (meaning standard input), sys.argv[0] is set to '-'. When -c command is used, sys.argv[0] is set to '-c'. When -m module is used, sys.argv[0] is set to the full name of the located module. Options found after -c command or -m module are not consumed by the Python interpreter\u2019s option processing but left in sys.argv for the command or module to handle.\n\n\u2022 Source Code Encoding:\n\n1\n2\n#!\/usr\/bin\/env python\n# -*- coding: utf-8 -*-\n\n\n### 1.2 Interactive Mode\n\nIn this mode it prompts for the next command with the primary prompt, usually three greater-than signs (>>>); for continuation lines it prompts with the secondary prompt, by default three dots (...). The last printed expression is assigned to the variable _.\n\n1\n2\n3\n4\n5\n6\n>>> 100.50 * (12.5)\/100\n12.5625\n>>> price + _\n113.0625\n>>> round(_, 2)\n113.06\n\n\u2022 start: python\n\u2022 quit:\n1. use end-of-file character (Control-D on Unix, Control-Z on Windows)\n2. use quit()\n\u2022 GNU readline library: support for the GNU readline library(which adds more elaborate interactive editing and history features). The current line can be edited using the conventional Emacs control characters.\n1. Line Editing\n\u2022 C-A: (Control-A) moves the cursor to the beginning of the line\n\u2022 C-E: to the end\n\u2022 C-B: moves it one position to the left\n\u2022 C-F: to the right. Backspace erases the character to the left of the cursor\n\u2022 C-D: the character to its right. C-K kills (erases) the rest of the line to the right of the cursor\n\u2022 C-Y: yanks back the last killed string\n\u2022 C-_: undoes the last change you made; it can be repeated for cumulative effect.\n2. History Substitution\n\u2022 C-P: moves one line up (back) in the history buffer\n\u2022 C-N: moves one down.\n\u2022 C-R: starts an incremental reverse search\n\u2022 C-S: starts a forward search\n3. Key Bindings(in ~\/.inputrc)\n\u2022 bind key: key-name: function-name, \"string\": function-name\n\u2022 set options: set option-name value\n\u2022 examples:\n\n# ~\/.inputrc\n\n# set vi-style editing:\nset editing-mode vi\n\n# Edit using a single line:\nset horizontal-scroll-mode On\n\n# Rebind some keys:\nMeta-h: backward-kill-word\n\"\\C-u\": universal-argument\n\n# make Tab be used for complete\nTab: complete\n\n\u2022 startup file: Python will execute the contents of a file identified by the PYTHONSTARTUP environment variable when you start an interactive interpreter.\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n28\n# Add auto-completion and a stored history file of commands to your Python\n# interactive interpreter. Requires Python 2.0+, readline. Autocomplete is\n# bound to the Esc key by default (you can change it - see readline docs).\n#\n# Store the file in ~\/.pystartup, and set an environment variable to point\n# to it: \"export PYTHONSTARTUP=~\/.pystartup\" in bash.\n\nimport atexit\nimport os\nimport rlcompleter\n\nhistoryPath = os.path.expanduser(\"~\/.pyhistory\")\n\ndef save_history(historyPath=historyPath):\n\nif os.path.exists(historyPath):\n\natexit.register(save_history)\n\n# use <Control-O> for complete\n\ndel os, atexit, readline, rlcompleter, save_history, historyPath\n\n\n\u2022 Alternatives: One alternative enhanced interactive interpreter that has been around for quite some time is IPython, which features tab completion, object exploration and advanced history management. It can also be thoroughly customized and embedded into other applications. Another similar enhanced interactive environment is bpython.\n\n## 2 Control Flow Statements\n\n\u2022 what is True: like in C, any non-zero integer value is true; zero is false. The condition may also be any sequence: anything with a non-zero length is true, empty sequences are false.\n\u2022 multiple assignment:\n1\n2\na, b=0, 1\na, b = b, a+b\n\n\nthe expressions on the right-hand side are all evaluated first before any of the assignments take place. The right-hand side expressions are evaluated from the left to the right.\n\n\u2022 indentation: indentation is Python\u2019s way of grouping statements. At the interactive prompt, you have to type a tab or space(s) for each indented line. When a compound statement is entered interactively, it must be followed by a blank line to indicate completion. Note that each line within a basic block must be indented by the same amount.\n\u2022 standard comparison operators: < (less than), > (greater than), == (equal to), <= (less than or equal to), >= (greater than or equal to) and != (not equal to)\n\u2022 in, not in: check whether a value occurs (does not occur) in a sequence\n\u2022 is, is not: compare whether two objects are really the same object\n\u2022 a < b == c: chained comparison, tests whether a is less than b and moreover b equals c.\n\u2022 and, or,not: between them, not has the highest priority and or the lowest. The Boolean operators and and or are so-called short-circuit operators: their arguments are evaluated from left to right, and evaluation stops as soon as the outcome is determined. When used as a general value and not as a Boolean, the return value of a short-circuit operator is the last evaluated argument.\n\nIt is possible to assign the result of a comparison or other Boolean expression to a variable. For example,\n\n1\n2\n3\n4\n>>> string1, string2, string3 = '', 'Trondheim', 'Hammer Dance'\n>>> non_null = string1 or string2 or string3\n>>> non_null\n'Trondheim'\n\n\nNote that in Python, unlike C, assignment cannot occur inside condition expressions.\n\n\u2022 Comparing Sequences and Other Types: Sequence objects may be compared to other objects with the same sequence type. The comparison uses lexicographical(\u8bcd\u5178) ordering: first the first two items are compared, and if they differ this determines the outcome of the comparison; if they are equal, the next two items are compared, and so on, until either sequence is exhausted. If two items to be compared are themselves sequences of the same type, the lexicographical comparison is carried out recursively. If all items of two sequences compare equal, the sequences are considered equal. If one sequence is an initial sub-sequence of the other, the shorter sequence is the smaller (lesser) one. Lexicographical ordering for strings uses the ASCII ordering for individual characters. Some examples of comparisons between sequences of the same type:\n\n1\n2\n3\n4\n5\n6\n7\n(1, 2, 3) < (1, 2, 4)\n[1, 2, 3] < [1, 2, 4]\n'ABC' < 'C' < 'Pascal' < 'Python'\n(1, 2, 3, 4) < (1, 2, 4)\n(1, 2) < (1, 2, -1)\n(1, 2, 3) == (1.0, 2.0, 3.0)\n(1, 2, ('aa', 'ab')) < (1, 2, ('abc', 'a'), 4)\n\n\nNote that comparing objects of different types is legal. The outcome is deterministic but arbitrary: the types are ordered by their name. Thus, a list is always smaller than a string, a string is always smaller than a tuple, etc. Mixed numeric types are compared according to their numeric value, so 0 equals 0.0, etc.\n\n### 2.1 if..elif..else.. Statements\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n>>> x = int(raw_input(\"Please enter an integer: \"))\n>>> if x < 0:\n... x = 0\n... print 'Negative changed to zero'\n... elif x == 0:\n... print 'Zero'\n... elif x == 1:\n... print 'Single'\n... else:\n... print 'More'\n...\nMore\n\n\nAn if \u2026 elif \u2026 elif \u2026 sequence is a substitute for the switch or case statements found in other languages.\n\n### 2.2 for..in.. Statements\n\nPython\u2019s for statement iterates over the items of any sequence (a list or a string), in the order that they appear in the sequence:\n\n1\n2\n3\n4\n5\n6\n7\n8\n>>> # Measure some strings:\n... words = ['cat', 'window', 'defenestrate']\n>>> for w in words:\n... print w, len(w)\n...\ncat 3\nwindow 6\ndefenestrate 12\n\n\nIf you need to modify the sequence you are iterating over while inside the loop, it is recommended that you first make a copy. Iterating over a sequence does not implicitly make a copy:\n\n1\n2\n3\n4\n5\n6\n>>> for w in words[:]: # Loop over a slice copy of the entire list.\n... if len(w) > 6:\n... words.insert(0, w)\n...\n>>> words\n['defenestrate', 'cat', 'window', 'defenestrate']\n\n\n### 2.3 while Statements\n\n1\n2\n3\n4\n5\n6\n>>> a, b = 0, 1\n>>> while b < 1000:\n... print b, # A trailing comma avoids the newline after the output\n... a, b = b, a+b\n...\n1 1 2 3 5 8 13 21 34 55 89 144 233 377 610 987\n\n\n### 2.4 break, continue and else in Loops\n\n\u2022 break: Like in C, breaks out of the innermost enclosing for or while loop.\n\u2022 continue: Like in C, continues with the next iteration of the loop:\n\u2022 else: codes in this block are excuted in the following situations:\n\u2022 the loop terminates through exhaustion of the list (with for)\n\u2022 the condition becomes false (with while)\n\nbut not when the loop is terminated by a break statement.\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n>>> for n in range(2, 10):\n... for x in range(2, n):\n... if n % x == 0:\n... print n, 'equals', x, '*', n\/x\n... break\n... else: # the else clause belongs to the for loop, not the if statement\n... # loop fell through without finding a factor\n... print n, 'is a prime number'\n...\n2 is a prime number\n3 is a prime number\n4 equals 2 * 2\n5 is a prime number\n6 equals 2 * 3\n7 is a prime number\n8 equals 2 * 4\n9 equals 3 * 3\n\n\n### 2.5 pass statements\n\ndo nothing.\n\n1\n2\n3\n>>> def initlog(*args):\n... pass # Remember to implement this!\n...\n\n\n## 3 Defining Functions\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n>>> def fib2(n): # return Fibonacci series up to n\n... \"\"\"Return a list containing the Fibonacci series up to n.\"\"\"\n... result = []\n... a, b = 0, 1\n... while a < n:\n... result.append(a) # see below\n... a, b = b, a+b\n... return result\n...\n>>> f100 = fib2(100) # call it\n>>> f100 # write the result\n[0, 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89]\n\n\n### 3.1 Default Argument Values\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\nwhile True:\nok = raw_input(prompt)\nif ok in ('y', 'ye', 'yes'):\nreturn True\nif ok in ('n', 'no', 'nop', 'nope'):\nreturn False\nretries = retries - 1\nif retries < 0:\nraise IOError('refusenik user')\nprint complaint\n\n\nImportant warning: The default value is evaluated only once. This makes a difference when the default is a mutable object such as a list, dictionary, or instances of most classes. For example, the following function accumulates the arguments passed to it on subsequent calls:\n\n1\n2\n3\n4\n5\n6\n7\ndef f(a, L=[]):\nL.append(a)\nreturn L\n\nprint f(1)\nprint f(2)\nprint f(3)\n\n\nThis will print\n\n1\n2\n3\n[1]\n[1, 2]\n[1, 2, 3]\n\n\nIf you don\u2019t want the default to be shared between subsequent calls, you can write the function like this instead:\n\n1\n2\n3\n4\n5\ndef f(a, L=None):\nif L is None:\nL = []\nL.append(a)\nreturn L\n\n\n### 3.2 Keyword Arguments\n\n1\n2\n3\n4\n5\ndef parrot(voltage, state='a stiff', action='voom', type='Norwegian Blue'):\nprint \"-- This parrot wouldn't\", action,\nprint \"if you put\", voltage, \"volts through it.\"\nprint \"-- Lovely plumage, the\", type\nprint \"-- It's\", state, \"!\"\n\n\nthe above function accepts one required argument (voltage) and three optional arguments (state, action, and type). This function can be called in any of the following ways:\n\n1\n2\n3\n4\n5\n6\nparrot(1000) # 1 positional argument\nparrot(voltage=1000) # 1 keyword argument\nparrot(voltage=1000000, action='VOOOOOM') # 2 keyword arguments\nparrot(action='VOOOOOM', voltage=1000000) # 2 keyword arguments\nparrot('a million', 'bereft of life', 'jump') # 3 positional arguments\nparrot('a thousand', state='pushing up the daisies') # 1 positional, 1 keyword\n\n\nWhen a final formal parameter of the form **name is present, it receives a dictionary containing all keyword arguments except for those corresponding to a formal parameter. This may be combined with a formal parameter of the form *name which receives a tuple containing the positional arguments beyond the formal parameter list. (*name must occur before **name.) For example, if we define a function like this:\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\ndef cheeseshop(kind, *arguments, **keywords):\nprint \"-- Do you have any\", kind, \"?\"\nprint \"-- I'm sorry, we're all out of\", kind\nfor arg in arguments:\nprint arg\nprint \"-\" * 40\nkeys = sorted(keywords.keys())\nfor kw in keys:\nprint kw, \":\", keywords[kw]\n\n\nIt could be called like this:\n\n1\n2\n3\n4\n5\ncheeseshop(\"Limburger\", \"It's very runny, sir.\",\n\"It's really very, VERY runny, sir.\",\nshopkeeper='Michael Palin',\nclient=\"John Cleese\",\nsketch=\"Cheese Shop Sketch\")\n\n\nand of course it would print:\n\n1\n2\n3\n4\n5\n6\n7\n8\n-- Do you have any Limburger ?\n-- I'm sorry, we're all out of Limburger\nIt's very runny, sir.\nIt's really very, VERY runny, sir.\n----------------------------------------\nclient : John Cleese\nshopkeeper : Michael Palin\nsketch : Cheese Shop Sketch\n\n\n### 3.3 Arbitrary Argument Lists\n\nFinally, the least frequently used option is to specify that a function can be called with an arbitrary number of arguments. These arguments will be wrapped up in a tuple. Before the variable number of arguments, zero or more normal arguments may occur.\n\n1\n2\ndef write_multiple_items(file, separator, *args):\nfile.write(separator.join(args))\n\n\n### 3.4 Unpacking Argument Lists\n\nWrite the function call with the *-operator to unpack the arguments out of a list or tuple:\n\n1\n2\n3\n4\n5\n>>> range(3, 6) # normal call with separate arguments\n[3, 4, 5]\n>>> args = [3, 6]\n>>> range(*args) # call with arguments unpacked from a list\n[3, 4, 5]\n\n\nIn the same fashion, dictionaries can deliver keyword arguments with the **-operator:\n\n1\n2\n3\n4\n5\n6\n7\n8\n>>> def parrot(voltage, state='a stiff', action='voom'):\n... print \"-- This parrot wouldn't\", action,\n... print \"if you put\", voltage, \"volts through it.\",\n... print \"E's\", state, \"!\"\n...\n>>> d = {\"voltage\": \"four million\", \"state\": \"bleedin' demised\", \"action\": \"VOOM\"}\n>>> parrot(**d)\n-- This parrot wouldn't VOOM if you put four million volts through it. E's bleedin' demised !\n\n\n### 3.5 Lambda Expressions\n\nSmall anonymous functions can be created with the lambda keyword.\n\n1\n2\n3\n4\n5\n6\n7\n8\n>>> def make_incrementor(n):\n... return lambda x: x + n\n...\n>>> f = make_incrementor(42)\n>>> f(0)\n42\n>>> f(1)\n43\n\n\nThe above example uses a lambda expression to return a function. Another use is to pass a small function as an argument:\n\n1\n2\n3\n4\n>>> pairs = [(1, 'one'), (2, 'two'), (3, 'three'), (4, 'four')]\n>>> pairs.sort(key=lambda pair: pair[1])\n>>> pairs\n[(4, 'four'), (1, 'one'), (3, 'three'), (2, 'two')]\n\n\n### 3.6 Documentation Strings\n\nThere are emerging conventions about the content and formatting of documentation strings.\n\n\u2022 The first line should always be a short, concise summary of the object\u2019s purpose. This line should begin with a capital letter and end with a period.\n\u2022 If there are more lines in the documentation string, the second line should be blank, visually separating the summary from the rest of the description.\n\u2022 The following lines should be one or more paragraphs describing the object\u2019s calling conventions, its side effects, etc.\n\nThe first non-blank line after the first line of the string determines the amount of indentation for the entire documentation string.\n\nHere is an example of a multi-line docstring:\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n>>> def my_function():\n... \"\"\"Do nothing, but document it.\n...\n... No, really, it doesn't do anything.\n... \"\"\"\n... pass\n...\n>>> print my_function.__doc__\nDo nothing, but document it.\n\nNo, really, it doesn't do anything.\n\n\n## Math\n\n\u2022 operators:\n\u2022 fundamental operations: +, -, *, \/\n\u2022 %: remainder\n\u2022 \/\/: explicit floor division discards the fractional part\n\u2022 **: power\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n>>> 17 \/ 3 # int \/ int -> int\n5\n>>> 17 \/ 3.0 # int \/ float -> float\n5.666666666666667\n>>> 17 \/\/ 3.0 # explicit floor division discards the fractional part\n5.0\n>>> 17 % 3 # the % operator returns the remainder of the division\n2\n>>> 5 * 3 + 2 # result * divisor + remainder\n17\n\n\u2022 type of numbers: int, float, Decimal, Fraction, complex\n\n## Strings\n\n\u2022 \"\" and '': The only difference between \"\" and '' is that within single quotes you don\u2019t need to escape \" (but you have to escape \\') and vice versa.\n\n\u2022 default output and print():\n\nIn the interactive interpreter, the output string is enclosed in quotes and special characters are escaped with backslashes. While this might sometimes look different from the input (the enclosing quotes could change), the two strings are equivalent. The string is enclosed in double quotes if the string contains a single quote and no double quotes, otherwise it is enclosed in single quotes. The print statement produces a more readable output, by omitting the enclosing quotes and by printing escaped and special characters:\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n>> '\"Isn\\'t,\" they said.'\n'\"Isn\\'t,\" they said.'\n>> print '\"Isn\\'t,\" they said.'\n\"Isn't,\" they said.\n>> s = 'First line.\\nSecond line.' # \\n means newline\n>> s # without print, \\n is included in the output\n'First line.\\nSecond line.'\n>> print s # with print, \\n produces a new line\nFirst line.\nSecond line.\n\n\u2022 character before strings:\n\u2022 raw strings: r'C:\\some\\name'(\u2018\\n\u2019 now not means newline)\n1\n2\n3\n4\n5\n>>> print 'C:\\some\\name' # here \\n means newline!\nC:\\some\name\n>>> print r'C:\\some\\name' # note the r before the quote\nC:\\some\\name\n\n\u2022 Unicode strings: u'This is a Unicoding\\u0020string\n1\n2\n3\n4\n>>> ur'Hello\\u0020World !' # It will only apply the above \\uXXXX conversion if there is an uneven number of backslashes in front of the small \u2018u\u2019\nu'Hello World !'\n>>> ur'Hello\\\\u0020World !' # The raw mode is most useful when you have to enter lots of backslashes, as can be necessary in regular expressions.\nu'Hello\\\\\\\\u0020World !'\n\n\u2022 multiple lines string: (In the following examples, the \\ in \"\"\"\\ is used to prevent End-of-lines, i.e. \\n)\nprint \"\"\"\\\nUsage: thingy [OPTIONS]\n-h Display this usage message\n-H hostname Hostname to connect to\n\"\"\"\n\n\u2022 string operators:\n\u2022 +: Concatenat strings(glued together). Two or more string literals next to each other are automatically concatenated.\n\u2022 *: Repeated strings.\n1\n2\n3\n4\n5\n>>> 3 * 'un' + 'ium'\n'unununium'\n>>> # This feature is particularly useful when you want to break long strings\n>>> text = ('Put several strings within parentheses '\n'to have them joined together.')\n\n\u2022 indexing: obtain individual characters. the first character having index 0. There is no separate character type; a character is simply a string of size one:\n1\n2\n3\n4\n5\n6\n>>> word = 'Python'\n>>> word[0] # character in position 0\n'P'\n>>> word[5] # character in position 5\n'n'\n\n\n\nIndices may also be negative numbers, to start counting from the right:\n\n1\n2\n3\n4\n5\n6\n>>> word[-1] # last character\n'n'\n>>> word[-2] # second-last character\n'o'\n>>> word[-6]\n'P'\n\n\u2022 slicing: obtain a substring\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n>>> word[:] # the value of word\n'Python'\n>>> word[0:2] # characters from position 0 (included) to 2 (excluded)\n'Py'\n>>> word[:2] + word[2:] # s[:i] + s[i:] is always equal to s\n'Python'\n>>> word[:2] # character from the beginning to position 2 (excluded)\n'Py'\n>>> word[-2:] # characters from the second-last (included) to the end\n'on'\n>>> word[4:42] # out of range slice indexes are handled gracefully when used for slicing\n'on'\n\n\u2022 Immutable: Python strings cannot be changed\n1\n2\n3\n4\n5\n6\n>>> word[0] = 'J'\n...\nTypeError: 'str' object does not support item assignment\n>>> word[2:] = 'py'\n...\nTypeError: 'str' object does not support item assignment\n\n\u2022 Unicode String:\n\u2022 u before string:\n1\n2\n>>> u\"abc\"\n'abc'\n\n\u2022 When a Unicode string is printed, written to a file, or converted with str(), conversion takes place using this default encoding.\n1\n2\n3\n4\n5\n6\n>>> str(u\"abc\")\n'abc'\n>>> str(u\"\u00e4\u00f6\u00fc\")\nTraceback (most recent call last):\nFile \"<stdin>\", line 1, in ?\nUnicodeEncodeError: 'ascii' codec can't encode characters in position 0-2: ordinal not in range(128)\n\n\u2022 To convert a Unicode string into an 8-bit string using a specific encoding, Unicode objects provide an encode() method that takes one argument, the name of the encoding. Lowercase names for encodings are preferred.\n\n1\n2\n>>> u\"\u00e4\u00f6\u00fc\".encode('utf-8')\n'\\xc3\\xa4\\xc3\\xb6\\xc3\\xbc'\n\n\u2022 If you have data in a specific encoding and want to produce a corresponding Unicode string from it, you can use the unicode() function with the encoding name as the second argument.\n\n1\n2\n>>> unicode('\\xc3\\xa4\\xc3\\xb6\\xc3\\xbc', 'utf-8')\nu'\\xe4\\xf6\\xfc'\n\n\n## Data Structures\n\n### Sequences\n\nSequences include str, unicode, list, tuple, bytearray, buffer, xrange\n\n#### Lists\n\n\u2022 define:\n1\n2\n3\n>>> squares = [1, 4, 9, 16, 25]\n>>> squares\n[1, 4, 9, 16, 25]\n\n\u2022 indexing:\n1\n2\n3\n4\n>>> squares[0] # indexing returns the item\n1\n>>> squares[-1]\n25\n\n\u2022 slicing:\n1\n2\n3\n4\n>>> squares[:]\n[1, 4, 9, 16, 25]\n>>> squares[-3:] # slicing returns a new list\n[9, 16, 25]\n\n\u2022 concatenation:\n1\n2\n>>> squares + [36, 49, 64, 81, 100]\n[1, 4, 9, 16, 25, 36, 49, 64, 81, 100]\n\n\u2022 mutable:\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n>>> cubes = [1, 8, 27, 65, 125] # something's wrong here\n>>> 4 ** 3 # the cube of 4 is 64, not 65!\n64\n>>> cubes[3] = 64 # replace the wrong value\n>>> cubes\n[1, 8, 27, 64, 125]\n>>> letters = ['a', 'b', 'c', 'd', 'e', 'f', 'g']\n>>> letters\n['a', 'b', 'c', 'd', 'e', 'f', 'g']\n>>> # replace some values\n>>> letters[2:5] = ['C', 'D', 'E']\n>>> letters\n['a', 'b', 'C', 'D', 'E', 'f', 'g']\n>>> # now remove them\n>>> letters[2:5] = []\n>>> letters\n['a', 'b', 'f', 'g']\n>>> # clear the list by replacing all the elements with an empty list\n>>> letters[:] = []\n>>> letters\n[]\n\n\u2022 nest lists:\n1\n2\n3\n4\n5\n6\n7\n8\n9\n>>> a = ['a', 'b', 'c']\n>>> n = [1, 2, 3]\n>>> x = [a, n]\n>>> x\n[['a', 'b', 'c'], [1, 2, 3]]\n>>> x[0]\n['a', 'b', 'c']\n>>> x[0][1]\n'b'\n\n\u2022 methods: print dir(list)\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n>>> a = [66.25, 333, 333, 1, 1234.5]\n>>> len(a)\n5\n>>> print a.count(333), a.count(66.25), a.count('x')\n2 1 0\n>>> a.insert(2, -1)\n>>> a.append(333)\n>>> a\n[66.25, 333, -1, 333, 1, 1234.5, 333]\n>>> a.index(333)\n1\n>>> a.remove(333)\n>>> a\n[66.25, -1, 333, 1, 1234.5, 333]\n>>> a.reverse()\n>>> a\n[333, 1234.5, 1, 333, -1, 66.25]\n>>> a.sort()\n>>> a\n[-1, 1, 66.25, 333, 333, 1234.5]\n>>> a.pop()\n1234.5\n>>> a\n[-1, 1, 66.25, 333, 333]\n\n##### Using Lists as Stacks\n\npush with list.append(), pop with list.pop():\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n>>> stack = [3, 4, 5]\n>>> stack.append(6)\n>>> stack.append(7)\n>>> stack\n[3, 4, 5, 6, 7]\n>>> stack.pop()\n7\n>>> stack\n[3, 4, 5, 6]\n>>> stack.pop()\n6\n>>> stack.pop()\n5\n>>> stack\n[3, 4]\n\n##### Using Lists as Queues\n\nLists are not efficient for queue. To implement a queue, use collections.deque(\u53cc\u7aef\u961f\u5217) which was designed to have fast appends and pops from both ends. enqueue with collections.deque.append(), dequeue with collections.deque.popleft():\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n>>> from collections import deque\n>>> queue = deque([\"Eric\", \"John\", \"Michael\"])\n>>> queue.append(\"Terry\") # Terry arrives\n>>> queue.append(\"Graham\") # Graham arrives\n>>> queue.popleft() # The first to arrive now leaves\n'Eric'\n>>> queue.popleft() # The second to arrive now leaves\n'John'\n>>> queue # Remaining queue in order of arrival\ndeque(['Michael', 'Terry', 'Graham'])\n\n##### filter(), map() and reduce()\n1. filter(function, sequence): Returns a sequence consisting of those items from the sequence for which function(item) is true. If sequence is a str, unicode or tuple, the result will be of the same type; otherwise, it is always a list. For example:\n\n1\n2\n3\n4\n>>> def f(x): return x % 3 == 0 or x % 5 == 0\n...\n>>> filter(f, range(2, 25))\n[3, 5, 6, 9, 10, 12, 15, 18, 20, 21, 24]\n\n2. map(function, sequence): Calls function(item) for each of the sequence\u2019s items and returns a list of the return values:\n\n1\n2\n3\n4\n>>> def cube(x): return x*x*x\n...\n>>> map(cube, range(1, 11))\n[1, 8, 27, 64, 125, 216, 343, 512, 729, 1000]\n\n\nMore than one sequence may be passed; the function must then have as many arguments as there are sequences and is called with the corresponding item from each sequence (or None if some sequence is shorter than another). For example:\n\n1\n2\n3\n4\n5\n>>> seq = range(8)\n>>> def add(x, y): return x+y\n...\n[0, 2, 4, 6, 8, 10, 12, 14]\n\n3. reduce(function, sequence): Returns a single value constructed by calling the binary function function on the first two items of the sequence, then on the result and the next item, and so on:\n\n1\n2\n3\n4\n...\n55\n\n\nIf there\u2019s only one item in the sequence, its value is returned; if the sequence is empty, an exception is raised.\n\nA third argument can be passed to indicate the starting value. In this case the starting value is returned for an empty sequence, and the function is first applied to the starting value and the first sequence item, then to the result and the next item, and so on:\n\n1\n2\n3\n4\n5\n6\n7\n8\n>>> def sum2(seq):\n...\n>>> sum2(range(1, 11))\n55\n>>> sum2([])\n0\n\n##### List Comprehensions\n1\n2\n3\n4\n5\n6\n>>> squares = []\n>>> for x in range(10):\n... squares.append(x**2)\n...\n>>> squares\n[0, 1, 4, 9, 16, 25, 36, 49, 64, 81]\n\n\nEqual to:\n\n1\nsquares = [x**2 for x in range(10)]\n\n\nAlso equal to:\n\n1\nsquares = map(lambda x: x**2, range(10))\n\n\nAnd a more complex example:\n\n1\n2\n>>> [(x, y) for x in [1,2,3] for y in [3,1,4] if x != y]\n[(1, 3), (1, 4), (2, 3), (2, 1), (2, 4), (3, 1), (3, 4)]\n\n\nAnd it\u2019s equivalent to:\n\n1\n2\n3\n4\n5\n6\n7\n8\n>>> combs = []\n>>> for x in [1,2,3]:\n... for y in [3,1,4]:\n... if x != y:\n... combs.append((x, y))\n...\n>>> combs\n[(1, 3), (1, 4), (2, 3), (2, 1), (2, 4), (3, 1), (3, 4)]\n\n\nIf the expression is a tuple (e.g. the (x, y) in the previous example), it must be parenthesized.\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n24\n25\n26\n27\n>>> vec = [-4, -2, 0, 2, 4]\n>>> # create a new list with the values doubled\n>>> [x*2 for x in vec]\n[-8, -4, 0, 4, 8]\n>>> # filter the list to exclude negative numbers\n>>> [x for x in vec if x >= 0]\n[0, 2, 4]\n>>> # apply a function to all the elements\n>>> [abs(x) for x in vec]\n[4, 2, 0, 2, 4]\n>>> # call a method on each element\n>>> freshfruit = [' banana', ' loganberry ', 'passion fruit ']\n>>> [weapon.strip() for weapon in freshfruit]\n['banana', 'loganberry', 'passion fruit']\n>>> # create a list of 2-tuples like (number, square)\n>>> [(x, x**2) for x in range(6)]\n[(0, 0), (1, 1), (2, 4), (3, 9), (4, 16), (5, 25)]\n>>> # the tuple must be parenthesized, otherwise an error is raised\n>>> [x, x**2 for x in range(6)]\nFile \"<stdin>\", line 1, in <module>\n[x, x**2 for x in range(6)]\n^\nSyntaxError: invalid syntax\n>>> # flatten a list using a listcomp with two 'for'\n>>> vec = [[1,2,3], [4,5,6], [7,8,9]]\n>>> [num for elem in vec for num in elem]\n[1, 2, 3, 4, 5, 6, 7, 8, 9]\n\n\nList comprehensions can contain complex expressions and nested functions:\n\n1\n2\n3\n>>> from math import pi\n>>> [str(round(pi, i)) for i in range(1, 6)]\n['3.1', '3.14', '3.142', '3.1416', '3.14159']\n\n\nThe initial expression in a list comprehension can be any arbitrary expression, including another list comprehension.\n\nConsider the following example of a 3x4 matrix implemented as a list of 3 lists of length 4:\n\n1\n2\n3\n4\n5\n6\n7\n>>> matrix = [\n... [1, 2, 3, 4],\n... [5, 6, 7, 8],\n... [9, 10, 11, 12],\n... ]\n>>> [[row[i] for row in matrix] for i in range(4)]\n[[1, 5, 9], [2, 6, 10], [3, 7, 11], [4, 8, 12]]\n\n\nEqual to:\n\n1\n2\n3\n4\n5\n6\n>>> transposed = []\n>>> for i in range(4):\n... transposed.append([row[i] for row in matrix])\n...\n>>> transposed\n[[1, 5, 9], [2, 6, 10], [3, 7, 11], [4, 8, 12]]\n\n\nAlso equal to:\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n>>> transposed = []\n>>> for i in range(4):\n... # the following 3 lines implement the nested listcomp\n... transposed_row = []\n... for row in matrix:\n... transposed_row.append(row[i])\n... transposed.append(transposed_row)\n...\n>>> transposed\n[[1, 5, 9], [2, 6, 10], [3, 7, 11], [4, 8, 12]]\n\n\nAnd almost equal to:\n\n1\n2\n>>> zip(*matrix)\n[(1, 5, 9), (2, 6, 10), (3, 7, 11), (4, 8, 12)]\n\n\n#### Tuples\n\nA tuple consists of a number of values separated by commas, for instance:\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n>>> t = 12345, 54321, 'hello!'\n>>> t[0]\n12345\n>>> t\n(12345, 54321, 'hello!')\n>>> # Tuples may be nested:\n... u = t, (1, 2, 3, 4, 5)\n>>> u\n((12345, 54321, 'hello!'), (1, 2, 3, 4, 5))\n>>> # Tuples are immutable:\n... t[0] = 88888\nTraceback (most recent call last):\nFile \"<stdin>\", line 1, in <module>\nTypeError: 'tuple' object does not support item assignment\n>>> # but they can contain mutable objects:\n... v = ([1, 2, 3], [3, 2, 1])\n>>> v\n([1, 2, 3], [3, 2, 1])\n\n\nTuples can contain mutable objects, such as lists(like above).\n\nCompare with Lists:\n\n\u2022 Tuples are immutable, and usually contain a heterogeneous(\u5f02\u8d28\u7684) sequence of elements that are accessed via unpacking or indexing (or even by attribute in the case of namedtuples)\n\u2022 Lists are mutable, and their elements are usually homogeneous(\u540c\u8d28\u7684) and are accessed by iterating over the list.\n\nA special problem is the construction of tuples containing 0 or 1 items(Carefully):\n\n1\n2\n3\n4\n5\n6\n7\n8\n>>> empty = ()\n>>> singleton = 'hello', # <-- note trailing comma\n>>> len(empty)\n0\n>>> len(singleton)\n1\n>>> singleton\n('hello',)\n\n\ntuple packing and sequence unpacking:\n\n1\n2\n>>> t = 12345, 54321, 'hello!'\n>>> x, y, z = t # t can also be replaced with [12345, 54321, 'hello!']\n\n\nNote that multiple assignment is really just a combination of tuple packing and sequence unpacking.\n\n### Unordered and Unique\n\n#### Sets\n\nA set is an unordered collection with no duplicate elements. Basic uses include membership testing and eliminating(\u6d88\u9664) duplicate entries. Set objects also support mathematical operations like union, intersection, difference, and symmetric(\u5bf9\u79f0) difference.\n\n{} or the set() function can be used to create sets. Note: to create an empty set you have to use set(), not {}(the latter creates an empty dictionary)\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n17\n18\n19\n20\n21\n22\n23\n>>> basket = ['apple', 'orange', 'apple', 'pear', 'orange', 'banana']\n>>> fruit = set(basket) # create a set without duplicates\n>>> fruit\nset(['orange', 'pear', 'apple', 'banana'])\n>>> 'orange' in fruit # fast membership testing\nTrue\n>>> 'crabgrass' in fruit\nFalse\n\n>>> # Demonstrate set operations on unique letters from two words\n...\n>>> b = set('alacazam')\n>>> a # unique letters in a\nset(['a', 'r', 'b', 'c', 'd'])\n>>> a - b # letters in a but not in b\nset(['r', 'd', 'b'])\n>>> a | b # letters in either a or b\nset(['a', 'c', 'r', 'd', 'b', 'm', 'z', 'l'])\n>>> a & b # letters in both a and b\nset(['a', 'c'])\n>>> a ^ b # letters in a or b but not both\nset(['r', 'd', 'b', 'm', 'z', 'l'])\n\n\nSimilarly to list comprehensions, set comprehensions are also supported:\n\n1\n2\n3\n>>> a = {x for x in 'abracadabra' if x not in 'abc'}\n>>> a\nset(['r', 'd'])\n\n\n#### Dictionaries\n\n*Dictionaries are indexed by keys, which can be any immutable types(strings, numbers and tuples that don\u2019t contain any mutable object).\n\nIt is best to think of a dictionary as an unordered set of key: value pairs, with the requirement that the keys are unique. A pair of braces creates an empty dictionary: {}. Placing a comma-separated list of key:value pairs within the braces adds initial key:value pairs to the dictionary; this is also the way dictionaries are written on output.\n\nThe keys() method of a dictionary object returns a list of all the keys used in the dictionary, in arbitrary order (if you want it sorted, just apply the sorted() function to it);\n\nThe values() method of a dictionary objects returns a list of all the values used in the dictionary.\n\nHere is a small example using a dictionary:\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n11\n12\n13\n14\n15\n16\n>>> tel = {'jack': 4098, 'sape': 4139}\n>>> tel['guido'] = 4127\n>>> tel\n{'sape': 4139, 'guido': 4127, 'jack': 4098}\n>>> tel['jack']\n4098\n>>> del tel['sape']\n>>> tel['irv'] = 4127\n>>> tel\n{'guido': 4127, 'irv': 4127, 'jack': 4098}\n>>> sorted(tel.keys())\n['guido', 'irv', 'jack']\n>>> 'guido' in tel\nTrue\n>>> tel.values()\n[4127, 4127, 4098]\n\n\nThe dict() constructor builds dictionaries directly from sequences of key-value pairs:\n\n1\n2\n>>> dict([('sape', 4139), ('guido', 4127), ('jack', 4098)])\n{'sape': 4139, 'jack': 4098, 'guido': 4127}\n\n\nIn addition, dict comprehensions can be used to create dictionaries from arbitrary key and value expressions:\n\n1\n2\n>>> {x: x**2 for x in (2, 4, 6)}\n{2: 4, 4: 16, 6: 36}\n\n\nWhen the keys are simple strings, it is sometimes easier to specify pairs using keyword arguments:\n\n1\n2\n>>> dict(sape=4139, guido=4127, jack=4098)\n{'sape': 4139, 'jack': 4098, 'guido': 4127}\n\n\n### Looping Techniques\n\nWhen looping through a sequence, the position index and corresponding value can be retrieved at the same time using the enumerate() function.\n\n1\n2\n3\n4\n5\n6\n>>> for i, v in enumerate(['tic', 'tac', 'toe']):\n... print i, v\n...\n0 tic\n1 tac\n2 toe\n\n\nTo loop over two or more sequences at the same time, the entries can be paired with the zip() function.\n\n1\n2\n3\n4\n5\n6\n7\n8\n>>> questions = ['name', 'quest', 'favorite color']\n>>> answers = ['lancelot', 'the holy grail', 'blue']\n>>> for q, a in zip(questions, answers):\n... print 'What is your {0}? It is {1}.'.format(q, a)\n...\nWhat is your name? It is lancelot.\nWhat is your quest? It is the holy grail.\nWhat is your favorite color? It is blue.\n\n\nTo loop over a sequence in reverse, first specify the sequence in a forward direction and then call the reversed() function.\n\n1\n2\n3\n4\n5\n6\n7\n8\n>>> for i in reversed(xrange(1,10,2)):\n... print i\n...\n9\n7\n5\n3\n1\n\n\nTo loop over a sequence in sorted order, use the sorted() function which returns a new sorted list while leaving the source unaltered.\n\n1\n2\n3\n4\n5\n6\n7\n8\n>>> basket = ['apple', 'orange', 'apple', 'pear', 'orange', 'banana']\n... print f\n...\napple\nbanana\norange\npear\n\n\nWhen looping through dictionaries, the key and corresponding value can be retrieved at the same time using the iteritems() method.\n\n1\n2\n3\n4\n5\n6\n>>> knights = {'gallahad': 'the pure', 'robin': 'the brave'}\n>>> for k, v in knights.iteritems():\n... print k, v\n...\nrobin the brave\n\n\nIt is sometimes tempting(\u8bf1\u4eba\u7684) to change a list while you are looping over it; however, it is often simpler and safer to create a new list instead.\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n>>> import math\n>>> raw_data = [56.2, float('NaN'), 51.7, 55.3, 52.5, float('NaN'), 47.8]\n>>> filtered_data = []\n>>> for value in raw_data:\n... if not math.isnan(value):\n... filtered_data.append(value)\n...\n>>> filtered_data\n[56.2, 51.7, 55.3, 52.5, 47.8]\n\n\n### The del statement\n\nThere is a way to remove an item from a list given its index instead of its value: the del statement. The del statement can also be used to remove slices from a list or clear the entire list. For example:\n\n1\n2\n3\n4\n5\n6\n7\n8\n9\n10\n>>> a = [-1, 1, 66.25, 333, 333, 1234.5]\n>>> del a[0]\n>>> a\n[1, 66.25, 333, 333, 1234.5]\n>>> del a[2:4]\n>>> a\n[1, 66.25, 1234.5]\n>>> del a[:]\n>>> a\n[]\n\n\ndel can also be used to delete entire variables:\n\n1\n>>> del a","date":"2020-09-26 18:04:42","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.24988961219787598, \"perplexity\": 7283.45076152637}, \"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-2020-40\/segments\/1600400244353.70\/warc\/CC-MAIN-20200926165308-20200926195308-00536.warc.gz\"}"} | null | null |
Star Wars is an American epic space opera media franchise, centred on a film series created by George Lucas. It depicts the adventures of characters "a long time ago in a galaxy far, far away".
The series has spawned an extensive media franchise including books, television series, computer and video games, theme park attractions and lands, and comic books, resulting in significant development of the series' fictional universe. Star Wars holds a Guinness World Records title for the "Most successful film merchandising franchise". In 2015, the total value of the Star Wars franchise was estimated at US$42 billion, making Star Wars the second-highest-grossing media franchise of all time.
In 2012, The Walt Disney Company bought Lucasfilm for US$4.06 billion and earned the distribution rights to all subsequent Star Wars films, beginning with the release of The Force Awakens in 2015. The former distributor, 20th Century Fox, was to retain the physical distribution rights for the first two Star Wars trilogies, was to own permanent rights for the original 1977 film and was to continue to hold the rights for the prequel trilogy and the first two sequels to A New Hope until May 2020. Walt Disney Studios currently owns digital distribution rights to all the Star Wars films, excluding A New Hope. On December 14, 2017, the Walt Disney Company announced its pending acquisition of 21st Century Fox, including the film studio and all distribution rights to A New Hope. | {
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{"url":"https:\/\/adel.princeton.edu\/","text":"My current research focuses on studying rotational dynamic in neural networks, natural and artificial. I am interested in using concepts and techniques from Lie theory and linear algebra to understand the geometry of neuronal transformation spaces, how they can be differentiated for various classes of tasks (specifically in the context of working memory) and how they evolve over time. We hope that the results of our studies open new horizons in understanding brain, as well as novel approaches for studying deep neural networks for AI.\n\nPrior to Princeton, I was at Zuckerman Mind Brain Behavior Institute, where I studied how neural networks represent the concepts they have learned. Specifically at the Qian Lab, we were studying such representations and their various topological and geometrical properties in recurrent neural networks of firing rate cells, specifically in the context of visual perception and its relationship with (working) memory.\n\nDuring my PhD, I worked on large-scale human-in-the-loop data analytics (HILDA). My research interests include a wide range of data management and analysis topics, from building frameworks and tools for large-scale HILDA to applications of data management techniques on Big Data. I worked as a part of AnHai Doan's group. We build frameworks for the next generation of entity matching, data integration and data cleaning tools.\n\nHere's my CV (as of Apr 2022).\n\nPh.D. in Computer Sciences, University of Wisconsin-Madison (2018).\nM.Sc. in Computer Sciences, University of Wisconsin-Madison (2015).\nM.Sc. in Information Technology Engineering, University of Tehran, Iran (2008).\nB.Sc. in Software Engineering, University of Tehran, Iran (2005).\n\n-----BEGIN GEEK CODE BLOCK-----\nVersion: 3.1\nGCS\/MU d-(+)@x s+: a C++$ULC++(+++)$\nP+ L++(++++) !E !W++ !N !o K--? w++\n!O M+ !V PS+ PE- Y+ !PGP !t !5 X- R\ntv+ b+ DI D+ G e++++ h r y++\n------END GEEK CODE BLOCK------","date":"2022-07-07 13:08:30","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\": 0.38855403661727905, \"perplexity\": 2572.2389967381405}, \"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-2022-27\/segments\/1656104692018.96\/warc\/CC-MAIN-20220707124050-20220707154050-00689.warc.gz\"}"} | null | null |
\section{Protasov-Comfort problem on minimally almost periodic group topologies}
Our main results provide a solution to Protasov--Comfort problem on minimally almost periodic group topologies on the abelian groups.
For the sake of completeness, we recall the relevant pair of notions, due to von Neumann \cite{vN}, in the abelian case.
A {\em character} of a topological abelian group $G$ is a continuous homomorphism $G \to \T$. A topological abelian group $G$ is called
\begin{itemize}
\item[(a)] {\em minimally almost periodic}, if every character $G \to \T$ is trivial;
\item[(b)] {\em maximally almost periodic}, if the characters $G \to \T$ separate the points of $G$.
\end{itemize}
It is easy to see that every abelian group admits a maximally almost periodic group topology (e.g., the discrete one).
Examples of minimally almost periodic abelian groups are not easy to come by. The first known examples were those borrowed from Analysis, namely vector topological spaces without non-trivial continuous functionals (more precisely, the underlying topological abelian group of such a space turns out to be minimally almost periodic \cite[Section 23.32]{HR}, example of vector spaces with this property can be found in \cite{Day}).
Nienhuys \cite{N} built a solenoidal (so, of size at least $\cont$) and monothetic minimally almost periodic group, which yields the existence of a minimally almost periodic group topology also on $\Z$ (see Section \ref{Nienhyus:section} for further details). All these examples are connected.
To the best of our knowledge, the first explicit and also quite simple example of a countable \map\ group was given by Prodanov \cite{Pr}. Later, Ajtai, Havas and Koml\' os \cite{AHK} provided minimally almost periodic group topologies on $\Z$ and some countably infinite direct sums of simple cyclic groups, deducing that every abelian group admits a group topology that is not maximally almost periodic.
In his review to their paper Protasov \cite{P} posed the natural question of whether {\em every} infinite abelian group admits a minimally almost periodic group topology.
In September 1989 Remus noticed the following counter-example to this question.
\begin{example}\label{Remus:example} (Remus)
{\em The group $ G = \Z(2)\times \Z(3)^\omega$ does not admit any \map\ group topology\/}
since the projection $p: \Z(2)\times \Z(3)^\omega\to \Z(2)$ is
a continuous character for every Hausdorff group topology on $G$.
Indeed, $\ker p = \{x\in G: 3x = 0\}$ is closed (and hence, also open) in every Hausdorff group topology on $G$.
\end{example}
Motived by this example, Comfort \cite[Question 3J.1]{Co} modified the original Protasov's question to the following
\begin{problem}\cite[Question 521]{Co}\label{Q:C}
Does every Abelian group which is not of bounded order admit a minimally almost periodic topological group topology? What about the countable case?
\end{problem}
Zelenyuk and Protasov \cite{ZP} introduced a new technique for building minimally almost periodic group topologies
on countable groups, using $T$-sequences. Applying this method, Gabriyelyan \cite{G3} obtained a description of the bounded abelian groups
admitting a minimally almost periodic group topology, by essentially showing that these are precisely the bounded groups whose leading Ulm-Kaplansky invariants are infinite.
Gabriyelyan \cite[Theorem 2]{G1} also proved that all finitely generated infinite abelian groups admit a minimally almost periodic group topology, and he recently extended this to all countably infinite unbounded groups \cite{G3}, resolving the second part of Question \ref{Q:C} (a short and self-contained proof of this result is given in Lemma \ref{Saak}). Nevertheless, Problem \ref{Q:C} remained open for all uncountable groups.
\section{Necessary conditions for the existence of a minimal almost periodic group topology}
\begin{proposition}\label{Necessary:condition} If $G$ is a minimally almost periodic abelian group, then every continuous homomorphism $f:G\to K$ of $G$ into a compact group $K$ is trivial. In particular, every proper closed subgroup $H$ of $G$ has infinite index in $G$.
\end{proposition}
Markov \cite{Mar1,Mar} says that subset $X$ of a group $G$ is {\em unconditionally closed\/} in $G$ if $X$ is closed in every Hausdorff group topology on $G$.
\begin{corollary}
\label{cor:necessary:condition}
If an abelian group $G$ admits a minimally almost periodic abelian group topology, then every proper unconditionally closed subgroup $H$ of $G$ has infinite index in $G$.
\end{corollary}
This corollary explains why the group $G$ from Example \ref{Remus:example} does not admit a minimally almost periodic group topology. Indeed, the subgroup $G[3]$ is unconditionally closed in $G$ and has index $2$.
It is natural to ask if the converse to Corollary \ref{cor:necessary:condition} also holds.
\begin{problem}
\label{Main:question}
If all proper unconditionally closed subgroups of a group $G$ have infinite index, does then $G$ admit a minimally almost periodic group topology?
\end{problem}
Due to the fundamental fact that all unconditionally closed subsets of abelian groups are algebraic \cite{DS-relection, DS_JA}, these sets are precisely the closed sets of the Zariski topology $\Zar_G$ of $G$ \cite{DS_JA}, having as closed sets the algebraic sets of $G$. Following \cite{DT}, call a group $G$ {\em Zariski-connected} (briefly, {\em $\Zar$-connected}) if $(G,\Zar_G)$ is connected.
Since the notion of an unconditionally closed subset of a group $G$ involves checking closedness of this set in {\em every\/} Hausdorff group topology on $G$, in practice it is hard to decide if a given subgroup of $G$ is unconditionally closed in $G$ or not. Our next proposition provides two equivalent conditions.
\begin{proposition}
\label{reformulation:of:Markov:condition} For an abelian group $G$, the following conditions are equivalent:
\begin{itemize}
\item[(a)] $G$ is {$\Zar$-connected};
\item[(b)] all proper unconditionally closed subgroups of $G$ have infinite index;
\item[(c)] for every $m\in\N$, either $mG =\{0\}$ or $|mG| \ge \omega$.
\end{itemize}
\end{proposition}
\begin{proof}
According to \cite{DS-relection, DS_JA}, a proper unconditionally closed subgroup $H$ of $G$ has the form $H = G[m]$ for some $m>0$. Since
$G/G[m]\cong mG$, the index of $H$ in $G$ coincides with $|G/H| = |G/G[m]|= |mG|$. This proves the equivalence of (b) and (c).
The equivalence of (a) and (b) is established in \cite[Theorem 4.6]{DS_JA}.
\end{proof}
If an abelian group $G$ is not bounded torsion, then
$G$ satisfies item (c) of Proposition \ref{reformulation:of:Markov:condition}, applying which we conclude that all proper unconditionally closed subgroups of $G$ have infinite index.
This shows that Problem \ref{Q:C} is a particular case of the more general Problem \ref{Main:question}.
The equivalence of items (a) and (b) in Proposition \ref{reformulation:of:Markov:condition} allows one to reformulate Problem \ref{Main:question} is follows: {\em Does every abelian $\Zar$-connected group admit a minimally almost periodic group topology?\/}
Based on Proposition \ref{reformulation:of:Markov:condition}, one can re-formulate the property of $\Zar$-connectedness for abelian groups of finite exponent
in terms of their Ulm-Kaplanski invariants:
\begin{proposition}
\label{Kirku*}
A non-trivial abelian group $G$ of finite exponent satisfies one (and then all) of the equivalent conditions of
Proposition \ref{reformulation:of:Markov:condition} if and only if all leading Ulm-Kaplanski invariants of $G$ are infinite.
\end{proposition}
\begin{proof} A proof of this statement can be found in \cite{CD} (more precisely, this is the equivalence (a$_2$) and (a$_3$) of \cite[Lemma 2.13]{CD}). For the sake of reader's convenience we include a proof here.
Write the group $G$ as in \eqref{eq:1}, and let $k = \prod_{q\in \pi(G)} q^{m_q}$ be the exponent of $G$. In order to compute the leading Ulm-Kaplanski invariant $\alpha_{p,m_p}$ for $p\in \pi(G)$, let
$$
k_p = \frac{k}{p}= p^{m_p-1}\cdot \prod_{q\in \pi(G)\setminus \{p\}} q^{m_q}.
$$
Then $k_pG \cong \Z(p)^{(\alpha_{p,m_p})}$, so
\begin{equation}
\label{eq:2}
1< |k_pG|=\begin{cases} p^{\alpha_{p,m_p}}\ \mbox{ if $\alpha_{p,m_p}$ is finite}\\
\alpha_{p,m_p}
\ \mbox{ if $\alpha_{p,m_p}$
is infinite}.
\end{cases}
\end{equation}
If $G$ satisfies item (c) of Proposition \ref{reformulation:of:Markov:condition}, then $|k_pG | \ge \omega$ by \eqref{eq:2}, so $\alpha_{p,m_p}\ge \omega$ as well.
Now suppose that all leading Ulm-Kaplanski invariants of $G$ are infinite.
Fix $m\in \N$ with $|mG| > 1$. There exists at least one $p \in \pi(G)$, such that $p^{m_p}$ does not divide $m$. Let $d$ be the greatest common divisor of $m$ and $k$. Then $mG = dG$, hence from now on we can assume without loss of generality that $m=d$ divides $k$. As $p^{m_p}$ does not divide $m$, it follows that $m$ divides $k_p$. Therefore, $k_p G$ is a subgroup of $mG$, and so $|mG| \geq |k_p G| = \alpha_{p,m_p} \geq \omega$ by \eqref{eq:2}.
This shows that, for every $m\in\N$, either $mG =\{0\}$ or $|mG| \ge \omega$. Therefore, $G$ satisfies item (c) of Proposition \ref{reformulation:of:Markov:condition}.
\end{proof}
\section{Characterization of abelian groups admitting a \map\ group topology}
\label{sec:main:results}
Our first theorem provides a final solution of Problem \ref{Q:C}.
\begin{theorem}\label{CH:theorem} Every unbounded abelian group admits a \map \ group topology.
\end{theorem}
The proof of Theorem \ref{CH:theorem} is postponed until Section \ref{proofs}.
The complementary bounded case is covered by the next theorem.
\begin{theorem} \label{new:corollary1}
For a bounded abelian group $G$ the following are equivalent:
\begin{itemize}
\item[(a)] $G$ admit a \map \ group topology;
\item[(b)] $G$ is {$\Zar$-connected};
\item[(c)] all leading Ulm-Kaplanskly invariants of $G$ are infinite.
\end{itemize}
\end{theorem}
The
equivalence of (a) and (c)
characterization of bounded abelian groups admitting a \map\ group topology
has been recently obtained by
Gabriyelyan \cite[Corollary 1.4]{G3}.
Since our proof of Theorem \ref{new:corollary1} based on the same ideas as that of Theorem \ref{CH:theorem} is also much shorter and simpler than Gabriyelyan's proof, we decided to provide it in Section \ref{proofs0} for the sake of completeness.
Our third theorem unifies Theorems
\ref{CH:theorem}
and
\ref{new:corollary1}, thereby providing a final solution to the general Problem \ref{Main:question}.
\begin{theorem}\label{main:theorem}
For an abelian group an abelian group $G$, the following are equivalent:
\begin{itemize}
\item[(a)] $G$ admits a \map \ group topology;
\item[(b)] $G$ is {$\Zar$-connected};
\item[(c)] all proper unconditionally closed subgroups of $G$ have infinite index;
\item[(d)] for every $m\in\N$, either $mG =\{0\}$ or $|mG| \ge \omega$.
\end{itemize}
\end{theorem}
\begin{proof}
Indeed, if $G$ is unbounded, then Theorem \ref{CH:theorem} and Proposition \ref{reformulation:of:Markov:condition} apply. If $G$ is bounded, then
Proposition \ref{reformulation:of:Markov:condition}, Proposition \ref{Kirku*} and Theorem \ref{new:corollary1} apply.
\end{proof}
Theorem \ref{main:theorem} answers positively also the following weaker versions of Problem \ref{Main:question} that also remained open:
\begin{itemize}
\item[(a)] \cite[Question 3.10]{G3} If an unbounded abelian group $G$ admits a compact group topology, does $G$ admit also a \map \ group topology?
\footnote{The group $ G = \Z(2)\times \Z(3)^\omega$ from Example \ref{Remus:example} obviously carries a compact group topology. This explains the restriction of unboundedness of $G$ in item (a).} What about the groups $\prod_p\Z(p)$, or $\Delta_p$, where $\Delta_p$ is the group of $p$-adic integers ?
\item[(b)] \cite[Question 4.1]{G3} Let $G$ be an uncountable abelian group of infinite exponent (for example, $G$ is an uncountable torsion-free abelian group). Does $G$
admit a \map \ group topology?
\item[(c)] \cite[Question 4.5]{G3} Does every uncountable indecomposable Abelian group admit
a \map \ group topology?
\end{itemize}
\medskip
Finally, Theorem \ref{CH:theorem} answers also the following question from \cite{G0,G3} generalizing Problem \ref{Q:C}:
\begin{itemize}
\item[(q)] \cite[Question 4.4]{G3} (also \cite[Problem 5]{G0})
Describe all infinite abelian groups $G$ that admit a \map \ group topology.
\end{itemize}
\medskip
In \cite{G}, the following two questions were raised. Let $H$ be a normal subgroup of a group $G$.
\begin{itemize}
\item[(1)] If we are given minimally almost periodic topologies for $H$ and $G/H$, is there always a minimally almost periodic topology for $G$ which generates those topologies on $H$ and $G/H$?
\item[(2)] If both $H$ and $G/H$ admit minimally almost periodic topologies, must $G$ admit a minimally almost periodic topology?
\end{itemize}
Our last theorem negatively answers both questions. Indeed, item (c) of the next theorem provides a stronger negative answer to (1), while
the conjunction of items (a) and (b) of this theorem obviously answers item (2) in the negative.
\begin{theorem}\label{Frank}
There exists a countable abelian group $G$ with a subgroup $H$ having the following properties:
\begin{itemize}
\item[(a)] $G$ does not admit a \map\ group topology;
\item[(b)] both $H$ and $G/H\cong H$ admit \map\ group topologies;
\item[(c)] there does not exist any group topology $\tau$ on $G$ such that both $H$ and $G/H$, equipped with the induced and the quotient topology of $\tau$ respectively, can be simultaneously minimally almost periodic.
\end{itemize}
\end{theorem}
\begin{proof}
Let $V = \Z(2)^{(\omega)}\oplus \Z(4)$, $G = \Z(2)^{(\omega)}\oplus V$ and $H = V[2]$.
(a) Since $G[2]=V[2]$ has index $2$ in $G$, it follows from Proposition \ref{Necessary:condition}
that $G$ admits no \map \ group topologies.
(b) Obviously, $H \cong G/H \cong\Z(2)^{(\omega)}$. Applying Theorem \ref{new:corollary1}, we conclude that $H \cong G/H$ admits a \map\ group topology.
(c) Assume that for some group topology $\tau$ on $G$ both $H$ and $G/H$ (equipped with the induced and the quotient topology of $\tau$, respectively) are
minimally almost periodic. Then $\tau$ itself is minimally almost periodic.
(A proof can be found, for example, in \cite[Theorem 2.2.10]{G}.) \footnote{For reader's convenience we propose
a short argument here. If $\chi: G\to \T$ is a character, then $\chi\restriction_H$ must be trivial by hypothesis. Therefore, there exists a character $\xi : G/H \to \T$ such that $\chi = \xi \circ q$, where $q: G\to G/H$ is the canonical homomorphism. As $\xi$ is trivial again by hypothesis, we conclude that $\chi$ is trivial.
This proves that $(G,\tau)$ is \map.} This contradicts item (a).
\end{proof}
\section{The realization problem of von Neumann's kernel}
For a topological abelian group $G$, the {\em von Neumann kernel} $n(G)$ of $G$ is the subgroup of all points of $G$ where each character of $G$ vanish.
Clearly, $G$ is minimally almost-periodic (maximally almost-periodic) precisely when $n(G) = G$ ($n(G) = \{0\}$, respectively).
\begin{definition}\label{Definition:potentially:vN}
Let $H$ be a subgroup of an abelian group $G$. We say that $H$ is a {\em potential von Neumann kernel} of $G$, if there exists a Hausdorff group topology $\tau$ on $G$ such that $n(G,\tau) = H$.
\end{definition}
The following problem, raised in \cite{G0}, is a generalization of the problem of finding a \map\ group topology on an abelian group considered in Section \ref{sec:main:results}; indeed, an abelian group $G$ admits a \map\ group topology if and only if $G$ is \pvN\ of itself.
\begin{problem}\label{Gen:Prob}\cite{G0}
Describe all potential von Neuman kernels of a given abelian group $G$.
\end{problem}
Gabriyelyan
resolved this problem for bounded abelian groups.
\begin{theorem}\label{corollary2} \cite[Theorem 1.2]{G3}
Let $G$ be a bounded abelian group and let $H$ be a subgroup of $G$. Then $H$ is a \pvN \ of $G$ if and only if
$G$ contains $\Z(k)^{(\omega)}$, where $k = exp(H)$ is the exponent of $H$.
\end{theorem}
Furthermore,
Gabriyelyan
resolved the following particular case of Problem \ref{Gen:Prob}.
\begin{theorem}
\cite[Theorem 1.3]{G3}
\label{Gab-2}
Every
bounded subgroup of an unbounded abelian group $G$ is a \pvN \ of $G$.
\end{theorem}
The following easy lemma
is
helpful
for
finding
a
necessary condition that all \pvN s must satisfy.
\begin{lemma}\label{very:easy:lemma} The \vNk\ of a topological group $G$ is contained in every open subgroup of $G$ and contains every \map\ subgroup of $G$.
\end{lemma}
\begin{proof}
If $H$ is an open subgroup of $G$, then $G/H$ is discrete, so it is maximally almost periodic. Since the characters of $G/H$ separate points of $G/H$,
we get $n(G)\subseteq H$. The second assertion is obvious.
\end{proof}
\begin{lemma}\label{lemma:necessary:condition} All \pvN s of an abelian group $G$ are contained in the intersection of all unconditionally closed subgroups of $G$ of finite index.
\end{lemma}
\begin{proof}
Let $N$ be an unconditionally closed subgroup of finite index of $G$ and let $H$ be a \pvN. To see that
$N \subseteq H$ equip $G$ with the group topology $\tau$ witnessing $H = n(G,\tau)$. Then $N$ is $\tau$-clopen,
so $H = n(G,\tau)\subseteq N$ by Lemma \ref{very:easy:lemma}.
\end{proof}
The above lemma makes it important to compute the intersection of all unconditionally closed subgroups of finite index of an abelian group $G$.
To this end we need the following definition.
\begin{definition}\label{lemma2}\cite[Definition 4.3]{DS_JA}
Let $G$ be an abelian group.
\begin{itemize}
\item[(i)] If $G$ is bounded, then the {\em essential order} $eo(G)$ of $G$ is the smallest positive integer $n$ such that $nG$ is finite.
\item[(ii)] If $G$ is unbounded, we define $eo(G) = 0$.
\end{itemize}
\end{definition}
The notion of the essential order of a bounded abelian group $G$, as well as the notation $eo(G)$, are due to Givens and Kunen \cite{GK}, although the definition in \cite{GK} is different (but equivalent) to this one.
\begin{theorem}\label{Theorem:JA}\cite[Theorem 4.6]{DS_JA}
Let $G$ be an abelian group and $n = eo(G)$. Then:
(i) $G[n]$ is a $\Zar_G$-clopen subgroup of $G$,
(ii) $G[n]$ coincides with the connected component $c_\Zar(G)$ of $0$ in $(G,\Zar_G)$.
\end{theorem}
By the definition of $eo(G)$, the subgroup $G[n]$ has finite index, and this is the smallest subgroup of the form $G[m]$ that has finite index.
So $c_\Zar(G) = G[n]$ coincides with the intersection of all unconditionally closed subgroups of $G$ of finite index.
Therefore,
from Lemma \ref{lemma:necessary:condition} and Theorem \ref{Theorem:JA} one obtains the following
necessary condition for a subgroup to be a \pvN.
\begin{corollary}\label{corollary3*}
Every
\pvN \ $H$ of an abelian group $G$
satisfies $H\subseteq c_\Zar(G)$.
\end{corollary}
In the next lemma we collect several equivalent forms of the necessary condition $H\subseteq c_\Zar(G)$ in the case when $G$ is bounded.
\begin{lemma}\label{lemma3} Let $G$ be a bounded abelian group and $m = eo(G)$. For every subgroup $H$ of $G$ TFAE:
(a) $H\subseteq c_\Zar(G) \ (=G[m]) $;
(b) $exp(H) | m$; or equivalently, $mH = 0$;
(c) $G$ contains $\Z(k)^{(\omega)}$, where $k = exp(H)$.
\end{lemma}
\begin{proof}
(a) and (b) are obviously equivalent. By \cite[Proposition 4.12]{DS_JA}, (b) is equivalent to (c).
\end{proof}
Our main theorem in this section shows that this necessary condition from Corollary \ref{corollary3*} is also sufficient for the realization of the \vNk.
\begin{theorem}\label{Main:Conjecture} A subgroup $H$
of an abelian group $G$ is a \pvN \ of $G$ if and only if $H\subseteq c_\Zar(G)$.
\end{theorem}
\begin{proof}
The necessity was proved in Corollary \ref{corollary3*}.
To prove the sufficiency, assume that $H\subseteq c_\Zar(G)$ and consider two cases.
\smallskip
{\em Case 1\/}. {\sl $H$ is bounded.\/}
If $G$ is unbounded,
then
$H$ is a potential \pvN\
by Theorem \ref{Gab-2}.
Suppose now that $G$ itself is bounded.
Since $H\subseteq c_\Zar(G)$ by our assumption, the implication (a)$\to$(c) of
Lemma \ref{lemma3}
allows us to conclude that
$G$ contains $\Z(k)^{(\omega)}$,
where $k=exp(H)$.
Now $H$ is a \pvN\ of $G$ by
Theorem \ref{corollary2}.
\smallskip
{\em Case 2\/}. {\sl $H$ is not bounded.\/}
In this case, we apply Theorem \ref{CH:theorem} to find
a \map \ group topology $\tau$ on $H$. Extend $\tau$ to a Hausdorff group topology
$\tau^*$ on $G$ by taking as a base of $\tau^*$ all translates $g+U$, where $g\in G$ and $U$ is a non-empty $\tau$-open subset of $H$. Since $H$ is $\tau^*$-open and $(H,\tau)$ is \map,
Lemma \ref{very:easy:lemma} implies $H = n(G,\tau^*)$.
Therefore, $H$ is a \pvN\ of $G$.
\end{proof}
Theorem \ref{Main:Conjecture} allows us to answer also the following
questions of Gabriyelyan that were open.
\begin{itemize}
\item[(a)] \cite[Question 4.2]{G3} If $H$ is a \pvN\ of an abelian group $G$, are then all subgroups of $H$ still \pvN s of $G$? In particular, if an unbounded
abelian group $G$ admits a \map\ group topology, is then every subgroup of $G$ a \pvN?
\item[(b)] \cite[Question 4.3]{G3} (also \cite[Problem 4]{G0})
Describe all infinite abelian groups $G$ such that every subgroup of $G$ is a \pvN.
\end{itemize}
Indeed, the affirmative answer to (a) immediately follows from Theorem \ref{Main:Conjecture}. This implies also an immediate answer of (b),
these are precisely the groups that admit a \map\ group topology.
The paper is organized as follows. In Section \ref{HM:construction} we recall some of the properties of the The \HM\ functorial construction of a pathwise connected and locally pathwise connected group $\HMi(G)$ depending on an arbitrary topological (abelian) group $G$.
In Section \ref{Sec:6} we collect necessary background on \map\ groups.
In Section \ref{proofs0} we
provide
the proof of Theorem \ref{new:corollary1}.
In Section \ref{extension:section} we consider extension of monomorphisms into $\HMi(\T)$.
In Section \ref{Dense:emd:HM(T)} we show that certain unbounded abelian groups (the Pr\"ufer groups and infinite direct sums of cyclic groups) admit a dense embedding in the group $\HMi(\T)$.
Using these embeddings and the fact that the group $\HMi(\T)$ is minimal almost periodic (Corollary \ref{HM(T)isMinAP}), we
resolve the countable case of Protasov-Comfort's problem in
Section \ref{Sec:10} and the general case
in Section \ref{proofs}.
\section{The \HM\ construction}
\label{HM:construction}
\label{Sec:5}
Let $G$ be a group, and let $I$ be the unit interval $[0,1]$. As usual, $G^I$ denotes the set of all functions from $I$ to $G$. Clearly, $G^I$ is a group under coordinate-wise operations. For $g\in G$ and $t\in (0,1]$ let $g_t\in G^I$ be the function defined by
$$
g_t(x)=
\left\{
\begin{array}{rl}
g & \mbox{if } x< t \\
e & \mbox{if } x\ge t,
\end{array}
\right.
$$
where $e$ is the identity element of $G$. Note that $G_t=\{g_t:g\in G\}$ is a subgroup of $G^I$ isomorphic to $G$ for each $t\in (0,1]$. Therefore, $\HMi(G)=\sum_{t\in (0,1]} G_t$ is a subgroup of $G^I$. It is straightforward to check that this sum is direct, so that
\begin{equation}
\label{direct:decomposition:of:HM(G)}
\HMi(G)=\bigoplus_{t\in (0,1]} G_t.
\end{equation}
When $G$ is a topological group, Hartman and Mycielski \cite{HM} equip $\HMi(G)$ with a topology making it pathwise connected and locally pathwise connected. Let $\mu$ be the standard probability measure on $I$. The {\em \HM\ topology\/} on the group $\HMi(G)$ is the topology generated by taking the family of all sets of the form
\begin{equation}
\label{basic:nghb:in:HM}
O(U,\varepsilon)=\{g\in G^I: \mu(\{t\in I: g(t)\not\in U\})<\varepsilon,
\end{equation}
where $U$ is an open neighbourhood $U$ of the identity $e$ in $G$ and $\varepsilon>0$, as the base at the identity function of $\HMi(G)$.
The next lemma lists three properties of the functor $G\mapsto \HMi(G)$ that are needed for our proofs.
\begin{lemma}
\label{Dima}
\begin{itemize}
\item[(i)] For every group $G$, the group $\HMi(G)$ is pathwise connected and locally pathwise connected.{HM}
\item[(ii)] If the group $G$ is divisible and abelian, then so is $\HMi(G)$.
\item[(iii)] If the group $G$ is first countable, then so is $\HMi(G)$.
\end{itemize}
\end{lemma}
\begin{proof}
(i) is proved in \cite{HM}; see also
\cite{DS_ConnectedMarkov}.
(ii) Since $G_t\cong G$ for each $t\in (0,1]$, from \eqref{direct:decomposition:of:HM(G)} we conclude that $\HMi(G)\cong G^{(\cont)}$. It follows from this that $\HMi(G)$ is divisible and abelian whenever $G$ is.
(iii) If $G$ is first countable, then the definition of the topology of $\HMi(G)$ easily implies that $\HMi(G)$ is first countable as well.
\end{proof}
In order to investigate the fine structure of the topological group $\HMi(G)$, we need an additional notation. Let $D$ be a subgroup of $G$. For each $t\in(0,1]$, $D_t=\{d_t:d\in D\}$ is a subgroup of $G_t$. Recalling \eqref{direct:decomposition:of:HM(G)}, we conclude that, for each non-empty subset $S$ of $(0,1]$, the sum
\begin{equation}
\label{decomposition:H(D,S)}
H(D,S)=\bigoplus_{t\in S} D_t
\end{equation}
is direct. Note that $\HMi(G)=H(G,(0,1])$.
\begin{lemma}\label{HM:groups}
Let $G$ be a topological group.
\begin{itemize}
\item[(i)] If $D$ is a subgroup of $G$ and $S$ is a non-empty subset of $(0,1]$, then the subgroup $H(D,S)$ is
algebraically
isomorphic to $D^{(S)}$; in particular, if $S$ is countable, then $H(D,S)\cong D^{(\omega)}$.
\item[(ii)] If $D$ is a dense subgroup of $G$ and $S$ is a dense subset of $(0,1]$, then the subgroup $H(D,S)$ of $\HMi(G)$ is dense in $\HMi(G)$.
\item[(iii)] If $D$ is a dense subgroup of $G$, then $\HMi(D)$ is a dense subgroup of $\HMi(G)$.
\end{itemize}
\end{lemma}
\begin{proof} (i) Since $D_t\cong D$ for each $t\in (0,1]$, from \eqref{decomposition:H(D,S)} we conclude that the subgroup $H(D,S)$ of $\HMi(G)$ is isomorphic to
$D^{(S)}$.
To prove (ii),
take an arbitrary element $x$ of $\HMi(G)$ and its open neighbourhood $U$. Fix a canonical representation $x=\sum_{i=1}^n x_i$, where $x_i =(g_i)_{t_i} \in G_{t_i}$ for some $g_i\in G$ and $t_i \in [0,1]$.
First, we use continuity of the group operation to choose open neighborhoods $V_i$ of each $x_i$ of the form $V_i= x_i + O(W_i,\varepsilon_i)$
such that $V_1 +\ldots + V_n\subseteq U$.
Second, we use density of $S$ in $[0,1]$ to pick $s_i\in (t_i-\varepsilon_i,t_i+\varepsilon_i)\cap S$ for all $i=1,\dots,n$.
Third, we use density of $D$ in $G$ to find $d_i\in (g_i+W_i)\cap D$ for all $i=1,\dots,n$.
Finally, we define $h_i:=(d_i)_{s_i}\in D_{s_i}$ for all $i=1,\dots,n$.
Since $s_1,s_2,\dots,s_n\in S$, it follows from \eqref{decomposition:H(D,S)} that
$h=\sum_{i=1}^n h_i\in H(D,S)$. Since
$h_i - x_i \in O(W_i,\varepsilon_i)$, we have $h_i\in V_i$ ($i=1,\dots,n$). Thus,
$h\in \sum_{i=1}^n V_i\subseteq U$.
Therefore, $h\in H(D,S)\cap U\not=\emptyset$.
Since $\HMi(D)=H(D,(0,1])$, (iii) follows from item (ii), in which one takes $S=(0,1]$.
\end{proof}
\begin{corollary}\label{corollary:Moscow2}
If $D$ is a dense subgroup of a topological group $G$, the group $\HMi(G)$ contains a dense subgroup isomorphic to $D^{(\omega)}$. In particular,
\begin{itemize}
\item[(a)] $\HMi(G)$ is separable, whenever $G$ is separable.
\item[(b)] $\HMi(G)$ is second countable, whenever $G$ is second countable.
\end{itemize}
\end{corollary}
\begin{proof} Let $S$ be an arbitrary countable dense subset of $(0,1]$. It follows from item (i) of Lemma \ref{HM:groups}
that $H(G,S)$ is isomorphic to $G^{(\omega)}$. Finally, the subgroup $H(G,S)$ of $\HMi(G)$ is dense in $\HMi(G)$ by item (ii) of Lemma \ref{HM:groups}.
(a) follows directly from the first assertion of the corollary.
(b) If $G$ is second countable, then $G$ is first countable and separable. By Lemma \ref{Dima}(iii), $\HMi(G)$ is first countable. Being a topological group, it is metrizable. On the other hand, (a) implies that $G$ is separable. Therefore, $\HMi(G)$ is second countable as well.
\end{proof}
\begin{corollary}
\label{HM(T):has:countable:base}
$ \HMi(\T)$ has a countable base.
\end{corollary}
\section{Background on \map\ groups}
\label{Sec:6}
In the next lemma we collect some permanence properties of the minimally almost periodic groups.
\begin{lemma} \label{easy:lemma}
The property of being minimally almost periodic is preserved by taking direct products, direct sums, dense subgroups and completions. In particular, if each $G_i$ admits a minimally almost periodic group topology, then also $\bigoplus_{i\in I} G_i$ and $\prod_{i\in I} G_i$ have the same property.
\end{lemma}
The next immediate corollary will be our main tool for producing \map \ group topologies on the abelian groups.
\begin{corollary}\label{easy:corollary} If an abelian group $G$ can be densely embedded into a \map \ group, then $G$ admits a \map \ group topology.
\end{corollary}
The next lemma shows that under a (mild) algebraic restraint on the group $G$ all connected group topologies of $G$ are minimally almost periodic.
\begin{lemma}\label{lemma1} Let $G$ be a connected abelian group with $r(G)<\cont$. Then $G$ is minimally almost periodic.
\end{lemma}
\begin{proof} Let $\chi: G\to T$ be a character of $G$. Then $r(\chi(G)) \le r(G) < \cont = r(\T)$. Hence, $\chi(G) \ne \T$. On the other hand, $\chi(G)$ is a connected subgroup of $\T$, so either $\chi(G) = \T$ or $\chi(G) = \{0\}$. This proves that $\chi(G) = \{0\}$; that is, $\chi = 0$. Therefore, $G$ is minimally almost periodic.
\end{proof}
For every abelian group $G$, the group $\HMi(G)$ is known to be \map; see \cite{DW}. We shall only need this fact for $G=\T$, as well as for torsion groups $G$.
This is why we decided to offer a short alternative proof of minimal almost periodicity of $\HMi(G)$ in these two special cases.
\begin{proposition}
\label{proposition:Matsuyama1}
For every topological abelian group $G$ having a dense torsion subgroup $t(G)$, the group $\HMi(G)$ is \map.
\end{proposition}
\begin{proof} Clearly, the group $\HMi(t(G))$ is torsion. It is abelian by Lemma \ref{Dima}(ii). Moreover, $\HMi(t(G))$ is connected by Lemma \ref{Dima}(i). Applying Lemma \ref{lemma1}, we conclude that $\HMi(t(G))$ is \map.
By Lemma \ref{HM:groups}(iii), the subgroup $\HMi(t(G))$ of $\HMi(G)$ is dense in $\HMi(G)$. Now Lemma \ref{easy:lemma} implies that $\HMi(G)$ is \map \ too.
\end{proof}
\begin{corollary}\label{HM(T)isMinAP}
$\HMi(\T)$ is \map.
\end{corollary}
\section{The bounded case: Proof of Theorem \ref{new:corollary1}}
\label{proofs0}
For the sake of this section only, let us call a group $N$ {\em nice\/} if it has the form $N = \bigoplus_{i=1}^n \Z(p^i)^{(\alpha_i)}$, where $p$ is a prime number, $n\in\N^+$, the cardinal $\alpha_n$ is infinite, while all cardinals $\alpha_1,\dots,a_{n-1}$ are finite (possibly zero).
\begin{lemma}
\label{map:topologies:on:nice:groups}
Every nice group admits a \map\ group topology.
\end{lemma}
\begin{proof} Let $N$ be a nice group as defined above. Then $N=F\oplus G$, where $F=\bigoplus_{i=1}^{n-1} \Z(p^i)^{(\alpha_i)}$ is a finite (possibly zero) group and $G=\Z(p^n)^{(\alpha_n)}$.
Fix a countable dense subset $S$ of $(0,1]$. The subgroup $H(G,S)$ of $\HMi(G)$ is dense in $\HMi(G)$ by Lemma \ref{HM:groups} (ii). By item (i) of the same lemma, $H(G,S)\cong G^{(\omega)}$. Since $\alpha_n$ is infinite, $G\cong G^{(\omega)}$. This allows us to fix an isomorphism $\xi:G\to H(G,S)$.
Let $T=(0,1]\setminus S$. Then $\HMi(G)=H(G,(0,1])=H(G,T)\oplus H(G,S)$. Since $H(G,T)\cong G^{(T)}\cong G^{(\cont)}\cong \Z(p^n)^{(\cont)}$, our assumption on $F$ allows us to fix a monomorphism $\eta:F\to H(G,T)$.
Clearly, the sum $\theta=\eta\oplus \xi: N=F\oplus G\to H(G,T)\oplus H(G,S)=\HMi(G)$ of the monomorphisms $\eta$ and $\xi$ is a monomorphism. In particular, $N\cong \theta(N)$.
Since $\xi(G)=H(G,S)$ is dense in $\HMi(G)$, so is $\theta(N)$. The group $\HMi(G)$ is \map\ by Proposition \ref{proposition:Matsuyama1}. Now it remains to apply Corollary \ref{easy:corollary}.
\end{proof}
From Lemmas \ref{easy:lemma} and \ref{map:topologies:on:nice:groups} one immediately obtains the following corollaries
\begin{corollary}
\label{direct:sums:of:nice:are;map}
A direct sum of nice groups admits a \map\ group topology.
\end{corollary}
\begin{corollary} \label{new:corollary}
A bounded abelian group admits a minimally almost periodic group topology whenever all its Ulm-Kaplanskly invariants are infinite.
\end{corollary}
\begin{lemma}
\label{nice:groups}
A bounded $p$-group having infinite leading Ulm-Kaplansky invariant is a finite direct sum of nice groups.
\end{lemma}
\begin{proof} Straightforward induction on the number of infinite Ulm-Kaplansky invariants of the group.
\end{proof}
\bigskip
\noindent {\bf Proof of Theorem \ref{new:corollary1}.}
We have to prove that a bounded abelian group admits a minimally almost periodic group topology if and only if all its leading Ulm-Kaplanskly invariants are infinite. Since the necessity is clear
from Proposition \ref{Necessary:condition},
we are left with the proof of the sufficiency.
Let $G$ be a bounded abelian group having all its leading Ulm-Kaplanskly invariants infinite. Since $G$ is a finite sum of $p$-groups, our assumption combined with Lemma \ref{nice:groups} allows us to claim that $G$ is a (finite) direct sum of nice groups. The conclusion now follows from Lemma \ref{direct:sums:of:nice:are;map}.
\qed
\section{Extention of monomorphisms into powers of $\HMi(\T)$}
\label{extension:section}
\begin{lemma}\label{embedding:lemma0}
Let $G$ be an abelian group and let $K$ be a divisible abelian group with $r_p(K) \ge |G|$ for every prime $p$ and $p=0$. If $H$ is a subgroup of $G$ and $j: H \to K$ is a monomorphism such that there exists a subgroup $K_1$ of $K$ with $K_1\cap j(H) = \{0\}$ and $K_1 \cong K$, then there exists a monomorphism $j': G \to K$ extending $j$.
\end{lemma}
\begin{proof} Since $K_1$ divisible and $K_1\cap j(H) = \{0\}$, we can write $ K = K_0\oplus K_1$, where the subgroup $K_0$ of $K$ contains $j(H)$.
Since, $K_1 \cong K$, $r_p(K_1) \ge |G|\ge |G/H|$ for every prime $p$ and $p=0$, so there exists a monomorphism $m: G/H \to K$. Extend $j: H \to K_0$ to a homomorphism $j_0: G\to K_0$ and define $j_1: G \to K$ to be the composition of $m$ and the canonical homomorphism $ G \to G/H$. Now define $j': G \to K = K_0\oplus K_1$ by $j'(g) = (j_0(g), j_1(g))\in K$. If $j'(g) = $ for some $g\in G$, then $j_0(g) = 0$ and $j_1(g) = 0$. The latter equality gives $g\in H$. Hence, $0= j_0(g) = j(g)$. Thus $g = 0$, as $j$ is a monomorphism.
\end{proof}
\begin{lemma}\label{New:claim} Let $\kappa$ be an infinite cardinal, $G$ be an abelian group with $|G|\leq \kappa$ and $H$ be a subgroup of $G$.
Then every monomorphism $j:H \to \HMi(\T)^\kappa$ can be extended to a monomorphism $j': G \to \HMi(\T)^\kappa$.
\end{lemma}
\begin{proof} From (\ref{direct:decomposition:of:HM(G)}) one deduces the algebraic isomorphisms
\begin{equation}\label{HM}
\HMi(\T) \cong \T^{(\cont)}\cong (\Q\oplus \Q/\Z)^{(\cont)}.
\end{equation}
Hence, $K = \HMi(\T)^\kappa$ is a divisible group with
\begin{equation}\label{ranks}
r_p(K) = 2^\kappa > |G| \ge r_p(G)\ \mbox{ for every }\ p \in \{0\}\cup \Prm.
\end{equation}
Since $|j(H)|\leq \kappa$, its divisible hull $D$ in $K$ satisfies $|D|\leq \kappa$. We can split $K = D \oplus D'$, where $D' \cong K/D$ is divisible, so completely determined by its $p$-ranks that are the same those of $K$, in view of (\ref{ranks}) and $|D|\leq \kappa$. Hence, $D'\cong \HMi(\T)$,
so Lemma \ref{embedding:lemma0} applies.
The extension of a monomorphism $j:H \to K$ to a monomorphism $j': G \to K$ can be obtained also by a direct application of \cite[Lemma 3.17]{DS_Forcing}.
\end{proof}
From (\ref{HM}) one deduces the algebraic isomorphism $\HMi(\T) \cong \HMi(\T)^\omega$. Hence, with $\kappa = \omega$ Lemma \ref{New:claim} gives:
\begin{corollary}\label{embedding:lemma}
Let $G$ be a countable abelian group. If $H$ is a subgroup of $G$ and $j: H \to \HMi(\T)$ is a monomorphism, then there exists a monomorphism $j': G \to \HMi(\T)$ extending $j$. \end{corollary}
\section{Countable groups admitting dense embeddings in $\HMi(\T)$}\label{Dense:emd:HM(T)}
The following lemma is a particular case of \cite[Lemma 4.1]{DS_HMP}.
\begin{lemma}
\label{simple:lemma}
Given a real number $\delta>0$, one can find $j\in\N$ such that for every integer $k\ge j$, each element $x\in\T$ and every
open arc $A$ in $\T$ of length $\delta$, there exists $y\in A\setminus\{0\}$ such that $ky=x$.
\end{lemma}
\begin{lemma}
\label{roots}
Given a non-empty open subset $W$ of $\HMi(\T)$, one can find an integer $j\in\N$ such that, for every $g\in \HMi(\T)$ and each integer $k\ge j$, there exists $h\in W$ satisfying $kh=g$.
\end{lemma}
\begin{proof} Since $W$ is non-empty, we can fix $w\in W$. Since $W$ is open in $\HMi(\T)$, there exist $\varepsilon>0$ and an open neighbourhood $U$ of $0$ in $\T$ such that $w+O(U,\varepsilon)\subseteq W$. Obviously, $U$ contains an open arc $L$ of length $\delta$ for sufficiently small $\delta$.
For this $\delta$, we apply Lemma \ref{simple:lemma} to find $j\in\N$ satisfying the conclusion of this lemma.
Let $g\in \HMi(\T)$. Since $g,w\in \HMi(\T)$, there exist $m\in\N$, $t_0, t_1,t_2,\dots,t_m\in\R$,
$x_1,x_2,\dots,x_m\in\T$ and $z_1,z_2,\dots,z_m\in \T$
such that
$0=t_0<t_1<t_2<\dots<t_{m-1}<t_m\le 1$, $g(s)=x_l$
and $w(s)=z_l$ whenever $1\le l\le m$ and $t_{l-1}\le s<t_l$.
Let $k\ge j$ be an integer. Since $j$ satisfies the conclusion of Lemma \ref{simple:lemma}, for every integer $l$ with $1\le l\le m$, we can find an element $y_l$ inside the arc $z_l+L$ such that $k y_l = x_l$.
Finally, define $h\in \HMi(\T)$ by letting $h(s)=y_l$ whenever $1\le l\le m$ and $t_{l-1}\le s<t_l$. Clearly, $k h = g$ by our construction.
Furthermore, since $y_l\in z_l+L\subseteq z_l+U$ for every $l=1,2,\dots,m$, it follows that $h\in w+O(U,\varepsilon)\subseteq W$.
\end{proof}
\begin{lemma}\label{MoscowLemma1}
For every prime number $p$, the Pr\"ufer group $\Z(p^\infty)$ is algebraically isomorphic to a dense subgroup of $\HMi(\T)$.
\end{lemma}
\begin{proof} By Corollary \ref{HM(T):has:countable:base}, $\HMi(\T)$ has a countable base. Let $\mathscr{B}=\{V_n:n\in\N^+\}$ be an enumeration of some countable base for $\HMi(\T)$ such that all $V_n$ are non-empty.
Let $g_0\in \HMi(\T)$ be an arbitrary element of order $p$. By induction on $n\in\N^+$ we shall define an element $g_n\in \HMi(\T)$ and an integer $i_n\in\N^+$ satisfying the following conditions:
\begin{itemize}
\item[(i$_n$)] $g_n\in V_n$;
\item[(ii$_n$)]
$p^{i_n} g_n=g_{n-1}$.
\end{itemize}
Assume that $g_{n-1}$ and $i_{n-1}$ satisfying (i$_{n-1}$) and (ii$_{n-1}$) have already been constructed. Applying Lemma \ref{roots}
to $W=V_n$, we select $j$ as in the conclusion of this lemma. Next we fix $i_n\in\N^+$ such that $p^{i_n}\ge j$. Applying Lemma \ref{roots} to
$g=g_{n-1}$ and $k=p^{i_n}$, we can find $g_n\in \HMi(\T)$ satisfying (i$_n$) and (ii$_n$). The inductive construction is complete.
Let $G$ be the subgroup of $\HMi(\T)$ generated by the set $\{g_n:n\in\N\}$. Since (i$_n$) holds for every $n\in\N^+$ and $\mathscr{B}$ is a base for $\HMi(\T)$, it follows that $G$ is dense in $\HMi(\T)$. Since $g_0$ has order $p$, and (ii$_n$) holds for all $n\in\N$, we conclude that $G$ is isomorphic to $\Z(p^\infty)$.
\end{proof}
For every $g\in \HMi(\T)$, we let
$$
S(g)=\{t\in[0,1]: g(t)\not=0\}.
$$
Let
$$
HM^*(\T)=\{g\in \HMi(\T): \sup S(g)<1\}.
$$
\begin{lemma}\label{MoscowLemma2} Given a real number $\eta< 1$ and a non-empty open subset $W$ of $\HMi(\T)$, one can find an integer $j\in\N$ such that,
for each integer $k\ge j$, there exists $h\in W$ satisfying the following conditions:
\begin{itemize}
\item[(i)] $kh=0$;
\item[(ii)] $h(s)\not=0$ for some $s\in [0,1]$ with $\eta<s$;
\item[(iii)] $h\in HM^*(\T)$.
\end{itemize}
\end{lemma}
\begin{proof} Since $W$ is non-empty, we can fix $w\in W$. Since $W$ is open in $\HMi(\T)$, there exist $\varepsilon>0$ and an open neighbourhood $U$ of $0$ in $\T$ such that $w+O(U,\varepsilon)\subseteq W$. Obviously, $U$ contains an open arc $L$ of length $\delta$ for sufficiently small $\delta$.
For this $\delta$, we apply Lemma \ref{simple:lemma} to find $j\in\N$ satisfying the conclusion of this lemma.
Since $w\in \HMi(\T)$, there exist $m\in\N$, $t_0, t_1,t_2,\dots,t_m$ and $z_1,z_2,\dots,z_m\in \T$ such that $0=t_0<t_1<t_2<\dots<t_{m-1}<t_m\le 1$
and $w(s)=z_l$ whenever $1\le l\le m$ and $t_{l-1}\le s<t_l$. By subdividing further if necessary, we may assume, without loss of generality, that $1-\gamma<t_{m-2}$ and $1-\varepsilon<t_{m-1}$.
Let $k\ge j$ be an integer. Since $j$ satisfies the conclusion of Lemma \ref{simple:lemma}, for every integer $l$ with $1\le l\le m-1$,
we can find a non-zero element $y_l$ inside the arc $z_l+L$ such that $k y_l = 0$.
Define $h\in \HMi(\T)$ by letting $h(s)=y_l$ whenever $1\le l\le m-1$ and $t_{l-1}\le s<t_l$, and letting $h(s)=0$ whenever $t_{m-1}\le s\le 1$. Since $y_l\in z_l+L\subseteq z_l+U$ for every $l=1,2,\dots,m-1$, and the measure of the interval $[t_{m-1},1]$ is less than $\varepsilon$ by the choice of $t_{m-1}$, it follows that $h\in w+O(U,\varepsilon)\subseteq W$.
Clearly, $k h = g$ by our construction, so (i) holds. Furthermore, (ii) holds by our choice of $t_{m-2}$. Indeed, defining $s=(t_{m-2}+t_{m-1})/2$, we get $\eta<t_{m-2}<s<1$ and $h(s)=y_{m-1}\not=0$. Finally, since $h(s)=0$ whenever $t_{m-1}\le s\le 1$, one has $\sup S(h)\le t_{m-1}<1$, which implies (iii).
\end{proof}
\begin{lemma}\label{MoscowLemma3} For every $n\in\N$, let $C_n$ be a cyclic group of order $a_n$.
If $\lim_{n\to\infty} a_n=\infty$, then $G=\bigoplus_{n\in\N} C_n$ is algebraically isomorphic to a dense subgroup of $\HMi(\T)$.
\end{lemma}
\begin{proof} By Corollary \ref{HM(T):has:countable:base}, $\HMi(\T)$ has a countable base. Let $\mathscr{B}=\{V_n:n\in\N^+\}$ be an enumeration of some countable base for $\HMi(\T)$ such that all $V_n$ are non-empty.
By induction on $n\in\N^+$ we shall define an element $g_n\in \HMi(\T)$ and an integer $i_n\in\N^+$ satisfying the following conditions:
\begin{itemize}
\item[(i$_n$)] $g_n\in V_n \cap HM^*(\T) $;
\item[(ii$_n$)] $a_ng_n= 0$;
\item[(iii$_n$)] $\langle g_n \rangle \cap \langle g_1,\ldots g_{n-1} \rangle = \{0\}$ when $n>1$.
\end{itemize}
With $\eta = 1/2$ and $W=V_1$ apply Lemma \ref{MoscowLemma2} to find an integer $j_1$ as in the lemma. Then choosing $i_1$ with $a_{i_1} \ge j_1$ there exists $g_1 \in V_1\cap HM^*(\T)$ with $a_{i_1} g_1= 0$ and $h(s) \ne 0$ for some $s > 1/2$. Assume that $n>1$ and $g_{n-1}\in V_{n-1}\cap HM^*(\T) $ with $a_{i_{n-1}} g_{n-1}= 0$ have already been constructed. Applying Lemma \ref{MoscowLemma2} to
$$
W=V_n\mbox{ and }\eta = \max \{\sup S(g_i): i=1,2,\ldots, n-1\}< 1
$$
we choose $j_n$ as in the conclusion of this lemma. Then choosing $i_n$ with $a_{i_n} \ge j_n$ there exists $g_n \in V_1\cap HM^*(\T)$ with $a_{i_n} g_n= 0$ and $g_n(s) \ne 0$ for some $s > \eta$. The latter property ensures that (iii$_n$) holds true, while (i$_n$) and (ii$_n$) are obviously satisfied. The inductive construction is complete.
Let $H$ be the subgroup of $\HMi(\T)$ generated by the set $\{g_n:n\in\N\}$.
Since (i$_n$) holds for every $n\in\N^+$ and $\mathscr{B}$ is a base for $\HMi(\T)$, it follows that $H$ is dense in $\HMi(\T)$.
For every $n\in\N$ denote by $C'_n$ the cyclic subgroup of $\HMi(\T)$ generated by $g_n$ and let $m_n = o(g_n)$ be its order.
Since $m_n| a_{i_n}$ by (ii$_n$), $C'_n$ is a cyclic group isomorphic to a subgroup of $C_n$. As (iii$_n$) holds for all $n\in\N$, we conclude that $H$ is isomorphic to $\bigoplus_{n\in\N} C'_n$. From $C_n'\leq C_{i_n}$ we deduce that $H$ is isomorphic to a subgroup of $G$. Let $j: H \to G$ be the monomorphism witnessing that.
Since $\HMi(\T)$ is divisible and since $G$ is countable, while $r_p(\HMi(\T)) = \cont$, there exists a monomorphism $\nu: G \to \HMi(\T)$ that composed with the monomorphism $j$ gives the inclusion $H \hookrightarrow \HMi(\T)$ (see Lemma \ref{embedding:lemma}). In particular, $\nu(G)$ contains the dense subgroup $H$, hence $\nu(G)$ is a dense subgroup of $\HMi(\T)$.
\end{proof}
\begin{lemma}
\label{basic:subgroup}
An unbounded torsion abelian group $G$ contains a subgroup algebraically isomorphic to one of the following three groups:
\begin{itemize}
\item[(i)] the Pr\"ufer group $\Z(p^\infty)$, for some prime number $p$;
\item[(ii)] the direct sum $\bigoplus _{n=1}^\infty \Z(p^n)^{(\alpha_n)}$,
where infinitely many ordinals $\alpha_n$ are non-zero;
\item[(iii)] the direct sum $\bigoplus_{p\in \pi}\Z(p)$ for a suitable infinite set $\pi$ of prime numbers.
\end{itemize}
\end{lemma}
\begin{proof}
If $G$ contains non-trivial divisible subgroups, then $G$ contains the Pr\"ufer group $\Z(p^\infty)$ for some prime number $p$. Thus (i) holds.
Assume now that $G$ is reduced, i.e., $G$ contains no non-trivial divisible subgroups.
For every $p\in \Prm$ let $G_p$ be the $p$-torsion subgroup of $G$ and let $\pi :=\{p\in \Prm: G_p\ne \{0\} \}$.
If $\pi$ is infinite, then pick an element $x_p\in G_p$ with $o(x_p) =p$ for each $p\in \pi$ and let $A$ be the subgroup generated by the set $\{ x_p : p\in \pi\}$. Then $A\cong \bigoplus_{p\in \pi}\Z(p)$, so (iii) holds.
Assume now that $\pi$ is finite.
Then for some prime $p\in \pi$ the group $G_p$ must be unbounded. According to \cite{Fuchs}, there exists a basic subgroup $B$ of $G_p$, i.e., a subgroup with the following three properties:
\begin{itemize}
\item[(a)] $B$ is a direct sum of cyclic groups, i.e., $B = \bigoplus _{n=1}^\infty \Z(p^n)^{(\alpha_n)}$;
\item[(b)] $B$ is pure, i.e., $p^n G_p \cap B = p^n B$ for every for some $n\in \N^+$;
\item[(c)] $G_p/B$ is divisible.
\end{itemize}
It is enough to show that $B$ is not bounded, which would yield (ii).
Suppose that $B$ is bounded; that is, $p^n B = 0$ for some $n\in \N^+$, then from (b) we deduce that $p^n G_p \cap B =\{0\}$. From (c) we deduce that
$$
(p^nG_p + B)/B=p^n(G_p/B) = G_p/B,
$$
hence $G_p = p^nG_p + B$. Since this sum $G_p = p^nG_p \oplus B$ is direct, (c) implies that $G_p$ contains a subgroup, namely $p^nG_p\cong G_p/B$ that is divisible group. Since $G_p$ is reduced, we conclude that $p^n G$ is trivial, so $G_p=B$ is bounded, a contradiction.
\end{proof}
\begin{theorem}
\label{embedding:countable:groups}
Let $G$ be a countable abelian group satisfying at least one of the following conditions:
\begin{itemize}
\item[(a)] the torsion part $t(G)$ of $G$ is unbounded;
\item[(b)] the rank of $G$ is infinite.
\end{itemize}
Then $G$ is algebraically isomorphic to a dense subgroup of $\HMi(\T)$.
\end{theorem}
\begin{proof}
It suffices to find a subgroup $H$ of $G$ which admits a dense embedding into
$\HMi(\T)$. Indeed, suppose that $H$ is such a group, and let $j:H\to \HMi(\T)$ be a monomorphism such that $j(H)$ is dense in $\HMi(\T)$.
By Corollary \ref{embedding:lemma}, $j$ can be extended to
a monomorphism $j': G\to \HMi(\T)$. Clearly, $j'(G)\cong G$.
Since $j(H)$ is dense in $\HMi(\T)$ and $j(H)\subseteq j'(G)\subseteq \HMi(\T)$, the subgroup $j'(G)$ is dense in $\HMi(\T)$.
To find a subgroup $H$ of $G$ which admits a dense embedding into
$\HMi(\T)$, we consider two cases.
\smallskip
{\em Case 1\/}. {\sl Item (a) holds\/}. Applying Lemma \ref{basic:subgroup} to $t(G)$, we conclude that $t(G)$ (and thus, $G$ as well) contains a subgroup $H$ algebraically isomorphic to one of the three groups listed in Lemma \ref{basic:subgroup}. In case (i) of Lemma \ref{basic:subgroup}, $H$ admits a dense embedding into $\HMi(\T)$ by Lemma \ref{MoscowLemma1}, while in cases (ii) and (iii) of Lemma \ref{basic:subgroup}, $H$ admits a dense embedding into $\HMi(\T)$ by Lemma \ref{MoscowLemma3}.
\smallskip
{\em Case 2\/}. {\sl Item (b) holds\/}.
In this case,
$G$ contains a subgroup $H$ algebraically isomorphic to $\Z^{(\omega)}$.
Since $\T$ contains a dense subgroup $D$ algebraically isomorphic to $\Z$,
Corollary \ref{corollary:Moscow2} implies that $\HMi(\T)$ contains a dense subgroup $N$ with $N\cong D^{(\omega)}\cong \Z^{(\omega)}\cong H$.
\end{proof}
\begin{remark}
\label{finitely:generated:are:not:dense}
It can be easily checked that {\em no finitely generated subgroup of $\HMi(\T)$ can be dense in $\HMi(\T)$\/}.
In particular, finite direct sums
of cyclic groups can never be dense in $\HMi(\T)$.
\end{remark}
\section{The countable case of Comfort-Protasov problem: Nienhuys group enters briefly}
\label{Nienhyus:section}
\label{Sec:10}
In this section we give
a short self-contained proof of the first part of Protasov-Comfort problem; namely, we show that {\em every countable unbounded abelian group admits a \map \ group topology\/}. To achieve this, we need to recall the construction of the minimally almost periodic and monothetic group $\Ni$ of Nienhuys \cite{N}, paying special attention only to the {\em algebraic\/} structure of $\Ni$.
Let $c_0(\T)$ be the subgroup of $\T^\omega$ consisting of all null sequences in $\T$ (equipped with the norm topology in $\T^\omega$).
As $c_0(\T)$ is divisible and contains $\T^{(\omega)}$, it follows that $r_0(c_0(\T)) = \cont$ and $r_p(c_0(\T))\geq \omega$ for every $p\in \Prm$.
On the other hand, if $x = (x_n) \in c_0(\T)$ and $px =0$, then $px_n= 0$ for all $n\in \N$. Therefore, $x_n \to 0$ implies that all but finitely many $x_n$ vanish, i.e., $x\in \T^{(\omega)}$. This proves that $c_0(\T)[p] = \T^{(\omega)}[p] = \T[p]^{(\omega)}$, i.e., $r_p(c_0(\T)) = \omega$.
The monothetic \map \ group $\Ni$, built by Nienhuys \cite{N}, is a quotient of $c_0(\T)$ with respect to a cyclic subgroup.
Hence, $\Ni$ is divisible and has the same $p$ ranks as $c_0(\T)$. Therefore, one has the algebraic isomorphisms
\begin{equation}\label{Nienhuys:equation}
\Ni \cong c_0(\T)\cong \Q^{(\cont)} \oplus (\Q/\Z)^{(\omega)}.
\end{equation}
\begin{lemma}
\label{Saak}
Every countable unbounded abelian group admits a \map\ group topology.
\end{lemma}
\begin{proof}
Let $G$ be a countable unbounded abelian group. We consider two cases.
\smallskip
{\em Case 1\/}. {\em Either item (a) or item (b) of Theorem \ref{embedding:countable:groups} holds\/}. In this case, the conclusion of Theorem \ref{embedding:countable:groups} guarantees that $G$ can be densely embedded into
the
group
$\HMi(\T)$. Since the latter group is \map\ by Corollary \ref{HM(T)isMinAP},
applying Corollary \ref{easy:corollary} we conclude that $G$ admits a \map\ group topology.
\smallskip
{\em Case 2\/}. {\em Neither item (a) nor item (b) of Theorem \ref{embedding:countable:groups} holds\/}. In this case, $G$ has finite rank. Since $t(G)$ is bounded and $G$ is unbounded, $G$ contains a subgroup $H$ such that $H\cong\Z$.
Since $\Ni$ is monothetic, we can
fix a dense embedding $j: H \to \Ni$. Since the divisible hull $D$ of $j(H)$ in $\Ni$ is isomorphic to $\Q$, it follows from (\ref{Nienhuys:equation}) that one can
split $\Ni = D \oplus D_1$, where $D_1\cong \Ni/D\cong \Ni$ is a divisible group. Now we can apply Lemma \ref{embedding:lemma0}
to extend $j$ to a dense embedding $j': G \to \Ni$.
Since $\Ni$ is \map, Corollary \ref{easy:corollary} implies that $G$ admits a \map\ group topology.
\end{proof}
\begin{remark}
\label{where:Nienhuys:is:used}
A careful analysis of the proof of Lemma \ref{Saak} shows that only the (countable) groups of finite non-zero rank require
the recourse to the Nienhuys group $\Ni$.
\end{remark}
\section{The general case: Proof of Theorem \ref{CH:theorem}}\label{proofs}
We recall here a fundamental notion from \cite{DGB}.
\begin{definition}
\label{w-divisible:reformulation}
\cite{DGB}
An abelian group $G$ is called {\em $w$-divisible\/} if $|mG|=|G|$ for all integers $m\ge 1$.
\end{definition}
\begin{lemma}
\label{w-divisible:embeddings}
Every uncountable $w$-divisible group is algebraically isomorphic to a dense
subgroup of $\HMi(\T)^G$.
\end{lemma}
\begin{proof}
Our aim is to find a family $\{G_i:i\in I\}$ of subgroups of $G$ such that $|I|=|G|$, $G$ contains its direct sum $H=\bigoplus_{i\in I} G_i$
and each $G_i$ admits a dense embedding into $\HMi(\T)$. Assuming this is done, one obtains a monomorphism
$j: H =\bigoplus_{i\in I} G_i\to \bigoplus_{i\in I}\HMi(\T) = \HMi(\T)^{(I)}$
such that $j(H)$ is dense in $\HMi(\T)^{(I)}$. Since $\HMi(\T)^{(I)}$ is dense in $\HMi(\T)^I$,
the subgroup $j(H)$ of $\HMi(\T)^I$ is also dense in $\HMi(\T)^I$.
Using Lemma \ref{New:claim}, this embedding can be extended to a monomorphism $j': G \to \HMi(\T)^{I}$, so that $j'(G)\cong G$ and $j'(G)$ is also dense in $\HMi(\T)^{I}$. Since $|I|=|G|$, the topological groups $\HMi(\T)^{I}$ and $\HMi(\T)^{G}$ are topologically isomorphic. This produces the desired
dense embedding of $G$ into $\HMi(\T)^G$.
According to
\cite[Theorem 3.6]{DS_ConnectedMarkov},
$G$ contains a direct sum $\bigoplus_{s\in S}H_s$ of unbounded groups with $|S| = |G|$. It is not restrictive to assume that, for each $s\in S$, either $H_s\cong \Z$ or $H_s$ is a countable unbounded torsion group. Let
$S_0 = \{s\in S: H_s \cong \Z\}$ and $S_1 = S \setminus S_0$.
Since $S$ is infinite, there exists $k = 0,1$ such that $|S|=|S_k|$.
\smallskip
{\em Case 1\/}. $k=0$.
Since $S_0$ is infinite (in fact, even uncountable), we can find a decomposition $S_0=\bigcup_{i\in I} T_i$ of $S$ into countably infinite pairwise disjoint sets $T_i$. For each $i\in I$, we let $G_i=\bigoplus_{t\in T_i} H_t$.
Since $H_t\cong \Z$ for every $t\in T_i$, it follows that
$G_i\cong \Z^{(\omega)}$ for every $i\in I$. Furthermore,
$G$ contains direct sum
$\bigoplus_{s\in S_0} H_s=\bigoplus_{i\in I}\bigoplus_{t\in T_i} H_t
=
\bigoplus_{i\in I}G_i$.
\smallskip
{\em Case 2\/}. $k=1$.
In this case we let $I=S_1$ and $G_i=H_i$ for all $i\in I=S_1$.
\smallskip
We claim that $\{G_i: i\in I\}$ is the desired family.
Indeed, $|I|=|G|$. By our construction, $G$ contains the direct sum
$H=\bigoplus_{i\in I} G_i$.
Finally, each $G_i$ is either a countable unbounded torsion group or the group $\Z^{(\omega)}$ of infinite rank. Applying Theorem \ref{embedding:countable:groups}, we conclude that $G_i$ admits a dense embedding into $\HMi(\T)$.
\end{proof}
\begin{corollary}\label{new:w-divisible:theorem}
Every $w$-divisible group admits a \map \ group topology.
\end{corollary}
\begin{proof} Let $G$ be a $w$-divisible group. If $G$ is
a bounded torsion group, then $G$ must be trivial
by Definition \ref{w-divisible:reformulation}. Clearly, the trivial group has a \map\ group topology. Therefore, from now on we shall assume that $G$ is not bounded torsion.
If $G$ is countable, then $G$ admits a \map\ group topology by Lemma \ref{Saak}.
If $G$ is uncountable,
then $G$ admits a dense embedding into $\HMi(\T)^G$ by Lemma \ref{w-divisible:embeddings}.
Since $\HMi(\T)$ is \map\ by Corollary \ref{HM(T)isMinAP}, its power
$\HMi(\T)^G$ is \map\ by Lemma \ref{easy:lemma}.
Now $G$ admits a \map\ group topology by Corollary \ref{easy:corollary}.
\end{proof}
We need the following lemma that can be obtained from \cite[Lemma 4.5]{DS_ConnectedMarkov} with $\sigma = \omega$:
\begin{lemma}
\label{homogeneous:split} \cite{DS_ConnectedMarkov}
Every unbounded abelian group $G$ admits a decomposition $G=N\oplus H$ such that $N$ is a bounded group with all its Ulm-Kaplanskly invariants infinite and $H$ is a $w$-divisible group.
\end{lemma}
\medskip
\noindent {\bf Proof of Theorem \ref{CH:theorem}.}
Let $G$ be an unbounded abelian group. By Lemma \ref{homogeneous:split}, we find a decomposition $G=N\oplus H$ such that $N$ is either trivial, or a bounded group with infinite Ulm-Kaplanskly invariants and $H$ is a
$w$-divisible group.
Applying Corollary \ref{new:corollary}, we conclude that $N$ admits a minimally almost periodic group topology
$\mathscr{T}_N$.
By Corollary \ref{new:w-divisible:theorem}, $H$ also admits a minimally almost periodic group topology $\mathscr{T}_H$.
Since both $(N,\mathscr{T}_N)$ and $(H,\mathscr{T}_H)$ are minimally almost periodic, so is their product $(N,\mathscr{T}_N)\times(H,\mathscr{T}_H)$, by Lemma \ref{easy:lemma}.
\qed
\begin{remark}
A careful analysis of the proofs in this section shows that our use of the Nienhuys group is restricted to the reference to Lemma \ref{Saak} in the proof
of Lemma \ref{new:w-divisible:theorem}.
Combining this with Remark \ref{where:Nienhuys:is:used}, we conclude that
the recourse to Nienhuys group $\Ni$ in the proof of our main results is necessary only for handling countable groups of finite non-zero rank.
\end{remark}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,900 |
A new DNA tool created by Michigan State University can accurately predict people's height, and more importantly, could potentially assess their risk for serious illnesses, such as heart disease and cancer.
For the first time, the tool, or algorithm, builds predictors for human traits such as height, bone density and even the level of education a person might achieve, purely based on one's genome. But the applications may not stop there.
Further applications have the potential to dramatically advance the practice of precision health, which allows physicians to intervene as early as possible in patient care and prevent or delay illness.
The research, featured in the October issue of Genetics, analyzed the complete genetic makeup of nearly 500,000 adults in the United Kingdom using machine learning, where a computer learns from data.
In validation tests, the computer accurately predicted everyone's height within roughly an inch. While bone density and educational attainment predictors were not as precise, they were accurate enough to identify outlying individuals who were at risk of having very low bone density associated with osteoporosis or were at risk of struggling in school.
Traditional genetic testing typically looks for a specific change in a person's genes or chromosomes that can indicate a higher risk for diseases such as breast cancer. Hsu's model considers numerous genomic differences and builds a predictor based on the tens of thousands of variations.
Using data from the UK Biobank, an international resource for health information, Hsu and his team put the algorithm to work, evaluating each participant's DNA and teaching the computer to pull out these distinct differences.
Hsu's team will continue to improve the algorithms, while tapping into larger, more diverse data sets. Doing this would further validate the techniques and continue to help map out the genetic architecture of these important traits and disease risks.
With greater computing power and decreasing costs around DNA sequencing, what was once thought to be five to 10 years out, is now a lot closer when it comes to this type of work, Hsu added.
April 18, 2019 - After a shark's bite, DNA kits could track down a culprit (or at least a species) Hawaii News NowUniversity of Hawaii researchers are developing a new tool to help them learn more about sharks around the islands.
April 17, 2019 - For African Americans, DNA Tests Reveal Just A Small Part Of A Complicated Ancestry KCURIt took nearly 30 minutes for Eric Depradine to extract a saliva sample from his dying grandmother. Depradine, 35, of Kansas City, wanted to have his. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,202 |
Award-winning communications experts in the built environment
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What has Theresa May done for Midlands housebuilders?
At the end of June, the outgoing prime minister, Theresa May reflected on her housing legacy during her time as Leader of the Conservatives as she delivered a keynote speech during Europe's largest Housing Festival, Housing 2019. But how has she really faired? And what has she done for the Midlands?
The West Midlands especially has bucked national trend to show strong growth in house building. Figures from the Office of National Statistics show that 10,640 new homes were started in the West Midlands Combined Authority (WMCA) area in 2018 – a seven per cent increase on 2017. Across England, the average increase was zero. The figures have been praised by the WMCA, which has pledged to deliver 215,000 new homes across its region by 2031.
Despite these new homes being built, it was the population figures of the UK's biggest cities that were hitting the headlines in June 2018. A new report, by Centre for Cities suggested that millions of people were flocking to city centres – with a rapid return to city living in places such as Birmingham and Leicester. Populations were measured in 2002 and again in 2015, with Leicester's city centre population increasing by 145% and Birmingham a whopping 163%, so what does this mean for rural house building development – and the need for 215,000 new homes?
Regardless of where people are living, whether that be in the surrounding areas outside of city centres, or elsewhere in the WMCA's three LEPs: Black Country (Dudley, Sandwell, Walsall and Wolverhampton), Coventry and Warwickshire, and Greater Birmingham and Solihull, the housing figures are excellent. Housing in Birmingham is up by 80% – streets ahead of London, which saw a decrease of a staggering 20%. With HS2 and the Commonwealth Games, does Birmingham and the Midlands show a higher desirability than London?
Figures by Rightmove's House Price Index showed that the price of property coming to market was within a whisker of a new record, in June 2019. There was a new all-time price high in the East Midlands, which pushed the national average to within £91 of a new record despite backdrop of political uncertainty.
During her speech at Housing 2019, the PM outlined the success of housing under her leadership, stating they had promised a million homes and duly delivered. The latest projections show that, by this autumn, a million homes will have been added to the national supply in less than five years. In her words, that is:
"a million homes for young families, for hardworking professionals, for downsizing retirees. A million homes giving more people the safety and security that many of us take for granted. A million homes that show that our promises are more than just words."
However, the legacy of Housing under Mrs May's premiership will certainly have a grey cloud over it. During the PM's speech, Ms May mentioned the Grenfell Tower fire just once and couldn't put a date on when all high-rise buildings shouldn't have Aluminium Composite Material (ACM) cladding.
In January 2019, the Ministry of Housing, Communities and Local Government released figures – as part of the Building Safety Programme, which reveal that there are still 437 high-rise buildings with ACM cladding systems. These are the numbers that people will remember. A million new homes is a fantastic record, but the numbers of those affected by that fateful day in June 2017, will continue to be scrutinised.
House building in the Midlands has been strong under Theresa May and now it will be down to Boris Johnson to continue the excellent work of the last three years.
Written by: Matthew Crisp
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"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,140 |
import {LightFilter, TYPE_POINT, TYPE_CONE} from './LightFilter.js'
declare var Sprite;
declare var $gameMap;
declare var Graphics;
const FLICKER = 7;
export default class LightProcessor{
_filterCache: {
[key: string]: LightFilter
};
_ambient: {r:number, g: number, b: number};
constructor(){
this._filterCache = {};
}
setAmbient(color: Object){
this._ambient = color;
}
_createOrGetFilter(numPoints: number, numCones: number){
const key = numPoints + ':' + numCones;
if(!this._filterCache[key]){
this._filterCache[key] = new LightFilter(numPoints, numCones);
}
return this._filterCache[key];
}
_lightX(sprite: Sprite){
return sprite.x;
}
_lightY(sprite: Sprite){
return Graphics.height - (-$gameMap.tileHeight()/2 + sprite.y);
}
_flicker(l: Object){
const r1 = l.radius;
if(l.flicker) return r1 + FLICKER;
else return r1;
}
_setupFilterLights(sprites: Array<Sprite>){
const lights = sprites.filter(s=>s._character._light.length !== 0);
let points = 0;
let cones = 0;
lights.forEach(sprite=>{
sprite._character._light.forEach(light=>{
switch(light.type){
case TYPE_POINT:
points++;
break;
case TYPE_CONE:
cones++;
break;
}
});
});
const filter = this._createOrGetFilter(points, cones);
lights.forEach(sprite=>{
sprite._character._light.forEach(light=>{
if(light.type === TYPE_POINT){
filter.setLight(--points,
this._lightX(sprite), this._lightY(sprite),
light.radius, this._flicker(light),
light.r, light.g, light.b);
}else if(light.type === TYPE_CONE){
let angle = this._dir(sprite);
filter.setCone(--cones,
this._lightX(sprite), this._lightY(sprite),
light.radius, this._flicker(light),
Math.cos(angle*Math.PI/180), Math.sin(angle*Math.PI/180),
light.angleMin*Math.PI/180, light.angleMax*Math.PI/180,
light.r, light.g, light.b);
}
});
});
let c = this._ambient;
filter.setAmbient(c.r, c.g, c.b);
return filter;
}
_quickReject(sprite: Sprite){
const x = this._lightX(sprite);
const y = this._lightY(sprite);
const radius = sprite._character._light.radius;
return x < -radius
|| x > Graphics.width + radius
|| y < -radius
|| y > Graphics.height + radius;
}
_dir(sprite: Sprite){
switch(sprite._character.direction()){
case 2:
return 270;
case 4:
return 180;
case 6:
return 0;
case 8:
return 90;
}
return 0;
}
update(baseSprite: Sprite, characterSprites: Array<Sprite>){
const sprites = [];
characterSprites.forEach(sprite=>{
if(sprite._character._light && !this._quickReject(sprite)){
sprites.push(sprite);
}
});
const filter = this._setupFilterLights(sprites);
const filters = baseSprite.filters.filter((f)=>!(f instanceof LightFilter));
filters.push(filter);
baseSprite.filters = filters;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,248 |
HomeMiddle East
Iran Blames The UK For Kashmir Mess; Says India Will Adopt A Fair Policy Towards Muslims
Iran's supreme leader Ayatollah Ali Khamenei urged Indian Prime Minister Narendra Mod to adopt a fair policy towards the people of Jammu and Kashmir while blaming the United Kingdom for all the mess in the region.
"We have good relations with the government of India, but the Indian government is expected to adopt a fair policy towards the noble people of Kashmir so that this region's Muslim people would not be oppressed," said Khamenei during a meeting with President Hassan Rouhani and his cabinet members on Wednesday.
According to Iranian media, Khamenei also described the current situation in Jammu and Kashmir and the disputes between Islamabad and New Delhi as a result of measures by the vicious British government before its pullout from the Indian Subcontinent.
Pakistan Says Voice Of Kashmiri People Heard At UNSC; India Calls Kashmir 'Internal Matter'
"The British have intentionally inflicted this wound on that region for the continuation of clashes in Kashmir," he added.
Tensions flared in the Kashmir valley after the Indian government revoked the special status of Jammu and Kashmir. By repealing Article 370 of the constitution, people from the rest of India will now have the right to buy property in Jammu and Kashmir and settle there permanently. Kashmiris see the move as an attempt to dilute the demographics of Muslim-majority Kashmir with Hindu settlers.
Previous articleImran Khan Says No Point Talking To India Anymore
Next articlePlease Prevent Impending Genocide Of Kashmir People: Pakistan PM Urges Global Community
China's Much Hyped Liaoning Aircraft Carrier 'Flawed'; US Commander Says Its 'Operational Restrictions' Exposed
More Powerful Than Rafales, France Tests New Generation Fighter (NGF) Engine Designed For 'Future Battles' | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,199 |
{"url":"http:\/\/alamos.math.arizona.edu\/~rychlik\/math263_old\/midterm_2_info.php","text":"### Last minute tips\n\n\u2022 Be able to set up sample spaces with pairs and tuples.\n\u2022 Be familiar with a deck of playing cards.\n\u2022 Know how to use Bayes formula in the context of medical testing, such as the breast cancer screening example on the slides.\n\u2022 Be able to calculate probabilities using density curves such as the normal curve, and \"triangular\" curve.\n\u2022 The formula for the correlation of two random variables ($$\\rho_{X,Y}$$ below) will not be required on the test.\n\u2022 The number of questions is exactly 18. Calculations are quick and easy.\n\n### General Information\n\n\u2022 Midterm 2 is an in-class test.\n\u2022 Midterm 2 will cover all topics of Chapter 3 and 4 and Section 2.6.\n\u2022 The number of questions is exactly 18.\n\u2022 The questions are primarily multiple-choice.\n\u2022 Several questions require calculator use.\n\u2022 Table A (normal distribution), Table B (random digits) and Table C (binomial distribution) will be provided if needed.\n\u2022 Please review old tests (2010 Midterm 2 and a portion of 2010 Midterm 3 in this folder) for a sample of problems that may be similar to some test questions.\n\u2022 You may find these notes on the three set Venn diagram and 2-way tables useful.\n\n### List of Chapter 3 topics covered\n\n\u2022 The three principles of experimental design.\n\u2022 Observational vs. experimental studies.\n\u2022 Identification of experimental units.\n\u2022 Identification of population.\n\u2022 Sampling techniques using table of random digits\n\u2022 Basic experimental designs:\n\u2022 Block (=Stratified)\n\u2022 Matched pair\n\u2022 Multi-stage\n\u2022 Lurking variables (including information in Section 2.6). Definition, identification, when to watch out for.\n\u2022 Confounding (including information in Section 2.6). Definition, identification, when to watch out for.\n\u2022 Bias and variability. Differentiating between the two.\n\u2022 Controlling bias. Controlling by randomization.\n\u2022 Problems when using anecdotal evidence.\n\u2022 Problems when using polling.\n\n### List of Chapter 4 topics covered\n\n#### Probability Models\n\n\u2022 Know the meaning of outcomes.\n\u2022 Be familiar with basic set theory: elements, sets, curly brace notation, pairs, tuples, union, intersection, complement.\n\u2022 Know the meaning of sample spaces and events.\n\u2022 Know the difference between outcomes and elementary events.\n\u2022 Be able to identify and construct sample spaces. Be able to describe sample spaces using set notation.\n\u2022 Be able to use union, intersection and complement to describe events described in plain English, using connectives such as \"or\", \"and\" and \"not\".\n\u2022 Be familiar with standard examples used in class such as multiple coin tosses, die tosses, free throws in basketball, picking M&M candy out of a jar (with and without replacement), tosses of a bottle cap, etc.\n\u2022 Be able to perform calculations of probabilities of events, based on laws of probability and set notation (union, intersection, complement).\n\u2022 Know the addition rule for disjoint events and its generalization, the Inclusion-Exclusion Principle for 2 and 3 events: $P(A\\cup B) = P(A) + P(B) \\quad\\text{if A\\cap B=\\emptyset}$ $P(A\\cup B) = P(A) + P(B) - P(A\\cap B) \\quad\\text{(always)}$ $P(A\\cup B \\cup C) = P(A) + P(B) + P(C) - P(A\\cap B) - P(A\\cap C) - P(B\\cap C) + P(A\\cap B \\cap C)$\n\u2022 Know the meaning of independence of events. Be able to apply the Multiplication Rule for independent events.\n\n#### Random variables\n\n\u2022 Understand the following definition: A random variable is a function on the sample space, with numerical values. Using mathematical notation, let $$S$$ be a sample space. Any function $$X:S\\to \\mathbb{R}$$ is a random variable (on the sample space $$S$$).\n\u2022 Understand the terminology of functions. Thus, if $$X:S\\to U$$ then $$S$$ is called the domain of $$X$$ and $$U$$ is called the range of $$X$$. Thus, for random variables the range is a subset of $$\\mathbb{R}$$ the real numbers. Note: $\\mathbb{R}=(-\\infty,\\infty)$ $\\mathbb{R}= ]-\\infty,\\infty[$ using two different conventions about denoting intervals.\n\u2022 The set of values of a random variable is the set of numbers $$X(s)$$ where $$s$$ is an outcome (an element of $$S$$). It is denoted $$X(S)$$ (\"X of the entire sample space\").\n\u2022 Know the definition of a discrete random variable: A function $$X:S\\to\\mathbb{R}$$ is a discrete random variable if the set of values of $$X$$ is either finite or countable infinite. Know that the definition in the book is incorrect, disallowing an infinite set of values. An example of a discrete random variable with an infinite set of values is the number of tosses of a coin before you see the first head. Thus,\n\u2022 If in the first toss you get $$Head$$, $$X=0$$.\n\u2022 If in the first toss is a $$Tail$$ but you get $$Head$$ on the second toss, $$X=1$$.\n\u2022 Thus, $$X$$ may be $$0, 1, 2, \\ldots$$.\n\u2022 It may be shown (using, for instance, tree-based calculations), that $P(X=k) = \\frac{1}{2^{k+1}}$ for $$k=0,1,2,\\ldots$$.\n\u2022 Know the definition of the probability function for discrete random variables. The probability function for a random variable assuming values $$x_1,x_2,\\ldots,x_n$$ with probabilities $$p_1,p_2,\\ldots,p_n$$ is: $f(x_i) = p_i$ The probability table of $$X$$ is simply a table that lists the values of this function: $\\begin\\left\\{array\\right\\}\\left\\{c|ccccc\\right\\} \\hline\\\\ x & x_1 & x_2 & x_3 & \\ldots & x_n \\\\ \\hline\\\\ p & p_1 & p_2 & p_3 & \\ldots & p_n\\\\ \\hline \\end\\left\\{array\\right\\}$ Since the set of values may be infinite (but countable), the table may be infinite, and it may be necessary to give a formula rather than listing values, as in the previous example.\n\u2022 Know the definition of a continuous random variable. The set of values of such a variable is an uncountable set such as an interval $$[a,b]$$ or $$[0,\\infty[$$. The probability $P(X=x)=0$ of a particular value $$x$$ is always zero for a continuous random variable. Therefore, to calculate probabilities related to continuous random variables requires the knownledge of the probability density function (\"density curve\"). If the formula for the density curve is $$y = f(x)$$ then the formula allowing us to compute the probability is: $P(a \\le X \\le b) = \\int_{a}^b f(x)\\,dx$ Thus, the probability is the area under the curve.\n\u2022 Note that $P(a \\le X \\le b) = P(a < X \\le b)= P(a \\le X < b) = P(a < X < b)$ for continuous random variables. This definitely not the case for discrete random variables (why?).\n\u2022 Probability distribution functions, probability tables, cumulative distributions, expected value, variance, standard deviation. Note that the cumulative distribution function of a random variable $$X$$ is defined by: $F(x) = P(X \\le x)$ This formula is valid, regardless of whether $$X$$ is discrete or continuous. However, the calculation is different in these two cases: $F(x) = \\sum_{y\\leq x} P(X=y)$ for discrete variables, where the summation is over $$y$$ which $$X$$ actually assumes. For continuous random variables, $F(x) = \\int_{-\\infty}^x f(y) \\,dy$ where $$f(x)$$ is the \"density curve\". Thus, this is the area under the curve $$w=f(y)$$ and to the left of the line $$y=x$$. Note that Table A tabulates $$F(x)$$ for the normal density curve: $f(x) = \\frac{1}{\\sqrt{2\\pi}} e^{-\\frac{1}{2} x^2}$\n\u2022 Calculation of the random variable expected value (also called the mean). Be familiar with the formula: $\\mathbb{E}X = \\mu_X = \\sum_{i=1}^n x_i p_i = \\sum_{x} x\\, P(X_i = x)$ where the summation extends over all values of $$X$$. The set of values could be an infinite, but countable set, such as natural numbers. In this case, the formula is an infinite series: $\\mathbf{E}X = \\mu_X = \\sum_{i=1}^\\infty x_i p_i = \\sum_{x} x\\, P(X_i = x)$ i.e. formally $$n=\\infty$$. For example, the number of tosses before the first head is seen in a potentially infinite sequence of coin tosses is: $\\sum_{k=0}^\\infty k \\cdot \\frac{1}{2^{k+1}} = 1.$ Obtaining this result requires some familiarity with infinite series.\n\u2022 Know that $$X+Y$$ is only defined if $$X$$ and $$Y$$ share the same sample space. In calculus, you have learnt that two functions may be added only if they have the same domain. This is the same principle.\n\u2022 The rule for the mean of a sum of random variables: $\\mu_{X+Y} = \\mu_{X}+\\mu_{Y}$ Know that $$X$$ and $$Y$$ do not have to be independent for this rule to hold.\n\u2022 The rule for the variance of independent random variables: $\\sigma_{X+Y}^2 = \\sigma_X^2 + \\sigma_Y^2$ if variables $$X$$ and $$Y$$ are independent.\n\u2022 Another variance formula: $\\sigma_{a\\,X+b}^2 = b^2\\,\\sigma_{X}^2$ where $$a$$ and $$b$$ are constants.\n\u2022 Combined rule for expected values: $\\mu_{a\\,X+b} = a\\,\\mu_X + b$ where $$a$$ and $$b$$ are numbers.\n\u2022 Combined rule for variances: $\\sigma_{a\\,X+b}^2 = a^2\\,\\sigma_X^2$ where $$a$$ and $$b$$ are numbers.\n\u2022 Know the meaning of independence of random variables: for all $$x$$ and $$y$$ $P(X=x\\;\\text{and}\\;Y=y) = P(X=x)\\,P(Y=y)$\n\u2022 Familiarity with correlation for random variables: $\\rho = \\rho_{X,Y} = \\mathrm{corr}(X,Y)=\\mathbb{E}\\left(\\left(\\frac{X-\\mu_X}{\\sigma_X}\\right) \\left(\\frac{Y-\\mu_Y}{\\sigma_Y}\\right)\\right) = \\sum_{i=1}^m\\sum_{j=1}^n \\left(\\frac{x_i-\\mu_X}{\\sigma_X}\\right) \\left(\\frac{y_j-\\mu_Y}{\\sigma_Y}\\right)\\cdot p_{ij}$ where $p_{ij} = P(X=x_i\\;\\text{and}\\;Y=y_j)$ (NOTE: Cannot use product rule because $$X$$ and $$Y$$ possibly are not independent; if they were, $$\\rho_{X,Y}=0$$) Here, we avoided excessive subscripts by using common notation for the expected value of a variable: $\\mathbb{E}X = \\mu_X$\n\u2022 Kow the meaning of the formula: $\\sigma_{X+Y}^2 = \\sigma_{X}^2 + \\sigma_{Y}^2 + 2\\,\\rho_{X,Y}\\,\\sigma_X\\sigma_Y$\n\u2022 Note that the formula for $$\\rho$$ is not in the book, although it appears to be used in several examples. Be able to use it when $$X$$ and $$Y$$ assume very few values (2 or 3). Here is an example: For a certain basketball team it was determined that the probability of scoring a hit in the first free shot is $$0.8$$. However, the probability of scoring on the second shot depends on whether the player scored on the first shot. If the player scored on the first shot, the probability of scoring on the second shot remains $$0.8$$. However, if the player missed on the first shot, the probability of scoring on the second shot is only $$0.7$$. This lower performance is known in the sports as \"choking\". Let $$X$$ be a random variable which represents the number of points scored on the first shot (0 or 1), and let $$Y$$ be the number of points scored on the second shot. Please answer the following questions:\n\u2022 Find the four probabilities: $p_{ij} = P(X=i\\;\\text{and}\\;Y=j)$ for $$i,j=0,1$$. That is, fill out the following table: $\\begin\\left\\{array\\right\\}\\left\\{c|c|c\\right\\} i\\backslash j & 0 & 1 \\\\ \\hline\\hline\\\\ 0 & p_\\left\\{00\\right\\} & p_\\left\\{01\\right\\}\\\\ 1 & p_\\left\\{10\\right\\} & p_\\left\\{11\\right\\}\\\\ \\hline\\hline \\end\\left\\{array\\right\\}$ The above table is the joint probability function or joint distribution of $$X$$ and $$Y$$.\n\u2022 Find $$\\mu_X$$ and $$\\mu_Y$$.\n\u2022 Find $$\\sigma_X$$ and $$\\sigma_Y$$.\n\u2022 Find $$\\mathrm{corr}(X,Y)$$. Note that in this case: $\\begin\\left\\{eqnarray\\right\\} \\mathrm\\left\\{corr\\right\\}\\left(X,Y\\right) &=&\\frac\\left\\{1\\right\\}\\left\\{\\sigma_X\\,\\sigma_Y\\right\\} \\times \\\\ &&\\bigg\\left[ \\left(0-\\mu_X\\right)\\left(0-\\mu_Y\\right)\\,p_\\left\\{00\\right\\}\\\\ &+&\\left(0-\\mu_X\\right)\\left(1-\\mu_Y\\right)\\,p_\\left\\{01\\right\\}\\\\ &+&\\left(1-\\mu_X\\right)\\left(0-\\mu_Y\\right)\\,p_\\left\\{10\\right\\}\\\\ &+&\\left(1-\\mu_X\\right)\\left(1-\\mu_Y\\right)\\,p_\\left\\{11\\right\\}\\bigg\\right] \\end\\left\\{eqnarray\\right\\}$\n\u2022 Find the probability function (table) for the random variable $$Z$$, the total score of two free throws:$$Z=X+Y$$. Find $$\\mu_Z$$ and $$\\sigma_Z$$ directly.\n\u2022 Verify the equation $\\sigma_{Z}^2 = \\sigma_{X}^2+\\sigma_{Y}^2+2\\rho_{X,Y}\\,\\sigma_X\\,\\sigma_Y$\n\u2022 Please do finish the calculations above!\n\u2022 You may also use the above example to practice conditional probabilities and the Bayes' formula. For example, if it is known that a player scored on the second throw, what is the probability that she\/he missed on the first throw?\n\u2022 Know that independence of random variables $$X$$ and $$Y$$ implies $$\\mathrm{corr}(X,Y)=0$$. Know that $$\\mathrm{corr}(X,Y)=0$$ \u00a0 does not imply independence of $$X$$ and $$Y$$. However, $$\\mathrm{corr}(X,Y)=\\pm 1$$ implies that $$Y=a\\,X+b$$ for some constants $$a$$ and $$b$$. Moreover the sign of $$a$$ is the same as the sign of $$\\mathrm{corr}(X,Y)$$. This is perfect linear dependence. NOTE: A similar result was true for sample correlations.\n\u2022 Note that $$\\mathrm{corr}(X,Y)$$ is only defined when $$\\sigma_X>0$$ and $$\\sigma_Y>0$$. In particular, $$X$$ and $$Y$$ must assume at least two different values.\n\u2022 Know that there is a mean $$\\mu_X$$ of a random variable and sample mean $\\bar{x}=\\frac{1}{n}\\sum_{i=1}^n x_i$ The former is a parameter of the population, and the latter is a statistic (a property of the sample).\n\u2022 Similarly, the variance $$\\sigma_X^2$$ is a parameter and the sample variance $s_x^2 = \\frac{1}{n-1}\\sum_{i=1}^n (x_i -\\bar{x})^2$ which is a statistic.\n\u2022 Also, the correlation $$\\rho_{X,Y}$$ is a parameter, while the sample correlation $$r=r_{xy}$$ is a statistic. Recall the formula for the sample correlation: $r=r_{xy}= \\frac{1}{n-1}\\sum_{i=1}^n \\frac{x_i-\\bar{x}}{s_x}\\frac{y_i-\\bar{y}}{s_y}$ Compare with the formula for $$\\mathrm{corr}(X,Y)$$, which involves double summation.\n\n### Conditional probability\n\n\u2022 Know the meaning of $$P(A\\,|\\,B)$$\n\u2022 Know the formula: $P(A\\,|\\,B) = \\frac{P(A\\cap B)}{P(B)}$\n\u2022 Know the rules of probability for conditional probabilities. The function $$P'(A) = P(A|B)$$ satisfies all Probability Rules for fixed $$B$$. For example, $P(A\\cup C|B) = P(A|B) + P(C|B) - P(A\\cap C|B)$ Thus, if you learned a rule for ordinary, non-conditional probability, there is a corresponding rule for the conditional probability.\n\u2022 Know the Law of Alternatives also known as the Total Probability Formula. Let $$C_1,C_2,\\ldots,C_n$$ be mutually disjoint events (\"causes\"): $C_i\\cap C_j = \\emptyset\\quad\\text{when i\\neq j}$ and exhaustive events: $C_1\\cup C_2\\cup \\ldots \\cup C_n = S$ Then for every event $$A$$ (\"consequence\"): $P(A) = \\sum_{i=1}^n P(A|C_i)\\,P(C_i)$\n\u2022 Know the Bayes' Formula: $P(A|B) = \\frac{P(B|A)P(A)}{P(B)}$\n\u2022 An alternative form of the Bayes' Formula for the probability of the cause, given a known consequence: $P(C_i|A) = \\frac{P(A|C_i) P(C_i)}{\\sum_{j=1}^nP(A|C_j) P(C_j)}$\n\u2022 Know how to apply Bayes' Formula to common examples discussed by the book and slides.\n\u2022 You may find it useful to read the following article on Bayes' Theorem\n\u2022 The Monty Hall Problem provides an interesting example of an application of conditional probability. This example is often used in job interviews.","date":"2018-04-20 16:33:00","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 3, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 1, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9057466983795166, \"perplexity\": 306.23426297289825}, \"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-2018-17\/segments\/1524125944479.27\/warc\/CC-MAIN-20180420155332-20180420175332-00368.warc.gz\"}"} | null | null |
Home/Community/Season 5/Episode 11/News/Page 1
News for Episode 11: G.I. Jeff
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'Community' video: 5 seasons of movie references in one video
Apr 9, 2014 | By Zap2It | 0
"Community" references a lot of movies. From "Pulp Fiction" to "My Dinner with Andre," the films mentioned on the show run the gamut of years and styles -- sometimes in the same episode.In honor of this eclectic mix of movie love, a truly passionate fan -- Anne Thomas -- of the show has made a super-cut of all "Community" movie shout-outs from the series. The password? It's "movietime."The question is: Did any movies get missed? The only one we caught was "For a Few Dollars More," the sequel to "A Fistful of Dollars" -- the inspirations for the second iteration of paintball on "Community."Leave any other omissions in the comments section below!"Community" Season 5 airs Thursdays at 8 p.m. on NBC.... //blog.zap2it.com/frominsidethebox/2014/03/community-video-five-seasons-of-movie-references-in-one-video.html
'Community' Season 5 video: '80s cartoon PSA from 'G.I. Jeff'
"Community" Season 5 has a treat for former 1980s cartoon watchers: In the upcoming "G.I. Jeff" episode, the characters become G.I. Joe characters and even includes a PSA at the end. Watch the ridiculous video here.For this PSA, Britta (as "Buzz Kill") and Abed (as "Fourth Wall") teach a couple of kids about the evils of graffiti. Sort of. In case you're not familiar with this particular form of "educational" television, many cartoons in the 1980s and 1990s -- including "G.I. Joe" -- ended their episodes with brief skits like this. Characters from the stories would step out of their world to tell normal kids about what was right and good in this world. Some of them were almost as funny as this "Community" video. For example, one time the "Thundercats" warned a couple of youngsters that sheltering from a thunderstorm under a tall tree was a bad idea. The more you know can... //blog.zap2it.com/frominsidethebox/2014/03/community-season-5-video-80s-cartoon-psa-from-gi-jeff.html
NBC's 'Community' Previews Upcoming Animated 'G.I. Joe' Episode (Video)
Apr 7, 2014 | By The Wrap | 0
During Wednesday's PaleyFest at L.A.'s Dolby Theater, the cast and producers of NBC's "Community" introduced a first look at its animated homage to "G.I. Joe" airing Thursday, April 3 at 8/7c. The episode re-imagines the study group as military heroes just like the '80's cartoon. Creator and executive producer Dan Harmon predicted that the episode, which celebrates the toy companies and their memorable toys, will be "ratings dynamite." Read more... //www.thewrap.com/paleyfest-nbc-community-gi-joe-episode-preview
Community Season 5 Episode 11 "G.I. Jeff"
Apr 4, 2014 | By maximiliano | 0
Community "G.I. Jeff" Season 5 episode 11 airs Thursday, April 3 2014 on NBC (8-8:30 p.m. ET). Episode Synopsis: Community Season 5 Episode 11 "G.I. Jeff" – The study group gets 'animated' in the vein of the 1980s 'G.I. Joe' series. Read More... //www.tvequals.com/2014/04/03/community-season-5-episode-11-g-i-jeff/
'Community' Season 5, episode 11 gets animated for 'G.I. Jeff'
"Community" is no stranger to animation. Prior to Season 5's "G.I. Jeff" (airing Thursday, April 3), the bizarre comedy had already aired two mainly animated episodes: "Abed's Uncontrollable Christmas" (stop-motion) and "Digital Estate Planning" (8-bit computer graphics).For the Season 5 version of animation, however, episode 11 is taking the "G.I. Joe" route in tribute to the 1980s animated series.What's going on here? "This [episode] has a very specific story reason for it," executive producer Chris McKenna tells EW.com. "I don't want to give too much away, but our characters find themselves in a 'Wizard of Oz'-type journey through the world of 'G.I. Joe' ... There's a mystery that Jeff in particular has to get to the bottom of."Considering that Jeff Winger would have been about the right age to enjoy "G.I. Joe" cartoons back in the day, that does make sense.Exactly how much "G.I. Joe" will make it into "G.I. Jeff"? "We do a typical... //blog.zap2it.com/frominsidethebox/2014/03/community-season-5-episode-11-gets-animated-for-gi-jeff.html
Tweets from Community | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,577 |
Oklahoma City Thunder Employees Helped Federal Authorities Identify Suspect in Capitol Siege, Leading to Several Charges
Jerry LambeMar 1st, 2021, 5:53 pm
Federal authorities investigating individuals involved in the Jan. 6 riots at the U.S. Capitol received an assist from an unlikely source this week, with employees of the NBA's Oklahoma City Thunder helping investigators identify a woman accused of entering the Capitol Complex unlawfully.
Danielle Nicole Doyle—who was previously employed by the Thunder—became at least the second Oklahoman to be charged in connection with the Capitol siege; two of Doyle's former colleagues saw her in videos posted online and contacted the FBI.
According to the criminal complaint filed in Washington, D.C. federal court, the first witness to contact the bureau, identified only as "Witness 1," was Doyle's coworker "when they both worked for a professional sports team in Oklahoma City." Witness 1 recognized Doyle after a friend sent a video of the riots. A second witness ("Witness 2"), who appears to currently work for the Thunder, identified Doyle during an interview with federal authorities after she and several other Thunder employees "circulated" video footage of Jan. 6 amongst themselves.
"Witness 2 advised that Witness 2 works for the same professional sports team in Oklahoma City and had previously worked with Danielle Nicole Doyle, when Doyle worked for the professional sports team," the criminal complaint stated. "Witness 2 recalled that following the events at the U.S. Capitol on January 6, 2021, employees of the professional sports team circulated a video that CNN had aired. The video was of individuals inside the U.S. Capitol during the breaching of the Capitol. Witness 2 obtained a copy of the video and identified Doyle as one of the individuals in the video."
The videos allegedly show Doyle climbing through a broken window to enter the Capitol building and walking down an interior staircase in the building known as the "Supreme Court Chambers stairs." She appeared to film herself with a cell phone.
Law&Crime reached out to the Oklahoma City Thunder regarding the circumstances of the case against Doyle, but we did not receive a response by the time of publication.
Doyle faces charges of knowingly entering or remaining in a restricted building or grounds without lawful authority, knowingly engaging in disorderly or disruptive conduct in a restricted building or grounds, and violent entry and disorderly conduct on Capitol grounds.
Read the full criminal complaint below:
USA v Doyle Affidavit Complaint by Law&Crime on Scribd
[image via criminal complaint]
Capitol RiotsDanielle Nicole DoyleOklahoma CityOklahoma City Thunder | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,475 |
namespace Gestion {
partial class linq_test {
/// <summary>
/// Required designer variable.
/// </summary>
private System.ComponentModel.IContainer components = null;
/// <summary>
/// Clean up any resources being used.
/// </summary>
/// <param name="disposing">true if managed resources should be disposed; otherwise, false.</param>
protected override void Dispose(bool disposing) {
if (disposing && (components != null)) {
components.Dispose();
}
base.Dispose(disposing);
}
#region Windows Form Designer generated code
/// <summary>
/// Required method for Designer support - do not modify
/// the contents of this method with the code editor.
/// </summary>
private void InitializeComponent() {
this.dataGridView1 = new System.Windows.Forms.DataGridView();
this.button1 = new System.Windows.Forms.Button();
this.textBox1 = new System.Windows.Forms.TextBox();
this.textBox2 = new System.Windows.Forms.TextBox();
this.textBox3 = new System.Windows.Forms.TextBox();
this.label1 = new System.Windows.Forms.Label();
this.label2 = new System.Windows.Forms.Label();
this.label3 = new System.Windows.Forms.Label();
this.insBT = new System.Windows.Forms.Button();
this.updBT = new System.Windows.Forms.Button();
this.delBT = new System.Windows.Forms.Button();
this.button2 = new System.Windows.Forms.Button();
((System.ComponentModel.ISupportInitialize)(this.dataGridView1)).BeginInit();
this.SuspendLayout();
//
// dataGridView1
//
this.dataGridView1.ColumnHeadersHeightSizeMode = System.Windows.Forms.DataGridViewColumnHeadersHeightSizeMode.AutoSize;
this.dataGridView1.Location = new System.Drawing.Point(13, 13);
this.dataGridView1.Name = "dataGridView1";
this.dataGridView1.Size = new System.Drawing.Size(305, 306);
this.dataGridView1.TabIndex = 0;
//
// button1
//
this.button1.Location = new System.Drawing.Point(504, 292);
this.button1.Name = "button1";
this.button1.Size = new System.Drawing.Size(80, 23);
this.button1.TabIndex = 1;
this.button1.Text = "Test Select";
this.button1.UseVisualStyleBackColor = true;
this.button1.Click += new System.EventHandler(this.button1_Click);
//
// textBox1
//
this.textBox1.Location = new System.Drawing.Point(477, 24);
this.textBox1.Name = "textBox1";
this.textBox1.Size = new System.Drawing.Size(100, 20);
this.textBox1.TabIndex = 2;
//
// textBox2
//
this.textBox2.Location = new System.Drawing.Point(477, 64);
this.textBox2.Name = "textBox2";
this.textBox2.Size = new System.Drawing.Size(100, 20);
this.textBox2.TabIndex = 3;
//
// textBox3
//
this.textBox3.Location = new System.Drawing.Point(477, 103);
this.textBox3.Name = "textBox3";
this.textBox3.Size = new System.Drawing.Size(100, 20);
this.textBox3.TabIndex = 4;
//
// label1
//
this.label1.AutoSize = true;
this.label1.Location = new System.Drawing.Point(342, 31);
this.label1.Name = "label1";
this.label1.Size = new System.Drawing.Size(79, 13);
this.label1.TabIndex = 5;
this.label1.Text = "Numero Cliente";
//
// label2
//
this.label2.AutoSize = true;
this.label2.Location = new System.Drawing.Point(342, 71);
this.label2.Name = "label2";
this.label2.Size = new System.Drawing.Size(44, 13);
this.label2.TabIndex = 6;
this.label2.Text = "Nombre";
//
// label3
//
this.label3.AutoSize = true;
this.label3.Location = new System.Drawing.Point(342, 110);
this.label3.Name = "label3";
this.label3.Size = new System.Drawing.Size(70, 13);
this.label3.TabIndex = 7;
this.label3.Text = "Limite Credito";
//
// insBT
//
this.insBT.Location = new System.Drawing.Point(332, 152);
this.insBT.Name = "insBT";
this.insBT.Size = new System.Drawing.Size(80, 23);
this.insBT.TabIndex = 8;
this.insBT.Text = "Insert";
this.insBT.UseVisualStyleBackColor = true;
this.insBT.Click += new System.EventHandler(this.insBT_Click);
//
// updBT
//
this.updBT.Location = new System.Drawing.Point(418, 152);
this.updBT.Name = "updBT";
this.updBT.Size = new System.Drawing.Size(80, 23);
this.updBT.TabIndex = 9;
this.updBT.Text = "Update";
this.updBT.UseVisualStyleBackColor = true;
this.updBT.Click += new System.EventHandler(this.updBT_Click);
//
// delBT
//
this.delBT.Location = new System.Drawing.Point(504, 152);
this.delBT.Name = "delBT";
this.delBT.Size = new System.Drawing.Size(80, 23);
this.delBT.TabIndex = 10;
this.delBT.Text = "Delete";
this.delBT.UseVisualStyleBackColor = true;
this.delBT.Click += new System.EventHandler(this.delBT_Click);
//
// button2
//
this.button2.Location = new System.Drawing.Point(389, 292);
this.button2.Name = "button2";
this.button2.Size = new System.Drawing.Size(80, 23);
this.button2.TabIndex = 11;
this.button2.Text = "Test Select";
this.button2.UseVisualStyleBackColor = true;
this.button2.Click += new System.EventHandler(this.button2_Click_1);
//
// linq_test
//
this.AutoScaleDimensions = new System.Drawing.SizeF(6F, 13F);
this.AutoScaleMode = System.Windows.Forms.AutoScaleMode.Font;
this.ClientSize = new System.Drawing.Size(601, 327);
this.Controls.Add(this.button2);
this.Controls.Add(this.delBT);
this.Controls.Add(this.updBT);
this.Controls.Add(this.insBT);
this.Controls.Add(this.label3);
this.Controls.Add(this.label2);
this.Controls.Add(this.label1);
this.Controls.Add(this.textBox3);
this.Controls.Add(this.textBox2);
this.Controls.Add(this.textBox1);
this.Controls.Add(this.button1);
this.Controls.Add(this.dataGridView1);
this.Name = "linq_test";
this.Text = "linq_test";
this.Load += new System.EventHandler(this.linq_test_Load);
((System.ComponentModel.ISupportInitialize)(this.dataGridView1)).EndInit();
this.ResumeLayout(false);
this.PerformLayout();
}
#endregion
private System.Windows.Forms.DataGridView dataGridView1;
private System.Windows.Forms.Button button1;
private System.Windows.Forms.TextBox textBox1;
private System.Windows.Forms.TextBox textBox2;
private System.Windows.Forms.TextBox textBox3;
private System.Windows.Forms.Label label1;
private System.Windows.Forms.Label label2;
private System.Windows.Forms.Label label3;
private System.Windows.Forms.Button insBT;
private System.Windows.Forms.Button updBT;
private System.Windows.Forms.Button delBT;
private System.Windows.Forms.Button button2;
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 5,748 |
Elliot Minor for special November/ December 2010 Solaris Acoustic album and UK/ Ireland tour
Posted: 9th October 2010
Elliot Minor - Solaris Acoustic
Elliot Minor have just announced details of an acoustic version of their 2009 album 'Solaris', complete with a DVD of the Kerrang Live Special.
The Symphonic rockers from York will release the album and DVD on 8th November, available either in CD/ DVD form from their website directly (www.elliotminor.com) and as of December via iTunes (audio only) as a download. The album and tour was perhaps prompted by the recent departure of keyboard player Ali Paul this August.
Of course, with a new album to plug, comes a tour - and it's (of course) an acoustic tour, allowing the band to showcase the newly acoustic versions of their 'Solaris' songs. The tour kicks-off in Birmingham on 12th November, moving to their hometown for a date at York's Stereo on 13th, finishing on 11th December in Glasgow, with further dates in London, Belfast, Dublin's recently-opened Workman's Club and a few more in-between (see below for the full list and tickets).
Tickets will cost just £10 and go on sale Monday 11th October from - now, pay attention: London (8AM), Bristol (10AM) and the others at 9AM.
Saturday, 9th Oct 2010
Elliot Minor
Cheap Days/ Nights Out!
Under a Tenner
Under a Fiver
JimmyEatWorld
WeWillRockYou
Oct 2014:The Week Ahead: The Specials, Jamie T, Netsky, The Courteeners, Counting Crows, Maceo Parker, The Levellers with The Selecter,
This Week:Fat Friday: Paul Weller, Katy B, Katherine Jenkins, Elliot Minor, Korn, Embrace, Frank Skinner and more
Nov 2010:The Week Ahead from 8th Nov 2010: Goldfrapp, Gorillaz, MIA, Hot Chip , LCD Soundsystem, Paramore plus more
Jul 2010:Elliot Minor set to play major dates - UK tour confirmed for September 2010 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 88 |
Married Couple Die of Coronavirus Four Minutes Apart Holding Hands for the Last Time
Patrick Hill
A family in North Carolina is grieving the loss of 2 family members as, after a month-long battle, a husband and wife passed just 4 minutes apart.
This is utterly heartbreaking.
The past few months have been hard.
As the pandemic continues, so does the devastation it leaves.
Far too many lives have been lost.
And families split apart, unable to see loved ones in their last moments.
Despite all the hard work of the incredible medical staff, some truly heartbreaking stories of loss have emerged.
But this week, there's one story that stood out.
A heartbreaking story...
That really shows the power of love.
As the wedding vows go...
A married couple will love each other, in both sickness and in health.
However, not all marriages stick strictly to their vows.
It is quite unusual in today's day and age for a married couple to stay together longer than a decade.
The statistics aren't great...
Figures show that the average U.S. marriage only lasts around 7 years... Which doesn't really give a couple much time to fully commit to their wedding vows.
However, some couples are in it for the long run...
And remain faithful to one another, despite what life throws in their path.
They devote their lives to each other.
And stay by each other's side until the very end.
One married couple has recently gone viral for this exact reason after they died just 4 minutes apart...
But it is amid the most heartbreaking of circumstances, due to the pandemic.
Johnny Lee and wife, Cathy Darlene Peoples, were together for 50 years.
Johnny, 67, was a sergeant in the U.S. Army and worked for the NC Department of Corrections, while wife Cathy, 65, worked as a teaching assistant and a lab technician.
The couple had been married 48 happy years and spent much of their life together.
But unfortunately, around a month ago tragedy hit.
Whilst just days away from Cathy's retirement...
The couple began suffering from coronavirus.
The couple's son, Shane People,s explained that his parents gradually declined.
Speaking to WBTV he said: "It was mainly the fever and loss of taste. My dad starting showing symptoms two days later. About two weeks later they were both put in the ICU. Everything just went south, everything just got worse."
Staff at the Novant Health Rowan Regional Medical Center made preparations to bring the couple together where possible, once they knew they were not getting better.
And 30 days after battling the virus, Johnny and Cathy passed, holding hands, just 4 minutes apart.
Shane said "The next day they put them in the same room, same ICU room, they put their hands together, the nurses gathered around and they passed within 4 minutes of each other."
In a post on Facebook Shane wrote, "The lives of Mom and Dad were stolen by a virus that many joke about on a daily basis or just straight out believe it's a hoax of some sort. Both of them took this pandemic seriously and still got sick, still died...My parents weren't just a blessing for me, my brother, my sister, our spouses, and our children. They were a blessing to every person that met them."
"They both loved their family very much and did anything and everything they could possibly do for them. I had some pretty darn awesome parents."
A private memorial service was held for the family.
In an obituary online it read, "Johnny loved coaching youth sports, playing music and building the family tree. Darlene enjoyed crafting, listening to music and playing cards. They both enjoyed fishing and spending quality time with family and friends."
Shane also added to WBTV "…they died together holding hands and walked into Heaven holding hands."
They are survived by their 3 children, 9 grandchildren, and numerous relatives.
Rest in Johnny and Cathy.
Our thoughts are with their friends and family at this difficult time. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,070 |
Q: Replacing dumb quotes with smart quotes and keeping caret position in contenteditable div I'm trying to turn some dumb quotes into curly quotes on the fly in a contenteditable div (as seen here), using the following:
$("#editor").on("keyup", ".content", function () {
$(".content").html(function() {
var start = this.selectionStart, end = this.selectionEnd;
return $(this).html()
.replace(/'\b/g, "\u2018") // opening single
.replace(/\b'/g, "\u2019") // closing single
.replace(/"\b/g, "\u201c") // opening double
.replace(/\b"/g, "\u201d") // closing double
.replace(/--/g, "\u2014") // em-dash
.replace(/\b\u2018\b/g, "'"); // handle conjunctions
this.setSelectionRange(start, end);
});
// other stuff happens here...
});
The return bit works fine on its own, but moves the caret position back to the start of the dive after every keystroke, which is obviously not desirable. But trying to keep the caret position (using some code seen here) throws Unreachable 'this' after 'return' in JSHint and so doesn't actually do anything in the browser. Can someone point me in the right direction here, please?
A: I always tend to keep this in a variable to avoid scope problems. I also don't really understand why you try so hard to encapsulate all in anonymous function. Try changing your code as following:
$("#editor").on("keyup", ".content", function () {
var target = this;
var selection = window.getSelection();
var range = selection.getRangeAt(0);
var startOffset = range.startOffset;
var endOffset = range.endOffset;
var html = $(target).html();
var newHtml = html.replace(/'\b/g, "\u2018") // opening single
.replace(/\b'/g, "\u2019") // closing single
.replace(/"\b/g, "\u201c") // opening double
.replace(/\b"/g, "\u201d") // closing double
.replace(/--/g, "\u2014") // em-dash
.replace(/\b\u2018\b/g, "'"); // handle conjunctions
var delta = html.length - newHtml.length;
$(target).html(newHtml);
var newRange = document.createRange();
newRange.setStart(range.startContainer, startOffset - delta);
newRange.setEnd(range.startContainer, endOffset - delta);
selection.removeAllRanges();
selection.addRange(newRange);
// other stuff happens here...
});
I corrected the code, making it work with contenteditable under Chrome. The behaviour of resetting caret position to the start does not occur with IE apparently. Required knowledge was found at:
https://developer.mozilla.org/en-US/docs/Web/API/Selection
Edit2: I fixed issue with "--" and prepared a fiddle. All works fine in Chrome and IE as I tested. http://jsfiddle.net/mcoo/8oyn41b1/
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,600 |
Q: Element not interactable Error : When script is executed via jenkins, But works fine from my local When i execute my script locally it works fine but, when i execute the same code via jenkins it says element not interactable.Is there any solution for this issue?
I already have explicit wait conditions in my script.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 3,381 |
using System;
using System.Collections;
using System.IO;
using System.Text;
namespace Acklann.Mockaroo
{
/// <summary>
/// Provides extension methods.
/// </summary>
public static class Helper
{
/// <summary>
/// Returns a MD5 hash of the specified <see cref="Schema"/>.
/// </summary>
/// <param name="schema">The schema.</param>
/// <returns></returns>
public static string ComputeHash(this Schema schema)
{
var md5 = System.Security.Cryptography.MD5.Create();
byte[] hash = md5.ComputeHash(Encoding.UTF8.GetBytes(schema.ToString()));
return BitConverter.ToString(hash);
}
internal static string ComputeHash(byte[] data)
{
var md5 = System.Security.Cryptography.MD5.Create();
byte[] hash = md5.ComputeHash(data);
return BitConverter.ToString(hash);
}
internal static string ComputeHash(string filePath)
{
if (!File.Exists(filePath)) return string.Empty;
var md5 = System.Security.Cryptography.MD5.Create();
byte[] hash = md5.ComputeHash(File.ReadAllBytes(filePath));
return BitConverter.ToString(hash);
}
internal static bool IsNullOrEmpty(this ICollection list)
{
return list == null || list.Count < 0;
}
internal static string CreateDirectory(string filePath)
{
string dir = Path.GetDirectoryName(filePath);
if (!Directory.Exists(dir)) Directory.CreateDirectory(dir);
return filePath;
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 8,993 |
{"url":"https:\/\/physics.stackexchange.com\/questions\/269410\/what-is-the-relationship-between-the-magnetic-units-oersted-and-tesla","text":"What is the relationship between the magnetic units oersted and tesla?\n\nHow are the units oersted and tesla related? For example, how would you express $20\\:\\mathrm{Oe}$ in tesla?\n\n\u2022 @ Ganesh, Oersted and Tesla dont talk to each other as each of them used to measure two physically different quantity. \u2013\u00a0AMS Jul 22 '16 at 16:44\n\nThey are technically units for incommensurate quantities, but in practice this is often just a technicality. The magnetic field that makes sense ($B$) is measured in teslas (SI) or gauss (CGS), and the magnetic field that people spoke about 100 years ago ($H$) is measured in amps per meter (SI, also equivalent to a number of other things) or oersteds (CGS).\n\nTo go between the two unit systems, we have \\begin{align} 1\\ \\mathrm{G} & = 10^{-4}\\ \\mathrm{T}, \\\\ 1\\ \\mathrm{Oe} & = \\frac{1000}{4\\pi} \\mathrm{A\/m}. \\end{align} To go between the two magnetic fields, we have \\begin{align} \\frac{B}{1\\ \\mathrm{G}} & = \\mu_r \\frac{H}{1\\ \\mathrm{Oe}} & \\text{(CGS)}, \\\\ B & = \\mu_r \\mu_0 H & \\text{(SI)}, \\end{align} where $\\mu_r$ is the dimensionless relative permeability of the medium ($1$ for vacuum and pretty much any material other than strong magnets) and $\\mu_0 = 4\\pi \\times 10^{-7}\\ \\mathrm{H\/m}$ (henries per meter) is the vacuum permeability.\n\nTherefore a $1\\ \\mathrm{Oe}$ corresponds to $10^{-4}\\ \\mathrm{T}$ in non-magnetic materials.\n\nOne caveat is that there are cases where $B$ and $H$ are not so simply related. If you are interested in their directions and not just magnitudes, then in some materials $\\mu_r$ is actually a tensor and can rotate one field relative to the other. In this case the relation is still linear. In worse cases (e.g. ferromagnets) the relationship is not linear and cannot be expressed in the forms presented above. At least the $\\mathrm{G} \\leftrightarrow \\mathrm{T}$ and $\\mathrm{Oe} \\leftrightarrow \\mathrm{A\/m}$ relations always hold.\n\n\u2022 Worth noting there are still plenty of experimentalists who talk about $H$ because $H$ is directly related to current and therefore under your control, while $B$ includes the magnetic response of the medium and is usually more complicated. \u2013\u00a0rob Jul 22 '16 at 19:37\n\u2022 @rob any word on whether they use CGS or SI units? You'd how that all experiments now use SI, but with astronomy there to set the example, you never know. \u2013\u00a0Emilio Pisanty Jul 22 '16 at 23:32\n\u2022 @EmilioPisanty I can't think off the top of my head but I may have an opportunity to confirm soon. I expect \"gauss\" for both $B$ and $H$, which in free space isn't such a bad way to go. \u2013\u00a0rob Jul 23 '16 at 0:15\n\u2022 @rob gauss for $H$? That's the sort of stuff that makes cgs such a weird \"almost\" on being nice and consistent and stuff. What's with having independent but semi-commensurate units for quantities that ought to have the same dimensionality, anyway? Man, cgs is weird. \u2013\u00a0Emilio Pisanty Jul 23 '16 at 2:03\n\u2022 @chris white what does '1 Oe corresponds to 10^\u22124 T in non-magnetic materials' means.Does this indicates that 1Oe equals to 1G \u2013\u00a0Ganesh Jul 23 '16 at 4:13\n\nThis is a relatively tricky one, because it involves the differences between the $\\mathbf B$ field and the $\\mathbf H$ field in the SI and CGS systems, and those relationships change in the different systems. In short:\n\n\u2022 Oersteds are used to measure the $\\mathbf H$ field in CGS units.\n\n\u2022 Teslas are used to measure the $\\mathbf B$ field in SI units.\n\n\u2022 In the SI system, the two fields are related via $\\mathbf B=\\mu_0(\\mathbf H+\\mathbf M)$ where $\\mu_0$ is the vacuum permeability and $\\mathbf M$ is the magnetization (volumetric density of magnetic dipole moment).\n\n\u2022 In a linear medium $\\mathbf M=\\chi_\\mathrm m \\mathbf H$, for $\\chi_\\mathrm m$ the material's dimensionless magnetic susceptibility, and the fields are related by $\\mathbf B=\\mu_0(1+\\chi_\\mathrm m)\\mathbf H=\\mu_0\\mu_r\\mathbf H=\\mu\\mathbf H$.\n\n\u2022 In the SI system, the $\\mathbf H$ field is measured in amperes per meter.\n\n\u2022 In the CGS system (gaussian and EMU units) the two fields are related via $\\mathbf B=\\mathbf H+4\\pi\\mathbf M$.\n\n\u2022 In a linear medium the magnetic polarization is also $\\mathbf M=\\chi_\\mathrm m \\mathbf H$ (but with a different susceptibility, $\\chi_\\mathrm m^\\mathrm{(CGS)}=\\frac{1}{4\\pi}\\chi_\\mathrm m^\\mathrm{(SI)}$), and $\\mathbf B=(1+4\\pi\\chi_\\mathrm m)\\mathbf H=\\mu\\mathbf H$, where now the material's permeability and relative permeability coincide.\n\n\u2022 Not all magnetic materials are linear. In particular, permanent magnets are best thought of as having a permanent, fixed magnetization $\\mathbf M$. In these cases the magnetic susceptibility and permeability is undefined inside the material.\n\n\u2022 As you might note, CGS has two distinct units, the oersted and the gauss, for two quantities of the same physical dimensionality. I'm not quite sure why people felt the need for two such units, but it's apparently one of the quirky reasons why using CGS units makes you \"cool\".\n\n\u2022 To repeat the caveat that Chris mentions, even if a material is linear it might still be inhomogeneous (i.e. have a spatially dependent $\\mu$), in which case the $\\mathbf B\\leftrightarrow\\mathbf H$ relationship changes from place to place, and it might still be anisotropic, in which case $\\mathbf B$ and $\\mathbf H$ can point in different directions and you only have a linear relationship between their components, $B_j=\\sum_k \\mu_{jk}H_k$, and $\\mu$ jumps from a scalar to a rank-two tensor (a.k.a. a matrix). In that case, you can still use the relationships below to relate the components and magnitudes of the fields, but you should doubly so heed the advice on sticking to a single system.\n\n\u2022 Thankfully, the SI and CGS systems do agree at least on the relative permeability of linear materials.\n\nThe easy part of the answer is that in vacuum the two units can be uniquely identified. In this case a CGS $\\mathbf H$ field strength of $1$ oersted coincides with a CGS $\\mathbf B$ field strength of $1$ gauss, which is exactly $10^{-4}\\:\\mathrm T$.\n\nOn the other hand, in a magnetic material, i.e. anywhere where $\\mathbf M\u22600$, the conversion factor will depend on the material.\n\n\u2022 If the material is not linear, then there is no way to relate the two, because you do not have enough information to relate the $\\mathbf B$ and $\\mathbf H$ fields.\n\n\u2022 If you know the material is linear and has a relative permeability $\\mu_r$ (equal to its CGS permeability), then a $\\mathbf H$ field of $1$ oersted corresponds to a CGS $\\mathbf B$ field of $\\frac{1}{\\mu_r}$ gauss and therefore an SI $\\mathbf B$ field of $(10^{-4}\/\\mu_r)$ tesla.\n\nIf you want to interpret a CGS $\\mathbf H$ field, a safer choice is to translate it into an SI $\\mathbf H$ field, for which a CGS $\\mathbf H$ field strength of $1$ oersted corresponds to an SI $\\mathbf H$ field strength of $10^{-4}$ amperes per meter.\n\nMore generally, though, if you're looking to convert between teslas and oersteds, my advice is: don't. Choose one one EM unit+formula system, and stick to it. If you need data on a material's properties that's in another system, convert that to the one you're using first thing. If you're looking to compare $\\mathbf B$ and $\\mathbf H$ field values, you had better have a good idea of exactly what the deal is with your material - and still, you should stick to one system.\n\nFrom a quick google search, it seems that Oersteds are used for defining magnetic field strength and Teslas are used for defining magnetic field strength in terms of flux density. They seem to not really be meant to be converted between, though you technically can (as evidenced by the other answers here).\n\nThis website and this website might be helpful to you.\n\nUpdate:\n\nThe other answers here cover the conversion very well.","date":"2020-02-25 12:27:10","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 2, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.860937774181366, \"perplexity\": 486.7448790497453}, \"config\": {\"markdown_headings\": false, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2020-10\/segments\/1581875146066.89\/warc\/CC-MAIN-20200225110721-20200225140721-00354.warc.gz\"}"} | null | null |
Do you feel that you pay too much for your TV service each month? If so, do you really watch all of the channels that you are subscribed to? If you don't watch all of those channels, it may be time to consider downsizing your subscription package. Did you know that some shows can be watched online without needing a subscription? There are a lot of ways to reduce the cost of your TV service without missing out on the shows that you enjoy watching. My blog is loaded with advice that can help you determine if it is time to make changes and help you decide which changes to make. | {
"redpajama_set_name": "RedPajamaC4"
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use byteorder::{LittleEndian, ReadBytesExt, WriteBytesExt};
use gump::GumpReader;
use image::Pixel;
use mul_reader::simple_from_vecs;
use std::io::Cursor;
fn example_gump_mul() -> GumpReader<Cursor<Vec<u8>>> {
let mut data = Cursor::new(vec![]);
data.write_u32::<LittleEndian>(3).unwrap(); //Row 1 offset
data.write_u32::<LittleEndian>(6).unwrap(); //Row 2 offset
data.write_u32::<LittleEndian>(7).unwrap(); //Row 3 offset
data.write_u16::<LittleEndian>(0).unwrap(); //Black
data.write_u16::<LittleEndian>(1).unwrap(); //One pixel
data.write_u16::<LittleEndian>(0xFFFF).unwrap(); //White
data.write_u16::<LittleEndian>(1).unwrap(); //1 pixel
data.write_u16::<LittleEndian>(0).unwrap(); //Black
data.write_u16::<LittleEndian>(1).unwrap(); //1 pixel
data.write_u16::<LittleEndian>(0xFFFF).unwrap(); //White
data.write_u16::<LittleEndian>(3).unwrap(); //3 pixels
data.write_u16::<LittleEndian>(0).unwrap(); //Black
data.write_u16::<LittleEndian>(1).unwrap(); //One pixel
data.write_u16::<LittleEndian>(0xFFFF).unwrap(); //White
data.write_u16::<LittleEndian>(1).unwrap(); //1 pixel
data.write_u16::<LittleEndian>(0).unwrap(); //Black
data.write_u16::<LittleEndian>(1).unwrap(); //1 pixel
let mul_reader = simple_from_vecs(vec![data.into_inner()], 3, 3);
GumpReader::from_mul(mul_reader)
}
#[test]
fn test_gump() {
let mut reader = example_gump_mul();
match reader.read_gump(0) {
Ok(gump) => {
let image = gump.to_image();
assert_eq!(image.width(), 3);
assert_eq!(image.height(), 3);
let transparent = (0, 0, 0, 0);
let white = (255, 255, 255, 255);
assert_eq!(image.get_pixel(0, 0).channels4(), transparent);
assert_eq!(image.get_pixel(1, 0).channels4(), white);
assert_eq!(image.get_pixel(2, 0).channels4(), transparent);
assert_eq!(image.get_pixel(0, 1).channels4(), white);
assert_eq!(image.get_pixel(1, 1).channels4(), white);
assert_eq!(image.get_pixel(2, 1).channels4(), white);
assert_eq!(image.get_pixel(0, 2).channels4(), transparent);
assert_eq!(image.get_pixel(1, 2).channels4(), white);
assert_eq!(image.get_pixel(2, 2).channels4(), transparent);
}
Err(err) => panic!("{}", err),
};
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,767 |
Q: How would you program this switchboard in your programming language of choice? These are the rules:
*
*A is true if B and C are false
*B is false if A is true
*C is false if B is true
The state of each is represented in code by a boolean variable or a closure that returns a boolean. At lease one of the variables must be true at all times.
I'm interested in clever ways to program this logic in different programming languages. I'm particularly interested in expressing the logic via bitwise operators, if that's possible. I'm not extremely interested in readability, although a concise and readable solution would be impressive.
If this question is too subjective for StackOverflow, could you please suggest a better forum?
A: Fairly universal
if(b === false && c === false)
a = true;
if(a === true)
b = false;
if(b === true)
c = false;
Guess it all depends on what controls a, b, and c.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,618 |
All categories Home Health Eco Dev
Craven County COVID-19 Update for January 15, 2021
According to the North Carolina Department of Health and Human Services (NCDHHS) COVID-19 Dashboard, Craven County has 6,240 COVID-19 cases as of 11:00 a.m. on January 15, 2021. A total of 439 Craven County residents have been hospitalized due t...
Additional Info... Home
Craven County Updates COVID-19 Vaccination Plan
The Craven County Health Department updated the local COVID-19 vaccination plan in accordance with the North Carolina Department of Health and Human Services (NCDHHS) plan that directly aligns with Centers for Disease Control (CDC) guidance. The CDC annou...
Craven County Moves to Phase 1b, Activates Online COVID-19 Vaccine Scheduler and Call Center
New Bern, N.C. - The Craven County Health Department has activated an online COVID-19 vaccine appointment scheduler and call center as Phase 1b begins today in Craven County. The call center will be in operation from 8:30 a.m. until 4:30 p.m. Monday throu...
Craven County COVID-19 Vaccination Plan
The Craven County Health Department has worked with local and state partners to implement a local COVID-19 vaccination plan in accordance with the North Carolina Department of Health and Human Services (NCDHHS) plan that directly aligns with Centers for D...
Craven Area Rural Transit System (CARTS) Fares Temporarily Suspended
The Craven County Board of Commissioners approved a temporary suspension of Craven Area Rural Transit System (CARTS) fares from January 1, 2021 through June 30, 2021. Fares will resume July 1, 2021. CARTS is the public transit system c...
Beatrice Riggs Smith Selected for District Three Craven County Board of Commissioner Seat
Beatrice Riggs Smith was selected by the Craven County Board of Commissioners at their December 14, 2020 work session to fill the Craven County District Three Craven County Board of Commissioners seat that became vacant after long-time Commissioner Johnni...
Rent and Utility Assistance Program
The Housing Opportunities and Prevention of Evictions (HOPE) Program is a new statewide initiative that may provide rent and utility assistance to eligible low- and moderate-income renters experiencing financial hardship due to the econo...
Recreation and Parks Office in Craven County is Relocating
The Craven County Recreation and Parks office will be relocating to Creekside Park, 1821 Old Airport Road in New Bern on Thursday, October 8, 2020 and Friday, October 9, 2020. The Creekside Park office will be open to the public on Mond... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,981 |
{"url":"https:\/\/math.stackexchange.com\/questions\/3119155\/integrate-orthogonal-function-over-solid-angle","text":"# Integrate orthogonal function over solid angle\n\nHow do I integrate a product of Legendre polynomials over a volume?\n\nSo I understand that bunch of complete basis orthogonal basis as well. i.e. for Legendre polynomial, $$\\int_{-1}^1P_n(x)P_m(x)dx=\\delta_m^n\\frac{2}{2n+1}$$. However, I'm not sure how this was applied\/transferred into the integration of solid angle. $$\\int_{4\\pi} P_n(cos(\\theta)) P_m(cos(\\theta)) d\\Omega$$.\n\nCan you provide a proof please?\n\nIn general, how does integration over sold angle work for the usual orthogonal basis? Especially, in ($$sin,cos$$ basis), (Hermite polynomials basis), and (Spherical harmonics basis), are there another form that followed the orthogonal condition as well?","date":"2019-09-21 23:21:02","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 3, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.9768484830856323, \"perplexity\": 1210.1392208848984}, \"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\/1568514574710.66\/warc\/CC-MAIN-20190921231814-20190922013814-00130.warc.gz\"}"} | null | null |
<table
mat-table
fxFill
[dataSource]="dataSource"
aria-label="List of HTTP Requests"
(keydown)="keyManager?.onKeydown($event)">
<ng-container matColumnDef="method">
<th mat-header-cell *matHeaderCellDef>Method</th>
<td mat-cell *matCellDef="let request">{{request.method}}</td>
</ng-container>
<ng-container matColumnDef="host">
<th mat-header-cell *matHeaderCellDef>Host</th>
<td mat-cell *matCellDef="let request" [attr.title]="request.host">{{request.host}}</td>
</ng-container>
<ng-container matColumnDef="pathquery">
<th mat-header-cell *matHeaderCellDef>Path + Query</th>
<td mat-cell *matCellDef="let request" [attr.title]="request.path + request.query">
<span style="padding-right: 1ch;">{{request.path}}</span><span>{{request.query}}</span>
</td>
</ng-container>
<ng-container matColumnDef="type">
<th mat-header-cell *matHeaderCellDef>Type</th>
<td mat-cell *matCellDef="let request">{{request.type}}</td>
</ng-container>
<ng-container matColumnDef="status">
<th mat-header-cell *matHeaderCellDef>Status</th>
<td mat-cell *matCellDef="let request">{{request.pending?"intercepted":request.hasResponse?"received":"sent"}}</td>
</ng-container>
<tr mat-header-row *matHeaderRowDef="displayedColumns; sticky: true"></tr>
<tr
mat-row
appRequestListItem
*matRowDef="let row; columns: displayedColumns; let i = index;"
[request]="row"
[disabled]="!row.visible"
[hidden]="!row.visible"
(click)="setActive(i)"
(focus)="setActive(i)"
[attr.tabindex]="isActive(i)?0:-1"
[attr.aria-selected]="isActive(i)"
[ngClass]="{'cdk-keyboard-focused': isActive(i)}">
</tr>
</table>
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,898 |
\section{Introduction}
\input{01_Introduction.tex}
\section{Related work} \label{prevWork}
\input{02_PrevWork.tex}
\section{Spatiotemporal aperture coding} \label{mask}
\input{03_coding.tex}
\section{The color-coded motion deblurring network} \label{CNN}
\input{04_CNN.tex}
\section{Experiments} \label{exp}
\input{05_exp.tex}
\section{Conclusion} \label{sum}
\input{06_sum.tex}
{\small
\bibliographystyle{ieee_fullname}
\subsection{Comparison to other coding methods}
In order to demonstrate our PSF estimation ability in the motion deblurring process vs. the motion direction sensitivity of the other methods, a scene with rotating spoke resolution target is simulated. Such scene contains motion in all directions and in various velocities (according to the distance from the center of the spoke target) simultaneously.
\begin{figure}[tb]
\def0.4{0.4}
\centering
\begin{tabular}{ c c c}
\includegraphics[width = 0.4\columnwidth]{Figures/spkWArr.png} &
\includegraphics[width = 0.4\columnwidth]{Figures/flt_spk_rec_rs.png}\\
{\small{(a) Rotating target}}&
{\small{(b) Flutter-shutter rec.}}\\
\includegraphics[width = 0.4\columnwidth]{Figures/prb_spk_rec.png} &
\includegraphics[width = 0.4\columnwidth]{Figures/our_spk_Rec.png}\\
{\small{(c) Parabolic motion rec.}}&
{\small{(d) Our rec.}}\\
\end{tabular}
\caption{\textbf{Simulation results of rotating target:} (a) rotating target and the reconstruction results for (b) fluttered-shutter, (c) parabolic motion camera and (d) our method.}
\label{fig:sim_res}
\end{figure}
The synthetic scene serves as an input to the imaging simulation for the three different methods (fluttered shutter, parabolic motion and ours). The fluttered shutter code being used (both in the imaging and reconstruction) is for motion to the right, in the extent of the linear motion of the outer parts of the spoke target. The parabolic motion takes place on the horizontal direction. Each imaging result is noised using AWGN with $\sigma=3$ to simulate a real imaging scenario in good lighting conditions (since the fluttered shutter coding blocks 50\% of the light throughput, the noise level of its image is practically doubled). Fig.~\ref{fig:sim_res} presents the deblurring results of the three different techniques.\footnote{The imaging simulations for the fluttered shutter and parabolic motion cameras were implemented by us, following the descriptions in \cite{Raskar_Flut_1,Levin_motionInvariant}. The fluttered shutter reconstruction is performed using the code released by the authors. The parabolic motion reconstruction is performed using the Lucy-Richardson deconvolution algorithm \cite{Richardson:72,Lucy:1974yx}, as suggested by the authors in Section 4.1 of \cite{Levin_motionInvariant}. As the authors stated, a little better performance can be achieved using the original algorithm used in \cite{Levin_motionInvariant}, but its implementation is not available. Moreover, probably much better results can be achieved for both methods using a CNN based reconstruction. However, the main issue in the current comparison is the sensitivity of the other coding methods to the motion direction, which is not related to the used reconstruction algorithm.}
The fluttered shutter based reconstruction restores the general form of the area with the corresponding motion coding (outer lower part, moving right), and some of the opposite direction (outer upper part, moving left), and fails on all other directions/velocities. This can be partially solved using a different coding that allows both PSF estimation and inversion. Yet, this introduces an estimation-invertibility trade-off. Note also that a rotating target is a challenging case for shift-variant PSF estimation, and thus, as can be seen, a restoration with incorrect PSF leads to poor results. Moreover, the noise sensitivity of this approach is apparent, as it blocks 50\% of the light throughput.
The parabolic-motion method achieves good reconstruction for the horizontal motion (both left and right) as can be seen in the upper and lower parts of the spoke (which move horizontally). Yet, notice that its performance are not the same for left/right (as any practical finite parabolic motion cannot generate a true motion invariant PSF). Also, both vertical motions are not coded properly, and therefore are not reconstructed well. Using our method, motion in all directions can be estimated, which allows a shift variant blind deblurring of the scene.
\subsection{Comparison to blind deblurring}
To analyze the advantages in motion-cues coding, our method is compared to the multiscale motion deblurring CNN presented by Nah \etal \cite{Nah_deepDeblur_gopro}. The test set of the GoPro dataset is used as the input. Since Nah \etal trained their model on sequences of between 7-13 frames, similar scenes were created using both our coding method and simple frame summation (as used in \cite{Nah_deepDeblur_gopro}, with the proper gamma-related transformations). Note that in our case, a spatial (diffraction related) blur is added with the motion blur, so our model is handling a more challenging task.
The reconstruction results are compared for several noise levels- $\sigma=[0,3]$ on a $[0,255]$ scale (the reference method was trained with $\sigma=2$). The measures on each motion length are averaged over the different noise levels, and the results are displayed in Table~\ref{Tab:goProStats}. As can be clearly seen, our method provides an advantage in the recovery error over the method of Nah \etal \cite{Nah_deepDeblur_gopro} in both PSNR and SSIM (visual reconstruction results are presented in the supplementary material). In small motion lengths, both methods provide visually pleasing restorations (though our method is more accurate in terms of PSNR/SSIM). Yet, as the motion length increases our improvement becomes more significant. This can be explained by the fact that the architecture used in \cite{Nah_deepDeblur_gopro} is trained using adversarial loss, and therefore inherent data hallucination occurs in their reconstruction. As the motion length gets larger, such data hallucination is less accurate, and therefore the reduction in their PSNR/SSIM performance is more significant. Our method employs the encoded motion cues for the reconstruction, therefore providing more accurate results.
Note also that our model is trained only on images generated using sequences of 9 frames, and the results for shorter/longer sequences are still better than the model from \cite{Nah_deepDeblur_gopro} that is trained on sequences in all this range. This clearly shows that our model has learned to extract the color-motion cues and utilize them for the image deblurring, beyond the specific extent present in the training data.
Note that in our dataset an additional diffraction-related spatial blur is added (as mentioned above), so in a case that a similar spatial blur is added to the original GoPro dataset (without the motion-color cues), our advantage over \cite{Nah_deepDeblur_gopro} is expected to be even larger. Note also that our network converges well and provides good performance for higher noise levels (as presented in the following), however, for a fair comparison to \cite{Nah_deepDeblur_gopro} we limit the noise level here to $\sigma=3$.
\begin{table}
\begin{center}
\begin{tabular}{|l|c|c|}
\hline
$N_{frames}$ & Nah \etal & Ours \\
\hline\hline
$N=7$ & $28.1/0.93$ & $\bf{30.9}/\bf{0.95}$ \\
$N=9$ & $27/0.91$ & $\bf{30}/\bf{0.94}$ \\
$N=11$ & $25.9/0.89$ & $\bf{28.9}/\bf{0.92}$ \\
$N=13$ & $24.9/0.87$ & $\bf{28}/\bf{0.91}$ \\
\hline
\end{tabular}
\end{center}
\caption{\textbf{Quantitative comparison to blind deblurring:} PSNR/SSIM comparison between the method presented in \cite{Nah_deepDeblur_gopro} and our method, for various lengths of motion ($N_{frames}$). Results are the average measures for various scenes and different noise levels. See the supplementary material for per noise level statistics and additional information.} \label{Tab:goProStats}
\end{table}
\begin{figure}[h]
\begin{center}
\includegraphics[width=0.6\linewidth]{Figures/Setup.jpg}
\end{center}
\caption{\textbf{The table-top experimental setup:} The liquid-lens and our phase-mask are incorporated in the C-mount lens. The micro-controller synchronizes the focus variation to the frame exposure using the camera flash signal.}
\label{fig:setup}
\end{figure}
\subsection{Table-top experiment}
\begin{figure}[tb]
\begin{center}
\includegraphics[width = 0.8\columnwidth]{Figures/ledZoom_1.png}
\end{center}
\caption{\textbf{Experimental validation of PSF coding:} a moving white LED captured with our camera, validates the required PSF encoding.}
\label{fig:exp_LEDs}
\end{figure}
Following the simulation results, a real-world setup is built (see Fig.~\ref{fig:setup}). A C-mount lens with $f=12[mm]$ is mounted on a 18MP camera with pixel size of $1.25[\mu m]$. A similar phase-mask to the one used in \cite{IEEE_depth} and a liquid focusing lens are incorporated in the aperture plane of the main lens. A signal from the camera indicating the start of the exposure (originally designed for flash activation) is used to trigger the liquid lens to perform the required focus variation (a detailed description of the experimental setup is presented in the supplementary material). The liquid lens is calibrated to introduce a focus variation equivalent to $\psi=[0,8]$ during exposure.
The first real-world test validates the desired PSF spatiotemporal encoding. Two white LEDs are mounted on a spinning wheel, and acts as point-sources, similar to the point sources simulated in Fig.~\ref{fig:dots}. A motion blurred image of the spinning LEDs is acquired, with the phase-mask incorporated in the lens and the proper focus variation during exposure. Zoom-in on one of the LEDs is presented in Fig.~\ref{fig:exp_LEDs}. The gradual color changes along the motion trajectory is clearly visible. The full image including both LEDs is presented in the supplementary material.
Following the PSF validation experiment, a deblurring experiment on moving objects is carried. In order to examine various motion directions and velocities at once, a rotating object is used. For reference, image of the same object is captured with a conventional camera (i.e. the same camera with a fixed focus and without the phase-mask), and deblurred using the multiscale motion blur CNN (Nah \etal \cite{Nah_deepDeblur_gopro}). Results on a rotating photo of Seattle's view are presented in Fig.~\ref{fig:stl}.
In addition to the rotating target test, a linearly moving object (toy train) is also captured, in the same configuration as the previous example. The results are presented in Fig.~\ref{fig:trn}. As can be clearly seen, our camera provides much better results in both cases. The full results of the experiments above along with additional experiments and demonstrations are provided in the supplementary material.
\begin{figure}[tb]
\def0.45{0.45}
\centering
\begin{tabular}{c c}
\includegraphics[width = 0.428\columnwidth]{Figures/dispOurRec_stl.jpg} &
\includegraphics[width = 0.45\columnwidth]{Figures/dispRefRec_stl.jpg}\\
\includegraphics[width = 0.428\columnwidth]{Figures/ourRec_zooms.jpg} &
\includegraphics[width = 0.45\columnwidth]{Figures/refRec_zooms.jpg}\\
{\small{(a) Our reconstruction}}&
{\small{(b) Nah \etal reconstruction \cite{Nah_deepDeblur_gopro}}}\\
\end{tabular}
\caption{\textbf{Seattle's view experiment:} reconstruction results of (top) rotating view of Seattle and (bottom) zoom-ins, using (a) out method and (b) Nah \etal reconstruction \cite{Nah_deepDeblur_gopro}.}
\label{fig:stl}
\end{figure}
\begin{figure}[tb]
\def0.9{0.9}
\centering
\begin{tabular}{c c}
\includegraphics[width = 0.9\columnwidth]{Figures/dispOurRec.jpg}\\
\includegraphics[width = 0.9\columnwidth]{Figures/ourRecCrop1.jpg}\\
{\small{(a) Our reconstruction}}\\
\includegraphics[width = 0.9\columnwidth]{Figures/dispRefRec.jpg}\\
\includegraphics[width = 0.9\columnwidth]{Figures/refRecCrop1.jpg}\\
{\small{(b) Nah \etal reconstruction (\cite{Nah_deepDeblur_gopro})}} \\
\end{tabular}
\caption{\textbf{Train experiment:} Recovery results of a moving train using (a) our method and (b) Nah \etal reconstruction \cite{Nah_deepDeblur_gopro}.}
\label{fig:trn}
\end{figure}
\subsection{U-Net model}
As discussed in the paper, our proposed architecture is based on the known U-net architecture \cite{Unet}. The U-net model includes several downsampling blocks, with their corresponding upsampling operations, which concatenate to their output the input of the same scale downsampling block, thus allowing multiscale processing. We use the U-net model available in \url{https://github.com/milesial/Pytorch-UNet}, with several modifications. A skip-connection is added between the input and the output, thus, letting the 'U' structure to estimate a 'residual' correction to the input blurred image (empirically, this change allows a much faster convergence).
The net contains four downsampling blocks and their corresponding four upsampling blocks, with additional convolutions in the input and output. Each convolution operation consists of $3\times3$ filters (with proper padding to keep the original input size, and without bias), followed by BatchNorm layer and a Leaky-ReLU activation. Each CONV block contains double Conv-BN-ReLU sequence. Downsampling blocks perform convolution and then a $2\times2$ max-pooling operation. Upsampling blocks perform a $\times2$ trainable upsampling (transpose-Conv layer), CONV block, and a concatenation with the corresponding downsampling block's output. The number of filters in each layer appears in Table~\ref{Tab:CNN_det}.
\begin{table}
\begin{center}
\begin{tabular}{|l|c|c|}
\hline
Layer & $N_{in}$ & $N_{out}$ \\
\hline\hline
$CONV_{in}$ & 3 & 64 \\
$DOWN_1$ & 64 & 128 \\
$DOWN_2$ & 128 & 256 \\
$DOWN_3$ & 256 & 512 \\
$DOWN_4$ & 512 & 512 \\
$UP_1$ & 1024 & 256 \\
$UP_2$ & 512 & 128 \\
$UP_3$ & 256 & 64 \\
$UP_4$ & 128 & 64 \\
$CONV_{out}$ & 64 & 3 \\
\hline
\end{tabular}
\end{center}
\caption{\textbf{CNN layers:} The number of filters in each convolution operation.} \label{Tab:CNN_det}
\end{table}
\subsection{Shallow model for ablation study}
As mentioned in the Section 4 in the paper, as part of the ablation study, a shallow model was trained using the same dataset. This test checked the impact of the model depth on the reconstruction performance, and specifically the multiscale operation effect. As mentioned in the paper, nominal performance is achieved with this network structure. However, it still achieves comparable results to a pure-computational model (like \cite{Nah_deepDeblur_gopro}), demonstrating the benefit of the aperture coding- the encoded cues are such strong guidance for the deblurring operation, that a very shallow model achieves comparable performance to a much larger one.
The shallow model is made of ten consecutive blocks, where each one contains Conv-BN-ReLU layers (without any pooling). A skip connection is made from the input to the output (in similarity to our U-Net structure), what leaves the inner layers to estimate only the residual correction the blurred image. The filter size in each block is $3\times3$, and each layer has 32 filters (besides the output layer, which has only three, to generate the final residual image).
\subsection{Comparison to other coding methods}
Following the comparison to other coding methods presented in Section~5.1 of the paper, the full results (including the intermediate images) are presented in Fig.~\ref{fig:sim_res_full}.
\begin{figure}[tb]
\def0.4{0.4}
\centering
\begin{tabular}{ c c c}
\includegraphics[width = 0.4\columnwidth]{Figures/spkWArr.png} &
\\
{\small{(a) Rotating target}}\\
\includegraphics[width = 0.4\columnwidth]{Figures/flt_spk.png} &
\includegraphics[width = 0.4\columnwidth]{Figures/flt_spk_rec_rs.png}\\
{\small{(b) Fluttered-shutter img.}}&
{\small{(c) Fluttered-shutter rec.}}\\
\includegraphics[width = 0.4\columnwidth]{Figures/prb_spk.png} &
\includegraphics[width = 0.4\columnwidth]{Figures/prb_spk_rec.png}\\
{\small{(d) Parabolic motion img.}}&
{\small{(e) Parabolic motion rec.}}\\
\includegraphics[width = 0.4\columnwidth]{Figures/our_spk_Img.png} &
\includegraphics[width = 0.4\columnwidth]{Figures/our_spk_Rec.png}\\
{\small{(f) Our img.}}&
{\small{(g) Our rec.}}\\
\end{tabular}
\caption{\textbf{Simulation results of rotating target:} (a) rotating target and the intermediate images vs. reconstruction results for (b,c) fluttered-shutter, (d,e) parabolic motion camera and (f,g) our method.}
\label{fig:sim_res_full}
\end{figure}
\subsection{Outdoor experimental results}
A table-top experimental setup that demonstrates our spatiotemporal aperture coding method is built (see details in the main paper and previous section). As mentioned in Section~5.3 of the paper, an experimental validation of the PSF encoding is performed, using a rotating wheel with two white LEDs (simulating point sources). The full image of the LEDs is presented in Fig.~\ref{fig:LEDs}. The color coded motion of each LED is clearly visible; the order of the colors indicates the direction, and the extent of the color trace is a cue for the object velocity.
\begin{figure}[tb]
\centering
\includegraphics[width = 1\columnwidth]{Figures/ledCrop.png}
\caption{\textbf{PSF encoding validation experiment:} two rotating white LEDs simulating point sources captured using our camera. The color coded motion trace indicates both direction and velocity.}
\label{fig:LEDs}
\end{figure}
Following the results presented in the paper, additional outdoor scenes (of plants moving in the wind) are presented here (see Fig.~\ref{fig:out_1}-\ref{fig:out_3})\footnote{Note that the experimental camera sensor size is $4912\times3684$ pixels, while the test set images (of the GoPro dataset) are $1280\times720$ pixels, therefore the motion extent scale in pixel terms is different.}. As it is very complex to generate controlled motion 'in the wild', only our method is used, without a reference conventional camera. In such scenes, the motion occurs in every direction, and in various velocities. One can see (especially in the zoom-ins on Fig.~\ref{fig:out_1}) that our CNN is able to reconstruct the objects moving in different velocities and directions, while also deblurring the static parts (which also get some blur in the coding process). Notice that we do not re-train the reconstruction network, but rather use the same used in all other experiments (that was trained on the GoPro dataset with the color-coded motion simulation). Thus, in areas where the motion extent is quite large, the reconstruction performance decrease. The reason is that our model is trained with data that contains a limited velocities range; inside this range, the reconstruction results are good, while in areas with motion beyond this limit, the reconstruction ability is limited. This is an inherent trade-off in our method- as the velocities range in the dataset gets larger, the CNN can handle a longer motion extent. However, the cost is that some compromise is done in the reconstruction performance of the slower motion range.
\begin{figure*}[tb]
\def0.85{0.85}
\centering
\begin{tabular}{c c}
\includegraphics[width = 0.85\columnwidth]{Figures/out_15_crops.jpg}&
\includegraphics[width = 1\columnwidth]{Figures/out_15_zooms.jpg}\\
\includegraphics[width = 0.85\columnwidth]{Figures/out_17_crops.jpg}&
\includegraphics[width = 1\columnwidth]{Figures/out_17_zooms.jpg}\\
\end{tabular}
\caption{\textbf{Outdoor experiment:} (left) full outdoor scenes with marks on magnified areas, and zoom-ins on (middle) intermediate image and (right) reconstruction results.}
\label{fig:out_1}
\end{figure*}
\begin{figure*}[tb]
\def1.7{1.7}
\centering
\begin{tabular}{c c}
\includegraphics[width = 1.7\columnwidth]{Figures/chk_15_Img_gc.jpg}\\
\includegraphics[width = 1.7\columnwidth]{Figures/chk_15_Rec_gc.jpg}\\
\end{tabular}
\caption{\textbf{Outdoor experiment, full images:} (top) intermediate image and (bottom) reconstruction results.}
\label{fig:out_2}
\end{figure*}
\begin{figure*}[tb]
\def1.7{1.7}
\centering
\begin{tabular}{c c}
\includegraphics[width = 1.7\columnwidth]{Figures/chk_17_Img.jpg}\\
\includegraphics[width = 1.7\columnwidth]{Figures/chk_17_Rec.jpg}\\
\end{tabular}
\caption{\textbf{Outdoor experiment, full images:} (top) intermediate image and (bottom) reconstruction results.}
\label{fig:out_3}
\end{figure*}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,165 |
Health policy can be defined as the "decisions, plans, and actions that are undertaken to achieve specific health care goals within a society." According to the World Health Organization, an explicit health policy can achieve several things: it defines a vision for the future; it outlines priorities and the expected roles of different groups; and it builds consensus and informs people.(read more at wiki) Here we introduce best colleges in the field of health policy and management where Harvard University is ranked top and Johns Hopkins University is positioned on second place.
We briefly compare those top colleges with various factors such as tuition, admission, graduation, enrolment, and more. The full comparison for each factors with tables and charts is described at health policy and management colleges comparison page.
First, we compare the best health policy and management schools with general characteristics.
Next, we compare the tuition and other costs at best Health Policy and Management colleges. Tuition is an amount of money charged to students for instructional services. Tuition may be charged per term, per course, or per credit. In-state tuition rate applies to students who meet the state's or institution's residency requirements and out-of-tuition rate is for students who do not meet the ins-state residency requirements.
This page contains comparison data on the number of full-time, first-time undergraduate students and all undergraduate students who receive different types of student financial aid, including grants and loans, from different sources at each best Health Policy and Management school.
This paragraph compares the number of students enrolled in best Health Policy and Managementschools. Next table counts total students headcount by school level and attendance status. Undergraduate full-time student is a student enrolled for 12 or more semester credits , or 12 or more quarter credits, or 24 or more contact hours a week each term. For graduate students, it includes students enrolled for 9 or more semester credits, or 9 or more quarter credits, or a student involved in thesis or dissertation preparation that is considered full time by the institution.
Next, we compare the graduation rate between best Health Policy and Management schools. The graduation Rate is calculated as the total number of completers within 150% of normal time divided by the revised cohort minus any allowable exclusions. For example, 6 years rate is applied for 4 year schools. For detail information of your desired school, follow links on the school name.
Once again, we note that the full comparison for each factors with tables and charts is described at health policy and management colleges comparison page. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,248 |
\section{Introduction}\label{section 1}
For $e\in\{-1,0\}$ and $n\in\mathbb{Z}_+$ let $\mathcal{B}(e,n)$ be the Gieseker-Maruyama moduli space of stable rank 2 algebraic vector bundles with Chern classes $c_1=e,\ c_2=n$ on the projective space $\mathbb{P}^3$. R.~Hartshorne \cite{H-vb} showed that $\mathcal{B}(e,n)$ is a quasi-projective scheme, nonempty for arbitrary $n\ge1$ in case $e=0$ and, respectively, for even $n\ge2$ in case $e=-1$, and the deformation theory predicts that each irreducible component of ${\mathcal B}(e,n)$ has dimension at least $8n-3+2e$.
In case $e=0$ it is known by now (see \cite{H-vb}, \cite{ES}, \cite{B1}, \cite{CTT}, \cite{JV}, \cite{T1}, \cite{T2}) that the scheme $\mathcal{B}(0,n)$ contains an irreducible component $I_n$ of expected dimension $8n-3$, and this component is the closure of the smooth open subset of $I_n$ constituted by the so-called mathematical instanton vector bundles. Historically, $\{I_n\}_{n\ge1}$ was the first known infinite series of irreducible components of $\mathcal{B}(0,n)$ having the expected dimension $\dim I_n=8n-3$. In \cite[Ex. 4.3.2]{H-vb} R.~Hartshorne constructed a first infinite series $\{\mathcal{B}_0(-1,2m)\}_{m\ge1}$ of irreducible components $\mathcal{B}_0(-1,2m)$ of $\mathcal{B}(-1,2m)$ having the expected dimension $\dim\mathcal{B}_0(-1,2m)=16m-5$.
The other infinite series of families of vector bundles of dimension $3k^2+10k+8$ from
${\mathcal B}(0,2k+1)$ was constructed in 1978 by W.~Barth and K.~Hulek \cite{BH}, and G.~Ellingsrud and S.~A.~Str\o mme in \cite[(4.6)-(4.7)]{ES} showed that these families are open subsets of irreducible componens distint from the instanton components $I_{2k+1}$.
Later in 1985-87 V.~K.~Vedernikov \cite{V1} and \cite{V2} constructed two infinite series of families of bundles from
${\mathcal B}(0,n)$, and one infinite family of bundles from ${\mathcal B}(-1,2m)$. A more general series of rank 2 bundles depending on triples of integers $a,b,c$, appeared in 1984 in the paper of A. Prabhakar Rao \cite{Rao}. Soon after that, in 1988, L.~Ein \cite{Ein} independently studied these bundles and proved that they constitute open parts of irreducible components of ${\mathcal B}(e,n)$ for both $e=0$ and $e=-1$.
A new progress in the description of the spaces ${\mathcal B}(0,n)$
was achieved in 2017 by J.~Almeida, M.~Jardim, A.~Tikhomirov and
S.~Tikhomirov in \cite{AJTT}, where they constructed a new infinite series of irreducible components $Y_a$ of the spaces ${\mathcal B}(0,1+a^2)$ for $a\in\{2\}\cup\mathbb{Z}_{\ge4}$. These components have dimensions
$\dim Y_a=4\binom{a+3}{3}-a-1$ which is larger than expected.
General bundles from these components are obtained as cohomology bundles of rank 1 monads, the middle term of which
is a rank 4 symplectic instanton with $c_2=1$, and the lefthand and the righthand terms are $\op3(-a)$ and $\op3(a)$, respectively.
The aim of present article is to provide two new infinite series of irreducible components ${\mathcal M}_n$ of ${\mathcal B}_(e,n)$, one for $e=0$ and another for $e=-1$ which in some sense generalizes the above construction from \cite{AJTT}.
Namely, in case $e=0$ we construct an infinite series $\Sigma_0$ of irreducible components ${\mathcal M}_n$ of ${\mathcal B}(0,n)$, such that a general bundle of ${\mathcal M}_n$ is a cohomology bundle of a monad of the type similar to the above, the middle term of which is a rank 4 symplectic instanton with arbitrary second Chern class. The first main result of the article, Theorem \ref{Thm A}, states that the
series $\Sigma_0$ contains components ${\mathcal M}_n$ for all big
enough values of $n$ (more precisely, at least for $n\ge146$). The series $\Sigma_0$ is a first example, besides the instanton series $\{I_n\}_{n\ge1}$, of the series with this property. (For all the other series mentioned above the question whether they contain components with all big enough values of second Chern class $n$ is open.)
In case $e=-1$ we construct in a similar way an infinite series $\Sigma_1$ of irreducible components ${\mathcal M}_n$ of ${\mathcal B}(-1,n)$, such that a general bundle of ${\mathcal M}_n$ is a cohomology bundle of a monad of the type similar to the above, in which the lefthand and the righthand terms are $\op3(-a-1)$ and $\op3(a)$, respectively, and the middle term is a twisted symplectic rank 4 bundle with first Chern
class -2. The second main result of the article, Theorem \ref{Thm B}, states that $\Sigma_1$ contains components ${\mathcal M}_n$ asymptotically for almost all big enough values of
$n$. (A precise statement about the behaviour of the set
of values of $n$ for which ${\mathcal M}_n$ is contained in $\Sigma_1$ is given in Remark \ref{Rem B1}).
Now give a brief sketch of the contents of the article.
In Section \ref{section 2} we study some properties of pairs $([{\mathcal E}_1],[{\mathcal E}_2])$ of mathematical instanton bundles
and prove the vanishing of certain cohomology groups of their
twists by line bundles $\op3(a)$ and $\op3(-a)$ (see Propositon \ref{Prop 1}). The direct sum $\mathbb{E}=
{\mathcal E}_1\oplus{\mathcal E}_2$ is then used in Section \ref{section 3}
as a test rank 4 symplectic instanton bundle. This bundle and
its deformations are used as middle terms of anti-self-dual monads of the form $0\to\op3(-a)\to\mathbb{E}\to\op3(a)\to0$,
the cohomology bundles of which provide general bundles of
the components ${\mathcal M}_n$ of of the series $\Sigma_0$ (see Theorem \ref{Thm A}). In Section \ref{section 4} we study
direct sums $\mathbb{E}={\mathcal E}_1\oplus{\mathcal E}_2$ of vector bundles, ${\mathcal E}_i$ are the bundles from the R.~Hartshorne series $\{{\mathcal B}_0(-1,2n)\}_{n\ge1}$ mentioned above. We prove
certain vanishing properties for cohomology of twists of
${\mathcal E}_i$ (see Proposition \ref{Prop 2}). These properties are then used in Theorem \ref{Thm B} in the construction of general vector bundles of coomponents ${\mathcal M}_n$ of the series
$\Sigma_1$. In Section \ref{section 5} we give a list of
components ${\mathcal M}_n\in\Sigma_0$ for $n\le20$ and of
components ${\mathcal M}_n\in\Sigma_1$ for $n\le40$.
\noindent
\textbf{Conventions and notation}.
\begin{itemize}
\item Everywhere in this paper we work over the base field of complex numbers $\mathbf{k}=\mathbb{C}$.
\item $\mathbb{P}^3$ is a projective 3-space over $\mathbf{k}$.
\item For a stable rank 2 vector bundle $E$ with $c_1(E)=e$, $c_2(E)=n$ on $\p3$, we denote by $[E]$ its isomorphism class in ${\mathcal B}(e,n)$.
\end{itemize}
\noindent
\textbf{Acknowledgements}.
A.~Tikhomirov was supported by the Academic Fund Program at the National Research University Higher School of Economics (HSE) in 2018-2019
(grant no. 18-01-0037) and by the Russian Academic Excellence Project "5-100". He also acknowledges the hospitality of the Max Planck Institute for Mathematics in Bonn, where this work was partially done during the winter of 2017.
\vspace{5mm}
\section{Some properties of mathematical instantons}
\label{section 2}
\vspace{2mm}
Let $a$ and $m$ be two positive integers, where $a\ge2$, and
let $\varepsilon\in\{0,1\}$. In this section we prove the following proposition about mathematical instanton vector bundles which will be used in the proof of Theorem
\ref{Thm A}.
\begin{proposition}\label{Prop 1}
A general pair
\begin{equation}\label{2 inst}
([{\mathcal E}_1],[{\mathcal E}_2])\in I_m\times I_{m+\varepsilon},
\end{equation}
of instanton vector bundles satisfies the following conditions:
\begin{equation}\label{not equal}
[{\mathcal E}_1]\ne[{\mathcal E}_2];
\end{equation}
for $i=1,\ m\le a+1,$ respectively, $i=2,\ m+\varepsilon\le a+1$,
\begin{equation}\label{vanish 1}
h^1({\mathcal E}_i(a))=0,
\end{equation}
\begin{equation}\label{vanish 2}
h^2({\mathcal E}_i(-a))=0,\ \ \ if\ \ \ a\ge12;
\end{equation}
for $i=1,\ m\le a-4,\ a\ge5,$ respectively, $i=2,\ m+\varepsilon\le a-4,\ a\ge5,$
\begin{equation}\label{vanish 2a}
h^2({\mathcal E}_i(-a))=0;
\end{equation}
for $j\ne1$,
\begin{equation}\label{vanish 3}
h^j({\mathcal E}_1\otimes{\mathcal E}_2)=0.
\end{equation}
\end{proposition}
\begin{proof} It is clearly enough to treat the case $i=1$, as the case $i=2$ is treated completely similarly.
Consider two instanton vector bundles such that
the condition (\ref{not equal}) can be evidently achieved. Show that the condition (\ref{vanish 1}) can also be satisfied for general bundles $[{\mathcal E}_1]\in I_m$ and $[{\mathcal E}_2]\in I_{m+\varepsilon}$. For this, consider a smooth
quadric surface $S\in\p3$, together with an isomorphism
$S\simeq\p1\times\p1$, and let
$Y=\overset{m+1}{\underset{i=1}{\sqcup}}l_i$ be a union of $m+1$ distinct projective lines $l_i$ in $\p3$ belonging to one of the two rulings on $S$. Considering $Y$ as a reduced scheme, we have
${\mathcal I}_{Y,S}\simeq\op1(-m-1)\boxtimes\op1$. Thus the exact triple $0\to{\mathcal I}_{S,\p3}\to{\mathcal I}_{Y,\p3}\to{\mathcal I}_{Y,S}\to0$ can be rewritten as
\begin{equation}\label{triple of ideals}
0\to\op3(-2)\to{\mathcal I}_{Y,\p3}\to\op1(-m-1)\boxtimes\op1\to0.
\end{equation}
Tensor multiplication of (\ref*{triple of ideals}) by $\op3(a+1)$, respectively, by $\op3(a-3)$
yields an exact triple
\begin{equation}\label{triple(a)}
0\to\op3(a-1)\to{\mathcal I}_{Y,\p3}(a+1)\to\op1(a-m)\boxtimes\op1
(a+1)\to0.
\end{equation}
\begin{equation}\label{triple(-a)}
0\to\op3(a-5)\to{\mathcal I}_{Y,\p3}(a-3)\to\op1(a-4-m)\boxtimes
\op1(a-3)\to0.
\end{equation}
By the K\"unneth formula $h^1(\op1(a-m)\boxtimes\op1(a+1))=0$ for $a\ge2$ and $m\le a+1$, and (\ref{triple(a)}) implies that
\begin{equation}\label{vanish 4}
h^1({\mathcal I}_{Y,\p3}(a+1))=0.
\end{equation}
Now consider an extension of $\op3$-sheaves of the form
\begin{equation}\label{trC}
0\to\op3(-1)\to{\mathcal E}_1\to{\mathcal I}_{Y,\p3}(1)\to0.
\end{equation}
Such extensions are classified by the vector space $V=\mathrm{Ext}^1({\mathcal I}_{Y,\p3}(1),\op3(-1))$, and it is
known that, for a general point $\xi\in V$ the extension sheaf
${\mathcal E}_1$ in (\ref{trC}) is a locally free instanton sheaf from $I_m$(see, e. g., \cite{NT}) called a {\it 't Hooft instanton}. Now tensoring the triple \eqref{trC} with $\op3(a)$ and passing to cohomology, in view of
\eqref{vanish 4} we obtain (\ref{vanish 1}) for $i=1$.
To prove \eqref{vanish 2a}, assume that $m\le a-4$; then
similar to \eqref{vanish 4} we obtain using \eqref{triple(-a)}:
\begin{equation}\label{vanish 5}
h^1({\mathcal I}_{Y,\p3}(a-3))=0,\ \ \ m\le a-4.
\end{equation}
Tensoring \eqref{trC} with $\op3(a-4)$ we obtain the triple
\begin{equation}\label{trC new}
0\to\op3(a-5)\to{\mathcal E}_1(a-4)\xrightarrow{\varepsilon}{\mathcal I}_{Y,\p3}(a-3)\to0.
\end{equation}
From \eqref{vanish 5} and \eqref{trC new} it follows that
\begin{equation}\label{vanish 6}
h^1({\mathcal E}_1(a-4))=0,\ \ \ m\le a-4.
\end{equation}
This together with Serre duality for ${\mathcal E}_1$ yields \eqref{vanish 2a} for $i=1$.
To prove \eqref{vanish 2}, consider a smooth quadric surface $S'\subset\p3$, together with an isomorphism $S'\simeq\p1\times\p1$, and let $Z=\overset{d}{\underset{i=1}{\sqcup}}\tilde{l}_i$ be a union of $d$ distinct projective lines $\tilde{l}_i$ in $\p3$, belonging to one of the two rulings on $S'$, where $1\le d\le 5$. Considering $Z$ as a reduced scheme, we have
${\mathcal I}_{Z,S'}\simeq\op1(-d)\boxtimes\op1$.
Without loss of generality we may assume that $Z\cap Y=
\emptyset$ and that $Z$ intersects the quadric surface $S$ treated above in $2d$ distinct points $x_1,...,x_{2d}$ such that the points $\mathrm{pr}_2(x_i),\ i=1,...,2d,$ are also distinct, where $\mathrm{pr}_2: S\simeq\p1\times\p1\to\p1$ is the projection onto the second factor. Tensoring the exact triple \eqref{triple of ideals} with $\op3(a-3)$ and restricting it onto $Z$ we obtain a commutative diagram of exact triples
\begin{equation}\label{diagr 1}
\xymatrix{
& 0 & 0 & 0 &\\
0\ar[r] &{\mathcal O}_Z(a-5)\ar[r]\ar[u] &
{\mathcal O}_Z(a-3)\ar[r]\ar[u] & \underset{1}{\overset{2d}{\oplus}}~\mathbf{k}_{x_j}\ar[u]\ar[r] & 0\\
0\ar[r]& \op3(a-5)\ar[r]\ar^-f[u] & {\mathcal I}_{Y,\p3}(a-3)\ar[r]
\ar_-{g}[u] &\op1(a-m-4)\boxtimes\op1(a-3)
\ar^-h[u]\ar[r] & 0,}
\end{equation}
where $f,\ g$ and $h$ are the restriction maps.
The sheaf $\ker f={\mathcal I}_{Z,\p3}(a-5)$ similar to \eqref{triple(a)} satisfies the exact triple
\begin{equation*}\label{triple(b)}
0\to\op3(a-5)\to\ker f\to\op1(a-d-5)\boxtimes\op1(a-5)\to0.
\end{equation*}
Passing to cohomology of this triple we obtain in view of the conditions $1\le d\le 5$ and $a\ge12$ that $h^1(\ker f)=0$,
i. e.
$$
h^0(f):\ H^0(\op3(a-5))\to H^0({\mathcal O}_Z(a-5))
$$
is an epimorphism. On the other hand, we have: i) $a-m-4\ge0$,
ii) $a-3\ge2d-1$, since $a\ge12$ and $d\le5$, and iii) the points $\mathrm{pr}_2(x_i),\ i=1,...,2d,$ are distinct. Therefore,
$$
h^0(h):\ H^0(\op1(a-m-4)\boxtimes\op1(a-3))\to H^0(\overset{2d}{\underset{1}{\oplus}}~\mathbf{k}_{x_j})
$$
is also an epimorphism. Whence by the diagram \eqref{diagr 1}
we obtain an epimorphism
\begin{equation}\label{h0(g) epi}
h^0(g):\ H^0({\mathcal I}_{Y,\p3}(a-3))\twoheadrightarrow H^0({\mathcal O}_Z(a-3)).
\end{equation}
Now consider the
$g\circ\varepsilon:\ {\mathcal E}_1(a-4)\twoheadrightarrow{\mathcal O}_Z(a-3)$, where
$\varepsilon$ is the epimorphism in the triple \eqref{trC new} and set $E:=\ker(g\circ\varepsilon)\otimes\op3(4-a)$.
Thus, since ${\mathcal O}_Z=\overset{d}{\oplus}{\mathcal O}_{\tilde{l}_i}$, we have an exact triple:
\begin{equation}\label{triple with l's}
0\to E(a-4)\to{\mathcal E}_1(a-4)\xrightarrow{g\circ\varepsilon}
\overset{d}{\underset{1}{\oplus}}{\mathcal O}_{\tilde{l}_i}(a-3)\to0.
\end{equation}
From the triple \eqref{trC new} it follows that $h^0(\varepsilon):H^0({\mathcal E}_1(a-4))\to H^0({\mathcal I}_{Y,\p3}(a-3))$ is an epimorphism, hence by \eqref{h0(g) epi} $h^0(g\circ\varepsilon):H^0({\mathcal E}_1(a-4))
\to H^0(\overset{d}{\oplus}{\mathcal O}_{\tilde{l}_i}(a-3))$ is also
an epimorphism. This together with (\ref{triple with l's}) and
\eqref{vanish 6} yields that
\begin{equation}\label{vanish 7}
h^1(E(a-4))=0.
\end{equation}
Note that from \eqref{triple with l's} it follows also that
\begin{equation}\label{c2(E)}
c_2(E)=c_2({\mathcal E}_1)+d=m+d\le a+1,
\end{equation}
since $d\le5$ and $m\le a-4$.
Now show that
\begin{equation}\label{E in closure}
[E]\in\bar{I}_{m+d},
\end{equation}
where $\bar{I}_{m+d}$ is the closure of $I_{m+d}$ in the Gieseker-Maruyama moduli scheme $M(0,m+d,0)$ of semistable rank 2 coherent sheaves with Chern classes $c_1=c_3=0$ and
$c_2=m+d$. (Recall that $M(0,m+d,0)$ is a projective scheme
containing ${\mathcal B}(0,m+1)$ as an open subscheme - see e. g., \cite{H-vb}, \cite{HL}.) It is enough to treat the case
$d=2$, since the argument for any $d\le5$ is completely similar. Consider the triple \eqref{triple with l's} and denote by $E'_0$ the kernel of the composition
$$
{\mathcal E}_1\xrightarrow{g\circ\varepsilon}
{\mathcal O}_{\tilde{l}_1}(1)\oplus{\mathcal O}_{\tilde{l}_2}(1)
\xrightarrow{\mathrm{pr_1}}{\mathcal O}_{\tilde{l}_1}(1).
$$
We then obtain an exact triple
\begin{equation}\label{tr E E'0}
0\to E\to E'_0\xrightarrow{\varepsilon'}{\mathcal O}_{\tilde{l}_2}(1)
\to0.
\end{equation}
Now we invoke one of the main results of the paper \cite{JMT}
according to which the sheaf $E'_0$ lies in the closure $\bar{I}_{m+1}$ of $I_{m+1}$ in the Gieseker-Maruyama moduli scheme $M(0,m+1,0)$. This implies that there exists a punctured curve $(C,0)\in\bar{I}_{m+1}$ and a flat over $C$ coherent ${\mathcal O}_{\p3\times C}$-sheaf $\mathbb{E'}$ such that the sheaf $E'_t:=\mathbb{E'}|_{\p3\times\{t\}}$ is an instanton bundle from $I_{m+1}$ for $t\ne0$ and coincides with $E'_0$ for $t=0$. Now, without loss of generality, after possible shrinking the curve $C$, one can extend the epimorphism $\varepsilon'$ in \eqref{tr E E'0} to an epimorphism
$$
\mathbf{e}:\mathbb{E'}\twoheadrightarrow{\mathcal O}_{\tilde{l}_2}(1)
\boxtimes{\mathcal O}_C
$$
such that $\mathbf{e}\otimes\mathbf{k}(0)=\varepsilon'$.
Set $\mathbb{E}=\ker\mathbf{e}$ and denote $E_t=\mathbb{E}|_{\p3\times\{t\}}$, $t\in C$. As for $t\ne0$ the sheaf $E'_t$ is an instanton bundle from $I_{m+1}$, and
it fits in an exact triple $0\to E_t\to E'_t\to{\mathcal O}_{\tilde{l}_2}(1)\to0$, the above mentioned result
from \cite{JMT} yields that $[E_t]\in\bar{I}_{m+2}$ for $t\ne0$. Hence, since $[E_t]\in\bar{I}_{m+2}$ is projective,
it follows that $E_0\in\bar{I}_{m+2}$. Now by construction
$E_0\simeq E$. Thus, $[E]\in\bar{I}_{m+2}$, i. e. we obtain
the desired result \eqref{E in closure} for $d=2$.
Formula \eqref{vanish 2} now follows from \eqref{vanish 7} for a general ${\mathcal E}$ by Semicontinuity and Serre duality.
To prove the vanishing \eqref{vanish 3}, consider the triple
\eqref{trC} twisted by ${\mathcal E}_2$:
\begin{equation}\label{trC times E2}
0\to{\mathcal E}_2(-1)\to{\mathcal E}_1\otimes{\mathcal E}_2
\to{\mathcal E}_2\otimes{\mathcal I}_{Y,\p3}(1)\to0,
\end{equation}
and the exact triple
$0\to{\mathcal E}_2\otimes{\mathcal I}_{Y,\p3}(1)\to{\mathcal E}_2(1)\to
\oplus_{i=1}^{m+1}({\mathcal E}_2|_{l_i})\to0.$
Since ${\mathcal E}_2$ is an instanton bundle, it follows that
\begin{equation}\label{vanish inst}
h^2({\mathcal E}_2(1))=0,\ \ \ h^2({\mathcal E}_2(-1))=0.
\end{equation}
On the other hand, without loss of generality, by the Grauert-M\"ulich Theorem \cite[Ch. 2]{OSS} we may assume
that ${\mathcal E}_2|_{l_i}\simeq\op1$. This together with the last exact triple and the first equality \eqref{vanish inst} yields
$h^2({\mathcal E}_2\otimes{\mathcal I}_{Y,\p3}(1))=0$. Therefore, in view of \eqref{trC times E2} and the second equality \eqref{vanish inst} we obtain the equality \eqref{vanish 3} for $j=1$.
Last, this equality for $j=0,3$ follows from \eqref{not equal} and the stability of ${\mathcal E}_1$ and ${\mathcal E}_2$.
\end{proof}
\begin{remark}\label{rem A}
Note that, under the conditions of Proposition \ref{Prop 1}, the equalities (\ref{vanish 3}) together with Riemann-Roch yield
\begin{equation}\label{h1 12}
h^1({\mathcal E}_1\otimes{\mathcal E}_2)=8m+4\varepsilon-4.
\end{equation}
\end{remark}
\vspace{5mm}
\section{Construction of stable rank two bundles with even determinant}
\label{section 3}
\vspace{5mm}
We first recall the notion of symplectic instanton.
By a \textit{symplectic structure} on a vector bundle $E$ on a scheme $X$ we mean an anti-self-dual isomorphism $\theta: E \xrightarrow{\sim}E^{\vee},\ \theta^{\vee}=-\theta$, considered modulo proportionality. Clearly,
a symplectic vector bundle $E$ has even rank:
\begin{equation*}\label{rk E}
\rk E=2r,\ \ \ r\ge1,
\end{equation*}
and, if $X=\p3$, vanishing odd Chern classes:
\begin{equation*}\label{odd c_i}
c_1(E)=c_3(E)=0.
\end{equation*}
Following \cite{AJTT}, we call a symplectic vector bundle $E$ on $\p3$ a \textit{symplectic instanton} if
\begin{equation}\label{def of sympl inst}
h^0(E(-1))=h^1(E(-2))=h^2(E(-2))=h^3(E(-3))=0,
\end{equation}
and
\begin{equation*}\label{c_2=n}
c_2(E)=n,\ \ \ n\ge1.
\end{equation*}
Consider the instanton bundles ${\mathcal E}_1$ and ${\mathcal E}_2$ introduced in Section \ref{section 2} (see Proposition \ref{Prop 1}).
Since $\det{\mathcal E}_1\simeq\det{\mathcal E}_2\simeq\op3$, there are symplectic structures $\theta_i:\ {\mathcal E}_i\xrightarrow{\simeq}{\mathcal E}_i^{\vee},\ i=1,2,$ which yield a symplectic structure on the direct sum
$\mathbb{E}={\mathcal E}_1\oplus{\mathcal E}_2$:
\begin{equation}\label{sympl str}
\theta=\theta_1\oplus\theta_2:\ \mathbb{E}={\mathcal E}_1\oplus{\mathcal E}_2 \xrightarrow{\simeq}{\mathcal E}_1^{\vee}\oplus{\mathcal E}_2^{\vee}=
\mathbb{E}^{\vee}.
\end{equation}
Clearly, $\mathbb{E}$ is a symplectic instanton.
Now assume that ${\mathcal E}_1$ and ${\mathcal E}_2$ are chosen in such a way that there exist sections
\begin{equation}\label{empty intersectn}
s_i:\ \op3\to{\mathcal E}_i(a), \ \ \ such\ that\ \ \ \dim(s_i)_0=1,\ \ i=1,2,\ \ \ (s_1)_0\cap(s_2)_0=\emptyset.
\end{equation}
(Such ${\mathcal E}_1\in I_m,\ [{\mathcal E}_2]\in I_{m+\varepsilon}$ always exist; for instance, two general 't Hooft instantons ${\mathcal E}_1$ and ${\mathcal E}_2$ satisfy the property (\ref{empty intersectn}) already for $a=1$, hence also for $a\ge2$.) The condition (\ref{empty intersectn}) implies that
the section
\begin{equation}\label{subbundle}
s=(s_1,s_2):\ \op3(-a)\to\mathbb{E}
\end{equation}
is a subbundle morphism, hence its transpose
\begin{equation*}\label{ta}
{}^ts:=s^{\vee}\circ\theta:\ \mathbb{E}\to\op3(a)
\end{equation*}
is a surjection. As $\theta$ in (\ref{sympl str}) is symplectic,
the composition ${}^ts\circ s:\op3(-a)\to\op3(a)$ is also symplectic. Since
$\op3(\pm a)$ are line bundles, it follows that ${}^ts\circ s=0$. Therefore the complex
\begin{equation}\label{monad}
K^{\cdot}:\ \ \ 0\to\op3(-a)\xrightarrow{s}\mathbb{E}\xrightarrow{{}^ts}
\op3(a)\to0
\end{equation}
is a monad and its cohomology sheaf
\begin{equation}\label{coho}
E=\frac{\ker({}^ts)}{\mathrm{im}(s)}
\end{equation}
is locally free. Note that, since the instanton bundles ${\mathcal E}_1$ and ${\mathcal E}_2$ are stable, they have zero spaces of global sections, hence also $h^0(\mathbb{E})=0$, and
\eqref{monad} and \eqref{coho} yield $h^0(E)=0$, i. e.
$E$ as a rank 2 vector bundle with $c_1=0$ is stable.
Besides, since $c_2(\mathbb{E})=c_2({\mathcal E}_1)+c_2({\mathcal E}_2)=2m+\varepsilon$,
it follows from \eqref{monad} that
$c_2(E)=2m+\varepsilon +a^2$. Thus,
\begin{equation*}\label{E in B}
[E]\in{\mathcal B}(0,2m+\varepsilon +a^2),
\end{equation*}
and the deformation theory yields that, for any irreducible
component ${\mathcal M}$ of ${\mathcal B}(0,2m+\varepsilon +a^2)$,
\begin{equation*}\label{def theory ineq}
\dim{\mathcal M}\ge1-\chi(\mathcal{E}nd~E)
=8(2m+\varepsilon +a^2)-3.
\end{equation*}
Note that, since ${\mathcal E}_1$ and ${\mathcal E}_2$ are instanton bundles, for $a\ge2$ one has $h^1({\mathcal E}_i(-a))=
h^j({\mathcal E}_i(a))=0,\ i=1,2,\ j=2,3$, hence by \eqref{sympl str}
\begin{equation}\label{h1bbE(-a)=0}
h^1(\mathbb{E}(-a))=0,\ \ \ a\ge2,
\end{equation}
\begin{equation}\label{h2,3bbE(a)=0}
h^j(\mathbb{E}(a))=0,\ \ \ j=2,3,\ \ \ a\ge2.
\end{equation}
Similarly, in view of \eqref{vanish 1},
\begin{equation}\label{h1E(a)=0}
h^1(\mathbb{E}(a))=0,\ \ \ m+\varepsilon\le a+1,\ \ \ a\ge2.
\end{equation}
This together with \eqref{h2,3bbE(a)=0} and Riemann-Roch yields:
\begin{equation}\label{h0bbE(a)=}
h^0(\mathbb{E}(a))=\chi(\mathbb{E}(a))=
4\binom{a+3}{3}-(2m+\varepsilon)(a+2),\ \ \
m+\varepsilon\le a+1,\ \ \ a\ge2.
\end{equation}
Next,
\begin{equation}\label{decomp EndE}
\mathcal{E}nd~\mathbb{E}\simeq\mathbb{E}\otimes
\mathbb{E}\simeq S^2\mathbb{E}\oplus\wedge^2\mathbb{E},
\end{equation}
and it follows from (\ref{sympl str}) that
\begin{equation}\label{decomp S^2E}
S^2\mathbb{E}\simeq S^2\mathcal{E}_1\oplus(\mathcal{E}_1\otimes\mathcal{E}_2)\oplus S^2\mathcal{E}_2,\ \ \
\wedge^2\mathbb{E}\simeq \wedge^2\mathcal{E}_1\oplus(\mathcal{E}_1\otimes\mathcal{E}_2)\oplus \wedge^2\mathcal{E}_2.
\end{equation}
Now, since $\mathcal{E}nd~\mathcal{E}_i\simeq{\mathcal E}_i\otimes
{\mathcal E}_i\simeq S^2{\mathcal E}_i\oplus \wedge^2{\mathcal E}_i,\ \wedge^2{\mathcal E}_i\simeq\op3,\ i=1,2,$ it follows from \cite{JV} that
$h^1(\mathcal{E}nd~\mathcal{E}_1)\simeq h^1(S^2{\mathcal E}_1)=8m-3,\ h^1(\mathcal{E}nd~\mathcal{E}_2)\simeq h^1(S^2{\mathcal E}_2)=8m+8\varepsilon-3,$ and
$h^j(\mathcal{E}nd~\mathcal{E}_i)=h^j(S^2{\mathcal E}_i)=0,\ i=1,2,\ j\ge2$. This together with \eqref{decomp EndE}-(\ref{decomp S^2E}), (\ref{vanish 1}) and (\ref{h1 12}) implies that
\begin{equation}\label{h1 S2bbE}
h^1(\mathcal{E}nd~\mathbb{E})=32m+16\varepsilon-14,\ \ \
h^1(S^2\mathbb{E})=24m+12\varepsilon-10,
\end{equation}
\begin{equation}\label{hi S2bbE}
h^i(\mathcal{E}nd~\mathbb{E})=h^i(S^2\mathbb{E})=0,\ \ \ i\ge2.
\end{equation}
Now assume that
\begin{equation}\label{basic cond}
either\ \ \ 5\le a\le 12,\ 1+\varepsilon\le m+\varepsilon\le a-4,\ \ \ or\ \ \
a\ge12,\ 1+\varepsilon\le m+\varepsilon\le a+1.
\end{equation}
It follows from (\ref{vanish 2}), (\ref{vanish 2a}) and
\eqref{sympl str} that
\begin{equation}\label{h2E(-a)=0}
h^2(\mathbb{E}(-a))=0.
\end{equation}
Consider the total complex $T^{\cdot}$ of the double komplex $K^{\cdot}\otimes K^{\cdot}$, where $K^{\cdot}$ is the monad (\ref{monad}):
\begin{equation*}\label{total T}
\begin{split}
& T^{\cdot}:\ \ \ 0\to\op3(-2a)\xrightarrow{d_{-2}}
2\mathbb{E}(-a)\xrightarrow{d_{-1}}\mathbb{E}\otimes
\mathbb{E}\oplus2\op3\xrightarrow{d_0}
2\mathbb{E}(a)\xrightarrow{d_1}\op3(2a)\to0,\\
& \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ E\otimes E=\frac{\ker(d_0)}
{\mathrm{im}(d_{-1})}.
\end{split}
\end{equation*}
Following Le Potier \cite{LP}, consider the
symmetric part $ST^{\cdot}$ of the complex $T^{\cdot}$:
\begin{equation*}\label{sym of monad}
ST^{\cdot}:\ \ \ 0\to\mathbb{E}(-a)\xrightarrow{\alpha}S^2\mathbb{E}\oplus
\op3\xrightarrow{{}^t\alpha}\mathbb{E}(a)\to0,\ \ \ \ \ \ \
S^2E=\frac{\ker({}^t\alpha)}{\mathrm{im}(\alpha)},
\end{equation*}
where $\alpha$ is the induced subbundle map.
The inclusion of complexes $ST^{\cdot}\hookrightarrow T^{\cdot}$
induces commutative diagrams
\begin{equation}\label{diagr D1}
\xymatrix{
0\ar[r] & \mathbb{E}(-a))\ar[r]\ar@{_{(}->}[d] &
\ker({}^t\alpha)\ar[r]\ar@{_{(}->}[d] & S^2E
\ar[r]\ar@{_{(}->}[d] & 0\\
0\ar[r] & \mathrm{im}(d_{-1})\ar[r]& \ker(d_0) \ar[r] & E\otimes E\ar[r] & 0,}
\end{equation}
\begin{equation}\label{diagr D2}
\xymatrix{
0\ar[r] & \ker({}^t\alpha)\ar[r]\ar@{_{(}->}[d] &
S^2\mathbb{E}\oplus\op3\ar[r]\ar@{_{(}->}[d] & \mathbb{E}(a)
\ar[r]\ar@{_{(}->}[d] & 0\\
0\ar[r] & \ker(d_0)\ar[r]& \mathbb{E}\otimes\mathbb{E}\oplus
2\op3 \ar[r] & \mathrm{im}(d_0)\ar[r] & 0,}
\end{equation}
and an exact triple
\begin{equation}\label{exact ker}
0\to\op3(-2a)\xrightarrow{d_{-2}}2\mathbb{E}(-a)
\to\mathrm{im}(d_{-1})\to0
\end{equation}
Passing to cohomology in \eqref{diagr D1}-\eqref{exact ker} and
using \eqref{h1bbE(-a)=0}, \eqref{h2E(-a)=0}, \eqref{h1E(a)=0}
and the equality $h^0(S^2\mathbb{E})=0$, we obtain an equality $h^0(\mathrm{coker}\alpha)=1$ and an exact sequence
\begin{equation}\label{seq S2E}
0\to H^0(\mathbb{E}(a))/\mathbb{C}\to
H^1(S^2E)\xrightarrow{\mu}H^1(S^2\mathbb{E})\to0,
\end{equation}
which fits in a commutative diagram
\begin{equation}\label{diagr D1}
\xymatrix{
0\ar[r] & H^0(\mathbb{E}(a))/\mathbb{C})\ar[r] &
H^1(S^2E)\ar[r]^-\mu\ar@{=}[d] & H^1(S^2\mathbb{E})
\ar[r]\ar@{_{(}->}[d] & 0\\
& & H^1(E\otimes E)) \ar[r] & \mathbb{E}\otimes
\mathbb{E}. &}
\end{equation}
From \eqref{h0bbE(a)=}, \eqref{h1 S2bbE} and \eqref{h1 S2E} it
follows that
\begin{equation}\label{h1 S2E}
h^1(S^2E)=h^0(\mathbb{E}(a))+24m+12\varepsilon-11=
4\binom{a+3}{3}+(2m+\varepsilon)(10-a)-11.
\end{equation}
Note that, since $E$ is a stable rank-2 bundle, $H^1({\mathcal E} nd~E)=H^1(S^2E)$ is isomorphic to the Zariski tangent space $T_{[E]}{\mathcal B}(0,2m+\varepsilon +a^2)$:
\begin{equation}\label{Kod-Sp1}
\theta_{E}:\ T_{[E]}{\mathcal B}(0,2m+\varepsilon +a^2)\stackrel{\sim}{\to}
H^1({\mathcal E} nd~E)=H^1(S^2E).
\end{equation}
(Here $\theta_{E}$ is the Kodaira-Spencer isomorphism.)
Thus, we can rewrite \eqref{h1 S2E} as
\begin{equation}\label{dim Zar tang sp}
\dim T_{[E]}{\mathcal B}(0,2m+\varepsilon +a^2)=
4\binom{a+3}{3}+(2m+\varepsilon)(10-a)-11.
\end{equation}
We will now prove the following main result of this section.
\begin{theorem}\label{Thm A}
Under the condition \eqref{basic cond}, there exists an irreducible family ${\mathcal M}_n(E)\subset{\mathcal B}(0,n)$, where $n=2m+\varepsilon +a^2$, of dimension given by the right hand side of \eqref{dim Zar tang sp} and containing the above constructed
point $[E]$. Hence the closure ${\mathcal M}_n$ of ${\mathcal M}_n(E)$ in
${\mathcal B}(0,n)$ is an irreducible component of ${\mathcal B}(0,n)$.
The set $\Sigma_0$ of these components ${\mathcal M}_n$ is an infinite series distinct from the series of instanton components $\{I_n\}_{n\ge1}$ and from the series of components described in \cite{Ein} and \cite{AJTT}.
Furthermore, at least for each $n\ge146$ there exists an irreducible component ${\mathcal M}_n$ of ${\mathcal B}(0,n)$ belonging to the series $\Sigma_0$.
\end{theorem}
\begin{proof}
According to J. Bingener \cite[Appendix]{BH}, the equality $h^2({\mathcal E} nd\mathbb{E})=0$ (see \eqref{hi S2bbE})
implies that there exists (over $\mathbf{k}=\mathbb{C}$)
a versal deformation of the bundle $\mathbb{E}$, i. e. a smooth variety $B$ of dimension $\dim B=h^1({\mathcal E} nd\mathbb{E})$, with a marked point $0\in B$, and a locally free sheaf $\boldsymbol{{\mathcal E}}$ on $\p3\times B$ such that $\boldsymbol{{\mathcal E}}|_{\p3\times\{0\}}\simeq\mathbb{E}$ and the
Koaira-Spencer map $\theta:\ T_{[\mathbb{E}]}B\to
H^1({\mathcal E} nd\mathbb{E})$ is an isomorphism. For $b\in B$ denote $E_b:=\boldsymbol{{\mathcal E}}|_{\p3\times\{b\}}$ and consider in $B$ a closed subset
\begin{equation*}
U=\{b\in B\ |\ E_b\ is\ a\ symplectic\ instanton\}.
\end{equation*}
By definition, $U=\tilde{U}\cap B^*$, where
$\tilde{U}=\{b\in B\ |\ E_b\ is\ a\ symplectic\ bundle\}$ is a closed subset of $B$ and
\begin{equation*}
\begin{split}
&B^*=\{b\in B\ |\ E_b \ satisfies\ the\ vanishing\ conditions\ \eqref{def of sympl inst}\ and\ the\ condition\ \\
& h^0(E_b)=h^i(E_b(-a))=h^j(E_b(a))=h^k(S^2E_b)=0,\ i=1,2,\ j\ge1,\ k=0,2,3\}
\end{split}
\end{equation*}
is an open subset of $B$ by the Semicontinuity. (Here $a$ is taken from \eqref{basic cond}). Since $\mathbb{E}$ is symplectic, so that
${\mathcal E} nd\mathbb{E}\simeq S^2\mathbb{E}\oplus\wedge^2\mathbb{E}$,
it follows from \cite{R} that the Kodaira-Spencer map $\theta$ yields an isomorphism $\theta:T_{[\mathbb{E}]}U=T_{[\mathbb{E}]}\tilde{U}\stackrel{\sim}{\to} H^1(S^2\mathbb{E})$.
Thus, $U$ is a smooth variety of dimension
$$
\dim U=h^1(S^2\mathbb{E})=24m+12\varepsilon−10.
$$
(We use Riemann-Roch and the vanishing of $h^i(S^2\mathbb{E}),\ i\ne1$.)
Let $p:\p3\times B\to B$
be the projection. By the Base-Change and the vanishing conditions defining $B^*$, respectively, $U$ the sheaf ${\mathcal A}:=p_*(\boldsymbol{{\mathcal E}}\otimes\op3(a)\boxtimes{\mathcal O}_U)$ is a locally free sheaf of rank $\chi(\mathbb{E}(a))=h^0(\mathbb{E}(a))$ given by \eqref{h0bbE(a)=}. Hence $\pi:\ \tilde{X}=\mathbf{Proj}
(S^{\cdot}_{\op3}{\mathcal A}^{\vee})\to U$ is a projective bundle
with the Grothendieck sheaf ${\mathcal O}_{\tilde{X}/U}(1)$ and a
morphism
$\mathbf{s}:\ \op3\boxtimes{\mathcal O}_{\tilde{X}/U}(-1)\to
\tilde{\pi}^*(\boldsymbol{{\mathcal E}}\otimes\op3(a)
\boxtimes{\mathcal O}_U)$
defined as the composition of canonical evaluation morphisms
$\op3\boxtimes{\mathcal O}_{\tilde{X}/U}(-1)\to\tilde{p}^*\pi^*{\mathcal A}
\to\tilde{\pi}^*(\boldsymbol{{\mathcal E}}\otimes\op3(a)\boxtimes
{\mathcal O}_U)$, where $\tilde{X}\xleftarrow{\tilde{p}}\p3\times
\tilde{X}\xrightarrow{\tilde{\pi}}\p3\times U$ are the induced
projections.
Let $X=\{x\in\tilde{X}\ |\ \mathbf{s}^{\vee}|_{\p3\times\{x\}}\ $is surjective $\}$. This is an open dense subset of the smooth irreducible variety $\tilde{X}$ since it contains the point $x_0=(s:\ \op3\to\mathbb{E}(a))$ given in \eqref{subbundle}.
Hence $X$ is smooth and irreducible. In addition, since $\boldsymbol{{\mathcal E}}$ is a versal family of bundles, it follows
that $X$ is an open subset of the Quot-scheme
$\mathrm{Quot}_{\p3\times B/B}(\boldsymbol{{\mathcal E}},P(n))$, where
$P(n):=\chi(\op3(a+n))$. Therefore, by \cite[Prop. 2.2.7]{HL}
in view of \eqref{h1E(a)=0} there is an exact triple
\begin{equation}\label{tangent seq}
0\to H^0(\mathbb{E}(a))/\mathbb{C}\to T_{x_0}X\xrightarrow{d\pi}T_{[\mathbb{E}]}B\to0,
\end{equation}
which is obtained as the cohomology sequence
\begin{equation}\label{coho seq}
0\to H^0(\mathbb{E}(a))/\mathbb{C}\to H^1({\mathcal H} om(F,\mathbb{E}))\to H^1({\mathcal E} nd~\mathbb{E})\to0
\end{equation}
of the exact triple
$0\to{\mathcal H} om(F,\mathbb{E})\to{\mathcal H} om(\mathbb{E},
\mathbb{E})\to\mathbb{E}(a)\to0$ obtained by applying the functor ${\mathcal H} om(-,\mathbb{E})$ to the exact triple
$0\to\op3(-a)\xrightarrow{s}\mathbb{E}\to F\to0$,
where $F:=\mathrm{coker}(s)$.
Next, since $\boldsymbol{{\mathcal E}}$ is a versal family of bundles, it follows that $\mathbf{E}=\boldsymbol{{\mathcal E}}|_{\p3\times U}$ is a versal family of symplectic instantons. Hence, denoting
$Y=U\times_BX$, we extend the exact triple \eqref{tangent seq}
to commutative diagram
\begin{equation}\label{diagr 4}
\xymatrix{
0\ar[r] & H^0(\mathbb{E}(a))/\mathbb{C}\ar[r] &
T_{x_0}X\ar[r] & T_{[\mathbb{E}]}B\ar[r] & 0\\
0\ar[r] & H^0(\mathbb{E}(a))/\mathbb{C}\ar[r]\ar[r]\ar@{=}[u] & T_{x_0}Y \ar[r]^-{d\pi}
\ar@{^{(}->}^-{i_Y}[u] & T_{[\mathbb{E}]}U
\ar@{^{(}->}^-{i_U}[u]\ar[r] & 0,}
\end{equation}
where $i_Y$ and $i_U$ are natural inclusions. (Note that, under the Kodaira-Spencer isomorphisms $\theta:T_{[\mathbb{E}]}U)
\xrightarrow{\simeq}H^1(S^2\mathbb{E})$ and $T_{[\mathbb{E}]}B
\xrightarrow{\simeq}H^1(\mathbb{E}\otimes\mathbb{E})\simeq
H^1({\mathcal E} nd~\mathbb{E})$ the rightmost inclusions in diagrams
\eqref{diagr D1} and \eqref{diagr 4} coincide.)
Consider the modular morphism
$$
\Phi:\ Y\to{\mathcal B}:={\mathcal B}(0,2m+\varepsilon +a^2),\ \
(b,s)\mapsto\bigg[\frac{\mathrm{Ker}
({}^ts)}{\mathrm{Im}(s)}\bigg],
$$
where, as before, $s:\op3(-a)\to E_b$ is a subbundle morphism.
Its differential $d\Phi$ composed with the Kodaira-Spencer map
$\theta_E$ from \eqref{Kod-Sp1} is a linear map
\begin{equation*}
\phi=\theta_E\circ d\Phi:\ T_{x_0}Y\to H^1(S^2E)=H^1(E\otimes E).
\end{equation*}
Now from the functorial properties of the Kodaira-Spencer maps
$\phi$ and $\theta$ it follows that the triple \eqref{seq S2E} and the lower triple in the diagram \eqref{diagr 4} fit in a
commutative diagram
\begin{equation*}
\xymatrix{
0\ar[r] & H^0(\mathbb{E}(a))/\mathbb{C}\ar[r]\ar[r] & H^1(S^2E) \ar[r]^-{\mu} & H^1(S^2\mathbb{E})\ar[r] & 0\\
0\ar[r] & H^0(\mathbb{E}(a))/\mathbb{C}\ar[r]\ar[r]\ar@{=}[u] & T_{x_0}Y \ar[r]^-{d\pi}\ar^-{\phi}[u] & T_{[\mathbb{E}]}U
\ar^-{\theta}_-{\simeq}[u]\ar[r] & 0.}
\end{equation*}
This implies that $\phi$ is an isomorphism, so that, since $Y$
is smooth at $x_0$ and irreducible, ${\mathcal M}_n(E)=\Phi(Y)$ is an open subset of an irreducible component ${\mathcal M}_n$ of ${\mathcal B}(0,n)$, of dimension given by \eqref{dim Zar tang sp}.
It is easy to check that the dimension $\dim{\mathcal M}_n$ given by
\eqref{dim Zar tang sp}, with $m,\varepsilon$ and $a$ subjected to the condition \eqref{basic cond}, satisfies
the strict inequality $\dim{\mathcal M}_n>8n-3=\dim I_n$.
This shows that the series $\Sigma_0$ is distinct from $\{I_n\}_{n\ge1}$. To distinguish $\Sigma_0$ from
from the series of components described in \cite{Ein}, it is enough to see that the spectra of general bundles of these two series are different. (We leave to the reader a direct verification of this fact.)
Note that, for each $a\ge12$ we have $1\le m\le a+1$ and $0\le\varepsilon\le1$, so that $n=2m+\varepsilon +a^2$
ranges through the whole interval of positive integers $(a^2+2,(a+1)^2+1)\subset\mathbb{Z}_{+}$. Hence, $n$ takes
at least all positive values $\ge12^2+2=146$. This shows that
for each $n\ge146$ there exists an irreducible component ${\mathcal M}_n\in\Sigma_0$.
Last, remark that, for the series of components ${\mathcal M}_n$ described in \cite{AJTT}, $n$ takes values $n=1+k^2,\ k\in\{2\}\cup(4,\infty)$. Hence this series is distinct from $\Sigma_0$.
Theorem is proved.
\end{proof}
\vspace{5mm}
\section{Construction of stable rank two bundles with odd determinant}
\label{section 4}
\vspace{5mm}
In this section we will construct an infinite series of stable vector bundles from ${\mathcal B}(-1,2m)$, $m\in\mathbb{Z}_+$. It is known from \cite[Example 4.3.2]{H-vb} that, for each $m\ge1$ there exists an irreducible component ${\mathcal B}_0(-1,2m)$ of ${\mathcal B}(-1,2m)$,
of the expected dimension
\begin{equation}\label{dim B0}
\dim{\mathcal B}_0(-1,2m)=16m-5,
\end{equation}
which contains bundles ${\mathcal E}_1$ obtained via the Serre constructions as the extensions of the form
\begin{equation}\label{Serre-1}
0\to\op3(-2)\to{\mathcal E}_1\to{\mathcal I}_{Y,\p3}(1)\to0,
\end{equation}
where $Y$ is a union of $m+1$ disjoint conics in $\p3$.
Below we will need the following analogue of the Proposition \ref{Prop 1}.
\begin{proposition}\label{Prop 2}
Let $a,m\in\mathbb{Z}_+$, $a\ge2$, and
let $\varepsilon\in\{0,1\}$. A general pair
\begin{equation}\label{2 inst a}
([{\mathcal E}_1],[{\mathcal E}_2])\in {\mathcal B}_0(-1,2m)\times {\mathcal B}_0(-1,2(m+\varepsilon)),
\end{equation}
of vector bundles satisfies the following conditions:
$$
[{\mathcal E}_1]\ne[{\mathcal E}_2];
$$
for $i=1,\ a\ge2m+4,$ respectively, for $i=2,\ a\ge2(m+\varepsilon)+4$,
\begin{equation}\label{vanish 1a}
h^1({\mathcal E}_i(a))=0,
\end{equation}
\begin{equation}\label{vanish 2a a}
h^2({\mathcal E}_i(-a))=0;
\end{equation}
\begin{equation}\label{vanish 2b}
h^1({\mathcal E}_i(-a))=0;
\end{equation}
\begin{equation}\label{vanish 3a}
h^j({\mathcal E}_1(1)\otimes{\mathcal E}_2)=0,\ \ \ \ j\ne1.
\end{equation}
\end{proposition}
\begin{proof}
Let $Y=\sqcup_1^{m+1}C_i$ be a disjoint union of conics $C_i=l_i\cup l'_i$ decomposable into pairs of distinct lines $l_i,l'_i$, such that\\
(i) there exist two smooth quadrics $S\simeq\p1\times\p1$
and $S'\simeq\p1\times\p1$ with the property that
$l_1,...,l_{m+1}$, respectively, $l'_1,...,l'_{m+1}$ are the lines of one ruling on $S$, respectively, on
$S'$; for instance, denoting $Y_0=l_1\sqcup...\sqcup l_{m+1}, \ Y'=l'_1\sqcup...\sqcup l'_{m+1},$
we may assume that
$$
{\mathcal O}_S(Y_0)\simeq\op1(m+1)\boxtimes\op1,\ \ \
{\mathcal O}_{S'}(Y')\simeq\op1(m+1)\boxtimes\op1;
$$
(ii) the set of $m+1$ distinct points $Z=(Y'\cap S)\setminus(Y_0\cap Y')$ satisfies the condition that
$pr_1(Z)$ is a union of $m+1$ distinct points, where
$pr_1:S'\to\p1$ is the projection.
We then have a diagram similar to \eqref{diagr 1}:
\begin{equation}\label{diagr 6}
\xymatrix{
& 0 & 0 & 0 &\\
0\ar[r] &{\mathcal O}_{Y'}(a-4)\ar[r]\ar[u] &
{\mathcal O}_{Y'}(a-3)\ar[r]\ar[u] & {\mathcal O}_{Z}(a-3) \ar[u]\ar[r] & 0\\
0\ar[r]& \op3(a-4)\ar[r]\ar^-f[u] & {\mathcal I}_{Y_0,\p3}(a-2)\ar[r]
\ar_-{g}[u] &\op1(a-m-3)\boxtimes\op1(a-2)
\ar^-h[u]\ar[r] & 0.}
\end{equation}
Under the assumptions $a\ge2m+4$ and $m\ge2$ the cohomology of the lower triple of this diagram yields
\begin{equation}\label{h1(IY0)}
h^1({\mathcal I}_{Y_0,\p3}(a-2))=0.
\end{equation}
Next, similar to \eqref{triple(a)} we have an exact triple
$0\to\op3(a-6)\to{\mathcal I}_{Y,\p3}(a-4)\to\op1(a-5-m)\boxtimes\op1
(a-4)\to0$ which implies that $h^1({\mathcal I}_{Y,\p3}(a-4))=0$ since $a-5-m\ge0$ for $a\ge2m+4$ and $m\ge1$. Since
${\mathcal I}_{Y,\p3}(a-4)=\ker f$, it follows that
\begin{equation}\label{h0(f) is surj}
h^0(f):\ H^0(\op3(a-4))\to H^0({\mathcal O}_{Y'}(a-4))\ \ \ is\ surjective.
\end{equation}
On the other hand, since $a-3-m\ge m+1=h^0(Z)$, from the above condition (ii) on $Z$ it follows that
$h^0(h):\ H^0(\op1(a-m-3)\boxtimes\op1(a-2))\to H^0({\mathcal O}_{Z}(a-3))$ is surjective. This together with \eqref{h0(f) is surj} and diagram \eqref{diagr 6} yields
that
$h^0(g):\ H^0({\mathcal I}_{Y_0,\p3}(a-2))\to H^0({\mathcal O}_{Y'}(a-3))$
is surjective. Since $\ker g\simeq{\mathcal I}_{Y,\p3}(a-2)$, it follows by \eqref{h1(IY0)} that
\begin{equation}\label{h1(IY)}
h^1({\mathcal I}_{Y,\p3}(a-2))=0.
\end{equation}
Now, twisting the triple \eqref{Serre-1} by $\op3(a-3)$ and
using \eqref{h1(IY)} we obtain $h^1({\mathcal E}_1(a-3))=0$, hence by Serre duality $h^2({\mathcal E}_1(-a))=0$. Besides, $h^1({\mathcal E}_1(a-3))=0$ clearly implies $h^1({\mathcal E}_1(a))=0$
since $a\ge2m+4.$ Now, by Semicontinuity, this yields
\eqref{vanish 1a} and \eqref{vanish 2a} for a general $[{\mathcal E}_1]\in{\mathcal B}_0(-1,2m)$. The same equalities are clearly true for $i=2$.
Next, since $a\ge2$, it follows that $h^0({\mathcal O}_{C_i}(1-a))=0$ for any conic $C_i\subset Y$, hence the cohomology of the triple $0\to{\mathcal I}_{Y,\p3}(1-a)\to\op3(1-a)\to
\oplus_{i=1}^{m+1}{\mathcal O}_{C_i}(1-a)\to0$ yields $h^1({\mathcal I}_{Y,\p3}(1-a))=0$; this together with \eqref{Serre-1} and the Semicontinuity yields \eqref{vanish 2b} for $i=1$ and similarly for $i=2$.
Last, the equalities \eqref{vanish 3a} are proved similarly to \eqref{vanish 3}.
\end{proof}
\begin{remark}\label{rem B}
Note that, under the conditions of Proposition \ref{Prop 2}, the equalities (\ref{vanish 3a}) together with Riemann-Roch yield
\begin{equation}\label{h1 12 a}
h^1({\mathcal E}_1(1)\otimes{\mathcal E}_2)=16m+8\varepsilon-6.
\end{equation}
\end{remark}
Now, to construct new series of components of ${\mathcal B}(-1,4m+2\varepsilon)$, we proceed along the same lines as in Section \ref{section 3}. We first introduce the notion
of a twisted symplectic structure on a vector bundle.
By a \textit{twisted symplectic structure} on a vector bundle $E$ on $\p3$ we mean an isomorphism
$\theta: E \xrightarrow{\sim}E^{\vee}(-1)$ such that $\theta^{\vee}(1)=-\theta$, considered modulo proportionality. (Here by definition $\theta^{\vee}(1):= \theta^{\vee}\otimes\mathrm{id}_{\op3(1)}$.) Clearly,
a vector bundle $E$ with twisted symplectic structure has even rank:\ \ $\rk E=2r,\ r\ge1.$
Consider the vector bundles ${\mathcal E}_1$ and ${\mathcal E}_2$ introduced in Proposition \ref{Prop 2}.
Since $\det{\mathcal E}_1\simeq\det{\mathcal E}_2\simeq\op3(-1)$, there are twisted symplectic structures $\theta_i:\ {\mathcal E}_i\xrightarrow{\simeq}{\mathcal E}_i^{\vee}(-1),\ i=1,2,$ which yield a twisted symplectic structure on the direct sum
$\mathbb{E}={\mathcal E}_1\oplus{\mathcal E}_2$:
\begin{equation}\label{twisted sympl str}
\theta=\theta_1\oplus\theta_2:\ \mathbb{E}={\mathcal E}_1\oplus{\mathcal E}_2 \xrightarrow{\simeq}{\mathcal E}_1^{\vee}(-1)\oplus{\mathcal E}_2^{\vee}(-1)=\mathbb{E}^{\vee}(-1).
\end{equation}
Now assume that ${\mathcal E}_1$ and ${\mathcal E}_2$ are chosen in such a way that there exist sections
\begin{equation}\label{empty intersectn a}
s_i:\ \op3\to{\mathcal E}_i(a+1), \ \ \ s.t.\ \ \ \dim(s_i)_0=1,\ \ i=1,2,\ \ \ (s_1)_0\cap(s_2)_0=\emptyset.
\end{equation}
(Such ${\mathcal E}_1\in{\mathcal B}_0(-1,2m),\ [{\mathcal E}_2]\in {\mathcal B}_0(-1,2(m+\varepsilon))$ always exist, since already for $a=1$, hence also for $a\ge2$ two general bundles of the form \eqref{Serre-1} satisfy the property (\ref{empty intersectn a}).) The assumption (\ref{empty intersectn a}) implies that
the section $s=(s_1,s_2):\ \op3(-a-1)\to\mathbb{E}$
is a subbundle morphism, hence its transpose
${}^ts:=s^{\vee}(-1)\circ\theta:\ \mathbb{E}\to\op3(a)$
is an epimorphism. As $\theta$ in (\ref{twisted sympl str}) is twisted symplectic, the composition ${}^ts\circ s:\op3(-a-1)\to\op3(a)$ is also twisted symplectic. Therefore, since
$\op3(a)$ and $\op3(-a-1)$ are line bundles, it follows that ${}^ts\circ s=0$, i. e. the complex
\begin{equation}\label{monad a}
K^{\cdot}:\ \ \ 0\to\op3(-a-1)\xrightarrow{s}\mathbb{E}\xrightarrow{{}^ts}
\op3(a)\to0,\ \ \ \ \ E=\frac{\ker({}^ts)}{\mathrm{im}(s)},
\end{equation}
is a monad and its cohomology sheaf $E$ is locally free. Note that, since the bundles ${\mathcal E}_1$ and ${\mathcal E}_2$ are stable, they have zero spaces of global sections, hence also $h^0(\mathbb{E})=0$, and \eqref{monad a}
yields $h^0(E)=0$, i. e. $E$ as a rank 2 vector bundle with $c_1=-1$ is stable. Besides, since $c_2(\mathbb{E})=c_2({\mathcal E}_1)+c_2({\mathcal E}_2)=4m+2\varepsilon$,
it follows from \eqref{monad a} that
$c_2(E)=4m+2\varepsilon +a(a+1)$. Thus,
$$
[E]\in{\mathcal B}(-1,4m+2\varepsilon +a(a+1)),
$$
and the deformation theory yields that, for any irreducible
component ${\mathcal M}$ of ${\mathcal B}(-1,4m+2\varepsilon +a(a+1))$,
$$
\dim{\mathcal M}\ge1-\chi(\mathcal{E}nd~E)=8(4m+2\varepsilon +a(a+1))-5.
$$
Now, as in \eqref{sym of monad}, consider the
symmetric part of the total complex of the double komplex
$K^{\cdot}\otimes (K^{\cdot})^{\vee}$, where $K^{\cdot}$ is the monad (\ref{monad a}):
\begin{equation}\label{sym of monad a}
0\to\mathbb{E}(-a)\xrightarrow{\alpha}S^2\mathbb{E}(1)
\oplus
\op3\xrightarrow{{}^t\alpha}\mathbb{E}(a+1)\to0,\ \ \ \
S^2E(1)=\frac{\ker({}^t\alpha)}{\mathrm{im}(\alpha)}.
\end{equation}
Here $\alpha$ is the induced subbundle map and $S^2E(1)$ is its cohomology sheaf. The monad (\ref{sym of monad a}) can be rewritten as a diagram of exact triples similar to \eqref{diagr 2}:
\begin{equation}\label{diagr 2a}
\xymatrix{
& & & 0 &\\
& & & \mathbb{E}(a+1)\ar[u]\\
0\ar[r]& \mathbb{E}(-a)\ar^-\alpha[r] & S^2\mathbb{E}(1)\oplus\op3\ar[r]&
\mathrm{coker}\alpha\ar[u]\ar[r] & 0\\
& & & S^2E(1)\ar[u]&\\
& & & 0.\ar[u]& }
\end{equation}
Note that, by \eqref{vanish 2b} and \eqref{twisted sympl str} one has
\begin{equation}\label{h1bbE(-a)=0 a}
h^1(\mathbb{E}(-a))=0,\ \ \ a\ge2,
\end{equation}
\begin{equation}\label{h2,3bbE(a)=0 a}
h^j(\mathbb{E}(a+1))=0,\ \ \ j=2,3,\ \ \ a\ge2m+3.
\end{equation}
Similarly, in view of \eqref{vanish 1a},
\begin{equation}\label{h1E(a)=0 a}
h^1(\mathbb{E}(a+1))=0,\ \ \ a\ge2m+3.
\end{equation}
This together with \eqref{h2,3bbE(a)=0 a} and Riemann-Roch yields:
\begin{equation}\label{h0bbE(a)= a}
h^0(\mathbb{E}(a+1))=\chi(\mathbb{E}(a+1))=
4\binom{a+3}{3}+2\binom{a+3}{2}-(2m+\varepsilon)(2a+5).
\end{equation}
Next,
\begin{equation}\label{decomp EndE a}
\mathcal{E}nd~\mathbb{E}\simeq\mathbb{E}(1)\otimes
\mathbb{E}\simeq S^2\mathbb{E}(1)\oplus\wedge^2\mathbb{E}(1),
\end{equation}
and it follows from (\ref{twisted sympl str}) that
\begin{equation}\label{decomp S^2E a}
\begin{split}
& S^2\mathbb{E}(1)\simeq S^2\mathcal{E}_1(1)\oplus(\mathcal{E}_1(1)\otimes
\mathcal{E}_2)\oplus S^2\mathcal{E}_2(1),\\
& \wedge^2\mathbb{E}(1)\simeq \wedge^2\mathcal{E}_1(1)\oplus(\mathcal{E}_1(1)\otimes
\mathcal{E}_2)\oplus\wedge^2\mathcal{E}_2(1).
\end{split}
\end{equation}
Now, since $\mathcal{E}nd~\mathcal{E}_i\simeq{\mathcal E}_i(1)\otimes
{\mathcal E}_i\simeq S^2{\mathcal E}_i(1)\oplus \wedge^2{\mathcal E}_i(1),\ \wedge^2{\mathcal E}_i\simeq\op3,\ i=1,2,$ it follows from \cite{JV} that
$h^1(\mathcal{E}nd~\mathcal{E}_1)\simeq h^1(S^2{\mathcal E}_1(1))=16m-5,\ h^1(\mathcal{E}nd~\mathcal{E}_2)\simeq h^1(S^2{\mathcal E}_2(1))=16(m+\varepsilon)-5,$ and
$h^j(\mathcal{E}nd~\mathcal{E}_i)=h^j(S^2{\mathcal E}_i(1))=0,\ i=1,2,\ j\ge2$. This together with \eqref{decomp EndE a}-(\ref{decomp S^2E a}), (\ref{vanish 1a}) and (\ref{h1 12 a}) implies that
\begin{equation}\label{h1 S2bbE a}
h^1(\mathcal{E}nd~\mathbb{E})=64m+32\varepsilon-22,\ \ \
h^1(S^2\mathbb{E}(1))=48m+24\varepsilon-16,
\end{equation}
\begin{equation*}\label{hi S2bbE a}
h^i(\mathcal{E}nd~\mathbb{E})=h^i(S^2\mathbb{E}(1))=0,\ \ \ i\ge2.
\end{equation*}
It follows from (\ref{vanish 2a a}) and
\eqref{twisted sympl str} that
\begin{equation}\label{h2E(-a)=0 a}
h^2(\mathbb{E}(-a))=0.
\end{equation}
Note that \eqref{h1bbE(-a)=0 a}, \eqref{h2E(-a)=0 a} and
\eqref{h1E(a)=0 a}, together with the diagram \eqref{diagr 2a} yield an equality $h^0(\mathrm{coker}\alpha)=1$ and an exact sequence:
$$
0\to H^0(\mathbb{E}(a+1))/\mathbb{C}\to
H^1(S^2E(1))\xrightarrow{\mu}H^1(S^2\mathbb{E}(1))\to0,
$$
hence by \eqref{h0bbE(a)= a} and \eqref{h1 S2bbE a} we have
\begin{equation}\label{h1 S2E a}
\begin{split}
& h^1(S^2E(1))=h^0(\mathbb{E}(a+1))+48m+24\varepsilon-17=\\
& 4\binom{a+3}{3}+2\binom{a+3}{2}-(2m+\varepsilon)(2a-19)
-17.
\end{split}
\end{equation}
Note that, since $E$ is a stable rank-2 bundle, $H^1({\mathcal E} nd~E)=H^1(S^2E(1))$ is isomorphic to the Zariski tangent space $T_{[E]}{\mathcal B}(-1,4m+2\varepsilon +a(a+1))$:
\begin{equation}\label{Kod-Sp1 a}
\theta_{E}:\ T_{[E]}{\mathcal B}(-1,4m+2\varepsilon +a(a+1))\stackrel{\sim}{\to}
H^1({\mathcal E} nd~E)=H^1(S^2E(1)).
\end{equation}
(Here $\theta_{E}$ is the Kodaira-Spencer isomorphism.)
Thus,
we can rewrite \eqref{h1 S2E} as
\begin{equation}\label{dim Zar tang sp a}
\begin{split}
& \dim T_{[E]}{\mathcal B}(-1,4m+2\varepsilon +a(a+1))=\\
& 4\binom{a+3}{3}+2\binom{a+3}{2}-(2m+\varepsilon)(2a-19)
-17.
\end{split}
\end{equation}
\begin{theorem}\label{Thm B}
For $m\ge1$, $\varepsilon\in\{0,1\}$ and $a\ge2(m+\varepsilon)+3$, there exists an irreducible family ${\mathcal M}_n(E)\subset{\mathcal B}(-1,n)$, where $n=4m+2\varepsilon +a(a+1)$, of dimension given by the right hand side of \eqref{dim Zar tang sp a} and containing the above constructed
point $[E]$. Hence the closure ${\mathcal M}_n$ of ${\mathcal M}_n(E)$ in
${\mathcal B}(-1,n)$ is an irreducible component of ${\mathcal B}(-1,n)$.
The set $\Sigma_1$ of these components ${\mathcal M}_n$ is an infinite series distinct from the series
$\{{\mathcal B}_0(-1,n)\}_{n\ge1}$ and from the series of Ein components described in \cite{Ein}.
\end{theorem}
The proof of this Theorem is completely parallel to the proof of Theorem \ref{Thm A}, with clear modifications due to the change from $c_1(E)=0$ to $c_1(E)=-1$.
It is easy to check that the dimension $\dim{\mathcal M}_n$ given by \eqref{dim Zar tang sp a}, with $m,\varepsilon$ and $a$
as in Theorem \ref{Thm B}, satisfies the strict inequality $\dim{\mathcal M}_n>8n-5=\dim{\mathcal B}_0(-1,n)$ (cf. \eqref{dim B0}).
This shows that $\Sigma_1$ is distinct from $\{{\mathcal B}_0(-1,n)\}_{n\ge1}$. To distinguish $\Sigma_1$ from
from the series of Ein components, it is enough to see that
the spectra of general bundles of these two series are different. (A direct verification of this fact is left to the reader.)
\begin{remark}\label{Rem B1}
Let ${\mathcal N}$ be the set of all values of $n$ for which ${\mathcal M}_n\in\Sigma_1$, i. e.
$$
{\mathcal N}=\{n\in2\mathbb{Z}_+\ |\ n=4m+2\varepsilon+a(a+1),\ where\ m\in\mathbb{Z}_+,\ \varepsilon\in\{0,1\},\ a\ge2m+\varepsilon+3 \},
$$
Then one easily sees that
$$
\lim\limits_{r\to\infty}\frac{{\mathcal N}\cap\{2,4,...,2r\}}{r}=1.
$$
\end{remark}
\vspace{5mm}
\section{Examples of moduli components of stable vector bundles with small values of $c_2$}\label{section 5}
The conditions imposed on the data $(m,\varepsilon,a)$ in Theorem \ref{Thm A}, respectively, Theorem \ref{Thm B} may not be satisfied for small values of these data. However, for some of small values of $(m,\varepsilon,a)$ the equalities \eqref{vanish 1}, \eqref{vanish 2}, \eqref{vanish 2a}, respectively, \eqref{vanish 1a}, \eqref{vanish 2a a}, \eqref{vanish 2b} are still true.
Hence, our construction of irreducible components ${\mathcal M}_n\in\Sigma_0$, where $n=2m+\varepsilon+a^2$, respectively,${\mathcal M}_n\in\Sigma_1$, where $n=4m+2\varepsilon+a(a+1)$,
given in Sections \ref{section 3} and \ref{section 4} is still true for these values of $(m,\varepsilon,a)$.
A precise computation of these values is performed via using the Serre construction \eqref{trC}, respectively, \eqref{Serre-1} for the pairs $([{\mathcal E}_1],[{\mathcal E}_2])$ from \eqref{2 inst}, respectively, from \eqref{2 inst a}. We thus provide the following list of irreducible components ${\mathcal M}_n\in\Sigma_0$ for $n\le20$ and, respectively, ${\mathcal M}_n\in\Sigma_1$ for $n\le40$.
\subsection{Components ${\mathcal M}_n\in\Sigma_0$ for $n\le20$}
By $\mathrm{Spec}(E)$ we denote the spectrum
of a general bundle $E$ from ${\mathcal M}_n$. (Below we use a standard notation $\mathrm{Spec}(E)=
(a^p,b^q,...)$ for the spectrum ($\underset{p}
{\underbrace{a......a}},~\underset{q}
{\underbrace{b......b}},...$).)
(1) $n=6,\ (m,\varepsilon,a)=(1,0,2)$. ${\mathcal M}_6$ is a component of the expected (by the deformation theory) dimension $\dim{\mathcal M}_6=45$, and
$\mathrm{Spec}(E)=(-1,0^4,1)$. This corresponds to the case
6(2) of the Table 5.3 of Hartshorne-Rao \cite{HR}.
(2) $n=7,\ (m,\varepsilon,a)=(1,1,2)$. ${\mathcal M}_7$ is a component of the expected dimension $\dim{\mathcal M}_7=53$, and
$\mathrm{Spec}(E)=(-1,0^5,1)$ (cf. \cite[Table 5.3, 7(2)]{HR}).
(3) $n=8,\ (m,\varepsilon,a)=(2,0,2)$. ${\mathcal M}_8$ is a component of the expected dimension $\dim{\mathcal M}_8=61$, and
$\mathrm{Spec}(E)=(-1,0^6,1)$ (cf. \cite[Table 5.3, 8(2)]{HR}).
(4) $n=9,\ (m,\varepsilon,a)=(2,1,2)$. ${\mathcal M}_9$ is a component of the expected dimension $\dim{\mathcal M}_9=69$, and
$\mathrm{Spec}(E)=(-1,0^7,1)$.
(5) $n=10,\ (m,\varepsilon,a)=(3,0,2)$. ${\mathcal M}_{10}$ is a component of the expected dimension $\dim{\mathcal M}_{10}=77$, and
$\mathrm{Spec}(E)=(-1,0^8,1)$.
(6) $n=11,\ (m,\varepsilon,a)=(3,1,2)$. ${\mathcal M}_{11}$ is a component of the expected dimension $\dim{\mathcal M}_{11}=85$, and
$\mathrm{Spec}(E)=(-1,0^9,1)$.
(7) $n=12,\ (m,\varepsilon,a)=(4,0,2)$. ${\mathcal M}_{12}$ is a component of the expected dimension $\dim{\mathcal M}_{12}=93$, and
$\mathrm{Spec}(E)=(-1,0^{10},1)$.
(8) $n=18,\ (m,\varepsilon,a)=(1,0,4)$. ${\mathcal M}_{18}$ is a component of the expected dimension $\dim{\mathcal M}_{12}=141$, and
$\mathrm{Spec}(E)=(-3,-2^2,-1^3,0^6,1^3,2^2,3)$.
\subsection{Components ${\mathcal M}_n\in\Sigma_1$ for $n\le40$}
${}$\\
(1) $n=24,\ (m,\varepsilon,a)=(1,0,4)$. ${\mathcal M}_{24}$ is a component of the expected dimension $\dim{\mathcal M}_{24}=187$, and
$\mathrm{Spec}(E)=(-4,-3^2,-2^3,-1^6,0^6,1^3,2^2,3)$.
(2) $n=34,\ (m,\varepsilon,a)=(1,0,5)$. ${\mathcal M}_{34}$ is a component of dimension $\dim{\mathcal M}_{34}=281$ larger than expected, and
$\mathrm{Spec}(E)=(-5,-4^2,-3^3,-2^4,-1^7,0^7,1^4,2^3,3^2,4)$.
(3) $n=36,\ (m,\varepsilon,a)=(1,1,5)$. ${\mathcal M}_{36}$ is a component of dimension $\dim{\mathcal M}_{34}=281$ larger than expected, and
$\mathrm{Spec}(E)=(-5,-4^2,-3^3,-2^4,-1^8,0^8,1^4,2^3,3^2,4)$.
(4) $n=38,\ (m,\varepsilon,a)=(2,0,5)$. ${\mathcal M}_{38}$ is a component of the expected dimension $\dim{\mathcal M}_{36}=299$, and
$\mathrm{Spec}(E)=(-5,-4^2,-3^3,-2^4,-1^9,0^9,1^4,2^3,3^2,4)$.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,476 |
\section{Introduction}
{\it Total descendent potentials}, also called {\it formal Gromov--Witten potentials}, are certain formal power series of the form
$$
\mathcal{F}(t^*_*,\varepsilon)=\sum_{g\ge 0}\varepsilon^{2g}\mathcal{F}_g(t^*_*)\in\mathbb C[[t^*_*,\varepsilon]],
$$
where $N\ge 1$, and $t^\alpha_a$, $1\le\alpha\le N$, $a\ge 0$, and $\varepsilon$ are formal variables, appearing in various curve counting theories in algebraic geometry including Gromov--Witten theory, Fan--Jarvis--Ruan--Witten theory, and the more recent theory of Gauged Linear Sigma Models. The number~$N$ is often called the {\it rank}. Typically, the coefficients of total descendent potentials are the integrals of certain cohomology classes over moduli spaces of closed Riemann surfaces with additional structures. The function $\mathcal{F}_g$ controls the integrals over the moduli spaces of Riemann surfaces of genus $g$. The simplest example of a total descendent potential is the Witten generating series $\mathcal{F}^W(t_0,t_1,t_2,\ldots,\varepsilon)$ of intersection numbers on the moduli space of stable Riemann surfaces of genus $g$ with $n$ marked points $\overline{\mathcal M}_{g,n}$. Note that here and below we omit the upper indices in the $t$-variables when the rank is $1$.
\medskip
There is a unified approach to total descendent potentials using the notion of a cohomological field theory (CohFT)~\cite{KM94} and the Givental group action~\cite{Giv01a,Giv01b,Giv04}. Briefly speaking, the generating series of correlators of CohFTs form the space of {\it total ancestor potentials}, and then using the lower-triangular Givental group action one gets the whole space of total descendent potentials (see e.g. \cite[Section~2]{Sha09} and~\cite{FSZ10}).
\medskip
There is a remarkable and deep relation between total descendent potentials and the theory of nonlinear PDEs. One of its manifestations is the following system of PDEs for the descendent potential in genus $0$ (see e.g. \cite[Section~2]{Sha09} and~\cite[Corollary 4.13]{FSZ10}):
\begin{align}
&\frac{{\partial}\mathcal{F}_0}{{\partial} t^1_0}=\sum_{a\ge 0}t^\alpha_{a+1}\frac{{\partial}\mathcal{F}_0}{{\partial} t^\alpha_a}+\frac{1}{2}\eta_{\alpha\beta}t^\alpha_0 t^\beta_0,\label{eq:closed string-0}\\
&\frac{{\partial}^3\mathcal{F}_0}{{\partial} t^\alpha_{a+1}{\partial} t^\beta_b{\partial} t^\gamma_c}=\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_a{\partial} t^\mu_0}\eta^{\mu\nu}\frac{{\partial}^3\mathcal{F}_0}{{\partial} t^\nu_0{\partial} t^\beta_b{\partial} t^\gamma_c},\quad 1\le\alpha,\beta,\gamma\le N,\quad a,b,c\ge 0,\label{eq:closed TRR-0}
\end{align}
called the {\it string equation} and the {\it topological recursion relations in genus $0$}, respectively. Here $(\eta_{\alpha\beta})=\eta$ is an $N\times N$ symmetric nondegenerate matrix with complex coefficients, the constants~$\eta^{\alpha\beta}$ are defined by $(\eta^{\alpha\beta}):=\eta^{-1}$, and we use the Einstein summation convention for repeated upper and lower Greek indices. Note that the system of equations~\eqref{eq:closed TRR-0} can be equivalently written~as
\begin{gather*}
d\left(\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_{a+1}{\partial} t^\beta_b}\right)=\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_a{\partial} t^\mu_0}\eta^{\mu\nu}d\left(\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\nu_0{\partial} t^\beta_b}\right),\quad 1\le\alpha,\beta\le N,\quad a,b\ge 0,
\end{gather*}
where $d \left( \cdot \right)$ denotes the full differential.
\medskip
There are equations similar to~\eqref{eq:closed TRR-0} in genus~$1$ (see e.g. \cite[Equation~(1.7)]{EGX00}):
\begin{gather}\label{eq:closed TRR-1}
\frac{{\partial}\mathcal{F}_1}{{\partial} t^\alpha_{a+1}}=\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_a{\partial} t^\mu_0}\eta^{\mu\nu}\frac{{\partial}\mathcal{F}_1}{{\partial} t^\nu_0}+\frac{1}{24}\eta^{\mu\nu}\frac{{\partial}^3\mathcal{F}_0}{{\partial} t^\mu_0{\partial} t^\nu_0{\partial} t^\alpha_a},\quad 1\le\alpha\le N,\quad a\ge 0.
\end{gather}
They are called the {\it topological recursion relations in genus $1$}. These equations imply that
\begin{gather}\label{eq:formula for the closed genus 1 potential}
\mathcal{F}_1=\frac{1}{24}\log\det(\eta^{-1}M)+G(v^1,\ldots,v^N),
\end{gather}
where the $N\times N$ matrix $M=(M_{\alpha\beta})$ is defined by $M_{\alpha\beta}:=\frac{{\partial}^3\mathcal{F}_0}{{\partial} t^1_0{\partial} t^\alpha_0{\partial} t^\beta_0}$, $v^\alpha:=\eta^{\alpha\mu}\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\mu_0{\partial} t^1_0}$, and $G(t^1,\ldots,t^N):=\left.\mathcal{F}_1\right|_{t^*_{\ge 1}=0}$ \cite{DW90} (see also \cite[Equation~(1.16)]{DZ98}). Equations similar to~\eqref{eq:closed TRR-0} and~\eqref{eq:closed TRR-1} exist in all genera, but their complexity grow very rapidly with the genus (see e.g.~\cite{Liu07} for some results in genus~$2$).
\medskip
One can see that equations~\eqref{eq:closed string-0},~\eqref{eq:closed TRR-0},~\eqref{eq:closed TRR-1} are universal, meaning that they do not depend on a total descendent potential. On the other hand, there is a rich theory~\cite{DZ01} of hierarchies of evolutionary PDEs with one spatial variable associated to total descendent potentials and containing the full information about these potentials. Conjecturally, for any total descendent potential~$\mathcal{F}$ there exists a unique system of PDEs of the form
\begin{gather}\label{eq:Dubrovin--Zhang system}
\frac{{\partial} w^\alpha}{{\partial} t^\beta_b}=P^\alpha_{\beta,b},\quad 1\le\alpha,\beta\le N,\quad b\ge 0,
\end{gather}
where $w^1,\ldots,w^N\in\mathbb C[[t^*_*,\varepsilon]]$, $P^\alpha_{\beta,b}$ are differential polynomials in $w^1,\ldots,w^N$, i.e., $P^\alpha_{\beta,b}$ are formal power series in $\varepsilon$ with the coefficients that a polynomials in $w^\gamma_x, w^\gamma_{xx}, \ldots$ (we identify $x=t^1_0$) whose coefficients are formal power series in $w^\gamma$, such that a unique solution of the system~\eqref{eq:Dubrovin--Zhang system} specified by the condition $w^\alpha|_{t^\beta_b=\delta^{\beta,1}\delta_{b,0}x}=\delta^{\alpha,1}x$ is given by $w^\alpha=\eta^{\alpha\mu}\frac{{\partial}^2\mathcal{F}}{{\partial} t^\mu_0{\partial} t^1_0}$. This system of PDEs (if it exists) is called the {\it Dubrovin--Zhang hierarchy} or the {\it hierarchy of topological type}. The conjecture is proved at the approximation up to $\varepsilon^2$~\cite{DZ98} and in the case when the Dubrovin--Frobenius manifold associated to the total descendent potential is semisimple~\cite{BPS12a,BPS12b}. The Dubrovin--Zhang hierarchy corresponding to the Witten potential $\mathcal{F}^W$ is the Korteweg--de Vries (KdV) hierarchy
\begin{align*}
\frac{{\partial} w}{{\partial} t_1}&=ww_x+\frac{\varepsilon^2}{12}w_{xxx},\\
\frac{{\partial} w}{{\partial} t_2}&=\frac{w^2w_x}{2}+\varepsilon^2\left(\frac{ww_{xxx}}{12}+\frac{w_xw_{xx}}{6}\right)+\varepsilon^4\frac{w_{xxxxx}}{240},\\
&\vdots\notag
\end{align*}
This statement is equivalent to Witten's conjecture~\cite{Wit91}, proved by Kontsevich~\cite{Kon92}.
\medskip
A more recent and less developed field of research is the study of the intersection theory on various moduli spaces of Riemann surfaces with boundary. Such a moduli space always comes with an associated moduli space of closed Riemann surfaces, and, thus, there is the corresponding total descendent potential $\mathcal{F}(t^*_*,\varepsilon)=\sum_{g\ge 0}\varepsilon^{2g}\mathcal{F}_g(t^*_*)$ of some rank~$N$. There is a large class of examples \cite{PST14,BCT18,ST19,Che18,CZ18,CZ19,Zin20} where the intersection numbers on the corresponding moduli space of Riemann surfaces with boundary of genus~$0$ are described by a formal power series $\mathcal{F}^o_0(t^*_*,s_*)\in\mathbb C[[t^*_*,s_*]]$ depending on an additional sequence of formal variable $s_a$, $a\ge 0$, and satisfying the relations
\begin{align}
&\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^1_0}=\sum_{a\ge 0}t^\alpha_{a+1}\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^\alpha_a}+\sum_{a\ge 0}s_{a+1}\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_a}+s_0,\label{eq:open string in genus 0,1}\\
&d\left(\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^\alpha_{a+1}}\right)=\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_a{\partial} t^\mu_0}\eta^{\mu\nu}d\left(\frac{{\partial} \mathcal{F}^o_0}{{\partial} t^\nu_0}\right)+\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^\alpha_a}d\left(\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_0}\right),&& 1\le\alpha\le N,&& a\ge 0,\label{eq:open TRR-0,t}\\
&d\left(\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_{a+1}}\right)=\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_a}d\left(\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_0}\right),&& && a\ge 0.\label{eq:open TRR-0,s}
\end{align}
Equation~\eqref{eq:open string in genus 0,1} is called the {\it open string equation}. Equations~\eqref{eq:open TRR-0,t} and~\eqref{eq:open TRR-0,s} are called the {\it open topological recursion relations in genus $0$}. The function~$\mathcal{F}^o_0$ is called the {\it open descendent potential in genus~$0$}.
\begin{remark}
The system of PDEs~\eqref{eq:open string in genus 0,1}--\eqref{eq:open TRR-0,s} implies that the function $\left.\mathcal{F}^o_0\right|_{t^*_{\ge 1}=s_{\ge 1}=0}$ satisfies the open WDVV equations (see \cite[Section~4]{Bur20}), which actually appear in some of the papers mentioned above. However, in~\cite{BB19} the authors presented a construction of an open descendent potential starting from an arbitrary solution of the open WDVV equations.
\end{remark}
Regarding higher genera, much less is known. However, conjecturally, the intersection theory on moduli spaces of Riemann surfaces with boundary of genus $1$ is controlled by formal power series $\mathcal{F}^o_1(t^*_*,s_*)\in\mathbb C[[t^*_*,s_*]]$ satisfying the relations
\begin{align*}
\frac{{\partial}\mathcal{F}_1^o}{{\partial} t^\alpha_{a+1}}=&\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_a{\partial} t^\mu_0}\eta^{\mu\nu}\frac{{\partial}\mathcal{F}^o_1}{{\partial} t^\nu_0}+\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^\alpha_a}\frac{{\partial}\mathcal{F}^o_1}{{\partial} s_0}+\frac{1}{2}\frac{{\partial}^2\mathcal{F}_0^o}{{\partial} t^\alpha_a{\partial} s_0},&& 1\le\alpha\le N,&& a\ge 0,\\
\frac{{\partial}\mathcal{F}_1^o}{{\partial} s_{a+1}}=&\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_a}\frac{{\partial}\mathcal{F}^o_1}{{\partial} s_0}+\frac{1}{2}\frac{{\partial}^2\mathcal{F}_0^o}{{\partial} s_a{\partial} s_0},&& && a\ge 0,
\end{align*}
called the {\it open topological recursion relations in genus~$1$}. In the case of the intersection theory on the moduli spaces of Riemann surfaces with boundary of genus~$g$ with~$k$ boundary marked points and~$l$ internal marked points~$\overline{\mathcal M}_{g,k,l}$, these relations were conjectured by the authors of~\cite{PST14} and proved in~\cite[Section~6.2.3]{BCT18} (a proof by other methods is obtained by J.~P.~Solomon and R.~J.~Tessler in a work in preparation). An evidence that the open topological recursion relations in genus $1$ hold for the open $r$-spin theory is also given in~\cite[Section~6.2.3]{BCT18}.
\medskip
An analog of the theory of Dubrovin--Zhang hierarchies for solutions of the system~\eqref{eq:open string in genus 0,1}--\eqref{eq:open TRR-0,s} was developed in~\cite{BB19}. Regarding higher genera, a very promising direction was opened by the series of papers~\cite{PST14,Tes15,Bur15,Bur16,BT17} (see also~\cite{ABT17}), where the authors studied the intersection numbers on the moduli spaces of Riemann surfaces with boundary of genus~$g$ with~$k$ boundary marked points and~$l$ internal marked points~$\overline{\mathcal M}_{g,k,l}$.
The main result of these works is the proof~\cite{BT17} of the Pandharipande--Solomon--Tessler conjecture~\cite{PST14} saying that the generating series
$$
\mathcal{F}^{o,PST}(t_*,s_*,\varepsilon)=\sum_{g\ge 0}\varepsilon^g\mathcal{F}^{o,PST}_g(t_*,s_*)\in\mathbb C[[t_*,s_*,\varepsilon]]
$$
of the intersection numbers satisfies the following system of PDEs:
\begin{align}
&\frac{{\partial}}{{\partial} t_p}\exp(\varepsilon^{-1}\mathcal{F}^{o,PST})=\frac{\varepsilon^{-1}}{(2p+1)!!}\left(L^{p+\frac{1}{2}}\right)_+\exp(\varepsilon^{-1}\mathcal{F}^{o,PST}),&& p\ge 0,\label{eq:Lax for PST,1}\\
&\frac{{\partial}}{{\partial} s_p}\exp(\varepsilon^{-1}\mathcal{F}^{o,PST})=\frac{\varepsilon^{-1}}{2^{p+1}(p+1)!}L^{p+1}\exp(\varepsilon^{-1}\mathcal{F}^{o,PST}),&& p\ge 0,\label{eq:Lax for PST,2}
\end{align}
where $L=(\varepsilon{\partial}_x)^2+2w$ is the Lax operator for the KdV hierarchy, and $w=\frac{{\partial}^2\mathcal{F}^W}{{\partial} t_0^2}$.
\begin{remark}
To be precise, we have presented a version of the Pandharipande--Solomon--Tessler conjecture, which is slightly different from the original one in two aspects. First of all, in~\cite{PST14} the authors considered a function $\widetilde{\mathcal{F}}^{o,PST}$ related to our function $\mathcal{F}^{o,PST}$ by $\widetilde{\mathcal{F}}^{o,PST}=\left.\mathcal{F}^{o,PST}\right|_{s_{\ge 1}=0}$. The function $\mathcal{F}^{o,PST}$ can be reconstructed from the function $\widetilde{\mathcal{F}}^{o,PST}$ using the system of PDEs
$$
\frac{{\partial}}{{\partial} s_p}\exp(\varepsilon^{-1}\mathcal{F}^{o,PST})=\frac{\varepsilon^p}{(p+1)!}\frac{{\partial}^{p+1}}{{\partial} s_0^{p+1}}\exp(\varepsilon^{-1}\mathcal{F}^{o,PST}),\quad p\ge 1.
$$
Second, the system of PDEs from~\cite[Conjecture~2]{PST14} determining the function $\widetilde{\mathcal{F}}^{o,PST}$ does not have the form of a system of evolutionary PDEs with one spatial variable. The fact that the presented version of the Pandharipande--Solomon--Tessler conjecture is equivalent to the original one was observed in~\cite{Bur16}.
\end{remark}
In this paper we study solutions of the open topological recursion relations in genus $1$. First, we find an analog of formula~\eqref{eq:formula for the closed genus 1 potential}. Then, using this formula, we construct a system of linear PDEs of the form similar to~\eqref{eq:Lax for PST,1} and~\eqref{eq:Lax for PST,2} such that the function $\exp(\mathcal{F}^o_0+\varepsilon\mathcal{F}^o_1)$ satisfies it at the approximation up to $\varepsilon$. An expectation in higher genera and a relation with a Lax description of the Dubrovin--Zhang hierarchies are also discussed.
\subsection*{Acknowledgements}
O.~B. is supported by Becas CONACYT para estudios de Doctorado en el extranjero awarded by the Mexican government, Ref: 2020-000000-01EXTF-00096. The work of A.~B. is funded within the framework of the HSE University Basic Research Program and the Russian Academic Excellence Project '5-100'.
\medskip
We are grateful to Oleg Chalykh for valuable remarks about the preliminary version of the paper.
\section{Closed and open descendent potentials in genus $0$}
In this section we recall the definitions of closed and open descendent potentials in genus~$0$ and the construction of associated to them systems of PDEs.
\subsection{Differential polynomials}
Consider formal variables $v^\alpha_i$, $\alpha=1,\ldots,N$, $i=0,1,\ldots$. Following~\cite{DZ01} (see also~\cite{Ros17}) we define the ring of {\it differential polynomials} $\mathcal{A}_{v^1,\ldots,v^N}$ in the variables $v^1,\ldots,v^N$ as the ring of polynomials in the variables~$v^\alpha_i$, $i>0$, with coefficients in the ring of formal power series in the variables $v^\alpha:= v^\alpha_0$:
$$
\mathcal{A}_{v^1,\ldots,v^N}:=\mathbb C[[v^*]][v^*_{\ge 1}].
$$
\begin{remark} It is useful to think of the variables $v^\alpha=v^\alpha_0$ as the components $v^\alpha(x)$ of a formal loop $v\colon S^1\to\mathbb C^N$ in the standard basis of $\mathbb C^N$. Then the variables $v^\alpha_1:= v^\alpha_x, v^\alpha_2:= v^\alpha_{xx},\ldots$ are the components of the iterated $x$-derivatives of the formal loop.
\end{remark}
The {\it standard gradation} on $\mathcal{A}_{v^1,\ldots,v^N}$, which we denote by $\deg$, is introduced by $\deg v^\alpha_i:= i$. The homogeneous component of $\mathcal{A}_{v^1,\ldots,v^N}$ of standard degree $d$ is denoted by $\mathcal{A}_{v^1,\ldots,v^N;d}$. Introduce an operator ${\partial}_x\colon\mathcal{A}_{v^1,\ldots,v^N}\to\mathcal{A}_{v^1,\ldots,v^N}$ by
$$
\partial_x := \sum_{i\geq 0} v^\alpha_{i+1}\frac{{\partial}}{{\partial} v^\alpha_i}.
$$
It increases the standard degree by $1$.
\medskip
Consider the extension $\widehat{\mathcal{A}}_{v^1,\ldots,v^N}:= \mathcal{A}_{v^1,\ldots,v^N}[[\varepsilon]]$ of the space $\mathcal{A}_{v^1,\ldots,v^N}$ with a new variable~$\varepsilon$ of standard degree $\deg\varepsilon:= -1$. Let $\widehat{\mathcal{A}}_{v^1,\ldots,v^N;d}$ denote the subspace of degree~$d$ of $\widehat{\mathcal{A}}$. Abusing the terminology we still call elements of the space $\widehat{\mathcal{A}}_{v^1,\ldots,v^N}$ \emph{differential polynomials}.
\subsection{Closed descendent potentials in genus $0$}
Let us fix $N\ge 1$, an $N\times N$ symmetric nondegenerate complex matrix $\eta=(\eta_{\alpha\beta})$, and an $N$-tuple of complex numbers $(A^1,\ldots,A^N)$, not all equal to zero. We will use the notation
$$
\frac{{\partial}}{{\partial} t^{1\!\! 1}_a}:=A^\alpha\frac{{\partial}}{{\partial} t^\alpha_a},\quad a\ge 0.
$$
\begin{definition}
A formal power series $\mathcal{F}_0\in\mathbb C[[t^*_*]]$ is called a {\it descendent potential in genus $0$} if it satisfies the following system of PDEs:
\begin{align}
&\sum_{a\ge 0} t^\alpha_{a+1}\frac{{\partial}\mathcal{F}_0}{{\partial} t^\alpha_a}-\frac{{\partial}\mathcal{F}_0}{{\partial} t^{1\!\! 1}_0}=-\frac{1}{2}\eta_{\alpha\beta}t^\alpha_0 t^\beta_0,&&&&\label{eq:string for desc-0}\\
&\sum_{a\ge 0} t^\alpha_a\frac{{\partial}\mathcal{F}_0}{{\partial} t^\alpha_a}-\frac{{\partial}\mathcal{F}_0}{{\partial} t^{1\!\! 1}_1}=2\mathcal{F}_0,&&&&\label{eq:dilaton for desc-0}\\
&\frac{{\partial}^3\mathcal{F}_0}{{\partial} t^\alpha_{a+1}{\partial} t^\beta_b{\partial} t^\gamma_c}=\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_a{\partial} t^\mu_0}\eta^{\mu\nu}\frac{{\partial}^3\mathcal{F}_0}{{\partial} t^\nu_0{\partial} t^\beta_b{\partial} t^\gamma_c},&& 1\le\alpha,\beta,\gamma\le N,&& a,b,c\ge 0,\label{eq:TRR for desc-0}\\
&\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_{a+1}{\partial} t^\beta_b}+\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_a{\partial} t^\beta_{b+1}}=\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_a{\partial} t^\mu_0}\eta^{\mu\nu}\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\nu_0{\partial} t^\beta_b},&& 1\le\alpha,\beta\le N,&& a,b\ge 0.\notag
\end{align}
We will sometimes call a descendent potential in genus~$0$ a {\it closed} descendent potential in genus~$0$ in order to distinguish it from an open analog that we will discuss below.
\end{definition}
\begin{remark}
Doing a linear change of variables, one can make $\frac{{\partial}}{{\partial} t^{1\!\! 1}_a}=\frac{{\partial}}{{\partial} t^1_a}\Leftrightarrow A^\alpha=\delta^{\alpha,1}$ in Equations~\eqref{eq:string for desc-0} and~\eqref{eq:dilaton for desc-0}. That is why authors often assume that $A^\alpha=\delta^{\alpha,1}$.
\end{remark}
\begin{remark}
For any total descendent potential $\mathcal{F}=\sum_{g\ge 0}\varepsilon^{2g}\mathcal{F}_g$ the function $\mathcal{F}_0$ is a descendent potential in genus~$0$. However, describing precisely which descendent potentials in genus $0$ can be extended to total descendent potentials is an interesting open problem.
\end{remark}
Define differential polynomials $\Omega^{[0]}_{\alpha,a;\beta,b}\in\mathcal{A}_{v^1,\ldots,v^N;0}$, $1\le\alpha,\beta\le N$, $a,b\ge 0$, by
$$
\Omega^{[0]}_{\alpha,a;\beta,b}:=\left.\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_a{\partial} t^\beta_b}\right|_{t^\gamma_c=\delta_{c,0}v^\gamma},
$$
and let
$$
(v^\top)^\alpha:=\eta^{\alpha\mu}\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\mu_0{\partial} t^{1\!\! 1}_0}\in\mathbb C[[t^*_*]],\quad 1\le\alpha\le N.
$$
Then we have (see e.g.~\cite[Proposition~3]{BPS12b})
\begin{gather}\label{eq:property of the closed two-point functions}
\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_a{\partial} t^\beta_b}=\left.\Omega^{[0]}_{\alpha,a;\beta,b}\right|_{v^\gamma=(v^\top)^\gamma}.
\end{gather}
This implies that the $N$-tuple of functions $\left.(v^\top)^\alpha\right|_{t^\gamma_0\mapsto t^\gamma_0+A^\gamma x}$ is a solution of the following system of PDEs:
$$
\frac{{\partial} v^\alpha}{{\partial} t^\beta_b}=\eta^{\alpha\mu}{\partial}_x\Omega^{[0]}_{\mu,0;\beta,b},\quad 1\le\alpha,\beta\le N,\quad b\ge 0,
$$
which is called the {\it principal hierarchy} associated to the potential~$\mathcal{F}_0$.
\subsection{Open descendent potentials in genus $0$}
Let us fix a closed descendent potential in genus $0$ $\mathcal{F}_0$.
\begin{definition}\label{definition:open descendent potential in genus 0}
An {\it open descendent potential in genus $0$} $\mathcal{F}^o_0\in\mathbb C[[t^*_*,s_*]]$ is a solution of the following system of PDEs:
\begin{align}
&\sum_{b\ge 0}t^\beta_{b+1}\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^\beta_b}+\sum_{a\ge 0}s_{a+1}\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_a}-\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^{1\!\! 1}_0}=-s_0,\label{eq:open string in genus 0,2}\\
&\sum_{b\ge 0}t^\beta_b\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^\beta_b}+\sum_{a\ge 0}s_a\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_a}-\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^{1\!\! 1}_1}=\mathcal{F}^o_0,\notag\\
&d\left(\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^\alpha_{p+1}}\right)=\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_p{\partial} t^\mu_0}\eta^{\mu\nu}d\left(\frac{{\partial} \mathcal{F}^o_0}{{\partial} t^\nu_0}\right)+\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^\alpha_p}d\left(\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_0}\right),\label{eq:open TRR for desc-0,t}\\
&d\left(\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_{p+1}}\right)=\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_p}d\left(\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_0}\right).\label{eq:open TRR for desc-0,s}
\end{align}
\end{definition}
\medskip
Consider a new formal variable $\phi$. Similarly to the differential polynomials $\Omega^{[0]}_{\alpha,a;\beta,b}$, let us introduce differential polynomials $\Gamma^{[0]}_{\alpha,a},\Delta^{[0]}_a\in\mathcal{A}_{v^1,\ldots,v^N,\phi;0}$, $1\le\alpha\le N$, $a\ge 0$, by
$$
\Gamma^{[0]}_{\alpha,a}:=\left.\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^\alpha_a}\right|_{\substack{t^\gamma_c=\delta_{c,0}v^\gamma\\s_c=\delta_{c,0}\phi}},\qquad \Delta^{[0]}_a:=\left.\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_a}\right|_{\substack{t^\gamma_c=\delta_{c,0}v^\gamma\\s_c=\delta_{c,0}\phi}},
$$
and let
$$
\phi^\top:=\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^{1\!\! 1}_0}\in\mathbb C[[t^*_*,s_*]].
$$
We have the following properties, analogous to the property~\eqref{eq:property of the closed two-point functions} (\cite[Section~4.4]{BB19}, \cite[Proposition~2.2]{ABLR20}):
\begin{gather*}
\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^\alpha_a}=\left.\Gamma^{[0]}_{\alpha,a}\right|_{\substack{v^\gamma=(v^\top)^\gamma\\\phi=\phi^\top}},\qquad\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_a}=\left.\Delta^{[0]}_{a}\right|_{\substack{v^\gamma=(v^\top)^\gamma\\\phi=\phi^\top}}.
\end{gather*}
This implies that the $(N+1)$-tuple of functions $\left.\left((v^\top)^1,\ldots,(v^\top)^N,\phi^\top\right)\right|_{t^\gamma_0\mapsto t^\gamma_0+A^\gamma x}$ satisfies the following system of PDEs:
\begin{align*}
&\frac{{\partial} v^\alpha}{{\partial} t^\beta_b}={\partial}_x\eta^{\alpha\mu}\Omega^{[0]}_{\mu,0;\beta,b},&& \frac{{\partial} v^\alpha}{{\partial} s_b}=0,\\
&\frac{{\partial}\phi}{{\partial} t^\beta_b}={\partial}_x\Gamma^{[0]}_{\beta,b},&& \frac{{\partial}\phi}{{\partial} s_b}={\partial}_x\Delta^{[0]}_b,
\end{align*}
which we call the {\it extended principal hierarchy} associated to the pair of potentials $(\mathcal{F}_0,\mathcal{F}^o_0)$.
\section{Open descendent potentials in genus $1$}
Here we introduce the notion of an open descendent potential in genus $1$ and prove two main results of our paper: Theorems~\ref{theorem:explicit formula in the open genus 1} and~\ref{theorem:linear PDEs up to genus 1}.
\subsection{Open descendent potentials in genus $1$}
Let us fix a pair $(\mathcal{F}_0,\mathcal{F}^o_0)$ of closed and open potentials in genus $0$.
\begin{definition}\label{definition:open descendent potential in genus 1}
An {\it open descendent potential in genus $1$} $\mathcal{F}^o_1\in\mathbb C[[t^*_*,s_*]]$ is a solution of the following system of PDEs:
\begin{align}
\frac{{\partial}\mathcal{F}_1^o}{{\partial} t^\alpha_{a+1}}=&\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_a{\partial} t^\mu_0}\eta^{\mu\nu}\frac{{\partial}\mathcal{F}^o_1}{{\partial} t^\nu_0}+\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^\alpha_a}\frac{{\partial}\mathcal{F}^o_1}{{\partial} s_0}+\frac{1}{2}\frac{{\partial}^2\mathcal{F}_0^o}{{\partial} t^\alpha_a{\partial} s_0},&& 1\le\alpha\le N,&& a\ge 0,\label{eq:open TRR-1,1}\\
\frac{{\partial}\mathcal{F}_1^o}{{\partial} s_{a+1}}=&\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_a}\frac{{\partial}\mathcal{F}^o_1}{{\partial} s_0}+\frac{1}{2}\frac{{\partial}^2\mathcal{F}_0^o}{{\partial} s_a{\partial} s_0},&& && a\ge 0.\label{eq:open TRR-1,2}
\end{align}
\end{definition}
\medskip
Consider an open descendent potential in genus $1$ $\mathcal{F}^o_1$. Define a formal power series $G^o\in\mathbb C[[v^*,\phi]]$ by
$$
G^o:=\left.\mathcal{F}^o_1\right|_{\substack{t^\alpha_a=\delta_{a,0}v^\alpha\\s_a=\delta_{a,0}\phi}}.
$$
\begin{theorem}\label{theorem:explicit formula in the open genus 1}
We have
\begin{gather}\label{eq:formula for the open genus 1 part}
\mathcal{F}^o_1=\frac{1}{2}\log\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} t^{1\!\! 1}_0{\partial} s_0}+\left.G^o\right|_{\substack{v^\gamma=(v^\top)^\gamma\\\phi=\phi^\top}}.
\end{gather}
\end{theorem}
\begin{proof}
Note that Equation~\eqref{eq:open string in genus 0,2} implies that
$$
\left.\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} t^{1\!\! 1}_0{\partial} s_0}\right|_{t^*_{\ge 1}=s_{\ge 1}=0}=1.
$$
Therefore, the logarithm $\log\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} t^{1\!\! 1}_0{\partial} s_0}$ is a well-defined formal power series in the variables~$t^*_*$ and~$s_*$. Also, Equations~\eqref{eq:string for desc-0} and~\eqref{eq:open string in genus 0,2} imply that
$$
\left.(v^\top)^\alpha\right|_{t^*_{\ge 1}=0}=t^\alpha_0,\qquad \left.\phi^\top\right|_{t^*_{\ge 1}=s_{\ge 1}=0}=s_0.
$$
Therefore, Equation~\eqref{eq:formula for the open genus 1 part} is true when $t^*_{\ge 1}=s_{\ge 1}=0$.
\medskip
Using the linear differential operators
\begin{align*}
&P^1_{\alpha,a}:=\frac{{\partial}}{{\partial} t^\alpha_{a+1}}-\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_a{\partial} t^\mu_0}\eta^{\mu\nu}\frac{{\partial}}{{\partial} t^\nu_0}-\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^\alpha_a}\frac{{\partial}}{{\partial} s_0},&&1\le\alpha\le N,&& a\ge 0,\\
&P^2_a:=\frac{{\partial}}{{\partial} s_{a+1}}-\frac{{\partial}\mathcal{F}^o_0}{{\partial} s_a}\frac{{\partial}}{{\partial} s_0},&& && a\ge 0,
\end{align*}
Equations~\eqref{eq:open TRR-1,1} and~\eqref{eq:open TRR-1,2} can be equivalently written as
\begin{align*}
&P^1_{\alpha,a}\mathcal{F}^o_1=\frac{1}{2}\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} t^\alpha_a{\partial} s_0}, && 1\le\alpha\le N, && a\ge 0,\\
&P^2_a\mathcal{F}^o_1=\frac{1}{2}\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} s_a{\partial} s},&& && a\ge 0.
\end{align*}
This system of PDEs uniquely determines the function $\mathcal{F}^o_1$ starting from the initial condition~$\left.\mathcal{F}^o_1\right|_{t^*_{\ge 1}=s_{\ge 1}=0}=G^o(t^1_0,\ldots,t^N_0,s_0)$. By Equations~\eqref{eq:TRR for desc-0},~\eqref{eq:open TRR for desc-0,t}, and~\eqref{eq:open TRR for desc-0,s}, we have
$$
P^1_{\alpha,a}(v^\top)^\beta=P^1_{\alpha,a}\phi^\top=P^2_a(v^\top)^\beta=P^2_a\phi^\top=0.
$$
Therefore, it remains to check that
\begin{align}
&P^1_{\alpha,a}\log\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} t^{1\!\! 1}_0{\partial} s_0}=\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} t^\alpha_a{\partial} s_0},\label{eq:proof of open genus 1 formula,1}\\
&P^2_a\log\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} t^{1\!\! 1}_0{\partial} s_0}=\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} s_a{\partial} s_0}.\label{eq:proof of open genus 1 formula,2}
\end{align}
\medskip
To prove Equation~\eqref{eq:proof of open genus 1 formula,1}, we compute
\begin{align*}
P^1_{\alpha,a}\log\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} t^{1\!\! 1}_0{\partial} s_0}=&\frac{1}{\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} t^{1\!\! 1}_0{\partial} s_0}}\left(\frac{{\partial}^3\mathcal{F}^o_0}{{\partial} t^\alpha_{a+1}{\partial} t^{1\!\! 1}_0{\partial} s_0}-\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_a{\partial} t^\mu_0}\eta^{\mu\nu}\frac{{\partial}^3\mathcal{F}^o_0}{{\partial} t^\nu_0{\partial} t^{1\!\! 1}_0{\partial} s_0}-\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^\alpha_a}\frac{{\partial}^3\mathcal{F}^o_0}{{\partial} s_0{\partial} t^{1\!\! 1}_0{\partial} s_0}\right)=\\
=&\frac{1}{\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} t^{1\!\! 1}_0{\partial} s_0}}\left[\frac{{\partial}}{{\partial} s_0}\underline{\left(\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} t^\alpha_{a+1}{\partial} t^{1\!\! 1}_0}-\frac{{\partial}^2\mathcal{F}_0}{{\partial} t^\alpha_a{\partial} t^\mu_0}\eta^{\mu\nu}\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} t^\nu_0{\partial} t^{1\!\! 1}_0}-\frac{{\partial}\mathcal{F}^o_0}{{\partial} t^\alpha_a}\frac{{\partial}^2\mathcal{F}^o_0}{{\partial} s_0{\partial} t^{1\!\! 1}_0}\right)}+\frac{{\partial}^2\mathcal{F}_0^o}{{\partial} t^\alpha_a{\partial} s_0}\frac{{\partial}^2\mathcal{F}_0^o}{{\partial} t^{1\!\! 1}_0{\partial} s_0}\right]=\\
=&\frac{{\partial}^2\mathcal{F}_0^o}{{\partial} t^\alpha_a{\partial} s_0},
\end{align*}
where the vanishing of the underlined expression follows from Equation~\eqref{eq:open TRR for desc-0,t}. The proof of Equation~\eqref{eq:proof of open genus 1 formula,2} is analogous. The theorem is proved.
\end{proof}
\subsection{Differential operators and PDEs}
Consider a differential operator $L$ of the form
$$
L=\sum_{i\ge 0}L_i(v^*_*,\varepsilon)(\varepsilon{\partial}_x)^i,\quad L_i\in\widehat{\mathcal{A}}_{v^1,\ldots,v^N;0}.
$$
Let $f$ be a formal variable and consider the PDE
\begin{gather}\label{eq:L-f PDE,1}
\frac{{\partial}}{{\partial} t}\exp(\varepsilon^{-1} f)=\varepsilon^{-1}L\exp(\varepsilon^{-1} f).
\end{gather}
Note that
$$
\frac{(\varepsilon{\partial}_x)^i\exp(\varepsilon^{-1}f)}{\exp(\varepsilon^{-1}f)}=Q_i(f_*,\varepsilon),\quad i\ge 0,
$$
where $Q_i\in\widehat{\mathcal{A}}_f$ can be recursively computed by the relation
$$
Q_i=
\begin{cases}
1,&\text{if $i=0$},\\
f_xQ_{i-1}+\varepsilon{\partial}_x Q_{i-1},&\text{if $i\ge 1$}.
\end{cases}
$$
\begin{remark}
Note that $Q_i$ does not depend on $f$ and is a polynomial in the derivatives $f_x,f_{xx},\ldots$ and $\varepsilon$. Moreover, if we introduce a new formal variable~$\psi$ and substitute $f_{i+1}=\psi_i$, $i\ge 0$, then~$Q_i$ becomes a differential polynomial of degree~$0$.
\end{remark}
We see that PDE~\eqref{eq:L-f PDE,1} is equivalent to the following PDE:
\begin{gather}\label{eq:L-f PDE,2}
\frac{{\partial} f}{{\partial} t}=\sum_{i\ge 0}L_i(v^*_*,\varepsilon)Q_i(f_*,\varepsilon).
\end{gather}
\medskip
Let us look at this PDE in more details at the approximation up to~$\varepsilon$.
\begin{lemma}
We have $Q_i=f_x^i+\varepsilon\frac{i(i-1)}{2}f_x^{i-2}f_{xx}+O(\varepsilon^2)$.
\end{lemma}
\begin{proof}
The formula is clearly true for $i=0$. We proceed by induction:
\begin{align*}
Q_{i+1}=&f_x Q_i+\varepsilon{\partial}_x Q_i=f_x^{i+1}+\varepsilon\left(f_x\frac{i(i-1)}{2}f_x^{i-2}f_{xx}+{\partial}_x\left(f_x^i\right)\right)+O(\varepsilon^2)=\\
=&f_x^{i+1}+\varepsilon\frac{(i+1)i}{2}f_x^{i-1}f_{xx}+O(\varepsilon^2).
\end{align*}
\end{proof}
Consider the expansion
$$
L_i(v^*_*,\varepsilon)=\sum_{j\ge 0}L_i^{[j]}(v^*_*)\varepsilon^j,\quad L_i^{[j]}\in\mathcal{A}_{v^1,\ldots,v^N;j}.
$$
We see that Equation~\eqref{eq:L-f PDE,2} has the form
\begin{gather}\label{eq:linear PDE - first order approximation}
\frac{{\partial} f}{{\partial} t}=\sum_{i\ge 0}L_i^{[0]}f_x^i+\varepsilon\sum_{i\ge 0}\left(L_i^{[1]}f_x^i+L_i^{[0]}\frac{i(i-1)}{2}f_x^{i-2}f_{xx}\right)+O(\varepsilon^2).
\end{gather}
\subsection{A linear PDE for an open descendent potential up to genus $1$}
Define differential operators $L_{\alpha,a}^{\mathrm{int}}$, $1\le\alpha\le N$, $a\ge 0$, and $L_a^{\mathrm{boun}}$, $a\ge 0$, by
\begin{align*}
&L_{\alpha,a}^{\mathrm{int}}:=\sum_{i\ge 0}\left(L_{\alpha,a,i}^{\mathrm{int};[0]}+\varepsilon L_{\alpha,a,i}^{\mathrm{int};[1]}\right)(\varepsilon{\partial}_x)^i,\\
&L_a^{\mathrm{boun}}:=\sum_{i\ge 0}\left(L_{a,i}^{\mathrm{boun};[0]}+\varepsilon L_{a,i}^{\mathrm{boun};[1]}\right)(\varepsilon{\partial}_x)^i,
\end{align*}
where
\begin{align*}
&L_{\alpha,a,i}^{\mathrm{int};[0]}:=\mathrm{Coef}_{\phi^i}\Gamma^{[0]}_{\alpha,a}\in\mathcal{A}_{v^1,\ldots,v^N;0},\\
&L_{\alpha,a,i}^{\mathrm{int};[1]}:=\mathrm{Coef}_{\phi^i}\left[\left(\frac{{\partial} G^o}{{\partial}\phi}\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial} v^\beta}-\frac{{\partial} G^o}{{\partial} v^\beta}\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi}+\frac{1}{2}\frac{{\partial}^2\Gamma^{[0]}_{\alpha,a}}{{\partial} v^\beta{\partial}\phi}\right)v^\beta_x+\frac{{\partial} G^o}{{\partial} v^\beta}\eta^{\beta\gamma}{\partial}_x\Omega^{[0]}_{\gamma,0;\alpha,a}\right]\in\mathcal{A}_{v^1,\ldots,v^N;1},\\
&L_{a,i}^{\mathrm{boun};[0]}:=\mathrm{Coef}_{\phi^i}\Delta^{[0]}_a\in\mathcal{A}_{v^1,\ldots,v^N;0},\\
&L_{a,i}^{\mathrm{boun};[1]}:=\mathrm{Coef}_{\phi^i}\left[\left(\frac{{\partial} G^o}{{\partial}\phi}\frac{{\partial}\Delta^{[0]}_a}{{\partial} v^\beta}-\frac{{\partial} G^o}{{\partial} v^\beta}\frac{{\partial}\Delta^{[0]}_a}{{\partial}\phi}+\frac{1}{2}\frac{{\partial}^2\Delta^{[0]}_a}{{\partial} v^\beta{\partial}\phi}\right)v^\beta_x\right]\in\mathcal{A}_{v^1,\ldots,v^N;1}.
\end{align*}
\begin{theorem}\label{theorem:linear PDEs up to genus 1}
The formal power series $v^\beta=\left.(v^\top)^\beta\right|_{t^\gamma_0\mapsto t^\gamma_0+A^\gamma x}$ and $f=\left.\left(\mathcal{F}^o_0+\varepsilon\mathcal{F}^o_1\right)\right|_{t^\gamma_0\mapsto t^\gamma_0+A^\gamma x}$ satisfy the system of PDEs
\begin{align}
&\frac{{\partial}}{{\partial} t^\alpha_a}\exp(\varepsilon^{-1} f)=\varepsilon^{-1}L_{\alpha,a}^{\mathrm{int}}\exp(\varepsilon^{-1} f),&& 1\le\alpha\le N,&& a\ge 0,\label{eq:internal equation}\\
&\frac{{\partial}}{{\partial} s_a}\exp(\varepsilon^{-1} f)=\varepsilon^{-1}L_a^{\mathrm{boun}}\exp(\varepsilon^{-1} f),&& && a\ge 0.\label{eq:boundary equation}
\end{align}
at the approximation up to $\varepsilon$.
\end{theorem}
\begin{proof}
Abusing notations let us denote the formal powers series $\left.\mathcal{F}^o_0\right|_{t^\gamma_0\mapsto t^\gamma_0+A^\gamma x}$, $\left.\mathcal{F}^o_1\right|_{t^\gamma_0\mapsto t^\gamma_0+A^\gamma x}$, $\left.(v^\top)^\alpha\right|_{t^\gamma_0\mapsto t^\gamma_0+A^\gamma x}$, and $\left.\phi^\top\right|_{t^\gamma_0\mapsto t^\gamma_0+A^\gamma x}$ by $\mathcal{F}^o_0$, $\mathcal{F}^o_1$, $v^\alpha$, and $\phi$, respectively. We can then write the statement of Theorem~\ref{theorem:explicit formula in the open genus 1} as
$$
\mathcal{F}^o_1=\frac{1}{2}\log\phi_s+G^o.
$$
\medskip
Let us prove Equation~\eqref{eq:internal equation} at the approximation up to~$\varepsilon$. We have
$$
{\partial}_x\left(\mathcal{F}^o_0+\varepsilon\mathcal{F}^o_1\right)=\phi+\varepsilon\left(\frac{1}{2}\frac{\phi_{xs}}{\phi_s}+{\partial}_x G^o\right).
$$
Therefore, by Equation~\eqref{eq:linear PDE - first order approximation}, we have to check that
\begin{align*}
&\frac{{\partial}}{{\partial} t^\alpha_a}\left(\mathcal{F}^o_0+\varepsilon\mathcal{F}^o_1\right)=\sum_{i\ge 0}L_{\alpha,a,i}^{\mathrm{int};[0]}\phi^i+\varepsilon\sum_{i\ge 0}L_{\alpha,a,i}^{\mathrm{int};[1]}\phi^i+\\
&\hspace{3.2cm}+\varepsilon\sum_{i\ge 0}L_{\alpha,a,i}^{\mathrm{int};[0]}\left(\frac{i(i-1)}{2}\phi^{i-2}\phi_x+i\phi^{i-1}\left(\frac{1}{2}\frac{\phi_{xs}}{\phi_s}+{\partial}_x G^o\right)\right)\Leftrightarrow\\
\Leftrightarrow&\frac{{\partial}}{{\partial} t^\alpha_a}\left(\mathcal{F}^o_0+\varepsilon\mathcal{F}^o_1\right)=\Gamma^{[0]}_{\alpha,a}+\varepsilon\left(\sum_{i\ge 0}L_{\alpha,a,i}^{\mathrm{int};[1]}\phi^i+\frac{1}{2}\frac{{\partial}^2\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi^2}\phi_x+\frac{1}{2}\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi}\frac{\phi_{xs}}{\phi_s}+\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi}{\partial}_x G^o\right)\Leftrightarrow\\
\Leftrightarrow&\frac{{\partial}\mathcal{F}^o_1}{{\partial} t^\alpha_a}=\frac{1}{2}\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi}\frac{\phi_{xs}}{\phi_s}+\left(\frac{1}{2}\frac{{\partial}^2\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi^2}+\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi}\frac{{\partial} G^o}{{\partial}\phi}\right)\phi_x+\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi}\frac{{\partial} G^o}{{\partial} v^\beta}v^\beta_x+\sum_{i\ge 0}L_{\alpha,a,i}^{\mathrm{int};[1]}\phi^i.
\end{align*}
\medskip
Using the definition of $L^{\mathrm{int};[1]}_{\alpha,a,i}$, we see that the last equation is equivalent to
\begin{align}
&\frac{{\partial}\mathcal{F}^o_1}{{\partial} t^\alpha_a}=\frac{1}{2}\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi}\frac{\phi_{xs}}{\phi_s}+\left(\frac{1}{2}\frac{{\partial}^2\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi^2}+\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi}\frac{{\partial} G^o}{{\partial}\phi}\right)\phi_x+\notag\\
&\hspace{1.3cm}+\left(\frac{{\partial} G^o}{{\partial}\phi}\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial} v^\beta}+\frac{1}{2}\frac{{\partial}^2\Gamma^{[0]}_{\alpha,a}}{{\partial} v^\beta{\partial}\phi}\right)v^\beta_x+\frac{{\partial} G^o}{{\partial} v^\beta}\eta^{\beta\gamma}{\partial}_x\Omega^{[0]}_{\gamma,0;\alpha,a}\Leftrightarrow\notag\\
\Leftrightarrow&\frac{{\partial}\mathcal{F}^o_1}{{\partial} t^\alpha_a}=\frac{1}{2}\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi}\frac{\phi_{xs}}{\phi_s}+\frac{1}{2}{\partial}_x\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi}+\frac{{\partial} G^o}{{\partial}\phi}{\partial}_x\Gamma^{[0]}_{\alpha,a}+\frac{{\partial} G^o}{{\partial} v^\beta}\eta^{\beta\gamma}{\partial}_x\Omega^{[0]}_{\gamma,0;\alpha,a}.\label{eq:intermediate equation for the proof of linear PDE}
\end{align}
On the other hand, we compute
\begin{align*}
\frac{{\partial}\mathcal{F}^o_1}{{\partial} t^\alpha_a}=&\left(\frac{1}{2}\log\phi_s+G^o\right)_{t^\alpha_a}=\frac{1}{2}\frac{\left(\phi_{t^\alpha_a}\right)_s}{\phi_s}+\frac{{\partial} G^o}{{\partial} v^\beta}\eta^{\beta\gamma}{\partial}_x\Omega^{[0]}_{\gamma,0;\alpha,a}+\frac{{\partial} G^o}{{\partial}\phi}{\partial}_x\Gamma^{[0]}_{\alpha,a}=\\
=&\frac{1}{2}\frac{{\partial}_x\left(\Gamma^{[0]}_{\alpha,a}\right)_s}{\phi_s}+\frac{{\partial} G^o}{{\partial} v^\beta}\eta^{\beta\gamma}{\partial}_x\Omega^{[0]}_{\gamma,0;\alpha,a}+\frac{{\partial} G^o}{{\partial}\phi}{\partial}_x\Gamma^{[0]}_{\alpha,a}=\\
=&\frac{1}{2}\frac{{\partial}_x\left(\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi}\phi_s\right)}{\phi_s}+\frac{{\partial} G^o}{{\partial} v^\beta}\eta^{\beta\gamma}{\partial}_x\Omega^{[0]}_{\gamma,0;\alpha,a}+\frac{{\partial} G^o}{{\partial}\phi}{\partial}_x\Gamma^{[0]}_{\alpha,a}=\\
=&\frac{1}{2}{\partial}_x\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi}+\frac{1}{2}\frac{{\partial}\Gamma^{[0]}_{\alpha,a}}{{\partial}\phi}\frac{\phi_{xs}}{\phi_s}+\frac{{\partial} G^o}{{\partial} v^\beta}\eta^{\beta\gamma}{\partial}_x\Omega^{[0]}_{\gamma,0;\alpha,a}+\frac{{\partial} G^o}{{\partial}\phi}{\partial}_x\Gamma^{[0]}_{\alpha,a},
\end{align*}
which proves Equation~\eqref{eq:intermediate equation for the proof of linear PDE} and, hence, Equation~\eqref{eq:internal equation} at the approximation up to~$\varepsilon$.
\medskip
The proof of Equation~\eqref{eq:boundary equation} is analogous.
\end{proof}
\section{Expectation in higher genera}
Consider a total descendent potential $\mathcal{F}(t^*_*,\varepsilon)=\sum_{g\ge 0}\varepsilon^{2g}\mathcal{F}_g(t^*_*)$ of some rank $N$.
\begin{expectation}\label{expectation}
Under possibly some additional assumptions, there exists a reasonable geometric construction of an open descendent potential in all genera $\mathcal{F}^o(t^*_*,s^*,\varepsilon)=\sum_{g\ge 0}\varepsilon^g\mathcal{F}^o_g(t^*_*,\varepsilon)$ satisfying the following properties:
\begin{itemize}
\item The functions $\mathcal{F}^o_0$ and $\mathcal{F}^o_1$ are open descendents potentials in genus $0$ and $1$, respectively (according to Definitions~\ref{definition:open descendent potential in genus 0} and~\ref{definition:open descendent potential in genus 1}).
\item The function $\mathcal{F}^o$ satisfies the open string equation in all genera
$$
\sum_{b\ge 0}t^\beta_{b+1}\frac{{\partial}\mathcal{F}^o}{{\partial} t^\beta_b}+\sum_{a\ge 0}s_{a+1}\frac{{\partial}\mathcal{F}^o}{{\partial} s_a}-\frac{{\partial}\mathcal{F}^o}{{\partial} t^{1\!\! 1}_0}=-s_0+C\varepsilon,
$$
where $C$ is some constant.
\item Consider formal variables $w^1,\ldots,w^N$. Then there exist differential operators $L_{\alpha,a}^{\mathrm{full},\mathrm{int}}$, $1\le\alpha\le N$, $a\ge 0$, and $L_a^{\mathrm{full},\mathrm{boun}}$, $a\ge 0$, of the form
\begin{align*}
&L_{\alpha,a}^{\mathrm{full},\mathrm{int}}=\sum_{i\ge 0}L_{\alpha,a,i}^{\mathrm{full},\mathrm{int}}(w^*_*,\varepsilon)(\varepsilon{\partial}_x)^i, && L_{\alpha,a,i}^{\mathrm{full},\mathrm{int}}\in\widehat{\mathcal{A}}_{w^1,\ldots,w^n;0},\\
&L_a^{\mathrm{full},\mathrm{boun}}=\sum_{i\ge 0}L_{a,i}^{\mathrm{full},\mathrm{boun}}(w^*_*,\varepsilon)(\varepsilon{\partial}_x)^i, && L_{a,i}^{\mathrm{full},\mathrm{boun}}\in\widehat{\mathcal{A}}_{w^1,\ldots,w^n;0},\\
&L_{\alpha,a,i}^{\mathrm{full},\mathrm{int}}=\left.L_{\alpha,a,i}^{\mathrm{int}}\right|_{v^\beta_b=w^\beta_b}+O(\varepsilon^2),\\
&L_{a,i}^{\mathrm{full},\mathrm{boun}}=\left.L_{a,i}^{\mathrm{boun}}\right|_{v^\beta_b=w^\beta_b}+O(\varepsilon^2),
\end{align*}
such that the formal power series $w^\beta=\left.\eta^{\beta\mu}\frac{{\partial}^2\mathcal{F}}{{\partial} t^\mu_0{\partial} t^{1\!\! 1}_0}\right|_{t^\gamma_0\mapsto t^\gamma_0+A^\gamma x}$ and $f=\left.\mathcal{F}^o\right|_{t^\gamma_0\mapsto t^\gamma_0+A^\gamma x}$ satisfy the system of PDEs
\begin{align}
&\frac{{\partial}}{{\partial} t^\alpha_a}\exp(\varepsilon^{-1} f)=\varepsilon^{-1}L_{\alpha,a}^{\mathrm{full},\mathrm{int}}\exp(\varepsilon^{-1} f),&& 1\le\alpha\le N,&& a\ge 0,\label{eq:full internal equation}\\
&\frac{{\partial}}{{\partial} s_a}\exp(\varepsilon^{-1} f)=\varepsilon^{-1}L_a^{\mathrm{full},\mathrm{boun}}\exp(\varepsilon^{-1} f),&& && a\ge 0.\label{eq:full boundary equation}
\end{align}
\end{itemize}
\end{expectation}
\medskip
Suppose that there exists a Dubrovin--Zhang hierarchy corresponding to our total descendent potential $\mathcal{F}$ (this is true when, for example, the associated Dubrovin--Frobenius manifold is semisimple). It is easy to show that if Expectation~\ref{expectation} is true, then the flows $\frac{{\partial}}{{\partial} t^\alpha_a}$ and $\frac{{\partial}}{{\partial} s_b}$ pairwise commute, which means that
\begin{align}
&\frac{{\partial} L^{\mathrm{full},\mathrm{int}}_{\alpha,a}}{{\partial} t^\beta_b}-\frac{{\partial} L^{\mathrm{full},\mathrm{int}}_{\beta,b}}{{\partial} t^\alpha_a}+\varepsilon^{-1}\left[L^{\mathrm{full},\mathrm{int}}_{\alpha,a}, L^{\mathrm{full},\mathrm{int}}_{\beta,b}\right]=0, && 1\le\alpha,\beta\le N, &&a,b\ge 0,\notag\\
&\frac{{\partial} L^{\mathrm{full},\mathrm{boun}}_a}{{\partial} t^\beta_b}+\varepsilon^{-1}\left[L^{\mathrm{full},\mathrm{boun}}_a, L^{\mathrm{full},\mathrm{int}}_{\beta,b}\right]=0, && 1\le\beta\le N, &&a,b\ge 0,\label{eq:full s-t compatibility}\\
&\left[L^{\mathrm{full},\mathrm{boun}}_a, L^{\mathrm{full},\mathrm{boun}}_b\right]=0, && && a,b\ge 0,\notag
\end{align}
where the derivatives $\frac{{\partial} L^{\mathrm{full},\mathrm{int}}_{\alpha,a}}{{\partial} t^\beta_b}$ and $\frac{{\partial} L^{\mathrm{full},\mathrm{boun}}_a}{{\partial} t^\beta_b}$ are computed using the flows of the Dubrovin--Zhang hierarchy. Note that Equation~\eqref{eq:full s-t compatibility} potentially gives a Lax description of the Dubrovin--Zhang hierarchy (see an alternative approach in~\cite{CvdLPS14}).
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 9,954 |
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I feel particularly proud of this years Incredible Egg campaign, which introduces the Dinner Egg. Hence the dark and moody vibes. I traveled to Barcelona with my producer, Taylor Vandegrift, and we put together a crew that truly blew my mind. I generally ask a lot of my crew to begin with, but even when we weren't sleeping much and working 24/7, everyone was looking for ways to make this production better. Our wardrobe stylist, Cris Quer, was so talented, fashioning custom made wardrobe for each character with such skill. Our set designer Dominique Aizpurua blew us away with an underwater coral reef with giant clam shells, and a hand painted ancient sarcophagus. Being a part of a creative team where everyone cares so much about what they do, and everyone is bringing unique talents to the project, was really a special experience.
As hard and as long as we worked, it never really felt like work. We had a lot of fun, and there were tears of laughter at many points. At the bottom of this post, there is a link to my Instagram Story with behind the scenes pictures and videos of some of the antics and pulled back perspective on how this all came together. Thanks for taking a look. I hope you enjoy the art, and a very big thank you to Energy BBDO and The Incredible Egg.
Contortionist Portrait for The Incredible Egg Dinner Egg ad campaign by John Keatley
Sheriff Portrait for The Incredible Egg Dinner Egg ad campaign by John Keatley
Angry Girl Portrait for The Incredible Egg ad campaign by John Keatley
Moses Portrait for The Incredible Egg Dinner Egg ad campaign by John Keatley
Bird Photographer Portrait for The Incredible Egg Dinner Egg ad campaign by John Keatley
Baseball Kid Portrait for The Incredible Egg ad campaign by John Keatley
Click HERE for a behind the scenes look at this project on my Incredible Egg Instagram Story. (clicking the image below is not working at the moment)
Over the past two years, I have been defining my visual voice, zeroing in on characters. Last year, I began looking at image grids to better understand color, wardrobe, and other patterns that may emerge from my work, helping me understand where I have been and where I want my future imagery to go.
This year, keeping a daily creative journal has revealed tremendous benefits, by forcing me to take daily steps in moving my work forward, leading to a greater understanding of myself, and the ability to communicate my vision clearly to others. Writing has taught me that I don't always need a big idea before I can begin exploring or creating. Just showing up and putting down ideas begins to reveal the story, one step at a time. Some days feel harder than others, but I am learning to trust my instincts, and I have never felt as excited about my art as I do today.
The initial response to the 2018 Keatley Family Picture was what this year's concept was all about. Deep disappointment. Confusion as to why we seemingly played it safe with nothing more than a purple theme. And uncertainty as to whether this really is the Keatleys or not.
It is true, two of the four individuals in this picture are NOT Keatleys! There is little that I enjoy more than creating a strong response in others. You might even call me an instigator. But the choice to replace my wife and myself with our good friends had me laughing for days.
As production began on this concept, the question became "how closely do we try to resemble Nichelle and I?" We considered leaving Jess and Brian as their natural selves, as we often are disguised in wigs and costumes in our family pictures. But if felt too obvious, and this picture needed subtlety to soar. In the end, my hat tilt and Nichelle's dark hair were key to pulling this off.
Many thanks to our kids for going along with my crazy ideas. Thankfully they seem to understand that they were born into this! Thank you to Brian and Jess for nailing the subtleties. Also a big thank you to Lindsey Watkins and Kristen Bonnallie for the styling magic. Retouching by GILD Studios.
Irish National Lottery / Lotto Plus
I recently had the opportunity to travel to Dublin, Ireland to work with ROTHCO on the new Irish National Lottery campaign.
The Celebrations have been the Lotto house band for the last 30 years, and for 30 years, they have played every time there is a new winner. Not a bad gig, and life has been pretty good. Now, The Irish National Lottery has released a new game called Lotto Plus, and there are so many winners, the band never stops playing. They are exhausted, and at their breaking point.
This is a really big campaign in Ireland right now, and the images are apparently all over the country. I wish I could be there to see it myself, but thankfully I have a few snaps of some of the executions.
A very big thank you to Stephen Rogers for this incredible project. It was an absolute blast. Thank you also to my producer, Patrick Daly for going above and beyond. I can't wait to return to Ireland.
Retouching by GILD Studios.
Wild Expression Portraits for AT&T
One of the wonderful things about kids is you can always count on them to provide you with outside perspective. It's so easy to get locked into a routine as an adult, but kids have a way of knocking you out of your world and seeing things in a new light. That happened to me recently as I was going through portraits from my AT&T shoot with Possible. My daughter Isla saw this BTS picture and asked, "Daddy, why are you wearing trash bags at a photo shoot?"
To which I replied, "So that man can spit coffee on me."
With an even more confused look on her face she said, "Why would he do that!?"
I had to think about that for a second before I said,"Well, because that's my job." smiling proudly.
Her look at that point said it all, and in that moment I was able to appreciate and enjoy just how unusual my job is. And I wouldn't have it any other way. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,533 |
Table of Contents
THE SPOOK WHO SAT BY THE DOOR
The Spook
Who sat by the door
THE SPOOK
WHO SAT BY THE DOOR
a novel by
Sam Greenlee
WAYNE STATE UNIVERSITY PRESS Detroit
1
Today the computers would tell Senator Gilbert Hennington about his impending campaign for reelection. The senator knew from experience that the computers did not lie.
He sat separated from his assembled staff by his massive, uncluttered desk, the Washington Monument framed by the window to his rear. They sat alert, competent, loyal and intelligent, with charts, graphs, clipboards and reports at the ready. The senator swept the group with a steely gaze, gave Belinda, his wife and chief aide, a bright smile of confidence, and said :
"All right, team, let's have a rundown, and don't try to sweeten the poison. We all know this will be the closest one yet: what I want to know is how close ? Tom, kick it off."
"The campaign war chest is in excellent shape, chief : no major defectors."
"Good. I'll look over your detailed breakdown later. Dick?"
"I spent a week on Mad. Av. with both the PR boys and our ad agency. They both have good presentations ready for your approval, Senator. I think you'll be pleased."
"How do we shape up on TV, Dick? All our ducks in line ?"
"Excellent, Senator. You'll be on network television a minimum of three times between now and election day—just about perfect, no danger of overexposure."
"Have you licked the makeup thing yet, Dick?" asked Belinda Hennington. "A small detail but it probably cost one man the presidency. We don't want that to happen to us."
"No sweat, Mrs. Hennington. Max Factor came out with a complete new line right after that fiasco. I think we'll be using 'Graying Temples,' in keeping with our maturity image. As we all know, the youth bit is out nowadays. Fortunately with the senator we can play it either way."
"Good show, Dick," said the senator. "Harry ?" "I've run the results of our polls through the computers, both the IBM 436 and the Remington Rand 1401. Louis Harris gave us a random pattern sampling with peer-group anchorage ; Gallup a saturation vertical-syndrome personality study and NORC an ethnic and racial cross-section semiology. The results check out on both computers, although I'm programing a third as a safety-valve check-out.
"The computers have you winning the election, Senator, but by less than three thousand votes. A small shift and there goes the ball game."
The senator, startled and troubled, glanced nervously toward his wife. She gave him a smile of reassurance.
"Do the computers indicate a possible breakthrough," he asked, "with any of the peer groups ? How do we stand with the Jewish vote ?"
"You're solid with the Jews, Senator. Where you're in trouble is with the Negroes."
"The Negroes !" exclaimed Senator Hennington. "Why, I have the best voting record on civil rights on Capitol Hill. Just last year I broke the ADA record for correct voting on civil rights with 97.64."
"Our polls reveal a sharp decline just after your speech requesting a moratorium on civil-rights demonstrations. If we can regain most of the lost Negro percentile, Senator, we're home free."
"No use crying about a lack of voter loyalty. This calls for a 'think session.' Perhaps we should have our special assistant on minorities and civil rights sit in ; although I'm not sure how helpful he'll prove. Frankly, I'm disappointed by his performance so far."
"Judy," said the senator into his office intercom, "Think session in here. No calls, please, and cancel all morning appointments. And ask Carter Summerfield to join us, will you ?"
The senator turned to his wife as they awaited the arrival of Summerfield.
"Belinda, I'm beginning to have serious doubts about Summerfield, he hasn't come up with a fresh idea since he joined us, and I don't expect anything other than tired clichés from him today."
"He's fine in a campaign, Gil, that's where he'll shine. I don't think you ought to rely on him for theory."
"Perhaps you're right. I guess it's not brains we're looking for in him anyway."
"No," she smiled. "That's his least valuable commodity to us."
The senator swiveled his leather-covered chair half-round and gazed out at the Washington Monument.
"This question of the Negro vote could be serious. I never thought I'd ever be in trouble with those people. We have to come up with something which will remind them I'm the best friend they have in Washington, and soon."
Carter Summerfield had sat in his office all morning, worried and concerned. He sensed the senator was not pleased with his performance and could not understand why. Summerfield had sought desperately to discover what it was the senator wanted to hear in order that he might say it, and was amazed to find that the senator seemed annoyed when his own comments were returned, only slightly paraphrased. In all his career as a professional Negro, Summerfield had never before encountered a white liberal who actually wanted an original opinion from a Negro concerning civil rights, for they all considered themselves experts on the subject. Summerfield found it impossible to believe Senator Hennington any different from the others.
He had spent the morning searching for the source of the senator's displeasure until his head ached ; the handwriting was on the wall and Summerfield knew his job was at stake. He must discover the source of displeasure and remove it. Perhaps he should wear ready-made clothes? Had the senator somehow seen . him driving the Lincoln, rather than the Ford he always drove to the office? It was essential never to have a more impressive car than one's boss. He told all his newly integrated Negro friends that. Had anyone discovered the encounter with the white girl in Colorado Springs when he had accompanied the senator on a trip to the Air Force Academy? He had been certain he had acted with -the utmost secrecy and discretion. But he had known even then that it was a stupid move which might threaten his entire career.
Summerfield took two Gelusils and a tranquilizer and reached for the phone to inquire discreetly of his fellow integrated Negro friends if there was word on the grapevine of an opening for a man of his experience. The phone rang. It was Senator Hennington's secretary summoning him to the senator's office for a think session.
Smiling, as always when in the presence of whites, Summerfield entered the senator's office, his eyes darting from face to face for some sign concerning his present status. But the looks of the other members of the staff were no longer funereal and the senator greeted him with a warm smile as he motioned Summerfield to an empty chair, briefly inquiring about his wife and children.
"It seems, Carter," said the senator, "that we're in serious trouble with the Negro vote." Summerfield frowned in sympathy and concern. "We must come up with a fresh, dramatic and headlinecapturing act on my part which will prove to my colored constituents that I'm the best friend they have in Washington." He swept the room again with his steely gaze, Gary Cooper, back to the wall, but undaunted. "And we must do it today."
Summerfield nervously licked his lips. "How about calling a conference of the responsible Negro leaders to discuss your new civil-rights bill, Senator?"
The senator considered for a moment.
"I don't think so, Carter. To be perfectly frank,
I don't think the bill will pass this session. White backlash. "
"How about a fact-finding tour of the African countries ?" said Dick.
"No, I did that last year and still haven't kicked the dysentery I picked up on safari in Tanganyika."
"How about a speech attacking apartheid at Capetown University ?" asked Harry.
"I don't think South Africa would grant me a visa. "
"Gil," said the senator's wife, "why don't we accuse the Central Intelligence Agency of a discriminatory hiring policy ?"
"Segregation in CIA ?" "Yes. They have no Negro officers at all ; mostly menial and clerical help."
"Are you certain, Belinda? This could be what we're looking for. "
"I'm positive, but I'll check it out. We have a man in personnel over there, you know."
"Couldn't a charge of that nature prove counterproductive, Mrs. Hennington ?" asked Dick. "CIA's almost as untouchable as the FBI."
"Not since U-2 and the Bay of Pigs. And this should prove an irresistible combination for the press : cloak and dagger and civil rights."
"I'm inclined to agree, Belinda," said the senator, who was usually inclined to agree with his attractive wife. "What's the best way of springing this thing for maximum impact ?"
"Why not at the Senate Watchdog Committee hearings ?" said Tom.
"But the hearings are closed," said the senator.
"It wouldn't be the first time we've used closed hearings for a press leak, Gil. I'll brief Mark Townsend over lunch here in the office on the day of the hearings," said Belinda.
"Excellent," said Dick. "A political columnist of his stature is perfect."
' 'B.-ow, how do I play it in the hearings? Indignant, angry, or do I underplay ?"
"Dignified, I think, Senator," said Harry. " 'You're shocked and saddened that the agency in closest grips with the forces of godless communism is shackled by the chains of racial prejudice."
"Right," said Tom. "You say that America must utilize the talents of its entire citizenry, regardless of race, color or creed, in the cold war."
"They'll deny it at first," said Belinda, "then probably claim their personnel files are classified, but they'll back down when they get enough negative press coverage. They're very image-conscious nowadays."
Carter Summerfield sat looking interested, but carefully silent. Advising the senator how to criticize other whites was definitely not one of his functions.
"I can program one of the computers to provide statistics showing the increased efficiency of the armed forces since their integration," said Tom.
"If CIA does select a Negro, he'll be the bestknown spy since 007," said Harry.
"Well, he will find it a bit difficult after all the publicity he's going to get," said the senator.
"You mean, Gil," said his wife, "the publicity you're going to get."
The senator smiled.
"General," said Senator Hennington, addressing the director of the CIA, "it has come to my attention that there are no Negroes on an officer level in CIA. Would you care to comment on that?"
The other committee members looked at Senator Hennington with some shock. They knew he faced a close election in the fall, but this gambit was below the belt. The general, fighting to control his famous temper, replied icily.
"You know, of course, Senator, that our personnel files are highly classified."
"I'm aware of• that, General, but this meeting is closed and we are all cleared for that kind of information."
"It's not true that we don't have any colored at the Agency. Our entire kitchen staff, our maintenance section and drivers are all colored."
"My question, General, concerned Negroes on an officer level."
"Well, we don't have any colored officers."
"Do you think, General, that a policy of racially selective recruiting which excludes a full 10 percent of our population is a wise one ?"
"Yes. While I personally have no race prejudice, I feel Negroes are not yet ready for the highly specialized demands of intelligence work."
"Really ?" said Senator Hennington, smiling a smile of patronizing pity into the face of bigotry.
"It's a question of sociology rather than prejudice; a gap simply exists between the races which is a product of social rather than racial factors. "
"There are Negroes who have bridged that gap."
"If so, I would welcome them in CIA."
"I would suggest you make more of an effort to find them."
"Senator Hennington," said the committee chairman, in his rich, aristocratic southern drawl, "we all know that deceit, hypocrisy, duplicity are the everyday tools of our agents in the field. Much to their credit, the childlike nature of the colored mentality is ill-suited to the craft of intelligence and espionage."
"I'm afraid, Mr. Chairman," replied the senator, once again entering into the charade concerning race he had conducted with his southern friend for well over a decade, "I don't understand what you mean by 'the colored mentality."
"There is the question of cover," said the general. "An agent must be capable of fading into the background, adopting the guise of the person one cannot remember minutes after meeting him. Negroes in the field would be far too conspicuous."
"General, I'd rather not carry this conversation any further. I would appreciate a report in a month's time concerning the progress of the establishment of a merit-hiring policy at CIA."
The luncheon table had been trundled away and Belinda Hennington and the famous political columnist Mark Townsend sat in the conversation corner of the senator's office, sipping brandy from large snifters.
"Both the senator and I wanted to give you an exclusive on this, Mark," said Belinda.
"Thanks, Belinda. Are you sure this checks out? The government is supposed to be at the forefront in merit hiring."
"Not CIA. We're positive."
"I have a man at CIA—mind if I check him out on this?"
"Of course not ; you can use 'an undisclosed CIA source' in your lead."
"This could bring civil rights back into the headlines. It's been suffering from overexposure lately."
"You're right, Mark. The public has tired of the same old thing : fire hoses, cattle prods, dogs on the one hand and singing, marching and praying on the other. Civil rights could use a good public relations man."
"When will the senator make an official state?
"I should guess after the wire services and television pick up your beat. About three days, I should think."
"Sounds about right. Where will you conduct the press conference ?"
"Right here. The Washington Monument makes a good backdrop for the television cameras ; almost a Hennington trademark."
Townsend had left and Belinda was sipping a well-earned scotch-on-the-rocks when the senator returned. She mixed her husband a drink as he sank into the leather chair behind his big desk.
"How did it go today, dear?" she asked.
"Couldn't have gone better, honey. I'm certain this is it."
"Yes, dear, by this time next week we'll have the Negro vote wrapped up again." She handed the senator his drink and rested one sleek hip on the polished mahogany of the desk. "Never take the voters for granted. Even Negroes react eventually, you know."
"I'd have thought the CIA would have been more alert on a thing like this. If they'd had even
one Negro officer, my charges would have fallen as flat as your sister's souffés."
"If it hadn't been CIA, it would have been someone else. We're not likely to run out of institutions to accuse of segregation in our lifetime, darling." They smiled at one another affectionately.
That November the senator won his reelection comfortably, the Negro vote accounting for more than his margin of victory.
2
Freeman watched the class reunion from a corner of the common room of the CIA training barracks. It was a black middle-class reunion. They were black bourgeoisie to a man, black nepotism personified. In addition to those who had recruited themselves upon receiving notice that the CIA was now interested in at least token integration, five were relatives or in-laws of civil-rights leaders, four others of Negro politicians. Only Freeman was not middle class, and the others knew it. Even had he not dressed as he did, not used the speech patterns and mannerisms of the Chicago ghetto slums, they would have known. His presence made them uneasy and insecure; they were members of the black elite, and a product of the ghetto streets did not belong among them.
They carefully ignored Freeman and it was as he wished ; he had no more love for the black middle class than they for him. He watched them establishing the pecking order as he sat sipping a scotch highball. It was their first day in the training camp after months of exhaustive screening, testing, security checks. Of the hundreds considered, only the twenty-three present in the room had survived and been selected for preliminary training and, constantly reminded of it since they had reported, they pranced, posed and preened in mutual and self-admiration. To be a "Negro firster" was considered a big thing, but Freeman didn't think so.
"Man, you know how much this twelve-year-old scotch cost me in the commissary? Three bills and a little change! Chivhead Regal! As long as I can put my mouth around this kind of whiskey at that price, I'm in love with being a spy."
"You know they call CIA agents spooks? First time we'll ever get paid for that title."
"Man, the fringe benefits—they just don't stop coming in! Nothing to say of the base pay and stuff. We got it made."
"Say, baby, didn't we meet at the Penn Relays a couple of years ago? In that motel on the edge of Philly? You remember that chick you was with, Lurlean? Well, she's teaching school in Camden now and I get a little bit of that from time to time. Now, man, don't freeze on me. I'm married, too, and you know Lurlean don't give a damn. I'll tell her I saw you when we get out of here."
Where'd you go to school, man ? Fisk? I went to Morris Brown. You frat? Q? You got a couple brothers here, those two cats over there. What you major in? What your father do? Your mother working, too? Where your wife go to school ? What sorority? What kind of work you do before you made this scene? How much bread you make? Where's your home? What kind of car you got? How much you pay for that suit? You got your own pad, or you live in an apartment? Co-op apartment? Tell me that's the new thing nowadays. Clue me in. You got color TV? Component stereo, or console?
Drop those names: doctors I have known, lawyers, judges, businessmen, dentists, politicians, and Great Negro Leaders I have known. Drop those brand names: GE, Magnavox, Ford, GM, Chrysler, Zenith, Brooks Brothers, Florsheim. Johnny Walker, Chivas Regal, Jack Daniels. Imported beer. Du Pont carpeting, wall-to-wall. Wall-to-wall drags with split-level minds, remote control color TV souls and credit-card hearts.
Play who-do-you-know and who-have-you-screwed. Blow your bourgeois blues, your nigger soul sold for a mess of materialistic pottage. You can't ever catch Charlie, but you can ape him and keep the gap widening between you and those other niggers. You have a ceiling on you and yours, your ambitions; but the others are in the basement and you will help Mr. Charlie keep them there. If they get out and move up to your level, then what will you have?
They eyed Freeman uneasily; he was an alien in this crowd. Somehow, he had escaped the basement. He had moved up to their level and he was a threat. He must be put in his place. He would not last, breeding told, but he should know that he was among his betters.
The tall, good-looking one with the curly black hair. and light skin approached Freeman. He was from Howard and wore his clothes Howard-style, the cu mess pants stopping at his ankles. His tie was very skinny and the knot almost unnoticeable, his shoulder-padding nonexistent. He had known these arrogant, Chicago niggers like Freeman be-
fore, thinking they owned Howard's campus, moving in with their down-home ways, their Mississippi mannerisms, loud laughter, no manners, elbowing their way into the fraternities, trying to steal the women, making more noise than anyone else at the football games and rallies. One of those diddy-bop niggers from Chicago had almost stolen his present wife.
"Where you from, man? You don't seem to talk much."
"No, I don't."
"Don't what ?"
"Don't talk much. I'm from Chicago."
"Chicago? Where you from before that? Wayback, Georgia, Snatchback, Mississippi? You look like you just got off the train, man. Where's the paper bag with your sack of fried chicken ?"
Freeman looked at him and sipped his drink.
"No, seriously, my man, where you from? Lot of boys here from the South; how come you got to pretend? I bet you don't even know where State Street and the Loop is. How you sneak into this group? This is supposed to be the cream, man. You sure you don't clean up around here?"
Freeman stood up slowly, still holding his drink. The tall one was standing very close to his armchair and had to step back when Freeman rose.
"Baby, I will kick your ass. Go away and leave me to hell alone."
The tall one opened his mouth to speak; a fraternity brother sidled up, took his arm and led him away. Freeman freshened his drink and sat down in front of the television set. After a lull, the black middle-class reunion resumed.
He had not made a mistake, he thought. All niggers looked alike to whites and he had thought
it to his advantage to set himself apart from this group in a way that would make the whites overlook him until too late. They would automatically assume that the others—who looked and acted so much like their black representatives and spokesmen who appeared on the television panels, spoke in the halls of Congress, made the covers of Time and Life and ran the Negro newspapers and magazines, who formed the only link with the white world—would threaten to survive this test. Both the whites and these sadistic niggers, Freeman thought, would ignore him until too late. And, he thought, Whitey will be more likely to ignore a nigger who approaches the stereotype than these others who think imitation the sincerest form of flattery.
He smiled when he thought about walking into his friend's dental office that day.
"Hey, Freebee, what's happening, baby? Ain't seen you in the Boulevard Lounge lately. Where you been hiding? Got something new on the string?"
"No, been working. Look, you know the cap you put on after I got hung up in the Iowa game? I want a new one. With an edge of gold around it."
"Gold ? You must be kidding. And where you get that refugee from Robert Hall suit?"
"That's where I bought it. I'm going out to Washington for a final interview panel and I want to please the crackers." His friend nodded. He understood.
Freeman did not spend much time socializing with the rest of the Negro pioneers, those chosen to be the first to integrate a segregated institution. He felt none of the gratitude, awe, pride and arrogance of the Negro "firsts" and he did not think after the first few days that many of them would be around very long; and Freeman had come to stay.
They had calisthenics in the morning and then six hours of classes. Exams were scheduled for each Saturday morning. They were not allowed to leave the area, but there was a different movie screened each night in a plush, small theater. There was a small PX, a swimming pool, a bar and a soda fountain. There was a social area at each end of the building in which they lived that included pool tables, ping-pong, a television room with color TV, chess and checker sets. There was a small library, containing technical material related to their classes and light fiction, magazines and periodicals. There was a music room with a stereo console containing an AM-FM receiver and with records consisting mostly of show tunes from Broadway hits of the last decade. There were Coke machines. It was like a very plush bachelor officers' quarter.
There were basketball courts, badminton courts, a nine-hole golf course, squash courts, a gym, a 220-yard rubberized track, a touch-football field. After the intensive screening which they had undergone prior to their selection, none of the rest thought that the classes and examinations were anything more than window dressing. They settled down to enjoy their plush confinement during the training period after which they would be given offices in the vast building in Langley, Virginia, down by the river.
Freeman combined a program of calisthenics, weight training,- isometrics, running and swimming, which never took more than an hour, usually less than half that time. He would watch television or read until dinner, take an hour's nap and then study until midnight.
No one at the training camp, white or colored, thought it strange that Freeman, a product of the Chicago ghetto, where Negroes spend more time, money and care in the selection of their wardrobe than even in Harlem, should be so badly dressed. Or that, although he had attended two first-rate educational institutions, he should speak with so limited a vocabulary, so pronounced an accent and such Uncle Tom humor. They put it down to the fact that he had been an athlete who had skated through college on his fame. Freeman did not worry about the whites because he was being exactly what they wished. The Negroes of the class would be ashamed of him, yet flattered by the contrast; but there might be a shrewd one among them.
There was only one. He approached Freeman several times with penetrating questions. The fraternity thing put him off.
"You a fraternity man, Freeman ?" he asked once over lunch.
"Naw. I was once because of the chicks. You had to have that pin, you know. Almost as good as a letter in football. But I thought that kinda stuff was silly. I used to be a Kappa."
He looked at Freeman coldly. "I'm still a Kappa," he said. He finished lunch and never spoke to Freeman again.
Midway through the fourth week, three of the group were cut. They were called into the front office and informed that their grades were not up to standard, and that same evening they were gone. Panic hit the group and there were several conferences concerning what should be done. Several long-distance phone calls were made, three to politicians, five to civil-rights bureaucrats. The group was informed that they were on their own and that after the time, energy, money and effort that had gone into their integration, they should feel obligated to perform up to the highest standards. Freeman had received the best grades in each of the exams, but no one was concerned with that fact.
Two others left the following weekend, although their grades were among the highest in the group. Freeman guessed correctly that it was for homosexuality and became convinced that in addition to being bugged for sound, the rooms were monitored by closed-circuit TV. He was right. The telephones, even the ones in the booths with coin boxes, were bugged as well. The general received a weekly report regarding the progress of the group. It appeared that those intellectually qualified could be cut on physical grounds. They were already lagging at the increasing demands of the morning calisthenics and were not likely to survive the rigors of hand-to-hand combat. The director of the school confidently predicted that not one of the Negroes would survive the ten weeks of the school, which would then be completely free for a new group of recruits presently going through preliminary screening. It was to the credit of Freeman's unobtrusive demeanor that the school's director did not even think of him, in spite of his excellent grades and physical condition, when making his report to the general. If he had, he might have qualified his report somewhat.
The general instructed his school's director to forward complete reports to the full senatorial committee. He intended to head off any possible criticism from Senator Hennington. He could not know that the senator was not in the least concerned with the success or failure of the Negro pioneers to integrate the Central Intelligence Agency. He had won his election and for another six years he was safe.
"When this group is finished, I want you to begin screening another. Don't bother to select Negroes who are obviously not competent; they have already demonstrated their inability to close the cultural gap and no one is in a position seriously to challenge our insistence not to lower standards for anyone. It will cost us a bit to flunk out six or eight a year, but we needn't worry about harassment on this race thing again in the future if we do. It's a sound investment," said the general. He was pleased and again convinced that he was not personally prejudiced. Social and scientific facts were social and scientific facts. He ate a pleasant meal in his club that evening and noted that there were both white and colored present. The whites were members and guests; the Negroes served them. The general did not reflect that this was the proper order of things. He seldom approved of the rising of the sun, either.
Two more were cut for poor marksmanship. Freeman had obtained an ROTC commission at college and had served in Korea during the police action. He was familiar with all of the weapons except the foreign ones, and a weapon is a weapon. Only the extremely high cyclic rate of the Schmeisser machine pistol bothered him and that did not last very long.
"Mr. Freeman," the retired marine gunnery sergeant said, "that is an automatic weapon and designed to be fired in bursts. Why are you firing it single-shot '?"
"It's to get its rhythm, Sergeant. I couldn't control the length of the bursts at first and I was wasting ammo, but I think I have it now."
The sergeant knew that Freeman had been an infantryman, and marines, in spite of what they claim, have at least a modicum of respect for any fighting man. "OK, Mr. Freeman. Show me what you mean. Targets one through five, and use only one clip."
"Call the bursts, Sergeant."
"Three. Five. Five . . e" He called the number of rounds he wanted in rapid succession, as fast as Freeman could fire them. There were rounds left for the final target and, on inspection, they found that one five-round burst had been six instead.
"That is very good shooting, Mr. Freeman. Were you a machine gunner in Korea ?"
"No. I was in a heavy weapons company for a while and got to know MG's fairly well, but I spent most of my time in a line infantry company. I like automatic weapons, though. I learned it's not marksmanship but firepower that wins a fire fight. I want to know as much as I can about these things."
"All right, Mr. Freeman, I'll teach you what I know. You can have all the extra practice and ammo you want. Just let me know a day ahead of time and I'll set it up. We'll leave the Schmeisser for a while and start with the simpler jobs, and then work up. Pistols, too?"
"Yes, I'd like that, Sergeant. And I'd rather your maintenance section didn't clean them for me.
I'd rather do it myself. No better way I know of to learn a weapon than to break it down, clean and reassemble it."
The gunnery sergeant nodded his head and something rather like a smile crossed his face.
Freeman read everything in the library on gunnery, demolition, subversion, sabotage and terrorism. He continued to head the class in examination results. There was much more study among the group now and they eyed one another uneasily, wondering who would be the next to go. They had no taste for returning to the jobs they had left : civil-rights bureaucracies, social welfare agencies, selling insurance, heading a playground in the ghetto, teaching school—all of the grinding little jobs open to a nonprofessional, middle-class Negro with a college degree. Long after Freeman retired, between midnight and one, his program not varying from the schedule he had established during the first week, the rest of the class studied far into the night. The group was given the army physical aptitude test, consisting of squat jumps, push-ups, pull-ups, sit-ups and a 300-yard run.
Freeman headed the gro.up with a score of 482 out of a possible 500. The men finishing second and third to him in academics were released when they scored less than 300 points. There were only two other athletes in the group, one a former star end at Florida A and M, the other a sprinter from Texas Southern. They were far down on the list academically, although they studied each night until dawn. It was just a matter of time before they left. In two months there were five of the group left, including Freeman. Hand-to-hand combat rid them of two more.
The instructor was a Korean named Soo, but Calhoun, his supervisor, was an American from North Carolina. The niggers would leave or Calhoun would break their necks. He broke no necks, but he did break one man's leg and dislocated another's shoulder. He was surprised and angered to find that Freeman had studied both judo and jujitsu and had a brown belt in the former and a blue stripe in the latter. He would throw Freeman with all the fury and strength he could muster, each time Freeman took the fall expertly. He dismissed the rest of the class one day and asked Freeman to remain.
"Freeman, I'm going to be honest with you. I don't think your people belong in our outfit. I don't have anything against the rest of the group ; I just don't think they belong. But you I don't like." "Well, I guess that's your hang-up."
"I don't like your goddamn phony humility and I don't like your style. This is a team for men, not for misplaced cottonpickers. I'm going to give you a chance. You just walk up to the head office and resign and that will be it. Otherwise, we fight until you do. And you will not leave this room until I have whipped you and you walk out of here, or crawl out of here, or are carried out of here and resign. Do I make myself clear?"
"Yes, whitey, you make yourself clear. But you ain't running me nowhere. You're not man enough for that." Freeman felt the adrenalin begin coursing through his body and he began to get that limp, drowsy feeling, his mouth turning dry. I can't back away from this one, he thought.
"Mr. Soo will referee. International judo rules. No chops, kicks or hand blows. Falls and chokeholds only. After a fall, you get three minutes' rest and we fight again and I keep throwing you, Freeman, until you walk out of this outfit for good."
"Mr. Soo?"
They bowed formally and circled one another, each reaching gingerly for handholds on the other's jacket. He had fifteen pounds on Freeman and wore a black belt; but a black belt signifies only that the wearer has studied judo techniques enough to instruct others. The highest degree for actual combat is the brown belt Freeman wore. Calhoun was not a natural athlete and had learned his technique through relentless and painstaking practice. His balance was not impressive and he compensated with a wide stance. Freeman figured his edge in speed all but nullified his weight disadvantage. He had studied Calhoun throughout the courses; he had watched him when he demonstrated throws and when he fought exhibition performances with Soo. Freeman was familiar with his technique and habits and knew that he favored two throws above all others, a hip throw and a shoulder throw, both right-handed.
He came immediately to the attack. Freeman avoided him easily, feeling him out, testing his strength. Calhoun was very strong in the shoulders and arms, but as slow as Freeman had anticipated. He compensated by bulling his opponent and keeping him on the defensive.
Calhoun tried a foot sweep to Freeman's left calf, a feint, then immediately swung full around for the right-handed hip throw. Freeman moved to his right to avoid the sweep, as the North Carolinian had wanted, then, when Calhoun swung into position for the hip throw, his back to Freeman, Freeman simply placed his hand on his
back and, before he could be pulled off balance and onto the fulcrum of Calhoun's hip, pushed hard with his left hand, breaking contact. It had been a simple and effective defensive move, requiring speed and expert timing. They circled and regained their handholds on each other's jackets. After a few minutes of fighting, realizing that he was outspending, Calhoun began bulling Freeman in an effort to exhaust him.
Soo signaled the end of the first five-minute period. They would take a three-minute rest. By now, Freeman knew his opponent.
You'd be dangerous in an alley, thought Freeman, but you hung yourself up with judo. Karate, or jujitsu, maybe, to slow me down with the chops and kicks. But there is just no way you can throw me in judo, white boy. He wondered whether to fight, or to continue on the defense. He looked at Calhoun, squatting Japanese-style on the other side of the mat, the hatred and contempt naked on his face. No, he thought, even if I blow my scene, I got to kick this ofay's ass. When you grab me again, whitey, you are going to have two handfuls of 168 pounds of pure black hell. He took slow, deep breaths and waited for the three minutes to end.
Soo nodded to them, they strode to the center of the mat, bowed and reached for one another.
Freeman changed from the standard judo stance, with feet parallel, body squared away and facing the opponent, to a variation; right foot and hand advanced, identical to a southpaw boxing stance. It is an attacker's stance, the entire right side being exposed to attack and counter from the opponent. Freeman relied on speed, aggressiveness, natural reflexes and defensive ability to protect himself in the less defensive position. He wanted only one thing: to throw this white man. He moved immediately to the attack.
Freeman tried a foot sweep, his right foot to Calhoun's left, followed up with a leg throw, _osotogare,_ then switched from right to left, turning his back completely to his opponent, whose rhythm he had timed, and threw him savagely with a right-handed hip throw.
Calhoun lay there and looked at Freeman in surprise. He got slowly to his feet, rearranging his judo jacket and retying his belt. Freeman did the same, then, facing him, he bowed as is the tradition. Calhoun remained erect, staring at Freeman coldly. Freeman maintained the position of the bow, hands on thighs, torso lowered from the
"Calhoun-san. You a judoka. You will return bow of Freeman-san," hissed Soo. Reluctantly, Calhoun bowed. They returned to their places on the mat, squatting Japanese style, waiting for the three minutes to end. Freeman wondered if he could keep from killing this white man. No, he thought, he's not worth it and it would really blow the scene. But he does have an ass-kicking coming and he can't handle it. This cat can't believe a nigger can whip him. Well, he'll believe it when I'm through .
Soo signaled them to the center of the mat.
Freeman methodically chopped Calhoun down. He threw him with a right-foot sweep, a lefthanded leg throw, another hip throw and finally a right-handed shoulder throw. Calhoun, exhausted by now, but refusing to quit, reacted too slowly and landed heavily on his right shoulder, dislocating it. Soo forced the shoulder back into the
socket and the contest was finished. Saying nothing, they bowed formally and Freeman walked slowly to the locker room. It was the end of the day, Friday, and he would have the weekend to recuperate. He would need it.
Calhoun asked for an overseas assignment. Within three days he left for leave at his family home in North Carolina, then disappeared into the Middle East.
Freeman would have to be more careful; there were holes in his mask. He would have to repair them.
3
The director of the school reported to the general on the new group. "General, I'm afraid that there is at least one who might stick it."
"I thought you told me only two weeks ago that there was hardly a chance that any of that group would last."
"That's correct, General. Even with the facts on paper before me as I made the report, I somehow forgot that the man existed. He has a way of fading into the background. You can't remember his face, or what he looks like, or what he has said, even minutes after you have spoken to him. But the records speak for themselves. He leads the class in everything and his marks are above average, although not by much, for our regular classes."
"Who is the man ?"
"His name is Freeman, sir."
"Is that the one who had the altercation with Calhoun?"
"Yes sir, and Soo reports that he could have 28
killed Calhoun. Soo thinks him one of the finest natural Judaists of his experience."
"But Calhoun is rated one of the best men we have in hand-to-hand combat."
"That's true, General; but in all honesty to Calhoun, this man has had several years' experience as a Judaist. He has fought in the Midwest regional championships for the last three years, qualifying for the national finals, but refusing the trip because of work commitments. Our physical instructor considers him one of the finest natural athletes he has ever seen."
"True, they do make good athletes. Great animal grace and reflexes. I'm disappointed, however, in Calhoun. He should never have lost his head, challenged the man. Soo indicates that there was no provocation. He's just too dedicated to the agency. Can't stand the idea of standards being lowered. Can't say that I disagree with him. Well, he has a chance to prove himself in Yemen." The general picked up some papers from his desk in dismissal.
When the director of the school had almost reached the door, the general lifted his head. "This man Freeman, if he survives this group, is to go into the incoming class. He might do well among his own people, but competing with whites might be another cup of tea. He is not to be treated unfairly, but he is not to be given any advantage, either." The general motioned the director back to the chair. "What kind of man would you say
"Well, slow-witted, a plodder. He studies five hours a day, seven days a week. Only in athletics does he seem to do things naturally. Physically, he could be expected to react instantly and efficiently, but in a mental crisis, I don't know."
"I thought as much. Then there is no question of his going into the field. Even if he does survive the group, we cannot subject the lives of other agents in the field to his deficiencies. We will find something for him here in headquarters. Thank you, that will be all. Keep me posted on this group."
Two weeks before the scheduled ending of the training, Freeman was alone. He watched the last of the group leave. No one bothered to say goodbye; Freeman had no friends among them and his continued presence in the camp was an insult. He settled himself in front of the television set and pressed the remote-control button and watched the white fantasy world in full color.
Freeman finished the course and was congratulated in a small ceremony in the director's office. There were to be three weeks before the new group arrived and he elected to remain in the camp, except for a few days leave. He drove to an all-Negro housing development just across the Anacostia River and applied for a one-bedroom apartment.
He then drove to New York and checked into the Hotel Theresa in Harlem. He made the rounds of the Harlem bars the first evening, before heading downtown to the jazz joints. His Robert Hall suit and too-pointy shoes were in the hotel closet, his gold-edged tooth cap in a plastic container in the bathroom, a plain one replacing it. He wore black-rimmed glasses of plain glass, cordovan bluchers, a button-down shirt of English oxford and a dark sharkskin suit from J. Press.
He saw Thelonious Monk at the Five Spot, the band with Johnny Griffn, Charlie Mingus at the Village Gate. He saw Three penny Operons in the Village and Five-Finger Exercise, The Night of the Iguano and I Can Get It for You Wholesale on Broadway. He visited the newly-opened Guggenheim and decided that Wright had goofed, but he enjoyed the Kandinskys. He visited the galleries on Fifty-seventh Street and the Museum of Modern Art. His second night, he found a sixfoot, compatible whore who knew the night clerk at his hotel. He tipped the night clerk and the bellboy on duty, and each night after that, he would meet her in the bar between two and three and they would spend the night in his hotel room.
He purchased sixteen books, but deferred delivery until further notice. The CIA Freeman would not have read those books.
Freeman left his suit, shoes, shirt, tie and tooth cap in a bag, with instructions that they be delivered to the storage company that stored the rest of his clothing, records, books and paintings that had no business in his new existence. He would establish a New York base later. He had pondered the danger of leading a double life and decided that the strain of squadron would have to be eased somehow from time to time. The few days in New York, doing the simple things he had done, had convinced him more than ever that this was important. He might be the CIA Tom in Washington, but for a few days elsewhere he would have to become Freeman again. He did not think that, even if he ran into his CIA colleagues in New York, identification would be a danger; niggers all look alike to whites, anyway, and no one would connect the New York Freeman with the Freeman who would pioneer integration in one of the most powerful governmental institutions in the United States.
He left early Sunday morning, the top of the car down; he put the Morgan onto the New Jersey Turnpike and headed south for the nation's capital. The Morgan would have to go, he decided. It did not fit his painstakingly created image. He was sorry about that as he listened to the exhaust note over the roar of the wind and the car radio, but a Ford or Chevrolet would be better; perhaps even an Oldsmobile or a Buick. He would be expected to own a car a bit outside his means, a bit more expensive and flashy than those of his peers at the agency. It was not difficult to conform to the image whites desired, since they did most of the work. They saw in most Negroes exactly what they most wanted to see; one need only impressionistically support the stereotype. Whites were fools and one had constantly to fight in order not to underestimate their power and danger, because a powerful and dangerous fool is not to be underestimated. Add the elements of hypocrisy and fear and one had an extremely volatile combination. It was a combination that could easily blow the country, even the world, apart. In the army Freeman had learned to respect but not fear the potential danger of explosives ; rather, he had learned how to use them.
He did not, as did the African diplomats, have difficult y with lunch on Highway 1 because he did not bother to try to get a meal in the greasy spoons and truck spots that dotted the highway like cancerous growths of chrome and neon. He stopped along the highway and ate a lunch he had packed the night before from a delicatessen. A premixed martini over the rocks from a thermos, a cold chicken with potato salad, a mixed salad with oil and vinegar dressing and a small bottle of Chablis, which had chilled in the cracked ice of the
cooler. Finished, he discarded the ice, repacked the container and lay back on the army blanket he had spread under a tree and slept for an hour. He reached Washington just ahead of the incoming weekend training and was in camp in time to shower and catch Ed Sullivan on television after a light dinner in the camp dining room.
An agent, as Freeman had anticipated, had checked his movements in New York, checking routinely at the hotel and questioning the whore with whom Freeman had slept. They would not find much, since Freeman knew that the people with whom he had talked would have seen nothing out of the ordinary in either his dress or behavior. Besides, Harlem Negroes, particularly hotel employees and prostitutes, seldom tell white men very much, especially those who look and act like cops, regardless of what they claim to be.
Free of classes, Freeman increased his personal study and was able to spend time each day working with Soo on the mat. He added karate to his repertoire of judo and jujitsu and spent time as well on the range, firing pistols, rifles and shotguns, as well as his more favored automatic weapons.
Freeman spent each weekend in Washington and became convinced that it is one of the squarest towns in the world. Within walking distance of the immaculate, white, neoclassical center lie some of the worst ghetto slums in the United States. The bigots on Capitol Hill need look no further than a few hundred yards to convince themselves of the inherent inferiority of Negroes and, controlling the capital like a colonial fiefdom, they can ensure that things will not change racially.
He found a whore on U Street and would spend time with her each weekend, in a bed in a hotel in the ghetto. She was questioned as well.
"Look, honey, what kinda cop you say you
"I'm not a police officer."
"You look like a cop to me, baby. How come you asking me stuff about this cat? He in trouble? He a nice John. What make you think I'm going to tell you anything?"
"I'm not a policeman, but I do have police friends ."
"Like that, huh? I wondered when you start leaning on me. You ofays never no different, no matter how you look. And you college cats with the smiles and pink baby faces the worst. OK, what you want to know?"
He slipped her a twenty. She picked it up from the table, never taking her eyes off his face.
"Have you ever known him to take dope? Marijuana Heroin ?"
"Naw, he don't even smoke much. After he eats, after we done turned a trick, maybe three or four cigarettes all the time we together. He ain't no junkie, baby."
"Does he have a tendency to boast, brag?"
"Him? Naw, he don't talk about himself at all. I thought he was a baseball player, even football, although he a little bit small for that; but he could be, he ain't nothing but muscle and prick. If he got any fat on him, I ain't found it."
"Does he gamble ?"
"Naw. One night I took him up to a place. Cat owed me some bread and I had to collect. Little poker, little craps. He just watch and when we through we leave.
"Tell you something 'bout that man, though. People don't give him no shit. He move quiet, don't say nothing, but I seen some bad cats move around him. This cat owed me the bread. He don't want to pay it. I don't tell him nothing about—what you say his name, Freeman? I don't say nothing about him being my man. I just took him there before we make it to the hotel and maybe he want some action, I get a cut. The cat owed me start to get a little off the wall about the bread, then he look at Freeman. He just standing there watching the crap game, quiet, ain't bothering nobody, but this cat owes me the bread look at him and then look at me and, baby, he give me the bread. And that cat one of the baddest men in D.C. He make two of Freeman, but he don't want no part of him. He ain't my man and it wasn't his scene, but I thought about it and if I had trouble that night, he be in it.
"And he ain't romantic about chicks on the block. Never comes on like a social worker: 'How a nice girl like you get into this shit.' 'Let me take you out of it, baby.' None of that shit. I'm a whore and he knows it, but he treats me liKe a queen. I think he putting me on for a long time and then I get to like it. He can put me on as long as he wants. I dig the way he treats me. I'm a whore. He knows it, I know it, but when I'm with him, he makes me feel like a queen."
"What about his sexual habits? Anything, well, unusual ?"
She looked at him through narrowed eyes. "So that your scene? Well, baby, twenty bills ain't enough for no freaks. Come again. Little more bread, dig? I know some chicks don't deal in nothing but freakish tricks, but that ain't my scene. And don't start no shit about the Man. I ain't on junk and I can always get my hat and make it to Baltimore, dig?"
He gave her another twenty, blushed furiously and damned the woman for sensing what no one else knew, that he enjoyed this part of his job, to his shame.
"Nothing fancy, little straight up and down, mostly. Blow jobs ain't his scene. He don't mind a little head now and then, but before the deal goes down, he wants to make it straight. No way-out positions or jazz, whips and wet towels, no gimmicks. He just like to screw, baby.
"Now, don't be disappointed. Now, I could tell you 'bout some real freaks. I got a trick, white boy like you, come down here once a week, just to listen. Get his cookies every time, You want to hear some scenes, honey? Or maybe you want me to get a show together. Straight, gay, or both. Anything you want, baby."
They were sitting in the back booth of a long, narrow and very dark bar on U Street, She stroked his thigh and confirmed his excitement. This was the easiest trick she had in a week. She wondered how much she could milk him for.
"Would you say that he might have homosexual tendencies ?"
She threw her pretty black head back and laughed. "Him? Man, you wasting your time there. He wouldn't make that scene for nothing. If you got a thing for him, look someplace else. Shit, I ought to know what I'm talking about, I make it that way myself and it takes one to know one.
"Anybody could turn me straight, it could be him. I mean, he don't need no fake scenes and he
just want a good, professional job ; but sometimes he really turns me on and I ain't had no thing for a man in years. No, baby, he ain't gay.
"But, if that your scene, I know somebody you might dig." She continued to stroke his thigh. "Or, baby, I could do things for you."
She leaned over, whispering in his ear and told him in detail what she could do. He backed into the corner of the booth, but continued to listen. Suddenly, he gripped her thigh with one hand and groaned. She moved away, looked at him in contempt, and then she lit a cigarette. He lay back against the corner of the booth until the color returned to his face, a thin film of perspiration on it. She crossed her legs and moved her foot in time to Sonny Stitt on the jukebox. She liked this joint; they had good sounds here.
He sat up and wiped his face with a handkerchief, looking around nervously. No one was paying the slightest attention to the booth.
She motioned toward the back of the bar. "The john back there." He started to slide across the bench of the booth.
"You forgetting something, honey." She held out her hand. He reached quickly into his pocket and stuffed crumpled bills into her hand. She looked at them, nodded and moved from the booth to let him pass. She smoothed her red dress, thinking that his grip on her thigh might leave a bruise. She walked to the bar and ordered a drink, fixed her hair in the mirror and straightened her red dress on her shoulders.
She had never worn red before, she had been told all her life that she could not, because she was too black, but Freeman had told her that she should wear it because she was a Dahomey queen.
She had gone to the library to find out what he had meant because he wouldn't explain, and asked for the book he had written down for her. She had found that he was talking about Africa and at first had been angry. But there was the picture of a woman in the book that had looked enough like herself to startle her, hair kinky and shortcropped, with big earrings in her ears. She had taken the book out of the library and painfully read it in its entirety. Then she bought a red dress and, later, several others when she found the tricks liked it, but mostly because Freeman liked her in red and said so. She wore big round and oval earrings like the queen in the picture, but she could not bring herself to wear her hair short and kinky; but sometimes she would look at the picture and see herself there and for the first time in her life, she began to think that she might be beautiful, as he said.
She returned to the booth and sipped her new drink. The trick returned from the john, all policelike and white-man strong. He thanked her for her information and cautioned her not to say anything to Freeman. He could not look her in the eye. She blew smoke into his face and he left. She would need no more tricks that night and she had done better than for a full weekend with the paddy-boy.
She walked to the phone booth and called her chick.
"Honey, got a scene working. I won't be able to make it tonight, big bread. See you tomorrow, huh ?
"Of course I miss you. Be cool, baby. Mama will make it up tomorrow. Bye, now."
She walked out into U Street and took a cab to a small efficiency apartment that she kept for herself. Even her woman did not know of its existence and she used it when she wanted to be alone. She wanted to be alone tonight and to think about Freeman. She might invite him to her apartment the coming weekend instead of making their usual hotel routine. Men were not her scene, she knew, but she liked that man. She knew soon after she entered the apartment that she would invite him there.
4
Each morning Freeman would drive his secondhand Corvair to the big building, descend three levels, unlock the door of his office and begin work. He ate in the cafeteria just often enough not to appear aloof, the rest of the time he would bring his own lunch in his briefcase and eat in his windowless room, listening to rhythm and blues or jazz from the Negro radio station in Washington on a transistor radio. He was the top secret reproduction section chief and, being the only man in the section, he was little more than the highest-paid reproduction clerk in Washington. Nevertheless, he was the only Negro officer in CIA.
He was promoted after a year in his cubbyhole. He had been making a run of two hundred of a three-page report dealing with the methods of bribery of Central American union leaders when there was a knock on the door. He stopped the Ditto machine, wiped his ink-stained hands on a piece of waste and opened the door to the general's secretary, Doris, a tall, golden girl from California with big breasts she considered a gift from the gods.
"Dan, the general needs someone to give a senator a guided tour of the building. Do you think you can handle it? I usually do it, or one of the other secretaries, but we're up to our ears getting ready for the closed budget hearings next week. Now, if you don't think you can do it, say so and I'll try to get someone else."
"I think I can handle it."
"Good, then, we can go right up to the general's office." She was standing very close to Freeman and when she turned she allowed one of her jutting breasts to brush his arm. She sat on the edge of his desk while he washed his hands in a basin in the corner.
"Don't you like me?"
"Why do you ask that ?" he said.
"Well, you don't seem very friendly when you bring top secret documents up to the office. Sometimes you don't even seem to know that I'm around." She pouted prettily.
"Well," said Freeman carefully, "it's just that I have a great deal on my mind, the responsibility of running the entire top secret reproduction, you know. I want to make good and sometimes I'm preoccupied with my duties. You know how it is. Actually, I think you're very intelligent and attractive. Everyone says so. To be the general's executive secretary at your age is quite an accomplishment; all of the other brass have secretaries who are much older. I'll try not to be so preoccupied the next time I bring some things to your office."
He put on his coat and held the door open for her and she brushed him again as she passed, He turned out the light and checked the door carefully to see that it was locked. They walked to the elevator and he pressed the button to summon it. "You're very ambitious, aren't you?" she said.
"Yes, I want to be the best reproduction section chief they've had here."
"Oh, but you are. Everyone says what a good job you're doing; much better than anyone ever expected. You know, I'm ambitious, too. I don't want to be a secretary all my life. I've applied for lateral entry into the officers' corps and if I pass the oral exams next year, I'll get an overseas assignment."
"I'm sure you'll pass." The elevator arrived and they entered it. He pressed the button for the top floor of the building.
"Well, I hope so," she said. "I'll probably start out under embassy cover, you know, handling cryptography and administration, but I hope eventually to be an operative agent."
"Well, I'm certain you'll be the prettiest spy wherever they assign you." They left the elevator, Doris managing to brush him again, and walked to the double doors of the general's suite of offices. They entered and Doris settled herself behind her desk in the outer office and waved to the door behind.
"Go right in, Dan. The general's expecting you." She thrust her breasts against the powder-blue cashmere sweater she was wearing and was smiling to herself when Freeman walked to the door of the general's office. He knocked on the door and entered.
The general waved him to a chair next to his
massive desk. "Sit down, Freeman. Been hearing good things about you. Like being with our team ?" "Yes sir."
"Good. I think you know the building and the agency pretty well. Think you could take a group on a guided tour? As you know, we don't give many tours around here and are not staffed up like some of the other publicity-hound agencies. It's the Senate committee, or most of them, and we try to keep them happy if we can. Think you're up to
"Yes sir."
"Good, then. Report to Morgan."
It was a group of three senators and three congressmen. The senator to whom Freeman owed his job was among them. Senator Hennington was pleasantly surprised to find Freeman his guide. He knew that the rest of the group had flunked out, but as long as there was one Negro, the CIA was integrated. The senator considered it another victory in his long battle in the field of civil rights. He was pleased to be faced with the fruit of his efforts.
"This, gentlemen, is our communications center, the nerve center of the agency. It operates around the clock, receiving and sending messages, from and to the far reaches of the earth. If the communications for one day were laid end to end, they would reach to Denver and back again to Washington.
"The center, of course, operates around the clock and is fully protected down here from anything except a pinpoint, direct hit of an H-bomb of at least one hundred megatons. The entire city of Washington could be destroyed and our communications center would continue to operate, carrying out its vital task as watchdog of the Communist conspiracy.
"In the next room is our computer system, capable of turning the raw material of intelligence, coming in through the communications center, into viable facts ; the kind of facts that keep us at least one step ahead of the Reds. That one crucial step ahead, gentlemen, has probably meant peace until now for the free world. Lose it by even half, and we may well lose our freedom with it."
Freeman showed them through the building, climaxing the tour with a demonstration in the indoor firing range, himself firing alongside the regular firing master.
"That was certainly fine shooting, Freeman. Had you done a great deal of shooting before you came into the agency ?"
"No sir, but the agency demands that we all qualify at least as marksman and a very high percentage check out as experts."
"That's very interesting. Tell me, where did you go to school ? Howard ? Fisk ?"
"No sir. Michigan State and the University of Chicago."
"Really? They have some fine football teams at Michigan State. I'll never forget their Rose Bowl victory a few years back. Tremendous comeback in the last quarter. Were you out there for that
one
"Yes sir."
"Senator, Freeman played on that Rose Bowl team," said Morgan.
"Is that right? You know, I played football for Dartmouth a few years back." He turned to Morgan, CIA congressional liaison officer, and asked : "Mr. Morgan, do you suppose Mr. Freeman could join us for lunch ?"
"Of course, Senator."
They completed the tour and had lunch in the executive dining-room, the senator insisting that Freeman sit next to him. The senator was deft in handling Negroes. As he often told members of his own staff, it was important to make them feel comfortable, to treat them as equals. This was very difficult for the senator, since he really did not feel that there were very many people equal to himself, white or black. The senator had a polished, practiced humility that wore well, however. His wit, charm and good looks more often than not left people feeling that he was a regular guy; and although the senator did not feel himself a regular guy, he would have been pleased at the accolade, since Americans regularly elected "regular guys" to move.
"Tell me, Mr. Freeman, do you like working for the agency?" The senator had found that calling Negroes "Mr." often had a magical effect on the relationship.
"Oh, yes sir. I think it's an extremely important job and I'm proud to be a small part of the team."
"What were you doing before you joined the agency
"I was in social work in Chicago. I worked with street gangs in the slums."
"That's very interesting, Mr. Freeman. Did you enjoy the work ?"
"Yes sir." Freeman used "sir" with whites as often as possible. He found that it had a magical effect on the relationship.
"Well, I think that kind of work is important. As I said in a speech on the Senate floor last year in our hearings on juvenile delinquency, it is extremely important that we establish some human and sympathetic contact with juvenile delinquents, school drop-outs and other youthful members of the culturally deprived."
"I couldn't agree more, Senator."
"Yes. I favor a national study on the problem of juvenile delinquency and crime; particularly among the culturally deprived."
"Excellent idea, Senator. We certainly need another study of that problem."
"Then, of course, an increase in the construction of low-cost public housing, a training program for high-school dropouts and the introduction of scientific methods of rehabilitation in our reform schools.
"I'll send you a copy of the speech, Mr. Freeman. Unfortunately, the program was not adopted, but I have high hopes of some federal action in this area in the not too distant future. I was discussing this very thing with the president at luncheon just a couple of weeks ago."
"Yes sir."
"Did you find an increase of dope addiction among juveniles during your work with them ?"
"Yes, Senator, I did."
"Yes, that is a major problem. We must strengthen our laws in that area. Make the penalties stiffer; but I definitely favor regarding the addict as a sick person, rather than a criminal." "I couldn't agree more, sir."
"And, of course, there is the problem of increased illegitimate pregnancies among the youth of the culturally deprived. I certainly don't take the extreme stand of those who advocate sterilization, but advice concerning birth-control devices should certainly be made available and in some cases, they should be provided. Dependent on the individual case, of course, and tightly controlled."
"Definitely, Senator."
"There is a great deal of work to be done in this area. We have to roll up our sleeves and get to work. I told the president that last week and he agreed emphatically. I expect great progress in this area within the next few years."
The luncheon ended and the congressional party spent a few minutes in the office of the director. The senator lingered when the party began to leave for Capitol Hill.
"General, I was extremely impressed with Freeman. An excellent choice, I think. I had an interesting and enlightening conversation with him concerning his former field, juvenile delinquency. He gave me some excellent tips; things I might be able to use eventually in a Senate speech, or perhaps in an article for the New Republic. I think that taking him on as your assistant was an extremely wise move. It's fine to see integration in action."
Freeman was appointed the next week as special assistant to the director. He was given a glassenclosed office in the director's suite. His job was to be black and conspicuous as the integrated Negro of the Central Intelligence Agency of the United States of America. As long as he was there, one of an officer corps of thousands, no one could accuse the CIA of not being integrated.
Freeman couldn't have been in a better position for what he intended to accomplish. He had access to most of the general's briefings, attended many of his meetings. He traveled with the general increasingly and his face became familiar throughout the agency. He took target practice three times a week, practiced judo, jujitsu and karate and had one weekly session of boxing with a retired former middleweight contender.
His routine seldom varied and the periodic checks by the security section confirmed this. He continued his trips to New York. He would check into the Hotel Theresa in Harlem, then leave for his apartment, shaking any possible tail. A shower, a shave, a Beefeater's martini, and Freeman the Tom became Freeman the hipster.
No one ever blew Freeman's cover. They accepted at face value what he appeared to be, because he became what they wanted him to be. Working for the agency, in the agency, Freeman was the best undercover man the CIA had.
5
Freeman saw his Dahomey queen at least once a week. She told him of the interview with the agent, but he did not want to switch whores and complicate things: finding a mistress among black Washington society promised to be tedious, unrewarding, and a potential threat to his carefully worked-out cover. His girl, Joy, came to Washington almost monthly and twice she met him in New York midtown hotels, since he would not reveal his soul hole even to her.
Late one spring evening they were lying in his bed in Washington. They had eaten a seafood dinner in a restaurant not far from his apartment, just north of the junction of the Anacostia and Potomac rivers; they had been seated by the kitchen as usual. Later they had listened to Sonny Stitt in a small jazz club just off U Street in the heart of the big Washington ghetto. They made love when they returned but had not slept and lay silently sipping scotch and smoking, listening to the music of a late-night jazz station 49
from the transistor radio which stood on the bed table.
Joy arose to one elbow and gazed into Freeman's face. "Dan, I think it's time to have a talk about the two of us. This kind of thing can't go on forever. It's time I started thinking about a home, family, security."
"OK," he said, "let's get married."
"Dan, you know I'd love to marry you, have your children, but this part of you, your bitterness, your preoccupation with the race thing—it frightens me, shuts me out. I feel threatened."
Freeman sat up in bed and looked at Joy in some surprise. "But why should you feel threatened? Hell, the way I feel doesn't even threaten whitey." "Dan, how much longer are you going to stick with this job? You haven't had a promotion in four years and you're the only Negro officer they have."
"Once I prove myself, they'll recruit more Negroes; I'm certain of it. We can't all join the demonstrations ; some of us have to try quietly to make integration work."
"Are you going to prove yourself by taking a bunch of bored housewives on guided tours?"
They were on shaky ground and Freeman had to be careful ; Joy knew him too well and one false move, a statement which didn't ring true, and he might expose himself. He arose, walked to the dresser to light a cigarette, regarding her in the mirror as he did so.
"I'm hoping I can move into something else soon; something more substantial." He returned to the bed, sat on its edge and lit a cigarette for her. "If I left now, before they began hiring other Negroes, I'd always think I'd given up. It's not easy continuing with this jive job, but it's little enough sacrifice for the cause of integration."
"Baby, I'm sorry, but I can't sacrifice my life for a cause. I admire the way you feel, but I fought too hard to get out of the slums and you continue to identify with the slum people you left behind."
"I never left them behind."
She placed her hand on his knee and smiled gently. "Honey, whether you admit it or not, the day you left Chicago for college, you left the block and the people on it. Besides, what's wrong with wanting to live in a decent neighborhood, to want the best for our kids ?"
"Who do you think pays for those nice things if not the people we ought to be helping because nobody ever gave them a chance to help themselves ? Joy, have you forgotten you came off those same streets ? Except for your college degree, those people are just like you."
"Not me, baby! I left that behind me: all those hot, stinky rooms, those streets full of ghosts. Junkies, whores, pimps, con men. The crooked cops, the phony, fornicating preachers. And the smells: garbage, stale sweat, stale beer, reefers, wine and funk. That bad, hand-me-down meat from the white supermarket, the price hiked up and two minutes this side of turning a buzzard's stomach. I've had that shit and going back won't change things."
"Somebody has to try and change things."
"You can't change whitey. He needs things just the way they are, like a junkie needs shit. Whitey's hooked with messing with niggers and you want him to go cold turkey. It's not going to happen. We can be happy, Dan; we can be anything we want."
"Whitey won't let you be what you want to be, they put you in a pigeonhole marked nigger. How can I be happy that way? There's no way I can spin a middle-class cocoon thick enough for them not to penetrate any time they choose. Even if I could, what about the rest ?"
"We have our own lives to live."
"I can't live in this country like an animal, I'm a man." He was restlessly pacing the room. She leaned toward him, the sheet falling away from her breasts, and he had a moment of panic looking at her, knowing that he might lose her.
"You don't have to live like an animal. If you really must spite whites, do it by succeeding. You can do more for your people by offering them an example to follow than by burying yourself in that building across the river."
"You mean hire myself out for a higher price to sit by a more impressive door ?"
"It doesn't have to be that way ; you don't have to think any cooperation with whites is a sellout. There are dozens of responsible positions available."
She wants me to tell her it will be all right if I make enough money. A man ought to be able to protect his woman, make her feel secure, but how long will it take for her to hate me as a man once I've traded in my balls? A showpiece spade is a showpiece spade, no matter how many times he gets his picture in the papers or how much bread he makes.
"Joy, I don't have to stay in Washington, I can return to Chicago." It was a bit soon for what he planned, but he had to try and keep her, keep them both. "There's an opening with the private
foundation I started out with in street-gang work."
"You're going back into social work? All the good positions for Negroes are filled now. I thought you might finish law school, maybe go into politics."
She wants that title, he thought, Mrs. Lawyer Freeman, then Mrs. Congressman Freeman. "I can go to law school; the kids only hang out at night."
"Stop kidding yourself, Dan : you don't want to go to law school, you never did like it, and you hate Negro lawyers. You hate all the Negro middle class because you think they don't do enough to help other Negroes. You forget something, honey ; I'm middle class, too, but you're still on the block in spirit. You've made your choice and I have a right to make mine."
He looked at her but she dropped her head and stared morosely at the glowing tip of her cigarette, its smoke lightly veiling her face.
"Yes," he said softly, "you have."
"I'm not coming to Washington anymore. I'm going to get married."
He picked up his drink and took a sip. The ice had melted and it was weak, watery and warm.
"The doctor or the lawyer?"
"The doctor," she answered.
He drew a deep breath, let it out slowly.
"Seems like a nice cat." He thought of her never being his again and thrust the thought from his mind. He listened to the radio, Miles Davis playing a ballad. It didn't help, it was from a record Joy had given him as a present. How many other things had she given to him in their years together, how much of her was a part of him? Suddenly, he was afraid she would cry.
"I'm sorry, Dan, but I'm not getting any younger, and . . e"
"It's all right, baby," he said, taking her cigarette and snuffing it out in an ashtray. "I guess it had to happen one day. Look, this is our last night together; let's say goodbye right." He reached for her.
She sent him an invitation to the wedding and he sent them a wedding present, but he did not go to Chicago for the ceremony because he thought you could carry being civilized too far.
Freeman had met Joy years before in East Lansing, Michigan, when they were both students at Michigan State University. They were both slumbered, bright, quick and tough and considered a college degree the answer to undefined ambitions. They had much in common: they were both second-generation immigrants of refugee families from the Deep South. Their grandparents had migrated as displaced persons to the greater promises of the urban North; Joy's grandfather from Alabama to the Ford plant during the first war; Freeman's to the Chicago stockyards about the same time. Both Joy and Freeman had been born during the bleak depression years and had known the prying, arrogant social workers, the easily identifiable relief clothing, the relief beans, potatoes, rice and raisins wrapped in their forbidding brown paper bags. But poverty had done different things to them.
Joy had become determined she would never be poor again ; Freeman that one day to be black and
poor would no longer be synonymous. She regarded his militant idealism and total identification to his race first with amusement, then irritation and finally, growing concern. Joy had no intention of becoming her black brother's keeper. Slowly, she convinced Freeman he could best use his talents to help Negroes as a lawyer dedicated to the cause of civil rights. He could join the legal staff of one of the established civil-rights bureaucracies; one day argues precedent-making cases before the Supreme Court.
She convinced him and he began preparing himself for law school while working toward an undergraduate degree in sociology. Life was being very kind to Joy; she had never felt she would marry the man she loved. But she knew she would have to be very careful because Freeman could be a very stubborn man and the mere idea of his becoming a member of the black bourgeoisie was enough to enrage him. Joy intended not only that he become a member, but one of the leaders. She felt that she could manage this essentially unmanageable man because he loved her. The greatest potential danger was that she loved him as well, but she thought that she could control that emotion. She would have to, because there was far too much at stake.
Joy made an unfortunate strategic error. She insisted that Freeman attend the national convention of the civil-rights organization they thought he would join. Because she had to work that summer to replenish her wardrobe for the fall, Freeman went to the convention alone. He returned bitter and disillusioned.
"Baby, there ain't no way I can •work for those motherfuckers. They don't give a damn about any niggers except themselves and they don't really think of themselves as niggers.
"You ought to hear the way they talk about people like us. Like, white folks don't really have much to do with the scene. It's that lower-class niggers are too stupid, lazy, dirty and immoral. If they weren't around, all them dirty, conk headed niggers with their African and down-home ways, why, everything would be swinging for the swinging black bourgeois bureaucrats, their high-yellow wives, their spoiled brat kids, and their white liberal mistresses. Integration, shit! Their definition of integration is to have their kids the only niggers in a white private school ; their wives with a well-paying job in an otherwise all-white firm and balling white chicks looking for some African kicks.
"And look at what they're trying to do. When did you ever see them raising hell with a lily-white union so that people like your father can get a job they're qualified for; or try to get those so-called building inspectors to do their jobs so people in the slums can live a little better, or get involved with any kind of nigger that wasn't just like themselves."
Joy was concerned, but not too much so; she figured that she could gradually smooth things over, but she underestimated Freeman's natural distrust and contempt for the Negro middle class; he remained adamant. Their fights about their future increased in frequency and intensity.
Freeman took frequent trips with the track team and it was not until he returned unexpectedly early from the annual Pacific Coast Conference-Big Ten dual meet, which had been held in Palo Alto, that he found that Joy had been spending each weekend he had been on the road
with the track team, in Detroit. He drove to Detroit that evening, but she was not in when he called and he left a message with her mother. He called a former roommate who was doing graduate work at Wayne State and dropped his bags at his apartment. Freeman went to a little bar near his friend's home. He and Joy went there often because the bartender had been a potential all-American at Michigan until injured in the Army game and the owner of the bar had played football at Illinois. The bar had a small, tasteful combo ; piano, bass, drums and electric guitar and sometimes a horn. He sat at the bar talking to the bartender, sipping a Ballantine's ale and listening to the music. He did not see Joy when she came in until he looked up and caught her reflection in the mirror behind the bar.
She was sitting in a booth almost directly behind where he sat and she had not noticed him there. She was with a tall, light-skinned Negro named Frank, who had graduated from Michigan two years before. He was the Negro quota at a local medical school and his father was a prominent society doctor •who made most of his money —tax free—selling dope to jazz musicians and performing abortions for Negro debutantes in Detroit.
Freeman watched them in the mirror and like lovers they touched one another in that way lovers think is casual. The bartender watched him closely and Freeman smiled that he intended to cool it. He was about to leave, hoping that Joy would not see him, when she looked up and their eyes met in the mirror. He nodded, smiled and lifted his drink in salute toward her reflection. He stood and walked to her table and talked small talk with her and her date. Freeman had met him often in Ann Arbor
6
It had something of that, Theodora thought, recalling the sudden remarkable blotting out of sound which she'd first experienced as her tafi swung under the Archgate and into the close yesterday morning.
'We have to find what our culture most lacks and provide it. We have no business aping the restless world. Those forces of spirituality and piety which have down the ages hallowed and sanctified the place see it now with no more luminosity than an airport.' Canon Millhaven leaned forward across the gear stick and held Theodora's gaze. 'I have spoken,' her tone was hushed, 'with the ancient dead of our cathedral. They are not pleased with us.'
There were few types of eccentricity which Theodora had not met in the course of her ecclesiastical childhood, but, though she'd heard tell of it, this was the first time she had personally encountered talking with the dead. Of course there had been that canon of St Paul's and doubtless others.
'Do you think the Janus might restore the numinous?' Theodora was hesitant.
'In a very low level way,' said Canon Millhaven dismissively. 'The pagan gods are not, of course, important in themselves.' She sounded like an oldfashioned landowner describing the tenantry. 'Nevertheless a Janus, looking both ways, is by no means negligible, Miss Braithwaite. Duality,' she exclaimed, 'so important in all the best religions.'
'Inner and outer, dark and light, heaven and earth,' Theodora murmured almost to herself. 'But we can decide though,' she went on more confidently. 'We can choose how to value it, the Janus, I mean, what to let it do to us. Wouldn't you say?'
'We shall need to be very careful, very alert and on our guard,' Canon Millhaven answered briskly. 'Beware in particular, I should say, of the new dean.'
Canon Millhaven sat back in her seat and slipped the car into gear. 'The only truly religious life lived round here,' she said dismissively, 'is down there.' She waved out of the driver's window.
Theodora caught a glimpse, half familiar, of three Nissen huts and some allotments beyond the railway
'In the Hollow,' said Canon Millhaven as she hit the main road.
'Erica Millhaven talks to the dead,' said Theodora when she rang Geoffrey before setting out for the dean's party.
'You don't surprise me,' said Geoffrey. 'I've got adult baptism at the door. I must rush. Keep me in the picture. '
Across the close in the Deanery kitchen, Nick Squires was putting four King's School boys through their paces as waiters before letting them loose on the dean's guests. He'd waited at clergy parties himself when he was their age, two years ago.
'Look, Beddows, you're supposed to be offering it, not trying to hit them with it. They've got a choice, you know. They don't have to eat the stuff.' He took the plate from the grinning Beddows and demonstrated. There was a round of applause.
The kitchen in the basement was large, warm, clean, and stacked with food and drink. All had been beautifully organised by the head verger. The dean had indicated his wishes for this, his first venture into hospitality in his new dignity, and Tristram Knight had seen to it. Nick was much impressed by Tristram. He tries to inquire as to whether a Negro in their midst might offend anyone. Only a relative number of replies in the affirmative were received, confirming the general's pride in the progress of race relations. Freeman was often used as a liaison between the general and Senator Hennington and their mutual relations improved considerably. The senator would often invite Freeman to lunch in the Senate dining-room. The senator liked to lunch on the Hill with a Negro at least two or three times a month and often would be stuck with one who looked white, a wasted effort in image-making. Nowadays the presence of Negroes in the Senate dining-room seldom evoked any dramatic response from the southern senators, as had been the case early in the senator's career, thus taking much of the drama and pleasure from the adventure, but the senator's reputation as a flaming liberal crusader for human rights remained intact and Freeman made his small contribution, making only one minor faux pas by once requesting a wine in a good French accent. The senator did not notice and Freeman made a mental note that knowing anything at all about wines was not part of his image. The senator, flattered by Freeman's feigned ignorance and naiveté, told everyone that Freeman was an extremely intelligent man.
Freeman moved through Washington like an invisible man. He was an occasional, though not frequent guest at Georgetown cocktail parties for African diplomats. He was seldom invited to sit-down dinners, not because the Georgetowners objected to eating with Negroes—they all did it several times a year—but to save him the embarrassment of which fork and spoon to use for which course.
His blackout from Washington black society, the most snob-ridden of a snob-ridden class in America, was total. It was as he wished. While Freeman could regard whites with a certain objectivity and controlled emotion, the black middle class and their mores sent him up the wall. He dated seldom and if there was a recurrence of the relationship, it was usually confined to bed. The Washington Negro women found much to fault Freeman in dress and cool, but little to complain about in bed. He seldom maintained a liaison more than a few months at a time. He still used the Dahomey queen, now moved up the whore's social ladder to the point where she was a high-priced call girl with a clientele consisting mainly of southern congressmen; she specialized in several sadistic variations, for which she charged extra. Her relationship with Freeman had become warm and personal, in spite of the fact that she knew nothing about him. She enjoyed his company, and increasingly she enjoyed sleeping with him. Men would never be her scene, but she knew by now that Freeman offered no threat and made no attempt to use the power her sexual enjoyment gave him. It intrigued her and she enjoyed the relationship they had.
Freeman studied the reports of the guerrilla fight in Algeria, particularly as confined to urban centers; the guerrilla war against the Huks in the Philippines; the guerrilla war against the Malayan Communists; the tactics of the Viet Cong; the theories of Giap and Mao Tse-tung.
He accompanied the general to Saigon four times as the war escalated. He stayed in the Hotel Caravelle, the air conditioning so high that you had to wear a jacket or a sweater. He had cocktails on the roof terrace among the correspondents who were trying to turn the day's ration of rumors, gossip and USIS and PIO propaganda into a meaningful dispatch. He ate in Vietnamese restaurants, several of the good French restaurants and a Japanese restaurant, set in a Japanese designed garden, across town from the hotel. He learned to walk slowly the other way when a crowd surged toward the scene of carnage created by a terrorist, to watch if a cyclist had an egg more lethal than those laid by hens in the basket of his bike.
He bought a set of Danish-style stainless ware in Japan, a Japanese camera in Hong Kong, Thai silk in Bangkok. He had suits made by Jimmy Chen in Kowloon, which he had shipped to his New York apartment. He became adept with chopsticks and learned to make love in several languages. He found that in Asia he was not as hip in bed as he had imagined and took a postgraduate course in sex, oriental style. He was enjoying the job, its prerogatives and putting on so many white people, but the general saved him from any addiction in that direction.
They were driving back to Langley from the Capitol when the general decided to take his assistant to lunch. He almost requested that the driver stop at his club. Although the club had no colored members, it was no oddity to see dark faces there several times a year. Only recently there had been a Nobel Prize winner, an operatic star, a federal judge and the president of Howard University. But Freeman might be uncomfortable in the impressive surroundings of the club. The general ordered the driver to take them to a good, moderately priced steak house instead.
They were ushered in and taken to an excellent table. It was the first time in several visits that Freeman had not been seated by the toilet or kitchen. Freeman ordered a martini on the rocks, the general bourbon and branch water while they awaited their steaks.
"Dan, I want to say how pleased I am at the way you've fitted into the agency. To be frank, I had my reservations at first, but they've been completely eliminated by your performance. If more of your people were like you, there would be much less difficult y." The general took a sip of his drink, rolled it on his tongue a moment before swallowing, then made a tent of his fingers.
"Honest sweat and toil. Pull yourself up by the bootstraps like the immigrants. These demonstrations and sit-ins stir up needless emotion. Your people must demonstrate a respect for law and order, earn the respect and affection of whites. Take yourself as an example: a fine natural athlete; no denying you people are great athletes."
"Yes," said Freeman, "and we can sing and dance, too."
"Right. In the fields of sports and entertainment, you're unsurpassed. But you must admit there's still a social and cultural gap to be closed. The Africans did develop centuries later than the Europeans, you know.
"As I've said, you're a fine athlete, but I think you'd have to admit your intellectual shortcomings. It will take generations, Dan. It's not a question of prejudice, but rather one of evolution." Their steaks arrived and Freeman fought to keep his down. He had no appetite now and he willed his hands not to tremble. He had not ordered wine because he thought the Spartan general might consider it an affectation. He sipped the cool Carlsberg, taking a deep breath as he replaced the glass. He forced himself to look at the general, masking his seething emotions. He surprised himself by summoning a smile.
"I'm encouraged after knowing you, Dan. You're a credit to your race. Perhaps your generation will achieve more by example than your civil-rights leaders can hope to accomplish through turmoil and agitation."
Somehow Freeman got through the meal. He excused himself as the general ate his dessert of apple pie a la mode and was violently ill in the toilet. He washed his face and saw how pale he had become in the washroom mirror, but he didn't think anyone as color-blind as the general would notice.
The general allowed Freeman to leave the office early that day and he reached his apartment ahead of rush-hour training. He mixed a stiff scotch and lay sipping it in a very hot tub while listening to Dinah Washington. Well into his second drink, he watched the six o'clock news: the big mounted cops charging the children with cattle prods, the police dogs and fire hoses, one long shot of a little girl bowled over by the battering stream of water and spun along in the gutter like a bowling pin, the freedom songs and praying, all the wasted, martyred faith and courage of a people who wouldn't quit. It had to be channeled where it would be most effective, where it might make whitey back off. It was time to stop procrastinating, time to do what he had to do; he had all the training he needed.
He walked into his bedroom, made a list of social welfare agencies in Chicago and wrote letters of inquiry concerning job openings. He discovered his hands had stopped trembling and that he was calmer than in years.
He requested an appointment with the general the next day. Freeman told him he had been profoundly impressed by the general's talk at lunch the day before. He felt he could make a greater contribution to his people by returning to Chicago and working among them and the general had shown him the way. The general reluctantly agreed that perhaps following his own advice might indeed be the best thing for Freeman to do. The general walked Freeman to the door of his office while heartily booming clichés concerning Freeman's grand sense of duty. He closed the door behind Freeman, frowned briefly, walked to his desk and phoned his director of training and personnel, informing him that they would need another Negro sometime within the
next year.
He hung up and returned to the direction of the cold war.
7
Freeman left the director's office and walked directly to the elevator, nodding briefly at the director's secretary on the way out. He pressed the button calling the elevator to his floor and inspected the attaché case he had been presented as his going-away present. It was serviceably large, of saddle-stitched leather, with brass fittings. It contained the few things he had cleared from his desk in the office which had been his for years in the suite of offices behind the armed guards and door marked, simply, "Director."
The elevator arrived and he rode it down, walked across the vast marble hall to the entrance and waited at the door for the director's limousine. He had known that it would not be awaiting him at the steps and he showed no surprise, no anger. He had waited many years for what he had to do and a few minutes more for a car was no problem. It might take him many more years to do what he had planned for so long, and, an impatient man, he had carefully schooled himself in patience.
He stood at the door looking out into the huge courtyard of the building and out toward the trees that screened the building from the Potomac River, toward the city of Washington, hidden by the trees. The courtyard was bright with the Virginia spring sunshine, unveiled by factory fallout. He could not hear the birds behind the big glass doors of the entranceway, but he knew that they would be singing in the trees that lined the river. He had often brought his lunch to eat among the trees there, listening to the birds, watching the slow current of the river, sitting among the pines and sycamore, recalling the smells and sounds of summers spent in a boy-scout camp in Michigan. A city boy from Chicago, he had never lost his awe and love of the woods; their sights and sounds. The building squatted vast and ugly, a marble and granite conglomeration of the worst of neoclassical and government-modern architecture, an ugly abscess created by bulldozers and billions in the midst of the once-beautiful north Virginia woods. A cancerous abscess, he thought, sending out its tendrils of infection tens of thousands of miles.
He stepped through the door into the sunshine and listened to the sounds from the woods, placing the case at his feet. He was dressed in quiet bad taste, his suit a bit too light, his cuffs a bit too deep, lapels a bit too wide, shoulders a shade too padded, tie too broad, trousers too wide at the knee and ankle, socks too short. He wore large airplane-type sunglasses, his hair was closely cropped and there was a thin surrounding of gold around a front tooth. His suit was a bit too cheap and his wristwatch, of eighteen-carat gold, a bit too expensive. He walked with a gangling shuffle, his head tilted slightly toward one shoulder and there was always a smile on his face, even when alone in the building in which he worked, broadening and flashing the thin gold when people approached. He was very well liked and would be missed. Waiting for the director's car, he never once glanced back at the building in which he had spent a great part of the last five years of his life.
The black Cadillac limousine swung into the drive and stopped just ahead of where Freeman stood ; the Negro chauffeur made no effort to get out to open the door. Freeman knew that he wouldn't open the door for him and patiently walked to the car, opened the door for himself and climbed into the air-conditioned interior. The driver started moving before Freeman was seated, throwing him awkwardly into the far corner of the rear seat. Smiling gently, Freeman disentangled himself and leaned back into the foam-rubber cushions, looking out at the Potomac River as they sped towards Washington, through the bulletproof glass.
"You really going through with it? You really quitting?
"Yes," said Freeman.
"And they ain't pushing you out. I thought so at first, but they ain't pushing you out. I didn't think cats like you ever quit a scene like you got. I see a lot like you in Washington, but I never knew one to quit on his own. Your kind love it here. You don't even quit for more money someplace else. It just don't make sense. You really going back to that job you had, like they say? Working with street gangs in Chicago ?"
"Yes, but this time I'll be in charge of the program."
"I shoulda known! A title, a goddamn title, the only damn thing you cats dig more than money. 'Special assistant to the director' wasn't enough ; now you going to be a director yourself, or did they think up something more fancy The driver snorted in disgust.
Freeman said nothing and they proceeded along the riverside road in silence. They moved briefly through Arlington and then onto the drive and past the cemetery, the big statue of the marines raising the flag on their right. They went across the bridge and past the pompous Lincoln Memorial, the phallic Washington Monument at the end of the reflection pool, and just short of the next bridge, the Jefferson Memorial. The cherry trees were in bloom, a blaze of pink against the green, the blue sky above, with big, fat cumulus clouds floating marsh-mallow-like against the bright, blue sky. They turned right on Pennsylvania Avenue and drove to the White House where the guard waved the car into the drive.
In front of the White House, the driver again did not bother with the door and Freeman let himself out and walked into the Office Annex. He was met at the office door by a pudgy, red-faced man with nervous mannerisms.
"Dan, how are you? Right on time, right on time. We'll be going in to see the president in— he looked at his watch and then glanced up at the clock on the wall behind the desk of the secretary "—exactly ten minutes. Now, we go in between the boy-scout delegation from East Bengal and just before his monthly tea with some of the congressmen's wives. Split-second timing around here, you know, and you know how he is if there is a foul-up. We have four minutes with him and then it's finished. He'll give you a little present, say a few words and then you pose for pictures. Should get some play in the Chicago press, which won't do you any harm and I'll send copies to you in Chicago next week. I don't know what his present will be; he's pretty cagy about that kind of thing, but you know what kind of response to give: little surprise, gratitude, thanks and tell him how sorry you are to be leaving the team. But I don't have to coach you, Dan. You know what to do, hey, boy?
"You know, we're all going to miss you around here. Remember that time he gave a rebel yell in the Taj Mahal and you smoothed things over with the Indian press? We haven't forgotten that, boy, around here, not a little bit. You were always in there swinging in the clutch. Yeah, sorry to see you go. Keep in touch now, you hear?
"Make yourself to home. I got to go check on those congressmen's wives. Be right back and then we can go on in."
Freeman sat in a big leather armchair and looked at the big four-color portrait of the president on the wood-paneled wall behind the secretary. In a few minutes the red-faced man returned.
"All right, let's go now. We stand outside his office until the boy scouts come out; then we go right in. You know the drill."
They waited at the door until the scouts left beaming, each holding a multi-purpose jackknife. They entered the office of the president.
The president was seated in his rocking chair. His .aide bent over and whispered in his ear as Freeman approached. The president arose and extended a hand, thrusting his own quickly far up toward Freeman's thumb so that it couldn't be squeezed, a trick learned from countless campaigns and handshakes.
"Well, Foreman, mighty glad to see you again. Sit down, sit down." He had heard Freeman's name incorrectly. The aide looked at Freeman in a moment of panic. Freeman ignored his misspoken name and the aide relaxed in gratitude.
"Well now, son, they tell me you're leaving us. Sure we can't get you to change your mind? The general, now, tells me he's mighty sorry to see you go. Says you do a good job over there by the river, and the general doesn't hand out praise very easy, you know."
"Yes sir, I know, and I'm flattered that he wants me to stay, but I'm afraid I have to leave. I've given it a great deal of thought and conscience calls. You see, I'll be going right back where I grew up to try and use my education to help kids who are like I used to be."
"Well, Foreman, that's a right fine attitude, but don't you think you might be making a bigger contribution for those very people here in Washington? Offer some inspiration for them to achieve, to emulate you? You know, local boy makes good."
"That's a point well taken, Mr. President, but I'd rather make my efforts in a more personal way: my small contribution to our Great Society.
"Well now, that's good! Very good! We could use you around here maybe speech writing. I like that : "a small contribution to our Great Society." The president turned to his aide. "Put that down, Smitty; want to see it in my next speech." He squinted and gazed out into space. "As your president it is my humble pleasure to be able to make a small contribution to our Great Society. Got that, Smitty ?
"We could use more of your idealism around here, Foreman. I certainly wish you the best of luck. I spoke to your mayor last week and they plan to do more in that area. I'm sure he could find room for you in his new commission."
"Well, Mr. President, I already have a position with a private social welfare agency, one I worked for before I came into the government."
"Well now, Foreman, never underestimate the good the government does for the people, even though I am a firm believer in private enterprise, as you know."
"Yes sir, I recognize the great contributions the government has made toward the lives of the individual citizen and that it is at the head of the war on poverty, but there have to be increased efforts by private agencies and individuals."
"You're right, of course. We can't have people dependent on the government to take care of all of their problems, now can we ?"
"No sir, and I always ask not what my. country can do for me, but what I can do for my country." The president's smile tightened, his aide hastily thrust a package into his hand. "Foreman, here's a little memento I'd like to let you have, a little token of my appreciation of your efforts in your country's behalf."
Freeman opened the package. It was a multipurpose pocket knife, identical to those handed to the East Bengali boy scouts. Freeman wondered if the congressmen's wives would find pocket knives useful. He smiled his thanks.
"You remember that time we were on that tour of the East and all that fuss they made because I gave a little old rebel yell in the Taj Mahal ? Well, you sure handled those Indian newsmen well. But I never could understand why they were so riled." "Well, sir, it's a tomb, you know. It was a little like someone being disrespectful in the Alamo." "Well, now that you put it that way. But I meant no disrespect."
"Yes sir, I told them that." His aide gave him a signal and the president rose and grasped Freeman's hand.
"Down my way, when you give a man a sharp instrument as a present, you have to give him something in return so that the friendship isn't broken."
"Yes sir. Here you are." Freeman slid off his tie clasp and gave it to the president. There was a cold silence, the president's grin frozen on his face. His aide hurried Freeman to the door.
Recovering, the president called out: "Now, if you get down my way, you stop in to see me, you hear?" The president frowned down at the tie clasp in his big hand. It was in the shape of a PT boat.
Freeman walked out into the bright spring sunshine, paused on the steps of the White House and looked out at the training on Pennsylvpnia Avenue. He walked to the side of the building to the parking lot. The chauffeur was leaning against the fender of the shiny black Cadillac, smoking a cigarette. He looked up at Freeman.
"Well, big-time, the Man say you can have the car as long as you want it today. Where you want weighed the fact that she did not agree with him against the fact that he was a bishop and she in deacon's orders. 'The author of "A View from a Pew" seemed to feel that the cathedral's ministry had fallen below an acceptable standard in fairly straightforward ways,' she said. 'It failed, in his opinion, to generate an atmosphere of prayer and seemed to celebrate the worldly and political aspects of the appointment.'
She'd compromised. She'd said what she thought but in a tone so gentle that she might have been agreeing with him. It was a measure of his quality that he listened to both words and tone. He paused. Then he replied almost curtly. 'If the author has serious points to make he should have signed it with his own name.'
Theodora was so much in agreement with this that she was caught off guard by his next move.
'I think it would help both the author himself as well as us to know who it is. I wondered if you would feel able to exercise your talent for detection in doing that. '
The bishop had reached his point.
'Me?' Theodora was astonished.
'With your journalistic experience,' the charming smile had returned. 'For the good of the church and of course the writer,' he concluded.
This was preposterous. This gentle, courteous man wanted to prevent free speech and cared more about the appearance of the church in the world than its spiritual reality.
'You will help us, will you not, Miss Braithwaite?'
They are none of them above image-making, Theodora thought with sudden disgust. Then she remembered his kindness in taking her under his wing and introducing her to the dean. She intuited, too, his real gentleness in human relationships. It was just possible that he saw a pastoral need to minister to anyone who could write as bitterly as the author of 'A View from a Pew'. 'Why not ask the editor who writes these pieces?' It was a forlorn hope she knew.
'That has of course been tried. He won't divulge his source. What we need is a discreet piece of private inquiry. For the church.'
He knew he had got her. She had eight generations of Anglican clerical ancestors who had all, more or less, done what their bishops had required of them. 'I suppose I could ask around a bit.' She knew she sounded ungracious.
The bishop absolved her of any lack of grace. He shone the light of his smile upon her. 'Splendid. I knew I could rely on you. Your father was such a dear man. Supper,' he said hungrily and led the way forward.
No further untoward incident occurred. At a quarter before midnight the party broke up. Many clergy would have early ashes services at which to officiate on the morrow, Ash Wednesday. The diocesan bishop and his wife had to catch an early plane to America. Theodora retrieved her coat from the dean's study and joined the guests trooping out into the bitter March night. In the middle of the close could be seen the mound of earth marking the archaeological activities of the earlier part of the day. Someone had draped the Janus in black polythene which the gusty wind had partially removed. It flapped suddenly and loudly as though indecorously, importunately demanding their attention. The cathedral clock chimed midnight had conned them all and in his own way had been the best of the spooks and they might never know it. For five years he had been the CIA nigger and his job had been to sit by the door.
8
Freeman was toward the head of the line waiting to board the Friday noon flight from New York to Chicago. He had closed his New York apartment; shipping those things to Chicago he chose to keep. He had listened to some jazz, attended several plays and shed his old cover as a snake sheds its skin. He boarded the plane and seated himself, moving with the grace and economy of a fit and well-trained athlete; gone the insecure shuffle, the protective, subservient smile, the ill-fitting clothes. The new Freeman, J. Pressed and Brooks button-downed, seated himself after placing his carry-on bag beneath the seat, fastened his seat belt and opened a little magazine with a psychedelic cover to an article urging the legalization of pot.
When aloft, he asked the stewardess if he might mix his own martini ; he had once ordered a martini on a flight and found that the label proudly claimed a mixture of an eleven-to-one ratio. Freeman, particularly the new Freeman, was a four-to-one man, on-the-rocks with a lemon twist. By the time he had finished his drink and eaten the plastic lunch, the plane was beginning its descent to O'Hare Field.
He moved forward to the terminal gate and caught sight of his new boss. Freeman accepted his outthrust hand.
"Dan, how are you? Welcome back to Chicago ; glad to have you aboard."
"Thanks, Mr. Stephens, it's good to be back."
"Dan," he smiled, with the air of a man bestowing a gift, "call me Steve." They moved down the wide tunnel with the crowd, toward the huge main hall of the terminal building.
"Right, Steve," Freeman smiled back.
"Did you receive that last batch of material I mailed you on the foundation and its work ?"
"Yes, I thought the brochure superb."
"The best PR firm in Chicago did that; my old advertising connections come in handy. That brochure will prove invaluable in fund raising. Never underestimate the power of advertising, I always say.
"Any baggage to claim, Dan ?"
"No, I only brought this bag with me. I shipped everything else ahead of me. It should be in my new apartment now."
"Fine. There's no need to stop in the office, we'll drop you at your apartment. Now, you take as long as you need settling in ; no pressing need to report into the office before late next week."
"I'll be in Monday morning, Steve." They walked through the automatic doors to the exit and stood waiting at the curb. Stephens motioned to a Ford station wagon across the way. Imprinted on its front doors was the legend: South Side Youth Foundation.
"I remembered you and Perkins worked together years ago when you both worked for the mayor's youth commission, so I had him drive me down to pick you up." The car slid to a smooth stop in front of them and when they were seated, Freeman leaned over to shake the hand of the slim Negro seated behind the wheel. "Perk, what's shakin', baby ?"
"Hey, Dan ; good to have you back." He moved the car smoothly into the outgoing training and south on Dan Ryan Parkway.
"Dan," said Stephens, withdrawing pipe and tobacco pouch, "you have your work cut out for you here, but since you're familiar with our operation, it should shorten the breaking-in period. Your first job in social welfare was with the foundation, wasn't it ? Before I took over."
"Yeah, we worked out of a storefront on Fifty-third Street."
"We've come a long way since those days," said Stephens, proudly. "Our new quarters are worth more than a million and we've had several sizable grants since I took charge."
"How's the contact with the street gangs ?"
"As well as could be expected, except for the Cobras."
"They won't let our street workers near them ; call us 'whitey's flunkies,' " said Perkins.
"We're hoping your appointment might change the image, Dan." Stephens toyed nervously with his unlit pipe. "A meaningful contact with the Cobras and we could get that Ford grant the next day."
"They• were on a real Afro kick for a while," said Perkins. "We thought they might have some connection with one of the Afro-nationalist groups, but it doesn't seem so."
"But we can't discount that possibility; it's one of the things we'd like to know about the Cobras, Dan," said Stephens. They had moved through the hole cut for the expressway through the bowels of the main post office and now they were moving south again on the Outer Drive.
"They're potentially the most dangerous gang in Chicago," said Perkins.
"The Cobras will be my personal responsibility ; I'd like their file to study over the weekend, Steve, if I may."
"Fine, Dan. I'll have Perkins deliver it to your apartment later this afternoon."
"Perk, I want to check out the Cobra turf tomorrow. Pick me up at my place at ten o'clock; Saturday night should be a good time."
"You know the Cobra turf, don't you, Dan ?" asked Perkins.
"I ought to, I grew up over there. I was the Cobra warlord when I was a kid. They used to call me Turk."
"I never knew that, Dan," said Stephens.
"Yeah, the gang goes on. Street gangs and churches are about the only durable social institutions in the ghetto."
"You've certainly come a long way since your days as a Cobra, Dan," Stephens said proudly, as if personally responsible. They stopped in front of the pretentious entrance of Freeman's apartment building.
"I'll see you first thing Monday morning, Steve. I'll see you right after that, Perk, and I want a meeting with the street workers at three in the afternoon."
"Good to see you taking charge this way, Dan. How about lunch on Monday?"
"Fine, Steve, see you then." Freeman felt things had gone well and he anticipated no trouble from Stephens. He entered the building, took the elevator to his floor, entered his new apartment and began perfecting his new cover.
In less than a month the apartment said everything he wanted about the new Freeman. Cantilevered bookshelves covered the wall of one end of the living room. He drank and served Chivas Regal, Jack Daniels black label, Beefeaters gin, Remy Martin, Carlsberg, Heineken's, La Batts and Ballantine. He had matching AR speakers in teak cabinets, Garrard changer with Shure cartridge and a Fisher solid-state amp with seventy-five watts power per channel. A Tandberg stereo tape recorder, a color television set, which could be played through the stereo system, and videotape completed the system.
There was a Bokhara prayer rug on the plastic parquet floor, wall-to-wall nylon carpeting in the bedroom and wall-to-wall terry cloth in the bathroom. His glasses were of crystal, his beer mugs pewter, his salad bowl Dansk and his women phony.
He slipped on his cover like a tailored suit, adjusting here, taking in there until it was perfect and every part of him, except a part of his mind which would not be touched, was in it and of it. He found that most people did most of the work as far as his cover was concerned : they wanted him to be the white-type, uptight Negro of "rising aspirations" and desperate upward mobility. He chose
his wardrobe with sober, expensive care ; opened a number of charge accounts and slid into barely comfortable debt.
He fell into step with others like himself, safe, tame, ambitious Negroes, marking time to a distant drummer, the beat hypnotic, un-syncopated, the smiles fixed on their faces, heads held high to pretend the treadmill did not exist and that their frantic motion was progress. More white than whites; devout believers in the American dream because fugitives from the American nightmare. The yawning chasm of ghetto misery at their Brooks Brothers backs, they trod its edge warily, their panic hidden behind bright smiles and the sharpened wiles to tell the Boss Man what he wanted to hear.
Freeman attended dull cocktail parties, becoming a bachelor to invite, a prized escort. He was Playboy-suave, witty, well-dressed and never drunk or disorderly. He could talk Time-Life Newsweek or Sunday New York Times-Manchester Guardian-New Statesman or little magazine-New York Review of Books. He could talk Antonioni, Truffaut, Polanski, Hitchcock, New Wave; football, basketball, track and boxing. He had a way of making a comment and making it sound as if the listener had said it. He could flirt with the women without angering their men ; make the fairies feel at ease and turn down their propositions without bruising their ego.
He ran into Joy in the bar of the Parkway ballroom and made a rendezvous for an afternoon the following week at his apartment. He told himself that he would not be there when she arrived, right up until the time she walked into his apartment,
bronze, lovely, bewigged and smelling of Arpége.
She walked gracefully around the apartment, inspecting it minutely, stroking, feeling, touching while he mixed their drinks. He handed her her drink and she smiled catlike at him, taking a sip.
"Rob Roy. Honey, you never forget anything, do you? It's wonderful to have you • back. The grapevine is saying all kinds of things about your new job."
"They wanted to slide me into the number two slot, prop up some white boy, but I wasn't having any. "
"Whatever happened to dedicated Dan? I knew a time when position wouldn't have been as important as making a contribution. You've changed, honey."
"No reason why I can't make a contribution and make some money at the same time."
"How much is the job worth?"
"I'll be able to live on it."
"I used to be in the game, too, remember? I could make a pretty good guess. The poverty program really inflated salaries; everybody's getting more money in social welfare nowadays."
"Yeah, everybody but the poor."
"Next year the president will be handing out Washington appointments to buy the black vote and with that much time in your new job, you'll be in a good position for one. More money and in Washington where the cost of living is lower." She leaned forward for a light, watching him shrewdly over her cigarette.
"Maybe I shouldn't have been so impatient; I always knew you'd change, you like nice things as much as I do. We're really two of a kind, Dan.
You're just more romantic about it than I am, and more idealistic. You always did get your nose open about slogans to save the world."
"Too late now, baby, you have a husband."
"After a fashion. His drinking is worse and he'd spend all his money on women if I let him. I don't care how many he has as long as he's cool with it and I have enough money to have whatever I want. His latest scene is a white girl on the North Side. And from Alabama, can you top that?"
"You had him followed?"
"Sure. I don't care what he does as long as I know it. We'll have to have you by sometime soon."
"No thanks. You know we never dug one another. "
"Why should you be bugged ? You know he never forgot we had a thing? He'd have a fit if he knew I was here." He listened to her chatter about her apartment, her hired help, her new car and wardrobe and thought she had come a long way from the girl from the Detroit slums he had known years ago in East Lansing. Her voice was polished, her mannerisms and gestures assured, she wore her expensive clothes with easy grace. She wore an expensive wig and he found himself wondering if she removed it when she made love. Later, he found that she did.
In order to confirm that the Cobras were the best organization for his plans, Freeman studied them carefully from a distance. He talked casually with anyone in the neighborhood willing to discuss the Cobras. He carefully built personal dossiers of each key gang member; in particular the gang leaders, Dean, Scott and Davis. He gradually worked out their chain of command and was pleased to find it efficient and effective. The rigid discipline of the gang impressed him more than anything else; discussion was permitted, even encouraged, but once a decision was made by one of the commanders, that decision stood. For most of the gang members, the Cobras provided the only family they had ever known; offering protection, affection, a sense of belonging, a refuge and haven from the unremitting hostility of the outside world.
Once certain of the Cobras, Freeman made his move. He casually walked into their poolroom headquarters not far from south State Street one cool evening. He walked to an empty table in the rear, selected a sixteen-ounce cue, rolled it on the table to test it for warp and began running balls. The owner of the poolroom walked the length of the room to Freeman's table.
"Look, mister," he said, "I think it might be better for you and for my place if you took one of the other tables."
"I like this one," said Freeman, without looking up from the table.
"This table is used by the leaders of the King Cobras." He waited expectantly. Freeman continued to run balls and did not answer. The man sighed and walked away shaking his head.
Freeman hadn't had a pool cue in his hands in years. He had spent much of his adolescence in poolrooms and it came back quickly, the easy stroke, wrist relaxed, the follow-through, English to leave the ball ready for the next shot. When he had run the balls, he tapped the butt of the cue on the floor and the owner walked to the table and racked the balls. He broke the varied colored pyramid, studied the table and began shooting bank.
He noticed the sudden silence in the room while studying a difficult shot. Without looking up, he shot a bank shot twice the length of the table. Before the ball could drop into the left-hand corner pocket, a black hand intercepted it, held it a moment until Freeman looked up, then carefully replaced it on the green table.
Freeman looked at them across the length of the table, three Cobras staring at him silently. He returned the stare. The room was silent, no click and clatter of ivory ball against ivory ball, the chatter and banter of the players gone from the smoky room. They stood silent and dangerous, just beyond the light suspended above the table. Freeman chalked his cue, still staring at them, then lined up his next shot.
Freeman knew them, Do-Daddy Dean, the gang leader, Sugar Hips Scott, secretary-treasurer, and tall and deadly Stud Davis, the warlord and at nineteen oldest of the three.
"This our table." The voice came from the shadows, low, soft and dangerous.
"I want to talk with you."
"No talk, man, move out. No talk for social workers, man."
"You know who I am ?"
"Yeah, we know who you are."
"Let's step outside and talk. It won't take long, just a few minutes."
"You don't want to talk, man; you just want to go home. Better that way. I don't ask no more, done asked too much already."
Freeman motioned to a door in the rear which opened onto an alleyway. "I'll wait for you out there." He walked out into the darkness and waited, facing the door.
They came out silently, spread out to arm's length and without preliminaries, moved in.
Freeman moved swiftly, crabwise in a circle to his left, bringing himself closer to the man on his left and away from the other two. He stabbed toward his eyes with stiffened fingers and when he covered, Freeman gave him a stiff-fingered jab, elbow and wrist locked, his arm in close to his body, to his opponent's solar plexus. When he doubled, Freeman pushed him into the legs of the one in the middle and in one motion moved to meet Stud Davis. Davis feinted a left and crossed a right, catching Freeman high on the head. Freeman countered with a left and right, which Davis slipped, then he kicked Davis hard in the ankle. One of the others was rising, the one he had not hit and Freeman whirled and gave him a judo chop just below the left ear, dropping him. He turned to Davis and threw him with a foot sweep to his good ankle. Freeman moved to the door of the poolroom, shut it, returned and sat on an upended Coke-bottle crate. When they stirred, he had them covered with a snub-nosed Smith & Wesson .38 revolver.
"Don't move. Just sit there and listen."
"Next time we see you, social worker, we have that, too."
"You ain't goin' to shoot nobody, small-time, and you sure as hell ain't goin' to shoot me. Now you shut up or I'll pistol-whip all the black off your ass.
"You think you bad, the King-ass Cobras. Nobody messes with you. You make Molotov cocktails and burn and loot supermarkets. Yeah, I
know what you were doing on the West Side in July when they had the riots over there. Three supermarkets, a pawnshop, two furniture stores and a television store. That color TV in the poolroom came from there."
They stared at him in sullen silence.
"The big-time Cobras. Lifting cameras and small-time shit from pawnshops, sniping at cops with pistols and .22 rifles from rooftops. You know how much chance you have of hitting anything with weapons like that from that range at night? What the hell you trying to prove?" He looked at Stud Davis. "The fuzz wasted three snipers and wounded eight others and the only casualties they had was from getting hit by bricks and bottles. You're damn lucky they didn't hit any of the Cobras.
"OK, you want to hurt whitey, you want to mess with Mr. Charlie, then stop playing a bunch of punk games. You didn't do any more damage than a mosquito on an elephant's ass !" Freeman paused, he had their attention now. "You really want to fuck with whitey, I'll show you how !"
9
"Gentlemen," Stephens said in a board meeting in his best advertising-executive manner, "I have splendid news." He paused dramatically, sweeping the room with his smiling gaze, allowing the suspense to build. "Dan has made contact with the Cobras."
There was a sudden silence. Stephens, puzzled at this reaction, looked uneasily around the room. He patted his pocket handkerchief, fingered his pipe and cleared his throat. Freeman sat quietly ; he was not puzzled.
"Dan, this contact with the Cobras," Stephens had said just prior to the meeting, "will really sell the board members on you. Not," he added hastily, "that they aren't already. But no one expected a contact with the Cobras so soon after you joined us ; this is really good news."
Freeman watched their ambivalence fill the room like a white velvet fog. White liberals to a man, with the single exception of Burkhardt, the hardheaded business member. Among them, Roger Thompson, professor of sociology at the University of Chicago, a professional white liberal who had devoted a career to proving that the inequities of Negroes were social and cultural, rather than racial; Stephens, an amateur white liberal turned pro. They can forgive a nigger almost anything other than competence, thought Freeman behind his mask. They want their choice to have been an act of charity for a Negro not quite up to the job; they want me to fumble, stumble, turn to them for help. They would like the Washington Freeman, he was a good boy. Part of them wants me to vindicate their choice of a spade for the position, but another part wants me to prove once again that it is spade incompetence, not white racism, which is responsible for the scene. They'll use me, but they'll never like me.
"That is good news, Steve," said Professor Thompson. "But are you certain this is a positive contact?"
Stephens, sensing difficulties, threw the ball to Freeman. "Dan?"
Freeman regarded a point midway across the polished mahogany conference table, frowning in thought; then he leaned forward sincerely and looked deeply into Professor Thompson's contact lenses.
"Of course, Professor, it is too early to be positive; I advocate caution at this point and I'm moving slowly. But, I made the first contact with the gang leaders several weeks ago in their poolroom headquarters; I've played pool with them almost nightly and they've visited my apartment several times since."
"Visited your apartment? Well, that is a real foot in the door. What are your immediate plans ?" "Standard procedure for a while; gain their trust and confidence, offer them an adult male to trust and admire, since they are almost all from broken homes and have none otherwise." He motioned to Stephen's colored secretary, who moved around the table gracefully, placing a folder in front of each member.
"You'll recognize that concept as your own, Professor. I've taken the liberty of including copies of the article you did concerning it for Atlantic a few years back, in these folders. In addition, there are biographic character sketches of the three gang leaders, Dean, Davis and Scott, plus shorter ones of several other key gang members. There is the table of organization of the gang, including the chain of command and a brief history of the Cobras as a ghetto social institution. My own plans for the gang, subject, of course, to change, are also included."
He watched them look through the folder from behind his Grecian mask, a black Prometheus among the gods, who had stolen the secret of fire from Olympus by the Potomac and was teaching its use to his people. Not the fire next time, he thought, but the fire right now. How long before they chained him, to let the black and white vultures tear at his liver?
Professor Thompson looked up. "I notice you plan a sports program for the Cobras. Hasn't that approach been pretty much discredited in reaching street-corner society ?"
"In general, yes, Professor; however, I think it might tend to work with the Cobras. As you know, they emphasize sports far more than usual for a street gang. Almost every member is a pretty good athlete and they still call themselves the Cobras ASC, Athletic and Social Club. And the gang leaders were all star athletes until they dropped out of high school.
"I thought I might start with a program which would appeal to both their combativeness and athletic ability." He paused. "I'm seriously considering organizing a judo club."
"Judo?" exclaimed Burkhardt. "Isn't that just asking for trouble? Think of what they could do with that knowledge in gang fights."
"On the contrary, Mr. Burkhardt; the judoka has a strict discipline, intrinsic in the training, that he will avoid a fight whenever possible ; and when offered no choice, restrain himself." Then, striking at Burkhardt's soft spot, a fanaticism concerning guns and marksmanship, he continued :
"It's almost identical to the principle employed in weapons training; by teaching gun discipline and a respect for the weapon and its potential, you tend to reduce, rather than increase, the possibility of its indiscriminate use."
"Yes, I see what you mean, Freeman. Certainly I've seen the positive effect of weapons training on antisocial types when I was in the army. It might work, at that."
"I actually got the idea from your article in American Rifleman," said Freeman.
"It was in the August issue," said Burkhardt, surprised, pleased and flattered. "You read it?"
"I've made it a point to read everything both you and Professor Thompson have written ; it gives me an insight in your approach to the foundation's work." Burkhardt relaxed and smiled at Freeman for the first time in his memory. Got your nose open now, you little motherfucker, and it would really be beautiful if I could talk you into coaching a rifle team made up of the Cobras ; but that can wait.
They discussed the Cobras for some time and then adjourned the meeting. Later, over -luncheon, Stephens expressed his pleasure.
"A masterful presentation, Dan; you'd do well in advertising if you ever left the social welfare game. And, I couldn't be more pleased in your finally breaking through with Burkhardt. I think we've earned another round of martinis." He motioned to the waitress.
In the early stages of his training and organization, Freeman often used the street for his classes. He would stand casually on the street corner, "hanging out," and pass along his CIA-bred knowledge.
"You already said, Turk, it's firepower, not marksmanship in a fire fight ; so how come all this jazz about range and wind and that shit asked Scott one afternoon.
"Because everybody has to double up as snipers. Even the cats who can't shoot will have nuisance value. Nobody likes to walk around in the dark and not know when somebody is going to shoot at him."
"Anyway, like Turk say, ain't nothing to figurin' range in a city. The length of a block, distance between lamppoles and telephone poles is all standard," said Dean.
"In an attack you'll be on battle sights, three hundred yards. It will handle anything you're likely to be shooting at; but for sniper work you try to figure the range to the inch." He paused, to make sure he had their attention.
"If you were working as a sniper from that corner building and we were the target, how would you figure the range?" He let them puzzle it a minute, then asked their resident mathematician :
"Sugar"
"I guess I'd estimate by sight from up there." "NG; no good. Range estimation at night is always tricky, and with artificial light, too." He shook his head. "Hell, man, it's simple geometry. You know the height of the building."
"Yeah," said Scott, "yeah, and I know the distance from the base of the building to here."
"So, what's the distance of the third side of the triangle Smiling broadly, Scott gave the range to the inch.
"That's all right for him, Turk, but what about the rest of us ?" Stud Davis asked.
"Nothing to it, Stud. Hips can run it down for you in ten minutes."
"Man, I never could do anything with math in school."
"That's because you never had educational motivation before. Takeover, Sugar, and don't let 'em go until they know how to do it ; then you each pass it along to the cells. Easy." He walked away.
Freeman continued to use the streets for his training, but as the weather grew cooler, he would use the poolroom after closing, or his own apartment. His close relationship with the gang leaders was completely in keeping with his cover and only added to his prestige as an active, energetic and imaginative social worker. He gradually gained their confidence and growing affection and the masks they wore for the outside world were loosened for Freeman ; their personality traits becoming increasingly clear.
They had been gifted athletes and had delayed dropping out of high school only because of their love of sports. Dean, the smallest one of them, had not played football but had been that rarest of American Negro athletes, a distance runner. Few Negroes in the United States have had any taste for duplicating in a distance run the exact kind of pain and endurance they face in their daily lives. But Dean ran as he lived, methodically, impassively, the intensity he hid so well flashing only at the finish of a race, when, head back, teeth bared, he would sprint the last three or four hundred yards, punishing his black wiry body, regardless of how far he might be in front or how hopeless his pursuit of a leading opponent. But he seldom lost, except in open meets against older, more experienced men. "Ain't but one place in a race," he would say. "First place. Second and third don't count." He had a sharp, methodical mind and was temperamentally opposed to making hasty decisions. "Lemme think about it," he would say when a problem arose, and sometimes days later, when others had forgotten, he would present his opinion and it would usually be accepted without question.
One night in the poolroom after it had closed, Freeman was discussing the table of organization of the Cobra underground.
"Turk," said Dean, "that the same we always been."
"Right," said Freeman. "The only change is here, where you had a duplication." He pointed to the blackboard. "The organization was already together and there's no need to change the chain of command or the T. & O. Ordinarily, it would take three to five years to organize an underground organization of this type from scratch, but the Cobras have always been an underground." "How do you mean ?" asked Dean.
"What do you think?" countered Freeman. "We discussed the characteristics of an underground revolutionary movement last week."
"Well, secrecy. Not even all the gang members know how big the Cobras is, and the fuzz sure as hell don't. We got organization and . . . what's that word you used ?"
"Motivation ?"
"Yeah, motivation. And, we got discipline and we got balls."
"Right! And there are at least five or six other gangs who fit that description, not as well as the Cobras, but well enough for recruitment once our training is complete. Got any idea who they might
"Let's see; the Comanches, the Apaches, the Blood Brothers on the Wes' Side," said Stud Davis. "The Crusaders and the Tigers," said Dean.
"The Tigers !" said Davis. "Man, they don't do nothing but give parties ; they even got chicks in the gang, the Tigerettes."
"You ever try to crash one of them parties ?" asked Dean. "Man, the chicks as bad as they
"Right," said Freeman. "And we can use women; they can often go places and do things men can't do. There are a couple of other possibilities, but those gangs top the list. How many do you figure if we recruit them all, including the Cobras?"
"More than five hundred," said Scott.
"Yeah, and what do you think five hundred well-trained revolutionaries can do to this town?" "Turn it inside out," said Davis. They looked at one another, impressed.
"Shit, Turk," said Dean, "you don't play, do
"I sure as hell don't. And not just Chicago, but every city with a ghetto, which is every major city in the north. The members of the recruitment and training cell move out in January, their cities are already selected as well as the first gangs they contact."
"You're gon' turn this country upside down," said Dean.
"Wrong. We're going to turn this country upside down."
"You really dig the Cobras, don't you, Daddy?" Freeman asked him one day as they stood in front of the poolroom late one lazy warm afternoon : the people were moving slowly through the streets, standing in clusters talking softly, occasionally waving to a friend, sometimes detaching themselves from one group to join another—the ghetto sounds, smells and colors, white teeth in dark faces below the sunglasses which hid them from the cold, hard world, protection for people born to the sun but forced to live in the sunless garbage heap of a sad, sunless, sick society.
"Yeah, the Cobras the best. Everybody know that. You know, one of the cats moved to New York and they know our rep way out there. Cats start gettin' bad 'cause he knew on the block—you know, to find out if he got any balls. 'Where you from, nigger? Think you bad ?' That kind of shit.
And he told 'em he from Chicago and he a Cobra and, man, he owned the block after that. He the leader of a gang there now."
"Groovy," said Freeman, always alert to potential recruits. "If they check out, maybe we can use them for our New York scene. Harlem or Bedford-Stuyvesant ?"
"Harlem."
"Go on, how'd you become a Cobra ?"
"I never thought I'd be one. You know, I was really little and skinny then and I wasn't worth a shit in sports. I got better all of a sudden, but I was never as good as Stud or Hips or Pretty Willie; them cats can do anything. I was good with my hands, though." He smiled. "You know if you can't fight out here in the streets, you sure better be able to run and I never been too fast." He did not have great straightaway speed, but he did possess the quality coaches call quickness, great speed of hands and the ability to change direction instantly. It had made Dean an excellent play-making basketball guard in spite of his lack of height.
"I was only 'bout ten or eleven and the Cobras used to like to watch me fight. Sometimes they set up a fight with somebody. I had good moves even then. Seem like I always knew how to hook. You know that a tough thing for a little kid to do; the other punches easy, but a real good hook ain't easy for a little kid, but I had it. They used to talk 'bout my left hand and make bets and stuff. Next thing I know they make me the mascot and then they start using me to wiggle through windows and things when they hit a store or something and then I get to be a Cobra." He said it as
if it were the most important thing that had ever happened to him.
"You ever think there might not be any Cobras one day?"
Dean looked surprised, then a bit shocked. "What you mean?"
"I mean we could all get wasted, every one of us." Freeman's gaze swept the street and he motioned to the people there. "I mean all of us. Nobody knows what whitey might do when the deal goes down."
"I never thought about that." Dean looked at the people in the street with new interest.
"Does it scare you ?" asked Freeman.
"I don't know. Does it scare you ?" He looked at Freeman intently.
"Hell, yeah. But that ain't nothing new. I been scared all my life. There ain't a nigger living who doesn't know fear ; we live in it all our lives, like a fish in water. We just have to learn how to use it." Dean took out a pack of cigarettes and offered Freeman one. He waved his refusal and watched two men across the street deep in an argument. A crowd was gathering, waiting for the explosion.
Freeman pointed to them : "They're scared ; that's why they might kill one another in a minute. Watch 'em."
They were serious now, not loud-talking anymore, their voices low and deadly and already they had dropped that right foot back and stood aslant one another in a position for instant combat. They kept their hands ready for attack or defense, each awaiting some move from the other. Damn near the deadliest people in the world, thought Freeman, but killing each other all these years instead of the people who put all that fear and anger inside them. Finally, one said something and the other laughed and they both stood there, their heads flung back and threw their dark laughter to the sky. The crowd joined in and others threw in jokes and asides, the smiles broad in their faces, the tension suddenly gone and giving the laughter its force and power. The two men strode into a bar and the crowd moved on to other diversions.
"They got a blues about that—'laughing to keep from cryin',' it says. And that about says it. A man could get to be a philosopher listening to the blues. If we ever forget how to laugh we're finished."
"You really dig spades, don't you, Turk?"
"Yeah. I feel about spades the way you feel about the Cobras ; they're my family. I got a family of twenty-five million people." He smiled. "Ain't no way I can ever be lonely."
But he was lonely, his cover, his plans had forced him into himself and his loneliness ate at him like a cancer. Always the iron control, even when drunk, the cover everything, himself nothing, afraid even when he cried out in orgasm that he might give something away.
Only with the Cobras could he be open, but he knew he used them as well. Sometimes he looked at their faces, trying to etch their features into his memory, knowing that many, if not all of them, would not survive; knowing also, that they had no future otherwise: the endless waiting in the lines, unemployment lines with sullen blank, black faces at their end, the white man boss in his office, memoranda on his desk to get tougher, cut down the unemployment rolls, force them out into the
streets and into the other lines leading to an employment office where there were jobs only for white faces. Junk, jail or junk and jail and more junk available for those with the price in the county jail; the guards the safest pushers in the city—"they can't bust you in the slam"—the junkies' joke between heroin nods. An aborted marriage to a favorite fuck and another black baby with no hope and no future in the land of milk and honey, replete with goodies in white plastic cases for God's own children with white plastic faces.
They moved toward the poolroom, walking slowly through the warm streets.
"That laughter, baby, is the best insurance whitey has. Once we stop laughing and singing, stop killing each other and start killing him— forget it !"
He was amazed at how quickly they learned. He pushed them ruthlessly, relentlessly and each time they reached the saturation point, they swelled to absorb more. He had figured on eighteen months to two years, but gradually he felt sure they could begin the next summer. He did not allow himself to think of it after so many years of lonely planning; that perhaps now it was only a matter of months.
They reached the poolroom and walked inside, past the never silent television set. He glanced at the set and watched Ernie Banks wiggle his spikes solidly into the dirt of the batter's box and set his long, lithe body for the pitch.
They walked to the table in the rear and began to shoot.
"Turk, lot of the cats I-A ; what happen wher they get called ? They go underground ?"
"No. Shit, the army is our postgraduate course, the best training ground we have; they can teach us the skills we need in a fraction of the time I can. When they get a call, have them check with you; then we go over what we need and they go and volunteer for the next call. That way they can pick their training and not get stuck in some unit that's a waste for us. Armor is a waste, unless it's armored infantry. You know what we need : skill with small arms, demolitions, small unit tactics . He paused and smiled. "There's a certain poetry in whitey training us to mess with him. Every black cat in Vietnam is a potential asset for our thing. Whitey knew what he was doing when he wouldn't let us fight in the Second World War." He paused to line up a shot, stroked and missed. He watched Dean as he chalked his cue.
"You know, American white folks got more nerve than anybody; they call them gooks and us niggers, out there in Vietnam and in Korea when I was there. And they don't see any reason why the gooks and niggers shouldn't kill one another for whitey's benefit. The hang-up is that's exactly what's happening. You can't criticize success, now, can you ?"
"They pretty smart, ain't they ?" smiled Dean.
"Don't know whether they're smart or we're stupid. Maybe a little bit of both."
The discipline, always good, tightened and the pride increased, no longer flamboyant, but quieter, cooler. The Cobras walked tall and ready, afraid of nothing. They even lost interest in pot.
"Tell the cats to turn on every now and then ; they get like suburban boy scouts and everybody will get suspicious," he told Dean. Then he smiled,
remembering pot had become as popular in the suburbs as swallowing goldfish or college panty raids had once been.
They drilled endlessly in weapons training, unarmed combat, demolition theory. He taught them the theory of resistance to torture.
"Most people can't stand up under it and there's no shame if you can't; but if you give in too quick, they won't believe you and they'll keep it up. You have to hold out long enough to convince them the info you give out is for real; otherwise, the scene goes on and you haven't gained a thing.
"What you have to remember is the cats who torture enjoy it ; they don't want you to quit too soon, so you have to hold on long enough for them to get their kicks, then tell them what they want to hear. Just whitey all over again. You know when he's putting you down, he don't dig it if you take low too soon."
They gradually built up a network of safe houses, arms caches and message drops. Only Freeman knew the entire network; the rest were divided between Dean, Scott and Davis. He tried to figure a way for training with handguns, then dismissed the idea as one which would attract too much attention. Besides, the kind of fighting he had in mind would find that kind of weapon almost useless.
Professor Thompson, who was writing a book on the activities of the foundation, decided to add a chapter concerning Freeman's work in contacting and defusing the Cobras. Freeman was becoming known as a bright young man in social work. He was named as an eligible bachelor in Ebony's annual listing, pictured leaning suavely against his Lotus. Since he was careful to direct credit toward Stephens, remaining modest and unassuming, refusing to flirt with the white women at the integrated Hyde Park cocktail parties, arriving early and leaving early, before the potent cocktails might provoke an unpleasant scene, always appearing with women dark enough not to be mistaken for white, he was pointed out as an example of a Negro who was making it in American society.
No one could imagine that Freeman, tame, smug and self-satisfied, would ever rock the boat; much less suspect that he planned to sink it.
10
The first time Freeman hinted what he planned for them, Stud picked it up immediately. "Man, you know the first time shit like that start happenin' they going come lookin' for the Cobras." "No they won't," replied Freeman.
"No? How come? We about the only gang over here with that kinda balls, and the fuzz know it." "They won't come looking for you because junkies don't do what we're going to do and starting very soon, the King Cobras is going to be the most turned-on gang in Chicago."
The street-gang wars in New York in the late fifties, which threatened the entirety of the city and which were increasingly difficult for the police to control, ended primarily for one reason: junk. Junkies don't fight; except in desperation for a fix. The word spread among the police that the King Cobras were toying with "horse" ; with some relief the police waited for the addiction to take hold. It would mean an increase in petty theft, but as long as it did not spill outside of the ghetto and it was nigger stealing from nigger, it meant no extra work.
The symptoms were there : upon being searched, there was an increasing amount of gang members with hypo scars on the forearm, the pupils dilated by drugs. Police were informed when receiving their cut from dope pushers that the Cobras were becoming excellent customers. The scars were from unsterilized hypodermic needles, the dilated pupils from pot. The heroin was refined and resold in New York at a slight loss, or a profit, depending on whether the narcotics men had busted an incoming shipment from abroad. The Cobras overnight became a no bopping gang ; still dangerous when their turf was invaded, but no longer aggressive. Police duty in their area became almost tame, consisting of the collection of graft, confinement of crime and violence to the area and the head whippings which are routine for any cop working in the ghetto.
Freeman's contact with the gang had been duly noticed and he was personally commended for his positive influence at a meeting of the board of directors of the foundation. The pacifying influence of heroin was not mentioned. Freeman did not assign a street worker to the Cobras, but worked with them himself. He often wondered at the reaction of the Board had they known what his program of rehabilitation actually entailed.
The gang organization was perfect for his purpose. The Cobras had already been an underground organization, visible only in their moments of violence, closed off to the outside world otherwise. Freeman built on the original organizational structure, following the same chain of command, utilizing the already rigid discipline and introducing refinements borrowed from the French Maquis, the Israeli, Cypriot and Yugoslav underground movements. The training proceeded more rapidly than he had anticipated, since for the first time in most of their lives the members of the King Cobras had educational motivation.
Stud remained Freeman's and the Cobra's field commander and Dean revealed why he had become the gang leader. He had a tendency to shuck the world, to pretend mental stagnation while in actuality he possessed a methodical, analytical mind, much more cautious and steady than that of the mercurial Davis. Freeman had to fight against hero worship since he was the only outsider who had ever demonstrated a genuine interest in them as human beings.
"Look, goddamnit, there ain't no head niggers in this outfit ! The whole scene is designed to have the man below move up at least three steps whenever necessary. If I go, Do-Daddy takes over and right on down the line. Every nigger revolt in the history of America has ended when they wasted the head nigger. That won't happen with us.
"I am the Man and you do what I say, but when they get to me, don't look back, dig? No tears, no flowers, just keep fucking with whitey."
He spent a great deal of time with his three lieutenants in his apartment, and when they relaxed in the new surroundings they asked exhaustive questions about each of his paintings, carvings and sculptures. One day he pointed out quietly that the majority of his collection had been created by men who were not white. They devoured whatever he told them about the history of their people in America. Somehow, they had attained a pride in themselves as human beings based on their contempt and hatred of the white man and a stubborn pride in their toughness and defiance. Now he told them that theirs was a history and tradition of which they could be proud.
"Man, we took everything they laid on us— slavery, lynchings, segregation, the worst damn jobs, the worst damn food—and come up swinging. We been on our ass, but never on our knees and that's what the white man don't understand. He thinks it's the same thing."
He told them about their history, where in Africa their ancestors had lived, the cultural heritage the white man had destroyed, forbidding them their language, their music, their sculpture, carving and painting, their dancing and giving them nothing in return.
"No, not even their jive white man's religion. Less than one-sixth of the freed men upon emancipation had been allowed to become Christians and they justified slavery on the basis of converting the heathen."
He told them of their heroes and heroines, of the countless slave revolts, Nat Turner, Denmark Vesey, Sojourner Truth, countless other names they had not been taught in the propaganda that passes for American history. He remembered his own introduction to the history of the Negro in America by a proud black teacher of music, who in defiance of the rules devoted time to teach them some pride in themselves and their people; who taught them to distrust the image the white man thrust at them in the movies, on the radio and in their history texts. He remembered how he had to descend to the bowels of the library in East Lansing, passing the books written by the "name" American historians in his search for the
truth about his people. Now he passed it along. Each cell meeting devoted part of its time to the history of the American Negro.
"We got no time for hate; it's a luxury we can't afford. Whitey just stands in the way of freedom and he has to be moved any damn way we can; he doesn't mean any more than that."
All the time he spoke to them of their mission he was aware of the inconsistencies in his arguments, the humor and horror in what they intended, the hell they would create for their people in the struggle for their freedom. He wondered how long this would last, how long before he began believing each word as if the word of God ? Would it be when he finally had the power of life and death, when the resources of the city and perhaps the nation were devoted to finding and destroying him ? How long before he saw himself as a black Messiah ?
Freeman knew from the personnel records at CIA that an agent became burned-out when he no longer had an identity as a distinct personality; after the erosion of the years of cover, constantly becoming someone different, finally wore away what he actually was, until he no longer knew what he was. Freeman had been playing roles for whites and finally for everyone. How long before the edges of his cover and those of his personality would blur, merge, and he could no longer tell where one began and the other ended ?
That was why it was an advantage to be black. There were millions of peoples and races in Europe whose centuries of subservience made them culturally perfect as raw material for spying. The nigger was the only natural agent in the United States, the only person whose life might depend, from childhood, on becoming what whites demanded, yet somehow remaining what he was as an individual human being, Freeman had had years of practice at the game before adopting the cover he had assumed at the CIA and the present cover as a playboy of the mid-western world. That might save him, but it was important nevertheless that he establish an organization which would survive him. Once they discovered him, he would disappear. There would be no martyr-making trials and no more public assassinations as with Malcolm X. He would just disappear and the white man, confident in the eternal passivity and stupidity of the Negro, except for rare individual exceptions, would assume that the organization would die with Freeman. It must not be so.
He read them their poets and their literature. He played their music. They knew the doo-waps, rock and roll, rhythm and blues, but jazz was something played for white folks in clubs in the Loop or in concert halls. He wondered how some of the jazz musicians would feel if they knew how far away many of their people felt they were. He remembered the days when Negroes in Chicago were allowed to work, before the stockyards moved west to the right-to-work-law states and before the steel mills and other plants automated, before the railroads cut back and laid off. There was jazz on the South Side, played by Negroes for Negroes. Of all those clubs, only the Drexel Hotel lounge survived. He used expense-account money to take them there to hear Miles Davis. Miles had always played the South Side, in the Pershing Ballroom and the smoky bars on Sixty-third before they went rock and roll, when he was blowing so much stuff the whites couldn't understand it until one day, after years of neglect, they rediscovered him and he asked what was all the noise, he always played like that.
They listened to Miles's records and to Lady Day, Pres, Monk, Diz and the rest, and began hearing things in jazz that they had heard in rhythm and blues. They listened to down-home blues. He became the father and big brother they had never had and, although he fought it, he returned their affection and love. He knew this could be a fatal weakness, but he could not help it. He watched them grow now that they had a purpose, and found it sad that there was nothing for them in this land where their ancestors had lived for hundreds of years, except destruction ; but in that negative urge to strike out against oppression they had found something that freed them from their fears and the doubts about themselves and their color. He watched them grow and become more human and knew that their new-found humanity would not inhibit their becoming the killers he was training them to be.
He taught them all the tricks of the trade and knew that if he had organized a network as quickly, silently and efficiently in the field, he would have been commended. They had not been tested, but he had no doubt about their performance once they were. Already they were impatient to begin. He told them they would begin operating during the coming summer under cover of the inevitable rioting. He knew as surely as the sun rises that the cops who occupied the ghetto would not give up their pastime of whipping niggers' heads in spite of directions from above concerning less physical means of arrest and investigation in the ghetto. They were not convinced, those thickheaded cops, that niggers would tolerate no more head whippings. The cops felt sure that the answer to the riots and the increasing lack of respect for the uniform was more blood, more stitches, more battered kidneys. Leopards do not change their spots and ghetto cops do not stop whipping heads. There would be riots during the coming summer and his network would be there experiencing their baptism of fire.
"The whole world will be watching us and if we can survive even a year they'll look like fools. And nothing bugs the American white man more than looking like a fool."
"Well, you know I don't mind making whitey look like a fool," Dean said. "And you know I'm with you all the way. But do you really think we got a chance of winning?"
"Who said anything about winning? We don't have to win ; what we have to do is get down to the nitty-gritty and force whitey to choose between the two things he seems to dig more than anything else: fucking with us and playing Big Daddy to the world."
"Makes sense, Turk. But how we going to force him to choose ?" asked Scott.
"By tying up so many of his troops here in the States and stretching him so thin economically that he discovers he can't do both at the same time. And don't ever think he's not so hooked on messing with us that he won't do anything to keep the scene the way it's always been: us on our ass and him digging it. He could go all the way : barbed wire, concentration camps, gas ovens; a 'final solution' of the Negro problem." He paused to listen to the stereo; Billie Holiday was singing "God Bless the Child."
'Pappa may have and Mamma may have, but God bless the child that's got his own.' Dig that? We got to get our own ! But before we can, we got to get that black nigger pride working for us. We put it away like sharecroppers put away their best most precious things.
"You know how the people are who come up from the South. Down inside that big trunk with the big lock, wrapped up inside tissue paper and cotton batting, and maybe nowadays in foam rubber and saran wrap, they got at least one thing they brought along from down South, one thing they couldn't leave behind, that they cherish above all else. It's too precious to even take out every now and then to look at and feel, but they know it's there.
"Well, that nigger pride is like that. We wrapped it up and put it away inside that trunk, inside a cedar chest, wrapped up in paper and cotton, mothballs, with a big lock on the outside, waiting for a day when we'd need it. We got to get inside that trunk and get it out or find out if it's empty. White folks been trying to rob that trunk for centuries and maybe they did and we didn't know it, or it shriveled up and disappeared out of disuse. But if we can find it and use it, if we can get that black pride going for us, then nothing is going to stop us until we're free— nothing and nobody. They'll have to kill us or free us, twenty-five millions of us !
"It's gone on long enough. Now's the time to say: no more, baby; it stops right here. They got very short memories about what they been doing to us since slavery and if you don't believe we've had it made from the first time they brought us over, crammed into those boats, ask whitey.
"Remember all those plantation movies? You can see them every night on television nowadays." Stud Davis straightened, tilted an imaginary planter's hat to the correct angle on his head, puffed on a long imaginary cheroot.
"Boy, I say their, boy !" Davis said, waving expansively toward the seated Cobras, speaking in a parody of the southern cultured drawl of the Hollywood bourbon aristocrat. Scott, the comedian of the three, took his cue and, leaping to his feet, approached Stud in a sidewise shuffle.
"Now, who dat, say who dat, when I say who dat? Why, lawdy, if it ain't Mr. Beauregard, come home from de wah."
"Bless me, it's George, my faithful old slave retainer. But, George," said Beauregard-Davis, choking with emotion, "you're free now. You're not a slave, anymore."
"Free ? Dat bad, massa? I gwine need a doctor ?" "No, George, it means you're not my slave anymore; you can leave the plantation, go anywhere you wish."
"But I don't want to leave, Mr. Beauregard."
"But I can't pay you, George ; the carpetbaggers have ruined me."
"Well, now, don't worry none 'bout paying me. Why, I wouldn't rightly know what to do with money. I jes' want to stay right here with you, Mr. Beauregard, hee, hee." He turned and shuffled off. The others were weak with laughter.
"You don't think it wasn't like that, read your history books. But somewhere we scared them. Think about it : everyone they tried to work those fields in the hot sun dropped like flies and then they started bringing us over from Africa. It didn't matter how many of us died as long as we
lasted through a crop, because they figured on an endless supply.
"We down on that big plantation when they first start bringing the slaves over, the one in all the movies with the happy niggers who never want to be free. They bring in those niggers from Charleston or Savannah—first nigger slaves to be worked in that area. Cat owns the plantation is progressive, see, so he's conducting this southern scientific agricultural experiment."
They were all standing now, as if on the street corner talking trash, the commedia dell'arte of the ghetto.
"Yeah, man," said Stud, "MGM in technicolor. Big-ass house, look like the White House, big porch on the front, big white pillars."
"Yeah," said Scott. "Barefoot darky sittin' on the steps plunkin' a banjo."
"And singin' a spiritual," said Dean. "Never no blues in the movies; white folks scared of that, but a spiritual talkin' 'bout how good it going to be when they dead. They got a big magnolia tree in the yard."
"And big-ass double doors 'bout twenty feet high ; and inside, in the hall, a big chandelier and a winding staircase."
"Naw, man, a big curved double staircase."
"And downstairs," said Freeman, "Massa
Charlie waiting for his wife, dressed in white."
"Smoking a cigar this long and this skinny. Black string tie and a long coat with a center vent up to here."
"Yeah, cat with the side vents and long sideburns is the bad guy. And maybe he got on a fancy vest, yellow silk."
"Then the chick standin' up there on top of the stairs."
"She a blonde or maybe a redhead; the chick with black hair is the bitch who digs the cat and gets somebody to waste him, but she changes her mind and jumps in front of the good guy and gets burned instead."
"Yeah, the good chick ain't never got black hair. She got one of them little umbrellas with the rumes."
"They call 'em parasols."
"Yeah, little-ass parasol and a floppy hat with a big ribbon down to her ass."
"Wide skirt and she all pink, maybe. Pink dress, pink parasol, pink cheeks; pretty little pinktoe chick."
"She come down the staircase like she floatin'."
"And the head nigger hand the cat his big brim hat."
"Yeah, nigger think he into something, don't never smile. But his handkerchief-head old lady always sassin' the white folks and the head nigger don't dig it."
"Groovy," said Freeman, "you got the picture. They go out into the yard and the carriage is there. Two horses."
"Not black; that's for the bad cat. Brown. The carriage real shiny with red or yellow spokes on the wheels. Nigger with a straw hat in his hand holdin' the horses."
"They going out into the cotton fields to see how many niggers died since the sun come up ; you know, property inventory. They get out into them Technicolor fields and the niggers ain't dead, they singing, baby. Singing and laughing and them white folks don't know if they laughing at them," said Freeman.
"You think them white folks weren't scared ? You think that didn't flip them out? Out there in them fields where everybody else been dropping like flies, niggers singing and swinging. You think they didn't want to stop that song ; that they still don't have to stop that song? Well, baby, we're gonna sing and nobody's going to shut us up; we're going to sing a freedom song, loud enough for the whole world to hear. We're going to open up that trunk and get out that black pride." He was silent, lost in his thoughts. They watched him cross the room and mix a drink.
"You cats don't listen to spirituals anymore ; they taught you to be ashamed, but the message is there: 'Go down Moses and set my people free . . . " What people you think they were talking about?
"Man, it's all there if you listen. You can't find your history in the white man's books. If you want to know your history, listen to your music. Remember that spade poetry I read you last week ?" He recited :
"Why do we wait so long on the marble steps, blood falling from our open wounds ?
And why do our black faces search the empty
Is there something we have forgotten ?
Something we have lost wandering in strange lands?"
Freeman paused and looked at them. "Yeah, we lost something; we lost our freedom. And we forgot something, too. We forgot how to fight."
He looked at each of them in turn. "Waste the Comanches !"
"The Comanches? What do you mean, Turk ?" asked Dean.
"I've been holding you back too long. It's time to find out what you can do. You've been complaining about the Comanches saying you don't have balls anymore, invading your turf, messing with your chicks, calling you junkies and punks. You don't do something the fuzz got to get suspicious; no way you could get that tame so fast.
"Stud, you set it up, you ain't supposed to be on nothing except some heavy pot, anyway, so it will figure to be your scene. Put out the word you want to rumble. Daddy will supervise, Stud's field commander. Insist on no heats; just fists if you can get it, but that's not so important. Try not to kill anybody if you can help it, and don't forget what I taught you.
"OK, split. I got to get ready for a date."
They filed out of his apartment, excited and already discussing plans for the fight.
The Cobras and the Comanches met in a vacant lot flanking the elevated railroad tracks of the Santa Fe railroad. The fight lasted less than twenty minutes. There were less than ten Comanches still standing, willing and able to fight, by the time Stud recalled his fighters with a single shrill whistle of command. Stud reported to Freeman the following day.
"Was as easy as eatin' soul food, Turk, We wasted the cats so quick they still trying to figure out what happened. But I didn't get no kicks just runnin' things ; Daddy better for that kind of scene."
"You just bugged, Stud, because you couldn't get with it. You'll have plenty of fighting pretty soon. Right now, cool it. Anybody hurt ?"
"Naw. We cooled it, like you said. Pretty Willie didn't let a cat loose soon enough and think he broke his arm. Heard it crack. Man, that judo shit you teach us as good as a blade."
"Pretty Willie, huh?"
"Yeah. You know how he get when he in a fight. He the baddest cat in the Cobras, 'cept for me. When he really mean, you got to kill him to stop him. You know, I think the reason he so evil because he so light; I really think that cat want to be black as you can get. Always got the blackest chick he can find, won't look at a light-skin chick. Before he joined the gang and we started riding him 'bout being so light, you couldn't even talk about his color without a fight. He better now, but he still hung up."
"OK, Stud, cool it. Tell the cats I said they did a good job." Freeman turned and walked down the street, thinking about Pretty Willie du Bois.
Pretty Willie was a problem Freeman had to solve, because he was potentially one of the leaders of the underground ; one who might be trusted to command an underground unit in another city himself. The gang had helped in bringing it "right down front." Teasing him about the color of his skin, joking that he thought himself better than the rest because he was so pretty.
If Freeman could find some way to get him over that psychological hurdle, Willie would be a field officer as valuable as Stud Davis. He had to find the key. Pretty Willie had improved, as had all of the members of the gang recruited into Freeman's underground, but there was still too much of the brooding about his lack of pigment to
trust him to lead others. Freeman saw him as a potential leader of his top cell, his elite guard commander, second in command to Stud Davis.
Sugar Hips was an undeveloped natural mathematician, a genius with figures. In any except a ghetto school his genius would have been recognized and nurtured. In the overcrowded classrooms, where the sheer job of maintaining discipline precluded any except the most dedicated of teachers teaching, Sugar Hips Scott was a nuisance. He was far too far ahead of the other students, did his problems too quickly and was bored with the level of mathematics with which he was forced to deal. He had dropped out of school at sixteen, in disgust. He went through the geometry, algebra and calculus texts that Freeman got for him from the University of Chicago library with a speed and alacrity that amused Freeman and awed others of the Cobras.
There was not a day when Freeman did not curse the talent and intelligence within the Cobras that had been stifled. And for the simple reason, he thought time and again, that they are black.
Scott was his logistics officer, Davis his battle commander, Dean his second-in-command. Pretty Willie du Bois could be his best fighting commander, if he could just find the key to free him from his hang-up about his color. Within a week of the rumble, Freeman thought that he had discovered that key.
11
"We need a propagandist," said Freeman. "I hear Pretty Willie can write."
They nodded in agreement. "You want to see him now, Turk?" asked Dean.
"Yeah, where can I find him?"
"Probably in his pad in Hyde Park."
"I know where it is. I'll see you later."
He drove to Willie's pad; it was a basement apartment located in a half-court building. He walked down the short flight of steps and knocked on the door.
"Who it is?"
"Turk."
Willie opened the door. He was barefoot and wearing only a pair of tight gabardine slacks. "Come on in, Turk."
Freeman walked into the apartment. There was a long, narrow living room, a small bedroom with kitchen and bath in the rear. A tall redhead with close-cropped hair sat cross-legged on a blanket-covered mattress on the floor, smoking a reefer. She had no clothes on.
"Go home, bitch," Willie said. Without a word, she arose and walked toward the bedroom. "Wait a minute." She paused. "Turk, you want some of
Freeman, looking at the Japanese prints hung on the wall, ignored him. The redhead regarded Freeman woodenly. Willie dismissed her with a wave of his hand. She came from the bedroom shortly after dressing and left.
"A fucking freak," Willie said contemptuously.
"So what are you doing with her?"
"I like to fuck with her white mind."
"Looks like you're more shook up than she is." Freeman pointed to the prints. "Nice reproductions."
"Hokusai, Hiroshige, Utamaro." Willie pointed to the prints in turn, watching Freeman closely. Freeman walked to a canvas sling chair in the corner and sat.
"Daddy tells me you write." Willie's face closed. "I'd like to see some of your stuff. It's for the scene. "
Willie nodded and disappeared into the bedroom. He returned with a large attaché case, unlocked it and sat it at Freeman's feet. He went back into the bedroom while Freeman opened the case, and returned wearing a lightweight V-necked pullover. Freeman rummaged through the case. There were the portions of three novels and more than a dozen short stories of varying length.
"Any poetry? That'll tell me quicker what I want to know."
Willie reached into the case, pulled out a manila
folder and handed it to Freeman. He reclined on the mattress on the floor and watched Freeman silently.
"We need a propagandist," explained Freeman as he opened the folder. Willie regarded him with more interest. Freeman leafed through several of the poems. He looked up at Willie. "You always write in a blues rhythm ?"
"Right now, yeah. I've been trying to capture the way South Side spades talk, but for a long time it didn't work. Then I figured out they talk the way they sing, so I've been doing a lot of blues things. Most of that stuff I wrote singing." Willie pointed to a guitar case leaning in the corner.
"When I get an idea, I switch on the tape recorder and mess around with it until it sounds right; then I play it back, write it down and rework it on paper. My dialogue is much better than before I started doing that. I can hear those blues rhythms now when I'm hanging out on the block. Funny I never thought of it that way before— spades singing when they talk."
"Whitey says we're nonverbal. Nowadays, they're teaching us English as a foreign language."
"That's his hang-up. You want some coffee?" Willie rose.
"Long as it's not instant."
"I don't drink that shit. Come on in the kitchen." Freeman followed him into the kitchen and sat reading the poetry at the kitchen table while Willie busied himself brewing the coffee. He read one poem written as if a Negro on relief were speaking to a white social worker, "Anti-Poverty Blues." He read aloud :
"I sit here in my soul-hole, forgotten, like a can of black caviar on my ghetto grocer's shelf."
Freeman nodded, read some of the other poems, glanced through some short stories.
"OK, baby," he said, "you're the propagandist. Keep the stuff like this, ironic, understated, blues things written for black folks ; no shit about white devils, don't give whitey any stature by screaming what a big drag he is. Put him down and you build him up; the put-on is what we want. Keep the themes simple, few in number and then work out varieties on those ; like in the blues.
"Use blues rhythms, your poetry, doggerel, anything catchy that people will remember and pass along; we want to plug into the ghetto grapevine. You understand the kind of thing we need ?" Willie returned to the kitchen table with two mugs of steaming black coffee.
"Yeah," he said, "yeah, beautiful; I have some ideas already. I can let you see something next week. OK?"
"Groovy. Top security, though." Willie nodded, smiling to himself, thinking things over. "Who else would be good at this kind of thing?" Freeman asked.
"Ted, Tan-Dan and Charlie."
"Set up your own cell, you're in charge. Let Hips know what kind of equipment you need— mimeo machines, typewriters, paper, ink, office supplies. Keep your operation simple and lightweight, stuff we can move quickly and easily. Use the two-car garage on Indiana Avenue as your headquarters and scout around for alternative
sites in case you have to run. Work out a budget and get the money from Hips : get as much of the stuff hot as you can and buy the rest through a dummy so it can't be traced back to you by the fuzz. You know the routine.
"We'll need leaflets, handbills, homemade bumper stickers and scripts for propaganda broadcasts. We'll use cheap transistor tape recorders for that. I want you to work out a complete propaganda program with major themes, priority target groups and including counterpropaganda for what you anticipate whitey will say about us once we begin operating."
"First thing he'll do is cry Red."
"Right. But don't spend too much time denying we're Communists. Our people will know better and we really don't care what whites will think. Also, I want you to work out an international propaganda program for distribution to overseas publications in Europe, Africa and Asia; they're not likely to get our side from the 'free' white press. "
"How about artwork? Cartoons, stuff like that ?"
"Excellent. Who do we have?"
"Lots of the cats can draw; no trouble there. Songs, too ?"
"You have the idea. Anything to get our message across, but remember that a lot of this stuff will be done after we've gone underground and are being hunted. The more you can do now, the easier later."
"I can work up a rough outline of the whole program in about two weeks. OK"'
"Take as much time as you need and get in touch whenever you have questions."
Willie reached over and turned on the transistor radio which stood on the kitchen table; it was tuned to the station beaming its broadcasts to the ghetto. They sat silent awhile, listening to Dionne Warwick singing "Walk On By."
"Do most of the Cobras know about your
Hyde Park scene, Willie?"
"Sure, they seem to dig I'm living right in the middle of whitey; particularly since what we've been planning. Daddy and Stud come by all the time; sometimes together, sometimes alone. We lollygag, maybe turn on, or cook up some soul food; we had some red beans and rice last week was out of sight." He gestured with his hand.
"This is my cover. My mother wanted me to go to one of those Louisiana excuses for schools where the pecking order is based on the lack of pigment in your skin and the lack of kink in your hair. I cooled her out by enrolling at the University and she brags about her son being a genius."
"How much time have you spent in college,
Willie?"
"I seldom take a full load of credits; it gets in the way of hanging out with the boys. I guess if I added them up it would come to a little over two years."
"You intend to get a degree?"
"What for? What kind of job could I get with it? I hope you're not going to suggest I pass." His face closed again.
"I don't know what people mean by 'passing.' Being black or white in this country is a state of mind. You're black because you think black, feel black and act black. I know people who look like charcoal who are more white than whitey. But that's not the point; whoever told you a university is a vocational school, preparation for a job or career
"What else is it ?"
"That's what it is for most of the people in this country, but in other places people feel education has an intrinsic value all its own. In most other countries an educated man is respected whether he has a crying quarter. Do you know that in Arab countries people carry a folded newspaper and a pen showing in their pockets as symbols they can read and write? And some Africans will put on their business cards the fact they flunked out of Oxford or Cambridge? The important thing is that they were there; it's not nearly so important that they didn't stay. Black people are going to have to get like that because whites are never going to share the wealth with us."
"So what's the use of getting an education if you still have to stand in that unemployment
"Because you're a better man for having an education. If we ever break the vicious circle they thrust us into and really get some quality educational institutions in the ghetto, you know what they'll produce?"
"What?"
"The best educated poor people in this country ; but it'll be worth it. My grandmother used to say to me: 'Get that education, boy, it's the one thing the white man can't take away from you.' And she was right. She never asked me what I was going to be, she was just delighted I kept my head in those books. Those books were symbols of freedom to her and she was right about that, too."
Willie was thoughtfully silent for a few minutes, moving his crossed foot in time to the music.
"So why all the effort if things can't improve '?' ' "Oh, things can improve, but there will always be two countries here: one white, rich, fat and smug ; the other black, poor, lean and striving. It's not such a bad state to be in ; there's a whole world out there not getting schizo because they don't keep up with the Joneses. Things can be better without being white." "I can't figure you, Turk."
"What's to figure ?"
"What are you, a Muslim, nationalist, black-power advocate, or what ?"
"Willie," said Freeman and shook his head sadly, "why do you have to label me? Whitey will do it soon enough when the deal goes down. They'll come up with a lot of labels until they find the one which best puts down Uncle Tom and then all the rest will echo it. Why can't I just be a man who wants to be free, who wants to walk tall and proud on his own turf as a black man ? Why can't it be as simple as that ?"
"Maybe it's power. If the scene ever gets off the ground, you'll have a lot."
"Who needs it? You can't eat it, wear it or screw it. It won't even cure a hangover. How come you're in it, Willie ?"
"I want to mess with whitey."
"It won't be enough; it's negative, you'll need a more positive reason than that to get you through. You ever kill a man, Willie ?"
"No, but I've hurt some pretty bad in fights." "It's not the same. When the time comes that you lay a man out with his guts spilled out in the gutter, you'll be surprised how fast hate disappears. He'll be another human being and you'll have put an end to it. You'll need some love, or something, to keep you doing that. Unless you find out you enjoy it, and I don't think you will. You better think about that. Some of the cats will enjoy it. Stud will."
"What do you mean ?"
"He's a killer. He doesn't know it yet, but he is. I saw a lot in Korea. Once they get a taste of blood, nothing else gives them as much, booze, women, nothing. I don't really believe there's any justification for killing, but freedom comes as close as any."
Suddenly Willie realized something. He leaned across the table toward Freeman and placed a hand on his arm.
"You don't like any of this, do you ?"
Freeman looked into his coffee cup as if searching for his soul.
"No," he replied. He rose, walked to the sink and rinsed out his cup. "I have to split." They walked to the door.
"Come by again, Turk; I dug talking to you." "Sure, baby." He roughed Willie's curly hair. "And don't worry about not having enough pigment. As far as I'm concerned, you're one of the blackest cats we have."
He walked out into the cool evening. He felt he'd got through to Willie, that he'd planted a seed. Willie was involved with the movement now that he had a propaganda program to create, but Freeman would have to bring the color thing down front, get it out into the open once and for all and either win him or lose him forever. He thought he knew how he could do it.
He felt restless and decided to walk most of his rounds that night. He had settled into a routine on the job, cruising or walking the streets first to
judge the mood and atmosphere. Then he would stop in a favorite tavern for a beer before making his checks with his street workers. He walked Sixty-third Street that night after leaving Willie's apartment and stopped at the Boulevard Lounge.
"Hey, Dan, how you doing, baby? Friend of yours in the back," said the bartender.
"Hey, man. Friend ? Who ?"
"Pete Dawson. Just got back from California." "Groovy ! I've been looking for the cat since I got back to Chicago." He walked the length of the bar and entered a smaller room in the back, more intimate, with less modern and more comfortable furnishings. A small bar was tucked in one corner and Dawson sat there chatting with the bartender and three customers.
"Pete, baby! How you doing, man?"
"Freebee, good to see you ! I heard you were back in town." They shook hands and punched each other's shoulders, smiling broadly. "Sit down and have a taste. You still drink scotch, don't you ?" Dawson ordered drinks.
"Pete Dawson, the Sherlock of the South Side.
How's the detective business?"
"Beats the post office or driving a bus."
"What were you doing so long in California?" "I was on detached duty, attending a seminar on riot control. I'm in charge of the plainclothes section of the riot-control task force."
"Beautiful, man, you're really coming along. Get a promotion along with it ?"
"Does it snow in July in Chicago Dawson smiled to soften the bitterness in his voice. "No, I'm still a sergeant. I read a paper at the seminar, though, and everyone seemed to dig it. The FBI requested a copy."
"What on?"
"The Chicago approach to riot control. We think we've combined the best of several plans." "Sounds interesting. Could I take a look at it?" "Sure. Main point is we try to get plainclothes spade cops into the area once it's been sealed off, to try and cool the crowd. We use force only when we have no other choice and then we don't play around."
Freeman thought the paper could be useful and while thinking it, hesitated. What kind of cat have I become? I'm going to use a friend and I don't even blink an eye.
"Pete, why don't you give us a rundown at board meeting next month? I'm sure the directors will be interested. It'll be good for at least fifty bills. I'll try to get you a hundred. No sweat with your superior ?"
"No; he's about as straight as they can get. Gets more bugged than me that I can't get a promotion. Let me know when you want me to come by."
"Crazy. You seen any of my street workers ?"
"Yeah. Tell me they been working for a change since you took over. I saw Perkins on Sixty-third Street a few minutes ago. He usually stops in here for a brew about this time."
"They had it made before I got here. One cat was in night school and two others were working the night shift at the post office downtown. But that scene is over now. Perkins is a good one, though."
"How does it feel to be back in Chicago? Place look different ?"
"No, looks the same as when we were kids and you were an Apache and I was a Cobra. Good to be home."
"Yeah, Chicago gets into your blood."
"Yeah, like leukemia." Dawson laughed and switched the pistol on his belt to a more comfortable position.
"The Cobras and the Apaches. Man, we had us some rumbles, didn't we? I still got the scar from that time I got shot." They smiled together, remembering.
"Ever see any of the cats ?"
"Every now and then. Lot of 'em in the slam in Joliet. I had to bust some." Dawson looked soberly into his glass, drained it and motioned for another. "Man, that's not an easy thing to do, bust a friend."
Freeman tried to change the subject.
"Think there'll be a riot this summer?"
"Ain't there always? We've kept them small so far, but you can never tell."
"When do you think they might break loose, and where ?"
"Who can tell? A good time would be when I'm on vacation catching fish in Michigan."
"You know they wouldn't do it while you're gone. They're going to give you a chance to be a hero."
"You know the only spade heroes on this police force are dead."
"That's what I mean, baby. I'll be there when they make the posthumous citation."
"Thanks, old buddy, you so good to me."
"And I'll take care of your old lady, too, after you've gone."
"Which one?"
"The one looks like twenty miles of bad road ; the only one you got."
"You know the chicks always did prefer me to you."
"Prefer you going the other way." They smiled at one another.
"Freebee, good to have you back. Man, we can really swing together again."
"Like, really, baby. I better get to work. Cool it, man."
He left the bar thinking of how much fun he could have now that Dawson was back in Chicago. He also wondered how to recruit him.
12
It had been a harsh winter, with subzero temperatures a regular thing. There would be an occasional respite while the big, soft snow covered the city and for a short time covered the grime and dirt and ugliness of Chicago with its virginal whiteness, but within hours after the last flake fell, the virginal snow would be a greasy, dirtgrimed whorelike snow and then the temperature would drop and film the streets with mirrorlike glaze, turning the city snow into something that crunched underfoot like an old cereal in a new box labeled super and all-new. There was nothing super and all-new about Chicago and it is not a place for people who concern themselves with the weather, winter or summer. The wind would whip in from the lake, bearing airborne razors of ice that sliced the flesh. There were regular gray skies and little sun. The sky seemed to sit just above the Tribune Tower and it would sometimes descend to the city streets when the warm-air masses moved up across the plains 134
from the Gulf of Mexico to turn the city into a fog-bound, slushy swamp full of mud-splattered people who groped their way in the dense muck, mire and moody low-sitting cloud, like amoebas in search of a guide to nowhere.
When there was sun, it would come from afar in a hazy, cloudless sky, giving a harsh, cold and biting light, the lack of clouds permitting what little warmth remained to flee toward the planets above, the people below creating little clouds of their own as they breathed and gasped, moving through the brutal city. Because the weather was so menacing, the Cobras were not missed from their usual haunts and there was no need to interrupt training by having some of the gang members on the block. It was too cold to be there and the police and social workers did not worry where they might be since the word was that the Cobras were no longer a bopping gang. And since lower-class Negroes are visible only when convenient or menacing, the Cobras disappeared and no one concerned themselves with what they might be doing that cold and forbidding winter.
They were learning the lessons of the oppressed throughout history in striking back at their oppressors ; the linguistics of deception ; subterfuge, to strike when least expected and then fade into the background; to hound, harass, worry and weaken the strong and whittle away at the strength and power that kept them where they were. Just before the rumble near the railroad tracks, the winter ended as abruptly as it had begun and spring was in Chicago with no warning, the flowers blooming, the trees suddenly budding, the grass turning green, the dirty snow melting and disappearing into the sewers. Spring meant baseball and track, walks in the parks for young lovers and examinations for the students reluctant to remain in the libraries and overheated apartments with textbooks that had become symbolic of the prison of a nasty Chicago winter.
It was time for examinations for Freeman's small band of revolutionaries-in-training. They were becoming restless with the constant drills, the routines; they wanted to "get it on." The rumble had convinced Freeman that they had not softened and that they could be counted on to follow orders. There were two more tests that were necessary and the change in weather permitted them.
Freeman continued to contribute to his playboy image ; pretending to enjoy parties that bored him, dating women he did not like, flattering men he detested, doing and saying and acting things that sickened him. But there was never a hint that he was anything other than he appeared to be and those of his committed friends who were now active in the "movement" and who remembered Freeman as a tireless firebrand in the struggle for civil rights now regarded him with contempt as a hopeless sellout. They stopped asking him to attend meetings, contribute to their campaigns, man their picket lines or join their marches. They were his barometer and he judged his performance by their personal reaction to him. The women thought him an eligible bachelor, if a bit of a chaser. The men thought him harmless and appreciated that he did not try to steal their women ; Freeman thought that there was little to choose from among the black middle-class chicks available and that risking the wrath of an insecure middle-class Negro, whose only available
test of manhood was confined to the boundaries of his bed, was a waste of time and energy. Like Willy Loman, but for different reasons, it was important that Freeman be well liked, and he was.
He made speeches in the white suburbs concerning juvenile delinquency in the ghetto, as the executive director of his foundation. He knew that his speeches were intended for entertainment rather than enlightenment and he spiced them with the white man's statistics concerning Negro crime. He did not point out that Negro crime was largely confined to the ghetto, because he knew those nice white people wanted to feel threatened by the nasty Negroes in the ghettoes they never need see, except in the picture magazines, on television or when behind a mop, broom or tray. He was urbane, witty and fake-informative.
His Lotus was known over most of the South Side and although he had to put on its hardtop so as not to muss the hair or wigs of his dates, he enjoyed it very much, as much as any part of his cover. He cultivated the police and politicians and the members of his board of directors. Freeman was constantly pointed out as an example of what a Negro could accomplish if he tried hard enough. He was considered an example of Negro progress and no one concerned themselves with the increasing unemployment in the ghetto, the fact that Negroes continued to fall behind in national economic statistics. Freeman was a good salve for the nonexistent conscience of the white man ; that vacuum the editorials spoke of as having been aroused by the "Negro Revolt." Freeman told them what they wanted to hear and was just argumentative enough in cocktail parties to have
whites refer to him as "militant, but responsible." Freeman was the best Tom in town. His cover was as good as it ever figured to be and would probably not be blown before he could get his program under way. It was time to begin. He gathered his lieutenants at his apartment, plus a few other members of the Cobras, including Pretty Willie Du Bois.
"This summer is the scene, but we need a few things first; most of all bread. We been building up the war chest with what we been stealing, plus whatever we make peddling shit in New York, but we don't have near enough. Sugar Hips, how we
"We got a balance of $8432.86."
"Right. That ain't enough for what we have to do, so we have to get more. We take it from whitey and we get it from where he keeps it." Freeman spread plans on his cocktail table. "These are the floor plans for the bank in the shopping center on 115th and Halstead. We ought to be able to get a bit more than a hundred thousand dollars. They have a closed-circuit TV and automatic cameras, plus a very sophisticated alarm system. When they touch it off, it alerts the nearest precinct station, but makes no noise.
"There are only two guards, one on the floor, the other on a balcony above, behind one-way, bulletproof glass, with ports for firing to the floor below, but we hit the bank when it's crowded and he won't be able to fire for fear of hitting some innocent bystander. The only tight moment is when you move out and he might get a clear shot as you go through the door, but you'll move to the door with a couple of people as a shield, then shove them back in and make it. He'll be able to
hear you up there and make it clear that if he starts firing at any time, you start wasting the crowd.
"It's a short hop to the Negro section in Princeton Park and since it's all middle-class nice, the cops are not likely to search it. That's where we hide out, in the house of Pretty Willie's mother, who's in New Orleans for the Mardi Gras. Besides, they're not going to be looking for niggers, anyway, but for white men."
They looked at him with curiosity. He drained his bottle of Carlsberg before he spoke.
"Pretty Willie leads it. He takes Red Beans, Benny Rooster, PO' Monty and Pussy Head."
They looked at one another and smiled, except Pretty Willie. Freeman had named the lightestcolored members of the Cobras.
"No, goddamnit; I ain't white, I'm a nigger," said Willie.
"Sure, baby, we know that, but that day whitey is going to think you're white. They'll be looking for everybody except us. Niggers don't rob banks, man, you know that. With a gun in your hand, telling Mr. Charlie what to do, to give you his money, why, man, you got to be white," said Dean.
"Look, baby, you know we need the bread and you know none of the rest of us can do it. They must not have more than twenty niggers with accounts there and if five walk in at the same time, they're going to be suspicious. You the only one in the gang can do it because, although your soul is black, your skin is white," added Scott.
"No !" said Willie.
"Man," said Stud. "Don't you see how beautiful it is? I mean, you're turning him around. We all know you a nigger and so does whitey when you on the block with your skinny-brim hat, hangin' out with the cats. He never treat you any different from the rest of us, 'cause you ain't black? Hell no, he gives you a harder time. Now, you got a chance to turn it around."
Dean spoke up. "Turk, give us a chance to talk about it. Willie, you don't have to do it if you don't want to." He looked at Freeman for confirmation. Freeman nodded. "And, nobody going to put you down if you don't." He swept the room with hard eyes. "We all know you hung-up 'bout not being black, just like some hung-up 'cause they is black. So come on down to the poolroom and we talk about it a little. You don't want to do it, you don't do it and we do it some other way. OK, Turk ?" Freeman nodded again. "Crazy, let's split. Stud, you go see your ol' lady, 'cause you might get bugged and want to get physical and there ain't going to be no shit like that, dig. 'Sides, I ain't sure you could whip Pretty Willie, any-
They all laughed and relaxed, including Willie and Stud. They were both good enough not to have to worry about the difference. They finished the beer, took the bottles into the kitchen and filed out.
Three days later Freeman received a call from Dean.
"Turk, it all right. Pretty Willie do it. Funny thing, he kind of happy 'bout it now. Never seen the cat smile so much."
"OK, come on by tonight. We'll need about two weeks to get it down pat and it will have to be before the weather turns good. The worse, the better."
It only took them ten days of coaching to get the details right, but a warm-air mass moved up from the Gulf and turned Chicago into a muddy, messy swamp of slush and snow that would freeze at night in the grotesque shapes that had been molded by tires during the day. They waited a week before the snow came again and then the temperature dropped swiftly one moonless night, hovering around zero and turning the icy streets into a glassy surface that sent cars spinning if braked too suddenly. The sky was a slate gray the next morning, seeming to hover just above the rooftops, heavy and leaden, threatening to fall on the ugly city below.
They left for the shopping center at noon, by separate routes, and by that time the temperature had risen and a fine wind-blown snow slanted to cover the icy streets. Freeman drove a Mini-Minor with Scandinavian ice tires on the front wheels, the front-wheel drive gripping the ice firmly. It was modified to serve as a delivery truck for a small dry cleaner's, whose owner ate lunch at home and napped afterward. They had timed him and he never returned to work before three in the afternoon. They had stolen the car and changed the front tires. Freeman drove alone into the big parking lot not more than a hundred yards from the bank. The others arrived in a Willys station wagon, its four-wheel drive giving it stability on the icy streets. Freeman watched as they approached the bank separately. He could see into the bank through the big glass doors. There were
about three dozen people inside, other than the employees. Not many, but enough.
He watched Pretty Willie walk in last and without hesitation pull his gun. Mouths opened to scream and shout, but no sound reached Freeman outside. They had been lucky and one of the guards was at lunch. The other reached for his gun and Willie dropped him with a bullet in the thigh. Monty was the closest to the guard; he kicked his gun away and then kicked him in the head until he lay quiet. Two of the others vaulted the cashier's counter and began sending money into canvas laundry bags. Willie watched the ports above for the sign of a gun barrel, unaware that the guard was absent. No one else moved. First Pussy Head, then Rooster moved from around the counter, bags full of money in each hand. Freeman started the car and wheeled swiftly in front of the bank, the back of the small truck toward the doors. They moved out of the bank and into the truck and Freeman drove off quickly. Dean, across the street and using the hood of the Willys wagon as a gun rest, fired a deer rifle with scope sights high into the glass doors, pinning the occupants to the floor. He emptied the chamber and got into the truck, following the Mini north on Halsted. Freeman drove to Ninety-fifth and turned east, coming to rest in front of Willie's house after zigzagging to it from five blocks away. They moved swiftly into the house with the money and he drove away and left the car in a residential neighborhood and was picked up by Stud Davis in another car. They had left the stolen Mini in a white neighborhood.
In fifteen minutes Freeman was home sipping twelve-year-old scotch, waiting for the bathtub to fill with very hot water.
Later that evening he watched the news on television and heard the announcer describe the daring daylight robbery of the bank. They were all listed as Caucasian in the police description of the bandits. Freeman smiled. A nigger with a gun in a bank with a lot of money had to be white because niggers snatched purses and rolled drunks— any cop could tell you that—they just didn't rob banks.
Freeman now had an army and a treasury. He needed an arsenal and he knew where he could get that. Then he could start messing with white folks.
Spring came early as if in apology for the fierce winter, the sun bright in pale blue skies, fat white clouds floating overhead, the ore boats moving lazily to the steel mills at the lake's southern tip. The buds burst, the city turned green, and the grime did not seem so noticeable, the noise of the El not so annoying, the stench of the fumes from the buses not so poisonous. For a few days in spring and fall Chicago seems almost fit for human habitation.
Baseballs, footballs, basketballs filled the air in the ghetto, the spherical symbols of a possible escape from the ghetto cage. The junkies stood and sat in the warm sun, their dope-filled blood moving sluggishly in their veins, the ugly world taking on a warm glow, everything soft and pretty prior to their moving through the streets at night looking for loot to support their habit. The winos drank their sweet wine beneath the El tracks, their hoarse voices rising with laughter as the sweet alcohol filled them ; the unemployed who still
Mrs Riddable had not seized his point, he added, 'I'll have to see his chief constable about him. I've known Ronnie for years, of course."
He sometimes told Mrs Riddable facts of which she was already aware. It seemed to make them more factual if he told her them again.
'I shot him down in flames,' he went on complacently. 'As you know, I shoot from the hip.' His temper was restoring as he talked and chewed. 'I shot him down in flames and rode off into the sunset.' He concluded, new made over.
Stella stared into the candle flame and then let her eye travel beyond it. The Hollowmen and their guests were gathered for their evening. They were twelve in all. The members of the community had washed and changed, taken off their shoes and assembled in the kitchen Nissen hut, Stella's and Fresh's. They sat in a circle, some cross-legged, some on the slatted stools which Oliver had constructed. On a low round table in their midst a single tall candle burned.
Besides Oliver and herself there were Mr and Mrs Bean, Erica Millhaven and her tall guest Theodora Braithwaite, Sean and Kevin and their probation officer, a fair diminutive Scot called Gavin, Kevin's current girl, an Afro-Caribbean called Jewel, and Miriam and Matthew Rosen, accountants both and early supporters of the enterprises at the Hollow.
The daylight, after hanging around for a bit as though not quite knowing what to do with itself, had by seven o'clock in early March faded. Through the un-curtained window Stella could see the arc lamps of the railway and the encroaching building site. The evening was mild. The smell of damp earth came in through the open door and near at hand a blackbird experimented with his spring calls. Stella's fears, first about the guilt-ridden past, then about the future, food, (would it be all right, would there be enough?) came to her as they regularly did at this moment. But now after three years she was strong enough to catch them as they came and watch them as they dropped away into limbo. She brought her gaze back to the candle flame and let out her breath. In these small nightly pockets of calm and silence she knew she was safe: silence saved, weather saved, friends saved.
Erica Millhaven stared into the flame. Through it she saw Oliver and Stella sitting side by side on the other side of the bare room, their heads close together and silhouetted like a double cameo. Beyond them in the darkness near the walls she could distinguish the dignified faces of the ancient dead, her friends, her witnesses. Her eye returned to the centre of the flame and she saw within it with equanimity her own death.
Theodora kept her eye steadily on the flame. The silence filled her. The questions which had occupied her over the last two days fell away to be dealt with at the proper time. Now all that mattered was to find in the light of the candle silence, stillness, peace.
At eight o'clock the chimes of the cathedral clock drifted across suburb and marsh to the Hollow. One by one, as though reluctant to break the peace which silence had induced, the twelve began to stir. The legs of older members realigned themselves with their bodies none too nimbly. The lurcher bitch, who had been lying doggo in the shadow beside the boiler, rose, stretched, yawned and showed a lot of pink tongue and very white teeth. Stella went to the oven, Kevin stoked the boiler. Jewel carried bread to the table. Mr Bean collared Matt
CT A garage was less than three blocks south of the armory on Cottage Grove, no one thought it strange to see an empty bus on the street. Within an hour, the arms and ammunition were cached and Freeman was ready. He need only wait for the hot, humid summer and an arrogant, headwhipping cop to spark the riots. It was like waiting for the sun to shine in the Sahara Desert. Freeman did not think that there would be much searching of the ghetto for the arms because niggers didn't steal government property and defy the FBI any more than they robbed banks.
13
Spring ended abruptly. A hot, moist air mass moved up from the Gulf of Mexico across the plains and into Chicago, smothering the city and turning the night into a furnace, the brick buildings radiating the heat collected from the sun during the day. Life in the ghetto moved outside, onto the doorsteps of the houses, into the air-conditioned bars and the cinemas that sold cool air and Doris Daydreams. On the South Side there was Washington Park, and families moved at night into its cool greenness, sleeping on blankets under the stars until the first rays of sun, returning to their stifling rooms to snatch a few more minutes of sleep before meeting the hot, humid day. Beer, watermelon, ice cream, anything cool, but there was no way to leave the engulfing heat. The city lay gasping like a big beast. Tempers shortened, and the ghetto lay like a bomb waiting to explode.
Freeman thought that it would begin on the West Side, but it began on the South Side, not far from his home.
A cop killed a fifteen-year-old boy beneath the El tracks near Fiftieth Street. A group of them had been drinking beer when the cop ordered them on; someone threw a brick and the policeman drew his gun. He claimed that the boy advanced with a knife, others disputed his claim, but for the neighborhood the dispute became irrelevant. The word of the shooting spread and the streets filled with people. The wails of sirens sounded as riot-trained police sped to the area. Freeman's phone rang.
"Turk, this is Sugar. Daddy say to tell you it look like it's going to begin. They got people in the street."
"Right. Where are you ?"
"In the poolroom."
"Stay there until you hear from me. I'll call back in about an hour."
He called his office and told the girl on night duty to alert his street workers and said that he would call again within the hour. He got into his car and headed for Fifty-first Street.
He walked up to a squadron and spoke to a cop he had known in high school. "Hi, Bill, how does it look?"
"Hey, Dan. Not too good. Young cop panicked and burned a kid and people are bugged. You know what happens when you have this kind of heat for this long; anything will touch them off. We got cops moving around in there now, spades in plain clothes telling them to cool it and go home. It's the kids I'm worried about. If they get started, it could spread."
"Anything I can do to help?"
"You know some of the kids around here, don't you ? Tell me you pretty tight with the Cobras."
"Yeah, I know them."
"Well, one of your street workers is in there already. It wouldn't hurt if you went in, too. But get the hell out if it looks bad. We don't need any heroes."
"You think it might get bad ?"
"I hope not, but I think so. These people wouldn't spend ten minutes in a picket line, but they'd fight all night. We've been lucky so far, but you know Chicago, if it really goes it will make those other riots look like picnics."
"OK, Bill, I'll see what I can do."
"Get the hell out of there if they start raising hell. And check back with me every now and then. Dawson is in charge there. You know him." "Right. Take it easy, man."
"I just hope the brothers take it easy. Rioting won't help things any."
"The picket lines don't seem to have helped much, either."
"Yeah, and that's the bitch of it."
Freeman walked down Indiana Avenue. People milled about in the streets. The jitneys had stopped making their runs up and down the streets and were either elsewhere or parked on side streets. He had noticed signs already in the windows indicating which stores the Negro owned. Plainclothes police moved through the crowds, softly urging the people to return to their homes. Here and there men harangued knots of listeners about whitey and police brutality. The police were particularly nervous about rioting on the South Side because it was not a solid ghetto; there were whites in the big housing development near the hospital on Thirty-first Street which had replaced the slums there as part of urban renewal and it was separated from the Negroes by only the broadness of South Park Boulevard. There were whites a short distance away in Hyde Park. Rioting would not be easy to contain and might entail more than just the burning of stores and buildings.
Freeman moved through the crowds, listening and trying to gauge their mood. There was a chance that they might cool off and return to their homes. There was also a chance that they might explode in any minute. He spotted Stud Davis and walked toward him.
"Hey, Stud, what's happening?" "Nothing shakin', man." They walked out into the street away from the crowds.
"You been to the poolroom ?"
"Yeah. Daddy, Hips and Pretty Willie's there."
"Where are the rest ?"
"Out in the street, like the plan."
"You think they'll blow ?"
"Yeah, and I don't even think we'll have to start it. Some of the younger kids are ready to start raising hell. If a cop start leaning on somebody, it's gonna blow, but the fuzz been real cool so far."
"You seen my street worker ?"
"Perkins? Yeah, he was around here a few minutes ago; I think he's in the next block. I think he kinda scared."
"Yeah ? well, so am 1. Cool it."
He walked down the block, feeling the tension in the air, the look of expectancy everywhere, on the faces of the people, the children, the cops.
The cops were seasoned veterans, no rookies, and they would not act hastily; they had kept the white cops on the periphery.
Freeman turned west and then south and walked back to Fifty-first Street. The cops had ordered the bars and liquor stores closed. The whites who owned businesses had faded into the. night and the only white faces on the street were those of the police. Freeman was told to move on several times by nervous cops and had to show his identification twice as he strolled down the street toward the east, watching the police in riot helmets and the angry faces of the crowd. Everyone waited for something to happen. Television crews moved here and there with hand-held cameras and cameras mounted on the roof of their equipment trucks, their bright lights giving a harsh glare and casting grotesque shadows. The blue lights on top of the squad cars revolved and winked every few yards. Gas masks were distributed and boxes of tear-gas canisters opened. Ambulances and paddy wagons stood at the ready. Cops spoke busily into their car radios and handy-talkies. Everywhere they clutched their nightsticks tightly like magic wands which would give them strength, beauty and power.
The smells were still there ; barbecue, fried chicken, fried shrimps, greens, beans and corn bread. But the sounds of the ghetto were not there : the jukeboxes silent and no soul music from inside the bars and restaurants, barbecue joints or from the loudspeakers hung outside the record store ; none of the laughter, shouts to friends, the teasing arguments or quick, staccato bursts of angry profanity, no whistling, humming or someone singing to themselves. The clatter of the El was present as it moved south toward Englewood or Jackson Park stations, or north to Howard Street, but otherwise there were alien sounds: the crackle of the police radios and handy-talkies, the disembodied voices over bullhorns, the sullen murmur of the milling crowds and the keening songs of hate by the street speakers. The air hung hot and heavy and there was no hint of a sudden rain to break the heat wave.
Freeman turned north on Indiana Avenue for the second time that night and approached Perkins, his street worker, who was talking to Detective Sergeant Pete Dawson.
"Hey, Dawson, Perk. How do things look?"
"Not good," said Dawson. "It could blow any minute, or maybe not. We hope we can keep it sealed off if it does go. We're keeping the white cops out of here for now. The cat that wasted the kid was ofay. He didn't have any choice, the kid came at him with a knife. It was self-defense." He sounded as if he were trying to convince himself more than Freeman and Perkins. They all knew that the cop had emptied his gun and missed only once. Standard procedure was to try and drop an assailant with a shot in the thigh or leg. Dawson shook his head. "The cop shouldn't have gone into that dark alley alone. He's new down here; he's used to having people respect the uniform.
"We keep them moving, but as soon as we break up one group, another forms in the next block. This heat is a bitch, maybe they'll run out of energy and go home." They looked at one another. They all knew that it was hotter in those buildings than in the street.
"Anything we can do, Dawson, to help ?"
"Just what you been doing. Tell the kids to cool it, they'll listen to you and Perk. I don't think there will be any head whippin'; the cops in here have orders and they're all pretty cool. Tough, but cool and the people here know how far they can push them."
Freeman turned to Perkins. "How does it look to you, Perk ? Kids nervous ?"
"Yeah, everybody but the Cobras. They seem to take the whole thing as some kind of joke. They're sitting around in small groups coolin' it. Stud Davis is on the block, the other leaders seem to be shootin' pool."
"They're on junk," said Dawson. "Junkies don't care about this kind of scene."
"What about the others, Perk ?"
"Well, the Bearcats are out in force, but I don't think they'll start anything, although they might join in if something starts. But the Apaches might start trouble, they'd like to take over from the Cobras. And I've seen a few Comanches around. Not many though, since they sealed off the area." "OK, Perk. First, get down to the drugstore and phone the office. Get the rest of the street workers down here. Then keep moving around. Keep in close touch with the Apaches, with their leaders, tell them to cool it. I'll be around." He turned to Dawson. "Take it easy, baby."
"Yeah, man, nothing else to do." He walked off down the street.
Freeman moved among them, talking to them, listening sympathetically to their angry protests. He soothed them and told them to go home, that rioting would do no one any good and he did not shuck. You either work at a cover, or forget it. And he did not feel that he would have to do any-
thing to start the thing; he knew if not tonight, there would be some other night like this one. Whitey was stupid and stubborn about Negroes. He would not believe that Negroes would not continue to passively accept the pushing around that whites had come to think of as their birthright. The white cops on Fifty-first Street, he knew, felt that all they had to do was to charge the people here, make enough arrests of the ringleaders, bloody enough heads, a bit of tear gas, guns fired overhead and if need be, shoot a few niggers and they would tuck their tails between their legs and become silent and invisible once more. They would not believe that things had changed and that these people had had enough.
They listened to Freeman because they trusted him, although they did not agree with what he was saying that night because they were too angry and frustrated. They remembered now the waiting in vain for jobs that never materialized; the long humiliating lines to sign the papers thrust at them by arrogant clerks at the unemployment compensation office; the slights and insults of the social workers ; the welfare raids on their women's rooms in the middle of the night in an effort to catch a man there. The streets seemed hotter, dirtier, uglier, smellier than they had ever been and no one seemed to care. But tonight they cared, tonight these people were not invisible. tonight whitey knew they existed and there were among them those who wanted whitey to remember this night for years to come.
A bit after midnight they began to cool down, some to drift to their small, hot rooms for a beer and the late show on TV. The Negro cops in
their midst sensed the change with relief. And then it happened.
It was the dogs. The truck with the four dogs and their handlers was to have been parked outside the area, to be used only on orders from the commander of the riot task force, but someone made a mistake and the truck stopped at the intersection of Fifty-first and Indiana Avenue. The commander had left his command post there to check the roadblock at South Park. One of the dog handlers approached a captain of police.
"Hello, Captain. We just brought the dogs down. You want us to go in ?"
"Well, the commandant and the second-incommand aren't here right now. I dunno."
"How are things so far ?"
"Well, they got the colored cops in there. Been in for a few hours. Been kinda tense, but nothing big so far. They're trying to use persuasion." His mouth twisted at the word. "Things have sure changed since I joined the force. All this scientific crap the new commissioner put in. Hell, we can't fart if a computer don't say so. Send three squads of big Irish cops in there, a little tear gas and we can all go home and have a beer."
"Maybe they can use us in there, Captain. These dogs will move 'em. Never seen it to fail."
"Yeah, one thing a nigger is scared of, it's a dog. Go ahead in, but you better report to Dawson, he's a shine detective lieutenant. Thinks he's hell on wheels because he's got some college. You don't report to him, he'll be complainin' to the chief."
"I'm special duty, attached to headquarters downtown. I don't have to worry about no jig
lieutenants !" He walked away to the van, a big man who carried his vast belly well, like a pro tackle in his last years, using his bulk and experience to add a few more seasons toward his pension. He motioned to the others and they opened the van carefully, first talking to the dogs to calm them and then in fierce command voices ordering them one by one to the pavement to attach the leashes and remove the muzzles. There were three dark Alsatians and a Doberman. The other cops watched them with interest, the Negro police with flat expressions that said nothing.
The big cop in charge of the detail checked to see that they were ready and adjusted his riot helmet, then drew his club. They advanced across the intersection four abreast and north into Indiana Avenue. It was very quiet now, the eyes of the cops on the dogs and the noise of an El going south from the station a short distance away echoed loudly through the streets. They moved down the street the short distance in which the shops curled briefly around the corner before stopping abruptly where the residential section of the block began. The people on the porches and steps of the buildings saw them first, the others in the streets turned at their shouts.
"They bringing dogs in! Dogs! They bringin' dogs in !"
The cry was picked up and traveled quickly down the block. The cops continued to advance slowly, four abreast, the dogs straining on their leashes.
"Ofay cops with dogs !"
"What the hell they think this is, Alabama ?
"When they going to get the cattle prods and bull whips ?"
Some of the people on the steps of the buildings flanking the street disappeared into the hallway, making their way to the garbage cans on the back porches, searching them for bottles, going into the backyards and alleys to look for bricks. The people in the street moved back slowly before the advancing four.
"Git them motherfuckers! This ain't Birmingham, no damn dogs here ! Git 'em !"
The first bottle came from a porch, then another from a window on the second floor of a building. They shattered close to the four cops and their straining, snarling dogs. At a signal from the big cop they advanced a few yards at a run, scattering the people in the street before them.
"All right, get back, you people, break it up, go on home now. Move it." A volley of bricks and bottles answered him.
One young black boy stood silently on the sidewalk, not moving. The cop opposite him moved toward him and the Doberman started tearing at his pants leg. The cop ordered the dog off and the boy moved away, but slowly, a look of contempt on his face as he looked the cop in the eye. He had never taken his eyes off the face of the policeman, even as the dog lunged at him. He moved away now, watching the cop with that look, his right pants leg in tatters. The people moved back into the street as the four moved past and they were surrounded now, with a space between them and the black, angry faces that retreated before them, the black faces that followed to their rear.
It was not according to the book. There should be cops to their rear to ensure that the cleared streets remained so, but they were in now and the big cop was stubborn and unafraid. He moved on, holding the big dog easily, his club at the ready, ignoring the missiles thrown at him. If they moved within the proscribed distance, he would order the dog to attack. He moved ahead, staring at them steadily from beneath the visor of his riot helmet, ignoring the shouts, insults, bricks and bottles.
Suddenly someone thrust himself through the crowd and he had to order the dog to relax as he recognized the shield of a sergeant of detectives.
"What the hell are you doing in here with those damn dogs ?" shouted Dawson.
"I'm doing my job, that's what," the big cop retorted hotly. "I'm trying to help you. Whose side you on, anyway ?"
"Get those dogs out of here right now! Don't you know what dogs mean to these people ?"
"The captain told me to bring them in," he responded stubbornly.
"Well, I'm telling you to get them out. I'm in charge here. Where the hell is the chief? He said no dogs."
"I don't know where the chief is and I got my orders."
Dawson drew his pistol from his belt holder. "If you don't move them, you're going to have some dead dogs on your hands." He moved back and aimed at the animal. Four other Negro detectives had appeared, they drew their weapons also.
The big cop looked at them with hate. "OK, we're goin', but you'll hear from headquarters about this. Pullin' a gun on a brother officer !"
Dawson looked at him and said softly: "You're not my brother, buddy." He turned to the crowd as the uniformed police turned about and headed south toward Fifty-first Street with their dogs.
"It's OK now, the dogs are leaving, they're going back. Cool it, now. Go on home." But it was too late and Dawson sensed it. The bottles continued to fly, the voices loud in rage.
"They brought the dogs in, just like in Birmingham. It'll be the fire hoses and cattle prods next. They don't give a shit. It the same damn thing anywhere if you black."
"One of the dogs bit a child. A three-year-old girl. She bleeding something awful. They got her in the ambulance and won't let her mother near her ! The bastards !"
The word spread quickly about the dogs and hundreds of people converged on the street between Fifty-first and Fiftieth. Dawson and his men moved among them, the crowd jostling some and occasionally they had to strike back. Reluctantly, Dawson fired a red flare, the signal to rendezvous prior to pulling out. He had his men move the crowd back by force, creating a circle twenty feet in diameter.
"Dan, we're moving back as soon as all my men are here. We'll probably have to move the uniformed cops in now. I don't know if we can snuff it out, but we'll have to charge them. Talking's no good anymore. The dogs finished that."
"I can't leave without Perkins, Daws. It's all right, you're not responsible. I think it will be OK for another few minutes, as long as they don't mistake me for a cop." He saw Perkins move quickly into the circle, his face drawn and worried.
"Shit, Dan, they were going home, they were going home. It was going to be fine until they brought them dogs in. It was going to be just fine." He fought back the tears, his face full of exhaustion.
They formed a diamond pattern and forced their way back down the block, striking out when they had to, taking no joy in it. They would have to crack a lot of heads tonight, they knew, and except for the sadists and power hungry among them, they took no pride in attacking their own people.
Finally, they made it to the barricades on the corner and moved through them. The jeering crowd stopped a short distance away, filling the street. The detectives slumped against police cars and vans, some sat on curbs, perspiration dripping from their faces, drenching their clothes. The police stirred nervously, fingering their weapons, their boredom gone. Dawson strode to his car and yanked the microphone from the dash.
"Baker four to Baker one, Baker four to Baker one. Come in Baker one. Over."
"This is Baker one, Baker four. This is Baker one, Baker four. I read you. Over."
"Better get back here, Chief. Looks like it's gonna be bad. Over."
"What happened? I thought things were quieting down. Over."
"They were. It was almost finished, then they brought the dogs in. You know what they think about the dogs. Over." He looked very tired. Someone handed him a cup of coffee in a paper cup.
There was a pause and the radio hummed. "OK, Pete, hold on. I'll be right there. You at the CP? Over."
"Yeah, Chief. You better hurry. Over."
"Right. We're on the way now. Out." Dawson could hear the scream of the siren approaching from the east. He looked up at the man who had handed him the coffee. "Well, Dan, we did what we could."
"Yeah, Daws." Dawson looked over the barricades at the angry mob. He knew many of them, he had lived for years within blocks of here. He knew their anger and their pain, but there was nothing he could do. He was a cop. He would have a very busy night.
"You know, Dan, they were going home. Another hour and you could have been in the Boulevard Lounge tastin' and those people at home. Now look at it. This could be the worst yet. I've never seen them like that before. Why did they bring those dogs in? Why, man? Don't they know how they feel about dogs? How come they so stupid ? Everything was cool and they had to mess it up. Why do they always mess it up?"
"I don't know, Daws, but they always do."
14
The chief's car screeched into the intersection. Dawson walked slowly to the car and stood talking intently to the chief.
He turned and waved Freeman to the command post. "Chief, this is Dan Freeman."
"Oh, yes, you're with the foundation, aren't you ? Been doing a fine job, I hear. Taken a lot of pressure off our precincts down here. They've been trying the same program, but without much success, over on the West Side. Pete tells me you were in there talking to the teenage gangs. How does it look to you ?"
"When it starts, they'll be the most dangerous and the most destructive. They haven't been rumbling and so they have a 10t of latent hostility to get rid of. Now they have a chance to hit out and become neighborhood heroes as well. Usually the folks here are afraid of the gangs and hate them, but it might be different after tonight. I don't think the Cobras will be very active, 162
although some of them will probably attach themselves to other gangs."
"Hold on. I thought the Cobras were the worst gang in the city." The chief nodded at Dawson.
"They were, Chief." Dawson shook his head and said one word: "Junk." The chief nodded.
"I think most of the trouble will come from the Apaches. They want to take over the reputation of the Cobras around here. There are a couple of smaller gangs who will join in and there are some of the Comanches in there from the near West Side, from the low-cost housing project on the other side of the Dan Ryan Expressway.
"The Apaches will do the most damage, though. They won't be afraid of anything that moves and getting shot, beaten or arrested they consider a badge of honor. They're going to be hard to control. And some of the older men who have been out of work for a long time will probably join in. There are guns in there, too. Your men will have to watch out for snipers from the rooftops when they go in. I have my street workers on aroundthe-clock alert. When you think we can help, Dawson knows how to reach me. But they're not likely to listen to any of us now; not since the dogs."
The chief winced. "Thanks, Mr. Freeman. Can one of my men drop you anywhere ?"
"No thanks, my car is not far from here. Good luck. See you later, Daws."
"Yeah, Freebee. Easy."
Freeman drove his car away and parked it in Washington Park, then walked back to watch. It wouldn't be long before they went for the stores on Fifty-first and they would probably infiltrate the lines through the alleyways when the police moved into the neighborhood.
But they had already started on Forty-seventh Street. The word about the dogs had spread quickly throughout the cordoned-off quadrangle and they had burst into Forty-seventh Street before the cops there, relaxing because of the reports that things were quieting down, knew that the mood had changed. They moved close to the street through the darkened alleys, then burst out of alleys in several places. Some were caught and arrested, others beaten, but more and more poured into the street.
They went for the pawnshops first, smashing the windows and grabbing whatever they could from the inside until cops approached and then scrambled further down the street. By now there were hundreds in the streets, rampaging, smashing and finally burning and south of Forty-seventh, when the cops were ordered inside the quadrangle, they burst into Fifty-first as well.
They went to the blood-sucking stores as steel filings to a magnet : the pawnshops that contained their few prized possessions, pawned for a fix, a bottle, money to hit a number, to impress a girl friend, for a wedding, medicine, an abortion. They hit the furniture stores that made more money in interest by reselling the same set of jerry-built furniture to people who could not keep up the payments than if they had sold the junk outright. They hit the supermarkets and liquor stores and then any store that was owned by whites. They would break in the windows and scramble inside, break the lock on the rear door and begin to haul stuff out through the back. When the cops arrived they would leave and move on to the next place being looted.
Freeman watched them on Fifty-first, strolling down the raging street. Once a cop swung at him with a club, he dodged the first blow and flashed his badge and ID, but thought that he might have to defend himself. The white cops were lashing out at anything black, in terror and hate.
A small boy struggled down the street with a small Japanese portable television set. A man went by with a stack of six hats on his head, several ties draped around his neck and a case of bonded bourbon in his arms. A large fat woman waddled down the street with two large hams in each arm. Too slow, she was intercepted and clubbed down by a cop. Three boys in their teens fought savagely with two cops. One of the cops went to his knees and when he reached for his gun, the three disappeared through a broken store window and out into the alley to its rear.
The cops grabbed, clubbed, handcuffed and arrested. The police vans moved out full of men, women and children, many of them bloody. Firemen tried to uncoil a hose to put out a fire in a furniture store and were driven away by a group of teenagers. Everywhere there were angry shouts.
"Burn, baby, burn !"
' 'Get whitey!"
"Black power !"
A cop was hit in the face with a brick and helped to an ambulance by two others, blood streaming freely onto his dark blue uniform. A boy, running, slipped on broken glass and four cops were upon him, beating and poking savagely at his groin with their night sticks. He screamed and thrashed in pain and they dragged him, bloody and still screaming, to a paddy wagon. Smoke licked up toward the heavy, humid
sky. The police shouted, men, women and children shouted and screamed in anger, hate and pain. One large boy of about nineteen fought a cop, knocked him to the ground with a swinging right hand, then taking his club, beat him while on the ground; then, dashing away, he was shot only two steps from the sanctuary of the dark alley. He lay on the sidewalk moving feebly while the cops carried his victim to an ambulance.
A police car was overturned and set afire, others had their tires slashed. The army of occupation was being attacked everywhere, and since better equipped, having the better of it. But the police took their casualties from bricks, bottles, baseball bats and fists. Sporadic firing broke out inside and Freeman saw two cops brought back to the street, wounded from sniper's fire.
The violence spread, to Fifty-fifth to the south, Forty-third to the north, a section of Halstead Street on the West Side. Everywhere the pattern was the same, break, loot, burn and run. Hurt the cops if you could, run if you couldn't. Take anything of value, stash it at home and then move back into the streets for more. Smash any store owned by whitey; he takes your bread to the suburbs where you can't live, he cheats you, overcharges you, insults you and your women, he hates and despises you. Hit him where it will hurt most: in his pocketbook.
Stores were stripped, then set ablaze. Everywhere people moved, laden with loot. The cops, both white and black, exhausted, frustrated, no longer feared or in control, struck out whenever they cornered someone. Ghetto cops operate in
a state of near paranoia and now they had been pushed over the line. Their protection, their status, their power is based on fear and now these people no longer feared them and the status quo had to be resumed. Gunfire increased, even people who surrendered when challenged were beaten to the ground and dragged to the ever waiting vans. There were no longer any efforts to protect property; only efforts to restore law and order at any cost. The clubs were transferred to left hands and everywhere Freeman looked there were cops on the prowl with drawn guns. People lay sprawled in the streets and sidewalks and cops fired at every fleeting shadow.
Freeman moved down the middle of the street, slowly and smiling, his hands a bit from his sides to show the cops he carried no loot and had no weapons. It was no place now for a black man not in uniform. He thought it would be a hell of a time for him to die, walking innocently down this street, shot by a scared, hate-filled cop. It would be poetic justice, considering what he planned, and he smiled at the thought. He caught the acrid smell of tear gas, borne from inside by a brief, weak breeze. Smoke filled the street, there was broken glass everywhere, overturned cars, empty cartridge cases, discarded loot, an abandoned pair of handcuffs, a pair of nylon stockings lying in a pool of congealing blood. A puppy cowered beneath a car, whimpering. In the broken window of an electrical appliance store stood a color television console, too large to be carted away, sitting there like a fat, rich dowager on a garbage heap. The rest of the store was littered and empty of its wares. The big set was turned on and Freeman could see the rioting filmed live and in color. It was a bit out of focus and he walked over and adjusted a dial until it came in sharp and clear. He watched a moment, cops on the screen fired at an unseen sniper on a rooftop, the deep voice of the narrator sounding as if he were commenting on a high-budget Hollywood film about the Normandy invasion. Freeman thought the cops looked more heroic on the television set than there on the streets. He turned and walked away.
He was stopped three times and once he talked calmly and inched closer to an hysterical cop, who waved a pistol in his face. Before Freeman could disarm him, the cop turned to a wall and sobbed into his folded arms, tightly clutching his pistol and club. Freeman waved to another cop and was almost shot until the cop realized Freeman was not threatening a brother officer. They helped the sobbing policeman into an ambulance and Freeman walked on toward the park.
He walked down the middle of the street, past the television cables, cops, cars, cartridge boxes, trucks, television vans, ambulances, equipment and debris. He stopped, facing toward the park a short distance away, listening to the alien sounds. No music, he thought, no good here without the music. He thought of what he would be doing from now on, and of the years during which he had prepared himself. He had known that one day this moment would come. He wished it had not.
He breathed deeply, trying to shut out the foreign smells until he could grasp one which belonged on this street and finally he could. He
breathed deeply and savored the smell of barbecue.
He drove to his apartment and phoned Daddy at the poolroom.
"We still here like you say, Turk; been watchin' things on TV. Don't look good."
"It looks worse out there in those streets. Three more days of that and the people ought to be fed up enough with whitey to back us." "What if they stop before then?"
"That's the chance we have to take, but I don't think they will. Not after tonight. The cops will make sure of that. They're going crazy down there. Tomorrow they'll divide by ten to come up with casualty figures for the press. You heard from Stud?"
"Yeah, he called 'bout fifteen minutes ago. Man, he having a ball. Almost got burned twice and busted half a dozen times, but he still loose and raisin' hell."
"When he calls again tell him to get out of sight and lay low as soon as things get tight in there. They'll be looking for him when things quiet down. How about the others ?" There was a pause.
"They shot Tony, one of the young kids. They took him away in an ambulance. Stud say, so maybe he ain't dead. They only arrest two others. Stud say them young Cobras is a bitch, slippery and mean as hell."
"OK, I'll check the hospitals tomorrow morning about Tony. Look, you cats better not be seen in the street tonight."
"Naw, Turk, we gonna stay here tonight. We got beer and some food, a deck of cards and the TV. When we get tired we can sleep on the tables."
"OK, baby, I'll call again in the morning. If anything important happens, call me. And tell Stud not to stay down there too long. He's no good to us dead or in jail." He hung up and when he had finished soaking in a hot tub, realized that he was ravenously hungry. He ate a mushroom omelet and almost fell asleep before he had finished eating it.
He arose early the next morning and turned on the television set to catch the news while the coffee brewed. The mayor, in a brief statement, said that "early reports indicated that agitators and Communists were responsible for the rioting." The police commissioner called for a return to law and order and hoped that cooler heads might prevail among the more "responSible citizens of the riot-torn area." A prominent University of Chicago professor of sociology stated that the riots indicated "an increasing pattern of sado-masochistic tendencies as an outgrowth of increased urbanization" and that the rioting was "a parallel to the death wish demonstrated by the Jews in Nazi Germany since the rioters could only face eventual repression." Prominent Negro leaders, none of whom had been in the riot area in years in spite of the fact that most lived within walking distance, deplored the riots and advocated nonviolence as the only means of attaining progress for Negroes. A former vice-president said that rioters should be shown no mercy, regardless of color. The governor of Alabama said I told you so. The governor of Illinois announced that the National Guard had been alerted.
Freeman took the morning newspapers from his doorstep and read them over his first cup of coffee. A cartoonist pictured a bearded figure in sunglasses with exaggerated Negroid features, clad in T-shirt, blue jeans and sandals, attacking a figure labeled "Civil Rights," clad in a Brooks Brothers suit, necktie and hat, who looked like a white man colored dark. The editorial called for a return to nonviolent protest, then suggested that nonviolent protest was responsible for the rioting by fostering disrespect for law and order, and ended by predicting that white backlash would negate the great progress thus far achieved in civil rights.
Freeman's Senator Hennington predicted that the rioting would hurt the civil-rights bill presently on the Senate floor. The president said nothing; his press secretary praised the Negroes fighting for the freedom of the Vietnamese. Three prominent authors, two playwrights and four movie stars resigned from civil-rights organizations in protest. A Negro leader requested the installation of sprinklers on fire hydrants, a swimming pool in the West Side ghetto and the hiring of Negro pilots by Trans-World Airlines. Each of the Negro aldermen denounced "hoodlumism, lawlessness and violence" and praised the mayor for his achievements for the Negro population of Chicago. Their statements were identical because they had been written in the mayor's office of public relations.
The House of Representatives Un-American Activities Committee announced that they would hold hearings in Chicago to investigate Communist agitation among the rioters. The Chicago Tribune requested the National Guard; the Daily
News suggested that Negroes prove themselves worthy of full citizenship by becoming model noncitizens first; the Sun-Times requested an increase in federal antipoverty funds. Time-Life flew in a staff of sixty-three, Newsweek a staff of thirty-eight and the Times of London authorized a round-trip ticket by Greyhound bus for its New York stringer, to cover the riots.
The television networks thought that riots in the North might replace the now absent police dogs, fire hoses, cattle prods and mounted state troopers of the South in entertainment value. They each dispatched top-flight task forces to cover the rioting in full color. Antonioni announced plans in Rome to do a technicolor movie concerning the riots. It would involve one man's agony in trying to decide whether to throw a brick at the police and the entire movie would take place in a kitchenette apartment. Marcello Mastroianni would play the lead in blackface.
Mr. Stephens, chairman of the board of the foundation, phoned Freeman and asked that he be present at an emergency meeting of the board at 3 that afternoon. Freeman ate a large breakfast and drove to the riot area.
15
Freeman parked his car outside the cordoned-off riot area and walked through the check point after showing his ID to the cop stationed there. A short distance down the street he found Dawson seated in his patrol car, sipping a cup of coffee, his face drawn with fatigue. Freeman walked over to the car when he waved to him. "Hey, Freebee, what you doin' in here; don't look like much work for you for a while."
"Hi, Daws. Come over to check out the scene; got a board meeting this afternoon and they'll want a report."
"OK, hop in, I'm just going to cruise the area." Freeman seated himself next to Dawson. "Where's Turner ?" he asked.
"At a briefing downtown; they couldn't spare both of us. He'll bring me up to date this afternoon when he gets back." He eased the car into the streets, empty except for official vehicles cruising the neighborhood. The word had spread that the cops had become trigger-happy and the people were in their buildings or sitting silently in groups on the porches; few ventured casually into the streets.
"How long since you had some sleep, Daws?" "Oh, I been catnapping now and then. They talking about bringing in the National Guard. I don't like the idea of seeing military uniforms in here. And the unit they alerted is an all-white outfit, too. That won't help things at all, but they don't trust the colored guard unit. Or many of us spade cops, either; they been acting like we started it instead of putting it down. Man, the colored cops worked harder than anyone here." Dawson fell silent a moment, gazing through the windshield. The groups of people on the porches followed the car sullenly with their eyes. The odor of smoke hung heavy in the air and the police calls from the squad-car radio sounded loud because of the unusually quiet streets.
"You know, Dan, I grew up around here, but I wonder if I really know or understand these people." "Well," said Freeman, "that badge does put distance between you and the rest."
"Yeah," replied Dawson. "I saw some good friends out in the street last night; not just hoodlums like they say on TV, but some very straight cats, cats with jobs, families, responsibility."
"In a scene like that, everybody can get involved, Daws."
"It was like it was war and I was part of the enemy. Some I can understand, they were always in trouble, but what were the others thinking? It wasn't like they hated me, it was like I was in the way of something important they had to do. "What good will it do? It's not enough to say they had it tough; they didn't have it any tougher than we did. Without respect for law and order, we might as well be back in the jungle." Dawson parked the car on Fifty-first Street and they stared at a burned-out store, the burn smell heavy in the rapidly warming late-morning air.
"Shit, Daws, the ghetto's always been a jungle. You really think you can treat people like animals and not have them act like animals? You really believe you're just a cop and I'm just a social worker? Man, we're keepers of the zoo. You can't cage a whole race of people without asking for trouble."
"You agree with what they did ?" Dawson turned to face Freeman, one arm draped on the steering wheel.
"No, because it won't mean a thing. You watch how the white folks will turn it around. They'll have a flood of people down here running surveys and shit, all designed to find out how to maintain the racial status quo without paying riot dues; how to get the niggers to go back to sleep. Whitey won't get the message because he doesn't want to get it. Things will quiet down and we'll both go back to working for Mr. Charlie—keeping niggers in their place."
The police calls came through the radio with a metallic quality, echoing in the quiet street. The sound of an approaching El grew in the distance.
"I can't buy that, Freebee," said Dawson seriously. "Somebody has to do what we're doing and we're a lot better than most. Don't you think it's better to have a cop who grew up here, who knows the people ?" The El stopped at the station just down the street; the people aboard pushed
close to the windows to gaze raptly at the ravaged street below.
"Better for whitey. It takes a nigger to catch a nigger."
"The streets have to be safe."
"Whose streets? Were our streets safe for our women when we were kids? Whitey discovered Negro crime when it spilled out of the ghetto and began threatening him. They don't give a damn what we do to one another." Two big cops in full riot gear, carrying three-foot riot sticks, noticed Freeman in the squad car and approached it in case Dawson needed assistance. Dawson waved them away and started the car, moving west toward State Street.
"We have to clean up our own backyard before we can expect whites to help us."
"You come on like Time magazine, man. The conditions they force us into cause the crime, then they use the crime to justify the conditions." "If every Negro worked as hard as you have to get where you are, we wouldn't have a problem."
"Later for that jazz, Daws: run twice as fast to get half as far ; whitey sitting there with his hand on the controls of the treadmill laughing his ass off."
"I don't think you really believe that." They had turned into an alley and the backyards were crisscrossed with lines of wash, the back porches crowded with people. One man arose, walked to the railing of his porch and contemptuously spat at the squad car as it slowly moved down the alley. "It's not easy working and going to law school, but in a couple of years I'll have my
degree and I think it's worth the extra effort." He waved toward the crowded porches.
"If everyone out there devoted as much energy trying to improve themselves as they did last night burning and looting, they might be where we are."
"Where we are? Where the hell are we? How are we any different from them ?"
"You know we're not like them."
"You think we're different because you got a badge and I got a couple of degrees?"
"We gotta be."
"You know what whites call people like you and me in private?"
"What?"
"Niggers !"
"Cool it." He looked at Freeman with an inquiring smile. Freeman, working to control his anger, forced himself to return the smile. "I know you don't really believe that stuff, Dan. You're tired and upset about this scene, just like me." "Yeah, I guess you're right, Daws. It does bug me to see all that work go down the drain. When things quiet down again, I have to start right from scratch. How long you been on the force
"Almost fifteen years. I can retire with a full pension after twenty. I joined right after I dropped out of school when I got hurt and it finished me for football."
"How come you quit school? Even though you couldn't play anymore, they figured to continue your ride. Public relations for future recruiting." "Yeah, I know, but I just couldn't stay around after having been the big hero. I had two good
seasons, Dan, two very good ones. They were letting freshmen compete in the Big Ten at that time, remember? I made first string as a freshman and even missing two games the next season after I injured my knee, I made all-conference as a sophomore." He paused, remembering.
"You would have been a cinch for all-American if you hadn't missed those two games; even so, you got honorable mention for AP and third string for Look magazine."
"That's the hang-up, Dan, everything was so groovy and then, nothing. Man, I couldn't stay around, the crippled ex-football star. I stayed long enough to take final exams so I'd have the credits for the semester and then I just came back to Chicago without even saying goodbye to anyone. The school was straight, though, they paid for the operation on the knee and as soon as it was strong enough to pass the police physical exam, I joined the force and I've been with them ever since."
Freeman asked the next question carefully. "Aren't you about due for a promotion ?"
Dawson laughed bitterly. "You trying to put me on, Freebee? You know the quota is full up : one spade captain of uniformed police and one lieutenant of detectives. If either one drops dead, I'm in, otherwise, I'm stuck as a sergeant until one retires. They keep telling me how swinging it is to be a detective sergeant at my age, like they doing me a favor and I had the highest score on the exam when I made sergeant. But, it's not so bad, I could be doing a lot worse." Freeman thought Dawson was not quite ready yet, but his justified indignation at not having the rank he deserved might be just the wedge he needed to recruit him. And, Freeman thought, the things he would see when the guard came into the neighborhood would not enamor Dawson of white folks. He would bide his time, work on him quietly, plant the seeds and cultivate them over a period of time. Dawson would be a good man. He could take over if they ever blew my cover, Freeman thought. I'll just have to be patient a little bit longer; one day the white man will force him to make a choice, he can't straddle the fence forever. He looked pretty damn good against those dogs last night.
"Daws, why don't we go on down to Mamma Soul Food's on Forty-seventh and eat? I'm starving."
"Good idea, Freebee; some pork chops would knock me out."
They drove through the checkpoint and on to Forty-seventh. They stopped first for a drink in a dark, air-conditioned bar and watched the Cubs playing San Francisco on the television set there. They watched long enough to see Willie Mays double in two runs and then walked the short distance to the small restaurant, ordered and ate in silence, each comfortable in the old friendship and lost in their own private thoughts. When they had finished, Dawson drove Freeman to his car, parked just outside the riot area.
"Easy, Dan, don't let those board members string you out."
"You better hope not, because if they do, I'll join the force and take your job in a couple of years."
Dawson laughed as he drove off. "You might make a pretty good cop; we'd make a good team."
Freeman smiled and waved. Yeah, he thought, we could make a damn good team, but not working for Mr. Charlie.
He drove to his office, not far from the University in the Hyde Park district and on the opposite of Washington Park from the riot area. He entered the paneled reception room, decorated in Danish modern, and smiled at the pretty colored receptionist. "Afternoon, Mr. Freeman. Mr. Stephens wants to see you before the meeting; everybody's shook up today."
"Thanks, honey. I'll try not to lose my cool." He walked down the hall and past Stephens's office into the toilet. As he washed his hands he checked his mask in the mirror for slippage ; this was not the time to arouse suspicion. The face that returned his stare in the mirror was sincere, serious, concerned and just a bit worried that it might lose a five-figure salary because the natives had been restless last night. His mask was perfect. He dried his hands, adjusted his tie and walked down the hall to reassure the white folks.
The board members were talking nervously in Stephens's outer office; Stephens's secretary motioned Freeman into the inner office when he arrived.
"Hello, Dan. I'd like a brief report before we convene the meeting. Needless to say, most of the board members are rather agitated about this thing. I understand you were there for most of the worst of it. What happened?"
"Same old thing, the heat was getting to the cops and the people over there and there had been the usual increase in head whippings for the summer-time; then a cop shot a kid and upset a lot of people. They had almost quieted down when they brought dogs in and that was like touching a match to gasoline."
"How was it over there?"
"Pretty bad. Looks as if they might have to call in the guard. I don't think the police can handle it now."
"How about the street gangs ?"
"They're in it, but only one, the Apaches, are spearheading anything. No, it's not a hoodlum scene like they say; the whole neighborhood is involved, if not actively, then they support it." "What do they hope to gain, Dan? This kind of thing could make it worse for them instead of better." Stephens played nervously with a small, bulldog pipe and stared at Freeman intently.
"They don't hope to gain anything. They don't have any hope, anymore, they're convinced nobody gives a damn about them and their problems. They're fed up with being ignored, pushed around and taking crap and now they're kicking back."
"Revenge ?"
"No, not revenge, retaliation; there's a subtle but important difference."
For a moment Stephens's face opened and disapproval of Freeman showed there briefly, as if Freeman had been personally responsible for the rioting which now threatened their jobs. Stephens looked down quickly at his desk blotter, the blotter a pale shade of violet held in place with hand-worked Florentine leather. When he looked up again he had a warm, if somewhat tight, white liberal smile on his face.
"Dan, I've spoken to most of the board mem-
bers and they're very upset that the money we've spent to prevent exactly what happened last night has been wasted."
"I disagree, Steve. Street-gang activity among the rioters was at an absolute minimum and the Cobras weren't operating at all. The riot-control task force report will substantiate that."
Stephens leaned forward eagerly toward the first sign of hope that morning, like a dog who does not really believe the bone being held out invitingly will be offered.
"That's good news. Do you think the foundation can take any credit for that ?" His white liberal glow increased and Freeman fanned the embers.
"Of course. If it hadn't been for the foundation, those kids would have been directly involved and the damage infinitely worse."
"I hope you can confirm this in the meeting, Dan." Stephens licked his lips nervously, hitched his pocket handkerchief and patted it into place as if to be certain his heart still beat beneath it. "There's talk among the board to dissolve the foundation. Burkhardt has been particularly critical. You know what it would mean to have the foundation ended. Jobs like ours don't grow on trees."
Freeman allowed a moment of panic to show in his eyes, then looked out of the window as if to contemplate the horrors of job hunting. He thumbed through his masks and chose one of smiling confidence.
"I picked up a copy of the report of the chief of the riot task force first thing this morning." He opened the slim attaché case he had placed at his feet and offered the report to Stephens like a tranquilizer. He spoke in clipped, somewhat more white-type tones than usual. "It indicates that street-gang activity was a minor factor in the rioting and cites us by name as the major reason for the inactivity of the gangs as units in the rioting." Stephens leafed through the report.
"Good thinking, Dan; this report could prove invaluable. If we can come through this crisis unscathed, we should be a cinch for the Ford grant. And you know that means substantial raises in pay and allowances for both of us. And nationwide prestige as well. Yes, I think this report, plus your own oral presentation today, could turn the trick. We have to increase our budget, not kill the foundation." "Precisely !" said Freeman.
Stephens decided to fill his pipe; in moments of doubt, his pipe somehow made him feel wiser and more secure. "I talked to the chief on the phone this morning, Dan; he had high praise for your activity last night."
"Perk was there, too; he did a damn good job. I'd like it mentioned in his personnel file."
"Of course, Dan. Draft a statement and I'll endorse it. I think we'd better go into the conference room now. This will be a tough session."
"I don't know," said Freeman. "It's a lot tougher across the park."
Stephens opened the meeting and Freeman gave a brief report.
Burkhardt spoke up: "I might as well be frank about this thing, I'm beginning to have serious doubts about the worth of this foundation. We spend a good deal of money to prevent this very kind of thing and I wonder if we want to pour good money after bad." He wore heavy horn-
rimmed glasses too big for his nose and he pushed them back up with a nervous gesture of his right hand.
"The streets aren't safe. What's to prevent some of those hoodlums from coming into Hyde Park to continue their burning and looting? I've sent my wife and children to her mother's in Winnetka and I've loaded my shotgun, deer rifle and pistol !"
Stephens spoke soothingly. "We're all concerned about this kind of thing, Burk, but I don't think it's as bad as all that. Dan here reports that street-gang activity is at a surprising minimum and the commandant of the riot task force confirms it, and he has high praise for Dan here and our foundation.
"Furthermore, the King Cobras, the potentially most dangerous gang in the area, is not functioning in the rioting as a unit and that is the outfit which had our top priority when Dan was brought in."
"Well, that is good news; perhaps the papers have exaggerated a bit. But still, we are less than a mile from the major rioting. What's to prevent it spilling over into our neighborhood ?"
Professor Thompson, of the department of sociology and social welfare at the University of Chicago and a foremost expert on juvenile delinquency, spoke: "Well, Burk, we've found that people from lower-income groups, and particularly the culturally deprived and socially disadvantaged, seldom cross what we term natural urban barriers such as the Midway mall, an expressway, large, busy thoroughfare or Washington Park, for instance. The park acts as a natural
buffer and I think we need fear little in that
sense.
"Fortunately, there are no middle-class areas contiguous on the riot area and there is little chance for the rioters to single out people better off than they as targets of their discontent and frustration."
What you mean, baby, is that there are no white middle-class areas contiguous on the riot area, thought Freeman. The slum portion of the ghetto ends at Sixty-first. But Freeman didn't think they would attack other Negroes, even the black middle class.
"I would tend to agree with Steve, that our organization has performed well beyond the level to be expected. That our activity among the youth of the neighborhood could act as a restraining factor, even in this moment of dramatic social disorganization, is little short of remarkable. In fact, I intend doing a paper concerning our role among the rioters for presentation at the next national convention of the American Society of Sociologists, if Dan would be so kind as to contribute some of his own observations." What you want, Professor, is for me to write the damn thing for you so you can inject some sociological gobbledygook and call it your own, Freeman thought sourly ; but his expression never changed.
The board meeting continued and after Freeman indicated what was planned, once things quieted down in the riot area, Stephens adjourned the meeting. They filed out, stopping to shake Freeman's hand, smiling and calm now, no longer afraid because Freeman was a good nigger, their nigger and he would protect them from the others. Even Burkhardt, who had always been awkwardly brusque with Freeman, smiled, pumped his hand, patted him on the back, chattered small talk at him and suggested they have lunch together one day. That's what they hired me for, thought Freeman, the house slave to keep the dangerous, dumb, nasty field slaves quiet and now that they know that it's only niggers raising hell in their own neighborhood, they feel fine. He watched them from behind his mask, betraying nothing, smiling, saying the right things automatically; then they were gone and he was alone in the room with Stephens.
"Well, Dan, I think that went much better than we had any reason to expect. I'd reminded them all before you arrived that you had predicted trouble of this kind this summer and I think they were pleased at the report regarding juvenile activity last night during the rioting. I think there's a raise in this thing for you."
"What about the street workers ?"
"Of course, the street workers, too, but that will have to await the grant from Washington. But, I'm requesting an immediate increase in your salary at the next official board meeting and I can see no opposition on the horizon. How about dinner ?"
Freeman couldn't eat with this white man, not after last night. He knew he might lose his cool and blow his cover.
"I don't think so, Steve. I've got a lot of checking to do before the staff meeting tomorrow afternoon. As soon as things settle down over there I'm putting the entire staff of street workers into the area and we'll have to work out a crash program. "
"Good idea, Dan. Next week, then." As he turned to leave the conference room Freeman silently handed him his forgotten pipe.
Freeman drove the few blocks to his apartment, took a steak out of the freezer compartment to thaw, put a potato into the oven to bake and made a lettuce-and-tomato salad. He walked into the living room and turned up the air conditioner, then mixed himself a martini. He sat down and sipped the drink slowly, waiting for the steak to thaw, looking at the Javanese Buddha head of black volcanic stone.
He had never thought that he would enjoy what he would do and now that the time had arrived, he knew that he would not enjoy it at all, but that would not change things any. He wondered how many of them would be alive this time next year and he wondered how many more they would kill tonight. The papers had reported three killed, but he had seen five dead the night before on Fifty-first alone—target practice with live ammo and live targets. But then, the bull'seye of every target he had ever seen had been black.
which caught her attention. It was covered with bones, animal and human, arranged in patterns of circles and squares. The use of these examples of mortality for decoration struck Theodora as some sort of perversion to which she could put no name.
Sir Lionel busied himself with a silver spirit stove and kettle at the other end of the room. 'I see you admire my memento mori.'
'Unique, surely?'
'Possibly, though there is an account of something rather similar in a Venetian palazzo circa 1670 mentioned in Giovanni Sestini's Antiqua Classica. We're more squeamish nowadays. I expect there are laws against it. "Health and Safety at Work" like as not. I don't have to bother about that. I have no staff. The collection was started by my grandfather in a more robust age. My father added to it and I have put in the odd femur as they cropped up.'
'You've dug all over the world, Sir Lionel,' Theodora steered him towards what she hoped would be safer ground. 'I read with great pleasure your Digging A way. ' 'How kind of you to say so, my dear. Yes, I've dug in a great many places.' Sir Lionel returned to his chair and leaned over the back of it. He eyed Theodora as though she might be a likely site. The kettle began to sing and interrupted whatever more he might have had in mind. He limped off to make the tea.
Theodora wondered how she was going to get what she wanted out of Sir Lionel. She'd hacked out to Quecourt by the afternoon bus to see the bishop suffragan. Bishops made her nervous. She'd known them from her earliest years; suffragan, diocesan, colonial, south Indian, Russian Orthodox and Roman Catholic had all been regularly entertained at her father's house. She must have handed tea and biscuits to scores of them. But she still felt a slight irrational apprehension when she had to deal with them in the line of duty. Nevertheless she had got what she needed in the way of information from Bishop Clement with no difficulty at all. He had shown her the typescript of 'A View from a Pew'. As soon as she saw the typeface, childhood memories had flooded back. It was undoubtedly the product of an ancient Remington. There was just such a machine in the vergers' office. It would make perfect sense if one of the vergers had been responsible for the articles. Dennis she discounted on grounds of age, literacy and loyalty. That left Knight or Squires. She did not think she would have much difficulty, armed with the evidence, of finding out which it was. The bishop had said, after some thought, 'I'd like to know, if you find out but unless you feel it relates to poor Vincent's death, it may, as you suggest, be more tactful to let it go. Provided of course it doesn't happen again.' Theodora had been amused at so arbitrary a denial of free speech from so gentle a man.
She had walked the couple of hundred yards across the village green from the bishop's house to Sir Lionel's rather grander establishment wondering whether indeed the articles did have anything to do with Dean Stream's death. She wondered too quite how she was going to account for herself to Sir Lionel. She had no worries about being received. When she had rung at lunchtime to ask if she might see him for a moment, Sir Lionel had been warm in his invitation. But now she was here, how could she put the questions she needed to if her suspicions were to be verified?
'Will you have gleanings from the Janus dig?' she ventured the best Horatio Alger tradition. Put down all the spades too lazy to do what I've done. But whitey helps; he might tell you that you're "different" from the rest and in the next breath remind you you're all the same underneath your manicured fingernails and J. Press suit.
The ghetto had been a refuge in his childhood. He had had no close contact with whites until he entered a high school split almost fifty-fifty, white and black. But it was two schools, as rigidly separated by unspoken rule as if someone had painted signs saying white and colored. Freeman hadn't thought much about the whites; he had his own colored friends.
It happened during the beginning of his second year, just after the war with the black veterans returning from the "war of freedom" bitter and disillusioned. It happened on a bright, autumn Chicago day.
It was sweater weather, gray and the sun peeking through now and then and the wind toward the lake. The football season was to start that Saturday. The big red streetcar moved down Sixty-third Street under the El tracks like a big bug in a tunnel. It was crowded by the time it reached South Park Avenue. A nice crowd. A laughing, friendly, teasing brown and black crowd. When it reached Stewart and the crowd tumbled off, laughing and loud, there was another car on the corner across the street, an eastbound streetcar, and its crowd was mixed : black, brown and white.
The sun was out now and the students would be sitting on the low concrete wall in front of the school, some on the porch of the sweets shop across the street, the girls laughing soft and low, their teeth flashing white in their brown faces. The leaves should be dropping off the trees and a football silhouetted against the blue, gray and white sky, a blackjack game in the alley and the school cop ready to saunter conspicuously across the street a few minutes before the bell rang, to see that the gamblers weren't late.
It wasn't like that. It was frozen. The few smiles were frozen, the sullen looks and the uncertain frowns were frozen, the battered football lay on the sidewalk unnoticed and the laughter deep down in a pretty brown breast was frozen and the group of white faces on the steps of the frame house across the street from the school were frozen with bleak, determined looks. Then it started, from the porch, from the white faces, low and weak and then getting stronger—"Old Black Joe."
"I hear those darkies' voices gently calling . . ." No breathing in brown throats and then a slow movement to the curb and the bored cops in the squad car on the corner didn't look bored anymore. The singing faltered for a second and then the voices got stronger.
"I hear those darkies' voices gently calling . . ." In the street now and the squad car turning its engine over and no laughter and no smiles and now fear in pretty blue eyes, a defiant sneer above a black leather jacket and a sign pulled into view: WE WON'T GO TO SCHOOL WITH NIGGERS ANYMORE.
"I hear those voices gently calling . . ."
A ring around the porch now, a black and brown ring and the black leather jackets and blue eyes and blue jeans and white faces, black hair and blonde hair, on the porch, standing on
the steps now and hate, hate in white faces, in brown faces, in black faces.
A white handkerchief flashed in the sun, was wrapped around black knuckles, and another, the sun glinted on a knife blade and the street was full now and the squad car pulling out from the curb, and the cop talking into the microphone. "I hear those voices..."
White handkerchief around black knuckles and knives along their legs and no more song and no one seemed to breathe. A bottle shattered on a curb and a tense black boy moved toward the porch, the bottle's jagged neck in his fist.
"Look out. Gonna get me a goddamn ofay today."
The cops out of the car, then in the car, the black boy inside and the crowd moving around the car, pressing close and fear now in the fat, flaccid faces inside, not bored anymore. The car rocking, rocking, rocking in a sea of faces, black faces, brown faces, yellow faces. Fear and hate and curses and then a voice. A bass voice from a brown face, over the crowd noises, clear :
"I hear those darkies' voices gently calling . . ."
The crowd picking it up and the car rocking in rhythm.
"Let him out, white man. Let him out or we open this car like a sardine can and come get him. And we get you too, white man, we get you good."
"Get back, go on to school, you kids."
"I hear those voices . . ."
"Let him out."
The car rocking in a dark sea.
"Better let him go, Jim. They got some squads on the way."
The black boy back in the street, the crowd milling around. Then a movement toward the forgotten porch, empty now, the leather jackets, the long hair, the blue eyes and blue jeans gone and the sign on the steps—WE WON'T GO TO SCHOOL WITH NIGGERS ANYMORE.
No song and no smiles, the sun is out and it's sweater weather with the first football game Saturday, but no laughter from pretty brown throats, no blackjack game in the alley, no football against the sky, and the sun on a sign on the porch.
The martini was sour with the memory. He put it down and left the apartment. The stereo would shut itself off automatically. As he closed the door, he heard Broonzy: "If you white, all right. If you brown, stick around; but if you're black, get back, get back . . ."
He walked out into the warm air of the street and began walking, ignoring the car. He found himself on East Forty-seventh at Drexel Boulevard. He crossed, and answering the smell of broiling barbecue, he walked into a shop on the north side of the street and ordered a plate of short ribs with medium-hot sauce. He walked outside and headed west again, took the paper plate from the bag and holding it in his left hand, tore off a rib, eating as he walked. Except for the passing motorized National Guard patrols, the ghetto seemed normal again. The burning had been light this far east; west of South Park Boulevard it was much worse. He finished the ribs, coleslaw and French fries just the other side of Langley, wiped his hands on the paper napkin, then rolled the bones in the paper plate and bag; placed it in a garbage can.
He stopped in front of a record store and entered; there was an Otis Redding record playing on the speaker outside. The girl behind the counter was dancing to the music and Freeman joined her; they smiled at one another across the counter. He pointed to the 45 rpm record on the turntable when it had finished and the girl placed it in a bag for him.
"Sweet thing, what time you get off work ?" he said.
"Sorry, baby, but my old man's picking me up, you know, Uncle Tom ?"
"Well, I sure wouldn't want to mess with him; but every chick I see says that's her old man." He smiled.
"Well, baby, he could be any of 'em. He could even be my old man."
"Sure could, so treat him right."
"Honey, I been taking good care of him since the thing started. He was out in them streets night it started and he wasn't nothing but man." "Whole lot of men got made that night."
"Better believe it ; lots of cats walkin' tall out there and chicks diggin' it."
And she was right. There was pride and defiance in the ghetto in place of sullenness and despair. There were jokes about the National Guard everywhere. Freeman stood on a street corner and watched four men "loud-talking" for a small audience of their friends. Each would sustain a comic monologue as long as he could about the white soldiers of the National Guard and would have to move offstage for another when his comic inventiveness became exhausted.
Freeman moved on, still chuckling. Thirsty, he turned into a small dark bar.
He paused just inside the door, allowing his eyes to accustom themselves to the gloom. Recognizing a woman at the bar, he walked over to her.
"Hello, Mrs. Duncan. Mind if I join you ?"
"Fine."
He motioned to the bartender and ordered a Ballantine's ale. "How's your son? He hasn't been hanging out on the block and some of the boys are worried about him."
"Shorty don't have time to hang out; he got a job now, making good money. Just bought me a new color TV."
"The boys say he's not just pushing, but he's hooked."
"Now, he ain't really no junkie. I mean, sure he shoot up every now and then, but most of the kids do that nowadays; they don't seem to like whiskey much. He say he ain't hooked. I don't imagine he got more than a twenty-, thirty-dollara-week habit and that ain't no habit at all. Woman down the hall where I live, now she got a habit. If she don't get a fix when she need it, well, we all got to go down there and hold her down and clean up when she get sick and all. Sometime us people got to be with her day and night; we got to take turns. Now that's a habit. My son ain't got nothing like that, no sir !"
"Did you ever think he might wind up in jail ?" "Not 'less somebody put some heat on the precinct station and the fuzz need to bust somebody and then he won't have to do more'n five, six months and the people he work for be putting money away for him all that time. But they like him a lot, say he a smart boy, so I don't think they choose him to get busted, even if the heat's on."
"Does he ever think about going back to School
"He don't want no part of school. You know how them teachers is, they hate us and sometimes the colored ones the worst. How come they hate us?"
"Fear."
"What they got to be afraid of? Seem like they got everything going for them."
"Did you ever think Shorty might get some kind of regular employment ?"
"Mr. Freeman, don't start comin' on like a social worker. Everybody know you hip to our scene. You about the only social worker they got any time for. How he going to get a job? Time was, when I first come up from down South, there was plenty work, but there ain't nothing nowadays. Steel mills done automated and the stockyards moved to them states where they don't like unions."
"You mean the states with the right-to-work
"Yeah, them states don't like unions. And the factories moving, too. And down South! Ain't that something? We come up here to work in 'em and they move the factories back where we come from. Mr. Freeman, you know it ain't no work for us, 'less it's out there in them streets."
"Yeah," he said softly, almost to himself. He finished his beer, motioned the bartender and paid for hers as well. "Well, tell Shorty I said hello."
"I sure will. He just like the rest of the boys ; think you really swinging. Thanks for the drink." She returned her attention to television as he left the bar. He ran into Shorty, dapper in a lightweight suit, skinny-brim straw hat and wraparound shades.
"Hey, Turk, how you doing?"
"Cool, Shorty. Just saw your mother inside." "She been kinda worried about what's been happenin'. Think we can do anything to help
Freeman turned to look at the bar. "No, Shorty," he said as he moved away, "she doesn't need any help from us."
He slowly retraced his steps along Forty-seventh, walking home. It was dusk now, a few hours before curfew. The streets were full of people, the neon flashing, red, pink and blue on the dark skins of their faces. He listened to the music and sounds of his home, sniffed the smells. He turned south on Drexel Boulevard and walked along the almost deserted street toward Drexel Square. Automobiles made a wide circuit of the riot area in case there might be another outbreak. The hot air became hotter and more still, there was no breeze blowing at all; then there was a faint stirring from the lake to the east and the first faint smell of rain. He had just crossed Fiftieth Street when the first big drops came down splashing on the pavement still hot from the day's sun. The rain increased and there was first the musty smell of the concrete sidewalks, then the sharper smell of the asphalt and as it increased, the clean green smell of the trees and grass.
One thing about Chicago, he thought; even in the ghetto there are trees and grass. He smiled to himself. That's supposed to make being black and poor all right. When Watts happened, all them white folks saying, "What they rioting for? Why, they got palm trees in that slum !" A palm-treelined slum; it could only happen in America. It was raining harder now and Freeman walked easily through the now deserted street, the cool rain feeling good on his skin. Suddenly, he began to run, slowly at first and then, as his muscles became warm and loose, faster.
It was dark now, the streets beautiful in the faint light from the street lamps and apartment windows, the trees and grass a translucent green. He ran across the street onto the broad strip of grass which ran down Drexel. If a cop comes along now I'm a dead nigger. Any nigger running at night has got to have done something wrong. He smiled to himself and sprinted the last hundred yards to Drexel Square. He stood near the fountain listening to the water running in it, to the sound of the rain on the streets and pavement. The rain had awakened the clean smell of Washington Park just across Cottage Grove and from somewhere in the large kitchenette apartment building behind him he could hear someone playing James Brown.
He walked to Fifty-fifth Street and then east to his apartment building. It was still raining when he entered his apartment, his clothing soaked. He undressed in the bathroom and took a very hot shower, then went into the kitchen, wearing his terry-cloth robe. Even after the barbecue he was hungry and he prepared a cheese omelet and washed it down with Carlsberg beer. He walked into the living room and looked at the
unfinished martini he had abandoned when the apartment and his cover, the things he was doing, had closed in on him. He took the glass into the kitchen, poured out the warm martini and rinsed the glass.
He thought of Shorty's mother as he mixed himself a scotch highball, walked into the living room and put Muddy Waters on the turntable.
He sat listening to his Mississippi delta sound:
"I'm ready, ready's anybody can be . . ."
17
Freeman met with his staff, Dean, Scott, Stud and Pretty Willie, one night in the back of the poolroom. The riots had continued for four nights and the calling up of the National Guard seemed imminent, but the mayor was milking the riots for every bit of sympathy and publicity that he could muster; his picture was in every paper, there were shots of him on the television newscasts blaming the rioting on Communists and agitators, speaking in his flat, unattractively nasal voice. Politicians were falling over themselves to get into the spotlight and say their piece, having run out of anything new to say about Vietnam. Since the National Guard would be under the control of the governor, the mayor was reluctant to request its use to put down the riots, but there was increasing pressure to call in troops and Freeman figured they would be in within twenty-four to thirty-six hours at the most.
The rioting had leapfrogged from the twenty200
four-square-block area originally cordoned off and police chased groups in an area as far north as Roosevelt Road and as far south as Sixtyfirst, and the Negroes in the West Side ghetto, led by the Comanches, were getting into the act. The police by now were exhausted and resorting to guns more often. The official total of dead was now placed at twenty-eight, none of them white.
"OK, that's the scene. We make our move on Sunday, but since it looks as if the guard will move in tomorrow night or the day after, we'll have to move some of our equipment outside of the ghetto and stash it. If they impose a curfew and set up checkpoints, we'll have to slip out the best way we can.
"Daddy will handle the command post; we'll use Stud and Willie's attack teams, and Willie's will handle the demolition. Stud's team will provide cover. We don't want any shooting if we can help it, we'll have plenty of that later. Right now we want to hit whitey in his own territory and make him look silly. The white man can handle a put-down, but a put-on hangs him up.
"Now, the timing has to be perfect or it spoils the propaganda effect. We want a series of dramatic incidents to advertise our existence, and then we can get down to the dirty work we've been training for. We'll go over this thing in detail, just like with the bank job, every night until H-hour."
When Freeman left through the back entrance of the poolroom, they were huddled over the plans spread on a pool table.
He had been in the streets every night since the riots began. The activity was not as intense,
except at sporadic moments when a particular act by the cops would arouse people at a spot here and there in the area. Nevertheless, the people of the neighborhood continued to hit back whenever they could. The women shouted insults at the police, white or black, or ignored them as if they did not exist, even when following their orders. The arrests continued in an effort to break the rioting by jailing the "ringleaders and troublemakers." The police spent a great deal of time looking for Black Nationalists and Communists, but the riots had been going on for three days before they could produce a leaflet having any Marxist-Leninist overtones. The ambulances and vans, full of the arrested, moved regularly in and out of the area.
Emergency rations had to be brought in because each of the supermarkets had been stripped and many of them gutted by fire. The establishment began to retaliate. Social workers increased their harassment, but the white social workers had to be escorted by police, those few willing to venture into the riot area. Welfare checks were held up; the unemployment compensation office on Forty-seventh and South Park was closed for "renovations" ; adult literacy classes were discontinued; playgrounds were closed; the swimming pools in Washington Park were emptied; a curfew of ten o'clock was imposed. Bars, liquor stores and poolrooms were closed. People were stopped in the streets repeatedly by police and searched.
Freeman was known throughout the area by now, by both the police and people. He chatted with the street gangs, not preaching to them, asking few questions, listening when they wanted
to volunteer anything and they knew he would not carry tales concerning their raiding and looting to the police. Rioting had been largely reduced to the gangs and the older, more bitter and desperate men, unemployed and without hope. But the rest of the community did not object to their activity; they felt it was past time to let the white man know they existed. Freeman knew they no longer gave a, damn. Advocates of nonviolence were trucked in by the police from their normal "integrated" haunts to try and pacify the crowds and were met with general abuse and ridicule.
Each night they met in the closed poolroom and went over their plans for Sunday night. Two days after the first meeting, the National Guard moved into the area. The people were not impressed with them any more than with the police, but they had more weapons and armor. The commander of the guard unit held a press conference full of tough-guy clichés and indicated that he expected a return to law and order immediately, or else. He did not indicate what "or else" meant, but he was used to giving orders and having them obeyed. He did not expect to have any trouble with the inhabitants of the ghetto and was somewhat dismayed that the police had not been able to handle the riots, but he had become convinced that the police mollycoddled hoodlums. He thought that he and his men could show them how it was done.
No one was willing to believe that the rioting was a spontaneous uprising by people disgusted with their lot, least of all the mayor and the politicians. They continued to search frantically for Muslims, Communists, Black Nationalists and agitators of any political kind, stripe or spectrum and proudly displayed any they could arrest for incitement to riot or any literature, pamphlet or signs they could uncover. The mayor had often blamed Republicans for his trouble, but felt that Communists and agitators were more appropriate scapegoats for the rioting. A drab, colorless figure, he was known nationally as a powerful politician in his party, but he was almost a nonentity outside of Chicago. Now his picture was in papers everywhere and he began to think in terms of national office for the first time in his career.
The National Guard commander, Colonel Archie "Bull" Evans, shared a common trait among military officers—he was a loud-mouth fool. He had tried to stay in the regular army after the war, but could not bear giving up his gold major's leaves for the stripes of a staff sergeant and returned to his hardware store in suburban Aurora, joining the National Guard and in the ensuing years attaining the rank of full colonel, commanding an infantry National Guard regiment.
His regiment was sloppily trained and illdisciplined but because of the colonel's mania concerning spit and polish, impressive in garrison and on the parade grounds. The colonel himself was a model of military dandyism, as impressive in uniform as he was nondescript in mufti. He wore a beret of infantry blue and the infantryblue ascot, preknotted and with hidden snaps, at his throat. There were the green tabs of the combat officer on his epaulets, pinned there by the regimental insignia, his uniforms were tailor-made and of tropical worsted or tropical gabar-dine. He wore spit-shined paratrooper boots made for him by a bootmaker in London. He wore a pearl-handled, silver-plated Colt .38 and carried a malacca swagger stick. The colonel wore his hair in a close-cropped military brush and surrounded himself with short staff officers to minimize the effect of his five-foot-seven-and-a-half-inch height. He had a short, thick neck and powerful shoulders and arms and a deep chest and he loved to arm-wrestle larger men and beat them.
Colonel Evans's first press conference was a near disaster and the television crew was relieved that they were filming on tape instead of live because of his profanity. He told the members of the press that he intended teaching the people of the area respect for law and order and that there would be no further nursemaiding of them. He said that he did not believe in special treatment for Negroes. They should stop whining and start working and earn the respect of the white citizenry by demonstrating their worth. Within twenty-four hours he announced that things were back to normal and that "they are back on the bottom and we are back on top."
"Man," said Freeman after reading Evans's comments in the papers and observing him on television, "that cat couldn't be any better for us if we ordered him special."
Within thirty-six hours Evans had solidly united the inhabitants of the riot area, many of whom had tired of the rioting and disruption ; a public-relations feat of major proportions. Freeman observed the guard unit with an ex-infantry company commander's trained eye and noticed its defects beneath the glitter.
"It's a jive outfit, not much better trained than a boy-scout troop. They're a drag by National Guard standards. First sign of pressure, though, and they're going to be trigger-happy as hell. They'll probably waste a lot of people.
"But when we're ready, they won't be hard to get rid of at all. Man, that colonel is so stupid, I wonder how come he's not a general."
They had checked weapons and explosives in three large suitcases in the checkroom at Union Station. Three went to the checkroom wearing the red caps of the station's porters and checked out the suitcases. Freeman watched it all through the plate-glass window of the coffee shop across the big station floor. They got the suitcases with no trouble, passed the coffee shop without looking at Freeman and went up the stairs to the street above. It was a short distance from the train station to City Hall, their destination. There was a bit more than an hour, so Freeman ordered breakfast, eating it slowly and reading about the riots and Vietnam in a newspaper propped against a sugar bowl. He left when he had eaten and walked to an empty corner of the station, sat on a bench and pretended to be asleep. He had purchased a ticket on the train leaving for Memphis, in case a cop might think him a vagrant or bum catching some sleep in the cool station.
At a few minutes to three he entered one of the empty phone booths and dialed the number of the Chicago Sun-Times. He asked for Victor Feldman.
"Mr. Feldman, I have a beat for you. I understand you like that kind of thing. That is the
right word, beat? They stopped using scoop in the movies in the thirties, I understand."
"Who is this ?"
"Uncle Tom, the official spokesman for the Black Freedom Fighters of America. We are the Chicago chapter of the Mau Mau."
"Look, buddy, if this is some kind of gag, I'm not in the least bit amused."
"No gag, baby, and if your watch has a sweep second hand, you can time it. In exactly forty-eight seconds you will hear an explosion. Down there on the river, you should be able to hear it very well. Thirty-six seconds, now.
"The explosion will be that of the mayor's nice new office in the nice new city hall, which was built by very, very few black men because of those nasty building-trade unions, you know. When the smoke clears, the office will be something of a mess.
"I think you ought to be able to hear something now." Freeman could hear nothing inside the big railroad station, but he heard a muffed roar through the telephone receiver at his ear. "That's it, Mr. Feldman. Now, you can discount any speculation by the mayor or police that this was an assassination attempt that failed. We blew it when we wanted and as an advertisement that we mean business.
"We want the National Guard out of the ghetto, 01' we will run them out. Then we can talk about how to improve things for the people there, dig? I'll keep in touch, Mr. Feldman. It's not that we trust you, dig, but you like having a beat and you might print the truth, if it's exclusive. There hasn't been much truth printed
about what's been happening on the South Side, lately. Remember, we blew the mayor's office when we wanted, when it was empty and he was home in bed. Nobody would want to make that ass into a martyr. Bye now." He hung up, went outside and drove home. By four o'clock he was asleep.
The blowing of the mayor's office made the front pages of the papers the next morning, but only Feldman's column held anything concerning the phone call and it was played with humor, as if a crank call. It did not surprise Freeman, since the mayor, politicians, newspapers and whites in general would prefer to think it a Communist assassination attempt on the mayor. That was all right with Freeman, because the longer they looked for Communists, the longer they would be looking for whites. The United States Communist party used Negroes as showpieces and flunkies just like all other American institutions. The best cover they had was the white man's stereotypes concerning Negroes.
He called Feldman the next evening and chided him for not printing his phone call on the front page. He announced that the motorcar of one of the Negro aldermen would be painted yellow and white. "Yellow because he's got no balls and white because that's his favorite color. It's to let him know what his constituents think of him. Good night, Mr. Feldman."
Feldman called the alderman's home, apologized for awakening him and asked if his car was missing. Upon checking, the alderman found that the car was indeed missing and called the police. The police found it parked in a zone reserved for police vehicles just down the street from the
precinct station in the alderman's neighborhood. It had been painted yellow and white.
This prank received a small story on page four, as well as half of Victor Feldman's column. But the incidents spread throughout all of Chicago's ghettos by the grapevine and they were discussed everywhere. Jokes were told of how spades had air-conditioned the mayor's office by blowing a hole in the wall. The mayor continued his Communist kick and he was echoed by his black puppet alderman.
Other things began to happen, all predicted by "Uncle Tom" in his nightly phone calls to Feldman. Jeeps which were not well guarded ground to a halt on their patrols, their engines ruined by sugar placed in the gas tanks, other jeeps were found with all the tires sliced. Signs were painted late at night saying "Whitey go home" ; prowling military patrols found the frequencies of their radios jammed without warning and the jamming device would abruptly go off the air before it could be traced to its source. One such device did not go off the air and when traced by triangulation, was found in the trunk of the colonel's personal Buick Riviera. The occupation troops gradually became objects of humor and ridicule and their ineptitude, in spite of their spit and polish, became obvious to the mayor, governor and the city police who resented being replaced by what they considered amateur, parttime soldiers.
The morale of the troops, many of whom were losing pay from their jobs in having been called up, declined sharply and the ridicule of the people plus the constant harassment of practical jokes took their toll. Freeman set up another
raid on the most prominent of the radio stations beaming their broadcasts to the ghetto audience. Entering the West Side station wearing nylonstocking masks, they locked up a Negro disk jockey and two engineers in a storeroom and went on the air.
"Hey, hey, ol' bean and you, too, baby, this is Uncle Tom of the Freedom Fighters, interrupting your favorite program to bring you news of the urban underground dedicated to the cause that Mr. Charlie is a drag and to the inhabitants of occupied territory on the South Side.
"I'd dedicate this program to the happy members of the National Guard army of occupation as well, but I'm afraid we are fresh out of hillbilly music, the Rolling Stones and the Beatles, even if they are more popular than Christ.
"How are the happy National Guardsmen, soul brothers and sisters? I see all those pictures of those colorless cats playing football with you and handing out CARE packages, or is it just the same old surplus food you been getting, without so much publicity, from the welfare? Anyway, if you believe the newspapers, those are the sweetest white soldiers going—even better than the ones who hand out candy in Vietnam to the kids in the village they just burned. Like, if you believe our free press, they must be the most beloved army of occupation since the dawn of history.
"But we know better, don't we, baby? We know about that fourteen-year-old kid who was shot last week because he didn't move quick enough; about how law and order is being brought to the 'lawless' South Side, but the numbers and dope peddling still swings; about how
they spend more time looking for nonexistent Communists than they do polishing brass—and you know, baby, they are the brassest polishing army unit in the world.
"The news is the troops have to go—immediately, if not sooner, and if they don't we will kick them out, including Bull-Head Evans. Now, won't we, baby ?
"The Freedom Fighters, the Urban Underground of Black Chicago, now deliver this ultimatum: Get out by Sunday or be prepared for war !
"And now, soul brothers and sisters, before the next message from your sponsors, a few jams —some James Brown, Esther Philips and the sweet, swinging Supremes. Whitey, please go home. We don't want you living next door, either."
When the police arrived, they found the three locked in the storeroom, the propaganda being aired by tape and no one anywhere around. The black Freedom Fighters became the talk of the occupied riot area. Handbills appeared announcing their feats until now, including the capture of the radio station, the broadcast and the blowing up of the mayor's office. The bills promised more to come. Colonel Evans ordered the arrest of anyone found with a propaganda handbill in his or her possession and his troops to police the area and pick up the rest of the propaganda leaflets from the streets and alleys.
The radio station was guarded thereafter, but the propaganda broadcasts continued with the disk-jockey format. The first was broadcast from the alleyway not far from the El station on Fiftyfirst Street. Guardsmen found a Japanese portable tape recorder playing through a loudspeaker attached high in the steel supports of the El tracks. The first guardsman to climb and attempt to tear down the equipment found that it had been booby-trapped with a wire attached to the metal outdoor speaker which led to the third rail. The electric shock hurled him to the ground and he was hospitalized. The broadcast continued for a growing crowd until an electrician disconnected it. Each night at the same time another broadcast would be made from a similar booby-trapped setup; from the roof of an apartment building, a garbage can in an alley, hidden in a newsstand and once through the speakers of an army truck which patrolled the neighborhood announcing orders and dispensing National Guard propaganda which had been stolen and parked in an alley.
The guard was continuously harassed and neither arrest, investigation nor threats could suppress the growing opposition to the troops by both the underground of Freeman's and the populace at large. Evans was furious at being made look a fool by people he held in contempt. He blamed it all on the Communists. "You know they would have neither the guts nor the intelligence for this kind of thing if it were not Communist-inspired, led and directed. Let the Commies get away with this and the entire nation is endangered. We must make our stand here so that the forces of godless communism are not misled concerning our determination to oppose their plans to take over the free world."
The colonel was beginning to embarrass a great many people with his statements, even in a city like Chicago, and his public information officer was used increasingly by the press to water down statements. But he had also become a white hope. Colonel Evans had contempt for the repeated demands by the Freedom Fighters that he and his National Guard be removed or suffer the consequences, and waited with some impatience on the Sunday night of the deadline announced in the communications center of his command post. A bit after two in the morning. he retired to view a John Wayne movie on late-night television. The colonel liked John Wayne movies very much. His National Guard unit had taken over the Washington Park recreation center for their headquarters and command post. The old field house sits facing north and across from the large field containing baseball and softball diamonds, a cricket pitch used by the Negro West Indians of the ghetto and football fields. Separating the field house and the athletic fields is a large drive that cuts through the park from Fifty-fifth Street at Cottage Grove Avenue, ending at South Park Boulevard. The field house sits in front of the swimming-pool complex, built by the WPA during the depression, with two Olympic-size pools and a diving pool. Spectator stands flank the pool to the west, the east is flanked by an eight-foot iron fence and high foliage and to the south is a children's playground. The pool had been refilled for the recreation of the National Guard troops but, since drawing sniper fire, was not used at night. Freeman's reconnaissance revealed that the approach to the field house through the pool area was not guarded.
At 0015, after curfew, Freeman led two attack teams of five men each into Washington Park. A
stolen army ambulance stood at the curb facing north on South Park Boulevard, just adjacent to the children's playground which was south of the swimming-pool area. They entered through the playground, scaled the fence, and crossing the unlit and unguarded concrete area which held the two swimming pools and the diving pool, then entered the locker room of the field house, now the command post and personal quarters of the commander of the guard unit on ghetto occupation duty, Colonel "Bull" Evans. The locker room had been converted into a private office and bedroom for the colonel, with the small gymnasium above it used for the communications center for the outfit. Freeman slipped into the colonel's empty office, seated himself behind the colonel's desk, poured a generous shot of Jack Daniel black label from the bottle on the colonel's desk and watched an old John Wayne western on the television in the corner while awaiting the colonel's return from the communications center. Freeman figured that there must be important messages for the colonel to answer to make him miss the beginning of a western. He thought that the messages might have something to do with Uncle Tom and the Black Freedom Fighters of Chicago.
Upon entering the room, his eyes not yet accustomed to the dim light within, the colonel failed to notice the figure behind the desk until he was only a short distance away, his eyes on Wayne on television whipping three men in a barroom brawl. He was reaching for the bottle of bourbon when he noticed the long-barreled Colt target pistol, a silencer fitted to its barrel, pointed at him by Freeman, a black man, clad
in all-black and seated in his chair, behind his desk, a glass of his whiskey at the nigger's elbow. The colonel, ignoring the threat of the silenced pistol, reached, with the foolish instincts of John Wayne, for his holstered, pearl-handled revolver, filling his lungs for a bellow of outrage, cut short by an accurate judo chop behind his left ear by Stud Davis, who caught his limp body before he could fall.
They carried the colonel back the way they had come, and as silently; slim black figures moving swiftly and efficiently through the sultry, moonless Chicago night. They flung the colonel, inside a large mail sack, over the • fence like a sack of grain and then into the waiting ambulance, Pretty Willie seated as the driver.
The colonel awakened tied securely to a kitchen chair, his face made up in Jolson blackface with large red lips painted in. The colonel stared back into the dark, silent faces with a look of grim, haughty disdain that would have done justice to Wayne futilely captured by redskins. He watched one of his captors as he carefully placed a drop of liquid onto a sugar cube, and put it into a cup of coffee, then approached the colonel with the cup.
"What was that you put on the sugar ?"
"Just a little acid, daddy."
"Acid ! You're going to kill me."
"Oh, no, daddy," he said, as the colonel was held and he forced the coffee down his throat, "we're going to let you Live!"
The next evening, at the time of his usual call, Uncle Tom informed Feldman where the kidnapped colonel could be found at a particular time and place. The colonel was found alive as promised, seated on the edge of the reflection pool of the Fountain of Time, at the southeast corner of Washington Park. The colonel was dangling his bare feet ecstatically in the pool, a look of beatitude on his face.
Upon being medically examined, the colonel was found to be under the influence of LSD and said repeatedly that he had met the "most wonderful niggers in the world."
18
The National Guard unit reacted with predictable ferocity to the humiliation of their commander. Never a well-disciplined unit, they now roamed the streets with their weapons fully loaded, a round in the chamber and safeties off, eager for their first kill. The ghetto, with a survival instinct centuries old, faded into their homes and the guard turned their fury on the very streets where a black mob had roamed only a short time before. All undamaged stores were used for target practice, a handful bombed with grenades, under the assumption that they were colored owned. "We're evenin' things up," said one sergeant. Trigger-happy, panicky, smelling the hatred and contempt the ghetto Negroes felt for them, they fired at almost anything that moved and black casualties mounted.
At nightfall, the ghetto hit back and snipers roamed their turf in search of white targets. Poorly armed with pistols, shotguns, surplus WW Il M-1 carbines, they proved a poor match for even the ill-trained guard unit, but their effect on the morale of the guard was staggering. Panic increased and the guard's casualties came mainly from their shooting one another accidentally or in fright. A sniper was rumored to be on the roof or in one of the apartments of a three-story building and a half-track with four .50 caliber machine guns mounted was rolled to within fifty yards of the building and the big guns turned on it, pinning the occupants to the glass-strewn floors of their apartments for more than half an hour, killing one mother of four and wounding two others.
Freeman risked roaming the streets, gathering intelligence, noting the command posts, the disposition of the units, the state of their morale, their dwindling discipline.
"Turk, when do we go into action ?" asked Stud Davis. They were gathered in the poolroom late that night, Davis, Dean, Scott and Du Bois.
"Tonight. First the sniper teams to keep them panicky, then we hit hard in a few days with our attack teams." They leaned toward him in eager anticipation, ready after the long months of training, for their baptism of fire.
"I want all sniper teams in the field except the reserve. Sugar, you know their sectors of operation, so get the word out. I want the reserve in the field the following night, then operate from that point on with one-third of the teams at a time—so everyone has two nights of rest in between. Daddy, you work out the rotation ; Sugar, you figure out logistics and supply; Willie, get to work on propaganda leaflets to be distributed the day following our first hit." "Turk?" interjected Stud.
"Yeah, Stud?"
"Is it all right if I work solo? I don't mind leading an attack team, but for sniper work I'd rather work alone."
Freeman thought for a minute, then said :
"Tell you what, we'll work together the first night; I'll act as your cover, you command. If I dig your action, OK, but either way, the decision is final."
"OK with me, Turk."
"Right. Willie, you take command of Stud's team. Will that give you time for the leaflets?"
"Yeah, Turk. I'll set things up and the cell can handle it from there. They're damn good; in fact, they could function without me."
"That's the scene we want; nobody indispensable. Now, listen carefully and take notes. Daddy, call the weather station and get the exact time of sundown and sunrise for tomorrow night. I want everyone in a darkened room one hour prior to sundown to start on their night vision and all teams out of the field no later than one hour prior to sunrise. I also want wind direction, wind velocity, temperature and relative humidity transmitted to the teams for calculation for firing.
"A large breakfast is OK, but a light lunch and no liquids or stimulants of any kind after lunch. No more than one cigarette an hour until H-hour. One joint prior to an afternoon nap is all right. We all know the drill. Any questions?" There were none.
"All right, synchronize watches by radio or telephone tomorrow night at curfew, 2200. H-hour is one hour later, 2300. Move out!"
Stud Davis awakened the next morning precisely at seven, the hour he had decided to awaken the night before, without the aid of an alarm. He was alert the moment he opened his eyes. He lived in a single-room kitchenette with small bathroom and shower. It was in the rear of a large kitchenette apartment building and included a back porch overlooking the El tracks. The room was neat and spare; it had a combination gas range and refrigerator, a television set tucked into one corner, two chairs, a card table covered with oilcloth and a transistor radio seated on a small table near the head of the bed which doubled as a couch.
Stud arose, walked into the bathroom, rubbed his light beard and decided he did not need a shave. After washing, he returned to the room and slipped into a pair of tight-fitting jeans which he wore only at home. He put on a percolator, and while awaiting the coffee, ate a large bowl of cornflakes with sliced bananas while listening to his favorite radio station. He drank the coffee, then ate four fried eggs and bacon, washing it down with half a quart of milk. Not until then did he have his first cigarette of the day. He snubbed out the butt, washed the dishes and tidied the room, a spotless oasis in the smelly, dirty building. He had a thing about cleanliness.
He left his apartment and walked the few blocks to the poolroom for the final briefing before the night's action. He moved loose and easy, fighting the heavy drowsy feeling he had always felt before an important athletic contest because it was much too early; that afternoon was soon enough and he would allow it to grow gradually until the release of the first round fired. It had been much the same in each of the sports
in which he had excelled ; you controlled the feeling and used it, not the other way around.
He sat silent through the final briefing, offering no contribution and asking no questions. Freeman asked him to stay after the others had
"I'll be your cover, Stud, but the scene is yours tonight. As I said in the briefing just now, we'll be operating as a roving team in sector six. Our first target will be the quad fifty; after that targets of opportunity." Stud smiled tightly.
"I saw what that thing did to the building; I never thought a machine gun could do that much damage to bricks and concrete."
"It's a bitch of a weapon and multiplied by four you got something else. One of the little girls who was in that building screams constantly every time they take her off tranquilizers. I want that gunner."
"I saw him on TV; he looked like he was sorry the building was still standing," said Stud.
Freeman pointed to a map of the area.
"They're usually stationed here. We'll approach this way and fire from the roof of this building; that way they'll have to swing the gun mount a full 180 degrees to bring them to bear on our position after they spot us and by that time we should have got in our shots and been long gone.
"We both go after the gunner and after he's hit, you take targets from twelve to six o'clock and I take them from six to twelve. Don't get greedy and don't wait for them to locate us; when we get in our rounds, we move on. OK 'P "
Stud nodded.
"I don't doubt you can work solo, but I have to check you out this way so the cats don't think you get special treatment." Stud smiled broadly. "I'll see you at the rendezvous tonight. Easy,
Stud."
Stud picked up his dismantled weapon and ammunition, one of the .03 sniper rifles with scope. He returned to his apartment, cleaned and oiled it, checked its moving parts and the scope, then wrapped it in a lightly oiled cloth and placed it beneath his bed. He washed away all traces of oil from his hands, carefully cleaning his nails as he did so. He turned on his small television set, found nothing to his liking, then lay on the couch listening to the radio until time to fix lunch.
Burkhardt had risen to Freeman's bait and after working on a .22 rifle range several evenings with the gang leaders, all on their best behavior, 'sir"-ing him until dizzy, he agreed to coach a larger group. It was Stud who sold him, and Burkhardt was enthusiastic about his potential as a marksman.
"This boy Davis," he said to Freeman, "is the best natural rifleman I've ever seen. You're sure he's had no previous experience with a rifle?"
"I'm certain, Mr. Burkhardt," Freeman replied.
"Amazing !" said Burkhardt, watching Stud firing prone on the University .22 range. "Why, he's potential Olympic team caliber. I'm certain he'll be able to do well in the National Junior Championships in less than a year."
"If he sticks with it," said Freeman. "He loses interest in anything as soon as he's mastered it.
As soon as he really becomes good with a rifle, he probably won't even want to see one."
But Freeman was wrong. Stud Davis had finally found his métier, the one thing which quieted the fear he hid so well. He never saw a bull at the center of the target, but always a man, any man. In less than two months he was a better marksman with a rifle than Freeman.
To Burkhardt's chagrin, but not Freeman's surprise, the National Rifle Association refused their application for membership and the club broke up ; but ten of the Cobras had had several months of intensive training on a rifle range.
Scott and Dean left the poolroom together. "You goin' home now, Daddy?"
"Naw. I better check with all the leaders of the sniper teams, make sure everything groovy." "Turk said to cool it until tonight."
"It's all right; Turk will understand. Cool it." Scott walked home and spent the afternoon figuring the formulas for the flight trajectory of a .30 caliber bullet for various angles and ranges. Mathematics always relaxed him.
Freeman spent the morning at the office, left just before lunch, announcing he would tour the riot area and made a last-minute check of the positions to be attacked. They had not been moved.
He returned to his apartment, cooked and ate a light lunch, went to his bedroom and slept until late evening. He awakened to the first sound of the alarm and dressed in the dark. He met Stud at the appointed rendezvous, both dressed in dark clothing, including low-cut black sneakers, wearing wraparound sunglasses to protect their night vision. The other snipers moved in teams of three, one as decoy, the other two as fire team. The decoy was to shoot from opposite the fire team to draw fire, then disappear. The remaining two would then open fire from behind to catch the guardsmen exposed and firing in the opposite direction.
Without a word, Stud and Freeman moved silently over the rooftops and took their position above and beyond the quad fifties mounted on a half track, a WW Il vehicle, as was most of the guard's equipment, but deadly nevertheless.
Stud held up two fingers to indicate a range of two hundred yards. Freeman nodded his confirmation. Stud nodded again and they each slipped their arms through the slings, using the roof parapet as a support, jacked a round into the chamber and slipped off the safety. The gunner was seated on the mount smoking a cigarette and talking to one of the crew. Freeman placed the crosshairs of the scope on his middle and at the sound of Stud's shot, squeezed off his own. Both rounds hit and the gunner dropped to the street, dead before he hit. Stud shot him again as he lay in the street. They both emptied their clips at the scattering crew below, reloaded and disappeared as silently and swiftly as they had come. Across the rooftops came the scattered firing of other teams in action. The war had begun.
That night the Freedom Fighters killed six, wounded nineteen and received no casualties. Escalation by both sides could be expected. After five nights of sniper activity, Freeman ordered a two-day rest and then ordered his attack teams into the field.
"Same scene as with the snipers; all teams into the field the first night, then a one-third rotation after that. We don't need any heroes; hit and run, hit them when they least expect it, then disappear as soon as they deploy and return your fire. If they follow too aggressively, ambush them ; otherwise, vanish."
Stud Davis moved his attack team down the alley through the moonless night. He could hear the big rats scurry into hiding as they approached and return to their feeding as they passed. He had hunted the rats at night in these same alleys, two boys leading, one with an air rifle, the other with a flashlight. One would fix the rat with the beam of the light and the other would fire. Unless hit exactly right, the BB would stun, but not kill and the other boys would finish the rat with sticks. It was good sport and he had learned that the rats did not fear man, they just got out of your way because it was easier. The rats would often charge the group when cornered if the boy with the air rifle missed.
He was leading the maneuver squad of his team, the fire squad approaching the street which intersected the alley from the opposite block. He halted his men with a hand signal and looked out into the deserted street. It was several hours after curfew and the streets were silent and empty. He checked his watch, then motioned his men into cover at the alley mouth way and behind parked cars a short distance away. He saw that across the street the fire squad was already in place. His maneuver squad of five were each armed with a hunting knife, pistol, grenades, three with shotguns, the other two with semiautomatic carbines. The fire team were all equipped with weapons on full automatic.
He heard the patrol approach, two jeeps much too close together, containing four men each; the driver, a rifleman and two tending the mounted machine gun at the rear of the vehicle. Stud waited until they were about ten yards away, stood and threw a grenade. Four others followed suit and they all ducked for cover. The explosions followed quickly upon one another and the sharp staccato of automatic fire echoed the loud thumps of the grenades. One jeep lay on its side, the other had crashed a parked car. Stud signaled and the fire ceased. He led his maneuver team cautiously toward the ambushed patrol, their guns at the ready. He turned off the ignition of each jeep in turn and motioned his men to strip the bodies of guns and ammunition.
One boy of about nineteen, Stud's age, moved feebly in a pool of his own blood, shock and surprise on his face. Stud paused to look at his face. He might have played basketball against him. He bent to take his weapon and ammunition.
The boy asked, over and over: "Why? Why?
Stud bent close, so that the boy could hear, and said : "Because it's war, whitey."
They finished and in seconds the dark of the alley swallowed them up. The rats moved from their burrows sniffing the strange smells in the air, the bolder and more curious ones moving toward the compelling smell of blood. The rest returned to their feeding and the occupied ghetto was silent once again.
19
Colonel Evans had been relieved of his command after his kidnapping, although the newspapers indicated he was recovering from injuries received during his heroic escape from his hoodlum captors. The president, acting on requests from the governor of Illinois and the mayor of Chicago, ordered a brigade of the Eighty-second Airborne Division airlifted from Fort Bragg, North Carolina, to Chicago to relieve the National Guard unit.
Brigadier Scott, commander of the airborne brigade, met with Lieutenant Colonel Jensen, acting commander of the National Guard unit of occupation, in the guard's headquarters for a debriefing session.
"Dammit, Colonel," said the brigadier, "you mean military troops can suffer the kind of casualties you've taken from a bunch of unorganized hoodlums ?"
"It's far worse than that, sir," replied Jensen. "Someone has trained those bastards and they've been trained well. And they have military weapons, most of them automatic."
"How could they get their hands on weapons of that kind ?"
"It must be the gang who robbed the National Guard Armory," said a quietly dressed civilian who had been allowed to attend the debriefing. "There was no publicity because the press agreed to a blackout while we investigated.
"They have a pretty good arsenal : M-14 and -16 rifles and ammo, grenades and launchers, communications equipment, the works. In the crowded slums where they operate and you can't use your heavy weapons, they can match your firepower."
"The sons of bitches !" said the brigadier grimly. "And we thought it would be another operation like Detroit last summer. What kind of tactics are they using, Colonel ?"
"Hit-and-run attacks and ambushes ; they won't stand and fight. Usually groups of from five to eight operating in tandem, each with a section leader and the entire unit commanded by one man. They demonstrate knowledge of fire and maneuver, small-unit tactics and seem well disciplined."
"The Commies must be at the bottom of this. Where else could they get that kind of training?"
"You're probably right, General Scott," said the civilian. "I'm certain the Russians have smuggled in one or more of their top agitprop agents and I'm requesting counterinsurgency teams from Washington today."
"Agitprop ?"
"Agitation and propaganda, experts in the creation of chaos, insurgency and revolution." "They wouldn't dare," said the colonel.
"That's what we said about the Cuban missiles." The brigadier turned to his adjutant.
"I want the patrols doubled tonight, weapons
on full automatic and the orders are to shoot anything moving after curfew. Shoot to kill."
The members of Freeman's recruitment and training cells had been organizing and training units in twelve major cities since the first of the year. Freeman considered nine of the underground units combat ready, the remaining three marginal. He ordered operations in Chicago reduced to harassment and ordered his twelve other units into action. He awaited the effects of his escalation of the war.
They had left during the bitter cold of early January, the quick-witted hustlers of the gang, those most likely to think, adjust, survive on any ghetto street. They were to submerge themselves in new identities, establish a cover and recruit, organize and train those street gangs in their new cities which were most like the Cobras of Chicago.
They left with sealed orders for a temporary destination within a two-hundred-mile radius of Chicago. The orders indicated their new area of operation and contained cash, various documents of identification and the name of a bank where funds could be drawn under their new names. They moved into the teeming ghettos, their new homes, each a seed of rebellion planted in fertile soil.
Freeman had patiently built up an intelligence network: bus and taxi drivers, the drivers of mail trucks, postmen, delivery and messenger boys, anyone who would be able to circulate freely after the riots had begun ; garbage men, street cleaners. He recruited six medical students, two interns and a young surgeon, plus eight trained nurses. He was extremely careful in attempting to recruit anyone from the black mid-
dle class, but found to his surprise that a growing number shared the attitudes of their less fortunate black brothers. He felt that if the Ku Klux Klan could infiltrate the Chicago police force, there was no reason the Cobras couldn't as well. In a short time he recruited six uniformed policemen and two detectives. He still hoped to recruit Dawson as his chief double agent and second-in-command.
He knew that inevitably the cumbersome bureaucratic machinery of the nation's law-enforcement agencies would begin their search to destroy him.
The general personally accompanied the CIA counterinsurgency teams to Chicago to supervise the setup of their operation. Shortly after arrival, he met with the hand-picked members of the group, code named : Operation Law and Order.
"It will be SOP for this kind of operation : infiltrate a paid informer; identify and place on our payroll a defector as a double agent; divide, confuse, bore from within. Our primary target is their leader, undoubtedly a Soviet professional. Cut off the head and the snake will die.
"Whoever he is, we should give the man due professional credit for organizing ignorant slum dwellers into a fighting unit as effective as so far proven. Until we crush this insurgency, we operate around the clock."
Three evenings later, Freeman received a phone call at his apartment. It was a vaguely familiar woman's voice who asked if she could come to his apartment within the hour, but refused her name. She said that it was urgent. By now, Freeman was operating under strict security measures. He assumed that all except public phones chosen at random were bugged. He made
standard checks for a tail whenever he went to a rendezvous which could not be explained by his normal duties with the foundation. He therefore regarded the mysterious call with suspicion.
He took out a Colt Woodsman .22, checked the mechanism and loaded a clip. He generally mistrusted automatics and never left a clip fully loaded so as not to weaken the spring. He slid the clip into the butt, jacked a round into the chamber, made sure the safety was on, and sat listening to music while he awaited the arrival of his visitor.
A half-hour later, the bell rang and he walked to the door of his apartment, pressed the buzzer to let the visitor in from the outer hall. He checked the door to make sure that no one was already waiting for him and then cracked it enough to see the elevator. He held the gun at his side, the safety on, but ready to fire.
The elevator stopped at his floor and the door opened. She stepped out and looked for the number of his apartment, then approached his door. Freeman slid the Colt behind a bookcase near the door then opened it wide to greet her. He noticed how well the red dress went with the rich black of her skin.
"I'll be damned," he said, "the Dahomey queen." He opened the door wide as he smiled in greeting and stepped back to let her enter. She walked down the short hallway to the living room, turned in the middle of the room to smile at him shyly. She wore her hair natural, kinky, and the same short length all around her head, no straightening irons, no oil. There were large gold hoop earrings that emphasized the beauty of her slim long neck. She wore no lipstick, just a touch of makeup to bring out the almond shape of her eyes.
"Baby, you're beautiful. Sit down and let me fix you a drink. Johnny Walker black label, isn't it? I'll be right back and then you'll have to tell me what you've been doing since the last time I saw you and what you're doing here in Chicago." He returned with the drinks, then moved to the stereo. "Let's see if I can remember the jams you dig.
"So what are you doing here and how did you find me Freeman was on guard because he had never told her his name and she had never asked it ; they hadn't that kind of arrangement. Then he remembered, she did know his name. They had been stopped in Washington one night by a cop and there had been the usual scene; he had to show Freeman, himself, and far more important, Freeman's woman, that Freeman simply did not count as a man. Freeman had endured it until the policeman threatened to search her and he had intervened, showing him his ID. Until then, she had known him only as Dan and she had wondered aloud why a District of Columbia cop would call any Negro "Mr. Freeman, sir."
"Well, I got to be here in Chicago for a few days and I thought I'd look you up and say hello. There ain't that many Freemans in the book with your first name and I got you the third time around. You know, you used to talk about Chicago all the time. Never could figure out if it drug you or you dug the place; seemed sometime like both scenes.
"I ain't hustlin' no more. Got me a sponsor, turned me on to a down pad out in North East D.C. and he sponsors the whole scene. I even save bread out of what he lays on •me and I got
a part interest in a bar. When the scene is over, I won't have to hustle no more. Bought a little property, too. Oh, he a good trick, honey, and not too bad for an ofay." She paused and looked directly at him. "You know him. You used to work for him." Freeman sipped his drink and looked at her, saying nothing. "He's out here to step on the Freedom Fighters and the cat started it all."
"Let's dance, baby." He moved to the amplifier and turned the volume up, then brought her to the center of the room, holding her close, his mouth close to her ear. "If the room is bugged they won't pick us up from here, not with the music. All right," he said, "run it on down."
"I figure you might know some of those cats and they'd want to know what's happen' downtown." He held her a bit more tightly. He wondered if he would have to kill her. Had she guessed anything on her own, or had she been sent to him? "What makes you think I might know any of the F F?"
"OK, maybe you don't, but you could get the word out. You know the city and I don't. You the only one I know here and they need to know."
"Why you sticking your neck out?"
She leaned away from him and gazed at him fiercely. "I'm black, ain't I?" He nodded and pulled her back into his arms.
"Who's this cat I'm supposed to have worked
"The general. You know who he is and you know what he does. He thinks he pretty cool, but he talk more than he think. Course, he don't think I got enough brains to know what's goin' on but a little piece here, little piece there; put it together and you got something." She described the general. There was no doubt, she
knew him. "Met him at this big place some big brass from the Pentagon own over on the tidewater. They had a lot of chicks out, every kind, every color. Them Pentagon cats pretty freakish, everybody in D.C. know that. Great big pad, swimming pool, boat, place to shoot and shotguns and shit everywhere. About a dozen whores and maybe five or six studs. The general was there and we spent all the time together. He dug my action and he got a big thing for dark meat. He a little bit freakish, but nothing really way out.
"He been my sponsor ever since. Almost a year now. Well, baby, he really flippin' out about these Black Freedom Fighters, thinks it a Communist scene. They want to find the ofay behind it and the head nigger and stomp on 'em. Ain't gonna be no arrest and trial if they get 'em. . .
"Yeah, baby, you worked for him. For a long time I didn't think so—he talk 'bout you all the time. You know, you his nigger? First time he call you name, I think he talkin' about you, but the cat he talkin' about ain't nothing like I knew you; then it start addin' up, must be the same cat: Chicago, same looks, live in the same place. I looked you up in the phone book after that time the fuzz stop us and you lived out in Anacostia, but I didn't call you. Didn't think you'd dig that, but I just want to know where you live. Anyway, after awhile I knew it the same cat, you, and you got to be puttin' one of us on and it got to be the general. If you come on like a Tom all that time, you got to have a reason. Never know you to do nothin' without you know just what you want and why. So that's how I know how to find you and why I think you can hip the cats the general wants to waste."
He had her describe the setup as best she could. He could fill in the gaps. It was a typical stakeout, a bit more elaborate than most. They had manpower, communications, weapons, and from her description of the few she had seen, some of their top agents. And they would have more money than anyone else in the game. They always had bread and sometimes they got lazy because they had. But they wouldn't be lazy now.
Hunting down niggers would be a new scene.
"Who's in charge ?"
"I don't know," she said, "he ain't here yet. Coming in from overseas in a couple days. General says he the best agent they got, talk about him a lot. I'll try to find out."
"No, you better not see me again, unless I get to Washington sometime. Maybe one of these days . . ." He looked at her.
"Don't go square on me now. One thing you don't need is a whore. Even a ex-whore."
They continued to dance and he questioned her through Sonny Stitt and half of Ray Charles. Finally he was satisfied and they returned to the couch and he freshened their drinks.
She turned on the couch to look at him. "I don't have to go home tonight."
He reached out slowly and touched her hair. "Baby," he said, "ain't that kinky stuff pretty?"
20
Oakland blew first, then I-as Angeles, then, leapfrogging the continent, Harlem and South Philadelphia. After years of crying conspiracy, the witch hunters found, to their horror, there was a conspiracy afoot among the black masses. Every city with a ghetto wondered if they might be next. The most powerful nation in history stood on the brink of panic and chaos. The Freedom Fighters fought first the police, then the National Guard and finally, the elite troops of the army and marines. Within a week there were major guerrilla uprisings in eight major cities in the United States and efforts to eliminate them had proven futile.
Several days of heavy rain, followed by a cool air mass moving down from Canada, broke the Chicago heat. The city lay under a bright warm sun and at night there was a cool breeze from the lake. The birds began their southern journey and the leaves of the cottonwood, poplar and maple trees began to change color, some of them reluctantly releasing from the limbs to trace a lazy descent to the ground, to be gathered and burned at the curb and produce the pungent smell of autumn come to Chicago. At night the city's silence would be broken by the explosion of grenades, the staccato message of automatic fire. The FF moved easily and silently through the ghetto which offered them affection and support, their coloration finally protective.
The curfew had silenced the streets below and a cleansing breeze from the lake stirred through Freeman's apartment. Miles Davis, mute meeting mike in a sexless kiss, blew bittersweet chocolate tones through the speakers, "My Funny Vallentine" becoming a poignant poem of lonely love.
Freeman and Joy sat on the couch, the lights low, the dishes in the sink, an empty wine bottle in the trash, slowly sipping their drinks and Miles in the room, brooding, black and beautiful, saying his thing on his horn. In the sudden silence as the record changed, Freeman turned to Joy.
"Joy," he asked, "what's bugging you to-
"Those damn Freedom Fighters, as they call themselves. You can't go to a cocktail party nowadays without running into someone who has lost his integrated job.
"My husband," she said with bitterness, "has lost his staff appointment at the hospital, and he was the only Negro doctor in a white hospital in the city."
"What did he have to do with the riots ?"
"That's the point: decent, innocent people are suffering. Now they want to get rid of me at the store, I can smell it."
"They can't let you go, baby, you're the bestlooking buyer with any department store in the Loop."
"It's nothing to joke about." She took a long swallow of her drink. "People are losing jobs they worked and sacrificed to get, all because of ignorant niggers who know nothing but hate."
Freeman rose to put another record on the changer and John Coltrane's soprano sax gave an oriental flavor to "My Favorite Things."
"Can't you remember, Joy, how it was to live like they do ?"
"Don't defend those hoodlums. Those Freedom
Fighters," she spat out the words in contempt, "are shooting real guns."
"Didn't figure to take long before some of them realized there's no win in throwing a brick at a man with a gun." Joy took a slow sip of her drink, watching him carefully over the rim of her glass.
"Dan," she asked quietly, "are you mixed up in this?"
"Baby, you got to be kidding," he laughed. "You think I'm stupid enough to get involved in a scene like that? They don't stand a chance." "You used to talk about this kind of thing when we were in East Lansing. You insisted when everything else failed, Negroes would have to fight."
"Honey," he said reasonably, "I was a kid in college then. And besides, everything else hasn't failed. We're both examples of the kind of progress Negroes have made in the last few years." She searched his face for some clue and he told her what she wanted to hear, calmed her, reassured her.
"Look at all the Negroes we have in influential positions: a member of the presidential cabinet, a Supreme Court justice, a senator. And look at me," he smiled, his arm swept his Playboy pad, "I've been poor and, believe me, this is better." "Honey," she smiled, "I'm being silly. This thing has us all nervous and on edge." She rose gracefully and walked to the door leading to the bedroom. She turned and took off her wig.
"Fix us some more drinks while I get comfortable," and with a smile, she turned and walked into the bedroom. Freeman smiled with some relief. It had been close for awhile, but he was sure he had convinced her. A short time later, lying bathed in sweat, her head on his shoulder, he was certain.
It was hot the next afternoon, but the thick walls of the precinct police station made it• cool and comfortable inside and the desk sergeant sat, oblivious to the station smells, working the crossword puzzle in the Chicago Tribune. The door opened and a bit of the hot day outside followed a woman who seemed immune to the heat. She walked to the sergeant's desk much like the models he had seen displaying fashions on television and in the newsreels. She wore a small white hat, white gloves and a pale green summer suit of shantung silk. The sergeant could not identify the material, but he knew it was expensive, so he spoke to her with far more respect than was usual for the colored women who normally entered his domain.
"Yes, lady ; can I help you ?"
"I want to speak with someone," she replied,
"about 'Uncle Tom."
That evening, Freeman, entering his lobby, found a coded message in his mailbox. It was from one of his double agents on the police force, indicating the police planned a shakedown of the block in which the FF had one of its arms caches. Because there was not much time, he ignored telephone security and, hurrying to his apartment, he phoned from his bedroom, the room lit only by the light in the hallway.
"Daddy? Turk. Listen carefully. The fuzz are shaking down the block where we have a stash, safe house three. Move it right away to safe house six. Cancel tonight's hit if you have to, but get that stuff moved and check back with me." He hung up and light burst into the room from the floor lamp in the corner. In one motion, Freeman whirled, dropped to the floor next to the bed and reached for the pistol he kept beneath the pillow, but the gun was not there.
"Freeze, Freebee," ordered Dawson, "or should I call you 'Uncle Tom.' " He sat in the Saarinen womb chair next to the teak floor lamp, his service Smith & Wesson .38 Police Positive held with easy competence and resting on the right knee of his crossed legs.
"The heat's not under the pillow. I have that one and the rest you had stashed here. You got quite an arsenal for a playboy—that how you get so much trim ?"
"Uncle Tom? Not me, baby. You trying to put me on ?" and suddenly indignant: "What the hell you doing in my pad pointing a gun at me? I hope you got a seach warrant, man."
"Martial law, remember? I don't need a warrant."
"Now you fuzz can continue to do what you
always have in the ghetto, only for a change, it's legal. That must take all the kicks out of it." Freeman, his eyes on those of Dawson, began to rise carefully. He had fired on the police range with Dawson; he knew what he could do with that revolver.
"Like I said, Freebee: freeze." Dawson rose from his seat, the gun never wavering. "OK, now stand up very slow and keep your hands where I can see them. Good, now walk over to that wall. Hands as high as you can reach, flat against the wall, you know the drill. Now hold it and don't move." Freeman stood, his weight on his hands flat against the wall and to either side of a Saito woodblock he had purchased in Tokyo. Dawson approached him, deftly searched him, then retreated slowly to the womb chair.
"OK, you can sit on the bed." Freeman walked slowly to the bed and seated himself as close to Dawson as he dared.
"Man, you sure had my nose open, Freebee. Out of all the people in Chicago, I never figured you for Uncle Tom. Cool Dan Freeman, the South Side playboy, nothing on your mind except chicks, clothes, good whiskey and sports cars. A beautiful cover, now I think about it." "You think I'd risk all this for . . ."
"Save it. I don't know if you are or if you're not Uncle Tom and it's not my job to decide. I have enough evidence to take you in and that's what I'm going to do."
"What evidence ?"
"This reel of tape I found here. One of Uncle Tom's propaganda broadcasts you cats run all over the neighborhood from booby-trapped recorders."
"I don't know anything about that."
"You know, right up until I found this I didn't believe it. Didn't even bother to bring my partner along. After I searched the place I was going to wait for you and apologize for having to check out the lead."
"That's mighty white of you."
"Who's the white man running things ?"
"How come there's got to be an ofay running things?"
"You know the Communist party in the States is like any other white scene: a few showpiece spades in the name of integration, but whitey calling the shots."
"And, of course, a spade couldn't possibly have done this on his own ?"
"Don't put me on," Dawson laughed. "The FBI detachment working on this thing say it's the most sophisticated underground in the Western Hemisphere, the creation of an expert."
"And there sure ain't no spade experts, are there, Sergeant Dawson? Expertise is a white man's monopoly—they got a patent on it. You sure are brainwashed. Well, I got news for you : it's a spade scene! A spade scene, dig. You think because you've made a career out of kissing whitey's ass, every black man is in the same bag?"
"You're not going to bug me with namecalling. You don't look so swinging on the wrong end of this gun."
"Makes you a big man, don't it? A gun, a badge and a hunting license for niggers—issued by Mr. Charlie."
"I found out a long time ago, big-time, out there in those streets, that there are the people who get their heads whipped and the people who do the whipping and it didn't take long to figure out which I was going to be."
"You're a fuckin' hypocrite—all that shit about helping your people. You want it both ways, to be supercop and have spades dig you, too. Well, you can't be a cop without betraying your people and you can't be with your people without betraying your badge." He hit Dawson in his soft spot, desperately hoping he could recruit him.
"And you think you're going to change the system, one man ? There's no changing this system, not in our lifetime and maybe never and the only way to make it is get in the best spot you can find."
"I don't want to change this system, just get it off my back. I'm no fucking integrationist. Integrate into what? Whitey's welcome to his chrome-plated shit pile. I dig being black and the only thing I don't dig about being black is white folks messing with me."
"Who appointed you the savior of soul? What makes you more sensitive than anyone else? You think nobody else feels the way you do? You think there aren't days when I want to smash every white face I see? Or are you the only black man with a sense of outrage ?"
"Use your outrage, hit back. Join us, Daws ! We got cats on the force, but nobody with your in. Join us."
"On the force? You don't stop at anything, do you? And you been using the Cobras, them tracks in their arms got to be phony. How could you get kids involved in a thing like this ?"
"Who am I going to get involved, you? You're scared shitless you might miss a promotion, not qualify for a pension. We're a wasted generation, dehydrated by whitey. I got to those kids before whitey did; they're the only hope we have."
"How are the things you're doing any better than whitey ? How are you any better ?"
"Why should I be? I'll do any damn thing to be free. Yeah, I'm Uncle Tom. There ain't a damn thing nonviolent about me. Anything whitey can do to keep me on my ass, I can do double to be free and when I'm gone there are others to take my place."
"I've heard enough ! Let's go."
"Sure, man, cuff me." Freeman rose, Dawson still seated, and held his hands forward for the handcuffs.
Dawson reached for the cuffs in his back pocket and, as he did, the change of his weight in the foam rubber chair threw him off-balance for a moment and his gun barrel shifted. He saw it in Freeman's eyes even before he began the kick, and with the way of an athlete Dawson compensated for his imbalance: even as he moved further into the enveloping softness of the chair, he brought his gun up and fired without aiming. Freeman's kick sent Dawson over backward with the chair, the gun spinning away, but the shot had hit Freeman a glancing blow in the side, half spinning him. Noticing that Freeman had been hit, Dawson moved for the weapon on all fours, but Freeman, recovering, kicked him again in the side, sending the gun slithering across the room on the nylon carpeting. Dawson turned on his knees toward Freeman with his arms crossed in front of his face in a judo defense, ready to block, parry, or grasp a flailing leg; groggy, but still dangerous. Freeman faked a kick and when Dawson covered, chopped him hard on the junction of the neck with the edge of his hand. Dawson fell to all fours and another kick turned him over. Freeman, moving in, chopped him hard on either side of the neck and grasping his collar with either hand, one crossed over the other, he applied pressure until the muscles of his arms ached from the strain. He released him and knew before he searched for a pulse that Dawson was dead.
He squatted at the side of his dead companion, rocking back and forth on his heels. "Shit, Daws, shit. Why you, man, why did it have to be you ? Anybody else, anybody."
Abruptly he stood and when he looked at the dead body of his friend, his face was impassive, except for one fleeting moment. He walked to the phone and dialed.
"Daddy? This is Turk. You get that stuff moved? Good. Get out to my place right away, it's Condition Red. And bring one of the big mail sacks. I got something for you to move."
He walked to the bathroom and stripped to the waist to check the wound. It did not look good and was bleeding freely. He dusted it with sulfa powder, placed a gauze compress against it and a thick towel over that. Returning to the bedroom he put on a loose sports shirt and a cardigan. He checked himself in the mirror and there was no telltale bulge. He walked to the living room and put a stack of records on the stereo and mixed a stiff drink. Then he sat down in the dark room to await the arrival of his lieutenants.
Dawson was too good a cop not to report where he was going, or check in when he could. When they found his body, Freeman's cover would be blown and probably that of the Cobras as well. The pain began to eat into him and he thought that his cover might not matter anyway. A doctor would have to treat him and there would be no first-class surgical facilities they could use if necessary. They would all have to go underground now. At least he wouldn't have to kill Joy to protect the cover.
He called a doctor he had recruited and told him to come by a half hour before curfew. The doorbell rang and he opened it to let in Daddy, Scott, Stud and Pretty Willie. He motioned to the bedroom.
"In the bedroom. Take it out and dump it somewhere."
"It's Dawson !" said Dean. They crowded around the body.
"Not Dawson ?" said Stud. "He's your main man, how could you kill Dawson? It would be like me killing Daddy."
"And maybe one day you'll have to kill Daddy, or Daddy you. Yeah, I killed him because he got in the way; anybody who gets in the way has to go—nobody counts until we're free.
"Did you think we were playing games? Killing people we don't know and don't dig? Forty percent of those paratroopers out there are black; it's a badge of honor for a black man to wear those wings. They soldier and fight to earn what no man can earn : freedom.
"Dawson's one dead black generation and you might be another, but at least you won't be dying an inch at a time. No more of whitey's con man's integration games, freedom on the installment plan, interest collected daily. Freedom now! No more begging, pleading and silent suffering.
"Don't tell me who I killed or what it cost to do it ; if you can't pay the dues, then get out." He paused, gazing at Dawson's body. They watched him in silence. He looked up abruptly, searching their faces in turn ; he nodded in satisfaction.
"Condition Red. All attack teams in the field nationwide. Hit them everywhere you can, everywhere they're fat, smug and complacent; use his strength and cockiness. Hit him, disappear and hit him again. Hound, harass, fuck with his mind. Keep gettin' up and don't back down.
"Willie, I want you to take over the D.C. operation as soon as you can get there. Here's a phone number in Washington; call her, take her out and buy her a drink for me; she drinks Johnny Walker black label.
"All right, move out." He motioned to the body. "And take that with you." They stuffed Dawson's body in the mail sack and without a word, left the apartment, their thoughts on the fighting ahead.
They're not boys anymore, Freeman thought, staring at the door of the apartment after they had left. They don't need me anymore. You grow up fast in the ghetto and I helped them grow faster than most. I wonder how many of them will be alive or free this time next year?
He poured himself a stiff drink and thought that the doctor wouldn't be of much help. Hours later he sat in the darkened living room listening to the first of the shooting, the rapid crackle of automatic weapons, the spit of rifles, the explosion of grenades. The firing grew in intensity, in counterpoint to the music, Lady Day singing "God Bless the Child."
He sipped his drink and listened. "Say it, baby," he said aloud, "sing it like it is : 'God bless the child that's got his own. . . . ' Go on, you black-ass Cobras, go get your own."
Freeman smiled and the pain didn't matter anymore. In fact, for the first time in many years, he hardly hurt at all.
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Chien de garde: Canada's Stray Oscar Bid
Chien de garde (Family First)
Written and directed by Sophie Dupuis
Starring: Jean-Simon Leduc, Théodore Pellerin, Maude Guérin, Paul Ahmarani
Telefilm Canada threw us all for a loop Wednesday when they announced, without the usual warning or fanfare, the completely random selection of Sophie Dupuis' Chien de garde as our official submission in the Oscar race for Best Foreign Language Film. It's a random choice because Chien de garde marks the first time in a very long while that the pan-Canadian committee has put forward a film that had virtually no boost, platform, or exposure outside of Quebec. The film, which opened theatrically in La Belle Province in March and was dumped unceremoniously onto VOD for the rest of Canada sometime between then and now, actually drew some stellar reviews and garnered three of Quebec's Iris Awards in the categories of Best Actress (Maude Guérin), Best Film Editing, and Best Breakthrough Performer. (The film has a full-throttle performance by Théodore Pellerin).
A nitty gritty family drama set in the world of petty crime and featuring a roster of actors performing as if each one is in a different film, Chien de garde is bold, but all over the map. Dupuis introduces herself as a filmmaker who is willing to "go there" as she displays an ambitious hand behind the camera in this volatile family drama. Chien de garde stars Jean-Simon Leduc as JP, a young man who works as a debt collector for his cartoonish crooked uncle Dany (Paul Ahmarani) in Montreal's working class neighbourhood of Verdun. JP's 19-year-old brother Vince (Pellerin) is his far-too-eager apprentice, who has countless energy to burn while roughing people up and making sure folks pay one way or the other.
JP's desire to be good and go straight sparks Vince's wild side and the untrained young man becomes like a Tasmanian devil/mobster twink of OCD fury in the family home. Vince twirls, caresses his tattoos, and mopes about in his undies while he shouts, drinks, and riles up everyone, particularly their troubled mother (a strong Maude Guérin) who suffers from depression and battles with the bottle. He seduces and alienates everyone with flamboyant, over-the-op affectations, but nobody's really buying the routine.
Deep down Vince is a sad little pup, longing for acceptance from his family and helpless in his search for authority figures. Vince has an attachment to his mother that might even make Xavier Dolan gag as the young man cuddles up to his mom in bed every night, and in one case proudly announces his morning boner to world as she sleeps beside him, yet he turns on her violently when what she needs most is a hug. Chien de garde is virtually all about Pellerin's performance as he unleashes a bold dramatic energy that takes hold of the camera and never lets go of it. His work drew comparisons to Robert De Niro's breakout performance in Martin Scorsese's Mean Streets when the film opened in Quebec, and rightly so, and it will be the thing that makes or breaks the film for most, if not all, viewers. It is a wild, dynamic, and boisterous cocktail of passion, aggression, loneliness, ambiguous sexuality, and raw vulnerability—but some might find it exasperating. Pellerin's been making a name for himself in Canuck flicks like It's Only the End of the World, Boost and Never Steady, Never Still, and will gain far more notice in Joel Edgerton's gay conversion drama Boy Erased, and Chien de garde boldly announces him as one of Canada's next stars.
Pellerin overwhelms the film, though, despite Dupuis's best efforts to capture the finer details of Guérin or Leduc's performances in shots that linger after the actor's unbridled energy has left the frame. There's an overall unevenness to the film as Dupuis negotiates her star's performance with everything else around them. Handheld camerawork jitters a bit too self-consciously as the action follows Vince's reign of terror, while issues of pacing abound in a story that never quite goes anywhere.
Dupuis captures the nitty gritty of the neighbour as she shoots the characters' surroundings with an eye for hardship, violence, and poverty. Chien de garde is a sobering portrait of working class survivalism told in kitchens, dive bars, and back alleys. It's a bleak and difficult film, but one that also rewards with the freshness and roughness of its approach to a corner of Montreal we haven't quite seen before.
Chien de garde is a respectable, if flawed, debut feature for Dupuis as a director, but this Canadian film fan and Oscar junkie doesn't have much confidence in its likelihood to go the distance. Its selection makes no sense from a strategic perspective. The film lacks the zip, gravitas, and accessibility of Denys Arcand's return to form The Fall of theAmerican Empire, which seemed like a no-brainer as the frontrunner selection last week after playing to strong reviews at TIFF with awards heavyweight Sony Pictures Classics handling its US release. If we were sending the best eligible film, then Robin Aubert's blood-soaked zombie flick Les affamés (bought by Netflix in the States), would have been not only an audacious choice, but also a worthy one since it's the best film out of Quebec since Xavier Dolan's Mommy.
For an urgent portrait of young Montrealers Pascale Plante's Fake Tattoos is more accessible and profound. The Fireflies are Gone, Sebastian Pilote's unexpected but worthy winner for Best Canadian Feature at TIFF, might have been a safer choice with its slice of life realism and outstanding performance by Karelle Tremblay. Even Sadaf Foroughi's Ava would have been a bold choice—but also a smarter one since it actually had a chance to connect with broader audiences in theatres and in festivals.
I don't like to review a film within the frame of its awards potential, but that's now the discussion within which most people will see Chien de garde. It's a decent film, but hardly an Oscar-worthy one.
Labels: Best Foreign Lang Film, Canadian Film, chien de garde, Oscars
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Kicking Off TIFF with Previews of 'The Stone Speak... | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,144 |
Q: What are requirements of web page must be for sharing link in Facebook? For example, there is a page with a description, title and image.
Somehow the image is not displayed when I try share link in Facebook.
Possible, the picture should be a specific size or tags? How about meta tags title, desctiption
A: The recommended size for images in links is 1200X628.
Here is a reference to all facebook related image dimesnions.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,562 |
\section{Introduction}
Modern cryptographic systems essentially rely on the computational-complexity assumption for their security, whereas a quantum communication primitive known as a private quantum channel (PQC) achieves its safety under information-theoretic conditions. The PQC, first proposed by Ambainis \emph{et al.}~\cite{AMTW00}, provides a fundamental and perfectly \emph{secure} way to transmit a quantum state from a sender, Alice, to a receiver, Bob, by using pre-shared classical secret keys generated by a quantum key distribution (QKD) scheme. As a kind of completely positive and trace preserving map (i.e., quantum channels)~\cite{NC00,W13}, the PQC transforms any quantum state into a maximally mixed state (MMS) in a given Hilbert space.
Because the output of PQCs always fulfills the genuine maximally mixed state, which is a quantum state with a maximal von Neumann entropy (i.e., strong against any type of attack), they can be used to construct a secure quantum network or quantum internet~\cite{K08,PWE+15,VPZ+16,WEH18} with the help of quantum teleportation~\cite{BBC+93} as well as QKD protocols~\cite{BB84,E91,B92} for emerging quantum communication technologies. Another main feature of the PQC at the purely theoretical level is related to a phenomenon known as \emph{superadditivity} on quantum channel capacity problems~\cite{SY08,H09,LWZG09}. In particular, a PQC and its dual (i.e., its complementary PQC) reportedly form a subtle counter-example to the additivity, especially on the classical capacity~\cite{H09} on quantum channels (the PQCs), owing to quantum entanglement~\cite{HHHH09}.
In traditional geometry, it is well known that an infinite number of regular polygons exists in a two-dimensional plane and five regular polyhedra exist in three-dimensional space~\cite{C73}. Few recent efforts to relate the regular polytopes to quantum information theory, for example, a construction on Bell's inequalities~\cite{TG20,BK20} have been reported. Classifying or proving the existence of a higher dimensional ($d>3$) regular $d$-polytope is not a trivial problem. However, the regular 4-polytope (with a cell) was well classified by several mathematicians many years ago.
Here, we attempt to devise an approach to connect the structure of PQCs to higher dimensional regular polytopes, under the constraint of preserving the maximal output entropy. To this end, we need to exploit the notion of an isotropic (or unitarily invariant) measure on the unitary group, and two modifications of the Gell-Mann matrix~\cite{G62} and quantum Fourier transform~\cite{C02}. In this study, we highlight a new connection between qutrit-based PQCs and the regular 4-polytope by generalizing our previous research on qubit PQCs and regular polyhedra~\cite{JLC+18} equipped with an isotopic measure.
The remainder of this paper is organized as follows. In Sec.~\ref{pre}, we describe the basic concept of the PQC, and provide definitions for several relevant and meaningful quantities, such as the isotropic measure and the regular polytopes, especially for the convex regular 4-polytope. To relate the PQCs and polytopes, we introduce a new notion of a hypervector in Sec.~\ref{hypervec}. In Sec.~\ref{main}, we derive our main results (using two methods) on the relationship between qutrit-based PQCs and the regular 4-polytope, which is a generalization of qubit-based PQCs. Especially, in Sec.~\ref{qutrit}, we briefly argue a universal strategy for the connection over higher dimensional cases. Finally, discussions and remarks are presented in Sec.~\ref{conclusion}, and a few intriguing questions are raised for future work.
\section{Preliminaries} \label{pre}
\subsection{Concept of a private quantum channel} \label{definitions}
Here, we briefly review the mathematical definition and related results of private quantum channels. Before providing the details, we explain our notations. Let $\cl{B}(\bb{C}^d)$ denote the set of linear operators from the Hilbert space $\bb{C}^d$ to itself, and let $\tn{U}(d)\subset\cl{B}(\bb{C}^d)$ be the unit group on $\bb{C}^d$. Let us define a quantum channel as $\Lambda:\cl{B}(\bb{C}^d)\to \cl{B}(\bb{C}^d)$, which is a linear, completely positive, and trace-preserving map. For any quantum state $\rho\in\cl{B}(\bb{C}^d)$, the quantum channel is conveniently denoted as $\Lambda:\rho\mapsto\Lambda(\rho)$ in $\cl{B}(\bb{C}^d)$.
Generally, we consider a quantum channel $\Lambda:\cl{B}(\bb{C}^d)\to \cl{B}(\bb{C}^d)$ to be a $\varepsilon$-PQC (or approximate PQC)~\cite{J12} if it satisfies that
\begin{equation} \label{eq:apqc}
\left\|\Lambda(\rho)-\frac{{\mathbb{1}}}{d}\right\|_p\le\frac{\varepsilon}{d^{\frac{p-1}{p}}},
\end{equation}
where $\varepsilon$ is a small (but non-negative) real number, and ${\mathbb{1}}$ denotes the $d\times d$ identity matrix. The Schatten $p$-norm or matrix norm, $\|\cdot\|_p$, is defined by $\|M\|_p=\sqrt[p]{\mbox{$\textnormal{Tr}$}(M^\dag M)^{p/2}}=\left(\sum_js_j^p(M)\right)^{1/p}$, where $s_j(M)$ denotes the singular values of any matrix $M$. If the parameter $\varepsilon=0$, that is, $\Lambda(\rho)=\frac{{\mathbb{1}}}{d}$, we say that the map $\Lambda(\cdot)$ is a \emph{complete} PQC. To obtain Eq.~\eqref{eq:apqc}, we can straightforwardly combine the definitions over the operator norm~\cite{HLSW04} and the trace norm~\cite{DN06} by using McDiarmid's inequality.
\begin{figure}[t!]
\includegraphics[width=\linewidth]{Fig1.eps}
\caption{Schematic diagram of a PQC. We assume that Alice and Bob shared a secret key $K=[k_j]$ via quantum key distribution (QKD) protocols. Should Alice wish to encode a quantum state $\rho$ through the PQC $\Lambda$, she applies unitary operations, $U_{[k_j]}$, on the input state depending on the key set $[k_j]$. Thus, $\Lambda(\rho)$ is equivalent to the maximally mixed state $\tfrac{{\mathbb{1}}}{d}$. At the end of the PQC, the receiver Bob can recover the original quantum state $\rho$ by exploiting the inverse units over $\Lambda(\rho)$.}
\label{Fig1}
\end{figure}
The advantage of $\varepsilon$-PQC is that it is possible to reduce its cardinality in terms of the unitary operations required, from the optimal case ($d^2$) to the approximate regimes of $\cl{O}(d\log d)$~\cite{HLSW04} or $\cl{O}(d)$~\cite{A09}. Here, we notice that the dependence of the optimality of PQCs on the input dimension of the quantum state was determined by several groups~\cite{BR03,NK06,BZ07}. However, in this paper, we only consider the optimal schemes for matching regular polytopes. In addition, several considerations are known to exist on continuous-variable PQCs~\cite{B05,JKL15,JL16,BSZ20} as well as on a sequential version~\cite{JK15}. However, the main purpose of this work is to contribute to the construction of (secure) key-dependent PQCs (but satisfying a maximal entropy condition) over a set of unitaries provided by the key set $K=[k_j]:=\{k_1,k_2,\ldots\}$ for all $j$. For a given key set $K$, we can create a private quantum channel in the form of
\begin{equation} \label{eq:pqc}
\Lambda_K(\rho)=\frac{1}{|K|}\sum_{j=1}^{|K|}U_{k_j}\rho U_{k_j}^\dag,
\end{equation}
where $U_{k_j}$ are unitary operators induced from each component $k_j\in K$. For an example of the optimal case (i.e., $|K|=4$), the key set $K$ can be constructed from Pauli matrices, that is, $K=\{{\mathbb{1}},X,Y,Z\}$ with $X=\begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix},Y=\begin{pmatrix} 0 & -i \\ i & 0 \end{pmatrix},X=\begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix}$. Then, for every $\rho\in\cl{B}(\bb{C}^2)$, we have
\begin{equation*}
\Lambda_K(\rho)=\frac{1}{4}({\mathbb{1}}\rho{\mathbb{1}}^\dag+X\rho X^\dag+Y\rho Y^\dag+Z\rho Z^\dag)=\frac{{\mathbb{1}}}{2}.
\end{equation*}
Here, we can observe that the channel output $\Lambda_K(\rho)$ is the exact two-dimensional maximally mixed state. As mentioned above, key set $K$ can be obtained from QKD protocols. (See the scheme in Fig.~\ref{Fig1}).
In our previous study~\cite{JLC+18}, we found a five key set $\{K_P=4,K_H=6,K_O=8,K_D=12,K_I=20\}$ corresponding to PQCs in terms of $\{\cl{N}_P,\cl{N}_H,\cl{N}_O,\cl{N}_D,\cl{N}_I\}$, where the subscripts denote Pauli (or tetrahedron), hexahedron, octahedron, dodecahedron, and icosahedron, respectively. The main requirement these constructions need to meet is that they always preserve the maximal von Neumann entropy because all outputs of each of the PQCs are exactly the maximally mixed state in the Hilbert space $\bb{C}^2$. These constructions are followed by the extension of the Pauli matrices via a complex rotation matrix of
\begin{equation} \label{2d}
R_\tn{c}(\theta)=\begin{pmatrix} \cos\theta & -i\sin\theta \\ i\sin\theta & \cos\theta \end{pmatrix},
\end{equation}
where the angle $\theta\in[0,2\pi]$ is a real number. The main objective of this research is to generalize the qubit-based PQCs above to a qutrit-based PQC as well as to find a formulation on the higher dimensional cases.
\subsection{Isotropic measure} \label{measure}
To intuitively obtain the relationship between PQCs and regular polytopes, we need to review the notion of an isotropic measure on the unitary group $\tn{U}(d)$. The isotropic measure for quantum states is formally defined as follows ~\cite{A09}: For any quantum state $\rho\in\cl{B}(\bb{C}^d)$, a probability measure $\mu$ on the unitary group $\tn{U}(d)$ is said to be \emph{isotropic}, if it holds that
\begin{equation} \label{eq:iso}
\int_{\tn{U}(d)}U\rho U^\dag\tn{d}\mu=\frac{{\mathbb{1}}}{d}.
\end{equation}
In addition, a random vector $\vec{v}$ generated by $U\in \tn{U}(d)$ is known to be isotropic if its law is isotropic. Conceptually, this implies that the integration over all random vectors (generated by $U$) equates to zero (i.e., the center of mass).
In the case of a discrete measure, the structure of PQC in Eq.~\eqref{eq:pqc} corresponds to that of the exact isotropic measure. As an example, the set of Pauli matrices $\{{\mathbb{1}},X,Y,Z\}$ is isotropic and the set of corresponding random vectors, namely, $\{\vec{v}_{{\mathbb{1}}},\vec{v}_X,\vec{v}_Y,\vec{v}_Z\}$ is also isotropic. Thus, by definition, the sums of the actions of the Pauli matrices and random vectors are $\frac{{\mathbb{1}}}{2}$ and 0, respectively. We notice that the Haar measure on $\tn{U}(d)$ is also an isotropic measure.
In this study, we connect the private quantum channels with a key set $K$ to the regular polytopes beyond the low-dimensional cases through Eq.~\eqref{eq:iso}. However, not only is the extension quite complex, even in the case of four dimensions, but the higher dimensional polytopes are also not well defined.
Before discussing the relationship, we briefly review the regular polytopes in four-dimensional (Euclidean) space.
\subsection{Regular 4-polytope} \label{polytope}
All the classifications and proofs of existence of the regular $d$-polytope are very difficult problems in geometry~\cite{C73}. Here, we only take into account the regular \emph{convex} 4-polytope as a natural matching for the three-level quantum state (i.e., qutrit) because the geometric shape (in terms of the Bloch sphere) of any quantum state satisfies the convex set and a unit sphere of a given dimension.
The regular convex 4-polytope was first introduced by Schl\"{a}fli. Six types of convex-type polytopes that are four-dimensional analogues of the three-dimensional regular polyhedra (i.e., Platonic solids) exist. The existence of the regular convex 4-polytope, which is generally denoted by a Schl\"{a}fli symbol $[\alpha,\beta,\gamma]$, is constrained by cells (i.e., three-dimensional regular polyhedra) and dihedral angles (see Table~\ref{table:pqc} below). In addition, each polytope in geometry can be classified by intrinsic symmetric groups as in Table~\ref{table:pqc}, and is generally known as a Coxeter group~\cite{C73,J18}.
\begin{table*
\caption{Summary of regular 4-polytopes versus qutrit-based PQCs.}
\begin{ruledtabular}
\begin{tabular}{l c c c c c c c}
{} & {-} & Simplex ($S$) & Hypercube ($H$) & Tesseract ($T_1$) & Octaplex ($O$) & Dodecaplex ($D$) & Tetraplex ($T_2$) \\
\hline
{\bf Orthographics} & {Conj.} & \begin{minipage}{0.6in}\includegraphics[width=\textwidth]{5cell}\end{minipage} & \begin{minipage}{0.6in}\includegraphics[width=\textwidth]{8cell}\end{minipage} & \begin{minipage}{0.6in}\includegraphics[width=\textwidth]{16cell}\end{minipage} & \begin{minipage}{0.57in}\includegraphics[width=\textwidth]{24cell}\end{minipage} & \begin{minipage}{0.6in}\includegraphics[width=\textwidth]{120cell}\end{minipage} & \begin{minipage}{0.6in}\includegraphics[width=\textwidth]{600cell}\end{minipage} \\
{\bf Cell} & - & 5 & 8 & 16 & 24 & 120 & 600 \\
{\bf Symmetric group} & {-} & $A_4=[3,3,3]$ & $B_4=[4,3,3]$ & $B_4=[3,3,4]$ & $F_4=[3,4,3]$ & $H_4=[5,3,3]$ & $H_4=[3,3,5]$ \\
\hline
{\bf PQC $\Lambda$} & $\Lambda_L$ & $\Lambda_{S}$ & $\Lambda_{H}$ & $\Lambda_{T_1}$ & $\Lambda_{O}$ & $\Lambda_{D}$ & $\Lambda_{T_2}$ \\
\hline
{\bf Hypervector $\mathbf{V}$} & {-} & 5 & 8 & 16 & 24 & 120 & 600 \\
{\bf Basis vector $\vec{v}$} & {9} & 20 & 48 & 96 & 192 & 1440 & 12000 \\
{\bf Cardinality $|K|$} & 9 & 20 & 48 & 96 & 192 & 1440 & 12000 \\
{\bf Optimality} & $\bigcirc$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ & $\times$ \\
{\bf Security} & $\bigcirc$ & $\bigcirc$ & $\bigcirc$ & $\bigcirc$ & $\bigcirc$ & $\bigcirc$ & $\bigcirc$
\end{tabular}
\end{ruledtabular}
\label{table:pqc}
\end{table*}
\section{Hypervector and regular convex 4-polytope} \label{hypervec}
Let $\mathbf{V}$ be a set of vectors, that is, $\mathbf{V}_t=(\vec{v}_1,\vec{v}_2,\ldots,\vec{v}_t)$, and we term it a \emph{hypervector}. It exactly corresponds to a $j$-component set of a regular polyhedron. We notice that a hypervector is a vector in four-dimensional space, but $\vec{v}_j$'s are three-dimensional. As an example, $\mathbf{V}_4=(\vec{v}_1,\vec{v}_2,\vec{v}_3,\vec{v}_4)$ can be interpreted as a schematic in the form of
\begin{equation} \label{hypervector}
\begin{minipage}{0.6in}\includegraphics[width=\textwidth]{vec1}\end{minipage}\;\;\;\; \equiv\;\;\;\; \begin{minipage}{0.6in}\includegraphics[width=\textwidth]{vec2}\end{minipage},
\end{equation}
where $\vec{v}_t$'s are four vectors in the tetrahedron (i.e., one of the regular 3-polytopes). Interestingly, our hypervector $\mathbf{V}_4$ corresponds with the Pauli matrices in a one-to-one manner. In this context, we can naturally define five kinds of hypervectors, as there are only five kinds of regular polyhedra in the three-dimensional space, namely, $\mathbf{V}=\{\mathbf{V}_4,\mathbf{V}_6,\mathbf{V}_8,\mathbf{V}_{12},\mathbf{V}_{20}\}$.
By using the definition of the hypervector above, we can easily classify the regular convex 4-polytope in terms of $\mathbf{V}_t^{[s]}$ as follows: (The index $s$ indicates the $s$-cell in the regular 4-polytope.)
\begin{align}
\mathbf{V}_t^{[s]}&=\left\{\mathbf{V}_t^{[5]},\mathbf{V}_t^{[8]},\mathbf{V}_t^{[16]},\mathbf{V}_t^{[24]},\mathbf{V}_t^{[120]},\mathbf{V}_t^{[600]}\right\}.
\end{align}
It is useful to note that, fortunately, each cell index $\ell$ is intimately concerned with the symbol $j$ in the regular convex 3-polytope. This observation offers the possibility for us to count the number of unitary sets for PQCs in secure quantum communication.
\bigskip
\section{Relationship between qutrit-based PQCs and regular 4-polytope} \label{main}
In this section, we show that the qutrit-based PQCs can be related to the regular 4-polytope by using two strategies known from the Gell-Mann matrix ($\lambda_j$ with $j\in\{1,\ldots,8\}$) expansion and by applying quantum Fourier transform. The Gell-Mann matrix is a fundamental aspect of high-energy physics, and quantum Fourier transform is a core process in quantum algorithms. Although the Gell-Mann matrices are traceless and Hermitian in the $\tn{SU}(3)$ group, the modified Gell-Mann matrices have one exception in terms of the identity matrix (i.e., they are non-traceless). We also notice that one of the elements of the original Gell-Mann matrix (formally, $\lambda_8$) does not form a unitary matrix; however, our matrices all have unitary matrices. Therefore, our modified version could be more naturally considered as a generalization of the well-known Pauli matrix in $\tn{SU}(2)$.
\subsection{Generalized Gell-Mann matrices and qutrit-based PQCs}
As mentioned above, the PQCs over a single-qubit system are naturally constructed by a $2\times2$ unitary matrix ${R}_{\tn{c}}(\theta)\in\tn{SU}(2)$ in Eq.~\eqref{2d}, thus we can imagine a $3\times3$ unitary matrix in $\tn{SU}(3)$ for generating qutrit-based PQCs. To do this, we first consider the $3\times3$ orthogonal rotation matrix in $\tn{SO}(3)$ in the standard form of
\begin{widetext}
\begin{align} \label{3d}
R(\phi,\theta,\varphi)=\begin{pmatrix} \cos{\phi}\cos{\theta}\cos{\varphi}+\sin{\phi}\sin{\varphi} & -\cos{\phi}\cos{\theta}\sin{\varphi}+\sin{\phi}\cos{\varphi} & \sin{\theta}\cos{\phi} \\ -\sin{\phi}\cos{\theta}\cos{\varphi}+\cos{\phi}\sin{\varphi} & \sin{\phi}\cos{\theta}\sin{\varphi}+\cos{\phi}\cos{\varphi} & -\sin{\varphi}\sin{\theta} \\ -\sin{\theta}\cos{\varphi} & \sin{\theta}\sin{\varphi} & \cos{\theta} \end{pmatrix},
\end{align}
\end{widetext}
where each angle parameter is bounded by $0 \le \varphi < \pi$, $-\cfrac{\pi}{2}\le \theta \le \cfrac{\pi}{2}$, and $-\pi<\phi\le\pi$. Similar to the case of the two-dimensional complex rotation matrix ${R}_\tn{c}(\theta)$ in Eq.~\eqref{2d}, we can find a complex rotation matrix (or unitary matrix) $\check{R}_\tn{c}(\phi,\theta,\varphi)$ by extending the real rotation matrix in Eq.~\eqref{3d} with some complex numbers. However, this is not a simple task, and thus we have to change our intension of finding the complex rotation matrix to generalizing the well-known Gell-Mann matrices.
Here, we define a set of nine components in the $\tn{SU}(3)$ group, which can be modified from the original Gell-Mann matrix~\cite{G62}, and we call it the \emph{generalized} Gell-Mann matrix. All matrices are unitaries and Hermitian, and are explicitly given by
\begin{align} \label{gGM}
L_1&=\begin{pmatrix} 0 & 1 & 0 \\ 1 & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix},
L_2=\begin{pmatrix} 0 & -i & 0 \\ i & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix},
L_3=\begin{pmatrix} 0 & i & 0 \\ -i & 0 & 0 \\ 0 & 0 & 1 \end{pmatrix}, \nonumber\\
L_4&=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & 1 \\ 0 & 1 & 0 \end{pmatrix},
L_5=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & i \\ 0 & -i & 0 \end{pmatrix},
L_6=\begin{pmatrix} 1 & 0 & 0 \\ 0 & 0 & -i \\ 0 & i & 0 \end{pmatrix}, \\
L_7&=\begin{pmatrix} 0 & 0 & 1 \\ 0 & 1 & 0 \\ 1 & 0 & 0 \end{pmatrix},
L_8=\begin{pmatrix} 0 & 0 & i \\ 0 & 1 & 0 \\ -i & 0 & 0 \end{pmatrix},
L_9=\begin{pmatrix} 0 & 0 & -i \\ 0 & 1 & 0 \\ i & 0 & 0 \end{pmatrix}. \nonumber
\end{align}
By exploiting the generalized Gell-Mann matrix ($L$), we can construct an exact private quantum channel on the qutrit system. Formally, it is given by ($\forall \rho\in\cl{B}(\bb{C}^d)$)
\begin{equation} \label{ourPQC}
\Lambda_L(\rho)=\frac{1}{9}\sum_{j=1}^9L_j\rho L_j^\dag=\frac{{\mathbb{1}}}{3},
\end{equation}
where $L_j$'s are elements of the generalized Gell-Mann matrix. As mentioned above (see subsec.~\ref{definitions}), this construction is \emph{optimal} for the qutrit-based PQC~\cite{BR03,NK06,BZ07}. (Here, we notice that the matrix $L_j$ is equivalent to the key set component $k_j$.) The output of the channel $\Lambda_L$ is the exact maximally mixed state (i.e., $\frac{{\mathbb{1}}}{3}$), thus preserving its perfect secrecy. As an example, we can check that the resulting state of a qutrit, via $\Lambda_L$ in Eq.~\eqref{ourPQC}, is the maximally mixed state (see details in Appendix~\ref{app}). Thus, we conjecture that there exists another regular convex 4-polytope corresponding to the optimal PQC on the 3-level quantum system (see Table~\ref{table:pqc}).
Now, we need to consider the way in which to construct the PQCs for non-optimal cases (but preserving their security). We believe that other types of generalized Gell-Mann matrices exist that correspond to the PQCs of $\Lambda_S$, $\Lambda_H$, $\Lambda_{T_1}$, $\Lambda_O$, $\Lambda_D$, and $\Lambda_{T_2}$ with 20, 48, 96, 192, 1440, and 12 000-component basis vectors, respectively. (Note that the number of basis vectors is equivalent to the cardinality of the corresponding key set $K$.) To do this, we use the quantum Fourier transform strategy.
\subsection{Quantum Fourier transform and PQCs} \label{qutrit}
The quantum Fourier transform (QFT), which is a linear transformation over quantum states, was first discovered by Coppersmith~\cite{C02}, and it was employed for many efficient calculations on quantum algorithms (e.g., Shor's algorithm~\cite{S97}). The Walsh–Hadamard or Hadamard transform is a special case of QFT on two-level quantum systems. Generally, the $d$-dimensional QFT, $\tn{QFT}_d$, is defined as follows: For any quantum state $\ket{j}\in\bb{C}^d$, the QFT is a map in the form of
\begin{equation}
\tn{QFT}_d:\ket{j}\mapsto\frac{1}{\sqrt{d}}\sum_{k=0}^{d-1}\omega^{jk}\ket{k},
\end{equation}
where $\omega=e^{\frac{2\pi i}{d}}$. In principle, it is possible to expand the sum of $k$ with $d$ elements to $\ell$ with $D(\ge d)$, and we call it an extended QFT in the $D$-dimension ($\tn{QFT}_D$), that is,
\begin{equation}
\tn{QFT}_D:\ket{j}\mapsto\frac{1}{\sqrt{D}}\sum_{\ell=0}^{D-1}\omega^{j\ell}\ket{\ell}\equiv|\tilde{\ell}\rangle.
\end{equation}
Here, we notice that the total probability is conserved as 1, and each quantum state has a uniform probability distribution $\frac{1}{D}$. This extension enables us to find hypervectors as mentioned above (see subsec.~\ref{hypervec}). If we choose $D=20$, then $\tn{QFT}_D$ performs the following action, that is, for any $j$
\begin{equation*}
\tn{QFT}_{20}:\ket{j}\mapsto\frac{1}{\sqrt{20}}\sum_{\ell=0}^{19}\omega^{j\ell}\ket{\ell},
\end{equation*}
where $\ket{\ell}$ forms a basis vector over the Hilbert space of the output. After this QFT, we can easily convert each basis vector (by using clustering) into a hypervector in Eq.~\eqref{hypervector}. We depict this situation in Fig.~\ref{Fig2}, and we define this PQC as $\Lambda_S$ (see also Table~\ref{table:pqc}). We notice that the reason why the one-to-one correspondence is possible originates from the condition of the isotropic measure in Subsec.~\ref{measure}.
Conceptually, this approach allows us to obtain further generalizations for $\Lambda_H$, $\Lambda_{T_1}$, $\Lambda_O$, $\Lambda_D$, and $\Lambda_{T_2}$ as well as for high-dimensional PQC cases on any $d$-dimensional quantum state (i.e., qudit).
\begin{figure}[t!]
\includegraphics[width=\linewidth]{Corresp.eps}
\caption{One-to-one corresponding strategy as a result of extending the quantum Fourier transform to the hypervector. More precisely, a quantum state $\ket{j}$ can be transformed into a 20-component superposed state via $\tn{QFT}_{20}$, after which it is possible to match these to the hypervector $\mathbf{V}_4^{[5]}$ on the regular convex 4-polytope.}
\label{Fig2}
\end{figure}
\section{Conclusions} \label{conclusion}
In this study, we constructed and analyzed a private quantum channel (PQC) involving a three-level quantum system (i.e., qutrit) to the maximally mixed state, by exploiting two methods: the generalized Gell-Mann matrix and the modified quantum Fourier transform (QFT). For these constructions, we newly defined the notion of a hypervector on the regular convex 4-polytope, and found nine components of the generalized Gell-Mann matrix $L$ in the optimal case. Furthermore, we provided an expansion technique on QFT for non-optimal qutrit-based PQCs with the new notion of the hypervector. Here, a total of seven kinds of PQCs were presented, and each of the PQCs satisfied the security condition, that is, it produced the maximal von Neumann entropy in terms of $\frac{{\mathbb{1}}}{3}$. In fact, we can conclude that the power of the isotropic measure induces our results.
A few intriguing open questions still remain with regard to the private quantum channel itself or beyond. The first one is that the optimal case of the qutrit-based PQC exactly predicts the component of unitary operations; however, we do not know what it is in the class of the regular convex 4-polytope. The second question relates to the way in which we can apply our results to the PQCs to highlight the research on higher dimensional geometry or quantum databases. Finally, our work is expected to contribute to establishing contact with mathematicians who are well acquainted with the quantum information sciences.
\bigskip
\section{Acknowledgments}
This work was supported by the Basic Science Research Program through the National Research Foundation of Korea through a grant funded by the Ministry of Science and ICT (Grant No. NRF-2020M3E4A1077861) and the Ministry of Education (Grant No. NRF-2018R1D1A1B07047512).
\section{Appendix} \label{app}
We describe the optimal private quantum channel with nine key-sets $K$ in Eq.~(\ref{gGM}) of the qutrit system. As a representative and exact example, if we choose $\check{\rho}=\frac{1}{2}\begin{pmatrix}1 & 0 & 0 \\ 0 & 1 & 0 \\ 0 & 0 & 0\end{pmatrix}$, then we can easily calculate the output (i.e., the maximally mixed state $\frac{{\mathbb{1}}}{3}$) of the private quantum channel $\Lambda_{L}$ as follows:
\begin{widetext}
\begin{align*} \label{eq:pqc}
\Lambda_L(\rho)&=\frac{1}{9}\sum_{j=1}^{9}L_{j}\check{\rho} L_{j}^\dag \nonumber\\
&= \frac{1}{18}\Bigg[\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 0\end{pmatrix}+\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 0\end{pmatrix}+\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 0\end{pmatrix}+\begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}+\begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}+\begin{pmatrix} 1 & 0 & 0\\ 0 & 0 & 0 \\ 0 & 0 & 1\end{pmatrix}+\begin{pmatrix} 0 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}+\begin{pmatrix} 0 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}+\begin{pmatrix} 0 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}\Bigg] \\
&= \frac{1}{3}\begin{pmatrix} 1 & 0 & 0\\ 0 & 1 & 0 \\ 0 & 0 & 1\end{pmatrix}=\frac{{\mathbb{1}}_3}{3}.
\end{align*}
\end{widetext}
In addition, it is possible to straightforwardly calculate the output for another input state of the qutrit ${\rho}$. We omit the results.
\bigskip
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CONFIG=package/ubuntu
VERSION=$1
BUILDDIR=/tmp/xdispatch
# copy sources to clean target
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mkdir -p $BUILDDIR
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\section{Introduction}
\label{sec:intro}
Light-cone distribution amplitudes (DAs) are important
nonperturbative quantities which (within the framework of collinear
factorisation) parameterise in partonic terms the components of the
hadronic wavefunction that control hard exclusive processes. Such
processes provide hadron structure information complementary to that
obtained from hard inclusive reactions.
The structure functions for inclusive processes are more accessible
both experimentally and theoretically owing to their larger cross
sections and branching ratios, simpler final-state detection and more
straightforward factorisation properties. They do not specify the
phases and correlations which would constitute amplitude-level hadron
structure information, but probe instead the bound states' partonic
content. Deep-inelastic scattering processes, for example, are
controlled by the charge and momentum of the struck parton and are
insensitive to its relation to the other hadronic constituents. The
associated parton distribution functions (PDFs) are therefore
single-particle probabilities, which reveal nothing about the role of
particular Fock states or of correlations between quarks and gluons.
Distribution amplitudes, involved in exclusive processes, always
appear in convolutions and, unlike the PDFs, are not directly
measurable. These exclusive processes are dominated by specific
partonic configurations. The outgoing quarks and gluons are unlikely
to form a given final-state hadron unless either they are
approximately collinear with small transverse separation, or one of
the partons carries almost all of the hadron's momentum (the soft
overlap or Feynman mechanism). In the former case, the basis for
collinear factorisation~\cite{Lepage:1980fj}, hard gluon exchange must
occur to allow the struck or decaying parton to communicate with the
others, turning them to the final direction. Since more partons
require more hard gluons, exclusive cross-sections and decay rates are
dominated by the valence Fock state at leading-order in $Q^2$, up to
soft effects.
Hard exclusive processes are therefore controlled at leading-order by
the distribution amplitudes of leading-twist (an operator's twist is
the difference between its dimension and its spin): essentially the
overlap of the hadronic state with the valence Fock state in which,
for a meson, the collinear quark-antiquark pair have small transverse
separation and carry longitudinal momentum fractions $u$ and $\bar{u}
= 1 - u$. The pion's electromagnetic form factor at large $Q^2$, for
example, can be written as a convolution of distribution amplitudes
$\phi_\pi(u,Q^2)$ for the incoming and outgoing pions with a
perturbatively-calculable hard-scattering kernel. Higher-twist DAs
associated with power-suppressed contributions originate in, for
example, higher Fock states~\cite{Ball:1998sk}. We consider only
leading, twist-$2$, DAs in this paper.
The phenomenological importance of hard exclusive processes has grown
since collinear factorisation was first established for cases such as
the pion's electromagnetic form factor and the $\gamma\gamma^*\pi$
transition form factor around 30 years
ago~\cite{Lepage:1980fj,Chernyak:1977fk,Efremov:1979qk,Farrar:1979aw}.
Of particular note is the theoretical description of hadronic $B$
decays, which have been studied in detail by BaBar and Belle and will
be studied by LHCb and at super-$B$ factories in order to constrain
the CKM matrix and to understand CP violation. Factorisation is more
difficult to establish in $B$-physics because the hard collinear and
soft mechanisms contribute at the same order in $1/m_b$. Two
approaches have been developed. In the QCD factorisation framework it
has been shown that collinear factorisation can be applied to leading
order in $1/m_b$ to a large class of nonleptonic
$B$-decays~\cite{Beneke:1999br, Beneke:2000ry, Beneke:2001ev}.
Soft-collinear effective theory (SCET)~\cite{Bauer:2000yr,
Bauer:2001yt, Bauer:2002nz} aims to provide a unified theoretical
framework for the factorisation of both hard-collinear and soft
effects. In both cases, distribution amplitudes play an important role
as nonperturbative inputs in flavour physics.
In this paper we focus on the distribution amplitudes of the light
pseudoscalar and longitudinally-polarised vector mesons, since, as we
shall discuss in Sec.~\ref{subsec:moms}, their lowest moments are of
phenomenological interest and are calculable on the lattice. For
pseudoscalars, these quantities are relevant for decays such as $B
\rightarrow \pi\pi$ and $B \rightarrow \pi K$; they also appear in
light-cone sum rule (LCSR) expressions for the form factors of
semileptonic decays such as $B \rightarrow \pi l \nu$. For hard
exclusive processes involving the light vector mesons $\rho, K^*$ and
$\phi$, polarisation-dependence can reveal much about the underlying
dynamics, with the longitudinally- and transversely-polarised final
vector meson states often involving different aspects of weak
interaction physics~\cite{Braun:2003jg}. Examples are the exclusive
semileptonic $B \rightarrow \rho l \nu_l$, rare radiative $B
\rightarrow \rho\gamma$ or nonleptonic, e.g.\ $B \rightarrow \pi\rho$,
decays of $B$-mesons, which are important for extracting CKM matrix
elements.
\subsection{Definitions}
\label{subsec:defs}
Mesonic light-cone DAs are defined from meson-to-vacuum matrix
elements of quark-antiquark light-cone operators, which are non-local
generalisations of those used to define the decay constants. For
example, for pions
\begin{multline}
\bra0\overline q_2(z)\gamma_\nu\gamma_5\op{P}(z,-z)q_1(-z)
\ket{\pi(p)}_{z^2 =0} \equiv\\
i f_\pi p_\nu \int_0^1 du\;e^{i(u-\bar u)p\cdot z}\phi_\pi(u,\mu)
\end{multline}
and for longitudinally-polarised rho-mesons
\begin{multline}
\bra 0\overline q_2(z)\gamma_\nu \op{P}(z,-z)
q_1(-z)\ket{\rho(p;\lambda)}_{z^2=0} \equiv\\
f_\rho m_\rho p_\nu\frac{\varepsilon^{(\lambda)}\cdot z}{p \cdot z}
\int_0^1 du\;e^{i(u-\bar{u})p.z}\phi^{\parallel}_\rho(u,\mu)\,,
\end{multline}
where
\begin{equation}
\label{eq:pdef}
\op{P}(z,-z)=\op{P}\,\exp\Bigl(-ig\int_{-z}^{z}dw^\mu
A_\mu(w)\Bigr)
\end{equation}
is the path-ordered exponential needed to maintain gauge invariance,
$\mu$ is a renormalisation scale, $u$ is the momentum fraction of a
quark, $\bar u = 1-u$ and $\varepsilon^{(\lambda)}$ is the
polarisation vector for a vector meson with polarisation state
$\lambda$. The distribution amplitudes are normalised by
\begin{equation}
\int^1_0 du\;\phi(u,\mu) = 1\,.
\end{equation}
The definitions above involve the pion and rho-meson decay constants
defined by
\begin{align}
\bra0\overline q_2\gamma_\mu \gamma_5 q_1\ket{\pi(p)} &=
if_\pi p_\mu\,,\\
\bra0\overline q_2\gamma_\mu q_1\ket{\rho(p;\lambda)} &=
f_\rho m_\rho \varepsilon^{(\lambda)}_\mu\,.
\end{align}
The vector meson decay constant, $f_\rho$, and its coupling to the
tensor current, $f^T_\rho$, are of interest in their own right and we
have previously calculated~\cite{Allton:2008pn} the ratios
$f^T_V/f_V$, for $V\in\{\rho,K^*,\phi\}$ as part of our domain-wall fermion
(DWF) phenomenology programme.
\subsection{Moments}
\label{subsec:moms}
Moments of light-cone DAs are defined by:
\begin{equation}
\langle\xi^n\rangle_\pi(\mu) =
\int_0^1 du\, \xi^n\,\phi(u,\mu)\,,
\end{equation}
where $\xi \equiv u - \bar u = 2u - 1$ is the difference between
the longitudinal momentum fractions.
Since the moments are obtained from matrix elements of local
operators~\cite{Brodsky:1980ny} we can study them using lattice QCD.
The light-cone matrix elements which define the DAs themselves are not
amenable to standard lattice techniques, since in Euclidean space the
light-cone has been rotated to the complex direction. By expanding the
non-local operators on the light cone, we obtain symmetric, traceless
twist-$2$ operators. With the following conventions for continuum
covariant derivatives,
\begin{equation}
\ovra{D}_\mu = \ovra{\partial}_\mu+ig A_\mu , \quad
\ovla{D}_\mu = \ovla{\partial}_\mu-ig A_\mu , \quad
\ovlra{D}_\mu = \ovla{D}_\mu-\ovra{D}_\mu ,
\end{equation}
the expressions relating the moments of DAs to the corresponding local matrix
elements are:
\begin{widetext}
\begin{subequations}
\label{eq:dadef}
\begin{align}
\bra{0}\overline{q}(0)\gamma_\rho\gamma_5\ovlra{D}_\mu s(0)\ket{K(p)}
&=
\PFM_Kf_Kp_\rho \,p_\mu \,, \label{eq:k1def}\\
\bra{0}\overline{q}(0)\gamma_\rho\gamma_5\ovlra{D}_\mu\ovlra{D}_\nu
q(0)\ket{\pi(p)} \,
&=
-i\PSM_\pi f_\pi p_\rho p_\mu p_\nu \,, \label{eq:pi2def}\\
\bra{0}\overline{q}(0)\gamma_\rho\ovlra{D}_\mu s(0)\ket{K^*(p,\lambda)}
&=
\VFM_{K^*}f_{K^*}m_{K^*}\frac{1}{2}\left(p_\mu\varepsilon^{(\lambda)}_\nu
+ p_\nu\varepsilon^{(\lambda)}_\mu \right)\,, \label{eq:kstar1def}\\
\bra{0}\overline{q}(0)\gamma_\rho\ovlra{D}_\mu\ovlra{D}_\nu q(0)
\ket{\rho(p,\lambda)}
&=
-i \VSM_\rho \, f_\rho m_\rho
\frac{1}{3}\left(\varepsilon^{(\lambda)}_\rho p_\mu p_\nu +
\varepsilon^{(\lambda)}_\mu p_\nu p_\rho + \varepsilon^{(\lambda)}_\nu
p_\rho p_\mu \right) \,. \label{eq:rho2def}
\end{align}
\end{subequations}
\end{widetext}
The operators in the matrix elements above are all to be considered
symmetric and traceless in the free Lorentz indices. Meson-meson
rather than meson-vacuum matrix elements of the same operators lead to
moments of generalised parton distributions (GPDs).
Recent analyses, especially those based on QCD sum rules, deal instead
with the Gegenbauer moments, which arise from a conformal
expansion~\cite{Braun:2003rp, Ball:2003sc}, in which the conformal
invariance of (classical) massless QCD is used to separate
longitudinal and transverse degrees of freedom, analogous to the
partial wave expansion in ordinary quantum mechanics. All dependence
on the longitudinal momentum fractions is described by orthogonal
polynomials that form an irreducible representation of the collinear
subgroup of the conformal group, SL(2,$\mathbb{R}$). The
transverse-momentum dependence is represented as the scale-dependence
of the relevant operators and is governed by renormalisation-group
equations. The different `partial waves', labelled by different
conformal spins, do mix but not to leading-logarithmic accuracy.
Conformal spin is thus a good quantum number in hard processes up to
small corrections of order $\alpha_s^2$.
The asymptotic $Q^2 \rightarrow \infty$ DA is known from perturbative
QCD: $\phi_\mathrm{as} = 6u\bar{u}$. For the leading-twist
quark-antiquark DAs that we are interested in, the conformal expansion
can then be conveniently written as:
\begin{equation}
\phi(u,\mu) = 6u\bar{u}\biggl(1+
\sum_{n=1}^\infty a_n(\mu)\,C_n^{3/2}(2u-1)\biggr) \,
\end{equation}
where $C_n^{3/2}$ are Gegenbauer polynomials. To one-loop order the
Gegenbauer moments, $a_n$, renormalise
multiplicatively~\cite{Ball:2003sc}:
\begin{equation}
a_n(\mu) = a_n(\mu_0) \left( \frac{\alpha_s(\mu)}{\alpha_s(\mu_0)}
\right)^{(\gamma_{(n)}-\gamma_{(0)})/\beta_0} \,.
\end{equation}
The one-loop anomalous dimensions are:
\begin{equation}
\gamma_{(n)} = \gamma_{(n)}^\parallel =
C_F\biggl(1 - \frac{2}{(n+1)(n+2)} +4\sum_{j=2}^{n+1}1/j\biggr) \,,
\end{equation}
where $C_F=4/3$. Since the moments are positive and increase with $n$,
the effects of higher-order Gegenbauer polynomials are damped at
higher scales as the DAs approach their asymptotic form. The conformal
expansion can thus be truncated. Quantities such as the pion's
electromagnetic form factor, for example, are given by convolutions in
which the kernels are slowly-varying and the strongly-oscillating
Gegenbauer polynomials are washed out. The same conclusion is reached
by considering, rather than the conformal expansion, the
diagonalisation of the ERBL
equations~\cite{Lepage:1979zb,Lepage:1980fj, Efremov:1979qk,
Efremov:1978rn} which govern the evolution of the DAs much as the
DGLAP equations~\cite{Gribov:1972ri, Lipatov:1974qm,
Dokshitzer:1977sg, Altarelli:1977zs} govern the evolution of PDFs.
We can obtain values for the Gegenbauer moments from lattice
simulations since the Gegenbauer moments are combinations of
ordinary moments of equal and lower order, e.g.:
\begin{equation}
a_1 = \frac53\langle\xi^1\rangle, \quad
a_2 = \frac7{12}\left(5\langle\xi^2\rangle - 1\right) \,.
\end{equation}
\subsection{Status}
\label{subsec:status}
In this section, we summarise what is currently known about
leading-twist light-meson distribution amplitudes. For mesons of
definite G-parity, there is a symmetry under the interchange $u
\leftrightarrow \bar{u}$ of the two momentum fractions. In these
cases, the distribution amplitude is an even function of $\xi = u -
\bar{u}$ and the odd moments therefore vanish. Thus, $\PFM_\pi$,
$\VFM_\rho$ and $\VFM_\phi$ all vanish, while $\PFM_K$ and
$\VFM_{K^*}$ are SU(3)-flavour breaking effects.
Since $\PFM_K$ is the average difference between the fractions of
longitudinal momentum carried by the strange and light quarks,
\begin{equation}
\label{eq:define_a1K}
\PFM_K(\mu)= \int_0^1 du (2u-1)\,\phi_K(u,\mu)=\langle 2u-1\rangle \,,
\end{equation}
we may expect from the constituent quark model that the sign of
$\PFM_K = \frac{3}{5}a_1^K$ is positive and this is indeed the case.
In fact, $\PFM_K$ is an important SU(3)-breaking parameter and is
relevant for predictions of $B$-decay transitions such as $B
\rightarrow K,\;K^*$~\cite{Braun:2006dg}. For example, a light-cone
sum rule analysis leads to~\cite{Khodjamirian:2003xk}:
\begin{equation}
\frac{f^{BK}_+(0)}{f^{B\pi}_+(0)} =
\frac{f_K}{f_\pi}(1 + c_1a^K_1) + \dots \,,
\end{equation}
where $f^{BP}_+(0)$ is the vector $B\to P$ form factor at zero
momentum transfer and $c_1 \sim O(1)$. Other examples include the
ratio of the weak radiative decay amplitudes $B \rightarrow
\rho\gamma$ and $B \rightarrow K^*\gamma$, where the main theoretical
error originates from such SU(3)-breaking effects. The measured ratio
of these decay rates allows for a determination of the ratio of CKM
matrix elements $|V_{ts}|/|V_{td}|$.
There have been three main approaches to the study of DAs: extraction
from experimental data, calculations using QCD sum rules and lattice
calculations. The overall normalisations are given by local hadronic
matrix elements, essentially the decay constants, which have already
been discussed and are partly accessible experimentally, and partly
have to be calculated theoretically. The shapes of the leading-twist
distribution amplitudes, in the form of the Gegenbauer moments, can be
determined from experiments by analysing data on form factors such as
$F_{\gamma\gamma^*\pi}$, which was studied by the CLEO
experiment~\cite{Gronberg:1997fj}, and the pion's electromagnetic
form factor, $F_\pi^\mathrm{em}$~\cite{Bakulev:2003gx}. There is a
lack of sufficiently accurate data, however, and it is difficult to
avoid contamination from other hadronic uncertainties and higher twist
effects. As a result, the existing experimental constraints are not
very stringent.
Moments of DAs, then, must largely be determined from theory.
Lattice~\cite{Braun:2006dg, Braun:2007zr, Braun:2008ur,
Martinelli:1987si, Daniel:1990ah, DelDebbio:1999mq,
DelDebbio:2002mq} and sum rule~\cite{Shifman:1978bx, Shifman:1978by,
Chernyak:1983ej, Colangelo:2000dp} studies have usually focussed on
the second moment of the pion's distribution amplitude. However, the
early lattice results were largely exploratory while sum rule results
have an irreducible error of $\sim 20\%$ because it is not possible
properly to isolate the hadronic states.
The first moment of the kaon's distribution amplitude, for example,
has in the past been determined mainly from QCD sum rules, and
representative results include:
\begin{equation}
\label{eq:other}
a_1^K(1\gev)=
\begin{cases}
0.05(2) & \text{\cite{Khodjamirian:2004ga}}\\
0.10(12) & \text{\cite{Braun:2004vf}}\\
0.050(25) & \text{\cite{Ball:2005vx}}\\
0.06(3) & \text{\cite{Ball:2006fz}}
\end{cases}
\end{equation}
These results all have the expected sign, but the uncertainties are
around $50\%$. The reduction of such uncertainties is the chief
motivation of the lattice programme. In an earlier
publication~\cite{Boyle:2006pw,Boyle:2006xq}, we obtained
$\PFM_K(2\gev)\equiv 3/5 \,a_K^1\,(2\gev)=0.032(3)$. We note that in
addition to the UKQCD/RBC programme for the calculation of DA moments
on the lattice using $N_f=2+1$ domain-wall fermions, there is a
UKQCD/QCDSF programme using $N_f=2$ improved Wilson quarks
\cite{Braun:2006dg}. QCDSF have also published results for moments of
baryon DAs~\cite{Braun:2008ur}.
Lattice results for hadronic distribution amplitudes are considered in
a recent review of hadron structure from lattice QCD
in~\cite{Hagler:2009ni}.
The plan for the remainder of this paper is as follows. In
Sec.~\ref{sec:bare} we discuss the extraction of bare moments of
distribution amplitudes from Euclidean lattice correlation functions
(we use `bare' or `latt' to denote quantities before matching from the
lattice to the continuum). In Sec.~\ref{sec:numerics} we give the
details of our numerical calculations and present the bare results.
The renormalisation of those bare results is described in
Sec.~\ref{sec:renormalisation}. We then present our summary in
Sec.~\ref{sec:conclusion}.
\section{Bare Moments from Lattice Correlation Functions}
\label{sec:bare}
In this section, we describe our general strategy for the lattice
calculation of the unrenormalised lowest moments of light meson
distribution amplitudes. We obtain expressions for the first and
second moments $\PFM$ and $\PSM$ for pseudoscalar mesons and for the
longitudinal moments $\VFM$ and $\VSM$ for vector mesons, in terms of
Euclidean lattice correlation functions which can be computed by Monte
Carlo integration of the QCD path integral. In each case, we consider
a generic meson having valence quark content $\overline{q}_2q_1$,
where the subscripts indicate that the flavours of the two quarks may
be different. We will see below that we can obtain all of these
moments from ratios of two-point correlation functions and thus we
expect to benefit from a significant reduction of the statistical
fluctuations.
\subsection{Lattice Operators}
We now define the lattice operators used in the correlation functions
from which we extract the moments of the distribution amplitudes. We
use the following interpolating operators for the pseudoscalar and
vector mesons:
\begin{subequations}
\label{eq:interpolating}
\begin{align}
P(x) &\equiv \overline{q}_2(x)\gamma_5q_1(x), \\
V_\mu(x) &\equiv \overline{q}_2(x)\gamma_\mu q_1(x), \\
A_\mu(x) &\equiv \overline{q}_2(x)\gamma_\mu\gamma_5 q_1(x) \,.
\end{align}
\end{subequations}
Although we have written $P,\,V$ and $A$ as local operators in
Eq.~\eqref{eq:interpolating}, in the numerical simulations we use
smeared operators at the source of our correlation functions in order
to improve the overlap with the lightest meson states. Since the
effects of smearing cancel in the ratios constructed below, the
discussion in this section holds for both smeared and local
interpolating operators. We explain the details of our smearing
procedures in Sec.~\ref{subsec:simdet}. The operators in
Eqs.~\eqref{eq:dadef} from which the moments of the distribution
amplitudes are obtained are of course local operators.
In constructing the lattice operators of Eqs.~\eqref{eq:dadef}, we use
the following symmetric left- and right-acting covariant derivatives:
\begin{equation}
\ovra{D}_\mu\psi(x) =
\frac{1}{2a}\left[U(x,x{+}\hat\mu)
\psi(x{+}\hat\mu)- U(x,x{-}\hat\mu)\psi(x{-}\hat\mu)\right] ,
\label{eq:derdef}
\end{equation}
\begin{equation}
\overline{\psi}(x)\ovla{D}_\mu =
\frac{1}{2a}\left[\overline{\psi}
(x{+}\hat\mu) U(x{+}\hat\mu,x) - \overline{\psi}(x{-}\hat\mu)
U(x{-}\hat\mu,x)\right] ,
\end{equation}
where $U(x,y)$ is the gauge link going from site $x$ to site $y$ and
$\hat{\mu}$ is a vector of length $a$ in the direction $\mu$ ($a$
denotes the lattice spacing). The operators of interest are then
defined by
\begin{subequations}
\label{eq:ops}
\begin{align}
\op{O}_{\{\rho\mu\}}(x) &\equiv
\overline{q}_2(x)\gamma_{\{\rho}\ovlra{D}_{\mu\}}q_1(x) \,,\\
\op{O}_{\{\rho\mu\nu\}}(x) &\equiv
\overline{q}_2(x)\gamma_{\{\rho}\ovlra{D}_{\mu}\ovlra{D}_{\nu\}}q_1(x) \,,
\\
\op{O}^5_{\{\rho\mu\}}(x) &\equiv
\overline{q}_2(x)\gamma_{\{\rho}\gamma_5\ovlra{D}_{\mu\}}q_1(x) \,,\\
\op{O}^5_{\{\rho\mu\nu\}}(x) &\equiv
\overline{q}_2(x)\gamma_{\{\rho}\gamma_5\ovlra{D}_{\mu}\ovlra{D}_{\nu\}}q_1(x) \,,
\end{align}
\end{subequations}
where the braces in the subscripts indicate symmetrisation of the
enclosed Lorentz indices, $\{\mu_1\dots\mu_n\} \equiv
\sum_{\mathrm{perms}\;s} \{\mu_{s(1)}\dots\mu_{s(n)}\}/n!$.
\subsection{Operator Mixing}
\label{subsec:opmix}
In the continuum the operators in Eq.~\eqref{eq:ops} transform as
second- or third-rank tensors under the Lorentz group. On the lattice
however we must consider their transformation properties under the
hypercubic group ${\cal H}_4$ of reflections and $\pi/2$ rotations,
together with the discrete symmetries parity $P$ and
charge-conjugation $C$, where the possibilities for operator mixing
are increased. A detailed study of the transformations of these
operators under ${\cal H}_4$ has been performed
in~\cite{Gockeler:1996mu}.
The choice of Lorentz indices in the operators used in simulations is
important both to keep the operator mixing simple and also to enable
the extraction of matrix elements using as few non-zero components of
momentum as possible. The latter is to avoid the associated
discretisation effects and statistical degradation.
$\op{O}_{\{\rho\mu\}}$ and $\op{O}_{\{\rho\mu\}}^5$ renormalise
multiplicatively under ${\cal H}_4$ when $\rho\neq\mu$. In the
notation of~\cite{Mandula:1983ut}, these operators transform under the
$6$-dimensional $6^{(+)}$ (for $\op{O}_{\{\rho\mu\}}^5$) or $6^{(-)}$
(for $\op{O}_{\{\rho\mu\}}$) irreducible representations of ${\cal
H}_4$. The choice $\mu\neq\rho$ is the most convenient one for the
extraction of the first moment of the distribution amplitudes. Charge
conjugation symmetry combined with ${\cal H}_4$ ensures that there is
no mixing with operators containing total derivatives.
It is also possible to obtain the first moment from the four operators
$\op{O}_{\{\mu\mu\}}$ (or $\op{O}^5_{\{\mu\mu\}}$), which each
transform as four-dimensional reducible representations containing a
singlet. The three traceless operators transform as the
$3$-dimensional irreducible representation $(3,1)^{(+)}$ (without
$\gamma^5$) or $(3,1)^{(-)}$ (with $\gamma^5$). Subtracting the trace
involves the subtraction of a power divergence, so for the first
moment of the distribution amplitude of the $K$ and $K^*$ we avoid
this by evaluating the matrix elements of $\op{O}^5_{\{\rho\mu\}}$ and
$\op{O}_{\{\rho\mu\}}$ respectively with $\rho\neq\mu$.
Similarly for the second moment of the distribution amplitudes the
most convenient choice is to use $\op{O}^5_{\{\rho\mu\nu\}}$ or
$\op{O}_{\{\rho\mu\nu\}}$ with all three indices different, which
transform as the $(\overline{1/2,1/2})^{(+)}$ and
$(\overline{1/2,1/2})^{(-)}$ $4$-dimensional irreducible
representations respectively. Charge conjugation symmetry allows
mixing of $\op{O}^5_{\{\rho\mu\nu\}}$ and $\op{O}_{\{\rho\mu\nu\}}$
with operators containing total derivatives:
\begin{eqnarray*}
\op{O}^5_{\{\rho\mu\nu\}}(x)
&\text{ mixes with }
&\partial_{\{\rho}\partial_\mu\,\left(\overline{q}_2(x)\gamma_{\nu\}}\gamma_5
q_1(x) \right) \,,\\
\op{O}_{\{\rho\mu\nu\}}(x)
&\text{ mixes with }
&\partial_{\{\rho}\partial_\mu\,\left(\overline{q}_2(x)\gamma_{\nu\}}
q_1(x) \right) \,.
\end{eqnarray*}
The moments of the distribution functions are obtained from
non-forward matrix elements between a meson at non-zero four momentum
and the vacuum, so the total-derivative operators must be included in
the analysis.
\subsection{$\boldsymbol{\PFM_P}$ and $\boldsymbol{\PSM_P}$ from Correlation
Function Ratios}
To obtain the first and second moments of the pseudoscalar meson
distribution amplitude, $\PFM$ and $\PSM$, we consider the following
two-point correlation functions:
\begin{subequations}
\label{eq:cpdef}
\begin{align}
C_{A_\nu P}(t,\vec{p}) &\equiv \sum_{\vec{x}} e^{i \vec{p}\cdot\vec{x}}
\bra{0} A_\nu(t,\vec{x}) P^\dagger(0) \ket{0} \,,\label{eq:capdef} \\
C^5_{\{\rho\mu\}}(t,\vec{p}) &\equiv \sum_{\vec{x}} e^{i
\vec{p}\cdot\vec{x}} \bra{0} \op{O}^5_{\{\rho\mu\}}(t,\vec{x}) P^\dagger(0)
\ket{0} \,,\label{eq:c1pdef} \\
C^5_{\{\rho\mu\nu\}}(t,\vec{p}) &\equiv \sum_{\vec{x}} e^{i
\vec{p}\cdot\vec{x}} \bra{0} \op{O}^5_{\{\rho\mu\nu\}}(t,\vec{x}) P^\dagger(0)
\ket{0} \,.\label{eq:c2pdef}
\end{align}
\end{subequations}
For a generic pseudoscalar meson $P$, we define $Z_P \equiv
\bra{P(p)}P^\dagger\ket{0}$ and the bare decay constant by
$\bra{0}A_\nu\ket{P(p)} \equiv ip_\nu f^{\textrm{bare}}_P$. The
operators $P^\dagger(0)$ in Eqs.~\eqref{eq:cpdef} are smeared as
explained below. At large Euclidean times $t$ and $T - t$, the
correlation functions defined above tend towards:
\begin{widetext}
\begin{align}
C_{A_\nu P}(t,\vec{p}) &\rightarrow
\frac{Z_P f^{\textrm{bare}}_P e^{-E_PT/2} \sinh((t{-}T/2)E_P)}{E_P} \,
ip_\nu \,,\label{eq:C5r}\\
C^5_{\{\rho\mu\}}(t,\vec{p}) &\rightarrow
\frac{Z_P f^{\textrm{bare}}_P e^{-E_PT/2} \sinh((t{-}T/2)E_P)}{E_P} \,
ip_\rho ip_\mu\PFM^{\textrm{bare}} \,,\label{eq:C5rm}\\
C^5_{\{\rho\mu\nu\}}(t,\vec{p}) &\rightarrow
\frac{Z_P f^{\textrm{bare}}_P e^{-E_PT/2} \sinh((t{-}T/2)E_P)}{E_P} \,
ip_\rho ip_\mu ip_\nu \PSM^{\textrm{bare}} \,.\label{eq:C5rmn}
\end{align}
\end{widetext}
We can extract bare values for the first and second moments of the
pseudoscalar meson distribution amplitudes from the following ratios
of correlation functions:
\begin{subequations}
\label{eq:Pratio}
\begin{align}
R^P_{\{\rho\mu\};\nu}(t,\vec{p}) &\equiv
\frac{C^5_{\{\rho\mu\}}(t,\vec{p})}{C_{A_\nu P}(t,\vec{p})} \rightarrow
i\frac{p_\rho p_\mu}{p_\nu}\PFM^{\textrm{bare}} \,,\label{eq:PFMratio} \\
R^P_{\{\rho\mu\nu\};\sigma}(t,\vec{p}) &\equiv
\frac{C^5_{\{\rho\mu\nu\}}(t,\vec{p})}{C_{A_\sigma P}(t,\vec{p})}
\rightarrow -\frac{p_\rho p_\mu
p_\nu}{p_\sigma}\PSM^{\textrm{bare}} \,.\label{eq:PSMratio}
\end{align}
\end{subequations}
Keeping in mind the operator mixing outlined above, we obtain the
first moment from $R^P_{\{\rho4\};4}(t,\vec{p})$ (the index $4$
corresponds to the time direction) with $\rho = 1, 2$ or $3$ and a
single non-zero component of momentum, $|p_\rho| = 2\pi/L$. The second
moment is extracted from $R^P_{\{\rho\mu4\};4}(t,\vec{p})$ with at
least two non-zero components of momentum. We take $\rho, \mu = 1, 2$
or $3$ with $\rho \ne \mu$ and $|p_\rho| = |p_\mu| = 2\pi/L$. We
present more details in Sec.~\ref{subsec:results}.
Apart from isolating the moments of the DAs as much as possible by
cancelling $Z_P$, $f^{\textrm{bare}}_P$ and most of the energy
dependence from Eqs.~\eqref{eq:C5rm} and \eqref{eq:C5rmn}, the ratios
also simplify the effect of mixing with total derivative operators.
These operators have matrix elements proportional to \eqref{eq:C5r}
with which we build a ratio similar to \eqref{eq:PSMratio}. Hence the
contribution of the mixing term becomes trivial and does not have to
be computed explicitly. It enters as an additive constant when
renormalising the bare moments as we will discuss later.
\subsection{$\boldsymbol{\VFM_V}$ and $\boldsymbol{\VSM_V}$ from Correlation
Function Ratios}
The treatment of the vector meson's longitudinal distribution
amplitude is analogous. We consider the following two-point
correlation functions:
\begin{subequations}
\begin{align}
C_{V_\mu V_\nu}(t,\vec{p}) &\equiv \sum_{\vec{x}} e^{i \vec{p}\cdot\vec{x}}
\bra{0} V_\mu(t,\vec{x}) V_\nu^\dagger(0) \ket{0}, \\
C_{\{\rho\mu\}\nu}(t,\vec{p}) &\equiv \sum_{\vec{x}} e^{i
\vec{p}\cdot\vec{x}} \bra{0} \op{O}_{\{\rho\mu\}}(t,\vec{x}) V_\nu^\dagger(0)
\ket{0}, \\
C_{\{\rho\mu\nu\}\sigma}(t,\vec{p}) &\equiv \sum_{\vec{x}} e^{i
\vec{p}\cdot\vec{x}} \bra{0} \op{O}_{\{\rho\mu\nu\}}(t,\vec{x})
V_\sigma^\dagger(0) \ket{0}.
\end{align}
\end{subequations}
Again, the source operators $V^\dagger(0)$ are smeared. We define the
bare longitudinal decay constant of a vector meson $V$, with
polarisation index $\lambda$ and polarisation vector
$\varepsilon_\mu^{(\lambda)}$, by $\bra{0}V_\mu\ket{V(p,\lambda)}
\equiv f^{\textrm{bare}}_Vm_V\varepsilon_\mu^{(\lambda)}$. Then, at
large Euclidean times $t$ and $T - t$, the correlation functions
defined above may be written:
\begin{widetext}
\begin{subequations}
\begin{align}
C_{V_\mu V_\nu}(t,\vec{p}) &\rightarrow
\frac{-(f_V^\mathrm{bare}m_V)^2e^{-E_VT/2}\cosh((t-T/2)E_V)}{E_V}
\left(-g_{\mu\nu}+\frac{p_\mu p_\nu}{m_V^2} \right), \\
C_{\{\rho\mu\}\nu}(t,\vec{p}) &\rightarrow
\frac{-i(f_V^\mathrm{bare}m_V)^2e^{-E_VT/2}\VFMb\sinh((t-T/2)E_V)}{E_V}
\,\frac{1}{2} \left(-g_{\rho\nu}p_\mu -g_{\mu\nu}p_\rho +
\frac{2p_\rho p_\mu p_\nu}{m_V^2} \right), \\
C_{\{\rho\mu\nu\}\sigma}(t,\vec{p}) &\rightarrow
\frac{(f_V^\mathrm{bare}m_V)^2e^{-E_VT/2}\VSMb\sinh((t-T/2)E_V)}{E_V}
\,\frac{1}{3} \left(-g_{\rho\sigma}p_\mu p_\nu
-g_{\mu\sigma}p_\rho p_\nu -g_{\nu\sigma}p_\rho p_\mu + \frac{3p_\rho p_\mu
p_\nu p_\sigma}{m_V^2} \right),
\end{align}
\end{subequations}
where we have used the completeness relation for the polarisation
vectors of massive vector particles,
$\sum_\lambda\varepsilon_\mu^{(\lambda)}\varepsilon_\nu^{*(\lambda)} =
-g_{\mu\nu} + p_\mu p_\nu/m_V^2$. We extract bare values for the first
and second moments from the following ratios:
\begin{subequations}
\label{eq:Vratio}
\begin{align}
\label{eq:VFMratio}
R^V_{\{\rho\mu\}\nu}(t,\vec{p}) &\equiv
\frac{C_{\{\rho\mu\}\nu}(t,\vec{p})}{\frac{1}{3}\sum_i
C_{V_iV_i}(t,\vec{p}=0\;)}
\rightarrow
-i\VFMb\tanh((t-T/2)E_V) \,\frac{1}{2} \left(-g_{\rho\nu}p_\mu
-g_{\mu\nu}p_\rho + \frac{2p_\rho p_\mu p_\nu}{m_V^2} \right), \\
\label{eq:VSMratio}
R^V_{\{\rho\mu\nu\}\sigma}(t,\vec{p}) &\equiv
\frac{C_{\{\rho\mu\nu\}\sigma}(t,\vec{p})}{\frac{1}{3}\sum_i
C_{V_iV_i}(t,p_i=0,|\vec{p}|=\tpoL\;)}\nonumber\\
&\rightarrow \VSMb\tanh((t-T/2)E_V)
\,\frac{1}{3} \left(-g_{\rho\sigma}p_\mu p_\nu
-g_{\mu\sigma}p_\rho p_\nu -g_{\nu\sigma}p_\rho p_\mu + \frac{3p_\rho p_\mu
p_\nu p_\sigma}{m_V^2} \right),
\end{align}
\end{subequations}
\end{widetext}
where the index $i$ runs over spatial dimensions only. We obtain the
first moment from $R^V_{\{\rho4\}\nu}(t,\vec{p})$ at $\vec{p} = 0$ by
taking $\rho = \nu = 1, 2$ or 3. The second moment is obtained from
$R^V_{\{\rho\mu\nu\}\sigma}(t,\vec{p})$ by taking, for example, $\nu
= 4, \rho = 1, \mu = \sigma = 2$ and a single non-zero component of
$\vec{p}$ in the 1-direction.
\section{Numerical Simulations and Results}
\label{sec:numerics}
\subsection{Simulation Details}
\label{subsec:simdet}
Our numerical calculations are based upon gauge field configurations
drawn from the joint datasets used for the broader UKQCD/RBC
domain-wall fermion phenomenology programme. Configurations were
generated with $N_f=2+1$ flavours of dynamical domain-wall fermions
and with the Iwasaki gauge action, using the Rational Hybrid Monte
Carlo (RHMC)~\cite{Clark:2006fx} algorithm on QCDOC
computers~\cite{Boyle:2003ue,Boyle:2003mj,Boyle:2005gf} running the
Columbia Physics System (CPS) software~\cite{cps} and the
BAGEL~\cite{boyle-bagel-cpc,Bagel} assembler generator.
Our set of gauge configurations includes data with two different
volumes but at a single lattice spacing, thus giving us some
indication of the size of finite volume effects but no ability to
perform a continuum extrapolation. We therefore have an unavoidable
systematic error which is, however, formally of
$O(a^2\Lambda^2_\mathrm{QCD})$ due to the automatic $O(a)$-improvement
of the DWF action and operators. In the future, this limitation will
be overcome by performing the analysis with a dataset with a finer
lattice spacing with the same action (such a dataset is now available
and is currently being calibrated). In the meantime, following the
UKQCD/RBC procedure for these configurations~\cite{Allton:2008pn}, we
ascribe a $4\%$ uncertainty as the discretisation error on the
moments. For both volumes, we have a single dynamical strange quark
mass, close to its physical value. We use several independent
ensembles with different light-quark masses ($m_u=m_d$), all heavier
than those found in nature. The hadronic spectrum and other properties
of these configurations have been studied in detail and the results
have been presented in \cite{Allton:2007hx} (for the lattice volume
$(L/a)^3 \times T/a = 16^3 \times 32$) and \cite{Allton:2008pn} (for
the lattice volume $24^3 \times 64$). In both cases the length of the
fifth dimension is $L_s = 16$.
The choice of bare parameters in our simulations is $\beta = 2.13$ for
the bare gauge coupling, $am_s = 0.04$ for the strange quark mass and
$am_q = 0.03,\;0.02,\;0.01$ and, in the $24^3$ case only, $0.005$ for
the bare light-quark masses. A posteriori, the strange quark mass is
found to be about 15\% larger than its physical value. The lattice
spacing is found to be $a^{-1} = 1.729(28)\gev$~\cite{Allton:2008pn},
giving physical volumes of $(1.83\;\mathrm{fm})^3$ and
$(2.74\;\mathrm{fm})^3$. The lattice spacing and physical quark masses
were obtained using the masses of the $\pi$ and $K$ pseudoscalar
mesons and the triply-strange $\Omega$ baryon. The quark masses
obtained in the $24^3$ study are shown in Table~\ref{tab:24qmasses}.
Owing to the remnant chiral symmetry breaking, the quark mass has to
be corrected additively by the residual mass in the chiral limit,
$a\mres = 0.00315(2)$~\cite{Allton:2008pn}. The physical pion masses
are as follows:
\begin{equation}
m_\pi \simeq
\begin{cases}
670\mev & am_q=0.03\\
555\mev & am_q=0.02\\
415\mev & am_q=0.01\\
330\mev & am_q=0.005
\end{cases}
\end{equation}
\begin{table*}
\caption{\label{tab:24qmasses}Lattice scale and unrenormalised quark
masses in lattice units, from the $24^4$
lattices~\cite{Allton:2008pn}. Note $\tilde{m}_X \equiv m_X +
\mres$. Only the statistical errors are given here.}
\begin{ruledtabular}
\begin{tabular}{c|cccccc}
$a^{-1}$ [\Gev] & $a$ [{\rm fm}] & $am_{ud}$ &
$a\widetilde{m}_{ud}$ & $am_s$ & $a\widetilde{m}_s$ &
$a\widetilde{m}_{ud}:a\widetilde{m}_s$ \\\hline
$1.729(28)$ & $0.1141(18)$ & $-0.001847(58)$ & $0.001300(58)$ &
$0.0343(16)$ & $0.0375(16)$ & 1:28.8(4)\\
\end{tabular}
\end{ruledtabular}
\end{table*}
\begin{table}
\caption{\label{tab:16datasets}Parameters for our $16^3$ dataset,
which corresponds largely to that of~\cite{Allton:2007hx}. The
range and measurement separation $\Delta$ are specified in
molecular dynamics time units. $N_\mathrm{meas}$ is the number of
measurements for each source position $t_\mathrm{src}$. The total
number of measurements is therefore $N_\mathrm{meas}\times
N_\mathrm{src}$, where $N_\mathrm{src}$ is the number of different
values for $t_\mathrm{src}$. In the right-most column, XY-XY
denotes contraction of two quark propagators with X-type smearing
at source and Y-type smearing at sink: G = Gaussian wavefunction,
L = point.}
\begin{ruledtabular}
\begin{tabular}{c|cccccc}
$m_l$ & Range & $\Delta$ & $N_\mathrm{meas}$ &
$t_\mathrm{src}$ locations & Smearing\\
\hline
0.01 & 500--3990 & 10 & 350 & 0,\,8,\,16,\,24 & GL-GL \\
0.02 & 500--3990 & 10 & 350 & 0,\,8,\,16,\,24 & GL-GL \\
0.03 & 4030--7600 & 10 & 358 & 0,\,16 & GL-GL \\
\end{tabular}
\end{ruledtabular}
\end{table}
\begin{table}
\caption{\label{tab:24datasets}Parameters for our $24^3$ dataset,
which corresponds to the unitary part of the dataset
of~\cite{Allton:2008pn}. Columns as in Table~\ref{tab:16datasets}
with addition of H = gauge-fixed hydrogen S-wave smearing.}
\begin{ruledtabular}
\begin{tabular}{c|cccccc}
$m_l$ & Range & $\Delta$ & $N_\mathrm{meas}$ &
$t_\mathrm{src}$ locations & Smearing\\ \hline
0.005 & 900--4480 & 20 & 180 & 0,\,32,\,16 & HL-HL\\
0.01 & 800--3940 & 10 & 315 & 0,\,32 & GL-GL \\
0.02 & 1800--3580 & 20 & 90 & 0,\,32 & HL-HL \\
0.03 & 1260--3040 & 20 & 90 & 0,\,32 & HL-HL \\
\end{tabular}
\end{ruledtabular}
\end{table}
Measurements were performed using the UKhadron software package that
makes use of both the BAGEL DWF inverter~\cite{boyle-bagel-cpc,Bagel}
and elements of the SciDAC software library stack including the Chroma
LQCD library~\cite{Edwards:2004sx} and QDP++. The details are
summarised in Tables~\ref{tab:16datasets} and \ref{tab:24datasets}. We
restrict our analysis to the unitary data for which the valence and
sea quark masses are the same (partially-quenched data was used
extensively in the studies of the chiral behaviour of the spectrum and
decay constants in~\cite{Allton:2008pn}). On the $16^3$ lattice, our
dataset differs from that used in~\cite{Allton:2007hx} in that the
Markov chains have been extended for the heaviest light quark mass to
give additional statistics, using an improved algorithm that
decorrelated topology rather more quickly.
In order to improve the statistical sampling of the correlation
functions, on each configuration we have averaged the results obtained
from either $2$, $3$ or $4$ sources spaced out along a lattice
diagonal. In the $16^3$ case, for example, the sources used are at the
origin, at $(4,4,4,8)$, $(8,8,8,16)$ and $(12,12,12,24)$. Statistical
errors for observables are estimated using single-elimination
jackknife, with measurements made on the same configuration but at
different source positions put in the same jackknife bin because of
the correlations expected between them. In order to lessen the effect
of autocorrelations, we follow the same blocking procedures as
in~\cite{Allton:2007hx} and~\cite{Allton:2008pn}. In the $16^3$ case,
the span of the measurements in each block covers $50$ molecular
dynamics time units. In the $24^3$ case, for the $m_qa = 0.005$ and
$am_q = 0.01$ ensembles, each jackknife bin contains measurements from
every $80$ molecular dynamics time units, while for the $am_q = 0.02$
and $am_q = 0.03$ ensembles each bin contains measurements from every
$40$ molecular dynamics time units in order to have a reasonable
number of bins for the analysis.
We use source smearing to improve the overlap with the mesonic states,
either gauge-fixed hydrogen $S$-wavefunction
smearing~\cite{Boyle:1999gx} with radius $r = 3.5$ in lattice units or
gauge invariant Gaussian smearing~\cite{Allton:1993wc} with radius $r
= 4$.
\subsection{Results}
\label{subsec:results}
\begin{figure*}
\includegraphics[width=0.48\textwidth,
bb=90 420 436 658]{figs/K1ST16plateau.eps}
\hfill
\includegraphics[width=0.48\textwidth,
bb=90 420 436 658]{figs/K1STplateau.eps}
\caption{\label{fig:corrsK1}Results for $\PFM_K^\mathrm{bare}$ as a
function of the time, on the $16^3$ (left) and $24^3$ (right)
lattices. The shaded band shows the fit range, fitted value and
its error.}
\end{figure*}
In order to extract $\PFM_K$ from the ratio
$R^P_{\{\rho\mu\};\nu}(t,\vec{p})$ defined in \eqref{eq:PFMratio}, we
need the two correlation functions to be measured at $|\vec{p}| \ne
0$. Since we expect hadronic observables with larger lattice momenta
to have larger lattice artefacts and statistical errors, we restrict
the choice of indices to $\rho=\nu=4$ and $\mu=1,2$ or 3 with $|{\vec
p}\,|=2\pi/L$ (i.e., $p_\mu=\pm 2\pi/L$ with the remaining two
components of ${\vec p}$ equal to 0). $\PFM_K^\mathrm{bare}$ can then
be obtained from the ratio at large times:
\begin{equation}
R^P_{\{4 k\};\,4}(t,p_k=\pm 2\pi/L)=
\pm\, i\,\tpoL\,\PFM^\mathrm{bare}\,,
\end{equation}
with $|\vec{p}|=2\pi/L$ and $k=1,2,3$. The plots in
Fig.~\ref{fig:corrsK1} show our results for $\PFM_K^\mathrm{bare}$ as
a function of $t$ obtained from the ratio $R^P_{\{4
k\};\,4}(t,p_k{=}\pm 2\pi/L)$ for the four values of the light-quark
mass, combining results at $t$ with those at $T-t-1$. The results have
been averaged over the three values of $k$ and, in total, the $6$
equivalent lattice momenta with $|\vec{p}|=2\pi/L$.
To obtain $\PSM_{\pi,K}^\mathrm{bare}$ from the ratio
$R^P_{\{\rho\mu\nu\};\sigma}(t,\vec{p})$ defined in
\eqref{eq:PSMratio} we need two non-zero components of momentum, so we
use
\begin{multline}
R^P_{\{4 j k\};\,4}(t,p_j=\pm 2\pi/L, p_k = \pm 2\pi/L)=\\
-(\pm\tpoL)(\pm\tpoL)\PSM^\mathrm{bare}\,,
\end{multline}
with $|\vec{p}|=\sqrt{2}\,2\pi/L$, $k,j=1,2,3$ and $k \ne j$. We average
over all $4$ momentum combinations appropriate to each of the $3$
possible Lorentz index choices.
We may extract $\VFMb_{K^*}$ from the ratio
$R^V_{\{\rho\mu\}\nu}(t,\vec{p})$ defined in \eqref{eq:VFMratio} by
considering only zero-momentum correlation functions. In the
denominator, we average $C_{V_iV_i}(t,\vec{p}=0)$ over all $3$ spatial
directions. In the numerator, we average over
$C_{\{41\}1}(t,\vec{p}=0)$, $C_{\{42\}2}(t,\vec{p}=0)$ and
$C_{\{43\}3}(t,\vec{p}=0)$. Results are shown in
Fig.~\ref{fig:corrsV1}.
\begin{figure*}
\includegraphics[width=0.48\textwidth,
bb=90 420 436 658]{figs/Kstar1ST16plateau.eps}
\hfill
\includegraphics[width=0.48\textwidth,
bb=90 420 436 658]{figs/Kstar1STplateau.eps}
\caption{\label{fig:corrsV1}Results for $\VFMb_{K^*}$ as a function of the
time, on the $16^3$ (left) and $24^3$ (right) lattices. Symbols as in
Fig.~\ref{fig:corrsK1}}
\end{figure*}
$\VSMb_{K^*,\rho,\phi}$ is extracted from the ratio defined in
\eqref{eq:VSMratio} by averaging
$C_{V_iV_i}(t,p_i=0,|\vec{p}|=\tpoL\;)$ over all $4$ appropriate
momenta for all $3$ spatial directions in the denominator. In the
numerator we average over all possible combinations of
$C_{\{4ij\}i}(t,p_j = \pm \tpoL,|\vec{p}|=\tpoL)$ with $i \ne j$. In
principle we should include disconnected contributions in the $\phi$
correlation functions. We argue that these contributions are
Zweig-suppressed however and can therefore be neglected.
If picking the fit range was not straightforward, we considered the
correlation functions in the numerator and denominator separately. We
identified and excluded from our fits the region where the excited
states still contributed. We then chose the fit range aiming for a good
$\chi^2/\text{d.o.f.}$ and a stable fit with respect to small
variations of the lower bound of the range. Owing to the increasing
noise when $t$ gets larger, the fits are insensitive to the upper
bound of the fit range.
The $16^3$ and $24^3$ bare results are given in
Tables~\ref{tab:16results} and \ref{tab:24results} respectively,
complete with linear chiral extrapolations which, as we shall discuss
in the next section, can be justified using chiral perturbation theory
(at least in the pseudoscalar case).
\subsection{Quark Mass Extrapolations}
\label{subsec:chiral}
\begin{figure*}
\includegraphics[width=0.48\hsize,bb=169 418 497 660]{figs/XKaon1st.ps}
\hfill
\includegraphics[width=0.48\hsize,bb=169 418 497 660]{figs/XKstar1st.ps}
\caption{\label{fig:Kaon1st_chiral}Chiral extrapolations for
$\PFM_K^\mathrm{bare}$ and $\VFMb_{K^*}$. The extrapolation to the
physical point is shown by the vertical solid line, with
uncertainty, dominated by the uncertainty in the physical strange
mass, indicated by the dotted lines.}
\end{figure*}
In leading-order chiral perturbation theory~\cite{Chen:2003fp}, $\PFM_K$ is
proportional to $m_s - m_{u/d}$ without chiral logarithms:
\begin{equation}
\langle\xi^1\rangle_K = \frac{8B_0}{f^2}(m_s - m_{u/d})b_{1,2}\;,
\end{equation}
where $f$ and $B_0$ denote the usual chiral perturbation theory
parameters and $b_{1,2}$ is a Wilson coefficient introduced
in~\cite{Chen:2003fp}. Our data shows clearly the effects of SU(3)
symmetry breaking and is compatible with this expectation. We
therefore perform a linear extrapolation in $a(m_s - m_q)$ to the
physical point $a(m_s - m_{ud})$, as shown in
Fig.~\ref{fig:Kaon1st_chiral}. The second error quoted in the results
in the chiral-limit for the first moments in
Tables~\ref{tab:16results} and~\ref{tab:24results} is due to the
uncertainty in this physical point (determined using the quark masses
in Table~\ref{tab:24qmasses}). In this way we deal simultaneously with
the usual light-quark mass extrapolation and with the strange quark
mass extrapolation which is necessitated by our strange quark mass
being approximately $15\%$ too heavy. We have not constrained our fit
to vanish in the SU(3) limit.
A similar linear behaviour is seen for $\VFMb_{K^*}$ (see
Fig.~\ref{fig:Kaon1st_chiral}), so we follow the same extrapolation
procedure. We note a hint of a finite volume effect in the $K^*$ case
but not in the $K$ case, which is contrary to what we would expect.
Where we have $K^*$ results for both volumes at the same
light-quark mass, however, they agree within the statistical
uncertainties.
For the second moments, we also have some guidance from chiral
perturbation theory~\cite{Chen:2005js}; there is no non-analytic
dependence at 1~loop and we should fit linearly in $m^2_\pi$. The
dependence on the quark masses is very mild in any case and in fact
our results for the $\rho$, $K^*$ and $\phi$ agree within the
statistical errors. Therefore we perform a linear extrapolation in the
light quark masses and neglect the effect of the too-heavy strange
quark mass (see Fig.~\ref{fig:a-z}). We see no indication for finite
size effects in the second moments when we compare the data points on
the two different lattice volumes. They agree within their statistical
errors.
\begin{figure*}
\includegraphics[width=0.48\hsize,bb=108 417 435 657]{figs/XPion2nd.ps}
\hfill
\includegraphics[width=0.48\hsize,bb=108 417 435 657]{figs/XKaon2nd.ps}\\[1ex]
\includegraphics[width=0.48\hsize,bb=108 417 435 657]{figs/XRho2nd.ps}
\hfill
\includegraphics[width=0.48\hsize,bb=108 417 435 657]{figs/XKstar2nd.ps}\\[1ex]
\includegraphics[width=0.48\hsize,bb=108 417 435 657]{figs/XPhi2nd.ps}
\caption{Chiral extrapolations for $\PSM_{\pi}^\mathrm{bare}$,
$\PSM_{K}^\mathrm{bare}$, $\VSMb_{\rho}$, $\VSMb_{K^*}$ and
$\VSMb_{\phi}$. The physical value for $am_q + a\mres$ is
shown by the solid vertical line in each case.\label{fig:a-z}}
\end{figure*}
\begin{table*}
\caption{\label{tab:16results}Summary of results for the bare values
of the distribution amplitude moments on the $16^3$ lattices. The
chiral extrapolations are discussed in Sec.~\ref{subsec:chiral}, and
the errors are statistical and (in the first moment case) due to the
uncertainty in the physical point for the chiral extrapolation.}
\begin{ruledtabular}
\begin{tabular}{l|llllll}
$am_{ud}$ & 0.03 & 0.02 & 0.01 & 0.005 & $\chi$-limit \\
\hline
$\PSM_\pi^\mathrm{bare}$ & 0.110(2) & 0.109(2) & 0.113(4) & - & 0.112(5) \\
$\PFM_K^\mathrm{bare}$ & 0.00543(27) & 0.01174(71) & 0.0194(15) & - & 0.0228(14)(11) \\
$\PSM_K^\mathrm{bare}$ & 0.109(2) & 0.107(2) & 0.113(3) & - &
0.112(4) \\
$\VSMb_\rho$& 0.113(4) & 0.100(5) & 0.116(6) & - & 0.109(10) \\
$\VFMb_{K^*}$ & 0.00610(24) & 0.01275(51) & 0.0207(10) & - &
0.02443(96)(107) \\
$\VSMb_{K^*}$ & 0.111(4) & 0.101(4) & 0.113(4) & - & 0.110(6) \\
$\VSMb_\phi$& 0.109(3) & 0.100(3) & 0.109(3) & - & 0.107(5) \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\begin{table*}
\caption{\label{tab:24results}Summary of results for the bare values
of the distribution amplitude moments on the $24^3$ lattices}
\begin{ruledtabular}
\begin{tabular}{l|llllll}
$am_{ud}$ & 0.03 & 0.02 & 0.01 & 0.005 & $\chi$-limit \\
\hline
$\PSM_\pi^\mathrm{bare}$ & 0.103(9) & 0.104(6) & 0.114(3) & 0.121(9) & 0.125(7) \\
$\PFM_K^\mathrm{bare}$ & 0.00566(33) & 0.01254(72) & 0.01946(65) & 0.0231(15) & 0.02377(71)(110) \\
$\PSM_K^\mathrm{bare}$ & 0.103(8) & 0.106(4) & 0.112(2) & 0.113(6) & 0.117(5) \\
$\VSMb_\rho$& 0.110(9) & 0.093(10) & 0.112(3) & 0.120(13)& 0.118(7) \\
$\VFMb_{K^*}$ & 0.00619(35) & 0.0139(10)& 0.0225(13)& 0.0311(30) & 0.0281(13)(14) \\
$\VSMb_{K^*}$ & 0.109(12) & 0.095(8) & 0.108(3) & 0.117(5) & 0.118(7) \\
$\VSMb_\phi$& 0.108(7) & 0.097(7) & 0.105(2) & 0.107(3) & 0.107(4) \\
\end{tabular}
\end{ruledtabular}
\end{table*}
\section{Renormalisation of the Lattice Composite Operators}
\label{sec:renormalisation}
We now discuss the conversion of our bare lattice results to results
in the \MSbar\ scheme. To reduce systematic uncertainties we have
determined the renormalisation factors nonperturbatively in the
\ripm\ scheme, continuing the work in~\cite{Aoki:2007xm}, and convert
to \MSbar\ using $3$-loop continuum perturbation
theory~\cite{Gracey:2003mr, Gracey:2006zr}. We begin, however, with a
perturbative calculation of the renormalisation factors. The
perturbative results have been used previously
in~\cite{Donnellan:2007xr, Boyle:2008nj} and will provide a comparison
to the nonperturbative results. The contribution to the second moment
from mixing with a total-derivative operator is calculated
perturbatively only. We will see that this contribution is small and
is not accessible within the current nonperturbative scheme.
\subsection{Perturbative Renormalisation}
\label{sec:PR}
\begin{table*}
\caption{Constants needed for the perturbative renormalisation of the
first and second moment operators using domain-wall fermions and the
Iwasaki gauge action ($c_1=-0.331$). $M$ is the domain-wall height,
$c=\Sigma_1^\MSbar - \Sigma_1 + V^\MSbar - V$, $\cDD =
\Sigma_1^\MSbar - \Sigma_1 + \VDD^\MSbar - \VDD$ and
$\cdd=\Vdd^\MSbar-\Vdd$. $\Sigma_1$, $V$, $\VDD$ and $\Vdd$ are
dependent on the gauge and the infrared regulator: Feynman gauge and
a gluon mass are used here. $V$ was calculated
in~\cite{Boyle:2006pw}, while $\VDD$ and $\Vdd$ have been calculated
as part of this work.}
\label{tab:VvsM}
\begin{minipage}{.9\textwidth}
\begin{ruledtabular}
\begin{tabular}{D..1D..4D..4D..4D..4D..4D..4D..4}
\multicolumn1c{$M$} &
\multicolumn1c{$\Sigma_1$} &
\multicolumn1c{$V$} &
\multicolumn1c{$c$} &
\multicolumn1c{$\VDD$} &
\multicolumn1c{$\cDD$} &
\multicolumn1c{$\Vdd$} &
\multicolumn1c{$\cdd$} \\
\hline
0.1 & 4.6519 & -4.6297 & -0.9110 & -10.816 & 4.9838 & 0.5415 & 0.0279 \\
0.2 & 4.5193 & -4.5614 & -0.8468 & -10.698 & 4.9982 & 0.4285 & 0.1409 \\
0.3 & 4.4093 & -4.5101 & -0.7881 & -10.608 & 5.0179 & 0.3433 & 0.2262 \\
0.4 & 4.3158 & -4.4678 & -0.7369 & -10.533 & 5.0362 & 0.2729 & 0.2966 \\
0.5 & 4.2354 & -4.4311 & -0.6932 & -10.467 & 5.0509 & 0.2119 & 0.3575 \\
0.6 & 4.1665 & -4.3980 & -0.6574 & -10.407 & 5.0603 & 0.1573 & 0.4122 \\
0.7 & 4.1079 & -4.3673 & -0.6295 & -10.352 & 5.0639 & 0.1070 & 0.4625 \\
0.8 & 4.0593 & -4.3381 & -0.6101 & -10.300 & 5.0604 & 0.0597 & 0.5098 \\
0.9 & 4.0204 & -4.3097 & -0.5996 & -10.250 & 5.0489 & 0.0142 & 0.5552 \\
1. & 3.9915 & -4.2816 & -0.5988 & -10.200 & 5.0283 & -0.0303 & 0.5998 \\
1.1 & 3.9731 & -4.2529 & -0.6090 & -10.151 & 4.9970 & -0.0749 & 0.6443 \\
1.2 & 3.9664 & -4.2232 & -0.6321 & -10.100 & 4.9528 & -0.1205 & 0.6899 \\
1.3 & 3.9727 & -4.1916 & -0.6700 & -10.047 & 4.8933 & -0.1682 & 0.7376 \\
1.4 & 3.9943 & -4.1571 & -0.7261 & -9.9895 & 4.8147 & -0.2195 & 0.7889 \\
1.5 & 4.0343 & -4.1182 & -0.8050 & -9.9267 & 4.7119 & -0.2764 & 0.8458 \\
1.6 & 4.0974 & -4.0728 & -0.9135 & -9.8551 & 4.5771 & -0.3418 & 0.9112 \\
1.7 & 4.1905 & -4.0176 & -1.0618 & -9.7700 & 4.3989 & -0.4205 & 0.9899 \\
1.8 & 4.3249 & -3.9462 & -1.2676 & -9.6627 & 4.1572 & -0.5211 & 1.0905 \\
1.9 & 4.5209 & -3.8447 & -1.5651 & -9.5140 & 3.8125 & -0.6631 & 1.2325
\end{tabular}
\end{ruledtabular}
\end{minipage}
\end{table*}
The perturbative matching from the lattice to \MSbar\ schemes is
performed by comparing one-loop calculations of quark two-point one
particle irreducible ($1$PI) functions with an insertion of the
relevant bare lattice operator. This requires the evaluation of the
diagrams shown in Fig.~\ref{fig:ptdiags}, together with wavefunction
renormalisation factors, Fig.~\ref{fig:wavefnren}. For the
first-moment operator, we define
\begin{equation}
\label{eq:Z1ddef}
\firstmomentop^\MSbar(\mu) = Z_{\firstmomentop}(\mu a)
\firstmomentop^\latt(a)\,.
\end{equation}
For the second moment calculation we must take account of mixing with
a total derivative operator (c.f.\ Sec.~\ref{subsec:opmix}). Adopting
the notation
\begin{equation}
\ODD = \overline{\psi} \gamma_{\{\mu}\gamma_5 \ovlra{D}_\nu
\ovlra{D}_{\kappa\}} \psi,\quad
\Odd = \partial_{\{\nu}\partial_\kappa \overline{\psi} \gamma_{\mu\}}
\gamma_5 \psi,
\end{equation}
with all Lorentz indices distinct and symmetrised, we need to determine
\begin{equation}
\label{eq:Z2ddef}
\ODD^\MSbar(\mu) =
Z_{DD,DD}(\mu a) \ODD^\mathrm{latt}(a) +
Z_{DD,\partial\partial}(\mu a) \Odd^\mathrm{latt}(a).
\end{equation}
The renormalisation factors are given by
\begin{widetext}
\begin{align}
\label{eq:Zpt}
Z_{\firstmomentop}(\mu a) &= \frac1{(1-w_0^2)Z_w}
\left[ 1 + \bigG \left( -\frac{16}3 \ln(\mu a) + \Sigma_1^\MSbar
- \Sigma_1 + V^\MSbar - V \right) \right]\,,\\
\label{eq:ZDDDD}
Z_{DD,DD}(\mu a) &= \frac1{(1-w_0^2)Z_w}
\left[ 1 + \bigG \left( -\frac{25}3 \ln(\mu a) + \Sigma_1^\MSbar
- \Sigma_1 + \VDD^\MSbar - \VDD \right) \right], \\
\label{eq:ZDDdd}
Z_{DD,\partial\partial}(\mu a) &= \frac1{(1-w_0^2)Z_w} \,
\bigG \left( \frac53 \ln(\mu a) + \Vdd^\MSbar - \Vdd \right).
\end{align}
\end{widetext}
In the equations above $(1-w_0^2)Z_w$ is a characteristic
normalisation factor for the physical quark fields in the domain-wall
formalism. $Z_w$ represents an additive renormalisation of the large
Dirac mass or domain-wall height $M=1-w_0$, which can be rewritten in
multiplicative form at one-loop as
\begin{equation}
Z_w = 1+\bigG \,z_w,
\qquad
z_w = \frac{2w_0}{1-w_0^2}\,\Sigma_w.
\end{equation}
The one-loop correction $z_w$ becomes very large for certain choices of
$M$~\cite{Aoki:1998vv, Aoki:2002iq}, including that used in our numerical
simulations, so that some form of mean-field improvement is necessary, as
discussed below.
\begin{figure}
\hbox to\hsize{\hss
\includegraphics[width=0.8\hsize]{figs/ptdiags.eps}
\hss}
\caption{One-loop vertex diagrams evaluated in the perturbative renormalisation
of the $1$st and $2$nd moment operators.}
\label{fig:ptdiags}
\end{figure}
\begin{figure}
\hbox to\hsize{\hss
\includegraphics[width=0.4\hsize]{figs/wavefn.eps}
\hss}
\caption{One-loop diagrams for the quarks' wavefunction renormalisation.}
\label{fig:wavefnren}
\end{figure}
Terms with superscripts \MSbar\ in Eqs.~\eqref{eq:Zpt},
\eqref{eq:ZDDDD} and~\eqref{eq:ZDDdd} arise from the continuum
calculations, while unsuperscripted terms come from the
computations in the lattice scheme. To shorten some expressions below
we will define
\begin{align}
\label{eq:c}
c &= \Sigma_1^\MSbar - \Sigma_1 + V^\MSbar - V, \\
\label{eq:cDD}
\cDD &= \Sigma_1^\MSbar - \Sigma_1 + \VDD^\MSbar - \VDD, \\
\label{eq:cdd}
\cdd &= \Vdd^\MSbar - \Vdd.
\end{align}
The terms $\Sigma_1^\MSbar$ and $\Sigma_1$ come from quark wavefunction
renormalisation, while $V^\MSbar$, $\VDD^\MSbar$, $\Vdd^\MSbar$ and $V$, $\VDD$,
$\Vdd$ come from the one-loop corrections to the amputated two-point function.
They are given by ``vertex'' and ``sail'' diagrams, plus an operator tadpole
diagram in the lattice case. $\VDD^\MSbar$ and $\VDD$ can be isolated by
computing the one-loop correction with equal incoming and outgoing quark
momenta. Likewise $\Vdd^\MSbar$ and $\Vdd$ are found by setting the incoming and
outgoing quark momenta equal and opposite (the lattice tadpole diagram does not
contribute in this case). Using naive dimensional regularisation (NDR) in
Feynman gauge with a gluon mass infrared (IR) regulator,
\begin{align}
\Sigma_1^\MSbar &= \frac12,
&
V^\MSbar &= -\frac{25}{18},\\
\VDD^\MSbar &= -\frac{121}{72},
&
\Vdd^\MSbar &= \frac{41}{72}.
\end{align}
The lattice contributions are evaluated for domain-wall fermions with
the Iwasaki gluon action ($c_1=-0.331$), also choosing Feynman gauge
and using a gluon mass IR regulator. $\Sigma_1$ has been evaluated
in~\cite{Aoki:2002iq}, while we calculated the vertex term $V$ for the
first moment operator in~\cite{Boyle:2006pw}. Here we have calculated
the vertex terms $\VDD$ and $\Vdd$ for the second moment operator.
Perturbative calculations with domain-wall fermions are explained
in~\cite{Aoki:1998vv, Aoki:2002iq} and the form of the Iwasaki gluon
propagator can be found in~\cite{Iwasaki:1983ck}. Values for
$\Sigma_1$, $V$, $\VDD$ and $\Vdd$ are given as functions of $M$ in
Table~\ref{tab:VvsM}, along with $c$, $\cDD$ and $\cdd$. Chiral
symmetry of the domain-wall action implies that these results also
apply for the operators which are like those used here, but without
the $\gamma_5$. We note that the perturbative renormalisation factor
for the first moment operator using alternative fermion and gauge
formulations can be found in~\cite{Capitani:2005vb} (domain-wall
fermions and plaquette action),~\cite{Horsley:2005jk} (overlap
fermions and L\"uscher--Weisz action) and~\cite{Gockeler:2006nb}
(clover fermions and plaquette action). Second moment calculations
with clover and Wilson fermions have been performed
in~\cite{Gockeler:2006nb} and~\cite{Gockeler:2004xb} respectively (in
both cases using the plaquette action).
\begin{table}
\caption{Values for $z_w$, $z_w^\mathrm{MF}$ extracted from the
results in~\cite{Aoki:2002iq}, and $d_f$ extracted
from~\cite{Aoki:2003uf}.}
\label{tab:zwmf}
\begin{ruledtabular}
\begin{tabular}{D..1D..4D..4D..7}
\multicolumn1c{$M$} &
\multicolumn1c{$z_w$} &
\multicolumn1c{$z_w^\mathrm{MF}$} &
\multicolumn1c{$d_f$}\\
\hline
0.1 & -243.86 & -86.579 & -0.02303\\
0.2 & -113.29 & -39.501 & -0.01798\\
0.3 & -69.404 & -23.830 & -0.01497\\
0.4 & -47.077 & -15.949 & -0.01274\\
0.5 & -33.278 & -11.142 & -0.01090\\
0.6 & -23.648 & -7.8365 & -0.009315\\
0.7 & -16.300 & -5.3538 & -0.007896\\
0.8 & -10.263 & -3.3459 & -0.006589\\
0.9 & -4.9617 & -1.6078 & -0.005379\\
1.0 & 0.0 & 0.0 & -0.004261\\
1.1 & 4.9442 & 1.5902 & -0.003227\\
1.2 & 10.192 & 3.2748 & -0.002290\\
1.3 & 16.136 & 5.1900 & -0.001485\\
1.4 & 23.346 & 7.5350 & -0.0008650\\
1.5 & 32.784 & 10.648 & -0.0005360\\
1.6 & 46.322 & 15.194 & -0.0006566\\
1.7 & 68.294 & 22.720 & -0.001570\\
1.8 & 111.69 & 37.901 & -0.004014\\
1.9 & 241.55 & 84.270 & -0.01020
\end{tabular}
\end{ruledtabular}
\end{table}
Our numerical simulations use $M=1.8$. For this value of $M$, with the
Iwasaki gluon action, the one-loop coefficient in the physical quark
normalisation is $z_w \approx 112$ (extracted from $\Sigma_w$ in
Table~III of~\cite{Aoki:2002iq}), making it clear that mean-field
improvement is necessary. We follow the prescription given
in~\cite{Aoki:2002iq}. The first step is to define a mean-field value
for the domain-wall height,
\begin{equation}
\Mmf = M - 4(1-P^{1/4}) = 1.3029
\end{equation}
where $P=0.58813(4)$ is the average plaquette value in the chiral
limit in our simulations. The physical quark normalisation factor
becomes $\left[1-(\wmf)^2\right]Z_w^\mathrm{MF}$, with
\begin{equation}
\begin{aligned}
\label{eq:Zzwmf}
Z_w^\mathrm{MF} &= 1+\bigG z_w^\mathrm{MF},\\
z_w^\mathrm{MF} &= \frac{2\wmf}{1-(\wmf)^2}\,(\Sigma_w + 32\pi^2\Tmf)
= 5.2509,
\end{aligned}
\end{equation}
where $\Tmf=0.0525664$~\cite{Aoki:2002iq} is a mean-field tadpole
factor and $\Sigma_w$ is evaluated at $\Mmf$. Values for
$z_w^\mathrm{MF}$ as a function of $M$ are quoted in
Table~\ref{tab:zwmf}, extracted from the results
in~\cite{Aoki:2002iq}. Likewise, $\Sigma_1=3.9731$, $V=-4.1907$,
$\VDD=-10.045$ and $\Vdd=-0.1696$ are evaluated at $\Mmf$.
For the operator $\ODD$ with two covariant derivatives, mean-field improvement
introduces a factor $u_\mathrm{pt}/u$ where $u$ is the mean link (here taken to
be $u=P^{1/4}$) and
\begin{equation*}
u_\mathrm{pt}=1-\bigG\,8\pi^2\Tmf
\end{equation*}
is its perturbative expansion. For $\Odd$ with two ordinary
derivatives, in contrast, the extra factor is $u/u_\mathrm{pt}$. The
mean-field-improved matching factors are thus
\begin{widetext}
\begin{align}
\label{eq:Z-MF}
Z_{\firstmomentop}^\mathrm{MF}
&= \frac1{1-(\wmf)^2}\,\frac1{Z_w^\mathrm{MF}
\left[1+\bigG \left( -\frac{16}3 \ln(\mu a) + c^\mathrm{MF}
\right) \right]\\
\label{eq:ZDDDD-MF}
Z_{DD,DD}^\mathrm{MF}
&= \frac1u\,\frac1{1-(\wmf)^2}\,\frac1{Z_w^\mathrm{MF}
\left[ 1 + \bigG \left( -\frac{25}3 \ln(\mu a) +
\cDD^\mathrm{MF} - 8\pi^2\Tmf
\right) \right]\\
\label{eq:ZDDdd-MF}
Z_{DD,\partial\partial}^\mathrm{MF}
&= u\,\frac1{1-(\wmf)^2}\,\frac1{Z_w^\mathrm{MF}} \,
\bigG \left( \frac53 \ln(\mu a) + \cdd^\mathrm{MF}
\right)
\end{align}
\end{widetext}
with $c^\mathrm{MF} = -0.6713$, $\cDD^\mathrm{MF} - 8\pi^2\Tmf =
0.7408$ and $\cdd^\mathrm{MF} = 0.7391$. To evaluate these
expressions, we make two choices for the coupling. The first is a
mean-field improved coupling defined using the measured plaquette
value $P$, according to~\cite{Aoki:2003uf}
\begin{multline}
\label{eq:meas-plaq}
\frac1{g^2_\mathrm{MF}(\mu)} =
\frac P{g_0^2} + d_g + c_p +\frac{22}{16\pi^2}\,\ln(\mu a)\\
+ N_f \left[ d_f -\frac4{48\pi^2}\,\ln(\mu a)\right]
\end{multline}
where $N_f$ is the number of dynamical quark flavours. For the Iwasaki
gauge action with $c_1=-0.331$, the values $d_g=0.1053$ and
$c_p=0.1401$ are given in~\cite{Aoki:2002iq}, while values for $d_f$
as a function of $M$ were calculated in~\cite{Aoki:2003uf} and are
quoted in Table~\ref{tab:zwmf}. In our simulations, $\beta = 6/g_0^2 =
2.13$ with $N_f=3$ and $a^{-1}=1.729\gev$. The second choice is the
continuum \MSbar\ coupling, calculated as outlined in Appendix A
of~\cite{Aoki:2007xm}. At $\mu a=1$, we find $\alpha_\mathrm{MF} =
0.1769$ and $\alpha^\MSbar = 0.3138$. We use these two values to
evaluate the renormalisation factors above. We also evaluate the
mean-field improved expression for the axial vector current
renormalisation~\cite{Aoki:2002iq}, interpolating to our mean-field
$\Mmf$. The values are shown in Table~\ref{tab:ptrenorm}. The ratios
of the renormalisation factors, from which the factor
$1/(1-(\wmf)^2)Z_w^\mathrm{MF}$ cancels, are also shown in the table.
\begin{table*}
\caption{Perturbative renormalisation factors and their ratios for two
choices of the strong coupling, evaluated at $\mu a = 1$.}
\label{tab:ptrenorm}
\begin{ruledtabular}
\begin{tabular}{lccccccc}
& $Z_{\firstmomentop}^\mathrm{MF}$ & $Z_{DD,DD}^\mathrm{MF}$
& $Z_{DD,\partial\partial}^\mathrm{MF}$ & $Z_\mathrm{A}^\mathrm{MF}$
& $\frac{Z_{\firstmomentop}^\mathrm{MF}}{Z_\mathrm{A}^\mathrm{MF}}$
& $\frac{Z_{DD,DD}^\mathrm{MF}}{Z_\mathrm{A}^\mathrm{MF}}$
&
$\frac{Z_{DD,\partial\partial}^\mathrm{MF}}{Z_\mathrm{A}^\mathrm{MF}}$\\[1.1ex]
\hline
$\alpha_\mathrm{MF}$ & 0.9896 & 1.1604 & 0.0122 & 0.8009 &
1.2356 & 1.4488 & 0.0152\\
$\alpha^\MSbar$ & 0.9162 & 1.0966 & 0.0202 & 0.6934 &
1.3214 & 1.5815 & 0.0291
\end{tabular}
\end{ruledtabular}
\end{table*}
We take the mean value of the results with the two different choices
for the coupling as the best answer for the renormalisation factors.
The difference between the two choices will form the error. The
relevant factors for the perturbative renormalisation of the ratios in
Eqs.~\eqref{eq:Pratio} and~\eqref{eq:Vratio} are given in
Table~\ref{tab:PTZs}. Chiral symmetry here ensures that we do not have
to distinguish between vector and axial-vector operators. We note that
the contribution from the mixing term $Z_{DD,\partial\partial}$ is
smaller than the error on $Z_{DD,DD}$ itself.
\begin{table}
\caption{Perturbative renormalisation factors to match the
lattice results to \MSbar\ at $a\mu = 1$.}
\label{tab:PTZs}
\begin{minipage}{.5\hsize}
\begin{ruledtabular}
\begin{tabular}{D..5D..5D..6}
\multicolumn1c{$\frac{Z_{\firstmomentop}}{Z_\mathrm{A}}$} &
\multicolumn1c{$\frac{Z_{DD,DD}}{Z_\mathrm{A}}$} &
\multicolumn1c{$\frac{Z_{DD,\partial\partial}}{Z_\mathrm{A}}$}\\[1.1ex]
\hline
1.28(4) & 1.52(7) & 0.022(7)\\
\end{tabular}
\end{ruledtabular}
\end{minipage}
\end{table}
\subsection{Nonperturbative Renormalisation}
\label{sec:NPR}
In order to renormalise the correlation functions nonperturbatively
we make use of the Rome-Southampton
\ripm\ scheme~\cite{Martinelli:1994ty} which we now briefly review and
discuss refinements to~\cite{Aoki:2007xm}. The starting point and
definition of the \ripm\ scheme is a simple renormalisation condition
that can be imposed independently of the regularisation used, thus on
the lattice as well as in the continuum. This facilitates scheme
changes which is important for the matching to \MSbar. The
renormalisation condition has the form
\begin{equation}
\label{eq:npr:rencond}
\Lambda_\op{O}(p) = Z_\op{O}(\mu) Z_q^{-1}(\mu)\,
\Lambda^\text{bare}_\op{O}(p)\Bigr|_{p^2=\mu^2}
= \Lambda^\text{tree}_\op{O}(p),
\end{equation}
where $\Lambda_\op{O}$ ($\Lambda^\text{bare}_\op{O}$) is the
renormalised (bare) vertex amplitude. Together with the quark field
renormalisation $Z_q^{1/2}$, defined by $\psi =
Z_q^{1/2}\psi^\mathrm{bare}$, this defines the renormalisation
constant $Z_\op{O}$ for the operator $\op{O}$. The renormalisation
scale $\mu$ is set by the momentum of the external states entering the
vertex amplitude. In the original \ripm\ scheme these momenta are
\emph{exceptional}, that is equal incoming and outgoing quark momenta,
$p$ and $p'$. For some renormalisation factors it is advantageous to
use a non-exceptional symmetric choice of momenta $p^2=p'^2=q^2$,
where $q=p-p'$, leading to the distinct \rism\ scheme. This suppresses
unwanted infrared effects in the vertex amplitude, pion poles for
example, and suggests a better-behaved accompanying continuum
perturbation theory~\cite{Sturm:2009kb}. Exceptional momenta with
$q=0$ also cause matrix elements of operators with total derivatives
to vanish, making $Z_{DD,\partial\partial}$ inaccessible in our
nonperturbative analysis.
The vertex amplitude is constructed from the unamputated Green's
function
\begin{equation}
\label{eq:npr:unamp}
\begin{split}
G_\op{O}(p) &= \bret{\psi(p) \op{O}(0) \overline{\psi}(p)}\,,\\
\op{O}(0) &= \sum_{x,x'} \bar{\psi}(x) J_\op{O}(x,x') \psi(x')\,.
\end{split}
\end{equation}
The external quark lines need gauge fixing, for which we use Landau
gauge. The current $J$ has the appropriate Dirac structure and may be
non-local if the operator contains derivatives. For example, a single
right derivative $\ovra{D}_\nu$ in the vector case would correspond to
\begin{equation}
\label{eq:npr:J}
J_{\op{O}_{\rho\mu}}(x,x') = \gamma_\rho \, \frac{1}{2} \Bigl(
U(x,x') \delta_{x',x+\hat{\mu}} - U(x,x') \delta_{x',x-\hat{\mu}}
\Bigr)
\end{equation}
matching the definition in Eq.~\eqref{eq:derdef}.
The vertex amplitude itself is found after amputating the Green's
function and tracing with a suitable projector $P_\op{O}$
\begin{align}
\label{eq:npr:amproj}
\Lambda_\op{O}(p) &= \Tr \left[ \Pi_\op{O}(p) P_\op{O} \right]\\
\intertext{with}
\Pi_\op{O}(p) &= \bret{S(p)}^{-1} \bret{G_\op{O}(p)} \bret{S(p)}^{-1}\,.
\end{align}
We have used the quark propagator $S(p)$ and the angle brackets
indicate the gauge average. The projector $P_\op{O}$ depends on the
particular operator and includes an overall normalisation factor to
account for the colour and Dirac trace. In a simple example $P_\op{O}$
would isolate the tree-level contribution to the vertex amplitude; we
will detail our choices below. We have now defined the renormalisation
procedure and will turn to details of the implementation before
discussing the results.
\subsubsection{Momentum sources}
One refinement to our previous work~\cite{Aoki:2007xm} is the use of
momentum sources~\cite{Gockeler:1998ye}. In contrast to the point
sources used before, this effectively amounts to a volume average over
the lattice resulting in much smaller statistical
errors~\cite{Boyle:2008nj}. Starting from \eqref{eq:npr:unamp} the
Green's function in momentum space is
\begin{equation}
\label{eq:npr:pspace}
G_\op{O}(p) =
\sum_{x,x'} \bret{
\gamma_5 S^\dagger(p)_x \gamma_5\,
J_\op{O}(x,x')\,S(p)_{x'} }\,,
\end{equation}
where rather than use the quark propagator $S(x|y)$ obtained by
inverting the Dirac Matrix $M$ on a point source
\begin{equation}
\label{eq:npr:prop}
\sum_{x} M(x',x)S(x|y) = \delta_{x',y}\,.
\end{equation}
we use
$S(p)_x = \sum_y S(x|y) e^{\I py}$ which can be found by inverting
with a momentum source~\cite{Gockeler:1998ye}
\begin{equation}
\label{eq:npr:pprop}
\sum_x M(x',x)S(p)_{x} = e^{\I px'}\,,
\end{equation}
and is defined on all lattice sites corresponding to the off-shell
quarks used in the Green's function. The gain in statistical accuracy
is paid for with a separate inversion for every momentum used in the
simulation. However, this is more than compensated by a much reduced
number of necessary configurations. Limiting ourselves to a few
carefully chosen momenta, statistical fluctuations are reduced with
lower overall computational cost.
The momenta we use are first of all constrained to be within a range
$\Lambda_\text{QCD}\ll p^2 \ll 1/a$ for the
\ripm\ scheme~\cite{Martinelli:1994ty}. We use our previous
results~\cite{Aoki:2007xm} to identify suitable values and focus on
momenta which are expected to have reduced hypercubic lattice
artefacts by trying to limit $\sum p_\mu^4$ for fixed
$p^2$~\cite{Boyle:2008nj} (see
also~\cite{Boucaud:2003dx,deSoto:2007ht}). The values used are:
\begin{align*}
16^3 \times 32:
&\quad(1,1,2,3),\,(1,1,2,4),\,(1,2,2,1),\\
&\quad(1,2,2,3),\,(1,2,2,4)\\
\intertext{and}
24^3 \times 64:
&\quad(2,2,2,7),\,(2,2,2,8),\,(2,2,3,7),\\
&\quad(2,2,3,8),\,(2,3,3,7) ,
\end{align*}
where we have given $n_\mu^\tra$ for momenta $p_\mu=2\pi n_\mu/L$
(with $L\to T$ for time components).
\subsubsection{Projectors}
We extend the set of operators considered previously
in~\cite{Aoki:2007xm}. We now require operators with up to two
derivatives, $\op{O}^{(5)}_{\{\mu_1\dots\mu_n\}}$ $(n\leq3)$, making
the the necessary projectors $P_\op{O}$ slightly more involved than
for bilinears. Since we resort to readily available
calculations~\cite{Gracey:2003yr, Gracey:2003mr, Gracey:2006zr} for
the final conversion to \MSbar\ as well as to account for running, we
have to tailor the projectors to match the \ripm\ scheme and vertex
functions used in the continuum calculations. Decomposing the
amputated Green's function into terms allowed by Lorentz symmetry and
remembering that we are taking all indices to be distinct, we
find~\cite{Gracey:2003yr,Gracey:2003mr,Gracey:2006zr}
\begin{equation}
\label{eq:npr:grdeco}
G^{}_\op{O}(p) =
\Sigma_1(p) \,\gamma_{\{\mu_1} p_{\mu_2} \dots p_{\mu_n\}} +
\Sigma_2(p) \,p_{\mu_1} \dots p_{\mu_n} \slashed{p}\,.
\end{equation}
For simplicity we limit the discussion to the vector case here;
axial-vector operators are analogous. The \ripm\ scheme uses the
contribution from $\Sigma_1(p)$ only in \eqref{eq:npr:grdeco}. The
required projector $P_\op{O}$ will depend on the momentum entering the
Green's function and its (fixed) directions $\mu_i$ $(i=1\dots n)$. In
general, multiplying $G_\op{O}$ with $\gamma_{\mu_i}$ picks up
combinations of both terms $\Sigma_1$ and $\Sigma_2$. On the other
hand, projecting with $\gamma_\rho$ where $\rho \notin \{\mu_i\}$ is
only sensitive to $\Sigma_2$ (note that we have $n \leq 3$). Thus
multiplying with the difference of the two Dirac matrices with
appropriate normalisation and momentum factors ensures that the vertex
amplitude in \eqref{eq:npr:amproj} contains $\Sigma_1(p)$ only. There
are simpler special cases in which one or more components of the
momentum $p$ are zero, causing the second term in
\eqref{eq:npr:grdeco} to vanish. However, since we tried to choose our
momentum directions close to the diagonal of the lattice, we do not
have momentum components that are zero.
For fixed indices $\mu_i$ $(i=1\dots n)$ of the Green's function we
can construct $n$ different projectors $P_{\op{O},i}$ by starting from
any of the $\gamma_{\mu_i}$:
\begin{equation}
\label{eq:npr:projs}
P_{\op{O},i} =
\frac{\displaystyle
\gamma_{\mu_i} - \gamma_\rho\frac{\bar{p}_{\mu_i}}{\bar{p}_\rho}}
{\displaystyle
\mathcal{N}\,\prod^n_{j\ne i,j=1} \bar{p}_{\mu_j}}
\,, \text{ with }i=1\dots n\,.
\end{equation}
The normalisation $\mathcal{N}$ is chosen such that for the tree-level
vertex amplitude we find $\Lambda_\op{O}^\text{tree}(p) = 1$. The
index $\rho$ is different from any of the $\mu_i$ and such that its
momentum component $\bar{p}_\rho$ is as small as possible to reduce
discretisation errors. We use $\bar{p}_\mu = \sin p_\mu$ to better
account for lattice momenta. The case of axial-vector operators
$\op{O}^5$ is straightforward, with $\gamma_5$ inserted in the
appropriate places.
Combining the $n$ different $P_{\op{O},i}$ with the possible index
combinations of the Green's functions results in a total of $4$, $12$
and $12$ $(n=1,2,3)$ choices to compute the vertex amplitude
$\Lambda_\op{O}(p)$ in Eq.~\eqref{eq:npr:amproj}, all of which should
provide the same result for the final renormalisation constant in the
absence of lattice artefacts. Because of the different sized momentum
components in different lattice directions, the expected
discretisation errors vary depending on the directions selected by the
indices of the projector. We reflect these artefacts coming from
breaking continuum O(4) symmetry to lattice hypercubic symmetry in the
systematic error of our final results. With additional lattice
spacings and the use of partially-twisted boundary conditions, we
could eliminate hypercubic lattice artefacts in the continuum
limit~\cite{Arthur:2010ht,Arthur:2010hy}.
\subsubsection{Quark field renormalisation}
In general, the renormalisation condition Eq.~\eqref{eq:npr:rencond}
requires knowledge of the field renormalisation $Z_q$ to obtain
$Z_\op{O}$. However, in the present calculation only ratios of
renormalisation factors of operators with one, two or no derivatives
appear, Eqs.~\eqref{eq:Pratio} and \eqref{eq:Vratio}. Combining this
with our renormalisation condition leads to,
\begin{equation}
\label{eq:npr:zratio}
\frac{Z_{\op{O},n=2,3}(\mu)}{Z_{\op{O},n=1}(\mu)} =
\left.\frac{\Lambda^\text{bare}_{\op{O},n=1}(p)}
{\Lambda^\text{bare}_{\op{O},n=2,3}(p)}\right|_{p^2=\mu^2}\,,
\end{equation}
where the explicit $Z_q$ dependence drops out. As mentioned earlier,
we can use either the vector or axial-vector bilinears in this ratio
thanks to chiral symmetry. We follow our earlier
procedure~\cite{Aoki:2007xm} and average $\Lambda_{\gamma_\rho}$ and
$\Lambda^{(5)}_{\gamma_\rho}$ ($\Lambda_V$/$\Lambda_A$ in the
reference) to obtain our best answer. The analysis is also performed
with $\Lambda^{(5)}_{\gamma_\rho}$ only and the difference of the two
enters our systematic error.
\subsubsection{Results for renormalisation factors}
Compared to~\cite{Aoki:2007xm} the reduced statistical errors make
previously hidden systematic effects apparent and
quantifiable~\cite{Boyle:2008nj} and affect the way we extract the
renormalisation factors. We start by considering different projectors
for a fixed momentum $p_\mu$ of the external quarks, see
Fig.~\ref{fig:npr:disc}. The results should be independent of the
rotation and size of the momentum components used for the projector.
The smaller statistical errors now reveal a disagreement due to
lattice artefacts. We combine all choices for our best answer and
account for the spread in our systematic error, improving previous
estimates.
\begin{figure}
\includegraphics[width=\hsize,bb=89 513 298 665]{figs/1deriv.eps}\\[1ex]
\includegraphics[width=\hsize,bb=89 513 298 665]{figs/2deriv.eps}
\caption{Results for $\Lambda^\text{bare}_{\op{O},n=2}$
($\Lambda^\text{bare}_{\op{O},n=3}$) on the top (bottom) for a
fixed momentum $(ap)^2 = 1.78201$, $p^\tra=(2,2,3,8)$. The labels
above and below the plots show the indices of the Green's function
$\{\mu_i\}$ (top) and projector (bottom). The disagreement between
the different projections is due to lattice artefacts.}
\label{fig:npr:disc}
\end{figure}
Our general recipe to obtain the renormalisation factors follows. The
ratio of bare vertex amplitudes is extrapolated linearly to the chiral
limit $m_q \to -m_\text{res}$ for each momentum. Only in the chiral
limit can we remove the running of our data points and match them to a
continuum scheme. So by using~\cite{Gracey:2003yr, Gracey:2003mr,
Gracey:2006zr} we take our results from the \ripm\ scheme at scale
$\mu^2=p^2$ to a common scale $\mu^2=4\gev^2$ and convert to
\MSbar\ at that scale. The values thus obtained are then linearly
interpolated to $p^2 = (2\gev)^2$ within our momentum window to obtain
$Z_{\op{O},n=2,3}/Z_{\op{O},n=1}$ at a scale $\mu=2\gev$.
The central value is computed from the averaged values from all
projectors and index combinations. A standard bootstrap analysis
provides the statistical error which is inflated with
$\sqrt{\chi^2/\text{d.o.f.}}$ (the PDG scale-factor~\cite{pdg2010})
from the interpolation. Several effects are taken into account for the
systematic error. Lattice artefacts are the dominant effect. To
estimate those, we perform the analysis for all projectors separately
as indicated above and chose the highest and lowest result for each
momentum for the interpolation. From the two fits, the larger
deviation from the central value then constitutes the systematic error
from discretisation effects (labelled `spread' in the final table).
This is a conservative approach for the discretisation error. Taking
random choices of projectors (or rather their direction) for each
momentum and looking at the $1\sigma$ width of the range of results
for many of those picks would lead to a smaller error. We account for
missing higher order terms in the continuum perturbative calculation
via the slope of the momentum interpolation, using the difference of
our results at $p^2 = (2\gev)^2$ and $(0\gev)^2$, indicated by
`slope', as a measure. We note, however, that we cannot disentangle
perturbative and discretisation errors here and thus double count some
of the discretisation effects. Another source of systematic error is
the strange quark mass, kept fixed at $\ms=0.04$ in our simulation. We
deal with that as described at the end of section IV.F
in~\cite{Aoki:2007xm}, estimating an error from half the linear
dependence (slope) multiplied by the strange quark mass, $m_\text{s}$.
This error is labelled `$\Delta m_s$'. The last contribution to the
systematic error is from the chiral symmetry breaking evident when
comparing our vector and axial-vector operators~\cite{Aoki:2007xm,
Aoki:2009ka, Sturm:2009kb}. This is estimated by the difference of
the final results when taking the axial-vector bilinear $(n=1)$ or the
averaged vector and axial-vector bilinear for the ratio in
Eq.~\eqref{eq:npr:zratio} (labelled `$V-A$'). Adding the four
contributions in quadrature gives our systematic error.
To illustrate some of the steps mentioned above, we include in
Fig.~\ref{fig:npr:chirlim} two examples of the extrapolation to the
chiral limit. Shown are extrapolations for all our five momenta, for
the renormalisation factors for one and two derivatives.
\begin{figure}
\includegraphics[width=0.95\hsize,
bb=89 513 303 657]{figs/LV2_S24_P_AV.eps}\\[1ex]
\includegraphics[width=0.95\hsize,
bb=89 513 303 657]{figs/LV3_S24_P_AV.eps}
\caption{Linear extrapolations of the renormalisation factors to the chiral
limit. The top shows $Z^\ripm_{\op{O},n=2}$, the bottom plot is for
$Z^\ripm_{\op{O},n=3}$. The momenta are increasing from top to bottom and we
have $(ap)^2=1.2947,1.4392,1.6374,1.7820$ and $1.9801$.}
\label{fig:npr:chirlim}
\end{figure}
In Fig.~\ref{fig:npr:scale} we show the renormalisation factors before
and after we remove the running, again for one and two derivatives.
Once at a common scale, the data points are much flatter indicating
the validity of the scale conversion and the momentum window. Also
included is the linear fit and our final result.
\begin{figure}
\includegraphics[width=0.93\hsize,
bb=89 511 298 654]{figs/LV2_S24_M0p000_P0p0000pAV.eps}\\[1ex]
\includegraphics[width=0.93\hsize,
bb=89 511 298 657]{figs/LV3_S24_M0p000_P0p0000pAV.eps}
\caption{These plots show the scale dependent $Z$'s and $Z$'s for a
fixed scale $\mu=2\gev$ with the running successfully removed
(both in \ripm). The top (bottom) plot is for the one (two)
derivative case. Also included are the linear interpolation to the
final result with the statistical error indicated by the error
band. Our result at $\mu=2\gev$ is then shown with error bars for
the statistical and systematic errors.}
\label{fig:npr:scale}
\end{figure}
Our final values for the ratios of renormalisation factors are given
in Table~\ref{tab:npr:res}. These have been obtained using vector like
operators. Results from the axial-vector operators are almost
identical and show no low energy effects from breaking chiral
symmetry, as for bilinears. We note that the renormalisation factors
are significantly different from one and deviate substantially from
the perturbative results. Thus nonperturbative renormalisation looks
imperative here.
\begin{table*}
\caption{Final results for the renormalisation factors in \MSbar\ at
$\mu=2\gev$. Results are given for both lattice sizes with all
systematic errors. The perturbative results are also shown for
comparison.}
\label{tab:npr:res}
\begin{minipage}{.7\textwidth}
\begin{ruledtabular}
\begin{tabular}{l|cc|cc}
& \multicolumn{2}{c|}{$Z_{\op{O}_{\{\rho\nu\}}}/Z_A$} &
\multicolumn{2}{c}{$Z_{DD,DD}/Z_A$} \\
& $16^3\times 32$ & $24^3\times 64$ & $16^3\times 32$ & $24^3\times 64$ \\
\hline
central value & 1.54575 & 1.52893 & 2.06064 & 2.02800 \\
statistical error & 0.00249 & 0.00081 & 0.00482 & 0.00149 \\
spread & 0.02968 & 0.01809 & 0.03702 & 0.01534 \\
slope & 0.00470 & 0.00743 & 0.00097 & 0.02285 \\
$\Delta m_s$ & 0.00089 & 0.00232 & 0.00469 & 0.00992 \\
$V-A$ & 0.00723 & 0.00602 & 0.00938 & 0.00760 \\
total error & 0.03102 & 0.02061 & 0.03879 & 0.03026 \\
\hline
\multicolumn{2}{l}{best result} & 1.5289(8)(206) & & 2.028(1)(30) \\
\hline
\multicolumn{2}{l}{perturbative result} & 1.24(3) & & 1.45(5)
\end{tabular}
\end{ruledtabular}
\end{minipage}
\end{table*}
\subsection{Renormalised Results}
\label{sec:renresults}
We now use the renormalisation factors from the previous section to
convert our bare lattice results to \MSbar\ at $\mu=2\gev$. The local
matrix elements in Eq.~\eqref{eq:dadef} require the renormalisation
factors $Z_{\firstmomentop}$, $Z_{DD,DD}$ and
$Z_{DD,\partial\partial}$ as defined in Eqs.~\eqref{eq:Z1ddef} and
\eqref{eq:Z2ddef}. The first two are computed nonperturbatively while
for the last one we use the perturbative result. From
Eq.~\eqref{eq:Z2ddef} we see that the mixing term requires the
computation of a matrix element with an operator insertion of $\Odd$.
This is simplified since we use the ratios \eqref{eq:Pratio} and
\eqref{eq:Vratio} to extract the moments of the distribution
amplitudes. Within the ratios, the matrix element with the operator
$\Odd$ differs from the denominator only by the momentum factors and
thus does not have to be computed separately. It contributes a
constant shift to the result. To summarise:
\begin{subequations}
\label{eq:resum}
\begin{align}
\label{eq:resum1}
\bret{\xi^1}^\MSbar &= \frac{Z_{\firstmomentop}}{Z_A}
\bret{\xi^1}^\text{bare}\,,\\
\label{eq:resum2}
\bret{\xi^2}^\MSbar &= \frac{Z_{DD,DD}}{Z_A} \bret{\xi^2}^\text{bare} +
\frac{Z_{DD,\partial\partial}}{Z_A}\,.
\end{align}
\end{subequations}
With our best nonperturbative results from Table~\ref{tab:npr:res} and the
perturbative result for the mixing term from Table~\ref{tab:PTZs} (computed at
the same scale of $\mu=2\gev$, $\frac{Z_{DD,\partial\partial}}{Z_A}=0.027(8)$),
we arrive at the renormalised moments of the distribution amplitudes given in
Table~\ref{tab:final}.
\begin{table*}
\caption{Final results in the chiral limit in \MSbar\/ at
$\mu=2\gev$ for both of our lattice volumes. Here the first error
is statistical, the second includes systematic errors from $m_s$,
discretisation and renormalisation.}
\label{tab:final}
\begin{ruledtabular}
\begin{tabular}{l|ccccccc}
& $\PSM_\pi$ & $\PFM_K$ & $\PSM_K$ & $\VSM_\rho$ & $\VFM_{K^*}$ & $\VSM_{K^*}$ & $\VSM_\phi$ \\
\hline
$16^3 \times 32$ & 0.25(1)(2) & 0.035(2)(2) & 0.25(1)(2) & 0.25(2)(2) & 0.037(1)(2) & 0.25(1)(2) & 0.24(1)(1
\rule{0pt}{4mm}\\
$24^3 \times 64$ & 0.28(1)(2) & 0.036(1)(2) & 0.26(1)(2) & 0.27(1)(2) & 0.043(2)(3) & 0.25(2)(2) & 0.25(2)(1)
\end{tabular}
\end{ruledtabular}
\end{table*}
The contribution from the mixing term in Eq.~\eqref{eq:resum2} is
small so using the perturbative result for $Z_{DD,\partial\partial}$
is not a drawback. Even if the correction of a nonperturbative result
for $Z_{DD,\partial\partial}$ is as sizeable as for
$Z_{\firstmomentop}$ or $Z_{DD,DD}$, the overall contribution remains
comparable to our present error on $Z_{DD,DD}/Z_A$. Hence our results
are essentially renormalised nonperturbatively.
\section{Summary}
\label{sec:conclusion}
We have computed the first or first two lowest non-vanishing moments
of the distribution amplitudes of the $\pi$, $K$, $K^*$, $\rho$ and
$\phi$ mesons, using nonperturbative renormalisation of the lattice
operators, with final numbers given in Table~\ref{tab:final}. Apart
from the uncertainty in $m_s$ for the first moments, systematic errors
mainly come from the renormalisation procedure. Within the current
statistical errors on our data we do not see any finite size effects.
With only one lattice spacing we can also only estimate a formal
discretisation error of $O(a^2 \Lambda_\text{QCD}^2)\approx 4\%$ from
the $O(a)$-improved DWF action and operators; this is included in our
sytematic error. The result for $\PFM_K$ in Table~\ref{tab:final}
supercedes but is compatible with our earlier result
in~\cite{Boyle:2006pw,Boyle:2006xq}, which was obtained on the
$16^3\times32$ ensembles only and used perturbative renormalisation.
Converting the lowest moment of the kaon distribution amplitude to the
first Gegenbauer moment $a_K^1=0.061(2)(4)$, we find it in agreement
with sum rule results from Eq.~\eqref{eq:other} but with a much
reduced uncertainty. We compare our results to those from the QCDSF
Collaboration~\cite{Braun:2006dg} in Table~\ref{tab:other}
(preliminary results for the first moment of the vector meson
distribution amplitudes are also available from
QCDSF~\cite{Braun:2007zr}). The results for $\PFM_K$ differ
significantly. However, we observe that our measurements correspond to
pion masses in the range $330$--$670\mev$ and are for $2+1$ dynamical
flavours, whereas the QCDSF results are for pion masses around
$600\mev$ and higher, with $2$ dynamical flavours. For one data point
from each collaboration where the pion and kaon masses are comparable,
the $\PFM_K$ values differ by about one standard deviation. These
points occur for the smallest values of $m_K^2-m_\pi^2$ from each
collaboration; for larger values the points, and therefore slopes in
$m_K^2-m_\pi^2$, differ.
\begin{table*}
\caption{Comparison to other lattice results (both for \MSbar\/ at $\mu=2\gev$).}
\label{tab:other}
\begin{minipage}{.75\textwidth}
\begin{ruledtabular}
\begin{tabular}{l|cccc}
& $\PSM_\pi$ & $\PFM_K$ & $\PSM_K$ & $\VFM_{K^*}$ \\
\hline
this work ($24^3\times 64$)& 0.28(1)(1) & 0.036(1)(2) & 0.26(1)(1) & 0.043(2)(3
\rule{0pt}{4mm}\\
QCDSF~\cite{Braun:2006dg} & 0.269(39) & 0.0272(5) & 0.260(6)
\end{tabular}
\end{ruledtabular}
\end{minipage}
\end{table*}
We plan to improve our results in the near future by reducing the
systematic uncertainties. We will improve the nonperturbative
calculation of the renormalisation factors by including the total
derivative mixing term. We will also have an additional lattice
spacing allowing us to estimate the continuum results, including using
partially-twisted boundary conditions to remove hypercubic lattice
artefacts~\cite{Arthur:2010ht,Arthur:2010hy}. Increased statistics on
the $24^3\times64$ lattice should also improve our conclusions about
finite volume effects.
\begin{acknowledgments}
The calculations reported here used the QCDOC computers
\cite{Boyle:2005qc,Boyle:2003mj,Boyle:2005fb} at Edinburgh University,
Columbia University and Brookhaven National Laboratory (BNL). The
Edinburgh QCDOC system was funded by PPARC JIF grant
PPA/J/S/1998/00756 and operated through support from the Universities
of Edinburgh, Southampton and Wales Swansea, and from STFC grant
PP/E006965/1. At BNL, the QCDOC computers of the RIKEN-BNL Research
Center and the USQCD Collaboration were used. The software used
includes: Chroma~\cite{Edwards:2004sx}, QDP++ and the CPS QCD
codes~\cite{cps}, supported in part by the USDOE SciDAC program; the
BAGEL~\cite{Bagel} assembler kernel generator for many of the
high-performance optimized kernels; and the UKHadron codes.
We thank the University of Southampton for access to the Iridis
computer system used in the calculations of the nonperturbative
renormalisation factors (with support from STFC grant ST/H008888/1).
DB, MAD, JMF, AJ, TDR and CTCS acknowledge support from STFC Grant
ST/G000557/1 and from EU contract MRTN-CT-2006-035482 (Flavianet); RA
and PAB from STFC grants PP/D000238/1, PP/C503154/1 and ST/G000522/1;
PAB from an RCUK Fellowship.
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,322 |
2.1 There has been considerable conjecture and controversy worldwide about the health impact of wind turbines. Australia has been no exception. Here, as in many other countries, there is a clear disconnect: between the official position that wind turbines cause no harm to human health and the strong and continuing empirical, biological and anecdotal evidence of many people living in proximity to turbines suffering from similar physiological symptoms and distress.
2.2 In the course of this inquiry, as in others conducted by the Australian Parliament, the committee has received considerable anecdotal evidence that those living in close proximity to wind turbines have suffered adverse health impacts from the operation of these turbines. These complaints have not been isolated to a particular wind farm or a particular region. While evidence to the committee suggests that some wind turbines may not have had the alleged health impact that others seem to have caused, the committee has received health complaints from dozens of submitters living near wind turbines at various locations across several States.
2.3 The committee believes that these complainants deserve to be taken seriously. Those who have labelled 'wind turbine syndrome' as a communicated disease or a psychogenic condition have been too quick to judge. In so doing, they have unnecessarily inflamed the debate on the issue. This has understandably caused those who suffer adverse symptoms even greater distress.
in December 2014, acoustician Mr Steven Cooper found a correlation between infrasound emitting from turbines at Cape Bridgewater and 'sensations' felt, and diarised, by six residents of three nearby homes. Significantly, the report identified a unique infrasound 'wind turbine signature'.
2.5 The possible effect of infrasound from wind turbines on human health has been a theme of this inquiry. Acousticians have provided different perspectives to the committee on the possible effect of infrasound from turbines. What is most striking is the lack of any professional consensus on this issue and the range of arguments as to what would constitute an acceptable research project to test the hypothesis. Accordingly, the committee's interim report recommended the need for independent research into both audible and sub-audible sound from turbines and for this research to inform national sound standards.
The committee recommends the Commonwealth Government create an Independent Expert Scientific Committee on Industrial Sound responsible for providing research and advice to the Minister for the Environment on the impact on human health of audible noise (including low frequency) and infrasound from wind turbines. The IESC should be established under the Renewable Energy (Electricity) Act 2000.
The committee recommends that the National Environment Protection Council establish a National Environment Protection (Wind Turbine Infrasound and Low Frequency Noise) Measure (NEPM). This NEPM must be developed through the findings of the Independent Expert Scientific Committee on Industrial Sound. The Commonwealth Government should insist that the ongoing accreditation of wind turbine facilities under the Renewable Energy (Electricity) Act 2000 in a State or Territory is dependent on the NEPM becoming valid law in that State or Territory.
2.7 The committee has taken evidence from a number of people who reside in proximity to wind turbines who have complained of a range of adverse health impacts. These include tinnitus, raised blood pressure, heart palpitations, tachycardia, stress, anxiety, vertigo, dizziness, nausea, blurred vision, fatigue, cognitive dysfunction, headaches, nausea, ear pressure, exacerbated migraine disorders, motion sensitivity, inner ear damage and worst of all, sleep deprivation.
From my experience there is a subset of people who are terribly impacted very early on. Those people are the ones who tend to present with acute vestibular disorder type of symptoms—dizziness and motion sickness, which can be accompanied by extreme anxiety. Those people often just cannot last very long, and they move if they can.
At my farm, I experience severe adverse health effects such as vibration, heart palpitations, tinnitus, head pressure, headaches, sleep deprivation, anxiety, night sweats, nausea, itchy skin, cramps, and ear, nose and throat pain. Twice now I have experienced horrendous pain in my chest stabbing through to my backbone in between my shoulder blades. I contemplated calling an ambulance both times but could not move to do so because of the severity of the pain. Ten minutes later it had dissipated, leaving me with great stress and anxiety and feeling washed out. All these sensations leave me drained in the morning. I find it very hard to start work that day.
My husband experienced bolts of pressure which tallied up with pressure peaks measured by Les Houston (sic) 86 per cent of the time while my husband was blind to the acoustic measurements of the time. Refer to his recap statement. I suffer day and night from headaches, nose and ear pressure, nausea, heart palpitations and chest burning from vibrations through the floor, couch, chair and in bed all night.
After a short period of living with an operating wind farm, we had these products installed. I find that, because I work and reside in close proximity to the wind farm, I suffer sleep interruption, mild headaches, agitation and a general feeling of unease; however, this occurs only when the towers are turning, depending on the wind direction and wind strength. My occupation requires that I work amongst the wind towers during the day which means I suffer the full impacts of noise for days at a time without relief. The impacts are that we are not able to open our windows because of the noise at night and we are not able to entertain outside because of the noise.
In conclusion, if we did not have soundproof batts in VLam Hush windows [special window laminate designed to dampen noise], our house would not be habitable. In my opinion, towers should not be within five kilometres of residences, and I would personally not buy a house within 20 kilometres of a wind farm.
2.12 The committee notes that the Gares have received payment of $2 million over five years to host turbines and have reported serious adverse impacts. The committee notes, therefore, that their evidence is an 'admission against interest' and as such represents highly reliable evidence.
My ears—especially when I go to my Stud Farm Road property, I have ear pressure that can develop into a headache and rapid heartbeat. If I leave that area and go back to one of my other properties, that can settle back down.
Whether it is low-frequency noise and the infrasound combining with it, it seems worse when it is quiet. Around our house the yard is pretty well protected by trees. When it is relatively quiet around the house yard there is still a really soft drone that comes through and just gets into you. It is pretty hard to explain. There are probably a lot of people going through the same thing who will have the same trouble trying to explain it, especially to people who have not experienced it. The problem with it is, it also seems to affect different people over different periods of time.
I really believe that we just do not have enough information yet. But throughout the interviews, country by country, people described the same symptoms. Many times they used the same phrases to describe them and the same gestures—and they were not speaking English. There is a common thread here.
The [Board of Health] has been studying adverse health effects for the past 4 ½ years in the Shirley Wind Project. We have reviewed many peer reviewed studies, at least 50 medical complaints including ear pain, pressure, headache, tinnitus, vertigo, nausea, chest pain, chest pressure, loss of concentration, sleep deprivation and more, as well as more than 80 other complaints from citizens of Shirley Wind. There have been 2 formal studies of infrasound/low frequency noise by acousticians in 2012 and 2014. The latter study revealed symptom generating [Infrasound/Low Frequency Noise] at a distance of 4 ½ [miles].
there is an urgent need to monitor the health effects of people exposed to turbines over time and that has been missing virtually in all jurisdictions.
As the committee noted in its interim report (paragraph 1.13), it is disappointed that renewable energy advocates, wind farm developers and operators, public officials and academics continue to denigrate those who claim that wind turbines have caused their illhealth.
The committee notes that RATCH Australia provided a formal apology to the committee for comments made at the public hearing. This apology was accepted.
…the phenomenon of people claiming to be adversely affected by exposure to wind turbines is best understood as a communicated disease that exhibits many signs of the classic psychosocial and nocebo phenomenon where negative expectations can translate into symptoms of tension and anxiety.
The argument that adverse health reactions are the result of nocebo effects, ie a directly anticipated adverse reaction, completely fails to consider the many cases where communities have initially welcomed the introduction of wind turbines, believing them to represent a clean, benign form of low-cost energy generation. It is only after the wind-turbines are commissioned, that residents start to experience directly the adverse nature of the health problems that they can induce.
many of his assertions do not withstand fact check analyses.
the use of non-disclosure clauses and 'good neighbour agreements' legally restricts people from making adverse public statements or complaints.
It is the reporting which is largely at fault. The fact is that people are affected by this, and the numbers are in the thousands. I only have to look at the emails that cross my desk from all over the world. I get bombarded from the UK, Ireland, France, Canada, the United States, Australia, Germany. There are tonnes of these things out there but, because the system does not understand the problem, nor does it have a strategy, many of those complaints go unlisted.
2.24 Third, Professor Chapman has queried that if turbines are said to have acute, immediate effects on some people, why were there no such reports until recent years given that wind turbines have operated in different parts of the world for over 25 years. Several submissions to the committee have stated that adverse health effects from wind turbines do not necessarily have an acute immediate effect and can take time to manifest.
Low-frequency noise from large fans is a well-known and well-published issue, and wind turbines are simply large fans on top of a big pole; no more, no less. They have the same sort of physical characteristics; it is just that they have some fairly unique characteristics as well. But annoyance from low-frequency sound especially is very well known.
2.26 Fifth, Professor Chapman contends that there are apparently only two known examples anywhere in the world of wind turbine hosts complaining about the turbines on their land. However, there have been several Australian wind turbine hosts who have made submissions to this inquiry complaining of adverse health effects. Paragraphs 2.11–2.12 (above) noted the example of Mr Clive Gare and his wife from Jamestown. Submitters have also directed attention to the international experience. In Texas in 2014, twenty-three hosts sued two wind farm companies despite the fact that they stood to gain more than $50 million between them in revenue. The committee also makes the point that contractual non-disclosure clauses and 'good neighbour' agreements have significantly limited hosts from speaking out. This was a prominent theme of many submissions.
2.27 Sixth, Professor Chapman claims that there has been no case series or even single case studies of so-called wind turbine syndrome published in any reputable medical journal. But Professor Chapman does not define 'reputable medical journal' nor does he explain why the category of journals is limited to medical (as distinct, for example, from scientific or acoustic). The committee cannot therefore challenge this assertion. However, the committee does note that a decision to publish—or not to publish—an article in a journal is ultimately a business decision of the publisher: it does not necessarily reflect the quality of the article being submitted, nor an acknowledgment of the existence or otherwise of prevailing circumstances. The committee also notes that there exist considerable published and publicly available reports into adverse health effects from wind turbines.
Pierpont has made one genuine contribution to the science of environmental noise, by showing that a proportion of those affected have underlying medical conditions, which act to increase their susceptibility.
2.29 Seventh, Professor Chapman claims that no medical practitioner has come forward with a submission to any committee in Australia about having diagnosed disease caused by a wind farm. Again, Professor Chapman fails to define 'disease'. Nonetheless, both this committee, and inquiries undertaken by two Senate Standing Committees, have received oral and written evidence from medical practitioners contrary to Professor Chapman's claim.
2.30 Eighth, Professor Chapman claims that there is not a single example of an accredited acoustics, medical or environmental association which has given any credence to direct harmful effects of wind turbines. The committee notes that the semantic distinction between 'direct' and 'indirect' effects is not helpful. Dr Leventhall and the NHMRC describe stress, anxiety and sleep deprivation as 'indirect' effects, but these ailments nonetheless affect residents' health.
2.31 Finally, Professor Chapman queries why there has never been a complainant that has succeeded in a common-law suit for negligence against a wind farm operator. This statement is simply incorrect. The committee is aware of court judgements against wind farm operators, operators making out of court settlements or withdrawing from proceedings, injunctions or shutdown orders being granted against operators, and properties adjacent to wind turbines being purchased by operators to avoid future conflict. The committee also reiterates its earlier point that contractual non-disclosure clauses have discouraged legal action by victims.
I think that the most important aspect of wind turbine noise—which I said in the paper I published nearly 10 years ago—is the amplitude modulation. Work is now developing on that, and I believe that that is where the main answer should be given, in amplitude modulation, because this is what upsets people.
The systemic regulatory failure with respect to the way industrial and environmental noise pollution is regulated in Australia is not confined to wind turbine noise. As you would have seen from the submissions of the Wollar Progress Association; and residents living near the coalmines in the Upper Hunter region and residents of Lithgow impacted by coal fired power stations and extractor fan noise and vibration. Their stories, both with respect to the range and severity of symptoms and the way they are treated by the noise polluters and the government regulatory authorities, are all too familiar to the growing numbers of rural residents living near industrial wind power generators.
Once sensitised, residents affected by infrasound and low-frequency noise from coal fired power stations find they also react to wind turbines in the same way. The body and the brain do not care about the source of the sound and vibration. The reactions are involuntary and hardwired, and part of our physiological fight/flight response.
At the heart of this systemic regulatory failure of environmental noise pollution is the failure of the planning and noise pollution regulations, because they all fail to varying degrees to predict, measure and regulate the excessive noise and vibration in the lower frequencies—in the infrasound and low-frequency noise regions, specifically between 0.1 and 200 hertz. These regulations also permit levels of audible noise which are guaranteed to cause adverse impacts because they are so much higher than the very quiet background noise environments in rural areas. These rules are not fit for purpose, and guarantee that some residents will be seriously harmed.
There has been pretence that there is no evidence of harm at the levels of infrasound and low-frequency noise being emitted. This is untrue. There is an extensive body of research conducted by NASA and the US Department of Energy 30 years ago, which: established direct causation of sleep disturbance and a range of physiological effects euphemistically called 'annoyance'; acknowledged that people became sensitised or conditioned to the noise with ongoing exposure; and recommended exposure thresholds in order to ensure residents were protected from harm directly caused by this pulsing infrasound and low-frequency noise.
the Inagaki study in Japan which found physiological effects from aerodynamic sound from wind turbines. | {
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Q: Find matrix and Jordan basis of endomorphism Find matrix and Jordan basis of endomorphism $f \in L(\mathbb{R}[x]_3)$ for which $ker\ f = span(1,x)$, $f\circ f=f$, $f(x^2) = 1-x+x^2$, $f(x^3)=p$ and $p(1)=p'(1)=0$. From $ker\ f = span(1,x)$ and $f(x^2) = 1-x+x^2$ we know that matrix of f is $M = \begin{bmatrix}
0 & 0 & 1 & a_{14} \\
0 & 0 & -1 & a_{24} \\
0 & 0 & 1 & a_{34} \\
0 & 0 & 0 & a_{44}
\end{bmatrix}$. Let $p(x) = a + bx + cx^2 +dx^3$. From $p(1)=p'(1)=0$ we have $a + b + c +d =0$ and $b + 2c +3d= 0$, so $p(x) =a + bx + (-3a-2b)x^2+(2a+b)x^3$. How to finish it?
A: you really ought to insert your coefficients for $p(x)$ in the correct places in $M,$ and
you have ignored $$ M^2 = M. $$
| {
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} | 6,623 |
What is the meaning of the mercy seat ?
The Ark of the Covenant or "Ark of the Testimony" was the only article of furniture in the Most Holy. (Hebrews 9:2-4) The word testimony suggests that it illustrated Jehovah's plan. It represented the eternal purpose of God–his foreordained arrangement of grace for mankind in the Christ (Head and Body). It therefore represented Christ Jesus and his Bride, the "little flock."
The Ark was a rectangular box overlaid with gold. It contained the two Tables of the Law (Deuteronomy 31:26), Aaron's Rod that budded (Numbers 17:8), and the Golden Pot of Manna (Exodus 16:32). The Law showed how the Christ would meet in full all the requirements of God's perfect Law, and also that legal authority would be vested in him as the Law-executor.
Within the golden Ark was represented the glory to be revealed in the divine Christ: in the budded rod (God's chosen priesthood), in the tables of the Law (the righteous Judge), in the incorruptible manna in the golden bowl (immortality, the divine nature).
Above this Ark, and constituting a lid, was "The Mercy Seat"–a slab of solid gold, on the two ends of which were formed two cherubim. These cherubim had wings uplifted as if ready to fly, their faces looking inward toward the center of the plate on which they stood. Between the cherubim, on the "Mercy Seat," there was believed to be a bright, supernatural light representing Jehovah's presence.
The light, called the Shekinah glory, represented Jehovah himself as the Light of the universe. (Similarly, Christ is the Light of the world.) This is abundantly testified by many scriptures. "Thou that dwellest between the cherubim, shine forth." (Psalm 80:1; 1 Samuel 4:4; 2 Samuel 6:2; Isaiah 37:16)
As the Ark represented the Christ, so the "Mercy Seat," (glory-light and cherubim together) represented Jehovah God–"the Head of Christ is God." (1 Corinthians 11:3)
The slab of gold called the "MERCY SEAT" (or Propitiatory, because on it the Priest offered the blood of the sacrifices which propitiated or satisfied the demands of divine justice) represented the underlying principle of Jehovah's character–justice. God's throne is based or established upon Justice. "Righteousness and justice are the foundation of thy throne." Psalm 89:14; Job 36:17; 37:23; Isaiah 56:1; Revelation 15:3
The Apostle Paul uses the Greek word for Mercy Seat or Propitiatory (hilasterion) when referring to our Lord Jesus, saying–"Whom God hath set forth to be a Propitiatory
[or Mercy Seat]…to declare his righteousness…that he might be just and the justifier of him which believeth in Jesus." (Romans 3:25,26) It pleased God that in his well-beloved Son, our Lord Jesus, all of His own fullness should dwell, and be represented to mankind.
If you are more interested in this topic, we encourage you to read it here http://www.agsconsulting.com/htdbv5/indext.htm
2019-11-07T21:32:10+00:00September 14th, 2016|Church and The Bride of Christ, Miscellaneous Bible Questions, Mosaic Law, Tabernacle, Tithes|
How old should kids be in order to take communion? (United States)
What is the difference between born again and believer? (United Kingdom)
How are we sons of God? Do Christians through Jesus become a son of God as is Jesus?
Why is the genealogy in the Bible important to the story and the big picture of the Bible? (United States)
In Romans 9:10-13, the Bible says that God chose Jacob over Esau when they had not been born. And God also said that the older will be slave to the younger. God honored Jacob over Esau. Is this like saying, "all human beings are created equal, but some are more equal than others." (Liberia)
Is Star of David a satanic symbol? (Canada)
Please, why do preachers use the prophecies meant for the Jews for present Christians? Also, a preacher applied Isaiah 41:10 to people alive today using an antitypical Cyrus to deliver Israel. (Nigeria)
I have been studying a lot about Israel. The Bible refers to Israel as "her," why? (United States)
What is the place of the tithe under the New Covenant in the light of Hebrews chapter 6 and 7? (Nigeria)
What are spiritual blessings based on 2 Peter 1:3-4? (Jamaica)
What does the Bible say about when to evangelize and when not to evangelize? (United Kingdom)
Do you think the technology age will ever be able to make salvation demonstrable? (United States)
Can a Christian fight or kill to protect himself or others? (United States)
Where is Lilith found in the Bible, or is she the real wife of Adam? (United States)
Can law and grace mix? (Liberia)
What is the meaning of 1 Corinthians 12:2-4 (NKJV), 2 "You know that you were Gentiles, carried away to these dumb idols, however you were led. 3 Therefore I make known to you that no one speaking by the Spirit of God calls Jesus accursed, and no one can say that Jesus is Lord except by the Holy Spirit. 4 There are diversities of gifts, but the same Spirit." Who is this about? (India)
How might I find whether my baptism is acknowledged or accepted by God? Am I eligible for the 144,000 or the 2nd death? (India)
With a Biblical view, can a woman becomes a leader or a pastor of a church? (Indonesia)
What is the difference atonement for sins Old versus New Testament? (United States)
Which of the two Testaments (Old and New) is to be practiced in today's life? (Liberia)
What is the difference between the law of Moses and the law of God? (United States)
What is the difference between foreknowledge, election, and predestination?
Explain Genesis 3:22 (NIV), "And the LORD God said, 'The man has now become like one of us, knowing good and evil. He must not be allowed to reach out his hand and take also from the tree of life and eat, and live forever.'"
Will it be a sin to take the COVID-19 vaccination?
What does the Bible teach about Idols? (Philippines)
Please explain Hebrews 4:12 (NKJV), "For the word of God is living and powerful, and sharper than any two-edged sword, piercing even to the division of soul and spirit, and of joints and marrow, and is a discerner of the thoughts and intents of the heart." (Nigeria)
Please explain 1 Peter 4:6 (NKJV), "For this reason the gospel was preached also to those who are dead, that they might be judged according to men in the flesh, but live according to God in the spirit." (Togo)
What did Jesus mean when he said a man "must be born again" in John 3:5? (South Africa)
Why is it necessary to learn about dispensations? How does it affect your salvation? Or walk with God? (United States)
How do I get my spiritual blessings to manifest into the physical world? (United States)
What does the number 14 represent in the Bible? Reference? 18.2k views
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How old was David when he killed Goliath? 8.7k views
How many people are in heaven now? (United States)
Why do modern day Christians claim Haggai 2:9 as promise from God? (Nigeria) | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,336 |
Looking for that perfect Mother's Day gift idea? Look no further than a custom portrait experience, for that gift that Mum can treasure forever. Book one for yourself and your family, or one for mum, to capture that beautiful 'generations' shot. Book HERE, places limited and expire Saturday May 10th. | {
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Rapid CommunicationAccelerated Communication
Activity of 2-Substituted Lysophosphatidic Acid (LPA) Analogs at LPA Receptors: Discovery of a LPA1/LPA3Receptor Antagonist
Christopher E. Heise, Webster L. Santos, Ann M. Schreihofer, Brian H. Heasley, Yurii V. Mukhin, Timothy L. Macdonald and Kevin R. Lynch
Molecular Pharmacology December 2001, 60 (6) 1173-1180; DOI: https://doi.org/10.1124/mol.60.6.1173
Christopher E. Heise
Departments of 1Pharmacology (C.E.H., A.M.S., K.R.L.) and 2Chemistry (W.L.S., B.H.H., T.L.M.), University of Virginia, Charlottesville, Virginia; and 3Department of Medicine (Nephrology Division) (Y.V.M.), Medical University of South Carolina, Charleston, South Carolina
Webster L. Santos
Ann M. Schreihofer
Brian H. Heasley
Yurii V. Mukhin
Timothy L. Macdonald
Kevin R. Lynch
The physiological implications of lysophosphatidic acid occupancy of individual receptors are largely unknown because selective agonists/antagonists are unavailable currently. The molecular cloning of three high-affinity lysophosphatidic acid receptors, LPA1, LPA2, and LPA3, provides a platform for developing receptor type-selective ligands. Starting with an N-acyl ethanolamide phosphate LPA analog, we made a series of substitutions at the second carbon to generate compounds with varying spatial, stereochemical, and electronic characteristics. Analysis of this series at each recombinant LPA receptor using a guanosine 5′-O-(3-[35S]thio)triphosphate (GTP[γ35S]) binding assay revealed sharp differences in activity. Our results suggest that these receptors have one spatially restrictive binding pocket that interacts with the 2-substituted moieties and prefers small hydrophobic groups and hydrogen bonding functionalities. The agonist activity predicted by the GTP[γ35S] binding assay was reflected in the activity of a subset of compounds in increasing arterial pressure in anesthetized rats. One compound with a bulky hydrophobic group (VPC12249) was a dual LPA1/LPA3 competitive antagonist. Several compounds that had smaller side chains were found to be LPA1-selective agonists.
Lysophosphatidic acid (LPA; 1-acyl, 2-hydroxyl-sn-glycerol-3-phosphate) is a family of lysophospholipid mediators that elicit diverse biological responses including calcium mobilization, cytoskeletal rearrangements, and mitogenesis (Moolenaar, 1994). Transient rises in blood pressure in rats and guinea pigs after intravenous LPA injection have also been documented (Tokumura et al., 1978). LPA is released by activated platelets and accumulates in serum to low micromolar levels (Schumacher et al., 1979; Simon et al., 1982; Watson et al., 1985). The induction of platelet aggregation and fibroblast recruitment along with its mitogenic capabilities implicate this lipid as a wound healing hormone (Moolenaar, 1995). Another pathologic fluid containing substantial amounts of LPA is the malignant ascites characteristic of ovarian cancer (Xu et al., 1995). Interestingly, the LPA found in these two fluids differs in that LPA from ascitic fluid is reportedly enriched in 2-acyl LPA species (Xu et al., 1995). Study of this 2-acyl LPA isoform is made difficult, however, by its chemical instability (i.e., the rapid migration of the acyl chain to the thermodynamically favored 1 position in an aqueous environment).
LPA signals cells in part via a set of G protein-coupled receptors named LPA1, LPA2, and LPA3 (formerly Edg-2, Edg-4, and Edg-7)1 (Hecht et al., 1996; An et al., 1997b; Bandoh et al., 1999; Im et al., 2000b). These receptors share 50 to 55% identical amino acids and cluster with five other receptors (S1P1, S1P2, S1P3, S1P4, and S1P5, formerly Edg-1, Edg-5, Edg-3, Edg-6, and Edg-8) for the structurally-related lipid sphingosine 1-phosphate (S1P) (An et al., 1997a; Lee et al., 1998; van Brocklyn et al., 2000; Im et al., 2000a; Yamazaki et al., 2000). LPA1 is most associated with activation of Giα pathways and is expressed in oligodendrocytes (Allard et al., 1998; Weiner et al., 1998) and peripheral tissues (An et al., 1998), whereas LPA2 and LPA3 are associated most prominently with Gq/11α pathways (Bandoh et al., 1999; Im et al., 2000b). LPA2 mRNA is found in testis and peripheral blood leukocytes (An et al., 1998) whereas LPA3 mRNA has been localized to prostate, testes, pancreas, kidney, and heart (Bandoh et al., 1999; Im et al., 2000b).
The physiologic implications of occupation of individual LPA receptors are largely unknown, partly because of a lack of receptor subtype selective ligands. This paucity led us to design and test a series of 2-substituted ethanolamide derivatives varying in degrees of size, hydrophobicity, and stereochemistry. The parent compound of our series,N-acyl ethanolamide phosphate (NAEPA) has been shown to be nearly indistinguishable from LPA in evoking platelet aggregation (Sugiura et al., 1994) and GTP[γ35S] binding at LPA1 and LPA2 containing membranes (Im et al., 2000b) but is distinctly less active than LPA at recombinant LPA3 (Im et al., 2000b) or in depolarizing Xenopus laevis oocytes (Santos et al., 2000). A pair of 2-substituted NAEPA compounds have already been reported. The 2-carboxyl-containing compound (N-acyl serine phosphoric acid) has been documented to antagonize both LPA-driven platelet aggregation (Sugiura et al., 1994) and oocyte depolarization (Liliom et al., 1996) and is a partial agonist at mammalian LPA receptors (Hooks et al., 1998). The 2-methylene hydroxy-containing compound, which is an analog of 2-acyl LPA wherein the ester is replaced by an amide, was shown by us to activate recombinant LPA receptors in a stereoselective fashion, whereas mitogenic responses and platelet aggregation did not show this stereoselectivity (Hooks et al., 2001). Because of the interesting properties of the existing 2-substituted NAEPA compounds, we synthesized and examined an expanded series of these compounds.
In this article, we present an analysis of several more compounds from this series. In addition to confirming the strong preference of each recombinant LPA receptor for a single enantiomer, we found compounds that exhibited increased potency at the LPA1receptor relative to LPA. The predicted enantiomeric selectivity was observed in a variety of assays, including the hypertensive response in anesthetized rats. Additionally, a dual LPA1/LPA3 competitive antagonist was identified and characterized.
Experimental Procedures
Transient Expression in HEK293T Cells.
The appropriate receptor plasmid DNA (encoding mouse LPA1, human LPA2 or human LPA3) was mixed with equal amounts of expression plasmids (pcDNA3) encoding a mutated (C351F) rat Gi2α, cow β1, and γ2 proteins, and these DNAs were used to transfect monolayers of HEK293T cells (where 'T' indicates expression of the SV-40 virus large T antigen) using the calcium phosphate precipitate method (Wigler et al., 1977). After about 60 h, cells were harvested, and membranes were prepared, aliquoted, and stored at −70°C until use (Im et al., 2000b).
GTP[γ35S] Binding.
The GTP[γ35S] assay was performed as described previously (Im et al., 2000b). Membranes containing 5 μg of protein were incubated in 0.1 ml of GTP-binding buffer (50 mM HEPES, 100 mM NaCl, 10 mM MgCl2, pH 7.5) containing 5 μg of saponin, 10 μM GDP, 0.1 nM GTP[γ35S] (1200 Ci/mmol), and indicated lipid(s) for 30 min at 30°C. Samples were analyzed for membrane-bound radionuclide using a Brandel Cell Harvester (Gaithersburg, MD). The C351F mutation renders the Gi2α protein resistant to inactivation by pertussis toxin or the alkylating agent N-ethylmaleimide; in practice, however, background binding was sufficiently low to obviate these maneuvers.
Measurement of cAMP Accumulation.
Assays for cAMP accumulation were conducted on populations of 5 × 105 cells stimulated with 10 μM forskolin in the presence of the phosphodiesterase inhibitor 3-isobutyl-1-methylxanthine for 15 min at 30°C. cAMP was measured by automated radioimmunoassay.
Measurement of Intracellular Calcium.
We used a FLIPR (Molecular Devices, Inc., Menlo Park, CA) to measure intracellular calcium in A431 and HEK293T cells. A431 cells were seeded (∼50,000 cells/well) in 96-well, clear bottom black microplates (Corning Costar Corp., Cambridge, MA) and left overnight in CO2incubator at 37°C. HEK293T cells were treated likewise, but seeded onto poly(d-lysine) coated microplates (Becton Dickinson, Franklin Lakes, NJ). A431 cells were dye-loaded with 4 μM Fluo-3 AM ester (Molecular Probes Inc., Eugene, OR) in a loading buffer (1× HEPES-buffered saline, pH 7.4, containing 20 mM HEPES, 0.1% BSA, and 2.5 mM probenecid) for 1 h at 37°C. Cells were then washed four times with the loading buffer and exposed in the FLIPR to sets of compounds. HEK293T cells were loaded with 2 μM Fluo-4 AM ester (Molecular Probes Inc., Eugene, OR) in the same loading buffer without probenecid for 30 min and washed four times before being exposed to compounds in the FLIPR. In all cases, each concentration of each compound was tested in at least quintuplicate.
Determination of KI.
K I values for VPC12249 in experiments were determined by plotting the log of Dose Ratio-1 at each concentration of inhibitor against the log concentration of inhibitor. Thex-intercept of the linear transformation is equal to the inverse log of the K I.
Stable Expression in RH7777 Cells.
Rat hepatoma RH7777 cell monolayers were transfected with the mLPA1plasmid DNA using the calcium phosphate precipitate method and clonal populations expressing the neomycin phosphotransferase gene were selected by addition of Geneticin (G418) to the culture media. The RH7777 cells were grown in monolayers at 37°C in a 5% CO2/95% air atmosphere in growth media consisting of: 90% minimal essential medium, 10% fetal bovine serum, 2 mM glutamine, and 1 mM sodium pyruvate.
Cardiovascular Measurements.
All procedures were performed on male Wistar rats in accordance with National Institutes of Health and University of Virginia animal care and usage guidelines. Anesthesia was induced by 5% halothane (in 100% O2). Rats were intubated and artificially ventilated with 1.5 to 1.8% halothane in 100% O2 for surgical procedures. A femoral artery was cannulated to record mean arterial pressure (MAP) and heart rate (HR), and a femoral vein was cannulated to administer anesthetic agents. A femoral vein was cannulated for administration of lipids. The left splanchnic nerve was isolated via a retroperitoneal approach, and the segment distal to the suprarenal ganglion was placed on two Teflon-coated silver wires that had been bared at the tip (250 μm bare diameter; A-M Systems, Everett, WA). The nerve and wires were embedded in a dental impression material (polyvinysiloxane; Darby Dental Supply, Westbury, NY), and the wound was closed around the exiting recording wires.
On completion of surgery, the halothane anesthesia was terminated and was replaced by a α-chloralose (30 mg/kg solution in 3% sodium borate; 70 mg/kg initial bolus followed by hourly supplements of 20 mg/kg i.v.; Fisher Scientific, Pittsburgh, PA). Rats were allowed to stabilize for 45 min before tests began. End-tidal CO2 was monitored by infrared spectroscopy and was maintained between 3.5 and 4.0%. Body temperature (measured rectally) was maintained at 37°C.
All physiological variables were monitored on a chart recorder (model RS 3600; Gould, Valley View, OH) and simultaneously stored on a videocassette recorder via a digitizer interface (model 3000A; frequency range: DC-22 kHz; Vetter Digital, Rebersburg, VA) for off-line computer analysis. Data were analyzed with Spike 2 (Cambridge Electronics). The MAP was calculated from the pulse pressure measured by a transducer (Statham P10 EZ; Gould) connected to the brachial arterial catheter. The HR was determined by triggering from the pulse pressure (Biotach; Gould). Splanchnic nerve activity was filtered (10 Hz-3 kHz band pass with a 60-Hz notch filter), full-wave rectified, and averaged in 1-s bins. The femoral venous catheter (dead space, 100 μl) was loaded with each lipid and was flushed with 200 μl of saline to expel the drug.
Materials.
Chemicals for syntheses were purchased from Aldrich Chemical Company (Milwaukee, WI), Sigma Chemicals (St. Louis, MO), Advanced ChemTech Chemical Company (Louisville, KY), and/or NovaBiochem Chemical Company (Laufelfingen, Switzerland) and were used without further purification. GTP[γ35S] was purchased from Amersham Pharmacia Biotech (Piscataway, NJ), Fura-3 and Fura-4 AM were purchased from Molecular Probes (Eugene, OR), A431 and RH7777 cells were purchased from the American Type Culture Collection (Manassas, VA), and tissue culture media and serum was from Invitrogen (Carlsbad, CA). HEK293T cells were a gift from Dr. Judy White's laboratory (Dept. of Cell Biology, University of Virginia, Charlottesville, VA) while G protein β and γ DNAs were a gift from Dr. Doug Bayliss (Dept. of Pharmacology, University of Virginia). LPAs (1-oleoyl and 1-palmitoyl) and dioctyl glyceryl pyrophosphate were purchased from Avanti Polar Lipids (Alabaster, AL).
Using N-oleoyl ethanolamide phosphoric acid (NAEPA) as a lead structure, we synthesized a series of 2-substituted LPA analogs (Fig. 1). The details of the synthesis and analysis of the full set of compounds in this series are to be provided in a separate publication (W. L. Santos, C. E. Heise, K. R. Lynch, T. L. Macdonald, in preparation). Each compound was characterized by 1H NMR,13C NMR, and mass spectrometry.
Structures of 2 substituted ethanolamide phosphoric acid compounds. Compounds in the left column contain the denoted functional group at R 1 with a hydrogen atom at the R 2 position; those in the right column contain indicated functional group atR 2 with a hydrogen atom at theR 1 position. Compounds were synthesized, purified and analyzed as will be described in Santos et al. (in preparation).
The differential coupling of LPA1 versus LPA2 and LPA3, the lack of a reliable radioligand binding assay, and the near ubiquity of endogenous LPA responses prohibited the use of most common receptor assay techniques (i.e., measurements of adenylyl cyclase activity, calcium mobilization, and radioligand binding) to assess each compound's activity. Therefore, we adapted a GTP[γ35S] binding assay to measure the relative efficacies and potencies of each compound compared with LPA as described previously (Im et al., 2000b; McAllister et al., 2000). This assay isolates each recombinant LPA receptor and allows analysis of all three receptors using the same system. Note that membranes from HEK293T cells transfected with only G protein DNAs (ie, no receptor DNA) were devoid of LPA-stimulated GTP binding despite expressing endogenous LPA receptors (Fig. 2D).
GTP[γ35S] binding to HEK293T cell membranes in response to LPA and analogs with hydrophilic constituents. HEK293T cells were transfected with LPA1 (A), LPA2 (B), or LPA3 (C) receptor DNAs plus G protein DNAs or G protein DNAs alone (D) (see Experimental Procedures for details). Lipids were dissolved at 1 mM in aqueous 1.0% fatty acid-free BSA and diluted in aqueous 0.1% BSA. LPA analogs contained oleoyl (18:1) fatty acid chains. Closed characters represent R 1 enantiomers and open characters represent R 2 enantiomers; all sets of curves include a 1-oleoyl-(S)-LPA curve as a reference. Points are in triplicate and are representative of at least two experiments. Typical values for zero and 100% binding were 300 and 800 dpm/tube for LPA1, 300 and 1000 dpm/tube for LPA2, and 300 and 2000 dpm/tube for LPA3, respectively.
Many NAEPA compounds with various 2-substituents were synthesized and examined; those with methyl, ethyl, isopropyl, benzyl, methylene hydroxy, carbomethyl, methylene amino, and benzyl-4-oxybenzyl functionalities (Fig. 1) are reported herein. Because the 2 position is a prochiral site, both enantiomers (at R 1and R 2) of the eight compounds were synthesized. Three patterns were revealed when we tested the agonist compounds in this series at the three LPA receptors in the broken cell assay. First, each LPA receptor showed a marked selectivity (1 log order or more) for one enantiomer. This confirms and extends our previously reported observation of stereoselectivity by LPA receptors for NAEPA compounds containing the 2-carboxyl (Hooks et al., 1998) or 2-methylene hydroxy groups (Hooks et al., 2001). Second, those compounds with substitutions at the R1 position were invariably the more potent agonists (Fig. 2,3). Third, agonist potency decreases as the bulk of the substituent increases.
GTP[γ35S] binding to HEK293T cell membranes in response to LPA and analogs with hydrophobic substituents. HEK293T cells were transfected with LPA1 (A), LPA2 (B), or LPA3 DNAs (C) and G protein DNAs. Lipids were dissolved at 1 mM in aqueous 1.0% fatty acid-free BSA and diluted in aqueous 0.1% BSA. Open characters representR 1 enantiomers and closed characters represent 1-oleoyl-(S)-LPA curve as a reference. All compounds contain oleoyl (18:1) fatty acid chains. Points are in triplicate and are representative of at least two experiments. Typical values for zero and 100% binding were 300 and 800 dpm/tube for LPA1, 300 and 1000 dpm/tube for LPA2, and 300 and 2000 dpm/tube for LPA3, respectively.
The 2-substituted NAEPA compounds containing either hydrophilic (methylene hydroxy, carbomethyl, methylene amino) or hydrophobic moieties (methyl, ethyl, isopropyl, benzyl) exhibited agonist activity in the GTP[γ35S] binding assay (Figs. 2 and3). The smaller groups conferred greater potency, with the methyl (VPC12086), methylene hydroxy (VPC31143), and methylene amino (VPC12178) compounds being more potent than 1-oleoyl LPA at LPA1 (Figs. 2 and 3 and Table1). In addition, because the 2-substituent becomes bulkier, the efficacy was noticeably reduced at this receptor. In contrast, bulkier hydrophobic side chains, although less potent, were fully efficacious at the LPA2receptor (Fig. 3B and Table 1). As was observed with the LPA1 receptor, the small methyl and methylene amino groups conferred the highest potency at the LPA2 receptor, but none of these compounds proved more potent than 1-oleoyl LPA at this site. The LPA3 receptor exhibited much the same profile as the LPA2 receptor with regard to efficacies and potencies of compounds relative to LPA. However, the LPA3 receptor characteristically exhibited higher (1–2 log order) EC50 values for all compounds, including LPA. Presumably, the LPA3 receptor has an intrinsically lower affinity for LPA and LPA analogs (Fig. 3C and Table 1). Like the hydrophilic compounds, each LPA receptor showed strong stereoselectivity for a hydrophobic substituent in the R1 position (Fig. 1 and Table 1).
Agonist activities of 2-substituted N-oleoyl ethanolamide phosphoric acid compounds
We wished to determine whether the stereoselectivity predicted by the broken cell assay at recombinant LPA receptors extends to endogenous LPA responses. We have reported previously that this is not the case for LPA stimulation of tritiated thymidine incorporation as an index of mitogenesis (Hooks et al., 2001). In contrast, the rank order potencies of the compounds predicted by the GTP[γ35S] binding assay are mimicked by calcium mobilizing activities in a variety of cell lines including A431, HEK293, and MDA MB231 (data not shown). To investigate an LPA response in a physiologic context, we monitored MAP, heart rate, and postganglionic sympathetic tone in anesthetized adult rats as a function of LPA or LPA analog administration. LPA has been shown previously to increase blood pressure transiently in this model (Tokumura et al., 1978). As shown in Fig. 4, intravenous injection of three enantiomeric pairs of compounds resulted in a transient increase in MAP with the same pattern of stereoselectivity as observed with the in vitro assays. Concomitant with this rise in MAP was a decrease in heart rate and sympathetic output indicative of baroreceptor reflex response (Fig. 4).
Blood pressure changes in response to intravenous injection of LPA analogs to ventilated anesthetized rats. A, representative trace of change in MAP, HR, and sympathetic outflow [splanchnic nerve activity (SNA)] on injection of 50 pmol of compound VPC12178. B, same representative traces on administration of 50 pmol of the enantiomeric compound, VPC12048. C to E, measurement of change in MAP in response to administration of LPA analogs at varying concentrations. Points are in triplicate from three rats (C), or duplicate from two rats (D, E). VPC31143 and VPC31144 in (D) contain palmitoyl (16:0) fatty acid chains.
The compounds that were only slightly efficacious at LPA1 (e.g., the benzyl-containing VPC12084) were assayed for their ability to antagonize LPA-induced GTP[γ35S] binding. Although this compound did block LPA activity in the GTP[γ35S] binding assay (not shown), the benzyl compound (VPC12084) was revealed to posses appreciable agonist activity in assays with greater levels of amplification (e.g., whole cell assays of calcium mobilization or inhibition of cAMP accumulation) (Fig.5). In the course of exploring variations of the benzyl substituent, we found that a benzyl-4-oxybenzyl substituent in the same relative configuration (R 1, VPC12204) had reduced, but still measurable, agonist activity in whole cell assays (Fig. 5). However, its enantiomer [i.e., VPC12249, the compound with the benzyl-4-oxybenzyl substituent in the S(R 2; see Fig. 1) configuration] was completely devoid of agonist activity in the whole-cell assays (Fig. 5) and in the GTP[γ35S] binding assay (not shown).
Calcium mobilization traces and inhibition of cAMP accumulation assay using mouse LPA1 expressing RH7777 cells. Upper left shows a calcium transient in response to LPA in untransfected cells. Remaining calcium mobilization traces are from LPA1 receptor stably expressing RH7777 cells. Arrow indicates addition of denoted compound; ∗ indicates addition of digitonin (to 75 μM) followed by EDTA (to 5 μM) to measure total intracellular calcium. Bar graph is also from LPA1 receptor stably expressing RH7777 cells showing accumulation of cAMP data.
We tested VPC12249 for its ability to block LPA-induced GTP[γ35S] binding at each recombinant LPA receptor. As shown by the rightward, parallel shifts in the concentration response curves as a function of VPC12249 concentration, this compound is a surmountable antagonist at the LPA1 and LPA3 but not the LPA2, receptors (Fig.6). The K Ivalues for VPC12249 determined by Schild regression are 137 and 428 nM at the LPA1 and LPA3receptors, respectively, in this assay. The same activity was determined with human LPA1 using a recombinant baculovirus-infected insect Sf9 cell membrane preparation (M. D. Davis and K. R. Lynch, unpublished observations).
Effect of the benzyl 4-oxybenzyl compounds, VPC12204 and VPC12249, on GTP[γ35S] binding. GTP[γ35S] binding assays of LPA1- (A), LPA2- (B), and LPA3- (C, D) transfected HEK293T cell membranes showing LPA concentration response curves with increasing concentrations of VPC12204 or VPC12249. Points are in triplicate and are representative of at least two experiments. Insert represents Schild regression analysis of concentration response curves.
The antagonist activity measured in the broken cell assays was confirmed in whole-cell experiments wherein LPA-induced increases in free intracellular calcium in HEK293T cells were blocked. This cell type expresses the LPA1 and LPA3 but not LPA2 receptor genes as determined by RT-PCR (not shown). As documented by the concentration response curves shown in Fig.7A, increasing concentrations of VPC12249 resulted in parallel, rightward shifts in the LPA concentration response curves (K I = 132 nM). The extent of rightward shift observed in the same experimental protocol with A431 cells, which express the LPA2 as well as the LPA1 and LPA3 genes (RT-PCR not shown), was much smaller (K I = 1970 nM; Fig. 7C) as predicted from the lack of antagonist activity of VPC12249 at the calcium-mobilizing LPA2 receptor in the GTP-binding assay (see Fig. 6). The blocking action of VPC12249 was not a general postreceptor event, as shown by the lack of antagonism of ATP-evoked calcium transients (Fig. 7B). Inhibition of forskolin-induced increases in cAMP levels in RH7777 cells stably expressing LPA1 was also inhibited by VPC12249 (Fig. 7D).
Effect of the benzyl 4-oxybenzyl compound VPC12249 on calcium mobilization and inhibition of cAMP accumulation. A, calcium mobilization of HEK293T cells showing LPA concentration response curves with increasing concentrations of VPC12249. B, calcium mobilization of HEK293T cells showing ATP dose-response curve with and without 10 μM VPC12249. C, calcium mobilization of A431 cells showing dose response curves with increasing concentrations of VPC12249. D, inhibition of forskolin-driven rises in cAMP on LPA1 receptor stably expressing RH7777 cells showing LPA concentration response curve with increasing concentrations of VPC12249. Calcium data points are replicates of at least five and representative of two experiments. Inhibition of cAMP accumulation data points are in duplicate with 0% averaging 3 pmol/tube and 100% averaging 80 pmol/tube cAMP. Inset, Schild regression analysis of concentration response curves.
The lack of medicinal chemistry associated with the LPA1, LPA2, and LPA3 receptors prompted us to develop LPA receptor selective agonists and antagonists. Among the most fruitful series that we have synthesized are 2-substituted N-acyl ethanolamide-phosphates. The lead compound, N-palmitoyl ethanolamide phosphate (Sugiura et al., 1994), in which the glycerol moiety of LPA is replaced with ethanolamine, has been shown to be indistinguishable from LPA in potency and efficacy at the mammalian LPA1 and LPA2 receptors (Im et al., 2000b) but is less potent than LPA at the LPA3 receptor (Im et al., 2000b). Although the synthetic routes to the compounds in our series did not proceed through NAEPA, they all contain this backbone (Fig. 1).
We have adapted a GTP[γ35S] binding assay to analyze directly the activation of individual recombinant LPA receptors, which allowed determination of relative efficacies and potencies at each receptor with a common assay platform. The same concentration response curves were obtained regardless of whether the recombinant receptor used exogenous G proteins from various mammalian species (HEK293T cells; see Experimental Procedures) or endogenous G proteins (RH7777 cells, data not shown). More importantly, the rank-order potencies established with the broken-cell, membrane-based GTP[γ35S] binding assay were maintained in whole-cell assays of calcium mobilization and inhibition of cAMP accumulation as well as in blood pressure responses in whole animals. Thus, our primary assay for compound potency and efficacy is a valid predictor of activity at endogenous LPA receptors.
In our exploration of various 2-substituents of the NAEPA backbone, several trends became apparent. First, each LPA receptor showed a marked (1 log order or more) preference for one enantiomer. We first reported this differential activity (albeit at undefined LPA receptors) using the 2-carboxyl-containing compound N-acyl serine phosphoric acid (Hooks et al., 1998) and later with the 2-methylene hydroxy compound at LPA1, LPA2, and LPA3 (Hooks et al., 2001). This stereoselectivity was evidenced further in vivo by inducing transient increases in blood pressure in ventilated, anesthetized rats in a dose-dependent manner. The lack of stereoselectivity of mitogenic responses to LPA mimetics such as the 2-methylene hydroxy compound is central to our argument that such responses are, at least in some cell types, independent of LPA1–3 receptors (Hooks et al., 2001).
Second, most substitutions were well tolerated in that they resulted in agonist ligands (Table 1). Although most active compounds were partial agonists with reduced potency (relative to LPA), the methyl (VPC12086), methylene hydroxy (VPC31143), and methylene amino (VPC12178) compounds are notable in that they are more potent than LPA at the LPA1 receptor. A pattern observed with all three LPA receptors was that activity is conferred for hydrophobic or hydrophilic compounds only when the substituent is at theR 1 position and theR 2 position is a hydrogen atom (refer to Fig. 1 for structures; see also Figs. 2 and 3). In aggregate, these data indicate to us that each LPA receptor has one spatial region within the ligand binding site for hydrophobic or hydrophilic functionalities and is restrictive in nature by recognizing only smaller substituents as potent agonists.
Third, we note again that the LPA3 receptor, unlike the LPA1 and LPA2receptors, discriminates between unsaturated and saturated acyl groups (Bandoh et al., 1999; Im et al., 2000b). Although this article focuses on oleoyl compounds, we have examined a number of these compounds as palmitoyl forms and find consistently that the saturated analogs are less potent at only the LPA3 receptor. Furthermore, LPA3 seems to have a lower affinity for LPA and LPA analogs, as indicated by a rightward shift in all concentration response curves.
Several of the compounds described deserve special mention. The first of these is the methylene hydroxy-containing compound (VPC31143). This compound is an analog of 2-acyl LPA in which the ester is replaced by an amide, thus conferring chemical stability (i.e., chain migration is prevented). The LPA in malignant ascites has been reported to consist of substantial amounts of the 2-acyl species; this is the form of LPA that reportedly confers a greater biological activity on ovarian cancer cells (Xu et al., 1995). The 2-acyl LPA analog (VPC31143) was equipotent to 1-acyl LPA at the LPA3 receptor, less potent at the LPA2 receptor, but more potent at the LPA1 receptor, which supports the notion that 2-acyl LPA has a different biologic activity than 1-acyl LPA.
A second pair of compounds worth noting exhibit significantly greater potency at the LPA1 receptor compared with either the LPA2 or LPA3 receptors: the methylene hydroxy- and methylene amino-containing compounds (VPC31143 and VPC12178, respectively) are both fully efficacious and highly potent at the LPA1 receptor with at least a 10-fold lower EC50 value at this receptor type than at either the LPA2 or LPA3 sites (Table 1). To our knowledge, these compounds represent the first receptor type selective agonists and as such might prove useful for elucidating the receptor entities responsible for various effects in cultured cells and tissues.
A third interesting pair of compounds contain a benzyl-4-oxybenzyl substituent at the 2 position in either the R orS configuration (VPC12204 and VPC12249, respectively). These compounds are potent, dual LPA1/LPA3 receptor antagonists. VPC12204 is more potent at inhibiting LPA induced activation of LPA3 but retains the partial agonist activity of the parent benzyl compound (VPC12084) at LPA1 (Fig. 5). The enantiomer of this compound (VPC12249) is entirely devoid of agonist activity in our most sensitive assays. We do not know if VPC12249 is an inverse agonist, but its blockade is surmountable, and thus can be considered a competitive antagonist. The K I value calculated for VPC12249 at inhibiting calcium mobilization in HEK293T cells is ∼130 nM. These cells express the LPA1 and LPA3 receptors as judged by RT-PCR analysis; the calcium response to LPA probably flows through both receptors because pretreatment with pertussis toxin partially blunts but does not completely inhibit the resultant calcium transient (our unpublished observations). In addition, the calculatedK I value of VPC12249 at the LPA1 receptor from the GTP[γ35S] binding assay is nearly identical to that obtained for these whole cells (137 versus 132 nM). The greater than 10-fold higher K I value measured with A431 cells is expected in view of the low potency of VPC12249 in antagonizing LPA2 mediated responses in assays of the recombinant LPA2 receptor. Furthermore, the agreement in data from the membrane based GTP[γ35S] assay and the whole cell assays of VPC12249 lends further credence to the broken cell assay system for rapidly assessing the activity of novel chemical entities at recombinant LPA receptors.
A recent report from Tigyi and associates describes a dual LPA1/LPA3 receptor antagonist, dioctyl glycerol pyrophosphate (Fisher et al., 2001), although this compound is a much less potent blocker of the LPA1 site. Our laboratory is in the process of comparing this compound with VPC12249 in our assay systems. Our preliminary results support the contention that dioctyl glycerol pyrophosphate is a LPA3 receptor blocker, although we find it less potent than VPC12249.
In summary, we have continued to explore variations of theN-acyl ethanolamide phosphate backbone in search of novel chemical entities that might prove useful in probing LPA biology. We have found that compounds with small substituents at the second carbon atom are in general potent and efficacious agonists; indeed, in some cases, these compounds are more potent and efficacious than LPA, providing two LPA1 receptor type selective agonists. Our observation that efficacy decreased sharply with the bulk of hydrophobic 2-substituents led to the discovery of a dual LPA1/LPA3 competitive antagonist, VPC12249. We are profiling this molecule in a variety of in vitro and in vivo experimental systems as well as attempting to develop similar compounds with enhanced potency and receptor type selectivity.
We thank Dr. Shelley B. Hooks (Dept. of Pharmacology, University of North Carolina, Chapel Hill, NC) and Michael D. Davis (Dept. of Biochemistry, University of Virginia) for sharing unpublished data. We thank also Dr. Gabor Tigyi (Dept. of Physiology, University of Tennessee at Memphis) for sharing his manuscript before publication inMolecular Pharmacology. We are grateful to Dr. Patrice Guyenet (Dept. of Pharmacology, University of Virginia) for use of his laboratory's equipment to make the blood pressure measurements reported herein and to Dr. John Raymond (Dept. of Medicine/Nephrology, Medical University of South Carolina, Charleston, SC) for allowing us time on the FLIPR instrument.
Received August 7, 2001.
↵1 The IUPHAR subcommittee on lysophospholipid receptor nomenclature has recommended that henceforth the colloquial 'Edg' nomenclature be replaced with LPA (or S1P) subscript number, where the number indicates order of molecular cloning. Thus Edg-2 becomes LPA1, Edg-4 becomes LPA2, and Edg-7 becomes LPA3.
This work was supported by National Institutes of Health research grants R01-GM52722 and R01-CA88994 (to K.R.L. and T.L.M.) and predoctoral traineeship T32-GM07055 (to C.E.H. and W.L.S.) and by American Heart Association postdoctoral fellowship AHA-130274N (to A.M.S.). The FLIPR was purchased with National Institutes of Health shared equipment grant S10-RR13005 (to John R. Raymond, Medical University of South Carolina).
lysophosphatidic acid
NAEPA
N-acyl ethanolamide phosphoric acid
Edg
endothelial differentiation gene
human embryonic kidney
GTP[γ35S]
guanosine 5′-O-(3-[35S]thio)triphosphate
FLIPR
fluorimetric imaging plate reader
RT-PCR
reverse transcription-polymerase chain reaction
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Molecular Pharmacology December 1, 2001, 60 (6) 1173-1180; DOI: https://doi.org/10.1124/mol.60.6.1173
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Trauma risk management (TRiM) is a method of secondary PTSD (and other traumatic stress related mental health disorders) prevention. The TRiM process enables non-healthcare staff to monitor and manage colleagues. TRiM training provides practitioners with a background understanding of psychological trauma and its effects.
TRiM is a trauma-focused peer support system and the way it works is wholly compliant with the PTSD management guidelines produced by the National Institute for Health and Care and Excellence.
Trauma risk management Practitioners are trained to carry out an interview which identifies a number of risk factors which, when present, increase the likelihood that an individual may suffer poor longer term mental health as a result of a traumatic event. The initial TRiM interview takes place with an individual, 72 hours after a traumatic incident. People who score highly on this initial interview are provided with extra support by colleagues, and where appropriate, line managers. A follow up TRiM interview is then carried out approximately one month later to assess how well people have come to terms with the traumatic event at that point. Individuals who are found to have persistent difficulties at this point are encouraged and assisted to seek a professional assessment in order to access any specific treatment they require.
TRiM originated within the UK military after previously-used reactive single session models of post incident intervention, such as Critical Incident Stress Debriefing, were subject to scientific scrutiny and shown to not just lack effectiveness but also have the potential to do harm. Professor Neil Greenberg was one of the team at the forefront of developing peer-led traumatic stress support packages, now known as TRiM. He is an academic psychiatrist based at King's College London UK and is a consultant occupational and forensic psychiatrist.
Although it was first developed in the UK military, trauma risk management is now used by a range of public and commercial organisations. This includes charities, emergency services, security firms, risk management organisations, UK Government departments including the Foreign and Commonwealth Office, the oil and gas industry, transport organisations and media companies including the BBC.
Evidence and research
A large number of research papers have been published about the use of TRiM and its effectiveness and acceptance. This includes research about the use of TRiM by Cumbria Constabulary, following the Cumbria shootings in 2010. This research showed that the officers and staff who received a TRiM response fared better than their colleagues who did not and were less likely to be absent from work than colleagues who did not receive a TRiM intervention.
Research has shown that the use of TRiM may assist in increasing the psychological resilience of military personnel through the facilitation of social support.
A review of TRiM research was published in the Journal of Occupational Medicine in April 2015 and since that time a further 2017 paper has shown that TRiM improves help-seeking after traumatic incidents.
References
Post-traumatic stress disorder
Behaviorism
Aftermath of war
Military medicine
Military personnel
Military psychiatry
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Antonio Marziale Carracci (1583 – 8 April 1618) was an Italian painter. He was the natural son of Agostino Carracci.
Life
Carracci was born in the parish of Sta Lucia in Venice, probably in 1583, the product of an affair with a courtesan called Isabella, occurring on his father's first visit to Venice. Giovanni Battista Agucchi, a friend and protégé of Cardinal Odoardo Farnese tells us in a 1609 letter, raised along with Sisto Badalocchio and a near contemporary to Domenichino and Lanfranco. The main Baroque artist biographers of his time, Baglione, Bellori and Malvasia, make some note of him.
He first apprenticed with his father. Malvasia recalls his father admired a ''Madonna and Child'' that completed at the age of seventeen. Annibale's 1590s portrait of a boy resembles those in Malvasia's woodcut with 1678 '''Felsina Pittrice'''. He may also be the boy in the Brera portrait group, including Annibale's father Mastro Antonio.
When his father died in 1602, Antonio moved to Rome to work under his uncle Annibale to whom he developed a deep affection. He likely worked on the frescoes of the Galleria Farnese, on the lunettes for the Palazzo Aldobrandini chapel, and probably in the Herrera Chapel. After his uncle's passing he received commissions in Rome from Cardinal Tonti and Cardinal Peretti-Montalto, including the ''Stanza del Diluvio'' in the Quirinal Palace, and major altarpieces like ''Madonna and Child with Saints'' (in Berlin) and the Galleria Corsini's ''Nativity''.
The self-assurance of the letters he wrote to Cardinal Farnese and his father's old protector, Cardinal Spinola, after his uncle's death suggest they were written by Agucchi, who in his slightly later (September 12, 1609) letter to Dulcini all about Antonio, speaks of a career that is already well begun. Malvasia misquotes this letter and copies it as '[Antonio's work] seems that of a beginner' the MS actually says the opposite 'il suo fare non pare da principiante'. It is no surprise to see that Antonio speaks of the impegni that he already has, to read that he has learned enough 'tanto da tirarmi avanti da me stesso', and that he sees himself as maintaining the Carracci school in Bologna, and turning the studio to service to the Farnese 'in essa scuola, procurerò che s'avanzino in essa per dovere indirizzare l'opera l'oro al servizio di V.S. Ill.ma...'
Antonio inherited the studio against the claims of Annibale's surviving brother Giovanni Antonio, who contested the inheritance by throwing doubt on his nephew's paternity. Despite gaining commissions, those from the Cardinal Pietro Aldobrandini, the main patron after the Farnese, declined once the cardinal's uncle Clement VIII died in 1605. This also coincides with the onset of Annibale's incapacitating illness. The ascent of the Borghese Pope, Paul V, continued the decline of the studio. By report, Annibale could not receive Cardinal Scipione Borghese, slipping out of a back door when he came to the studio; but Antonio was introduced to the pope's Prodatario, Cardinal Tonti, also an avid collector. Tonti employed Antonio to paint four chapels in his church; other public commissions included a chapel in Santa Maria in Monticelli, work in Sant' Girolamo dei Schiavoni, Santa Maria in Trastevere, and Sant' Sebastiano fuori le Mura, and in Palazzo Mattei. Antonio and Guido Reni collaborated in the decoration of the Cappella dell' Annunciata in the Palazzo Quirinal, painting a frieze of the Stanza del Diluvio. He contributed, around 1616, to the Alexander frescoes done for Cardinal Peretti Montalto; other works were done for Marchese Giustiniani, Cardinal Orsino, the Ludovisi, Cavalier Sachetti, Dionigio Buonavia in Bologna; and of course he was under the protection of Cardinal Odoardo Farnese, who recognized him as his 'pittor di casa' and paid him a monthly stipend.
Antonio was well known for doing cabinet-size compositions often based on the great variety of graphic material in possession of the Carracci studio. His masterpiece, The Deluge was one of Cardinal Mazarin's most prized pictures, and influential for Poussin's early Battle paintings as well as his paintings of the Seasons for Louis XIV 40 years later. With his Reni-like refinement, articulation of narrative, space, gesture and expression, the Deluge was an important academic collaboration with Agucchi.
Antonio was greatly admired by contemporaries, such that made his early death a particularly tragic loss "nel morire che seguì nel 1618 mostrò tal contritione e sentimento che simil passaggi si vedono in pochi". Recently, his reputation has declined, and he has been described as a small-scale and awkward mimic of his elder's works. His friend Giulio Mancini said he not only inherited Annibale's studio, but also "Herede dei suoi disegni et arte fù Antonio...". Antonio, being as Guido put it, the 'ultima scintilla (del) valor Caraccesco' had in fact more invested in the Carracci studio than anyone else.
The original of the much copied composition of the Martyrdom of St Denis ('St Denis effrayant ses bourreaux') is evidently by Antonio, and the companion subject to his St Paul baptizing St Denis.
It was not an idle encomium on Mancini's part when he spoke that Antonio 'mostrò gran segni di dover venir grande'. All the same, it was easy for his identity to be subsumed to that of elder Carracci. Because he shared the same initials as his father and uncle, and because the Carracci often collaborated, paintings by them were often confused. When Padre Resta gave the drawing of the Deluge, which he had had from Bellori, to Max Emmanuel of Bavaria, he was pleased to hear that the Elector treasured the gift and kept it on a table at his bedside; but 'lo teneva per pittura d'Annibale'.
List of works (selection)
The Flood (Louvre)
Latona, with her children Apollo and Diana, turning the Lycian peasants into frogs Whitfield Fine Art, London
Landscape with Bathers Museum of Fine Arts, Boston
The Martyrdom of Saint Stephen National Gallery, London
Christ healing a blind man (Modena)
Lute Player (Modena)
Further reading
Clovis Whitfield, Antonio Carracci. Catalogue Raisonné of the paintings and drawings. (in preparation)
Clovis Whitfield, "Landscape paintings and drawings by Antonio Carracci" in Paragone, November 2006 pp. 3–20
Anne Sutherland Harris, "Annibale's Legacy: Proposals for Giovanni Angelo Canini and Antonio Carracci" in Master Drawings Vol. XLIII 2005, pp. 440–454
Clovis Whitfield, "A name for a ridiculous man – Rinaldo Coradini by the Carracci" in Festrift for Dr. Alfred Bader, 2004, pp. 239–244
Clovis Whitfield, "Antonio Carracci" in Studi di Storia dell'arte in onore di Denis Mahon, Milan 2000
Stephen M. Bailey, "Newly identified drawings by Antonio Carracci" in Master Drawings, XXXVII/3 (1999), pp. 277–286
Classicismo e Natura; La Lezione di Domenichino, Exhibition catalogue, ed. Sir Denis Mahon & C. Whitfield, Rome, Gallerie Capitoline 1996/97
C.Van Tuyll van Serooskerken " Two Preparatory drawings by Antonio Carracci" in Master Drawings, XXXI (1993), pp. 428–32
R. Zapperi, "I ritratti di Antonio Carracci" in Paragone, XXXVIII/449 (1987), pp. 3–22
E. Schleier, "Ancora su Aontonio Caracci e il ciclo di Alessandro Magno per il Cardinal Montalto" in Paragone XXXII/381 (1981), pp. 10–25
F. Frisoni, "Antonio Carracci: Riflessioni e aggiunte" in Paragone, XXXI/367 (1980), pp. 22–28
Notes
References
Antonio Marziale Carracci at the Catholic Encyclopedia
See also
Agostino Carracci, father
Annibale Carracci, uncle
Lodovico Carracci, father's cousin
1583 births
1618 deaths
Painters from Venice
16th-century Italian painters
Italian male painters
17th-century Italian painters
Italian Baroque painters | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 5,505 |
Q: Facebook Iframe App - Strange Redirection Problems When I'm doing a server-side redirection in a Facebook iframe application, I get the this strange Facebook Logo with a link. When I click it, I get redirected to site I set the redirection to in the first place. What happens here? Any "Click Protection" in place?
Thanks!
Redirection Code:
Tried Client Redirect
args = dict(client_id=FACEBOOK_APP_ID, redirect_uri=base_dir , scope='user_photos,email,offline_access')
return render_to_response('fb/nopermission.html', { 'javascript': "<script>window.top.location=\"" + "https://graph.facebook.com/oauth/authorize?" + urllib.urlencode(args )+ "\"</script>" } )
and Server Redirect:
args = dict(client_id=FACEBOOK_APP_ID, redirect_uri=base_dir , scope='user_photos,email,offline_access')
return HttpResponseRedirect(https://graph.facebook.com/oauth/authorize?" + urllib.urlencode(args))
Same Result in both cases
A: Arrgh! Facebook has such a shitty documentation that I hit upon zillion problems while developing for it.
This one is because you cannot redirect directly from with your app. use a href link and specify the target property to _top or use the javascript
window.top.location="redirecturl"
to workaround this bug.
A: For redirection purpose use this code:
<html><head>
<script type="text/javascript">
window.top.location.href = "<?php echo $loginUrl; ?>";
</script>
<noscript>
<meta http-equiv="refresh" content="0;url=<?php echo $loginUrl; ?>" />
<meta http-equiv="window-target" content="_top" />
</noscript>
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 4,918 |
{"url":"https:\/\/www.physicsforums.com\/threads\/empirical-proof-of-a-conjecture.249870\/","text":"# Empirical proof of a conjecture\n\n1. Aug 13, 2008\n\n### mhill\n\ncan a conjecture be proved by 'empirical' means (observation) ??\n\ni mean let us suppose that exists some functions named $$f_{i} (x)$$\n\nso $$\\sum _{n=0}^{\\infty} = \\sum _{p} f(p)$$\n\nthen an 'empirical' method would be to calculate the 2 sums and compare the error , let us suppose that the error made in the equation above is less or equal than 0.001\n\nso $$|\\sum _{n=0}^{\\infty} - \\sum _{p} f(p)| \\le 0.001$$\n\nthen , would this be simple coincidence or a fact that our conjecture is true ? , for example physicist and chemists work this way , as an approximation of a theory to our observed reality.\n\n2. Aug 13, 2008\n\n### mathman\n\nIt would help if your statements could be clarified. Your first sum is n=, but there is nothing being summed. Your second sum is for a function over p, without anything said about what p is.\n\n3. Aug 13, 2008\n\n### mhill\n\nUh.. excuse me , the sum on the left is made over f(n) , the sum on the right is over all 'primes' p\n\n$$\\sum _{n=0}^{\\infty}f(n) = \\sum _{p} f(p)$$\n\n4. Aug 13, 2008\n\n### frb\n\nof course it is not a proof, it can give an indication that the theorem might be true and give you a reason to find a formal proof.\n\nfor example\nthe difference of the partial sums might eventually be smaller than some epsilon, but it might also always be larger than some lower bound\n\nsome ideas seem very plausible but might have very pathological counterexamples.\nhttp:\/\/en.wikipedia.org\/wiki\/List_of_disproved_mathematical_ideas","date":"2017-11-18 18:49:27","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.7077397108078003, \"perplexity\": 748.5580179459057}, \"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-2017-47\/segments\/1510934805008.39\/warc\/CC-MAIN-20171118171235-20171118191235-00723.warc.gz\"}"} | null | null |
{"url":"https:\/\/socratic.org\/questions\/how-do-you-differentiate-f-x-xtanx-x-2-x-3-using-the-quotient-rule","text":"# How do you differentiate f(x)=(xtanx)\/(x^2-x+3) using the quotient rule?\n\nMar 29, 2018\n\n$- x \\cdot {\\left({x}^{2} - x + 3\\right)}^{-} 2 \\cdot \\left(2 x - 1\\right) \\cdot \\tan x + {\\left({x}^{2} - x + 3\\right)}^{-} 1 \\cdot \\tan x + x \\cdot {\\left({x}^{2} - x + 3\\right)}^{-} 1 \\cdot {\\sec}^{2} x$\n\nThere are 2 Answers first with the product rule the second with the quotient rule\n\n#### Explanation:\n\nyou can write the function like this: $f \\left(x\\right) = x \\cdot {\\left({x}^{2} - x + 3\\right)}^{-} 1 \\cdot \\tan x$\n\nnow consider:\n$g \\left(x\\right) = x$\n\n$h \\left(x\\right) = \\tan x$\n$v \\left(x\\right) = {\\left({x}^{2} - x + 3\\right)}^{-} 1$\n\n$\\frac{d}{\\mathrm{dx}} \\left(g \\left(x\\right) \\cdot h \\left(x\\right) \\cdot v \\left(x\\right)\\right) = g ' \\left(h\\right) \\cdot h \\left(x\\right) \\cdot v \\left(x\\right) + g \\left(x\\right) \\cdot h \\left(x\\right) \\cdot v ' \\left(x\\right) + g \\left(x\\right) \\cdot h ' \\left(x\\right) \\cdot v \\left(x\\right)$\n\n$g ' \\left(x\\right) = 1$\n\n$h ' \\left(x\\right) = {\\sec}^{2} x$\n\n$v ' \\left(x\\right) = - {\\left({x}^{2} - x + 3\\right)}^{-} 2 \\cdot \\left(2 x - 1\\right)$\n\nand by substituting in the relation above you get:\n$\\frac{d}{\\mathrm{dx}} x \\cdot {\\left({x}^{2} - x + 3\\right)}^{-} 1 \\cdot \\tan x = - x \\cdot {\\left({x}^{2} - x + 3\\right)}^{-} 2 \\cdot \\left(2 x - 1\\right) \\cdot \\tan x + {\\left({x}^{2} - x + 3\\right)}^{-} 1 \\cdot \\tan x + x \\cdot {\\left({x}^{2} - x + 3\\right)}^{-} 1 \\cdot {\\sec}^{2} x$\n\ny'=(g(x)f'(x)\u2212f(x)g'(x))\/(g(x))^2\nwhere $f \\left(x\\right) = x \\tan x$\n$g \\left(x\\right) = {x}^{2} - x + 3$\n$f ' \\left(x\\right) = x {\\sec}^{2} x + \\tan x$\n$g ' \\left(x\\right) = 2 x - 1$\n$y ' = \\frac{\\left({x}^{2} - x + 3\\right) \\left(x {\\sec}^{2} x + \\tan x\\right) - \\left(x \\tan x\\right) \\left(2 x - 1\\right)}{{x}^{2} - x + 3} ^ 2$","date":"2022-06-27 21:50:52","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 16, \"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\": 0.9814501404762268, \"perplexity\": 1841.9559915228097}, \"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-2022-27\/segments\/1656103341778.23\/warc\/CC-MAIN-20220627195131-20220627225131-00149.warc.gz\"}"} | null | null |
\section{Introduction}
\label{sec:Intro+Model}
Recent Bethe Ansatz studies of the one-dimensional Hubbard model with open
boundaries subject to boundary chemical potentials or magnetic fields
\cite{hschulz:85,assu:96,deyu:pp,shwa:97} have opened new possibilities to
apply the predictions of Boundary Conformal Field Theory
\cite{card:89,aflu:94,affl:94} for the asymptotics of correlation functions
to quantum impurity problems. As in the case of periodic systems,
\cite{frko:90,frko:91,kaya:91} the corresponding matrix elements cannot be
computed directly but must be extracted from the scaling behaviour of the
low--lying excited states. For generic filling and magnetization these
finite size spectra allow the identification of the contributions from two
massless bosonic sectors associated with the spin and charge excitations in
a Tomonaga-Luttinger liquid.
A crucial step in these studies of systems with open boundary conditions is
the correct interpretation of the finite-size spectra obtained from the
Bethe Ansatz. These spectra determine both the bulk correlation functions,
which are independent of the boundary fields, and the nonuniversal
dependence of boundary phenomena such as the orthogonality exponent or
X-ray edge singularities on the strength of the scatterer.
Hence both in the requirement of conformal invariance and in the analysis
of the finite-size spectra, the computation of correlation functions relies
on assumptions which need to be verified by a more direct method, most
notably by numerical calculations. Of course, comparison with exact
results is also of interest for the numerical calculation: predictions of
critical exponents for microscopic models allow the algorithms to be
improved, which can then, in turn, be expected to produce better and more
reliable results for more general systems.
These considerations motivate our study of the Friedel oscillations for the
single particle density and magnetization in the open Hubbard chain with
boundary chemical potentials, which is described by the Hamiltonian
\begin{eqnarray}
{\cal H} = &-& \sum_{\sigma,j=1}^{L-1} \left( c_{j,\sigma}^\dagger
c_{j+1,\sigma} + {\rm h.c.} \right) + U \sum_{j=1}^L n_{j,\uparrow}
n_{j,\downarrow} \nonumber \\
&-& p \left( n_{1,\uparrow} +
n_{1,\downarrow}+n_{L,\uparrow} +
n_{L,\downarrow} \right) ,
\label{ham:hubb}
\end{eqnarray}
where the lattice has $L$ sites, $c_{j,\sigma}^\dagger$ ($c_{j,\sigma}$)
creates (destroys) an electron on site $j$, $n_{j,\sigma} =
c_{j,\sigma}^\dagger c_{j,\sigma}$, and we have made the hopping integral
dimensionless so that the Coulomb repulsion $U$ and the on--site potential
$p$ are measured in dimensionless units.
Numerical calculations of critical exponents in low dimensional systems
such as magnetic chains or electronic systems have a long history \cite{}.
Due to the need to consider systems of sufficient size, many earlier
studies have used Quantum Monte Carlo methods to treat systems with
periodic boundary conditions (PBC). \cite{sopa:90} More recently, the
Density Matrix Renormalization Group (DMRG) \cite{whit.92,whit.93} has
become a new and powerful method especially suited to the study of one
dimensional systems with open boundary conditions (OBC).
\cite{hifu:98,meschmi:98} However systems with PBC have also been studied
with this method. \cite{qili:95,schmeck:96,shueni:96}
In the usual approach to the calculation of correlation functions with DMRG
one considers large systems and averages over an ensemble of correlation
functions located sufficiently far from the boundaries. In the
thermodynamic limit, this procedure removes the Friedel oscillations due to
the boundary and gives the bulk behaviour of the quantity in question.
Here we want to make use of the existence of the exact solution of Eq.\
(\ref{ham:hubb}) in two ways in order to prepare the way for further
extensions of the method: First, we use the quantities obtained from the
Bethe Ansatz to provide checks of the numerical method at large system
sizes. Second, we use the information contained in the oscillating
behaviour to obtain more reliable results for the critical exponents.
This paper is organized as follows. In Sec.\ \ref{sec:Meth}, we give a
short description of the Bethe Ansatz and DMRG methods, citing the relevant
results from the Bethe Ansatz and Conformal Field Theory (CFT) concerning
Friedel oscillations.
In Sec.\ \ref{sec:fr_osc}, we study the Friedel oscillations of the density
$N(x)$ and the magnetization $M(x)$. Combining the CFT results with those
for noninteracting fermions, we obtain conjectures for the explicit form of
the Friedel oscillations. After introducing the fit method used to obtain
the exponents and coefficients from the DMRG results, we check the
conjectures for two fixed densities and varying on--site interaction $U$.
In addition, we study the dependence of the exponents and coefficients on
the boundary potential $p$ at one density.
\section{Methods}
\label{sec:Meth}
\subsection{Bethe Ansatz}
\label{sec:BA}
The one-dimensional Hubbard model with OBC, Eq.\ (\ref{ham:hubb}), can be
solved using the coordinate Bethe ansatz. The symmetrized Bethe Ansatz
equations (BAE) determining the spectrum of ${\cal H}$ in the
$N_{e}=N_\uparrow+N_\downarrow$-particle sector read
\cite{hschulz:85,assu:96,deyu:pp,shwa:97}
\begin{eqnarray}
e^{ik_j 2L} B_c(k_j) =
\prod_{\beta=-N_\downarrow}^{N_\downarrow}
\frac{\sin(k_j)-\lambda_\beta+iu}{\sin(k_j)-\lambda_\beta-iu} \nonumber
\\ \nonumber \\
B_s(\lambda_\alpha) \prod_{j=-N_e}^{N_e}
\frac{\lambda_\alpha-\sin(k_j)+iu}{\lambda_\alpha-\sin(k_j)-iu} =
\prod_{\beta=-N_\downarrow \atop \beta \not= \alpha}^{N_\downarrow}
\frac{\lambda_\alpha-\lambda_\beta+2iu}{\lambda_\alpha-\lambda_\beta-2iu}
\nonumber \\
j=-N_e,\ldots,N_e \quad;\quad
\alpha=-{N_\downarrow},\ldots,{N_\downarrow}
\label{eq:bae}
\end{eqnarray}
where we have defined $u=U/4$ and identified the solutions $k_{-j}=-k_j$
and $\lambda_{-\alpha}=-\lambda_{\alpha}$ in order to simplify the BAE.
The boundary terms read
\begin{eqnarray}
B_c(k)&=&\left(\frac{e^{ik}-p}{1-p e^{ik}}\right)^2
\frac{\sin(k)+iu}{\sin(k)-iu} \,,\nonumber \\
B_s(\lambda)&=&
\frac{\lambda+2iu}{\lambda-2iu} \,.
\label{bound}
\end{eqnarray}
Since the BAE are already symmetrized and the solutions $k=0$ and
$\lambda=0$ have to be excluded, the energy of the corresponding eigenstate
of Eq.\ (\ref{ham:hubb}) is given by
\begin{equation}
E=1- \sum_{j=-N_e}^{N_e} \cos{}(k_j) .
\label{energy}
\end{equation}
In Refs. \onlinecite{assu:96,deyu:pp,shwa:97} the ground state and the
low--lying excitations were studied for small boundary fields. In Ref.
\onlinecite{befr:97} the existence of boundary states in the ground state
for $p>1$ was established. Bound states occur as additional complex
solutions for the charge and spin rapidities.
Here we will use the explicit form of the BAE, Eq.\ (\ref{eq:bae}), to
check the energy convergence of the DMRG results for finite $L$.
Furthermore, the expectation values of the density at the boundaries can be
calculated from the derivative of the energy with respect to $p$ (cf.\
Sec.\ \ref{sec:DMRG}) allowing another check of the numerics. Finally, the
value of the magnetization at the boundaries for vanishing $p$ can be
calculated with a slightly modified Bethe--Ansatz (i.e. with a magnetic
field at the boundary, see Refs. \onlinecite{deyu:pp,shwa:97}).
Using standard procedures, the BAE for the ground state and low--lying
excitations can be rewritten as linear integral equations for the densities
$\rho_c(k)$ and $\rho_s(\lambda)$ of real quasi-momenta $k_j$ and spin
rapidities $\lambda_\alpha$, respectively:
\begin{equation}
\left( \begin{array}{c} \rho_c(k) \\[5pt]
\rho_s(\lambda) \end{array} \right) =
\left( \begin{array}{c} {1 \over \pi}+{1 \over L}\hat{\rho}_c^0(k)\\[5pt]
{1 \over L}\hat{\rho}_s^0(\lambda) \end{array} \right)
+ K * \left( \begin{array}{c} \rho_c(k') \\[5pt]
\rho_s(\lambda') \end{array} \right)\
\label{eq:dnorm}
\end{equation}
with the kernel $K$ given by
\begin{equation}
K=
\left( \begin{array}{cc} 0 & \cos{}(k)\ a_{2u}(\sin(k)-\lambda') \\[5pt]
a_{2u}(\lambda-\sin(k')) &
-a_{4u}(\lambda-\lambda') \end{array} \right).
\label{eq:kernel}
\end{equation}
Here we have introduced $a_y(x)={1 \over 2\pi}\frac{y}{y^2/4+x^2}$, and
$f*g$ denotes the convolution $\int_{-A}^{A}dy f(x-y)g(y)$ with boundaries
$A=k_0$ in the charge and $A=\lambda_0$ in the spin sector. The values of
$k_0$ and $\lambda_0$ are fixed by the conditions
\begin{equation}
\int_{-k_0}^{k_0}dk \rho_c =
\frac{2\left[N_e-C_c\right]+1}{L}\quad
\label{eq:fix1}
\end{equation}
and
\begin{equation}
\int_{-\lambda_0}^{\lambda_0}d\lambda\rho_s =
\frac{2\left[N_\downarrow-C_s\right]+1}{L} ,
\label{eq:fix2}
\end{equation}
where $C_c$ $(C_s)$ denotes the number of complex $k$
$(\lambda)$--solutions present in the ground state.\cite{befr:97} In
addition to the boundary terms in Eq.\ (\ref{bound}), the driving terms
$\hat{\rho}_c^0$ and $\hat{\rho}_s^0$ depend on whether or not the complex
solutions are occupied or not. The explicit form can be found in Refs.
\onlinecite{assu:96,deyu:pp,shwa:97,befr:97}. The presence of these $1/L$
corrections leads to the shifts
\begin{equation}
\theta^c_p = {1 \over 2}\left(L \int_{-k_0}^{k_0}dk \hat{\rho}_c-1+2
C_c \right)
\label{eq:tpc1}
\end{equation}
\begin{equation}
\theta^s_p = {1 \over 2}\left(L \int_{-\lambda_0}^{\lambda_0}d\lambda
\hat{\rho}_s -1+2 C_s \right) ,
\label{eq:tpc2}
\end{equation}
where $\hat{\rho}_c$ and $\hat{\rho}_s$ denote the solution of Eq.\
(\ref{eq:dnorm}) without the $1/\pi$ driving term, i.e.\ the bulk system
solution.
Here we will be mainly interested in the exponents of the Friedel
oscillations, given in Table\ \ref{tab:gamma}. The quantity that
determines the critical exponents is the dressed charge matrix ${\bf Z}$:
\cite{frko:90,woyn:89}
\begin{equation}
{\bf Z}= \left( \begin{array}{cc} Z_{cc} & Z_{cs} \\
Z_{sc} & Z_{ss} \end{array} \right)=
\left( \begin{array}{cc} \xi_{cc}(k_0) & \xi_{sc} (k_0) \\
\xi_{cs} (\lambda_0) & \xi_{ss}(\lambda_0) \end{array} \right)^\top
\label{eq:z}
\end{equation}
which is defined in terms of the integral equation
\begin{equation}
\left( \begin{array}{cc} \xi_{cc}(k) & \xi_{sc}(k) \\
\xi_{cs}(\lambda) & \xi_{ss}(\lambda) \end{array} \right)=
\left( \begin{array}{cc} 1 & 0 \\ 0 & 1 \end{array} \right)+
K^\top *
\left( \begin{array}{cc} \xi_{cc}(k') & \xi_{sc}(k') \\
\xi_{cs}(\lambda') & \xi_{ss}(\lambda') \end{array} \right) \,. \
\end{equation}
In Ref. \onlinecite{card:84} it was shown that the $n$-point correlation
functions of the open boundary system are related to the $2n$-point
functions of the periodic boundary system. Thus the expectation value of
the local density in the open system, $\langle N(x) \rangle_o$ can be
extracted from the two--point correlation function $\langle N(z_1) N(z_2)
\rangle_p$ of the periodic system (see also Ref.\ \onlinecite{wavp:96}
where a spinless fermion model was considered). We can therefore use the
results obtained in Refs. \onlinecite{frko:90} and \onlinecite{frko:91} for
the density--density correlation function. As a function of $z_c=x+iv_ct$
and $z_s=x+iv_st$ (where $v_c$ and $v_s$ denote the Fermi velocities of the
charge and spin sector, respectively), the asymptotic form of $G_{nn}(x,t)=
\langle N(x,t) N(0,0) \rangle$ is
\begin{eqnarray}
&&G_{nn}(x,t)=
n_e^2
+ \frac{B_\downarrow \cos(2k_{F,\downarrow}x+\delta_\downarrow)}
{|z_c|^{2\gamma_{\downarrow,c}} |z_s|^{2\gamma_{\downarrow,s}}}
\nonumber \\
&+& \frac{B_\uparrow \cos(2k_{F,\uparrow}x+\delta_\uparrow)}
{|z_c|^ {2\gamma_{\uparrow,c}} |z_s|^ {2\gamma_{\uparrow,s}}}
+ \frac{B_n \cos(2(k_{F,\uparrow}+k_{F,\downarrow})x+\delta_n)}
{|z_c|^{2\gamma_{n,c}} |z_s|^{2\gamma_{n,s}}}
\nonumber , \\
\label{ddper}
\end{eqnarray}
with $k_{F,\uparrow,\downarrow}=\pi n_{\uparrow,\downarrow}$. The
exponents are displayed in Table\ \ref{tab:gamma}. Eq.~(\ref{ddper}) shows
the oscillating terms which are the most relevant ones asymptotically. For
vanishing magnetization the momenta $k_{F,\downarrow}$ and $k_{F,\uparrow}$
coincide, and one has to consider logarithmic corrections in $x$ (see Ref.
\onlinecite{gischu:89}) -- this case will not be considered below.
Following Cardy \cite{card:84} one has to replace $|z_c|^2 \rightarrow x$
and $|z_s|^2 \rightarrow x$ to obtain $\langle N(x) \rangle_o$ from Eq.\
(\ref{ddper}). The final result is
\begin{eqnarray}
\langle N(x) \rangle=
n_e &+& A_\downarrow \frac{\cos(2k_{F,\downarrow}x+\varphi_\downarrow)}
{x^{\gamma_{\downarrow,c}+\gamma_{\downarrow,s}}}
+ A_\uparrow \frac{\cos(2k_{F,\uparrow}x+\varphi_\uparrow)}
{x^ {\gamma_{\uparrow,c}+\gamma_{\uparrow,s}}}
\nonumber \\
&+& A_n \frac{\cos(2(k_{F,\uparrow}+k_{F,\downarrow})x+\varphi_n)}
{x^{\gamma_{n,c}+\gamma_{n,s}}} .
\label{dopen}
\end{eqnarray}
The correlation function $G_{\sigma \sigma}^z \sim \langle M(x,t)M(0,0)
\rangle$ with magnetization $M=N_\uparrow-N_\downarrow$ has the same
critical behaviour as $G_{nn}(x,t)$. Therefore $\langle M(x) \rangle$ has
the same form as $\langle N(x) \rangle$, but with different coefficients
$A_\alpha$.
\subsection{Density Matrix Renormalization Group}
\label{sec:DMRG}
The density matrix renormalization group method (DMRG)
\cite{whit.92,whit.93} has become one of the most powerful numerical
methods for calculating the low-energy properties of one-dimensional
strongly interacting quantum systems. The expectation values of equal-time
operators in the ground state, such as the local density or magnetization
which interest us here, can be calculated with very good accuracy on quite
large systems (on lattices of up to $L=400$ sites in this paper). As we
will see, access to such large system sizes is essential for the real space
fitting method used to extract the coefficients and exponents of the
Friedel oscillations (see Sec.\ \ref{sec:fit_meth}). In the DMRG, open
boundaries are also the most favourable type of boundary conditions
numerically: for a given number of states kept (which corresponds to the
amount of computer time needed) the accuracy in calculated quantitites such
as the ground state energy is, in general, orders of magnitude better for
open boundary conditions than for periodic boundary conditions.
\cite{whit.93,skne.97}
In this work the {\sl finite system} DMRG method is used: after the system
is built up to a given size using a variation of the {\sl infinite system}
method, a number of finite-system iterations are performed in which the
overall size of the system (i.e.\ the superblock) is kept fixed, but part
of the system (the system block) is built up. Optimal convergence is
attained by increasing the number of states kept on each iteration, and the
convergence of the exact diagonalization step is improved by keeping track
of the basis transformations and using them to construct a good initial
guess for the wavevector.\cite{whit.96} For all calculations shown in this
paper, we have performed 5 iterations with a maximum of $m=600$ states
kept. The resulting discarded weight of the density matrix was ${\cal
O}(10^{-7})$ and below.
We illustrate the convergence of the algorithm explicitly in Figs.
\ref{fig:delta_egz}--\ref{fig:delta_m1}. One finds that for all the
parameters which are used in this paper the ground-state energy per site is
accurate to ${\cal O}(10^{-6})$ or less, Fig.\ \ref{fig:delta_egz}, while
the expectation value of the density at the first site (or at the last
site, due to symmetry) is accurate to ${\cal O}(10^{-5})$, as can be seen
in Fig.\ \ref{fig:delta_n1}.
\narrowtext
\begin{table*}
\caption{Exponents $\gamma_{\alpha,c}$ and $\gamma_{\alpha,s}$ as
a function of the elements of the dressed charge matrix.
\label{tab:gamma}}
\renewcommand{\arraystretch}{1.2}
\begin{tabular}{ccc}
& $\gamma_{\alpha,c}$ & $\gamma_{\alpha,s}$ \\
\tableline
$\alpha=\downarrow$ & $(Z_{cc}-Z_{sc})^2$ & $(Z_{cs}-Z_{ss})^2$ \\
$\alpha=\uparrow$ & $Z_{sc}^2$ & $Z_{ss}^2$ \\
$\alpha=n$ & $Z_{cc}^2$ & $Z_{cs}^2$ \\
\end{tabular}
\end{table*}
\begin{figure}
\vbox{%
\centerline{\epsfig{file=fig01.eps,width=8cm}} \narrowtext
\caption{\label{fig:delta_egz}
The difference between the ground state energy per site calculated with
the DMRG, $E_0^{D}$, and the exact BA-energy, $E_0^{B}$, as a function of
the number of DMRG iterations for a typical system ($L=400$, $n_e=0.55$,
$n_\uparrow=0.35$, $U=10$ and $p=0$).}}
\end{figure}
\begin{figure}
\vbox{%
\centerline{\epsfig{file=fig02.eps,width=8cm}}
\caption{\label{fig:delta_n1}
The difference between the expectation value of the density at site 1
calculated with DMRG, $N_1^{D}$, and with BA, $N_1^{B}$, for the fillings
$n_e=0.55$ and $n_e=0.70$ without boundary fields.}}
\end{figure}
The magnetization shows an analogous behaviour but with results correct up
to ${\cal O}(10^{-4})$, Fig.\ \ref{fig:delta_m1}. It is also interesting
to note that for $U\le 10$ there is no strong $U$-dependence of the
accuracy of either the density or the magnetization expectation values.
We now want to examine the effect of switching on the boundary fields
simultaneously at the first and last sites on the quality of the DMRG
results. In order to do this, we compare the mean density $\langle N_1(p)
\rangle$ from DMRG calculations with Bethe Ansatz results calculated in the
thermodynamic limit. Within the Bethe Ansatz, the mean density is
calculated from the derivative of $\langle {\cal H} \rangle$ with respect
to the boundary field $p$.
\begin{figure}
\vbox{%
\centerline{\epsfig{file=fig03.eps,width=8cm}}
\caption{\label{fig:delta_m1}
The difference between the expectation value of the magnetization at site
1 calculated with DMRG, $M_1^{D}$, and with BA, $M_1^{B}$, for $n_e=0.55$
and $n_e=0.70$, without boundary fields.}}
\end{figure}
\begin{figure}
\vbox{%
\centerline{\epsfig{file=fig04.eps,width=8cm}}
\caption{\label{fig:n1ofp}
The expectation value of $\langle N_1\rangle$ for different values of the
interaction $U$ and boundary fields $p$ on a $L=100$ lattice with
$n_e=0.55$ and $n_\uparrow=0.35$. The solid lines represent the exact
solutions in the thermodynamic limit, while the symbols are DMRG
results.}}
\end{figure}
The numerical results, shown in Fig.\ \ref{fig:n1ofp} on an $L=100$ lattice
for electron density $n_e=0.55$, $n_\uparrow=0.35$ and for two values of
the interaction $U$, are again in very good agreement with the values for
the thermodynamic limit. The difference $N_1^{\rm D}-N_1^{\rm B}$ is now
${\cal O}(10^{-3})$. While this seems to be worse than the $p=0$ case, we
have neglected finite-size corrections to $\langle N_1 \rangle$ since we
have compared to thermodynamic limit Bethe Ansatz calculations. If we
explicitly take the $1/L$ corrections to the Bethe Ansatz values into
account, we find agreement to ${\cal O}(10^{-5})$. This value is already
smaller than the $1/L^2$-corrections so that finally we state that $\langle
N_1 \rangle$ is accurate to ${\cal O}(10^{-4})$.
\section{Friedel Oscillations}
\label{sec:fr_osc}
In general, the presence of an impurity or boundary in a one-dimensional
fermion system leads to Friedel oscillations in the density, which have the
general form
\begin{equation}
\delta \rho(x) \sim {\cos(2k_F x+\varphi) \over x^\gamma} \ ,
\label{eq:drho}
\end{equation}
where the exponent $\gamma$ depends on the interaction. In addition to
numerical studies of these oscillations for spinless fermions
\cite{schmeck:96} and Kondo--Systems \cite{shueni:96}, several theoretical
attempts have been made to clarify the role of interaction.
Using bosonization it is possible to obtain the asymptotic exponents as a
function of the interaction parameters and corrections to the power--law
behaviour of Eq.\ (\ref{eq:drho}).\cite{fago:95,eggra:95} CFT results were
used to calculate the interaction dependence of the exponent $\gamma$ for
interacting spinless fermions. \cite{wavp:96}
Here we start with noninteracting fermions to obtain some conjectures for
the connection between the explicit form of the Friedel oscillations and
Bethe Ansatz results. These conjectures will then be checked using the
DMRG results.
\subsection{Noninteracting Fermions}
By considering only spin-$\uparrow$ electrons without any boundary
potential, one can easily obtain the expectation value of the electron
density:
\begin{equation}
N(x)={N_\uparrow+{1 \over 2} \over L+1}-{1 \over 2 (L+1)}
\frac{\sin \left(2 \pi x {N_\uparrow+1/2 \over L+1} \right)}
{\sin\left({\pi x \over L+1}\right)}.
\end{equation}
In the limit $L \to \infty$ and $x \ll L$ the density is given by
\begin{eqnarray}
N(x) &=&n_\uparrow-{n_\uparrow-{1 \over 2} \over L}-{\sin \left(2 \pi x
(n_\uparrow-{n_\uparrow-{1 \over 2} \over L}) \right) \over 2 \pi x}
\nonumber \\
&=& n_\uparrow-{\theta^c_0 \over L}-{\sin \left(2 \pi x
(n_\uparrow-{\theta^c_0 \over L}) \right) \over 2 \pi x} ,
\label{free}
\end{eqnarray}
with $\theta^c_0$ defined in Eq.\ (\ref{eq:tpc1}). The density $N(x)$ can
also be calculated explicitly when the boundary field $p=1$. It then has
the same structure as Eq.\ (\ref{free}).
If one assumes that the Friedel oscillations in the interacting system have
an analogous structure to those in the non-interacting system, one can
combine Eqs.\ (\ref{dopen}) and (\ref{free}) to obtain the following
conjectures for the finite-size shifts of the average density, average
magnetization and the characteristic wave vectors in the {\sl interacting}
system:
\begin{equation}
\overline{n}=n_e-{\theta_n \over L}, \qquad
\overline{m}=m-{\theta_m \over L}
\label{eq:nmc}
\end{equation}
\begin{equation}
k_{F,\downarrow}=\pi \left(n_\downarrow-{\theta_\downarrow \over L}\right),
\qquad
k_{F,\uparrow}=\pi \left(n_\uparrow-{\theta_\uparrow \over L}\right)
\label{eq:kc}
\end{equation}
with $\theta_\downarrow=\theta^s_p, \theta_\uparrow= \theta^c_p-\theta^s_p,
\theta_n=\theta^c_p$ and $\theta_m= \theta^c_p-2\theta^s_p$.
\subsection{Fit procedure}
\label{sec:fit_meth}
Previously, several methods have been used to obtain asymptotic exponents
of correlation functions using numerical
data.\cite{sopa:90,qili:95,ogasushi:91} All of these methods use the
$L$--dependence of the fourier-transformed correlation functions near the
relevant peaks $k_\alpha$ $(\alpha=\uparrow,\downarrow,n)$ in Fourier space
to extract the exponents. Due to the fact that only systems with periodic
boundary conditions were considered, the $k_\alpha$ were all independent of
$L$. This $L$-independence seems to be crucial for these methods to work;
we were not able to extract a reasonable exponent with any of these methods
on a system with open boundary conditions.
Therefore, we fit the DMRG results for $N(x)$ and $M(x)$ to the real-space
test function
\begin{eqnarray}
\label{eq:fitfunc}
f(x)=\left\{ \begin{array}{c} \overline{n} \\
\overline{m} \end{array} \right\}
+\sum_{\alpha \in \{\uparrow,\downarrow,n\}}
{A_\alpha \sin(2 k_\alpha x +\varphi_\alpha) \over x^{\gamma_\alpha}}
\nonumber \\
+{A_\alpha \sin(2 k_\alpha x +\varphi_\alpha) \over (L+1-x)^{\gamma_\alpha}}
\end{eqnarray}
which explicitly includes the momenta as fit parameters. Here the second
term is included due to symmetry. There are a total of 13 fit parameters
in this function, a prohibitively large number to do a simultaneous fit of
all parameters. However, if we only consider systems in which the three
peaks in the Fourier--spectrum are well seperated, there is an effective
fit of 4 parameters to every peak. As we will see, the peak at $k_n$ is
supressed for small $U$, reducing the number of fit parameters to 9 for
only two peaks. The amplitudes $A_\alpha$ will be assumed to be positive,
with any sign given by the phase $\varphi_\alpha$. We fit to the
magnetization $M(x)$ and the density $N(x)$ independently.
The right side of Eq.\ (\ref{free}) is only valid for $x \ll L$. In
addition, the CFT results are valid only asymptotically for large
distances. As a consequence and compromise, we do not use the density
information on the first five and last five lattice points.
We perform the least squares fit in two stages. In the first stage, the
start parameters of the subsequent fit are determined using simulated
annealing. The final fit is performed using a combined Gauss--Newton and
modified Newton algorithm (using the NAG routine E04FCF). To estimate the
fit error, 10 fits are performed for each system with 10\% of the points
randomly excluded from each fit.
\subsection{Results}
\label{sec:results}
Before discussing the results for the Friedel oscillations in detail, we
make some general comments on the numerical results. As described in Sec.\
\ref{sec:fit_meth}, we calculate the quantities for the density and
magnetization oscillations by applying a 13-parameter fit. Fitting to this
many parameters requires the use of large system sizes. While the
numerical expense for the DMRG procedure grows linearly with the system
size for a fixed number of states kept, the accuracy in the energy and in
the local density and magnetization decreases with the system size,
especially in the Luttinger liquid regime. \cite{skne.97} We have compared
the accuracy of the DMRG results with the accuracy and convergence of the
fitting procedure for different lattice sizes from $L=300$ to $L=500$ and
have decided that $L=400$ yields optimal results for the amount of
computing power available. However, results within this range of system
sizes are in agreement to within DMRG and fitting errors.
Another important issue is the influence of the boundary potentials on the
fitting method and on the Friedel oscillations (discussed in Sec.\
\ref{sec:ne55p}). As the boundary potential $p$ is increased, bound states
will develop at site 1 and site $L$.\cite{befr:97} In order to avoid these
bound states in the fitting procedure, one has to enlarge the range in
which the local density $N(x)$ is disregarded from 5 (i.e. $x, L-x-1 \le
5$) at $p=0$ to about 20 at $p=9.9$.
The discussion of the next three sections will focus on comparing the
BA/CFT predictions for the different fit parameters with the DMRG results,
especially on checking the conjectures from Eqs.\ (\ref{eq:nmc}) and
(\ref{eq:kc}). We also compare the numerical results to the different
exactly known values for different limits such as the limit of
noninteracting fermions, $U\to 0$.
\subsubsection{Density $n_e=0.55$}
\label{sec:ne055}
We first examine the Fourier transform of the local electron density
$N(x)$, defined as
\begin{equation}
N(k,U)=\sum_{x=1}^L \cos\left(k(x-{1 \over 2})\right) N(x,U)
\label{eq:ft}
\end{equation}
with $k={ 2 \pi j \over L}$ and $j=0,\ldots,{L \over 2}-1$. (Due to
symmetry $N(k,U)$ vanishes for odd multiples of ${\pi \over L}$.) The
quantity $N(k,U)$ is displayed in a three-dimensional plot in Fig.\
\ref{fig:ftne55}(a). Distinct peaks at the three wave vectors,
$k_\uparrow$, $k_\downarrow$, and $k_n$ can clearly be seen. Note that we
have chosen $n_e$ and $n_\uparrow$ so that these three peaks are
well-separated. However, the peak at $k_n$ becomes very lightly weighted
and therefore ill-defined for $U<1$. In fact, we have found that it is not
possible to locate the third momentum $k_n$ for $U<1$ using the
13-parameter fit procedure described in Sec.\ \ref{sec:fit_meth}.
Therefore, we fit $N(x)$ using only 9 parameters for $U=0.1$ and $U=0.5$.
We display the fourier-transformed magnetization $M(k,U)$, defined
analogously, in Fig.~\ref{fig:ftne55}(b). Here the $k_n$ peak is even more
\end{multicols}
\widetext
\begin{figure}
\epsfxsize=0.9\textwidth \epsfbox{fig05.eps}
\caption{Fourier transformation of (a) the density $N(x)$ and (b) the
magnetization $M(x)$ for several values of $U$ at the density $n_e=0.55$
and $n_\uparrow=0.35$. (The $k=0$ values are excluded.)
\label{fig:ftne55}}
\end{figure}
\begin{multicols}{2}
\narrowtext
\begin{figure}
\vbox{%
\centerline{\epsfig{file=fig06.eps,width=8cm}}
\caption{
Difference $\theta_\alpha=(\alpha-\overline{\alpha})L$ for the density
($\alpha=n$) and the magnetization ($\alpha=m$) as a function of $U$ for
$n_e=0.55$. The solid lines are the Bethe Ansatz conjectures
$\theta_\alpha$.
\label{fig:mean55}
}}
\end{figure}
\noindent%
poorly defined, and, in fact, is at best barely discernible, even at large
$U$. Therefore, it is only possible to fit to two peaks using 9 parameters
in the entire region from $U=0$ to $U=10$. For these fit procedures, we
have found that the mean-squared deviation $\sigma^2 =
\sum_{x}(N(x)-f(x))^2$ is between ${\cal O} (10^{-7})$ and ${\cal
O}(10^{-6})$ for the density and between ${\cal O}(10^{-6})$ and ${\cal
O}(10^{-5})$ for the magnetization for all $U$ values. These limits will
apply to all the fit results shown in this paper.
In Fig.\ \ref{fig:mean55} we show the $1/L$ corrections of the mean values
$\overline{n}$ and $\overline{m}$ calculated with the DMRG and from Bethe
Ansatz using the conjectures in Eq.\ (\ref{eq:nmc}). One can see that
there is quite good agreement between the two calculations for all $U$.
The comparison of the exact asymptotic exponents at the different momenta
with the numerical results is one of the most interesting and important
features of this work because similar methods will then be able to be used
to calculate properties not directly predictable with CFT/BA, and to treat
systems that are not Bethe Ansatz solvable.
The exponents at $k_\downarrow$ and $k_\uparrow$, extracted from the
density as well as from the magnetization data, are shown in
Fig.~\ref{fig:exp55}. The difference between the fit--exponents and the
CFT-prediction is less than 2\% for the fits to the density and less than
3\% for the fits to the magnetization. As mentioned above, we can obtain
an exponent for the peak at $k_n$ from the density fit for $U \ge 1$ only,
due to its small weight especially for small $U$. At $U=1$ the large error
bars reflect a poor fit. Here we have only considered $U \le 10$, for
which $\gamma_n>\gamma_\uparrow$. A further increase in $U$ would lead to
a region where $\gamma_n<\gamma_\uparrow$. A crossover between these two
regions will be seen for the $n_e=0.70$ in the next section, for which it
occurs at a somewhat lower $U$.
\begin{figure}
\vbox{%
\centerline{\epsfig{file=fig07.eps,width=8cm}}
\caption{
Exponents for the three peaks as a function of $U$ fitted for (a) the
particle density and (b) the two peaks of the magnetization for
$n_e=0.55$. The solid lines denote the CFT predictions from the Bethe
Ansatz solution.
\label{fig:exp55}
}}
\end{figure}
Due to the fact that we obtain only three independent exponents from the
fitting procedures, it is not possible to determine all of the elements of
the dressed charge matrix ${\bf Z}$. In fact, only the following
combinations are relevant for the three exponents extracted:
$Z_{cc}^2+Z_{cs}^2,Z_{ss}^2+Z_{sc}^2$ and $Z_{cc}Z_{sc}+Z_{cs}Z_{ss}$.
\cite{ogasushi:91} It would be possible to uniquely determine all of the
independent elements of the dressed charge matrix with additional
information from, for example, any susceptibility \cite{frko:90} or from
another correlation function with a different set of critical exponents.
Relationships between the elements of the dressed charge matrix ${\bf Z}$
and the parameters of the Tomonaga--Luttinger model are given in Ref.
\onlinecite{peso:93}.
Within the framework of CFT, the amplitudes $A_\alpha$ are completely
undetermined. However, the form--factor approach \cite{less:97} may lead
to explicit results for the amplitudes in the future. For example, a
conjecture of Lukyanov and Zamolodchikov \cite{lukyza:97} concerning the
amplitude of the spin--spin correlation functions of the XXZ chain was
recently confirmed by a fit to DMRG results. \cite{hifu:98}
At this point, however, the fit results can only be compared to
noninteracting fermions ($U=0$), for which $A_\downarrow=A_\uparrow={1
\over 2 \pi}$. As can be seen in Fig.~\ref{fig:koeff55}, this value is
in relatively good agreement (4\% deviation) with the fit results. In
addition, $A_n=0$ for $U=0$, in agreement
\begin{figure}
\vbox{%
\centerline{\epsfig{file=fig08.eps,width=8cm}}
\caption{
Amplitudes $A_\alpha$ fitted to (a) the density and (b) the magnetization
as a function of $U$ for $n_e=0.55$. The solid lines are guides to the
eye. The arrow denotes the value for the noninteracting fermions (=$1
\over 2 \pi$). Note that the amplitudes are, in general, different for
the density and the magnetization.
\label{fig:koeff55}
}}
\end{figure}
\noindent%
with the $U=0$ extrapolated value in Fig.\ \ref{fig:koeff55}(a). The large
error bars in $A_n$ at $U=1$ are due to the difficulty in fitting the $k_n$
peak for small $U$. Since the fitting procedure seems to work well for the
exponents, and the amplitudes yield the correct $U=0$ limit, we feel that
the calculation of the amplitudes is under good control. This is therefore
the first determination of the qualitative as well as quantitive behaviour
of these amplitudes.
The exact position of the momenta $k_\uparrow$ and $k_\downarrow$ are
further fit parameters. The $1/L$ corrections to the {\it thermodynamic
value} are plotted in Fig.~\ref{fig:mom55}. The fit values agree well
with the Bethe Ansatz conjectures, except for the correction to
$k_{\uparrow,m}$, which deviates from the Bethe Ansatz value for $U<4$.
These deviations are probably due to problems with the fit. Note also that
the Bethe Ansatz results for $\theta_\alpha$ are correct only up to ${\cal
O}(1/L)$. The momentum $k_n$, which is not shown, is another independent
fit parameter. The fit error in $k_n$ extracted from the fit to the
density $N(x)$ is rather large for $U<4$. This is due to the fact that the
peak at $k_n$ is not well--defined enough in this region to obtain the
$1/L$ corrections to this momentum. Nevertheless the agreement between the
fit values and the Bethe Ansatz conjectures is very good for%
\begin{figure}
\vbox{%
\centerline{\epsfig{file=fig09.eps,width=8cm}}
\caption{
Difference $\theta_{\alpha,\beta}=(n_\alpha-{k_{\alpha,\beta} \over
\pi})L$ for $\alpha=\downarrow,\uparrow$ as a function of $U$ for the
density $n$ and magnetization $m$ for $n_e=0.55$. The solid lines denote
the Bethe Ansatz conjectures $\theta_\alpha$ [see Eq.\ (\ref{eq:kc})].
\label{fig:mom55}
}}
\end{figure}
\noindent
$U>4$.
\subsubsection{Density $n_e=0.70$}
\label{sec:ne070}
In this section, we examine the same quantities as in the previous section
at a density of $n_e=0.70$. A treatment of this density is interesting for
a number of reasons. Since we use the same numerical parameters for both
densities, we can examine the density dependence of the error in truncating
the Hilbert space using the DMRG. This density is also interesting because
the BA/CFT calculations predict that the crossover between the $\gamma_n$
and $\gamma_\uparrow$ exponents will take place within the range of
interaction treated here, $U=0\ldots 10$.
The Fourier transforms of the local density $N(x)$ and the magnetization
$M(x)$ are shown in Fig.~\ref{fig:ftne70}. Note that the momentum $k_n$
(wrapped back to the range $0$ to $\pi$) is now located between
$k_\downarrow$ and $k_\uparrow$. As can be seen in
Fig.~\ref{fig:ftne70}(a), the peak in $N(x)$ at $k_n$ is not well-defined
for $U<1$, so the $U=0.1$ and $U=0.5$ data are fitted using 9 parameters to
fit two peaks. However, the $k_n$ peak in $M(x)$ is now more well-defined
than for $n_e=0.55$, as can be seen in Fig.~\ref{fig:ftne70}(b), and it is
now possible to fit to all three momenta for $U \ge 2$. For smaller values
of $U$ ($U=0.1$, $U=0.5$ and $U=1$), a nine-parameter fit is again made to
two peaks.
The $1/L$ corrections to the mean values of the density and the
magnetization, shown in Fig.~\ref{fig:mean70}, are in very good agreement
with the BA conjectures, thereby providing a further confirmation of the
predictions of Eqs.\ (\ref{eq:nmc}) and (\ref{eq:kc}). The exponents
extracted from the fit are shown in Fig.~\ref{fig:exp70}. The expected
crossing of the two largest exponents at $U \approx 7$ can clearly be seen.
For $\gamma_\downarrow$ and $\gamma_\uparrow$ obtained from the density
fit, Fig.~\ref{fig:exp70}(a), the deviation from the CFT results is about
5\% at most, with the largest errors occuring for $U>6$, especially in
$\gamma_\uparrow$. As can be seen in Fig.~\ref{fig:ftne70}(a), the peak at
$k_\uparrow$ in $N(k,U)$ gets weaker for
\end{multicols}
\widetext
\begin{figure}[ht]
\epsfxsize=0.9\textwidth \epsfbox{fig10.eps}
\caption{Fourier transformation of (a) the density $N(x)$ and (b) the
magnetization $M(x)$ for several values of $U$ for the density $n_e=0.70$
with $n_\uparrow=0.55$ ($k=0$--values excluded). Note that the momentum
$k_n$ is located between $k_\downarrow$ and $k_\uparrow$.
\label{fig:ftne70}}
\end{figure}
\begin{multicols}{2}
\narrowtext
\noindent%
larger $U$, leading to a less effective fit. The agreement of the fitted
exponents $\gamma_n$ with the CFT predictions is much better, with a
deviation from the BA/CFT values of less than 1\% for $U>1$. The exponents
obtained from fits to the magnetization, Fig.~\ref{fig:exp70}(b), show
deviations of up to about 6\% from the BA/CFT results. For $U=9$ and
$U=10$, the deviation is largest and the exponents coincidentally take on
the same value. Again, this is probably due to larger errors in the fit
because the peak at $k_\uparrow$ becomes weaker at larger $U$. One can see
that the peaks at $k_\uparrow$ and $k_n$ are much less heavily weighted
than the peak at $k_\downarrow$ at large $U$.
The amplitudes $A_\downarrow$ and $A_\uparrow$ extracted from the fit to
the density, shown in Fig.~\ref{fig:koeff70}(a), decrease monotonically
with increasing $U$. The $U=0$ values agree with the exactly known value
of $1/2\pi$ to within about 4\%. The amplitude $A_n$, on the other hand,
increases with increasing $U$. Its $U\rightarrow 0$ extrapolation agrees
well with the value zero of the noninteracting fermions if the $U=1$ point,
which cannot be very accurately determined, is excluded. The behaviour and
even the quantitative values of all three coefficients are quite similar to
the $n_e=0.55$ case shown previously in Fig.~\ref{fig:koeff55}(a). The
amplitudes obtained from the fit to the magnetization $M(x)$ are shown in
Fig.~\ref{fig:koeff70}(b). The amplitude $A_\downarrow$ behaves similarly
to the $n_e=0.55$ case [Fig.~\ref{fig:koeff55}(b)] in that it increases
with increasing $U$, but $A_\uparrow$ shows different behaviour in that it
reaches a maximum at $U\approx 1$ and then decreases. Both fit amplitudes
yield the $U=0$ value of $1/2\pi$ to within 6\%. The amplitude $A_n$ for
the summed momenta, which could not be determined for $n_e=0.55$, increases
monotonically with $U$, and its $U\rightarrow 0$ extrapolation agrees well
with the value for noninteracting fermions, $A_n = 0$.
The $1/L$ corrections to the momenta $k_\downarrow$ fitted for the density
and the magnetization, shown in Fig.~\ref{fig:mom70}, are in good agreement
with the Bethe Ansatz conjecture. The agreement is also fairly good for
$\theta_{\uparrow,n}$, although the error of the fit is rather large for
small $U$. However, the $\theta_{\uparrow,m}$ fit results do not match well
with the conjecture. As we have seen in Fig.~\ref{fig:ftne70}(b), the
$k_n$ and $k_\downarrow$ peaks in $M(x)$ have much lower amplitudes than
the $k_\uparrow$ peak, leading to lower accuracy in the fitting procedure.
The fit results%
\begin{figure}
\vbox{%
\centerline{\epsfig{file=fig11.eps,width=8cm}}
\caption{
Difference $\theta_\alpha=(\alpha-\overline{\alpha})L$ for density
($\alpha=n$) and magnetization ($\alpha=m$) as a function of $U$ for
$n_e=0.7$. The solid lines are the Bethe Ansatz conjectures
$\theta_\alpha$.
\label{fig:mean70}
}}
\end{figure}
\begin{figure}
\vbox{%
\centerline{\epsfig{file=fig12.eps,width=8cm}}
\caption{
Exponents for the three peaks of (a) the particle density and (b) the
magnetization as a function of $U$ for $n_e=0.7$. The solid lines denote
the CFT predictions from the Bethe Ansatz solution.
\label{fig:exp70}
}}
\end{figure}
\noindent%
for the exponents, Fig.~\ref{fig:exp70}(b) also had a rather large
deviation from the CFT results in this regime. Thus it is not possible to
confirm or deny the conjecture concerning the shift of $k_\uparrow$ for
$M(x)$.
For $k_n$ the situation is even worse. Both density and magnetization fits
lead to large fit errors for $U<4$. The deviation from the Bethe Ansatz
conjectures is about $0.08$ in both fits, outside the range of the $1/L$
correction to $\theta_n$.
In summary, for $n_e=0.70$ the DMRG results for the mean density
(magnetization, respectively) and the exponents are in good agreement with
exact results from BA/CFT, further confirming Conformal Field Theory.
A detailed examination of the convergence of the DMRG shows that the
numerical accuracy is actually slightly worse than the $n_e=0.55$ case, but
this could be improved by increasing the number of states kept in the DMRG.
We therefore expect to be able to apply these techniques reliably to obtain
the boundary exponents and coeffients of other, non-Bethe-Ansatz solvable
one-dimensional models.
\begin{figure}
\vbox{%
\centerline{\epsfig{file=fig13.eps,width=8cm}}
\caption{
Amplitudes $A_\alpha$ fitted to (a) the density and (b) the magnetization
as a function of $U$ for $n_e=0.7$. The solid lines are a guide to the
eye. The arrow denotes the value $1\over 2 \pi$ of the noninteracting
fermions. Note that the amplitudes are, in general, different for the
density and the magnetization.
\label{fig:koeff70}
}}
\end{figure}
\begin{figure}
\vbox{%
\centerline{\epsfig{file=fig14.eps,width=8cm}}
\caption{
Difference $\theta_{\alpha,\beta}=(n_\alpha-{k_{\alpha,\beta} \over
\pi})L$ for $\alpha=\downarrow,\uparrow$ as a function of $U$ for the
density $n$ and magnetization $m$ for $n_e=0.7$. The solid lines denote
the Bethe Ansatz conjectures $\theta_\alpha$ [see Eq.\ (\ref{eq:kc})].
\label{fig:mom70}
}}
\end{figure}
\subsubsection{Effect of boundary potentials}
\label{sec:ne55p}
We now examine the effect of the boundary potential $p$ on the Bethe Ansatz
predictions. We set $n_e=0.55$, $n_\uparrow=0.35$, and $U=8$ and consider
six $p$ values, for which four qualitatively different Bethe Ansatz
solutions exist. For $p=-0.5,\, 0,\, 0.5$, no bound state is present in
the Bethe Ansatz ground state; we examine both repulsive and attractive
fields. The values $p=2.6$ and $p=6$ are in a region in which the Bethe
Ansatz has two complex $k$ solutions, one for each boundary potential at
site $x=1$ and $x=L$. The $p=6$ ground state configuration contains, in
addition, two complex $\lambda$ solutions. Finally, for $p=9.9$ there are
four complex $k$ and two complex $\lambda$ solutions, corresponding to a
bound pair of electrons at each end of the chain. Details of the structure
of the ground state as a function of $p$ are given in Ref.\
\onlinecite{befr:97}.
The $p$-dependence of $N(k,p)$ and $M(k,p)$ at $U=8$ is not as strong as
the $U$-dependence of $N(k,U)$ and $M(k,U)$ found previously. Since we
have chosen a fairly large $U$, the peaks in the density $N(x)$ have enough
weight to fit all three momenta.
As was the case for $p=0$, the peak at $k_n$ in the magnetization is not
pronounced enough to be fitted at any $p$.
The $1/L$ corrections to the mean values of the density and magnetization,
Fig.~\ref{fig:meanp55}, are again in very good agreement with the Bethe
Ansatz conjectures, showing that Eqs.\ (\ref{eq:nmc}) and (\ref{eq:kc}) are
valid even in the different physical regions described above. Within the
BA/CFT calculations, the values of the exponents are independent of the
boundary potentials $p$. This agrees with the DMRG results, which we do
not show here: the range of the exponents varies by at most 2\% from the
exact values for $p=0$ (after a larger number of lattice points are
discarded from the fits in order to avoid the bound states at the ends).
The amplitudes also have no significant $p$-dependence. The density fit
yields $A_\downarrow \approx A_\uparrow \approx 0.045$ and $A_n \approx
0.12$,%
\begin{figure}
\vbox{%
\centerline{\epsfig{file=fig15.eps,width=8cm}}
\caption{
Difference $\theta_\alpha=(\alpha-\overline{\alpha})L$ for density
($\alpha=n$) and magnetization ($\alpha=m$) as a function of $p$ for
$U=8$, $n_e=0.55$ and $n_\uparrow=0.35$. The solid lines are the Bethe
Ansatz conjectures for $\theta_\alpha$.
\label{fig:meanp55}
}}
\end{figure}
\begin{figure}
\vbox{%
\centerline{\epsfig{file=fig16.eps,width=8cm}}
\caption{
Difference $\theta_{\alpha,\beta}=(n_\alpha-{k_{\alpha,\beta} \over
\pi})L$ for $\alpha=\downarrow,\uparrow,n$ as a function of $p$ for the
density $n$ and magnetization $m$ for $U=8$, $n_e=0.55$ and
$n_\uparrow=0.35$. The solid lines denote the Bethe Ansatz conjectures
$\theta_\alpha$ [see Eq.\ (\ref{eq:kc})].
\label{fig:mom55p}
}}
\end{figure}
\noindent%
while the magnetization fit yields $A_\downarrow \approx 0.28$ and
$A_\uparrow\approx 0.21$. The absence of $p$-dependence at $U=8$ suggests
that the interaction dependence of the coefficients should be that of Fig.\
\ref{fig:koeff55}, independent of the boundary fields $p$.
The effect of the boundary potential $p$ on the shift of the positions of
the peaks is much larger than the effect of varying $U$ (compare
Fig.~\ref{fig:mom55p} with Fig.~\ref{fig:mom55}). Since the fit results
for all three $k$ values agree very well with the Bethe Ansatz conjectures,
the confirmation of Eq.~(\ref{eq:kc}) is even more compelling than it was
for the $U$--dependence.
\section{Summary and Conclusions}
\label{sec:Concl}
We have carried out a detailed comparison between the exact Bethe Ansatz
solution and Density Matrix Renormalization Group calculations for the
one-dimensional Hubbard model with open boundary conditions both with and
without an additional chemical potential at both ends. A direct comparison
of the ground state energies as well as the density and magnetization at
the ends of the chain has allowed us to estimate the accuracy of the DMRG
on the large system sizes used in this work.
We have then compared the behaviour of the Friedel oscillations in the
local density and local magnetization calculated directly using the DMRG
with Conformal Field Theory predictions for the asymptotic forms for which
the exponents can be calculated using the Bethe Ansatz. We have performed
this check for two different fillings, $n_e = 0.55$ and $n_e = 0.70$ for
the case without boundary potentials, $p = 0$. We have obtained results
consistent with the CFT predictions in all cases except those in which it
is clear that the accuracy of the fitting procedure breaks down. Such a
breakdown occurs when a particular peak in the fourier transform of the
density or magnetization becomes lightly weighted and thus poorly defined.
This occurs principally for the $k_n=(k_{F,\uparrow}+k_{F,\downarrow})$
peak, especially at small $U$ values. The good agreement between the CFT
forms and BA values of the exponents and the DMRG calculations provides
both a confirmation of the CFT predictions and a way to test the accuracy
of the DMRG and of the effectiveness of fitting procedures for the Friedel
oscillations.
In addition, we have proposed a relation between the $1/L$ corrected mean
values in the density and magnetization and the $1/L$ corrections occuring
in the BAE. This conjecture is supported by good agreement between mean
values obtained from the fit to the DMRG data and the BAE results.
We have been able to extract for the first time the interaction dependence
of the amplitudes for the Friedel oscillations, a property not possible to
calculate in the framework of the CFT, and have found the correct behaviour
in the $U\to 0$ limit.
Finally, we have turned on boundary chemical potentials at $n_e=0.55$ and
examined the $p$-dependence of the critical exponents, the amplitudes and
our conjectures for the behaviour of the mean density and magnetization.
The different $p$-regimes that we have considered yield qualitatively
distinct Bethe Ansatz solutions that are physically connected with the
formation of different types of bound states at the system boundaries. In
agreement with BA/CFT predictions, we have found that the critical
exponents are independent of $p$ and that the influence of $p$ on the
amplitudes is very weak. We have also found that our conjectures for the
$1/L$ corrected mean values of the density, magnetization and the
wave-vector hold in all of the physically different $p$-regimes.
The combination of analytical and numerical methods presented here has
yielded new insights into both. The success of the numerical techniques
will now allow the examination of more complicated systems that are not
exactly solvable. Through comparison with the DMRG calculations, we have
also been able to show that more information is contained in the BAE than
is obtained from a direct interpretation via Conformal Field Theory.
\acknowledgments
We would like to thank J.\ Voit for helpful discussions.
This work has been supported by the Deutsche Forschungsgemeinschaft under
Grant No.\ Fr~737/2--2 (G.B.\ and H.F.) and by the Deutsche
Forschungsgemeinschaft under Grant No.\ Ha~1537/14--1 (B.B.). R.M.N.\ was
supported by the Swiss National Foundation under Grant No.\ 20-46918.96.
\end{multicols}
\widetext
\begin{multicols}{2}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 958 |
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The Sound… The Feel… The Vibe… and The Look… Scotty produces some of the best consistently exceptional finishes and instruments that you can get. A real treasure! | {
"redpajama_set_name": "RedPajamaC4"
} | 5,034 |
Versuch steht für:
Ausprobieren einer von mehreren Möglichkeiten, siehe Versuch und Irrtum
Experiment, in der wissenschaftlichen Betätigung (präziser: Durchführung eines Experiments)
Versuch (StGB), eine unvollendete Straftat in der deutschen Rechtswissenschaft
Versuch (italienisches Strafrecht), ein Deliktsstadium im Strafrecht Italiens
Versuch (Rugby), in der Sportart Rugby eine Art, Punkte zu erzielen
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Siehe auch:
Versuchung | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,906 |
package me.kevinthorne.MQTTBlocks.blocks;
import java.io.File;
import java.io.IOException;
import java.net.URL;
import java.net.URLClassLoader;
import java.util.Date;
import java.util.jar.JarFile;
import me.kevinthorne.MQTTBlocks.BlockManager;
public class BlockLoader extends MQTTBlock {
private Date started;
private boolean firstLoop = true;
@Override
public void onEnable() {
started = new Date();
BlockManager.logInfo(this, "Block Loader Daemon Started");
}
@Override
public void onDisable() {
BlockManager.logInfo(this, "Block Loader Daemon Stopped");
}
public void run() {
this.name = getBlockConfig().getName();
try {
onEnable();
setRunning(true);
} catch (Exception e) {
e.printStackTrace();
}
while (isRunning()) {
update();
try {
Thread.sleep(getSleepTime() * 100);
} catch (InterruptedException ignored) {
}
}
try {
onDisable();
} catch (Exception e) {
e.printStackTrace();
}
}
@Override
public void update() {
// ComponentManager.logInfo(this, "Checking for new components...");
File[] jars = BlockManager.blockLocation.listFiles();
for (File jar : jars) {
// System.out.println("[started: " + started.toString()+ "] - " + jar.getName() + " - ["+new
// Date(jar.lastModified()).toString()+"]");
// System.out.println(started.before(new Date(jar.lastModified())));
if (started.before(new Date(jar.lastModified())) || firstLoop) {
firstLoop = false;
// System.out.println("Found jar file after start date, adding");
try {
JarFile jarFile = new JarFile(jar);
BlockConfigurationFile config;
try {
config = new BlockConfigurationFile(
jarFile.getInputStream(jarFile.getJarEntry("config.properties")));
} catch (NullPointerException ne) {
logError("Couldn't find config.properties for \"" + jar.getName() + "\"");
continue;
}
// System.out.println("Loaded config");
URL[] urls = {new URL("jar:file:" + jar + "!/")};
URLClassLoader cl = URLClassLoader.newInstance(urls);
Class<?> jarClass;
try {
jarClass = Class.forName(config.getMain(), true, cl);
} catch (ClassNotFoundException ex) {
logError("Couldn't find main for \"" + jar.getName() + "\" reload");
continue;
} catch (NullPointerException ne) {
logError("Couldn't find main for \"" + jar.getName() + "\" reload");
logError("\tconfig Main Method: " + config.getMain());
logError("\tClass Loader: " + cl);
continue;
}
// System.out.println("Finding component subclass");
Class<? extends MQTTBlock> componentClass;
try {
componentClass = jarClass.asSubclass(MQTTBlock.class);
} catch (ClassCastException ex) {
logError("Couldn't find Component subclass for " + jar.getName());
continue;
}
// System.out.println("Instantiating and registering");
try {
if (getParent().getBlocks().containsKey(config.getName())) {
// System.out.println("Removing Old Component...");
getParent().disableBlock(config.getName());
getParent().removeBlock(config.getName());
}
// System.out.print("Instantiating...");
MQTTBlock comp = componentClass.newInstance();
// System.out.println(" Done");
// System.out.println("Registering...");
getParent().addBlock(config, comp);
getParent().enableBlock(config.getName());
logInfo("Enabled new block \"" + config.getName() + "\"");
} catch (InstantiationException | IllegalAccessException e1) {
logError("Couldn't instantiate \"" + jar.getName() + "\"");
e1.printStackTrace();
continue;
}
jar.setLastModified(started.getTime());
// System.out.println("File completed");
jarFile.close();
} catch (IOException e) {
logger.severe("Couldn't find config for " + jar.getName());
continue;
}
} else {
continue;
}
}
}
@Override
public boolean onMessageReceived(String topic, Object mqttMessage, String message, int qos,
boolean isDuplicate, boolean isRetained, boolean fromHome) {
return true;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 2,614 |
Archives For Google
Killer Acquisition or Leveling Up: The Use of Mergers to Enter Adjacent Markets
Dirk Auer — 1 February 2023 — Leave a comment
In the world of video games, the process by which players train themselves or their characters in order to overcome a difficult "boss battle" is called "leveling up." I find that the phrase also serves as a useful metaphor in the context of corporate mergers. Here, "leveling up" can be thought of as acquiring another firm in order to enter or reinforce one's presence in an adjacent market where a larger and more successful incumbent is already active.
In video-game terminology, that incumbent would be the "boss." Acquiring firms choose to level up when they recognize that building internal capacity to compete with the "boss" is too slow, too expensive, or is simply infeasible. An acquisition thus becomes the only way "to beat the boss" (or, at least, to maximize the odds of doing so).
Alas, this behavior is often mischaracterized as a "killer acquisition" or "reverse killer acquisition." What separates leveling up from killer acquisitions is that the former serve to turn the merged entity into a more powerful competitor, while the latter attempt to weaken competition. In the case of "reverse killer acquisitions," the assumption is that the acquiring firm would have entered the adjacent market regardless absent the merger, leaving even more firms competing in that market.
In other words, the distinction ultimately boils down to a simple (though hard to answer) question: could both the acquiring and target firms have effectively competed with the "boss" without a merger?
Because they are ubiquitous in the tech sector, these mergers—sometimes also referred to as acquisitions of nascent competitors—have drawn tremendous attention from antitrust authorities and policymakers. All too often, policymakers fail to adequately consider the realistic counterfactual to a merger and mistake leveling up for a killer acquisition. The most recent high-profile example is Meta's acquisition of the virtual-reality fitness app Within. But in what may be a hopeful sign of a turning of the tide, a federal court appears set to clear that deal over objections from the Federal Trade Commission (FTC).
Some Recent 'Boss Battles'
The canonical example of leveling up in tech markets is likely Google's acquisition of Android back in 2005. While Apple had not yet launched the iPhone, it was already clear by 2005 that mobile would become an important way to access the internet (including Google's search services). Rumors were swirling that Apple, following its tremendously successful iPod, had started developing a phone, and Microsoft had been working on Windows Mobile for a long time.
In short, there was a serious risk that Google would be reliant on a single mobile gatekeeper (i.e., Apple) if it did not move quickly into mobile. Purchasing Android was seen as the best way to do so. (Indeed, averting an analogous sort of threat appears to be driving Meta's move into virtual reality today.)
The natural next question is whether Google or Android could have succeeded in the mobile market absent the merger. My guess is that the answer is no. In 2005, Google did not produce any consumer hardware. Quickly and successfully making the leap would have been daunting. As for Android:
Google had significant advantages that helped it to make demands from carriers and OEMs that Android would not have been able to make. In other words, Google was uniquely situated to solve the collective action problem stemming from OEMs' desire to modify Android according to their own idiosyncratic preferences. It used the appeal of its app bundle as leverage to get OEMs and carriers to commit to support Android devices for longer with OS updates. The popularity of its apps meant that OEMs and carriers would have great difficulty in going it alone without them, and so had to engage in some contractual arrangements with Google to sell Android phones that customers wanted. Google was better resourced than Android likely would have been and may have been able to hold out for better terms with a more recognizable and desirable brand name than a hypothetical Google-less Android. In short, though it is of course possible that Android could have succeeded despite the deal having been blocked, it is also plausible that Android became so successful only because of its combination with Google. (citations omitted)
In short, everything suggests that Google's purchase of Android was a good example of leveling up. Note that much the same could be said about the company's decision to purchase Fitbit in order to compete against Apple and its Apple Watch (which quickly dominated the market after its launch in 2015).
A more recent example of leveling up is Microsoft's planned acquisition of Activision Blizzard. In this case, the merger appears to be about improving Microsoft's competitive position in the platform market for game consoles, rather than in the adjacent market for games.
At the time of writing, Microsoft is staring down the barrel of a gun: Sony is on the cusp of becoming the runaway winner of yet another console generation. Microsoft's executives appear to have concluded that this is partly due to a lack of exclusive titles on the Xbox platform. Hence, they are seeking to purchase Activision Blizzard, one of the most successful game studios, known among other things for its acclaimed Call of Duty series.
Again, the question is whether Microsoft could challenge Sony by improving its internal game-publishing branch (known as Xbox Game Studios) or whether it needs to acquire a whole new division. This is obviously a hard question to answer, but a cursory glance at the titles shipped by Microsoft's publishing studio suggest that the issues it faces could not simply be resolved by throwing more money at its existing capacities. Indeed, Microsoft Game Studios seems to be plagued by organizational failings that might only be solved by creating more competition within the Microsoft company. As one gaming journalist summarized:
The current predicament of these titles goes beyond the amount of money invested or the buzzwords used to market them – it's about Microsoft's plan to effectively manage its studios. Encouraging independence isn't an excuse for such a blatantly hands-off approach which allows titles to fester for years in development hell, with some fostering mistreatment to occur. On the surface, it's just baffling how a company that's been ranked as one of the top 10 most reputable companies eight times in 11 years (as per RepTrak) could have such problems with its gaming division.
The upshot is that Microsoft appears to have recognized that its own game-development branch is failing, and that acquiring a well-functioning rival is the only way to rapidly compete with Sony. There is thus a strong case to be made that competition authorities and courts should approach the merger with caution, as it has at least the potential to significantly increase competition in the game-console industry.
Finally, leveling up is sometimes a way for smaller firms to try and move faster than incumbents into a burgeoning and promising segment. The best example of this is arguably Meta's effort to acquire Within, a developer of VR fitness apps. Rather than being an attempt to thwart competition from a competitor in the VR app market, the goal of the merger appears to be to compete with the likes of Google, Apple, and Sony at the platform level. As Mark Zuckerberg wrote back in 2015, when Meta's VR/AR strategy was still in its infancy:
Our vision is that VR/AR will be the next major computing platform after mobile in about 10 years… The strategic goal is clearest. We are vulnerable on mobile to Google and Apple because they make major mobile platforms. We would like a stronger strategic position in the next wave of computing….
Over the next few years, we're going to need to make major new investments in apps, platform services, development / graphics and AR. Some of these will be acquisitions and some can be built in house. If we try to build them all in house from scratch, then we risk that several will take too long or fail and put our overall strategy at serious risk. To derisk this, we should acquire some of these pieces from leading companies.
In short, many of the tech mergers that critics portray as killer acquisitions are just as likely to be attempts by firms to compete head-on with incumbents. This "leveling up" is precisely the sort of beneficial outcome that antitrust laws were designed to promote.
Building Products Is Hard
Critics are often quick to apply the "killer acquisition" label to any merger where a large platform is seeking to enter or reinforce its presence in an adjacent market. The preceding paragraphs demonstrate that it's not that simple, as these mergers often enable firms to improve their competitive position in the adjacent market. For obvious reasons, antitrust authorities and policymakers should be careful not to thwart this competition.
The harder part is how to separate the wheat from the chaff. While I don't have a definitive answer, an easy first step would be for authorities to more seriously consider the supply side of the equation.
Building a new product is incredibly hard, even for the most successful tech firms. Microsoft famously failed with its Zune music player and Windows Phone. The Google+ social network never gained any traction. Meta's foray into the cryptocurrency industry was a sobering experience. Amazon's Fire Phone bombed. Even Apple, which usually epitomizes Silicon Valley firms' ability to enter new markets, has had its share of dramatic failures: Apple Maps, its Ping social network, and the first Home Pod, to name a few.
To put it differently, policymakers should not assume that internal growth is always a realistic alternative to a merger. Instead, they should carefully examine whether such a strategy is timely, cost-effective, and likely to succeed.
This is obviously a daunting task. Firms will struggle to dispositively show that they need to acquire the target firm in order to effectively compete against an incumbent. The question essentially hinges on the quality of the firm's existing management, engineers, and capabilities. All of these are difficult—perhaps even impossible—to measure. At the very least, policymakers can improve the odds of reaching a correct decision by approaching these mergers with an open mind.
Under Chair Lina Khan's tenure, the FTC has opted for the opposite approach and taken a decidedly hostile view of tech acquisitions. The commission sued to block both Meta's purchase of Within and Microsoft's acquisition of Activision Blizzard. Likewise, several economists—notably Tommasso Valletti—have called for policymakers to reverse the burden of proof in merger proceedings, and opined that all mergers should be viewed with suspicion because, absent efficiencies, they always reduce competition.
Unfortunately, this skeptical approach is something of a self-fulfilling prophecy: when authorities view mergers with suspicion, they are likely to be dismissive of the benefits discussed above. Mergers will be blocked and entry into adjacent markets will occur via internal growth.
Large tech companies' many failed attempts to enter adjacent markets via internal growth suggest that such an outcome would ultimately harm the digital economy. Too many "boss battles" will needlessly be lost, depriving consumers of precious competition and destroying startup companies' exit strategies.
In Antitrust & Competition, Apple, Exclusionary Conduct, Facebook, Federal Trade Commission, Google, Mergers & Acquisitions, Platforms, Truth on the Market Android, Apple, gatekeeper, google, killer acquisition, mergers, Meta, Meta-Within, microsoft
Section 230 & Gonzalez: Algorithmic Recommendations Are Immune
Ben Sperry — 1 February 2023 — Leave a comment
In our previous post on Gonzalez v. Google LLC, which will come before the U.S. Supreme Court for oral arguments Feb. 21, Kristian Stout and I argued that, while the U.S. Justice Department (DOJ) got the general analysis right (looking to Roommates.com as the framework for exceptions to the general protections of Section 230), they got the application wrong (saying that algorithmic recommendations should be excepted from immunity).
Now, after reading Google's brief, as well as the briefs of amici on their side, it is even more clear to me that:
algorithmic recommendations are protected by Section 230 immunity; and
creating an exception for such algorithms would severely damage the internet as we know it.
I address these points in reverse order below.
Google on the Death of the Internet Without Algorithms
The central point that Google makes throughout its brief is that a finding that Section 230's immunity does not extend to the use of algorithmic recommendations would have potentially catastrophic implications for the internet economy. Google and amici for respondents emphasize the ubiquity of recommendation algorithms:
Recommendation algorithms are what make it possible to find the needles in humanity's largest haystack. The result of these algorithms is unprecedented access to knowledge, from the lifesaving ("how to perform CPR") to the mundane ("best pizza near me"). Google Search uses algorithms to recommend top search results. YouTube uses algorithms to share everything from cat videos to Heimlich-maneuver tutorials, algebra problem-solving guides, and opera performances. Services from Yelp to Etsy use algorithms to organize millions of user reviews and ratings, fueling global commerce. And individual users "like" and "share" content millions of times every day. – Brief for Respondent Google, LLC at 2.
The "recommendations" they challenge are implicit, based simply on the manner in which YouTube organizes and displays the multitude of third-party content on its site to help users identify content that is of likely interest to them. But it is impossible to operate an online service without "recommending" content in that sense, just as it is impossible to edit an anthology without "recommending" the story that comes first in the volume. Indeed, since the dawn of the internet, virtually every online service—from news, e-commerce, travel, weather, finance, politics, entertainment, cooking, and sports sites, to government, reference, and educational sites, along with search engines—has had to highlight certain content among the thousands or millions of articles, photographs, videos, reviews, or comments it hosts to help users identify what may be most relevant. Given the sheer volume of content on the internet, efforts to organize, rank, and display content in ways that are useful and attractive to users are indispensable. As a result, exposing online services to liability for the "recommendations" inherent in those organizational choices would expose them to liability for third-party content virtually all the time. – Amicus Brief for Meta Platforms at 3-4.
In other words, if Section 230 were limited in the way that the plaintiffs (and the DOJ) seek, internet platforms' ability to offer users useful information would be strongly attenuated, if not completely impaired. The resulting legal exposure would lead inexorably to far less of the kinds of algorithmic recommendations upon which the modern internet is built.
This is, in part, why we weren't able to fully endorse the DOJ's brief in our previous post. The DOJ's brief simply goes too far. It would be unreasonable to establish as a categorical rule that use of the ubiquitous auto-discovery algorithms that power so much of the internet would strip a platform of Section 230 protection. The general rule advanced by the DOJ's brief would have detrimental and far-ranging implications.
Amici on Publishing and Section 230(f)(4)
Google and the amici also make a strong case that algorithmic recommendations are inseparable from publishing. They have a strong textual hook in Section 230(f)(4), which explicitly protects "enabling tools that… filter, screen, allow, or disallow content; pick, choose, analyze or disallow content; or transmit, receive, display, forward, cache, search, subset, organize, reorganize, or translate content."
As the amicus brief from a group of internet-law scholars—including my International Center for Law & Economics colleagues Geoffrey Manne and Gus Hurwitz—put it:
Section 230's text should decide this case. Section 230(c)(1) immunizes the user or provider of an "interactive computer service" from being "treated as the publisher or speaker" of information "provided by another information content provider." And, as Section 230(f)'s definitions make clear, Congress understood the term "interactive computer service" to include services that "filter," "screen," "pick, choose, analyze," "display, search, subset, organize," or "reorganize" third-party content. Automated recommendations perform exactly those functions, and are therefore within the express scope of Section 230's text. – Amicus Brief of Internet Law Scholars at 3-4.
In other words, Section 230 protects not just the conveyance of information, but how that information is displayed. Algorithmic recommendations are a subset of those display tools that allow users to find what they are looking for with ease. Section 230 can't be reasonably read to exclude them.
Why This Isn't Really (Just) a Roommates.com Case
This is where the DOJ's amicus brief (and our previous analysis) misses the point. This is not strictly a Roomates.com case. The case actually turns on whether algorithmic recommendations are separable from publication of third-party content, rather than whether they are design choices akin to what was occurring in that case.
For instance, in our previous post, we argued that:
[T]he DOJ argument then moves onto thinner ice. The DOJ believes that the 230 liability shield in Gonzalez depends on whether an automated "recommendation" rises to the level of development or creation, as the design of filtering criteria in Roommates.com did.
While we thought the DOJ went too far in differentiating algorithmic recommendations from other uses of algorithms, we gave them too much credit in applying the Roomates.com analysis. Section 230 was meant to immunize filtering tools, so long as the information provided is from third parties. Algorithmic recommendations—like the type at issue with YouTube's "Up Next" feature—are less like the conduct in Roommates.com and much more like a search engine.
The DOJ did, however, have a point regarding algorithmic tools in that they may—like any other tool a platform might use—be employed in a way that transforms the automated promotion into a direct endorsement or original publication. For instance, it's possible to use algorithms to intentionally amplify certain kinds of content in such a way as to cultivate more of that content.
That's, after all, what was at the heart of Roommates.com. The site was designed to elicit responses from users that violated the law. Algorithms can do that, but as we observed previously, and as the many amici in Gonzalez observe, there is nothing inherent to the operation of algorithms that match users with content that makes their use categorically incompatible with Section 230's protections.
After looking at the textual and policy arguments forwarded by both sides in Gonzalez, it appears that Google and amici for respondents have the better of it. As several amici argued, to the extent there are good reasons to reform Section 230, Congress should take the lead. The Supreme Court shouldn't take this case as an opportunity to significantly change the consensus of the appellate courts on the broad protections of Section 230 immunity.
In Google, Platforms, Supreme Court, Truth on the Market algorithms, intermediary liability, platforms, roommates.com, section 230, Supreme Court
Brussels Effect or Brussels Defect: Digital Regulation in Emerging Markets
Geoffrey Manne & Dirk Auer — 20 December 2022
The blistering pace at which the European Union put forward and adopted the Digital Markets Act (DMA) has attracted the attention of legislators across the globe. In its wake, countries such as South Africa, India, Brazil, and Turkey have all contemplated digital-market regulations inspired by the DMA (and other models of regulation, such as the United Kingdom's Digital Markets Unit and Australia's sectoral codes of conduct).
Racing to be among the first jurisdictions to regulate might intuitively seem like a good idea. By emulating the EU, countries could hope to be perceived as on the cutting edge of competition policy, and hopefully earn a seat at the table when the future direction of such regulations is discussed.
There are, however, tradeoffs involved in regulating digital markets, which are arguably even more salient in the case of emerging markets. Indeed, as we will explain here, these jurisdictions often face challenges that significantly alter the ratio of costs and benefits when it comes to enacting regulation.
Drawing from a paper we wrote with Sam Bowman about competition policy in the Association of Southeast Asian Nations (ASEAN) zone, we highlight below three of the biggest issues these initiatives face.
To Regulate Competition, You First Need to Attract Competition
Perhaps the biggest factor cautioning emerging markets against adoption of DMA-inspired regulations is that such rules would impose heavy compliance costs to doing business in markets that are often anything but mature. It is probably fair to say that, in many (maybe most) emerging markets, the most pressing challenge is to attract investment from international tech firms in the first place, not how to regulate their conduct.
The most salient example comes from South Africa, which has sketched out plans to regulate digital markets. The Competition Commission has announced that Amazon, which is not yet available in the country, would fall under these new rules should it decide to enter—essentially on the presumption that Amazon would overthrow South Africa's incumbent firms.
It goes without saying that, at the margin, such plans reduce either the likelihood that Amazon will enter the South African market at all, or the extent of its entry should it choose to do so. South African consumers thus risk losing the vast benefits such entry would bring—benefits that dwarf those from whatever marginal increase in competition might be gained from subjecting Amazon to onerous digital-market regulations.
While other tech firms—such as Alphabet, Meta, and Apple—are already active in most emerging jurisdictions, regulation might still have a similar deterrent effect to their further investment. Indeed, the infrastructure deployed by big tech firms in these jurisdictions is nowhere near as extensive as in Western countries. To put it mildly, emerging-market consumers typically only have access to slower versions of these firms' services. A quick glimpse at Google Cloud's global content-delivery network illustrates this point well (i.e., that there is far less infrastructure in developing markets):
Ultimately, emerging markets remain relatively underserved compared to those in the West. In such markets, the priority should be to attract tech investment, not to impose regulations that may further slow the deployment of critical internet infrastructure.
Growth Is Key
The potential to boost growth is the most persuasive argument for emerging markets to favor a more restrained approach to competition law and regulation, such as that currently employed in the United States.
Emerging nations may not have the means (or the inclination) to equip digital-market enforcers with resources similar to those of the European Commission. Given these resource constraints, it is essential that such jurisdictions focus their enforcement efforts on those areas that provide the highest return on investment, notably in terms of increased innovation.
This raises an important point. A recent empirical study by Ross Levine, Chen Lin, Lai Wei, and Wensi Xie finds that competition enforcement does, indeed, promote innovation. But among the study's more surprising findings is that, unlike other areas of competition enforcement, the strength of a jurisdiction's enforcement of "abuse of dominance" rules does not correlate with increased innovation. Furthermore, jurisdictions that allow for so-called "efficiency defenses" in unilateral-conduct cases also tend to produce more innovation. The authors thus conclude that:
From the perspective of maximizing patent-based innovation, therefore, a legal system that allows firms to exploit their dominant positions based on efficiency considerations could boost innovation.
These findings should give pause to policymakers who seek to emulate the European Union's DMA—which, among other things, does not allow gatekeepers to put forward so-called "efficiency defenses" that would allow them to demonstrate that their behavior benefits consumers. If growth and innovation are harmed by overinclusive abuse-of-dominance regimes and rules that preclude firms from offering efficiency-based defenses, then this is probably even more true of digital-market regulations that replace case-by-case competition enforcement with per se prohibitions.
In short, the available evidence suggests that, faced with limited enforcement resources, emerging-market jurisdictions should prioritize other areas of competition policy, such as breaking up or mitigating the harmful effects of cartels and exercising appropriate merger controls.
These findings also cut in favor of emphasizing the traditional antitrust goal of maximizing consumer welfare—or, at least, protecting the competitive process. Many of the more recent digital-market regulations—such as the DMA, the UK DMU, and the ACCC sectoral codes of conduct—are instead focused on distributional issues. They seek to ensure that platform users earn a "fair share" of the benefits generated on a platform. In light of Levine et al.'s findings, this approach could be undesirable, as using competition policy to reduce monopoly rents may lead to less innovation.
In short, traditional antitrust law's focus on consumer welfare and relatively limited enforcement in the area of unilateral conduct may be a good match for emerging nations that want competition regimes that maximize innovation under important resource constraints.
Consider Local Economic and Political Conditions
Emerging jurisdictions have diverse economic and political profiles. These features, in turn, affect the respective costs and benefits of digital-market regulations.
For example, digital-market regulations generally offer very broad discretion to competition enforcers. The DMA details dozens of open-ended prohibitions upon which enforcers can base infringement proceedings. Furthermore, because they are designed to make enforcers' task easier, these regulations often remove protections traditionally afforded to defendants, such as appeals to the consumer welfare standard or efficiency defenses. The UK's DMU initiative, for example, would lower the standard of proof that enforcers must meet.
Giving authorities broad powers with limited judicial oversight might be less problematic in jurisdictions where the state has a track record of self-restraint. The consequences of regulatory discretion might, however, be far more problematic in jurisdictions where authorities routinely overstep the mark and where the threat of corruption is very real.
To name but two, countries like South Africa and India rank relatively low in the World Bank's "ease of doing business index" (84th and 62nd, respectively). They also rank relatively low on the Cato Institute's "human freedom index" (77th and 119th, respectively—and both score particularly badly in terms of economic freedom). This suggests strongly that authorities in those jurisdictions are prone to misapply powers derived from digital-market regulations in ways that hurt growth and consumers.
To make matters worse, outright corruption is also a real problem in several emerging nations. Returning to South Africa and India, both jurisdictions face significant corruption issues (they rank 70th and 85th, respectively, on Transparency International's "Corruption Perception Index").
At a more granular level, an inquiry in South Africa revealed rampant corruption under former President Jacob Zuma, while current President Cyril Ramaphosa also faces significant corruption allegations. Writing in the Financial Times in 2018, Gaurav Dalmia—chair of Delhi-based Dalmia Group Holdings—opined that "India's anti-corruption battle will take decades to win."
This specter of corruption thus counsels in favor of establishing competition regimes with sufficient checks and balances, so as to prevent competition authorities from being captured by industry or political forces. But most digital-market regulations are designed precisely to remove those protections in order to streamline enforcement. The risk that they could be mobilized toward nefarious ends are thus anything but trivial. This is of particular concern, given that such regulations are typically mobilized against global firms in order to shield inefficient local firms—raising serious risks of protectionist enforcement that would harm local consumers.
The bottom line is that emerging markets would do well to reconsider the value of regulating digital markets that have yet to reach full maturity. Recent proposals threaten to deter tech investments in these jurisdictions, while raising significant risks of reduced growth, corruption, and consumer-harming protectionism.
In Amazon, Antitrust & Competition, European Union, Google, International, International Trade, Patent, Platforms, Truth on the Market, United Kingdom abuse of dominance, antitrust, Competition law, Digital Markets Act, digital markets unit, Efficiencies, European Commission, google, India, Patents, Protectionism, South Africa, trade
AG Paxton's Google Suit Makes the Perfect the Enemy of the Good
Eric Fruits & Geoffrey Manne — 14 December 2022
Having just comfortably secured re-election to a third term, embattled Texas Attorney General Ken Paxton is likely to want to change the subject from investigations of his own conduct to a topic where he feels on much firmer ground: the 16-state lawsuit he currently leads accusing Google of monopolizing a segment of the digital advertising business.
The segment in question concerns the systems used to buy and sell display ads shown on third-party websites, such as The New York Times or Runner's World. Paxton's suit, originally filed in December 2020, alleges that digital advertising is dominated by a few large firms and that this stifles competition and generates enormous profits for companies like Google at the expense of advertisers, publishers, and consumers.
On the surface, the digital advertising business appears straightforward: Publishers sell space on their sites and advertisers buy that space to display ads. In this simple view, advertisers seek to minimize how much they pay for ads and publishers seek to maximize their revenues from selling ads.
The reality is much more complex. Rather than paying for how many "eyeballs" see an ad, digital advertisers generally pay only for the ads that consumers click on. Moreover, many digital advertising transactions move through a "stack" of intermediary services to link buyers and sellers, including "exchanges" that run real-time auctions matching bids from advertisers and publishers.
Because revenues are generated only when an ad is clicked on, advertisers, publishers, and exchange operators have an incentive to maximize the likelihood that a consumer will click. A cheap ad is worthless if the viewer doesn't act on it and an expensive ad may be worthwhile if it elicits a click. The role of a company running the exchange, such as Google, is to balance the interests of advertisers buying the ads, publishers displaying the ads, and consumers viewing the ads. In some cases, pricing on one side of the trade will subsidize participation on another side, increasing the value to all sides combined.
At the heart of Paxton's lawsuit is the belief that the exchanges run by Google and other large digital advertising firms simultaneously overcharge advertisers, underpay publishers, and pocket the difference. Google's critics allege the company leverages its ownership of Search, YouTube, and other services to coerce advertisers to use Google's ad-buying tools and Google's exchange, thus keeping competing firms away from these advertisers. It's also claimed that, through its Search, YouTube, and Maps services, Google has superior information about consumers that it won't share with publishers or competing digital advertising companies.
These claims are based on the premise that "big is bad," and that dominant firms have a duty to ensure that their business practices do not create obstacles for their competitors. Under this view, Google would be deemed anticompetitive if there is a hypothetical approach that would accomplish the same goals while fostering even more competition or propping up rivals.
But U.S. antitrust law is supposed to foster innovation that creates benefits for consumers. The law does not forbid conduct that benefits consumers on grounds that it might also inconvenience competitors, or that there is some other arrangement that could be "even more" competitive. Any such conduct would first have to be shown to be anticompetitive—that is, to harm consumers or competition, not merely certain competitors.
That means Paxton has to show not just that some firms on one side of the market are harmed, but that the combined effect across all sides of the market is harmful. In this case, his suit really only discusses the potential harms to publishers (who would like to be paid more), while advertisers and consumers have clearly benefited from the huge markets and declining advertising prices Google has helped to create.
While we can't be sure how the Texas case will develop once its allegations are fleshed out into full arguments and rebutted in court, many of its claims and assumptions appear wrongheaded. If the court rules in favor of these claims, the result will be to condemn conduct that promotes competition and potentially to impose costly, inefficient remedies that function as a drag on innovation.
Paxton and his fellow attorneys general should not fall for the fallacy that their vision of a hypothetical ideal market can replace a well-functioning real market. This would pervert businesses' incentives to innovate and compete, and would make an unobtainable perfect that exists only in the minds of some economists and lawyers the enemy of a "good" that exists in the real world.
[For in-depth analysis of the multi-state suit against Google, see our recent ICLE white paper "The Antitrust Assault on Ad Tech."]
In Antitrust & Competition, Google, Monopolization, Platforms, Truth on the Market adtech, Advertising, google, intermediaries, Publishing, state antitrust enforcement, Texas, two-sided markets
Imposed Final Offer Arbitration: Price Regulation by Any Other Name
Brian Albrecht — 7 December 2022
"Just when I thought I was out, they pull me back in!" says Al Pacino's character, Michael Corleone, in Godfather III. That's how Facebook and Google must feel about S. 673, the Journalism Competition and Preservation Act (JCPA).
Gus Hurwitz called the bill dead in September. Then it passed the Senate Judiciary Committee. Now, there are some reports that suggest it could be added to the obviously unrelated National Defense Authorization Act (it should be noted that the JCPA was not included in the version of NDAA introduced in the U.S. House).
For an overview of the bill and its flaws, see Dirk Auer and Ben Sperry's tl;dr. The JCPA would force "covered" online platforms like Facebook and Google to pay for journalism accessed through those platforms. When a user posts a news article on Facebook, which then drives traffic to the news source, Facebook would have to pay. I won't get paid for links to my banger cat videos, no matter how popular they are, since I'm not a qualifying publication.
I'm going to focus on one aspect of the bill: the use of "final offer arbitration" (FOA) to settle disputes between platforms and news outlets. FOA is sometimes called "baseball arbitration" because it is used for contract disputes in Major League Baseball. This form of arbitration has also been implemented in other jurisdictions to govern similar disputes, notably by the Australian ACCC.
Before getting to the more complicated case, let's start simple.
Scenario #1: I'm a corn farmer. You're a granary who buys corn. We're both invested in this industry, so let's assume we can't abandon negotiations in the near term and need to find an agreeable price. In a market, people make offers. Prices vary each year. I decide when to sell my corn based on prevailing market prices and my beliefs about when they will change.
Scenario #2: A government agency comes in (without either of us asking for it) and says the price of corn this year is $6 per bushel. In conventional economics, we call that a price regulation. Unlike a market price, where both sides sign off, regulated prices do not enjoy mutual agreement by the parties to the transaction.
Scenario #3: Instead of a price imposed independently by regulation, one of the parties (say, the corn farmer) may seek a higher price of $6.50 per bushel and petition the government. The government agrees and the price is set at $6.50. We would still call that price regulation, but the outcome reflects what at least one of the parties wanted and some may argue that it helps "the little guy." (Let's forget that many modern farms are large operations with bargaining power. In our head and in this story, the corn farmer is still a struggling mom-and-pop about to lose their house.)
Scenario #4: Instead of listening only to the corn farmer, both the farmer and the granary tell the government their "final offer" and the government picks one of those offers, not somewhere in between. The parties don't give any reasons—just the offer. This is called "final offer arbitration" (FOA).
As an arbitration mechanism, FOA makes sense, even if it is not always ideal. It avoids some of the issues that can attend "splitting the difference" between the parties.
While it is better than other systems, it is still a price regulation. In the JCPA's case, it would not be imposed immediately; the two parties can negotiate on their own (in the shadow of the imposed FOA). And the actual arbitration decision wouldn't technically be made by the government, but by a third party. Fine. But ultimately, after stripping away the veneer, this is all just an elaborate mechanism built atop the threat of the government choosing the price in the market.
I call that price regulation. The losing party does not like the agreement and never agreed to the overall mechanism. Unlike in voluntary markets, at least one of the parties does not agree with the final price. Moreover, neither party explicitly chose the arbitration mechanism.
The JCPA's FOA system is not precisely like the baseball situation. In baseball, there is choice on the front-end. Players and owners agree to the system. In baseball, there is also choice after negotiations start. Players can still strike; owners can enact a lockout. Under the JCPA, the platforms must carry the content. They cannot walk away.
I'm an economist, not a philosopher. The problem with force is not that it is unpleasant. Instead, the issue is that force distorts the knowledge conveyed through market transactions. That distortion prevents resources from moving to their highest valued use.
How do we know the apple is more valuable to Armen than it is to Ben? In a market, "we" don't need to know. No benevolent outsider needs to pick the "right" price for other people. In most free markets, a seller posts a price. Buyers just need to decide whether they value it more than that price. Armen voluntarily pays Ben for the apple and Ben accepts the transaction. That's how we know the apple is in the right hands.
Often, transactions are about more than just price. Sometimes there may be haggling and bargaining, especially on bigger purchases. Workers negotiate wages, even when the ad stipulates a specific wage. Home buyers make offers and negotiate.
But this just kicks up the issue of information to one more level. Negotiating is costly. That is why sometimes, in anticipation of costly disputes down the road, the two sides voluntarily agree to use an arbitration mechanism. MLB players agree to baseball arbitration. That is the two sides revealing that they believe the costs of disputes outweigh the losses from arbitration.
Again, each side conveys their beliefs and values by agreeing to the arbitration mechanism. Each step in the negotiation process allows the parties to convey the relevant information. No outsider needs to know "the right" answer.For a choice to convey information about relative values, it needs to be freely chosen.
At an abstract level, any trade has two parts. First, people agree to the mechanism, which determines who makes what kinds of offers. At the grocery store, the mechanism is "seller picks the price and buyer picks the quantity." For buying and selling a house, the mechanism is "seller posts price, buyer can offer above or below and request other conditions." After both parties agree to the terms, the mechanism plays out and both sides make or accept offers within the mechanism.
We need choice on both aspects for the price to capture each side's private information.
For example, suppose someone comes up to you with a gun and says "give me your wallet or your watch. Your choice." When you "choose" your watch, we don't actually call that a choice, since you didn't pick the mechanism. We have no way of knowing whether the watch means more to you or to the guy with the gun.
When the JCPA forces Facebook to negotiate with a local news website and Facebook offers to pay a penny per visit, it conveys no information about the relative value that the news website is generating for Facebook. Facebook may just be worried that the website will ask for two pennies and the arbitrator will pick the higher price. It is equally plausible that in a world without transaction costs, the news would pay Facebook, since Facebook sends traffic to them. Is there any chance the arbitrator will pick Facebook's offer if it asks to be paid? Of course not, so Facebook will never make that offer.
For sure, things are imposed on us all the time. That is the nature of regulation. Energy prices are regulated. I'm not against regulation. But we should defend that use of force on its own terms and be honest that the system is one of price regulation. We gain nothing by a verbal sleight of hand that turns losing your watch into a "choice" and the JCPA's FOA into a "negotiation" between platforms and news.
In economics, we often ask about market failures. In this case, is there a sufficient market failure in the market for links to justify regulation? Is that failure resolved by this imposition?
In Antitrust & Competition, Economic Theory, Facebook, Google, Intellectual Property, Platforms, Regulation, Truth on the Market arbitration, facebook, google, JCPA, Journalism, price controls, social media
Biden's Data Flows Order: Does It Comport with EU Law?
Mikolaj Barczentewicz — 30 November 2022
European Union officials insist that the executive order President Joe Biden signed Oct. 7 to implement a new U.S.-EU data-privacy framework must address European concerns about U.S. agencies' surveillance practices. Awaited since March, when U.S. and EU officials reached an agreement in principle on a new framework, the order is intended to replace an earlier data-privacy framework that was invalidated in 2020 by the Court of Justice of the European Union (CJEU) in its Schrems II judgment.
This post is the first in what will be a series of entries examining whether the new framework satisfies the requirements of EU law or, as some critics argue, whether it does not. The critics include Max Schrems' organization NOYB (for "none of your business"), which has announced that it "will likely bring another challenge before the CJEU" if the European Commission officially decides that the new U.S. framework is "adequate." In this introduction, I will highlight the areas of contention based on NOYB's "first reaction."
The overarching legal question that the European Commission (and likely also the CJEU) will need to answer, as spelled out in the Schrems II judgment, is whether the United States "ensures an adequate level of protection for personal data essentially equivalent to that guaranteed in the European Union by the GDPR, read in the light of Articles 7 and 8 of the [EU Charter of Fundamental Rights]" Importantly, as Theodore Christakis, Kenneth Propp, and Peter Swire point out, "adequate level" and "essential equivalence" of protection do not necessarily mean identical protection, either substantively or procedurally. The precise degree of flexibility remains an open question, however, and one that the EU Court may need to clarify to a much greater extent.
Proportionality and Bulk Data Collection
Under Article 52(1) of the EU Charter of Fundamental Rights, restrictions of the right to privacy must meet several conditions. They must be "provided for by law" and "respect the essence" of the right. Moreover, "subject to the principle of proportionality, limitations may be made only if they are necessary" and meet one of the objectives recognized by EU law or "the need to protect the rights and freedoms of others."
As NOYB has acknowledged, the new executive order supplemented the phrasing "as tailored as possible" present in 2014's Presidential Policy Directive on Signals Intelligence Activities (PPD-28) with language explicitly drawn from EU law: mentions of the "necessity" and "proportionality" of signals-intelligence activities related to "validated intelligence priorities." But NOYB counters:
However, despite changing these words, there is no indication that US mass surveillance will change in practice. So-called "bulk surveillance" will continue under the new Executive Order (see Section 2 (c)(ii)) and any data sent to US providers will still end up in programs like PRISM or Upstream, despite of the CJEU declaring US surveillance laws and practices as not "proportionate" (under the European understanding of the word) twice.
It is true that the Schrems II Court held that U.S. law and practices do not "[correlate] to the minimum safeguards resulting, under EU law, from the principle of proportionality." But it is crucial to note the specific reasons the Court gave for that conclusion. Contrary to what NOYB suggests, the Court did not simply state that bulk collection of data is inherently disproportionate. Instead, the reasons it gave were that "PPD-28 does not grant data subjects actionable rights before the courts against the US authorities" and that, under Executive Order 12333, "access to data in transit to the United States [is possible] without that access being subject to any judicial review."
CJEU case law does not support the idea that bulk collection of data is inherently disproportionate under EU law; bulk collection may be proportionate, taking into account the procedural safeguards and the magnitude of interests protected in a given case. (For another discussion of safeguards, see the CJEU's decision in La Quadrature du Net.) Further complicating the legal analysis here is that, as mentioned, it is far from obvious that EU law requires foreign countries offer the same procedural or substantive safeguards that are applicable within the EU.
Effective Redress
The Court's Schrems II conclusion therefore primarily concerns the effective redress available to EU citizens against potential restrictions of their right to privacy from U.S. intelligence activities. The new two-step system proposed by the Biden executive order includes creation of a Data Protection Review Court (DPRC), which would be an independent review body with power to make binding decisions on U.S. intelligence agencies. In a comment pre-dating the executive order, Max Schrems argued that:
It is hard to see how this new body would fulfill the formal requirements of a court or tribunal under Article 47 CFR, especially when compared to ongoing cases and standards applied within the EU (for example in Poland and Hungary).
This comment raises two distinct issues. First, Schrems seems to suggest that an adequacy decision can only be granted if the available redress mechanism satisfies the requirements of Article 47 of the Charter. But this is a hasty conclusion. The CJEU's phrasing in Schrems II is more cautious:
…Article 47 of the Charter, which also contributes to the required level of protection in the European Union, compliance with which must be determined by the Commission before it adopts an adequacy decision pursuant to Article 45(1) of the GDPR
In arguing that Article 47 "also contributes to the required level of protection," the Court is not saying that it determines the required level of protection. This is potentially significant, given that the standard of adequacy is "essential equivalence," not that it be procedurally and substantively identical. Moreover, the Court did not say that the Commission must determine compliance with Article 47 itself, but with the "required level of protection" (which, again, must be "essentially equivalent").
Second, there is the related but distinct question of whether the redress mechanism is effective under the applicable standard of "required level of protection." Christakis, Propp, and Swire offered a helpful analysis suggesting that it is, considering the proposed DPRC's independence, effective investigative powers, and authority to issue binding determinations. I will offer a more detailed analysis of this point in future posts.
Finally, NOYB raised a concern that "judgment by 'Court' [is] already spelled out in Executive Order." This concern seems to be based on the view that a decision of the DPRC ("the judgment") and what the DPRC communicates to the complainant are the same thing. Or in other words, that legal effects of a DPRC decision are exhausted by providing the individual with the neither-confirm-nor-deny statement set out in Section 3 of the executive order. This is clearly incorrect: the DPRC has power to issue binding directions to intelligence agencies. The actual binding determinations of the DPRC are not predetermined by the executive order, only the information to be provided to the complainant is.
What may call for closer consideration are issues of access to information and data. For example, in La Quadrature du Net, the CJEU looked at the difficult problem of notification of persons whose data has been subject to state surveillance, requiring individual notification "only to the extent that and as soon as it is no longer liable to jeopardise" the law-enforcement tasks in question. Given the "essential equivalence" standard applicable to third-country adequacy assessments, however, it does not automatically follow that individual notification is required in that context.
Moreover, it also does not necessarily follow that adequacy requires that EU citizens have a right to access the data processed by foreign government agencies. The fact that there are significant restrictions on rights to information and to access in some EU member states, though not definitive (after all, those countries may be violating EU law), may be instructive for the purposes of assessing the adequacy of data protection in a third country, where EU law requires only "essential equivalence."
There are difficult questions of EU law that the European Commission will need to address in the process of deciding whether to issue a new adequacy decision for the United States. It is also clear that an affirmative decision from the Commission will be challenged before the CJEU, although the arguments for such a challenge are not yet well-developed. In future posts I will provide more detailed analysis of the pivotal legal questions. My focus will be to engage with the forthcoming legal analyses from Schrems and NOYB and from other careful observers.
In Administrative Law, Data Privacy & Security, European Union, Facebook, Google, International, Truth on the Market European Commission, European Court of Justice, European Union, executive order, GDPR, Joe Biden, privacy
7 Big Questions About the Open App Markets Act
Lazar Radic — 1 November 2022
With just a week to go until the U.S. midterm elections, which potentially herald a change in control of one or both houses of Congress, speculation is mounting that congressional Democrats may seek to use the lame-duck session following the election to move one or more pieces of legislation targeting the so-called "Big Tech" companies.
Gaining particular notice—on grounds that it is the least controversial of the measures—is S. 2710, the Open App Markets Act (OAMA). Introduced by Sen. Richard Blumenthal (D-Conn.), the Senate bill has garnered 14 cosponsors: exactly seven Republicans and seven Democrats. It would, among other things, force certain mobile app stores and operating systems to allow "sideloading" and open their platforms to rival in-app payment systems.
Unfortunately, even this relatively restrained legislation—at least, when compared to Sen. Amy Klobuchar's (D-Minn.) American Innovation and Choice Online Act or the European Union's Digital Markets Act (DMA)—is highly problematic in its own right. Here, I will offer seven major questions the legislation leaves unresolved.
1. Are Quantitative Thresholds a Good Indicator of 'Gatekeeper Power'?
It is no secret that OAMA has been tailor-made to regulate two specific app stores: Android's Google Play Store and Apple's Apple App Store (see here, here, and, yes, even Wikipedia knows it).The text makes this clear by limiting the bill's scope to app stores with more than 50 million users, a threshold that only Google Play and the Apple App Store currently satisfy.
However, purely quantitative thresholds are a poor indicator of a company's potential "gatekeeper power." An app store might have much fewer than 50 million users but cater to a relevant niche market. By the bill's own logic, why shouldn't that app store likewise be compelled to be open to competing app distributors? Conversely, it may be easy for users of very large app stores to multi-home or switch seamlessly to competing stores. In either case, raw user data paints a distorted picture of the market's realities.
As it stands, the bill's thresholds appear arbitrary and pre-committed to "disciplining" just two companies: Google and Apple. In principle, good laws should be abstract and general and not intentionally crafted to apply only to a few select actors. In OAMA's case, the law's specific thresholds are also factually misguided, as purely quantitative criteria are not a good proxy for the sort of market power the bill purportedly seeks to curtail.
2. Why Does the Bill not Apply to all App Stores?
Rather than applying to app stores across the board, OAMA targets only those associated with mobile devices and "general purpose computing devices." It's not clear why.
For example, why doesn't it cover app stores on gaming platforms, such as Microsoft's Xbox or Sony's PlayStation?
Source: Visual Capitalist
Currently, a PlayStation user can only buy digital games through the PlayStation Store, where Sony reportedly takes a 30% cut of all sales—although its pricing schedule is less transparent than that of mobile rivals such as Apple or Google.
Clearly, this bothers some developers. Much like Epic Games CEO Tim Sweeney's ongoing crusade against the Apple App Store, indie-game publisher Iain Garner of Neon Doctrine recently took to Twitter to complain about Sony's restrictive practices. According to Garner, "Platform X" (clearly PlayStation) charges developers up to $25,000 and 30% of subsequent earnings to give games a modicum of visibility on the platform, in addition to requiring them to jump through such hoops as making a PlayStation-specific trailer and writing a blog post. Garner further alleges that Sony severely circumscribes developers' ability to offer discounts, "meaning that Platform X owners will always get the worst deal!" (see also here).
Microsoft's Xbox Game Store similarly takes a 30% cut of sales. Presumably, Microsoft and Sony both have the same type of gatekeeper power in the gaming-console market that Apple and Google are said to have on their respective platforms, leading to precisely those issues that OAMA ostensibly purports to combat. Namely, that consumers are not allowed to choose alternative app stores through which to buy games on their respective consoles, and developers must acquiesce to Sony's and Microsoft's terms if they want their games to reach those players.
More broadly, dozens of online platforms also charge commissions on the sales made by their creators. To cite but a few: OnlyFans takes a 20% cut of sales; Facebook gets 30% of the revenue that creators earn from their followers; YouTube takes 45% of ad revenue generated by users; and Twitch reportedly rakes in 50% of subscription fees.
This is not to say that all these services are monopolies that should be regulated. To the contrary, it seems like fees in the 20-30% range are common even in highly competitive environments. Rather, it is merely to observe that there are dozens of online platforms that demand a percentage of the revenue that creators generate and that prevent those creators from bypassing the platform. As well they should, after all, because creating and improving a platform is not free.
It is nonetheless difficult to see why legislation regulating online marketplaces should focus solely on two mobile app stores. Ultimately, the inability of OAMA's sponsors to properly account for this carveout diminishes the law's credibility.
3. Should Picking Among Legitimate Business Models Be up to Lawmakers or Consumers?
"Open" and "closed" platforms posit two different business models, each with its own advantages and disadvantages. Some consumers may prefer more open platforms because they grant them more flexibility to customize their mobile devices and operating systems. But there are also compelling reasons to prefer closed systems. As Sam Bowman observed, narrowing choice through a more curated system frees users from having to research every possible option every time they buy or use some product. Instead, they can defer to the platform's expertise in determining whether an app or app store is trustworthy or whether it contains, say, objectionable content.
Currently, users can choose to opt for Apple's semi-closed "walled garden" iOS or Google's relatively more open Android OS (which OAMA wants to pry open even further). Ironically, under the pretext of giving users more "choice," OAMA would take away the possibility of choice where it matters the most—i.e., at the platform level. As Mikolaj Barczentewicz has written:
A sideloading mandate aims to give users more choice. It can only achieve this, however, by taking away the option of choosing a device with a "walled garden" approach to privacy and security (such as is taken by Apple with iOS).
This obviates the nuances between the two and pushes Android and iOS to converge around a single model. But if consumers unequivocally preferred open platforms, Apple would have no customers, because everyone would already be on Android.
Contrary to regulators' simplistic assumptions, "open" and "closed" are not synonyms for "good" and "bad." Instead, as Boston University's Andrei Hagiu has shown, there are fundamental welfare tradeoffs at play between these two perfectly valid business models that belie simplistic characterizations of one being inherently superior to the other.
It is debatable whether courts, regulators, or legislators are well-situated to resolve these complex tradeoffs by substituting businesses' product-design decisions and consumers' revealed preferences with their own. After all, if regulators had such perfect information, we wouldn't need markets or competition in the first place.
4. Does OAMA Account for the Security Risks of Sideloading?
Platforms retaining some control over the apps or app stores allowed on their operating systems bolsters security, as it allows companies to weed out bad players.
Both Apple and Google do this, albeit to varying degrees. For instance, Android already allows sideloading and third-party in-app payment systems to some extent, while Apple runs a tighter ship. However, studies have shown that it is precisely the iOS "walled garden" model which gives it an edge over Android in terms of privacy and security. Even vocal Apple critic Tim Sweeney recently acknowledged that increased safety and privacy were competitive advantages for Apple.
The problem is that far-reaching sideloading mandates—such as the ones contemplated under OAMA—are fundamentally at odds with current privacy and security capabilities (see here and here).
OAMA's defenders might argue that the law does allow covered platforms to raise safety and security defenses, thus making the tradeoffs between openness and security unnecessary. But the bill places such stringent conditions on those defenses that platform operators will almost certainly be deterred from risking running afoul of the law's terms. To invoke the safety and security defenses, covered companies must demonstrate that provisions are applied on a "demonstrably consistent basis"; are "narrowly tailored and could not be achieved through less discriminatory means"; and are not used as a "pretext to exclude or impose unnecessary or discriminatory terms."
Implementing these stringent requirements will drag enforcers into a micromanagement quagmire. There are thousands of potential spyware, malware, rootkit, backdoor, and phishing (to name just a few) software-security issues—all of which pose distinct threats to an operating system. The Federal Trade Commission (FTC) and the federal courts will almost certainly struggle to control the "consistency" requirement across such varied types.
Likewise, OAMA's reference to "least discriminatory means" suggests there is only one valid answer to any given security-access tradeoff. Further, depending on one's preferred balance between security and "openness," a claimed security risk may or may not be "pretextual," and thus may or may not be legal.
Finally, the bill text appears to preclude the possibility of denying access to a third-party app or app store for reasons other than safety and privacy. This would undermine Apple's and Google's two-tiered quality-control systems, which also control for "objectionable" content such as (child) pornography and social engineering.
5. How Will OAMA Safeguard the Rights of Covered Platforms?
OAMA is also deeply flawed from a procedural standpoint. Most importantly, there is no meaningful way to contest the law's designation as "covered company," or the harms associated with it.
Once a company is "covered," it is presumed to hold gatekeeper power, with all the associated risks for competition, innovation, and consumer choice. Remarkably, this presumption does not admit any qualitative or quantitative evidence to the contrary. The only thing a covered company can do to rebut the designation is to demonstrate that it, in fact, has fewer than 50 million users.
By preventing companies from showing that they do not hold the kind of gatekeeper power that harms competition, decreases innovation, raises prices, and reduces choice (the bill's stated objectives), OAMA severely tilts the playing field in the FTC's favor. Even the EU's enforcer-friendly DMA incorporated a last-minute amendment allowing firms to dispute their status as "gatekeepers." While this defense is not perfect (companies cannot rely on the same qualitative evidence that the European Commission can use against them), at least gatekeeper status can be contested under the DMA.
6. Should Legislation Protect Competitors at the Expense of Consumers?
Like most of the new wave of regulatory initiatives against Big Tech (but unlike antitrust law), OAMA is explicitly designed to help competitors, with consumers footing the bill.
For example, OAMA prohibits covered companies from using or combining nonpublic data obtained from third-party apps or app stores operating on their platforms in competition with those third parties. While this may have the short-term effect of redistributing rents away from these platforms and toward competitors, it risks harming consumers and third-party developers in the long run.
Platforms' ability to integrate such data is part of what allows them to bring better and improved products and services to consumers in the first place. OAMA tacitly admits this by recognizing that the use of nonpublic data grants covered companies a competitive advantage. In other words, it allows them to deliver a product that is better than competitors'.
Prohibiting self-preferencing raises similar concerns. Why wouldn't a company that has invested billions in developing a successful platform and ecosystem not give preference to its own products to recoup some of that investment? After all, the possibility of exercising some control over downstream and adjacent products is what might have driven the platform's development in the first place. In other words, self-preferencing may be a symptom of competition, and not the absence thereof. Third-party companies also would have weaker incentives to develop their own platforms if they can free-ride on the investments of others. And platforms that favor their own downstream products might simply be better positioned to guarantee their quality and reliability (see here and here).
In all of these cases, OAMA's myopic focus on improving the lot of competitors for easy political points will upend the mobile ecosystems from which both users and developers derive significant benefit.
7. Shouldn't the EU Bear the Risks of Bad Tech Regulation?
Finally, U.S. lawmakers should ask themselves whether the European Union, which has no tech leaders of its own, is really a model to emulate. Today, after all, marks the day the long-awaited Digital Markets Act— the EU's response to perceived contestability and fairness problems in the digital economy—officially takes effect. In anticipation of the law entering into force, I summarized some of the outstanding issues that will define implementation moving forward in this recent tweet thread.
We have been critical of the DMA here at Truth on the Market on several factual, legal, economic, and procedural grounds. The law's problems range from it essentially being a tool to redistribute rents away from platforms and to third-parties, despite it being unclear why the latter group is inherently more deserving (Pablo Ibañez Colomo has raised a similar point); to its opacity and lack of clarity, a process that appears tilted in the Commission's favor; to the awkward way it interacts with EU competition law, ignoring the welfare tradeoffs between the models it seeks to impose and perfectly valid alternatives (see here and here); to its flawed assumptions (see, e.g., here on contestability under the DMA); to the dubious legal and economic value of the theory of harm known as "self-preferencing"; to the very real possibility of unintended consequences (e.g., in relation to security and interoperability mandates).
In other words, that the United States lags the EU in seeking to regulate this area might not be a bad thing, after all. Despite the EU's insistence on being a trailblazing agenda-setter at all costs, the wiser thing in tech regulation might be to remain at a safe distance. This is particularly true when one considers the potentially large costs of legislative missteps and the difficulty of recalibrating once a course has been set.
U.S. lawmakers should take advantage of this dynamic and learn from some of the Old Continent's mistakes. If they play their cards right and take the time to read the writing on the wall, they might just succeed in averting antitrust's uncertain future.
In Antitrust & Competition, Apple, Data Privacy & Security, European Union, Federal Trade Commission, Google, Monopolization, Platforms, Truth on the Market, Tying, Vertical Restraints aicoa, Android, App Store, Apple App Store, cybersecurity, data security, Digital Markets Act, EU, European Commission, ftc, gaming, klobuchar, microsoft, open access, Open App Markets Act, privacy, Richard Blumenthal, self-preferncing, Sony
The Case Against Self-Preferencing as a New Antitrust Offense
Giuseppe Colangelo — 22 September 2022
The practice of so-called "self-preferencing" has come to embody the zeitgeist of competition policy for digital markets, as legislative initiatives are undertaken in jurisdictions around the world that to seek, in various ways, to constrain large digital platforms from granting favorable treatment to their own goods and services. The core concern cited by policymakers is that gatekeepers may abuse their dual role—as both an intermediary and a trader operating on the platform—to pursue a strategy of biased intermediation that entrenches their power in core markets (defensive leveraging) and extends it to associated markets (offensive leveraging).
In addition to active interventions by lawmakers, self-preferencing has also emerged as a new theory of harm before European courts and antitrust authorities. Should antitrust enforcers be allowed to pursue such a theory, they would gain significant leeway to bypass the legal standards and evidentiary burdens traditionally required to prove that a given business practice is anticompetitive. This should be of particular concern, given the broad range of practices and types of exclusionary behavior that could be characterized as self-preferencing—only some of which may, in some specific contexts, include exploitative or anticompetitive elements.
In a new working paper for the International Center for Law & Economics (ICLE), I provide an overview of the relevant traditional antitrust theories of harm, as well as the emerging case law, to analyze whether and to what extent self-preferencing should be considered a new standalone offense under EU competition law. The experience to date in European case law suggests that courts have been able to address platforms' self-preferencing practices under existing theories of harm, and that it may not be sufficiently novel to constitute a standalone theory of harm.
European Case Law on Self-Preferencing
Practices by digital platforms that might be deemed self-preferencing first garnered significant attention from European competition enforcers with the European Commission's Google Shopping investigation, which examined whether the search engine's results pages positioned and displayed its own comparison-shopping service more favorably than the websites of rival comparison-shopping services. According to the Commission's findings, Google's conduct fell outside the scope of competition on the merits and could have the effect of extending Google's dominant position in the national markets for general Internet search into adjacent national markets for comparison-shopping services, in addition to protecting Google's dominance in its core search market.
Rather than explicitly posit that self-preferencing (a term the Commission did not use) constituted a new theory of harm, the Google Shopping ruling described the conduct as belonging to the well-known category of "leveraging." The Commission therefore did not need to propagate a new legal test, as it held that the conduct fell under a well-established form of abuse. The case did, however, spur debate over whether the legal tests the Commission did apply effectively imposed on Google a principle of equal treatment of rival comparison-shopping services.
But it should be noted that conduct similar to that alleged in the Google Shopping investigation actually came before the High Court of England and Wales several months earlier, this time in a dispute between Google and Streetmap. At issue in that case was favorable search results Google granted to its own maps, rather than to competing online maps. The UK Court held, however, that the complaint should have been appropriately characterized as an allegation of discrimination; it further found that Google's conduct did not constitute anticompetitive foreclosure. A similar result was reached in May 2020 by the Amsterdam Court of Appeal in the Funda case.
Conversely, in June 2021, the French Competition Authority (AdlC) followed the European Commission into investigating Google's practices in the digital-advertising sector. Like the Commission, the AdlC did not explicitly refer to self-preferencing, instead describing the conduct as "favoring."
Given this background and the proliferation of approaches taken by courts and enforcers to address similar conduct, there was significant anticipation for the judgment that the European General Court would ultimately render in the appeal of the Google Shopping ruling. While the General Court upheld the Commission's decision, it framed self-preferencing as a discriminatory abuse. Further, the Court outlined four criteria that differentiated Google's self-preferencing from competition on the merits.
Specifically, the Court highlighted the "universal vocation" of Google's search engine—that it is open to all users and designed to index results containing any possible content; the "superdominant" position that Google holds in the market for general Internet search; the high barriers to entry in the market for general search services; and what the Court deemed Google's "abnormal" conduct—behaving in a way that defied expectations, given a search engine's business model, and that changed after the company launched its comparison-shopping service.
While the precise contours of what the Court might consider discriminatory abuse aren't yet clear, the decision's listed criteria appear to be narrow in scope. This stands at odds with the much broader application of self-preferencing as a standalone abuse, both by the European Commission itself and by some national competition authorities (NCAs).
Indeed, just a few weeks after the General Court's ruling, the Italian Competition Authority (AGCM) handed down a mammoth fine against Amazon over preferential treatment granted to third-party sellers who use the company's own logistics and delivery services. Rather than reflecting the qualified set of criteria laid out by the General Court, the Italian decision was clearly inspired by the Commission's approach in Google Shopping. Where the Commission described self-preferencing as a new form of leveraging abuse, AGCM characterized Amazon's practices as tying.
Self-preferencing has also been raised as a potential abuse in the context of data and information practices. In November 2020, the European Commission sent Amazon a statement of objections detailing its preliminary view that the company had infringed antitrust rules by making systematic use of non-public business data, gathered from independent retailers who sell on Amazon's marketplace, to advantage the company's own retail business. (Amazon responded with a set of commitments currently under review by the Commission.)
Both the Commission and the U.K. Competition and Markets Authority have lodged similar allegations against Facebook over data gathered from advertisers and then used to compete with those advertisers in markets in which Facebook is active, such as classified ads. The Commission's antitrust proceeding against Apple over its App Store rules likewise highlights concerns that the company may use its platform position to obtain valuable data about the activities and offers of its competitors, while competing developers may be denied access to important customer data.
These enforcement actions brought by NCAs and the Commission appear at odds with the more bounded criteria set out by the General Court in Google Shopping, and raise tremendous uncertainty regarding the scope and definition of the alleged new theory of harm.
Self-Preferencing, Platform Neutrality, and the Limits of Antitrust Law
The growing tendency to invoke self-preferencing as a standalone theory of antitrust harm could serve two significant goals for European competition enforcers. As mentioned earlier, it offers a convenient shortcut that could allow enforcers to skip the legal standards and evidentiary burdens traditionally required to prove anticompetitive behavior. Moreover, it can function, in practice, as a means to impose a neutrality regime on digital gatekeepers, with the aims of both ensuring a level playing field among competitors and neutralizing the potential conflicts of interests implicated by dual-mode intermediation.
The dual roles performed by some platforms continue to fuel the never-ending debate over vertical integration, as well as related concerns that, by giving preferential treatment to its own products and services, an integrated provider may leverage its dominance in one market to related markets. From this perspective, self-preferencing is an inevitable byproduct of the emergence of ecosystems.
However, as the Australian Competition and Consumer Commission has recognized, self-preferencing conduct is "often benign." Furthermore, the total value generated by an ecosystem depends on the activities of independent complementors. Those activities are not completely under the platform's control, although the platform is required to establish and maintain the governance structures regulating access to and interactions around that ecosystem.
Given this reality, a complete ban on self-preferencing may call the very existence of ecosystems into question, challenging their design and monetization strategies. Preferential treatment can take many different forms with many different potential effects, all stemming from platforms' many different business models. This counsels for a differentiated, case-by-case, and effects-based approach to assessing the alleged competitive harms of self-preferencing.
Antitrust law does not impose on platforms a general duty to ensure neutrality by sharing their competitive advantages with rivals. Moreover, possessing a competitive advantage does not automatically equal an anticompetitive effect. As the European Court of Justice recently stated in Servizio Elettrico Nazionale, competition law is not intended to protect the competitive structure of the market, but rather to protect consumer welfare. Accordingly, not every exclusionary effect is detrimental to competition. Distinctions must be drawn between foreclosure and anticompetitive foreclosure, as only the latter may be penalized under antitrust.
In Antitrust & Competition, European Union, Exclusionary Conduct, Google, Platforms, Price Discrimination, Truth on the Market, Tying, United Kingdom abuse of dominance, antitrust, competition, consumer welfare standard, discrimination, EU General Court, European Commission, European Court of Justice, foreclosure, Google Shopping, Harm to Competition, italy, leveraging, maps, Search Engines, self-preferncing, tying
The Four Ways of Spending Data
Stephen Dnes — 28 July 2022
[TOTM: The following is part of a digital symposium by TOTM guests and authors on Antitrust's Uncertain Future: Visions of Competition in the New Regulatory Landscape. Information on the authors and the entire series of posts is available here.]
In Free to Choose, Milton Friedman famously noted that there are four ways to spend money[1]:
Spending your own money on yourself. For example, buying groceries or lunch. There is a strong incentive to economize and to get full value.
Spending your own money on someone else. For example, buying a gift for another. There is a strong incentive to economize, but perhaps less to achieve full value from the other person's point of view. Altruism is admirable, but it differs from value maximization, since—strictly speaking—giving cash would maximize the other's value. Perhaps the point of a gift is that it does not amount to cash and the maximization of the other person's welfare from their point of view.
Spending someone else's money on yourself. For example, an expensed business lunch. "Pass me the filet mignon and Chateau Lafite! Do you have one of those menus without any prices?" There is a strong incentive to get maximum utility, but there is little incentive to economize.
Spending someone else's money on someone else. For example, applying the proceeds of taxes or donations. There may be an indirect desire to see utility, but incentives for quality and cost management are often diminished.
This framework can be criticized. Altruism has a role. Not all motives are selfish. There is an important role for action to help those less fortunate, which might mean, for instance, that a charity gains more utility from category (4) (assisting the needy) than from category (3) (the charity's holiday party). It always depends on the facts and the context. However, there is certainly a grain of truth in the observation that charity begins at home and that, in the final analysis, people are best at managing their own affairs.
How would this insight apply to data interoperability? The difficult cases of assisting the needy do not arise here: there is no serious sense in which data interoperability does, or does not, result in destitution. Thus, Friedman's observations seem to ring true: when spending data, those whose data it is seem most likely to maximize its value. This is especially so where collection of data responds to incentives—that is, the amount of data collected and processed responds to how much control over the data is possible.
The obvious exception to this would be a case of market power. If there is a monopoly with persistent barriers to entry, then the incentive may not be to maximize total utility, and therefore to limit data handling to the extent that a higher price can be charged for the lesser amount of data that does remain available. This has arguably been seen with some data-handling rules: the "Jedi Blue" agreement on advertising bidding, Apple's Intelligent Tracking Prevention and App Tracking Transparency, and Google's proposed Privacy Sandbox, all restrict the ability of others to handle data. Indeed, they may fail Friedman's framework, since they amount to the platform deciding how to spend others' data—in this case, by not allowing them to collect and process it at all.
It should be emphasized, though, that this is a special case. It depends on market power, and existing antitrust and competition laws speak to it. The courts will decide whether cases like Daily Mail v Google and Texas et al. v Google show illegal monopolization of data flows, so as to fall within this special case of market power. Outside the United States, cases like the U.K. Competition and Markets Authority's Google Privacy Sandbox commitments and the European Union's proposed commitments with Amazon seek to allow others to continue to handle their data and to prevent exclusivity from arising from platform dynamics, which could happen if a large platform prevents others from deciding how to account for data they are collecting. It will be recalled that even Robert Bork thought that there was risk of market power harms from the large Microsoft Windows platform a generation ago.[2] Where market power risks are proven, there is a strong case that data exclusivity raises concerns because of an artificial barrier to entry. It would only be if the benefits of centralized data control were to outweigh the deadweight loss from data restrictions that this would be untrue (though query how well the legal processes verify this).
Yet the latest proposals go well beyond this. A broad interoperability right amounts to "open season" for spending others' data. This makes perfect sense in the European Union, where there is no large domestic technology platform, meaning that the data is essentially owned via foreign entities (mostly, the shareholders of successful U.S. and Chinese companies). It must be very tempting to run an industrial policy on the basis that "we'll never be Google" and thus to embrace "sharing is caring" as to others' data.
But this would transgress the warning from Friedman: would people optimize data collection if it is open to mandatory sharing even without proof of market power? It is deeply concerning that the EU's DATA Act is accompanied by an infographic that suggests that coffee-machine data might be subject to mandatory sharing, to allow competition in services related to the data (e.g., sales of pods; spare-parts automation). There being no monopoly in coffee machines, this simply forces vertical disintegration of data collection and handling. Why put a data-collection system into a coffee maker at all, if it is to be a common resource? Friedman's category (4) would apply: the data is taken and spent by another. There is no guarantee that there would be sensible decision making surrounding the resource.
It will be interesting to see how common-law jurisdictions approach this issue. At the risk of stating the obvious, the polity in continental Europe differs from that in the English-speaking democracies when it comes to whether the collective, or the individual, should be in the driving seat. A close read of the UK CMA's Google commitments is interesting, in that paragraph 30 requires no self-preferencing in data collection and requires future data-handling systems to be designed with impacts on competition in mind. No doubt the CMA is seeking to prevent data-handling exclusivity on the basis that this prevents companies from using their data collection to compete. This is far from the EU DATA Act's position in that it is certainly not a right to handle Google's data: it is simply a right to continue to process one's own data.
U.S. proposals are at an earlier stage. It would seem important, as a matter of principle, not to make arbitrary decisions about vertical integration in data systems, and to identify specific market-power concerns instead, in line with common-law approaches to antitrust.
It might be very attractive to the EU to spend others' data on their behalf, but that does not make it right. Those working on the U.S. proposals would do well to ensure that there is a meaningful market-power gate to avoid unintended consequences.
Disclaimer: The author was engaged for expert advice relating to the UK CMA's Privacy Sandbox case on behalf of the complainant Marketers for an Open Web.
[1] Milton Friedman, Free to Choose, 1980, pp.115-119
[2] Comments at the Yale Law School conference, Robert H. Bork's influence on Antitrust Law, Sep. 27-28, 2013.
In Amazon, Antitrust & Competition, Apple, Barriers to Entry, European Union, Google, International, Platforms, Symposia, Truth on the Market, United Kingdom Amazon, antitrust, Apple, competition, data, European Union, google, interoperability, microsoft, Milton Friedman, Robert Bork, self-preferncing, UK CMA
The Woman in the High Office
Dirk Auer — 27 July 2022
May 2007, Palo Alto
The California sun shone warmly on Eric Schmidt's face as he stepped out of his car and made his way to have dinner at Madera, a chic Palo Alto restaurant.
Dining out was a welcome distraction from the endless succession of strategy meetings with the nitpickers of the law department, which had been Schmidt's bread and butter for the last few months. The lawyers seemed to take issue with any new project that Google's engineers came up with. "How would rivals compete with our maps?"; "Our placement should be no less favorable than rivals''; etc. The objections were endless.
This is not how things were supposed to be. When Schmidt became Google's chief executive officer in 2001, his mission was to take the company public and grow the firm into markets other than search. But then something unexpected happened. After campaigning on an anti-monopoly platform, a freshman senator from Minnesota managed to get her anti-discrimination bill through Congress in just her first few months in office. All companies with a market cap of more than $150 billion were now prohibited from favoring their own products. Google had recently crossed that Rubicon, putting a stop to years of carefree expansion into new markets.
But today was different. The waiter led Schmidt to his table overlooking Silicon Valley. His acquaintance was already seated.
With his tall and slender figure, Andy Rubin had garnered quite a reputation among Silicon Valley's elite. After engineering stints at Apple and Motorola, developing various handheld devices, Rubin had set up his own shop. The idea was bold: develop the first open mobile platform—based on Linux, nonetheless. Rubin had pitched the project to Google in 2005 but given the regulatory uncertainty over the future of antitrust—the same wave of populist sentiment that would carry Klobuchar to office one year later—Schmidt and his team had passed.
"There's no money in open source," the company's CFO ruled. Schmidt had initially objected, but with more pressing matters to deal with, he ultimately followed his CFO's advice.
Schmidt and Rubin were exchanging pleasantries about Microsoft and Java when the meals arrived–sublime Wagyu short ribs and charred spring onions paired with a 1986 Chateau Margaux.
Rubin finally cut to the chase. "Our mobile operating system will rely on state-of-the-art touchscreen technology. Just like the device being developed by Apple. Buying Android today might be your only way to avoid paying monopoly prices to access Apple's mobile users tomorrow."
Schmidt knew this all too well: The future was mobile, and few companies were taking Apple's upcoming iPhone seriously enough. Even better, as a firm, Android was treading water. Like many other startups, it had excellent software but no business model. And with the Klobuchar bill putting the brakes on startup investment—monetizing an ecosystem had become a delicate legal proposition, deterring established firms from acquiring startups–Schmidt was in the middle of a buyer's market. "Android we could make us a force to reckon with" Schmidt thought to himself.
But he quickly shook that thought, remembering the words of his CFO: "There is no money in open source." In an ideal world, Google would have used Android to promote its search engine—placing a search bar on Android users to draw users to its search engine—or maybe it could have tied a proprietary app store to the operating system, thus earning money from in-app purchases. But with the Klobuchar bill, these were no longer options. Not without endless haggling with Google's planning committee of lawyers.
And they would have a point, of course. Google risked heavy fines and court-issued injunctions that would stop the project in its tracks. Such risks were not to be taken lightly. Schmidt needed a plan to make the Android platform profitable while accommodating Google's rivals, but he had none.
The desserts were served, Schmidt steered the conversation to other topics, and the sun slowly set over Sand Hill Road.
Present Day, Cupertino
Apple continues to dominate the smartphone industry with little signs of significant competition on the horizon. While there are continuing rumors that Google, Facebook, or even TikTok might enter the market, these have so far failed to transpire.
Google's failed partnership with Samsung, back in 2012, still looms large over the industry. After lengthy talks to create an open mobile platform failed to materialize, Google ultimately entered into an agreement with the longstanding mobile manufacturer. Unfortunately, the deal was mired by antitrust issues and clashing visions—Samsung was believed to favor a closed ecosystem, rather than the open platform envisioned by Google.
The sense that Apple is running away with the market is only reinforced by recent developments. Last week, Tim Cook unveiled the company's new iPhone 11—the first ever mobile device to come with three cameras. With an eye-watering price tag of $1,199 for the top-of-the-line Pro model, it certainly is not cheap. In his presentation, Cook assured consumers Apple had solved the security issues that have been an important bugbear for the iPhone and its ecosystem of competing app stores.
Analysts expect the new range of devices will help Apple cement the iPhone's 50% market share. This is especially likely given the important challenges that Apple's main rivals continue to face.
The Windows Phone's reputation for buggy software continues to undermine its competitive position, despite its comparatively low price point. Andy Rubin, the head of the Windows Phone, was reassuring in a press interview, but there is little tangible evidence he will manage to successfully rescue the flailing ship. Meanwhile, Huawei has come under increased scrutiny for the threats it may pose to U.S. national security. The Chinese manufacturer may face a U.S. sales ban, unless the company's smartphone branch is sold to a U.S. buyer. Oracle is said to be a likely candidate.
The sorry state of mobile competition has become an increasingly prominent policy issue. President Klobuchar took to Twitter and called on mobile-device companies to refrain from acting as monopolists, intimating elsewhere that failure to do so might warrant tougher regulation than her anti-discrimination bill:
In Antitrust & Competition, Antitrust's Uncertain Future, Apple, Barriers to Entry, Exclusionary Conduct, Exclusive Dealing, Google, Platforms, Symposia, Truth on the Market, Tying Android, antitrust, Apple, competition, google, klobuchar, open source, platforms, self-preferncing
Verses on Self-Preferencing
Ramsi Woodcock — 26 July 2022
About earth's creatures great and small,
Devices clever as can be,
I see foremost a ruthless power;
You, their ingenuity.
You see the beak upon the finch;
I, the beaked skeleton.
You see the wonders that they are;
I, the things that might have been.
You see th'included batteries
I, the poor excluded ones.
You, the phone that simply works;
I, restrain'd competition.
'Twould be a better world, I say,
Were all the options to abide—
All beaks and brands of battery—
From which the public to decide.
You say that man the greatest is
Because he dominates today,
But meteor, not caveman, drove
The ancient dinosaurs away.
If they were here when we were new,
We might the age not have survived;
They say some species could outwit
The sharpest chimps today alive.
Just so, I say, 'twould better be
Replacement batteries t'allow
For sales alone can prove what brands
Deserve el'vation to the Dow.
Designing batt'ries switchable
Makes their selection fully public;
It substitutes democracy
For an engineering logic.
For only buyers should decide
Which components to inter;
Their taste alone determines worth,
Though engineers be cleverer.
You: The meaning of component,
We can always redefine.
From batteries to molecules,
We can draw most any line.
Of cogs, thus, an infinitude;
Of time a finitude to lose.
You cannot interchange all parts,
Or each one carefully to choose.
Product choices corporate
We cannot all democratize,
At least so long as consumers
Wish to get on with their lives.
Exclusion therefore cannot we
Universally condemn;
Oft must we let the firm decide
Which components to put in.
Power, then, is everywhere—
What is is built on what is not—
And th'elimination of it
Is no cornerstone of thought.
Of components infinite
We must choose which few to free;
Th'criterion for doing that,
Abuse-of-power cannot be.
Power and oppression are,
In life and goods ubiquitous.
But value differentiates—
Build we antitrust on this.
Alone when letting buyers say
Which part into a product goes
Would make those buyers happier,
Must we interchange impose.
Batt'ry brand must matter much,
Else, we seriously delude,
To think consumers want to hear:
"We the batt'ries not include."
The same is true for Amazon.
If it knows which seller's best,
Let it cast the others out for us—
Give our scrolling bars a rest.
If Apple knows which app's a dud,
Let Apple cast it out as well.
Which app's a fraud and which a scam,
Smartphone users cannot tell.
If Google wants to show me how
To get from A to B to C,
I'd rather that she use her maps
Than search for others separately.
A rule against self-preferencing
No legal principle provides;
For what opposes power's role
Can't be neutrally applied.
What goes for all third-party sales
Goes for Amazon's front-end.
Self-preferencing alone prevents
My designing a new skin.
We cannot hire its warehouse staff;
We cannot choose its motor fleet;
We cannot source its cargo planes;
Or its trucks route through our streets.
But this is all self-preferencing;
And it cannot all be banned;
Unless we choice's value weigh,
We strike with arbitrary hand.
So say you and differ I:
'Twixt dinosaur and man must choose.
If one alone fits on this earth—
Wilt for man our power use?
These verses are based in part on arguments summarized in this blog post and this paper.
In Amazon, Antitrust & Competition, Antitrust's Uncertain Future, Apple, Google, Platforms, Symposia, Truth on the Market American Innovation and Choice Online Act, self-preferncing
AICOA Is Neither Urgently Needed Nor Good: A Response to Professors Scott Morton, Salop, and Dinielli
Thom Lambert — 25 July 2022
Earlier this month, Professors Fiona Scott Morton, Steve Salop, and David Dinielli penned a letter expressing their "strong support" for the proposed American Innovation and Choice Online Act (AICOA). In the letter, the professors address criticisms of AICOA and urge its approval, despite possible imperfections.
"Perhaps this bill could be made better if we lived in a perfect world," the professors write, "[b]ut we believe the perfect should not be the enemy of the good, especially when change is so urgently needed."
The problem is that the professors and other supporters of AICOA have shown neither that "change is so urgently needed" nor that the proposed law is, in fact, "good."
Is Change 'Urgently Needed'?
With respect to the purported urgency that warrants passage of a concededly imperfect bill, the letter authors assert two points. First, they claim that AICOA's targets—Google, Apple, Facebook, Amazon, and Microsoft (collectively, GAFAM)—"serve as the essential gatekeepers of economic, social, and political activity on the internet." It is thus appropriate, they say, to amend the antitrust laws to do something they have never before done: saddle a handful of identified firms with special regulatory duties.
But is this oft-repeated claim about "gatekeeper" status true? The label conjures up the old Terminal Railroad case, where a group of firms controlled the only bridges over the Mississippi River at St. Louis. Freighters had no choice but to utilize their services. Do the GAFAM firms really play a similar role with respect to "economic, social, and political activity on the internet"? Hardly.
With respect to economic activity, Amazon may be a huge player, but it still accounts for only 39.5% of U.S. ecommerce sales—and far less of retail sales overall. Consumers have gobs of other ecommerce options, and so do third-party merchants, which may sell their wares using Shopify, Ebay, Walmart, Etsy, numerous other ecommerce platforms, or their own websites.
For social activity on the internet, consumers need not rely on Facebook and Instagram. They can connect with others via Snapchat, Reddit, Pinterest, TikTok, Twitter, and scores of other sites. To be sure, all these services have different niches, but the letter authors' claim that the GAFAM firms are "essential gatekeepers" of "social… activity on the internet" is spurious.
Nor are the firms singled out by AICOA essential gatekeepers of "political activity on the internet." The proposed law touches neither Twitter, the primary hub of political activity on the internet, nor TikTok, which is increasingly used for political messaging.
The second argument the letter authors assert in support of their claim of urgency is that "[t]he decline of antitrust enforcement in the U.S. is well known, pervasive, and has left our jurisprudence unable to protect and maintain competitive markets." In other words, contemporary antitrust standards are anemic and have led to a lack of market competition in the United States.
The evidence for this claim, which is increasingly parroted in the press and among the punditry, is weak. Proponents primarily point to studies showing:
increasing industrial concentration;
higher markups on goods and services since 1980;
a declining share of surplus going to labor, which could indicate monopsony power in labor markets; and
a reduction in startup activity, suggesting diminished innovation.
Examined closely, however, those studies fail to establish a domestic market power crisis.
Industrial concentration has little to do with market power in actual markets. Indeed, research suggests that, while industries may be consolidating at the national level, competition at the market (local) level is increasing, as more efficient national firms open more competitive outlets in local markets. As Geoff Manne sums up this research:
Most recently, several working papers looking at the data on concentration in detail and attempting to identify the likely cause for the observed data, show precisely the opposite relationship. The reason for increased concentration appears to be technological, not anticompetitive. And, as might be expected from that cause, its effects are beneficial. Indeed, the story is both intuitive and positive.
What's more, while national concentration does appear to be increasing in some sectors of the economy, it's not actually so clear that the same is true for local concentration — which is often the relevant antitrust market.
With respect to the evidence on markups, the claim of a significant increase in the price-cost margin depends crucially on the measure of cost. The studies suggesting an increase in margins since 1980 use the "cost of goods sold" (COGS) metric, which excludes a firm's management and marketing costs—both of which have become an increasingly significant portion of firms' costs. Measuring costs using the "operating expenses" (OPEX) metric, which includes management and marketing costs, reveals that public-company markups increased only modestly since the 1980s and that the increase was within historical variation. (It is also likely that increased markups since 1980 reflect firms' more extensive use of technology and their greater regulatory burdens, both of which raise fixed costs and require higher markups over marginal cost.)
As for the declining labor share, that dynamic is occurring globally. Indeed, the decline in the labor share in the United States has been less severe than in Japan, Canada, Italy, France, Germany, China, Mexico, and Poland, suggesting that anemic U.S. antitrust enforcement is not to blame. (A reduction in the relative productivity of labor is a more likely culprit.)
Finally, the claim of reduced startup activity is unfounded. In its report on competition in digital markets, the U.S. House Judiciary Committee asserted that, since the advent of the major digital platforms:
"[t]he number of new technology firms in the digital economy has declined";
"the entrepreneurship rate—the share of startups and young firms in the [high technology] industry as a whole—has also fallen significantly"; and
"[u]nsurprisingly, there has also been a sharp reduction in early-stage funding for technology startups." (pp. 46-47.)
Those claims, however, are based on cherry-picked evidence.
In support of the first two, the Judiciary Committee report cited a study based on data ending in 2011. As Benedict Evans has observed, "standard industry data shows that startup investment rounds have actually risen at least 4x since then."
In support of the third claim, the report cited statistics from an article noting that the number and aggregate size of the very smallest venture capital deals—those under $1 million—fell between 2014 and 2018 (after growing substantially from 2008 to 2014). The Judiciary Committee report failed to note, however, the cited article's observation that small venture deals ($1 million to $5 million) had not dropped and that larger venture deals (greater than $5 million) had grown substantially during the same time period. Nor did the report acknowledge that venture-capital funding has continued to increase since 2018.
Finally, there is also reason to think that AICOA's passage would harm, not help, the startup environment:
AICOA doesn't directly restrict startup acquisitions, but the activities it would restrict most certainly do dramatically affect the incentives that drive many startup acquisitions. If a platform is prohibited from engaging in cross-platform integration of acquired technologies, or if it can't monetize its purchase by prioritizing its own technology, it may lose the motivation to make a purchase in the first place.
Despite the letter authors' claims, neither a paucity of avenues for "economic, social, and political activity on the internet" nor the general state of market competition in the United States establishes an "urgent need" to re-write the antitrust laws to saddle a small group of firms with unprecedented legal obligations.
Is the Vagueness of AICOA's Primary Legal Standard a Feature?
AICOA bars covered platforms from engaging in three broad classes of conduct (self-preferencing, discrimination among business users, and limiting business users' ability to compete) where the behavior at issue would "materially harm competition." It then forbids several specific business practices, but allows the defendant to avoid liability by proving that their use of the practice would not cause a "material harm to competition."
Critics have argued that "material harm to competition"—a standard that is not used elsewhere in the antitrust laws—is too indeterminate to provide business planners and adjudicators with adequate guidance. The authors of the pro-AICOA letter, however, maintain that this "different language is a feature, not a bug."
That is so, the letter authors say, because the language effectively signals to courts and policymakers that antitrust should prohibit more conduct. They explain:
To clarify to courts and policymakers that Congress wants something different (and stronger), new terminology is required. The bill's language would open up a new space and move beyond the standards imposed by the Sherman Act, which has not effectively policed digital platforms.
Putting aside the weakness of the letter authors' premise (i.e., that Sherman Act standards have proven ineffective), the legislative strategy they advocate—obliquely signal that you want "change" without saying what it should consist of—is irresponsible and risky.
The letter authors assert two reasons Congress should not worry about enacting a liability standard that has no settled meaning. One is that:
[t]he same judges who are called upon to render decisions under the existing, insufficient, antitrust regime, will also be called upon to render decisions under the new law. They will be the same people with the same worldview.
It is thus unlikely that "outcomes under the new law would veer drastically away from past understandings of core concepts…."
But this claim undermines the argument that a new standard is needed to get the courts to do "something different" and "move beyond the standards imposed by the Sherman Act." If we don't need to worry about an adverse outcome from a novel, ill-defined standard because courts are just going to continue applying the standard they're familiar with, then what's the point of changing the standard?
A second reason not to worry about the lack of clarity on AICOA's key liability standard, the letter authors say, is that federal enforcers will define it:
The new law would mandate that the [Federal Trade Commission and the Antitrust Division of the U.S. Department of Justice], the two expert agencies in the area of competition, together create guidelines to help courts interpret the law. Any uncertainty about the meaning of words like 'competition' will be resolved in those guidelines and over time with the development of caselaw.
This is no doubt music to the ears of members of Congress, who love to get credit for "doing something" legislatively, while leaving the details to an agency so that they can avoid accountability if things turn out poorly. Indeed, the letter authors explicitly play upon legislators' unwholesome desire for credit-sans-accountability. They emphasize that "[t]he agencies must [create and] update the guidelines periodically. Congress doesn't have to do much of anything very specific other than approve budgets; it certainly has no obligation to enact any new laws, let alone amend them."
AICOA does not, however, confer rulemaking authority on the agencies; it merely directs them to create and periodically update "agency enforcement guidelines" and "agency interpretations" of certain affirmative defenses. Those guidelines and interpretations would not bind courts, which would be free to interpret AICOA's new standard differently. The letter authors presume that courts would defer to the agencies' interpretation of the vague standard, and they probably would. But that raises other problems.
For one thing, it reduces certainty, which is likely to chill innovation. Giving the enforcement agencies de facto power to determine and redetermine what behaviors "would materially harm competition" means that the rules are never settled. Administrations differ markedly in their views about what the antitrust laws should forbid, so business planners could never be certain that a product feature or revenue model that is legal today will not be deemed to "materially harm competition" by a future administration with greater solicitude for small rivals and upstarts. Such uncertainty will hinder investment in novel products, services, and business models.
Consider, for example, Google's investment in the Android mobile operating system. Google makes money from Android—which it licenses to device manufacturers for free—by ensuring that Google's revenue-generating services (e.g., its search engine and browser) are strongly preferenced on Android products. One administration might believe that this is a procompetitive arrangement, as it creates a different revenue model for mobile operating systems (as opposed to Apple's generation of revenue from hardware sales), resulting in both increased choice and lower prices for consumers. A subsequent administration might conclude that the arrangement materially harms competition by making it harder for rival search engines and web browsers to gain market share. It would make scant sense for a covered platform to make an investment like Google did with Android if its underlying business model could be upended by a new administration with de facto power to rewrite the law.
A second problem with having the enforcement agencies determine and redetermine what covered platforms may do is that it effectively transforms the agencies from law enforcers into sectoral regulators. Indeed, the letter authors agree that "the ability of expert agencies to incorporate additional protections in the guidelines" means that "the bill is not a pure antitrust law but also safeguards other benefits to consumers." They tout that "the complementarity between consumer protection and competition can be addressed in the guidelines."
Of course, to the extent that the enforcement guidelines address concerns besides competition, they will be less useful for interpreting AICOA's "material harm to competition" standard; they might deem a practice suspect on non-competition grounds. Moreover, it is questionable whether creating a sectoral regulator for five widely diverse firms is a good idea. The history of sectoral regulation is littered with examples of agency capture, rent-seeking, and other public-choice concerns. At a minimum, Congress should carefully examine the potential downsides of sectoral regulation, install protections to mitigate those downsides, and explicitly establish the sectoral regulator.
Will AICOA Break Popular Products and Services?
Many popular offerings by the platforms covered by AICOA involve self-preferencing, discrimination among business users, or one of the other behaviors the bill presumptively bans. Pre-installation of iPhone apps and services like Siri, for example, involves self-preferencing or discrimination among business users of Apple's iOS platform. But iPhone consumers value having a mobile device that offers extensive services right out of the box. Consumers love that Google's search result for an establishment offers directions to the place, which involves the preferencing of Google Maps. And consumers positively adore Amazon Prime, which can provide free expedited delivery because Amazon conditions Prime designation on a third-party seller's use of Amazon's efficient, reliable "Fulfillment by Amazon" service—something Amazon could not do under AICOA.
The authors of the pro-AICOA letter insist that the law will not ban attractive product features like these. AICOA, they say:
provides a powerful defense that forecloses any thoughtful concern of this sort: conduct otherwise banned under the bill is permitted if it would 'maintain or substantially enhance the core functionality of the covered platform.'
But the authors' confidence that this affirmative defense will adequately protect popular offerings is misplaced. The defense is narrow and difficult to mount.
First, it immunizes only those behaviors that maintain or substantially enhance the "core" functionality of the covered platform. Courts would rightly interpret AICOA to give effect to that otherwise unnecessary word, which dictionaries define as "the central or most important part of something." Accordingly, any self-preferencing, discrimination, or other presumptively illicit behavior that enhances a covered platform's service but not its "central or most important" functions is not even a candidate for the defense.
Even if a covered platform could establish that a challenged practice would maintain or substantially enhance the platform's core functionality, it would also have to prove that the conduct was "narrowly tailored" and "reasonably necessary" to achieve the desired end, and, for many behaviors, the "le[ast] discriminatory means" of doing so. That is a remarkably heavy burden, and it beggars belief to suppose that business planners considering novel offerings involving self-preferencing, discrimination, or some other presumptively illicit conduct would feel confident that they could make the required showing. It is likely, then, that AICOA would break existing products and services and discourage future innovation.
Of course, Congress could mitigate this concern by specifying that AICOA does not preclude certain things, such as pre-installed apps or consumer-friendly search results. But the legislation would then lose the support of the many interest groups who want the law to preclude various popular offerings that its text would now forbid. Unlike consumers, who are widely dispersed and difficult to organize, the groups and competitors that would benefit from things like stripped-down smartphones, map-free search results, and Prime-less Amazon are effective lobbyists.
Should the US Follow Europe?
Having responded to criticisms of AICOA, the authors of the pro-AICOA letter go on offense. They assert that enactment of the bill is needed to ensure that the United States doesn't lose ground to Europe, both in regulatory leadership and in innovation. Observing that the European Union's Digital Markets Act (DMA) has just become law, the authors write that:
[w]ithout [AICOA], the role of protecting competition and innovation in the digital sector outside China will be left primarily to the European Union, abrogating U.S. leadership in this sector.
Moreover, if Europe implements its DMA and the United States does not adopt AICOA, the authors claim:
the center of gravity for innovation and entrepreneurship [could] shift from the U.S. to Europe, where the DMA would offer greater protections to start ups and app developers, and even makers and artisans, against exclusionary conduct by the gatekeeper platforms.
Implicit in the argument that AICOA is needed to maintain America's regulatory leadership is the assumption that to lead in regulatory policy is to have the most restrictive rules. The most restrictive regulator will necessarily be the "leader" in the sense that it will be the one with the most control over regulated firms. But leading in the sense of optimizing outcomes and thereby serving as a model for other jurisdictions entails crafting the best policies—those that minimize the aggregate social losses from wrongly permitting bad behavior, wrongly condemning good behavior, and determining whether conduct is allowed or forbidden (i.e., those that "minimize the sum of error and decision costs"). Rarely is the most restrictive regulatory regime the one that optimizes outcomes, and as I have elsewhere explained, the rules set forth in the DMA hardly seem calibrated to do so.
As for "innovation and entrepreneurship" in the technological arena, it would be a seismic shift indeed if the center of gravity were to migrate to Europe, which is currently home to zero of the top 20 global tech companies. (The United States hosts 12; China, eight.)
It seems implausible, though, that imposing a bunch of restrictions on large tech companies that have significant resources for innovation and are scrambling to enter each other's markets will enhance, rather than retard, innovation. The self-preferencing bans in AICOA and DMA, for example, would prevent Apple from developing its own search engine to compete with Google, as it has apparently contemplated. Why would Apple develop its own search engine if it couldn't preference it on iPhones and iPads? And why would Google have started its shopping service to compete with Amazon if it couldn't preference Google Shopping in search results? And why would any platform continually improve to gain more users as it neared the thresholds for enhanced duties under DMA or AICOA? It seems more likely that the DMA/AICOA approach will hinder, rather than spur, innovation.
At the very least, wouldn't it be prudent to wait and see whether DMA leads to a flourishing of innovation and entrepreneurship in Europe before jumping on the European bandwagon? After all, technological innovations that occur in Europe won't be available only to Europeans. Just as Europeans benefit from innovation by U.S. firms, American consumers will be able to reap the benefits of any DMA-inspired innovation occurring in Europe. Moreover, if DMA indeed furthers innovation by making it easier for entrants to gain footing, even American technology firms could benefit from the law by launching their products in Europe. There's no reason for the tech sector to move to Europe to take advantage of a small-business-protective European law.
In fact, the optimal outcome might be to have one jurisdiction in which major tech platforms are free to innovate, enter each other's markets via self-preferencing, etc. (the United States, under current law) and another that is more protective of upstart businesses that use the platforms (Europe under DMA). The former jurisdiction would create favorable conditions for platform innovation and inter-platform competition; the latter might enhance innovation among businesses that rely on the platforms. Consumers in each jurisdiction, however, would benefit from innovation facilitated by the other.
It makes little sense, then, for the United States to rush to adopt European-style regulation. DMA is a radical experiment. Regulatory history suggests that the sort of restrictiveness it imposes retards, rather than furthers, innovation. But in the unlikely event that things turn out differently this time, little harm would result from waiting to see DMA's benefits before implementing its restrictive approach.
Does AICOA Threaten Platforms' Ability to Moderate Content and Police Disinformation?
The authors of the pro-AICOA letter conclude by addressing the concern that AICOA "will inadvertently make content moderation difficult because some of the prohibitions could be read… to cover and therefore prohibit some varieties of content moderation" by covered platforms.
The letter authors say that a reading of AICOA to prohibit content moderation is "strained." They maintain that the act's requirement of "competitive harm" would prevent imposition of liability based on content moderation and that the act is "plainly not intended to cover" instances of "purported censorship." They further contend that the risk of judicial misconstrual exists with all proposed laws and therefore should not be a sufficient reason to oppose AICOA.
Each of these points is weak. Section 3(a)(3) of AICOA makes it unlawful for a covered platform to "discriminate in the application or enforcement of the terms of service of the covered platform among similarly situated business users in a manner that would materially harm competition." It is hardly "strained" to reason that this provision is violated when, say, Google's YouTube selectively demonetizes a business user for content that Google deems harmful or misleading. Or when Apple removes Parler, but not every other violator of service terms, from its App Store. Such conduct could "materially harm competition" by impeding the de-platformed business' ability to compete with its rivals.
And it is hard to say that AICOA is "plainly not intended" to forbid these acts when a key supporting senator touted the bill as a means of policing content moderation and observed during markup that it would "make some positive improvement on the problem of censorship" (i.e., content moderation) because "it would provide protections to content providers, to businesses that are discriminated against because of the content of what they produce."
At a minimum, we should expect some state attorneys general to try to use the law to police content moderation they disfavor, and the mere prospect of such legal action could chill anti-disinformation efforts and other forms of content moderation.
Of course, there's a simple way for Congress to eliminate the risk of what the letter authors deem judicial misconstrual: It could clarify that AICOA's prohibitions do not cover good-faith efforts to moderate content or police disinformation. Such clarification, however, would kill the bill, as several Republican legislators are supporting the act because it restricts content moderation.
The risk of judicial misconstrual with AICOA, then, is not the sort that exists with "any law, new or old," as the letter authors contend. "Normal" misconstrual risk exists when legislators try to be clear about their intentions but, because language has its limits, some vagueness or ambiguity persists. AICOA's architects have deliberately obscured their intentions in order to cobble together enough supporters to get the bill across the finish line.
The one thing that all AICOA supporters can agree on is that they deserve credit for "doing something" about Big Tech. If the law is construed in a way they disfavor, they can always act shocked and blame rogue courts. That's shoddy, cynical lawmaking.
So, I respectfully disagree with Professors Scott Morton, Salop, and Dinielli on AICOA. There is no urgent need to pass the bill right now, especially as we are on the cusp of seeing an AICOA-like regime put to the test. The bill's central liability standard is overly vague, and its plain terms would break popular products and services and thwart future innovation. The United States should equate regulatory leadership with the best, not the most restrictive, policies. And Congress should thoroughly debate and clarify its intentions on content moderation before enacting legislation that could upend the status quo on that important matter.
For all these reasons, Congress should reject AICOA. And for the same reasons, a future in which AICOA is adopted is extremely unlikely to resemble the Utopian world that Professors Scott Morton, Salop, and Dinielli imagine.
In Amazon, Antitrust & Competition, Antitrust's Uncertain Future, Apple, Error Costs, European Union, Facebook, Federal Trade Commission, Google, Monopolization, Platforms, Price Discrimination, Symposia, Truth on the Market, Twitter, Tying, Vertical Restraints American Innovation and Choice Online Act, antitrust, competition, error costs, platforms, price discrimination, self-preferncing | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,768 |
La avenida General Las Heras es una arteria vial de la Ciudad de Buenos Aires, Argentina.
Historia
En 1580 Juan de Garay estableció los límites urbanos de la Ciudad de Buenos Aires: al Este la barranca del Río de la Plata (Avenida Paseo Colón - Av. Alem), al oeste las actuales calles Salta y Libertad, al Sur la actual Avenida Independencia y al norte la calle Viamonte. Garay también repartió las tierras más allá del ejido tanto al norte como al sur del nuevo centro urbano. Hacia el sur el reparto abarcó desde el Riachuelo hasta la zona de Ensenada y Magdalena, mientras que hacia el norte, la distribución comenzó desde la actual Plaza San Martín (Retiro) hasta lo que es hoy el Partido de San Fernando. Entre cada chacra (de una legua de largo) debía correr un camino, así como también por su frente y fondo. Hacia el norte, el camino del fondo lo constituían las actuales avenidas Constituyentes y Fondo de la Legua, mientras que el camino del frente, denominado «Camino de Santa Fe» o «Camino del Bajo» lo conformaban sucesivamente las actuales avenidas Libertador, Las Heras, Santa Fe, Luis María Campos y nuevamente Libertador, siempre discurriendo al pie de la barranca, elevándose sobre esta después de pasar el Partido de Vicente López.
Toponimia
La actual avenida Las Heras era originalmente una huella o rastrillada casi costera (el Río de la Plata estaba a poca distancia) que a fines del siglo XVIII comenzó a ser llamada «Camino de Chavango» ya que al parecer en tiempos del virrey Vértiz por ese camino se trajeron desde el noroeste algunas crías de llamas (la cría de llama es denominada popularmente «chavango») con el infructuoso propósito de efectuar su crianza y aprovechamiento en la entonces pequeña ciudad de Buenos Aires; esto casi un siglo más tarde dio motivo a una humorada en la que participó Lucio V. Mansilla: ya a fines del siglo XIX; estando en gran parte edificada para entonces la zona recorrida por ese camino se le declaró avenida pero olvidando el antiguo nombre popular, ante esto Mansilla y otros amigos elevaron una queja pública diciendo que se «afrentaba con el olvido a la memoria del Coronel Chavango», los burócratas municipales de ese tiempo totalmente ignorantes de las cuestiones públicas atendieron solicitamente el reclamo ya que provenía de «gente decente» y buscaron obtener más información del supuesto «Cnel. Chavango» hasta que Mansilla y sus amigos les explicaron que les estaban tomando en broma para demostrarles su ignorancia.
El nombre actual de la avenida homenajea al militar patriota Juan Gualberto Gregorio de las Heras destacado en la luchas por la Independencia Argentina y gobernador de Buenos Aires.
Arbolado y modificaciones
Por la Avenida Las Heras corrió en abril de 1897 el primer tranvía eléctrico de Buenos Aires, un tramo experimental construido por el inglés Charles Bright para unir Plaza Italia con la esquina de la actual Avenida Scalabrini Ortiz. Ese mismo año se inauguraba el primer servicio público de tranvía eléctrico, un transporte que funcionaría hasta 1962.
La avenida fue a comienzos del siglo XX un amplio bulevar con cantero central, en el tramo desde la calle Sánchez de Bustamante hasta Plaza Italia. Adornado por frondosas tipas que llegaban a tapar toda la calzada, servía además para el ordenamiento de tranvías, luego colectivos. Ya en la década de 1920, se plantaron además plátanos en ambas veredas en el tramo desde Avenida Callao hasta el Jardín Botánico, en donde esto no fue necesario porque ya sus ejemplares cubrían la avenida hasta Plaza Italia.
En los años 40, se experimentó con el uso de adoquines de quebracho en búsqueda de minimizar el ruido que hacían los vehículos y el traqueteo que los dañaba con los clásicos adoquines de granito que se iban redondeando con el paso del tiempo. El sistema fracasó, y finalmente se adoptó el asfalto como revestimiento para las calles porteñas. Para la década de 1960, el cantero central había desaparecido totalmente para conseguir más carriles de circulación, en una avenida cuyo tránsito había aumentado fuertemente a medida que la zona pasaba de barrio de casas familiares a cada vez más edificios de departamentos de varios pisos. En esa misma época, suspendido el servicio del tranvía, los colectivos pasaron a ocupar la mayor parte de esta importante arteria.
En la actualidad, los plátanos centenarios sobreviven en el tramo desde avenida Callao hasta la calle Sánchez de Bustamante, aunque varios ejemplares fueron retirados, algunos reemplazados por otras especies arbóreas. En cambio, desde Sánchez de Bustamante hasta la calle República Árabe Siria, los viejos árboles fueron talados y reemplazados por árboles más jóvenes. Desde este punto hasta Plaza Italia, reaparece la densa arboleda centenaria.
Recorrido
Nace en la Plaza Vicente López (antiguamente llamada "Hueco de las Cabecitas", porque era un matadero de ovejas), partiendo de la Calle Montevideo, en el barrio de Recoleta. Durante sus primeros 200 metros es una calle angosta de un solo sentido de circulación, con una añeja arboleda.
Al cruzar la Avenida Callao su anchura aumenta y se hace doble mano. En Las Heras 1837 se encuentra el edificio anexo del Colegio de Escribanos, obra de 1997 del arquitecto Clorindo Testa, de estilo moderno y fachada de hormigón a la vista. En Las Heras 1855 funciona la Comisaría 17 de la Policía Federal; y en la esquina con la calle Ayacucho está la facultad de medicina Barcelo.
En Las Heras 2214 se encuentra un imponente edificio neogótico, proyectado por el arquitecto Arturo Prins en 1912 y jamás finalizado. Fue pensado como Facultad de Derecho, pero hoy es una sede de la Facultad de Ingeniería de la UBA. En la siguiente cuadra está la Plaza Emilio Mitre; y luego, el cruce con la Avenida Pueyrredón. Allí están la Plazoleta Reino de Tailandia, y la Avenida Gelly y Obes, que entra a un pequeño y exclusivo barrio residencial conocido como La Isla.
Entre las calles Agüero y Austria está la Plaza del Lector, acceso a la Biblioteca Nacional, otro edificio de Clorindo Testa, imponente estructura brutalista que se sostiene sobre cuatro columnas de grandes proporciones. Se construye allí el Museo del Libro y el Autor Argentino, antes del mismo se encuentra la Embajada de Paraguay. En la vereda opuesta, están una torre que pertenece a Telecom y la Iglesia de San Agustín, de estilo gótico. Entre las calles Austria y Sánchez de Bustamante está el predio del Hospital Rivadavia, inaugurado en 1887 y conformado por pabellones distribuidos en un amplio parque.
En Las Heras 3084 está la Escuela Primaria n.º 18 "Dr. Rafael Herrera Vegas", una de las muchas encargadas por el Estado al arquitecto Waldorp en la década de 1920, y junto a ella está la sede de la Academia Nacional de Medicina. Luego del cruce con la Avenida Coronel Díaz se extiende el Parque Las Heras, lugar donde se emplazó hasta 1962 la Penitenciaría Nacional. Ahí comienza el Barrio de Palermo, donde se alzan las Torres Alto Palermo, gemelas de 130 metros de altura. Dentro del parque está la Escuela Primaria n.º 26 "Adolfo Van Gelderen".
Entre las calles Aráoz y Scalabrini Ortiz está la Plaza Alférez Sobral, y el Instituto Santa Teresa de Jesús. Pasando la calle Ugarteche se alza el llamado Palacio de los Gansos, un gran edificio residencial de estilo racionalista, basamento revestido en piedra rústica y con un gran parque privado. A partir de la calle República Árabe Siria comienza el Jardín Botánico. La avenida continúa hasta Plaza Italia donde desemboca en la Avenida Santa Fe. Allí se encuentran la entrada principal del Zoológico de Buenos Aires y la sede de la Sociedad Rural Argentina y predio de exposiciones La Rural.
Cruces importantes y lugares de referencia
Recoleta
1600-1800: Tramo de mano única
1600: Calle Montevideo - Plaza Vicente López
1800-2700: Tramo de doble mano
1800: Avenida Callao - inicio de doble mano
2300: Avenida Pueyrredón - Plaza Emilio Mitre - Cementerio de la Recoleta - Estación Las Heras de la Línea H del subte
2600: Calle Austria - Hospital General de Agudos Bernardino Rivadavia
Palermo/Recoleta
2700-3200: Tramo de doble mano
Palermo
3200-4300: Tramo de doble mano
3200: Avenida Coronel Díaz - Parque Las Heras
3400: Calle Silvio Ruggieri - Hospital General de Agudos Dr. Juan A. Fernández
3700: Avenida Raúl Scalabrini Ortiz - Plaza Alférez Sobral
4300: Avenida Santa Fe - Avenida Sarmiento - Plaza Italia - Jardin Botánico Carlos Thays - Ecoparque Interactivo de Buenos Aires ex Zoológico - Predio ferial de la Sociedad Rural Argentina - Estación Plaza Italia de la Línea D del subte
Galería de imágenes
Referencias
Enlaces externos
Calles del barrio de Palermo (Buenos Aires)
Calles del barrio de Recoleta (Buenos Aires)
Epónimos de Juan Gregorio de Las Heras
Avenidas de la ciudad de Buenos Aires | {
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Advanced Disk Space Monitor is an essential piece of software for system administrators who wish to remotely monitor disk space on servers and high-performance workstations. This is the perfect solution to avoid unexpected lack of disk space on systems with growing logs, enlarging databases, data backups, caching data, web servers, etc.
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Supported formats include Wav and mp3 audio files.
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The server and desktop editions are combined into a single solution: Advanced Disk Space Monitor.
1. Added support for Terabytes (TB) and Petabytes (PB).
2. Added ability to edit subject of email reports.
3. Minor improvements in email reports.
5. Improved service installation and uninstallation.
6. Fixed bug disabling automatic startup with Windows.
7. Fixed issue causing "Send test email" button to become permanently disabled.
8. Changed names of some interface items.
9. Changed main icon and system tray icons.
10. Floppy disks and CDs are no longer supported.
Recommended update for all users.
Fixed an infinite loop accessing the settings file. | {
"redpajama_set_name": "RedPajamaC4"
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Nicolas Cage owned haunted house
Writer: Florey DM
Nic Cage buys a haunted house for the sake of horror novel writing.
23 Sep – Nicolas Cage previously owned the "most haunted house in America", the actor himself admitted while speaking in the Daily Mail newspaper's Event magazine.
Cage bought the LaLaurie Mansion in New Orleans in 2007 and lived in it for a short period of time before selling it in 2009. He was hoping to gain some inspiration in writing his American horror novel.
The house used to belong to Madame Lalaurie, a Louisiana-born 19th century socialite, who was also a notorious serial killer known for torturing and murdering her slaves.
Alas the house did not do much to his writing goal as Cage admits, "I didn't get too far with the novel."
Cage sold the haunted LaLaurie Mansion after living in it for two years.
The "Left Behind" actor is no stranger to spooky happenings. He recounted the time when someone once told him his son Weston, nine years old at the time, would be dead before the age of fifteen. But thankfully the anonymous person's premonition is wrong as Weston is now 23 and still very much alive.
He also isn't one to shy away from spooky happenings. Idris Elba, Cage's co-star on "Ghost Rider: Spirit of Vengeance", revealed during his recent 'Ask Me Anything' Q&A on Reddit that the 50-year-old actor once spent the night in Dracula's Castle.
The cast was shooting in Romania, Transylvania when Cage decided he wanted to channel the spooky energy in the house and slept there, only telling his co-star about it the following morning.
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Watch Nic Cage hunt down a dangerous assassin aboard a ship full of uncaged wild animals | {
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PubDate: 1790-05-31
Slug: letters/george-washington/to-his-excellency-george-washington-may-31-1790
Title: To His Excellency George Washington May 31, 1790
Date: Mon Aug 4 09:08:54 EDT 2014
From the original letter at the Library of Congress.
London, May 31, 1790
Sir
By Mr. James Morris, who sailed in the May Packet, I transmitted you a
letter from the Marquis de la Fayette, at the same time informing you that
the Marquis had entrusted to my charge the Key of the Bastille and a
drawing of that Prison as a present to your Excellency. Mr. J. Rutledge,
jun'r had intended coming in the ship Marquis de la Fayette and I had
chosen that opportunity for the purpose of transmitting the present —
but the Ship not sailing at the time appointed Mr. Rutledge takes his
passage, on the Packet, and I have committed to his care those trophies of
liberty which I know it will give you pleasure to receive — The french
Revolution is not only compleat but triumphant and the envious despotism of this
nation is compelled to own the magnanimity with which it has been conducted
—
The political hemisphere is again clouded by a dispute between England and
Spain the circumstances of which you will hear before this letter can
arrive — A Messenger was sent from hence the 6th Inst. to Madrid with very
peremptory demands, and to wait there only forty-eight hours — his return
has been expected for two or three days past — I was this morning at the
Marquis del Campo's but nothing is yet arrived — Mr. Rutledge sets off at
four o'clock this afternoon, but should any news arrive before the making
up the mail on Wednesday June 2 I will forward it to you under cover.
The views of this Court as well as of the Nation so far as they extend to
South America, are not for the purpose of Freedom but Conquest — They
already talk of sending some of the young Branches to Reign over them and
to pay off their National Debt with the produce of their mines — The
Bondage of those Countries will, as far as I can perceive, be prolonged by
what this Court has in contemplation.
My Bridge is arrived and I have engaged a place to erect it in — a little
time will determine its fate but I yet see no cause to doubt of its
success — tho' it is very probable that a War, should it break out will,
as in all new things, prevent its progress so far as regards profits —
In the partition in the Box, which contains the Key of the Bastille I
have put up half a dozen Razors, manufactured from cast-steel made at the
works where the bridge was constructed which I request you to accept as a
little token from a very grateful heart.
I received about a Week ago a letter from Mr. G Clymer It is dated the
4th Feby — but has been travelling ever since — I request you to
acknowledge it for me and that I will answer it when my Bridge is erected —
With much affection to all my Friends and many wishes to see them again
I am Sir your much obliged and Obedient humble Servant
THOMAS PAINE
| {
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Upstate veterans "wrapped" in Quilts of Valor
by: Ben Hoover
Posted: Nov 20, 2019 / 08:11 PM EST / Updated: Nov 21, 2019 / 05:55 AM EST
SPARTANBURG, S.C. (WSPA) – Three Upstate veterans were honored for their service in a special ceremony in Cleveland Park Tuesday night.
The Spartanburg chapter of the Quilts of Valor Foundation presented and wrapped the veterans in handmade quilts made by local volunteers.
Quilts of Valor was started in 2003 by a Blue Star Mom and since then thousands of quilters across the country have created quilts to cover warriors from all wars.
9 veterans receive quilts in "Folds of Honor" ceremony
One of the veterans, Army Lt. Col. (Ret.) John Ceasar Phillips, spoke with 7 News about this expression of gratitude and comfort.
"Oh, it's wonderful. It was a total surprise. In fact, tonight was a surprise. We thought it was Thursday and they showed up at a quarter til six and I hadn't eaten yet," Phillips said.
It's a good thing the event organizers fed the veterans and their families at the ceremony.
Organizers said South Carolina has led the nation with presenting the most quilts to veterans over the last several years.
by WSPA staff / Jan 15, 2021
RUTHERFORD COUNTY, NC (WSPA) - A man is wanted in Rutherford County on several charges, including assault and kidnapping.
25-year-old Mitchell Joseph Martin is wanted for a domestic violence assault, according to the Rutherford County Sheriff's Office.
Carolina's Family videos
Miracle Hill open as cold weather shelter on Christmas Eve, Christmas Day
More Carolina's Family | {
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} | 457 |
{"url":"https:\/\/nyates314.wordpress.com\/tag\/trig\/","text":"# Tag Archives: trig\n\n## Trigonometry with Algebra\u00a02\n\nSince this May may be my last month teaching math (for a while? forever? I doubt my departure from math will be permanent \u2026 more on the story behind this later), I thought I\u2019d get into the swing of things and connect back into the theme of my blog by declaring that May is Maryland\u2019s Math Madness Month! [To give due credit: this motto was originally thought up by my father as a way I could get kids excited about taking the Algebra HSAs, coming up now in less than two weeks.]\n\nTo kick off the month of math madness, I wish to ask other teachers of Algebra 2 with Trigonometry (A2T): how do you balance covering all of a (full-year or full-semester) Algebra 2 course in a reduced time-frame so that there is room in your course also for Trig?\n\nA2T Collage\n\nI suppose there are two main options for dealing with this addition of new material into an already full course: leaving out pieces of content, or teaching all the content at a faster pace.\n\nThis is something I think about every year I guess, but even moreso recently when helping students with their Algebra 2 Twilight make-up coursework and having discussions with the Precalculus teacher about what he will expect my students to bring with them to that class.\n\nThe Twilight students had question after question to answer about completing the square, which is a method I don\u2019t cover when teaching A2T. For me, solving quadratics by factoring, \u201cdoing the opposite\u201d, and using the quadratic formula is exhausting enough, both in the sense that it exhausts the techniques required to solve quadratics of every form, and in that my students are tired of so many methods without adding a fourth. My students have been know to complain that they are learning something new every day in my class (!). \u00a0Am I wrong for leaving out this method? Should I perhaps teach completing the square instead of the quadratic formula since it shows deeper understanding of the math involved? I don\u2019t have enough time to do both (plus the other two methods I mentioned, which are even more fundamental).\n\nSimilarly, I treat complex numbers very lightly (last year, with the 9++ snow days, I even skipped them!). And I hear on the web about some Algebra 2 teachers teaching rational functions, which I never even conceived of as an Algebra 2 topic, since gaining an abiding understanding of polynomials is challenge enough.\n\nSo I guess that lands me primarily on the side of leaving out content. A faster-paced curriculum would leave more students lost, and I do not have selection criteria for entering the class as some teachers might. Additionally, this relates to my philosophy of math teaching, that it\u2019s better to learn fewer things deeply than to shallowly cover everything. I try to focus my attention on the things that connect A2T to prior math and future math (e.g. \u201cdoing the opposite\u201d as equation-solving technique and function transformations), that connect it to other subjects, that engage students with project-based learning, and that highlight big picture concepts and skills.\n\nBut it\u2019s still a struggle, and I doubt myself (maybe I really should be teaching completing the square; maybe conic sections are more important than my Olympics research project and should replace it in my choice of topics). Especially since that Precalc teacher is counting on me to teach them certain things they will need when they arrive in that class (and just as the Calculus professor is counting on Precalc teachers to cover certain key topics).\n\nSo, A2T teachers, how do you deal with the pressure? Do you teach at hyperspeed, or what topics do you cut? [Other teachers feel free to weigh in too \ud83d\ude42 ]\n\nFiled under math, teaching\n\n## Algebra 2 \/ Trig\u00a0Skills\n\nHi! I\u2019m looking for some feedback on what skills I should use in my standards-based grading Algebra 2 with Trigonometry (A2T) course spring semester. I just threw together the following list as a rough draft, and do expect to edit it extensively over the coming week.\n\nA note on organization: I\u2019m planning to focus on algebra third quarter and analysis fourth quarter (did I miscategorize any skills?). Fourth quarter tends to be the shortest quarter, mainly due to our state standardized tests, so it has fewer skills. The first six skills each quarter are meant to be prerequisite skills from Algebra I, Geometry, or even middle-school math; I will hold students accountable for knowing and remembering these skills, but aside from a miniature review I do not intend to teach skills #1-6. I\u2019ve italicized skills that seem weak or a little too easy, and may be combined with others or even eliminated. I\u2019ve bolded skills that seem more advanced or non-essential and possibly not for every student. I\u2019m toying with the idea of requiring students demonstrate mastery of certain numbers of core skills and advanced skills to earn different letter grades.\n\nALGEBRA (especially of the second degree)!\n\n1. Graph a line given its equation\n2. Determine the equation of a line from its graph\n3. Solve a linear equation in any form\n4. Solve a system of two linear equations\n5. Manipulate algebraic expressions to simplify or expand\n6. Substitute numbers for variables in an algebraic expression\n7. Convert between scientific and standard notations of numbers\n8. Apply the rules of exponents to simplify an expression\n9. Find the degree and leading coefficient of a polynomial\n10. Determine the end behavior of a function\n11. Combine like terms to add or subtract polynomials\n12. Use the distributive property to multiply polynomials\n13. Use long division to divide polynomials\n14. Factor a monic quadratic expression\n16. Characterize the shape of a parabola given its equation\n17. Graph a parabola given its equation\n18. Determine the equation of a parabola given its graph\n19. Solve quadratic equations in factored form using the zero product property\n20. Solve quadratic equations in vertex form by isolating the variable\n22. Solve quadratic equations in standard form by completing the square\n23. Combine like terms to get an equation in standard form\n24. Apply quadratic equations to physics or other real-world scenarios\n25. Plot a complex number in the complex plane\n26. Add and subtract complex numbers\n27. Multiply complex numbers\n28. Determine the magnitude of a complex number\n29. Solve quadratic equations with complex roots\n30. Add and subtract matrices, and multiply a matrix by a scalar\n31. Multiply matrices\n32. Find matrix determinants\n33. Find the inverse of a matrix\n34. Solve matrix equations\n35. Solve systems of linear equations\n36. Graph linear inequalities\n37. Solve systems of linear inequalities by linear programming\n38. Plot the graph of various conic sections based on its equation\n39. Identify key characteristics of each conic section\n40. Explain the relationship among the conic sections\n\nANALYSIS (including TRIGONOMETRY)!\n\n1. Determine a function\u2019s domain, range, maxima, and minima\n2. Determine a function\u2019s zeros and y-intercepts\n3. Determine the end behavior of a function\n4. Convert between tables of values, equations, and graphs of functions\n5. Apply the Pythagorean Theorem to find unknown sides in right triangles\n6. Use trigonometry to find unknown sides & angles in right triangles\n7. Use laws of sines and cosines to find unknown sides & angles in any triangle\n8. Solve entire triangles given three pieces of information\n9. Apply triangular trigonometry to real-world scenarios\n10. Draw and measure angles in standard position\n11. Convert between degrees and radians\n12. Find coordinates and slope where an angle meets the unit circle\n13. Solve trigonometric equations (including multiple solutions)\n14. Use common trigonometric identities\n15. Sketch the graph of trigonometric functions\n16. Find period and amplitude of periodic functions\n17. Plot points using polar coordinates\n18. Convert between polar and Cartesian coordinates\n19. Plot various polar functions\n20. Evaluate exponential expressions\n21. Apply the rules of exponents to simplify an expression\n22. Identify characteristics of exponential functions\n23. Use exponential functions to model real-world scenarios\n24. Convert between exponential and logarithmic equations\n25. Evaluate logarithmic expressions\n26. Apply the rules of logarithms to simplify an expression\n27. Solve exponential and logarithmic equations\n28. Use logistic growth to model real-world scenarios\n29. Identify the effects of common function transformations\n30. Graph functions and their inverses\n31. Use inverse operations to solve equations\n32. Compose functions\n33. Discuss the effect of transforming, inverting, and composing functions on domain and range\n\nI\u2019m looking for feedback of any kind now, all the way from how to rephrase a skill better and more specifically, to what quintessential A2T skills are missing that I need to add right away. Also, are there some skills here that I can get rid of entirely? Even with the idea of prerequisite\/core\/advanced skills, 73 seems like way too many (I had only 32 skills this fall for geometry).\n\n[PS I promise a post up soon reflecting back on what worked and what didn\u2019t from semester 1\u2019s courses, and looking ahead to my plans for the new semester\u2019s courses, including how those plans reflect on SBG and PBL.]\n\nFiled under math, teaching\n\n## The Thrill of the\u00a0Chase\n\n\u201cWe need these experiences \u2014 noticing patterns, making conjectures, struggling with them mostly unaided by the teacher, researching appropriate related material, crafting a series of flawed-but-each-one-better-than-the-last arguments for why the conjecture could be true, creating a good logical argument and then tweaking it to be airtight \u2014 to be the essence of mathematics learning from a young age.\u201d\n\nAppropriately, after writing that yesterday, I got caught up this afternoon in a challenging math question posed by @CmonMattTHINK: \u201cAre there any triangles w\/ integer side lengths and exactly one 60\u00b0 angle?\u201d\n\nIf you want to think more about this question yourself, stop reading. I intend to describe my thought processes and answer here, in the interests of thinking through how to create or encourage those thought processes in students.\n\n#### Mathematical Problem-Solving and Proof\n\nThese two processes should not be separated, since they are so intimately connected.\n\nI began actually by stumbling halfway into a conversation of other people. @DrMathochist suggested using the law of cosines to create a Diophantine equation:\n\n$c^2 = a^2 + b^2 - ab.$\n\nThen @k8nowak discovered a triple of sides that worked: (3, 8, 7). Note that $3^2 + 8^2 - 3 \\cdot 8 = 9 + 64 - 24 = 49 = 7^2$ and that the resulting triangle does indeed have a 60\u00b0 angle. And @calcdave used the law of sines to express side $a$ in terms of side $b$ and the angle across from $b$ (with a sign error).\n\nNoticing Patterns. At this point, I embarked on my own quest to solve the problem. I noticed that clearly any integer multiples of (3, 8, 7) \u2014 like (6, 16, 14) \u2014 will also satisfy our equation (of course they will, since they will be similar to the above-linked triangle and therefore also have an angle of 60\u00b0). Thus, there are in fact infinitely many triangles that fit @CmonMattTHINK\u2019s question. Rather than stopping there with this answer to the original question, though, it makes sense to ask if there are any other triangles which are not multiples of (3, 8, 7) \u2014 that is, going forth we can limit our search to primitive triples wherein the greatest common divisor of the three sides is 1.\n\nFailed Proofs and Stumbling in Wrong Directions. Next, I am\u00a0embarrassed\u00a0to say I spent a while using the law of sines (along the track that @calcdave had suggested) to prove that $3=3.$ I should have recognized (but didn\u2019t) when I set two complicated trigonometric expressions equal that had just come from different sides of the law of sines. Then I tried factoring the Diophantine equation above to get $c^2 = (a + \\zeta b)(a + \\zeta^2 b)$, where $\\zeta = \\frac{-1+\\sqrt{-3}}{2}$\u00a0is a cube root of unity. Which led nowhere. Finally, I\u00a0desperately\u00a0tried looking at the continued fractions and rational approximations for some irrational numbers that were close to the ratios 3\/8, 3\/7, etc. No luck. The continued fractions I sought are shown below, using the software Pari\/GP.\n\nMore Patterns. At this point, I considered writing a program to do a brute force search for more triples. But more handy at the moment was Microsoft Excel, where I could display a number of combinations of $a,$ $b,$ and\u00a0$a^2 + b^2 - ab,$ and search visually for any that I saw to be perfect squares. I hoped to notice some patterns that could fuel my further work. I found (5, 8, 7), and in general saw that if $(a, b, c)$ is a triple, then $(b-a, b, c)$ will also be a triple (assuming $b>a$). Also encountered were (7, 15, 13) and (5, 21, 19), together with their complements (8, 15, 13) and (16, 21, 19). A piece of my spreadsheet is seen below.\n\nResearch. At this point, remembering back to the proof and generation of infinitely many primitive Pythagorean triples from the unit circle, I divided through the Diophantine equation by $c^2$ to get a planar curve $1 = x^2 + y^2 - x y$ on which to look for rational points $(x, y) = (a\/c, b\/c).$ I graphed this curve in GeoGebra, and plotted the points associated with the triples I had found, as seen here.\n\nI realized that the only rational points on this curve (ellipse?) that I need be concerned with were in the 45\u00b0 half-quadrant where you see the cluster of points, between (0,1) and (1,1). Clearly negative rational numbers are irrelevant to the sides of a triangle, and the points to the right of (1,1) on the downslope to (1,0) were just rearrangements of which side of the triangle got called $a$ or $b.$\n\nToward a Solution. OK, we\u2019re in the home stretch now. I pretty much just followed along with what I remembered of the Pythagorean-circle proof. I picked the simplest rational point that I knew was on the curve: (0,1), which doesn\u2019t correspond to any actual triangle, but will soon produce triangles by the method of drawing lines through it. We shall see shortly that lines with rational slope through (0,1) will intersect the curve at rational points (i.e. points where both $x$ and $y$ are rational, and thus yielding solutions to our goals of integer solutions of $c^2 = a^2 + b^2 - a b$ and triangles with angles of 60\u00b0).\n\nNote that a slope of 0 through (1,0) would go through (1,1), which represents the equilateral triangle where all angles are 60\u00b0. (1,1) is also the point of reflection symmetry, past which we need not go unless we care about the order of $(a,b).$ At (0,1) the curve has instantaneous slope $\\frac{dy}{dx} = \\frac{1}{2},$ so larger slopes than $\\frac{1}{2}$ will intersect the curve in other quadrants. Thus, choosing rational slope $m = p\/q \\in (0, \\frac{1}{2}),$ the line\u2019s equation using good-old point-slope form will be $y = m x + 1.$\n\nCombining this with the curve\u2019s equation and solving for $x,$ we get:\n\n$1 = x^2 + y^2 - x y$\n\n$1 = x^2 + (m x + 1)^2 - x (m x + 1)$\n\n$1 = x^2 + m^2 x^2 + 2 m x + 1 - m x^2 - x$\n\n$0 = x (x + m^2 x + 2 m - m x -1)$\n\n$0 = x + m^2 x + 2 m - m x -1$\n\n$1 - 2m = m^2 x - m x + x$\n\n$x = \\frac{1 - 2 m}{m^2 - m + 1}$\n\nNote that $x$ is nonzero because our triangle must have sides that are nonzero.\n\nAlthough $m$ is rational and constrained to the interval $(0,\\frac{1}{2}),$ we\u2019d actually like to express it in terms of natural numbers $p$ and $q,$ where we require that $gcd(p,q)=1$ and $2p < q.$ Leaving aside the messy algebra, we substitute $p\/q$ in for $m$ above, and get that:\n\n$x = \\frac{q^2 - 2 p q}{p^2 - p q + q^2},$ and\u00a0$y = \\frac{q^2 - p^2}{p^2 - p q + q^2},$ which correspond to sides of the triangle\n\n$(a, b, c) = (q^2 - 2 p q, q^2 - p^2, p^2 - p q + q^2).$\n\nI verified that these expressions do in fact satisfy the relation $c^2 = a^2 + b^2 - a b,$ thus completing the proof that there are infinitely many primitive triples of sides that form a triangle with one 60\u00b0 angle. [A few easily verifiable details have been left out here.] In fact, not only do we know there are infinitely many, but we can generate them at will. Just pick natural numbers $p$ and $q$ that have no common divisors and where $q$ is more than twice as big as $p.$ Then the triple given above, $(a, b, c) = (q^2 - 2 p q, q^2 - p^2, p^2 - p q + q^2),$\u00a0will form a triangle with a single angle of 60\u00b0.\n\nThe Role of Counterexample in Proof. In trying to prove that this method would always generate a primitive triple (which I believed at first), I actually found a counterexample! For example, $p = 1, q = 5$ yields $(a, b, c) = (15, 24, 21),$ which is a multiple of 3 times the primitive triple (5, 8, 7). But it doesn\u2019t seem that (5, 8, 7) can ever be generated using this method.\n\nI\u2019m pretty sure I have proved that this can only arise if the sum $p + q$ is divisible by 3. If so, this method will generate a triple of numbers thrice a primitive triple. That is, the numbers generated do indeed form a triangle with integer sides and a 60\u00b0 angle, but which triangle dilated by 1\/3 will still satisfy the same criteria. We can call these triple triples, since the triple generated is triple (i.e. three times) a primitive triple.\n\nIf the sum\u00a0$p + q$ is congruent to 1 or 2, modulo 3, then the triple generated will indeed be primitive.\n\nOpen Questions (Unknown by me, anyway).\n\n\u2022 Is there an alternative method which always generates primitive triples?\n\u2022 Can my method ever generate (5, 8, 7) or any other of the primitive triples for which a triple triple is generated?\n\nFiled under math, teaching\n\n## Vector Video\n\nI was thinking recently about how best to teach trigonometry, and especially how to teach the long and difficult processes of vector analysis and truss calculation.\u00a0 Now, it seems to me that these processes aren\u2019t really going to sink in unless a student practices them a number of times. But that sort of independent practice is difficult to achieve for such complex, many-step processes.\n\nNow, I can do guided practice of these processes by letting students lead me through a 20-minute example on the board.\u00a0 But then the student gets little individual time to work it out herself.\u00a0 If I try to have the students work a problem on their own too early, I\u2019ll be called in fifteen directions at once as each student gets stuck in a different place and needs one-on-one help.\u00a0 To varying degrees of success, I may try to scaffold a problem by doing some parts for a student or asking him leading questions; or I may list the steps in the algorithm on the front screen while students work on applying it.\n\nHowever, to get the students all the practice they need to get comfortable with these topics, they do need to work on them in homework too.\u00a0 And I cannot be there to scaffold the instruction; I can provide neither one-on-one assistance nor lead a classroom guided practice.\n\nIn an effort to help students with their homework in these areas, I decided to videotape myself working through a vectors example:\u00a0 that way they could see the process while working on a homework assignment, with the ability to pause to work on a similar part in their own example, and the ability to rewind to hear a step over again.\n\nI\u2019d appreciate any feedback, and especially constructive criticism, as I\u2019d like to make videos a recurring part of my teaching this year.\u00a0 I intend to make a truss calculations video quite soon, to follow up on these ideas.\n\n1 Comment\n\nFiled under engineering, math, teaching\n\n## Trigonometry Everywhere\n\nLooking back on the courses, units, and concepts\/skills I have taught, I think right-triangle trigonometry must be the topic I have taught most often.\n\nFive years ago, I student taught for several months in a western Massachusetts high school.\u00a0 The first unit I was given to prepare\/teach\/assess\/grade by myself (instead of just doing one or two of these tasks, assisting and learning from the teacher) was a month-long Trigonometry unit in a Geometry class.\u00a0 I prepared this page of notes (scribd, html) back then, which I still use with my students today.\n\nIn every year of my teaching career save the first, I have taught trigonometry of the right triangle, often in more than one course each year!\u00a0 I have taught it in Geometry, and in Algebra II with Trigonometry (wherein many students have forgotten much of what they learned in Geometry, even if I was their teacher for both classes :-[\n\nMost recently\u2013spring semester and now again for the fall semester of 2009\u2013I have been teaching right triangle trigonometry and its applications to vector analysis of forces on truss bridges & other structures, in an introductory engineering course.\u00a0 One difficulty is that I am teaching it to these students usually for the first time, before they encounter trigonometry in Geometry or another math class.\u00a0 My introductory engineering students are mainly ninth and tenth graders.\u00a0 Also, applying trigonometry to vectors and trusses requires additional mathematical sophistication.\u00a0 And so, even though I have taught trigonometry to more than ten classes in the past, I find these students struggling.\n\nTherefore, I ask:\u00a0 Does anyone know any good online resources about trigonometry that I could use with \/ share with my students?\u00a0 Or have any suggestions for making right triangle trigonometry more motivating and memorable?\n\n\u2013N","date":"2017-11-18 00:35:45","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 57, \"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\": 0.5943096280097961, \"perplexity\": 870.0781030633945}, \"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-2017-47\/segments\/1510934804125.49\/warc\/CC-MAIN-20171118002717-20171118022717-00455.warc.gz\"}"} | null | null |
\section{Introduction}
Let $(x_1, x_2, \ldots)$ be a positive sequence with finite sum $t = \sum_{i=1}^\infty x_i$. Its size-biased permutation (s.b.p) is the same sequence presented in a random order $(x_{\sigma_1}, x_{\sigma_2}, \ldots)$, where $\mathbb{P}(\sigma_1 = i) = \displaystyle \frac{x_i}{t}$, and for $k$ distinct indices $i_1, \ldots, i_k$,
\begin{equation}\mathbb{P}(\sigma_k = i_k| \sigma_1 = i_1, \ldots, \sigma_{k-1} = i_{k-1}) = \frac{x_{i_k}}{t - (x_{i_1} + \ldots + x_{i_{k-1}})}. \label{eqn:sbp} \end{equation}
An index $i$ with bigger `size' $x_i$ tends to appear earlier in the permutation, hence the name size-biased.
Size-biased permutation of a random sequence is defined by conditioning on the sequence values.
\subsection*{A brief history}
One of the earliest occurences of size-biased permutation is in social choice theory. Luce \cite{luce} studied distributions on the permutation on $n$ letters given by (\ref{eqn:sbp}) as a function of the $x_i$'s. These are the relative scores or desirabilities of the candidates, to be inferred through observing multiple rankings. This ranking model is now known as the Plackett-Luce model and it has wide applications \cite{plackett, plmeier, volkmer}.
Around the same time, biologists in population genetics were interested in inferring the distribution of alleles in a population through sampling. In these applications, $x_i$ is the abundance and $x_i/t$ is the relative abundance of the $i$-th species \cite{en78}. Size-biased permutation models the outcome of successive sampling, where one samples without replacement from the population and records the abundance of newly discovered species in the order that they appear. To account for the occurrence of new types of alleles through mutation and migration, they considered random abundance sequences and did not assume an upper limit to the number of possible types. Species sampling from random infinite sequence is sometimes known as \emph{size-biased random permutation}, a term coined by Patil and Taillie \cite{pt77}. The earliest work along this vein is perhaps that of McCloskey \cite{mc65}, who obtained results on the size-biased permutation of ranked jumps in a certain Poisson point process (p.p.p). The distribution of this ranked sequence is now known as the Poisson-Dirichlet $PD(0, \theta)$, and the distribution of its size-biased permutation is the $GEM(0, \theta)$; see Section \ref{subsec:parallel} for their definitions. This work was later generalized by Perman, Pitman and Yor \cite{pitmanyor}, who studied size-biased permutation of ranked jumps of a subordinator; see Section \ref{sec:sbp}.
The finite combinatorial version of size-biased sampling from $PD(0, \theta)$ is the Ewens sampling formula \cite{ew72}. Kingman \cite{ki78a} wrote: `One of the most striking results of recent theoretical research in population genetics is the sampling formula enunciated by Ewens and shown by (others) to hold for a number of different population models'. In studying this formula, Kingman initiated the theory of partition structure \cite{ki78a, ki78b}. Kingman showed that the Ewens sampling formula defines a particular partition structure by deletion of type; see \cite{ghp} for recent developments. Subsequent authors have studied partition structures and their representations in terms of exchangeable random partitions, random discrete distributions, random trees and associated random processes of fragmentation and coalescence, Bayesian statistics and machine learning. See \cite{csp} and references therein.
\subsection*{Organization}
This paper focuses on \emph{finite i.i.d size-biased permutation}, that is, the size-biased permutation of $n$ independent and identically distributed (i.i.d) random variables from some distribution $F$ on $(0, \infty)$. Our setting is a finite dimensional analogue of the size-biased permutation of ranked jumps of a subordinator studied in Perman-Pitman-Yor (PPY) \cite{pitmanyor}, as well as a special form of \emph{induced order statistics} \cite{david73, bhat74}; see Section \ref{sec:history} for a brief historical account of this field. We utilize this connection in our paper to arrive at new results. In Section \ref{sec:sbp} we study joint and marginal distribution of finite i.i.d size-biased permutation through a Markov chain. We draw connections between our settings and that of PPY \cite{pitmanyor}, and prove a converse of the stick-breaking representation of Patil and Taillie when $F$ is gamma (Proposition \ref{prop:pt}). Section \ref{sec:ios} we show that finite i.i.d size-biased permutation is a form of induced ordered statistics, and use this fact to derive previous distributional results. Comparisons to corresponding statements in Section \ref{sec:sbp} lead to a new beta-gamma identity when F is gamma (Corollary \ref{cor:newgb2}). As the sequence length tends to infinity, we derive asymptotics of the last $u$ fraction of finite i.i.d size-biased permutation in Section \ref{sec:asymp}, and that of the first few terms in Section \ref{sec:ppp}.
\subsection*{Notations} We shall write $gamma(a,\lambda)$ for a Gamma distribution whose density at $x$ is $\lambda^a x^{a-1}e^{-\lambda x}/\Gamma(a)$ for $x > 0$, and $beta(a,b)$ for the Beta distribution whose density at $x$ is $\displaystyle \frac{\Gamma(a+b)}{\Gamma(a)\Gamma(b)}x^{a-1}(1-x)^{b-1}$ for $x \in (0,1)$. For an ordered sequence $Y_n(k), k = 1, \ldots, n$, let $Y^{rev}_n(k) = Y_n(n-k+1)$ be the same sequence presented in reverse. For order statistics, we write $(Y^{\uparrow}(1), Y^{\uparrow}(2), \ldots)$ for the increasing sequence, and $(Y^{\downarrow}(1), Y^{\downarrow}(2), \ldots)$ for the decreasing sequence. Throughout this paper we use $(X_1, \ldots, X_n)$ for the underlying i.i.d sequence, and $(X_n[1], \ldots, X_n[n])$ for its size-biased permutation.
\section{Connections to the work of Perman-Pitman-Yor\cite{pitmanyor}.}\label{sec:sbp}
PPY considered the size-biased permutation of the sequence of ranked jumps of a \emph{subordinator} $\tilde{T}$, that is, a non-decreasing process with no drift component, right continuous paths, stationary independent increments, $\tilde{T}(0) = 0$, and
$$ \tilde{T}(s) = \sum_i X_i \mathbf{1}_{\{\sigma_i \leq s\}} $$
for $ 0 \leq s \leq 1$, where $\{(\sigma_i, X_i), i \geq 1\}$ is the countable random set of points of a Poisson point process (p.p.p) on $(0, \infty)^2$ with intensity measure $ds \Lambda (dx)$. The $X_i$'s are the jumps of the subordinator $\tilde{T}$.
They assumed $\Lambda(dx) = \rho(x)\, dx$ for some density $\rho$, $\Lambda(0, \infty) = \infty, \Lambda(1, \infty) < \infty$, $\int_0^1 x \Lambda(dx) < \infty$. So $T > 0$ is an infinitely divisible random variable with L\'{e}vy measure $\Lambda$ and no drift component in its L\'{e}vy-Khitchine representation (see, for example, \cite[\S 15]{kallenberg}).
Proposition \ref{prop:converge} states that the PPY setup is the limit in distribution of finite i.i.d s.b.p. This is an application of the principle that convergence of order statistics is equivalent to convergence of size-biased permutation; see \cite{gnedin} for a precise statement.
\begin{prop}[Gnedin \cite{gnedin}]\label{prop:converge}
Let $X = (X^{\downarrow}(1), X^{\downarrow}(2), \ldots)$ be the infinite sequence of ranked jumps of a subordinator, whose sum $T = \sum_{i\geq 1}X^\downarrow(i)$ is a positive, infinitely divisible, a.s. finite random variable whose L\'evy-Khitchine representation has no drift component. Let $(X^n, n \geq 1)$ be an i.i.d positive triangular array, that is, $X^n = (X^n_1, \ldots, X^n_n)$ where $X^n_i$ are i.i.d, a.s. positive, and $T_n = \sum_{i=1}^nX^n_i \stackrel{d}{\rightarrow} T$ as $n \to \infty$. Then as $n \to \infty$,
$$ \mbox{s.b.p of } X^n \,\, \stackrel{f.d.d}{\longrightarrow} \mbox{ s.b.p of } X $$
meaning convergence in distributoin of finite dimensional distributions.
\end{prop}
\begin{proof} Recall that the sequence of decreasing order statistics $(X^{n, \downarrow}(1),\ldots, X^{n, \downarrow}(n))$ converges in distribution to $X$ \cite[\S 15]{kallenberg}. Since $T_n, T > 0$ a.s. and $T_n \stackrel{d}{\rightarrow} T$, the normalized sequence $(X^{n, \downarrow}(1)/T, \ldots, X^{n, \downarrow}(n)/T)$ converges in distribution to $X/T$. The result follows from
\cite[Theorem 3]{gnedin}.
\end{proof}
One can obtain another finite version of the PPY setup by setting $\Lambda(0, \infty) <~ \infty$, but this can be reduced to finite i.i.d s.b.p by conditioning. Specifically, $\tilde{T}$ is now a compound Poisson process, where the subordinator waits for an exponential time with rate $\Lambda(0, \infty)$ before making a jump $X$, whose length is independent of the waiting time and distributed as $\mathbb{P}(X \leq t) = \Lambda(0, t]/\Lambda(0, \infty)$ \cite{bertoin99}. If $(X_1, X_2, \ldots)$ is the sequence of successive jumps of $(\tilde{T}_s, s \geq 0)$, then $(X_1, X_2, \ldots, X_N)$ is the sequence of successive jumps of $(\tilde{T}_s, 0 \leq s \leq 1)$, where $N$ is a Poisson random variable with mean $\Lambda(0, \infty)$, independent of the jump sequence $(X_1, X_2, \ldots)$. For $N > 0$, properties of the size-biased permutation of $(X_1, \ldots, X_N)$ can be deduced from those of a finite i.i.d size-biased permutation by conditioning on $N$.
\subsection{Joint distribution, Markov property and stick-breaking}\label{subsec:joint.ppy}
Recall that $(X_1, \ldots, X_n)$ is an i.i.d sequence from distribution $F$, and $(X_n[1], \ldots, X_n[n])$ is its size-biased permutation. Let $T_{n-k} = X_n[k+1] + \ldots + X_n[n]$ denote the sum of the last $n-k$ terms in an i.i.d size-biased permutation of length $n$. We now derive the joint distribution of the first $k$ terms $X_n[1], \ldots, X_n[k]$ when $F$ has density $\nu_1$. The parallels in PPY are discussed in Section ~\ref{subsec:parallel}.
\begin{prop}[Barouch-Kaufman \cite{bkauf}] For $1 \leq k \leq n$, let $\nu_k$ be the density of $S_k$, the sum of $k$ i.i.d random variables with distribution $F$. Then
\begin{align}
& \mathbb{P}(X_n[1] \in dx_1, \ldots, X_n[k] \in dx_k) \nonumber \\
=& \frac{n!}{(n-k)!}\left(\prod_{j=1}^kx_j\, \nu_1(x_j)\, dx_j\right)\, \int_0^\infty \nu_{n-k}(s) \prod_{j=1}^k(x_j + \ldots + x_k+s)^{-1}\, ds. \label{eqn:joint.xnk}
\end{align}
\end{prop}
\begin{proof}
Let $\sigma$ denote the random permutation on $n$ letters defined by size-biased permutation as in (\ref{eqn:sbp}). Then there are $\frac{n!}{(n-k)!}$ distinct possible values for $(\sigma_1, \ldots, \sigma_k)$. By exchangability of the underlying i.i.d random variables $X_1, \ldots, X_n$, it is sufficient to consider $\sigma_1 = 1$, $\ldots$, $\sigma_k = ~k$. Note that
$$\mathbb{P}\left((X_1, \ldots, X_k) \in dx_1\, \ldots dx_k, \sum_{j=k+1}^nX_j \in ds\right) = \nu_{n-k}(s) \, ds \prod_{j=1}^k \nu_1(x_j) \, dx_j.$$
Thus, restricted to $\sigma_1 = 1$, $\ldots$, $\sigma_k = ~k$, the probability of observing $(X_n[1], \ldots, X_n[k]) \in dx_1\, \ldots dx_k$ and $T_{n-k} \in ds$ is precisely
$$\frac{x_1\, dx_1}{x_1 + \ldots + x_k + s}\, \frac{x_2\, dx_2}{x_2 + \ldots + x_k + s} \cdots \frac{x_k\, dx_k}{x_k + s}\, \nu_{n-k}(s)\, \left(\prod_{j=1}^k\nu_1(x_j)\, dx_j\right) ds. $$
By summing over $\frac{n!}{(n-k)!}$ possible values for $(\sigma_1, \ldots, \sigma_k)$, and integrating out the sum $T_{n-k}$, we arrive at (\ref{eqn:joint.xnk}).
\end{proof}
Note that $X_n[k] = T_{n-k+1} - T_{n-k}$ for $k = 1, \ldots, n-1$. Thus we can rewrite (\ref{eqn:joint.xnk}) in terms of the joint law of $(T_{n}, T_{n-1}, \ldots, T_{n-k})$:
\begin{equation}\label{eqn:joint.T}
\mathbb{P}(T_n \in dt_0, \ldots, T_{n-k} \in dt_k) = \frac{n!}{(n-k)!}\left(\prod_{i=0}^{k-1} \frac{t_i - t_{i+1}}{t_i}\nu_1(t_i - t_{i+1})\right)\, \nu_{n-k}(t_k)\, dt_0\ldots dt_k.
\end{equation}
Rearranging (\ref{eqn:joint.T}) yields the following result, which appeared as an exercise in \cite[\S 2.3]{cyor}.
\begin{cor}[Chaumont-Yor \cite{cyor}]
The sequence $(T_n, T_{n-1},\ldots, T_1)$ is an inhomogeneous Markov chain with transition probability
\begin{equation}\label{eqn:T.tp}
\mathbb{P}(T_{n-k} \in ds| T_{n-k+1} = t) = (n-k+1)\, \frac{t-s}{t}\cdot\nu_1(t-s)\, \frac{\nu_{n-k}(s)}{\nu_{n-k+1}(t)}\, ds,
\end{equation}
for $k = 1, \ldots, n-1$. Together with $T_n \stackrel{d}{=} S_n$, equation (\ref{eqn:T.tp}) specifies the joint law in (\ref{eqn:joint.T}), and vice versa.
\end{cor}
An equivalent way to state (\ref{eqn:T.tp}) is that for $k \geq 1$, conditioned on $T_{n-k+1} = t$, $X_n[k]$ is distributed as the first size-biased pick out of $n-k+1$ i.i.d random variables conditioned to have sum $S_{n-k+1} = t$. This provides a recursive way to generate a finite i.i.d s.b.p: first generate $T_n$ (which is distributed as $S_n$). Conditioned on the value of $T_n$, generate $T_{n-1}$ via (\ref{eqn:T.tp}), let $X_n[1]$ be the difference. Now conditioned on the value of $T_{n-1}$, generate $T_{n-2}$ via (\ref{eqn:T.tp}), let $X_n[2]$ be the difference, and so on. Let us explore this recursion from a different angle by considering the ratio $\displaystyle W_{n,k} := \frac{X_n[k]}{T_{n-k+1}}$ and its complement, $\overline{W}_{n,k} = 1 - W_{n,k} = \displaystyle \frac{T_{n-k}}{T_{n-k+1}}$. For $k \geq 2$, note that
\begin{equation}\label{eqn:stickbreak}
\frac{X_n[k]}{T_n} = \frac{X_n[k]}{T_{n-k+1}}\, \frac{T_{n-k+1}}{T_{n-k+2}}\cdots \frac{T_{n-1}}{T_n} = W_{n,k}\prod_{i = 1}^{k-1}\overline{W}_{n,i}.
\end{equation}
The variables $\overline{W}_{n,i}$ can be interpreted as residual fractions in a \emph{stick-breaking} scheme: start with a stick of length 1. Choose a point on the stick according to distribution $W_{n,1}$, `break' the stick into two pieces, discard the piece of length $W_{n,1}$ and rescale the remaining half to have length 1. Repeating this procedure $k$ times, and (\ref{eqn:stickbreak}) is the fraction broken off at step $k$ relative to the original stick length.
Together with $T_n \stackrel{d}{=} S_n$, one could use (\ref{eqn:stickbreak}) to compute the marginal distribution for $X_n[k]$ in terms of the ratios $\overline{W}_{n,i}$. In general the $W_{n,i}$ are not necessarily independent, and their joint distributions need to be worked out from (\ref{eqn:T.tp}). However, when $F$ has gamma distribution, $T_n, W_{n,1}, \ldots, W_{n,k}$ are independent, and (\ref{eqn:stickbreak}) leads to the following result of Patil and Taillie \cite{pt77}.
\begin{prop}[Patil-Taillie \cite{pt77}]\label{prop:pt} If $F$ has distribution $gamma(a, \lambda)$ for some $a, \lambda > 0$, then $T_n$ and the $W_{n,1}, \ldots, W_{n,n-1}$ in (\ref{eqn:stickbreak}) are mutually independent. In this case,
\begin{align*}
X_n[1] &= \gamma_0 \beta_1 \\
X_n[2] &= \gamma_0 \bar{\beta}_1 \beta_2 \\
& \ldots \\
X_n[n-1] &= \gamma_0 \bar{\beta}_1 \bar\beta_2\ldots \bar\beta_{n-2}\beta_{n-1} \\
X_n[n] &= \gamma_0 \bar{\beta}_1 \bar\beta_2\ldots \bar\beta_{n-1}
\end{align*}
where $\gamma_0$ has distribution $gamma(an,\lambda)$, $\beta_k$ has distribution $beta(a+1, (n-k)a)$, $\bar\beta_k = 1 - \beta_k$ for $1 \leq k \leq n-1$, and the random variables $\gamma_0, \beta_1, \ldots, \beta_n$ are independent.
\end{prop}
\begin{proof} By assumption $S_k$ has distribution $gamma(ak, \lambda)$. One substitutes the density of $gamma(ak, \lambda)$ for $\nu_k$ in (\ref{eqn:T.tp}), and the result follows by direct computation.
\end{proof}
In the subordinator setting, McCloskey \cite{mc65} proved the analogue of Proposition \ref{prop:pt}, and Perman-Pitman-Yor \cite{pitmanyor} proved the converse, see Proposition \ref{prop:mccloskey} below. Using the same idea, we obtain the following converse to Proposition \ref{prop:pt}, which appears to be new.
\begin{cor}\label{cor:pt} If $T_n$ is independent of $X_n[1]/T_n$, then $F$ is $gamma(a, \lambda)$ for some $a, \lambda > 0$.
\end{cor}
\begin{proof} Lukacs \cite{lukacs} proved that if $X, Y$ are non-degenerate, positive independent random variables, then $X+Y$ is independent of $\frac{X}{X+Y}$ if and only if both $X$ and $Y$ have gamma distributions with the same scale parameter. Note that
$$
\mathbb{P}(X_n[1]/T_n \in du, T_n \in dt) = n\, u\, \mathbb{P}\left(\frac{X_1}{X_1 + (X_2+\ldots+X_n)} \in du, T_n \in dt\right)
$$
and
$$\mathbb{P}(X_n[1]/T_n \in du) = n\, u\, \mathbb{P}\left(\frac{X_1}{X_1 + (X_2+\ldots+X_n)} \in du\right).$$
Since $X_n[1]/T_n$ and $T_n$ are independent, $\frac{X_1}{X_1 + (X_2+\ldots+X_n)}$ is independent of $T_n$. The conclusion follows by applying Lukacs' theorem to the pair $X_1$ and $(X_2 + \ldots + X_n)$.
\end{proof}
\subsection{Parallels in PPY \cite{pitmanyor}} \label{subsec:parallel}
In the subordinator setting of PPY, let $\tilde{T}_k$ denote the remaining sum after removing the first $k$ terms of the size-biased permutation of the infinite sequence $(X^\downarrow(1), X^\downarrow(2), \ldots)$. The sequence $(\tilde{T}_0, \tilde{T}_1, \ldots)$ is a Markov chain with stationary transition probabilities \cite[equation 2.a]{pitmanyor}
$$
\mathbb{P}(\tilde{T}_{1} \in dt_1| \tilde{T}_0 = t) = \frac{t-t_1}{t}\cdot\rho(t-t_1)\, \frac{\nu(t_1)}{\nu(t)}\, dt_1.
$$
Conditionally given $\tilde{T}_0 = t_0, \tilde{T}_1 = t_1, \ldots, \tilde{T}_n = t_n$, the sequence of remaining terms in the s.b.p $(X[n+1], X[n+2], \ldots)$ is distributed as $(X^\downarrow(1), X^\downarrow(2), \ldots, )$ conditioned on $\sum_{i\geq 1}X^\downarrow(i) = t_n$, independent of the first $n$ size-biased picks \cite[Theorem 4.2]{pitmanyor}. The stick-breaking representation in (\ref{eqn:stickbreak}) now takes the form
\begin{equation}\label{eqn:stick2}
\frac{X[k]}{\tilde{T}_0} = W_{k}\prod_{i=1}^{k-1}\overline{W}_{i},
\end{equation}
where $X[k]$ is the $k$-th size-biased pick, and $W_{i} = \frac{X[i]}{\tilde{T}_{i-1}}$, $\overline{W}_{i} = 1 - W_{i} = \frac{\tilde{T}_i}{\tilde{T}_{i-1}}$. Proposition ~\ref{prop:pt} and Corollary \ref{cor:pt} parallel the following result.
\begin{prop}[McCloskey \cite{mc65} and PPY \cite{pitmanyor}]\label{prop:mccloskey}
The random variables $\tilde{T}_0$ and $W_{1}, W_2, \ldots$ in (\ref{eqn:stick2}) are mutually independent if and only if $\tilde{T}_0$ has distribution $gamma(a, \lambda)$ for some $a, \lambda > 0$. In this case, the $W_{i}$ are i.i.d with distribution $beta(1, a)$ for $i = 1, 2, \ldots$.
\end{prop}
We take a small detour to explain some results related to Proposition \ref{prop:pt} and \ref{prop:mccloskey}
on characterization of size-biased permutations. For a random discrete distribution prescribed by its probability mass function $(P_k) = (P_1, P_2, \ldots)$ with $\sum_iP_i = 1$, $P_i \geq 0$, let $(\tilde{P}_k)$ be the s.b.p. of $(P_k)$. One may ask when is a given distribution $(Q_k)$ the s.b.p of some distribution $(P_k)$. Clearly $(\widetilde{\tilde{P}}_k) \stackrel{d}{=} (\tilde{P}_k)$ for any $(P_k)$, thus this question is equivalent to characterizing random discrete distributions on $\mathbb{N}$ which are invariant under size-biased permutation (ISBP).
Pitman \cite[Theorem 4]{pitman96} gave a complete answer in terms of symmetry of a certain function of the finite dimensional distributions. Furthermore, Pitman proved a complete characterization of ISBP when $P_k$ can be written as the right hand side of (\ref{eqn:stick2}) with $W_{1}, W_2, \ldots$ independent. In this situation, apart from some limiting cases, $(P_k)$ is ISBP if and only if $W_{i}$ is distributed as $beta(1-\alpha, \theta + i\alpha)$, $i = 1, 2, \ldots$ for certain pairs of real numbers $(\alpha, \theta)$. The two main cases are $(0 \leq \alpha < 1, \theta > -\alpha)$ and $(\alpha = -a < 0, \theta = na)$ for some $n = 1, 2, \ldots$. In both settings, $(P_k)$ is known as the $GEM(\alpha, \theta)$ distribution. The abbreviation GEM was introduced by Ewens, which stands for Griffiths-Engen-McCloskey. The McCloskey case of Proposition \ref{prop:mccloskey} is $GEM(0, \theta)$, and the Patil-Taillie case of Proposition \ref{prop:pt} is $GEM(-a, na)$. The sequence obtained by ranking a $GEM(\alpha, \theta)$ sequence is called a Poisson-Dirichlet distribution with parameters $(\alpha, \theta)$ \cite{pitmanyor}. The $GEM$ distribution has a generative description known as the Chinese restaurant process \cite[\S 3]{csp} and has applications in Bayesian statistics and machine learning, see \cite{csp}.
\section{Connections to induced order statistics}\label{sec:ios}
When $n$ i.i.d pairs $(X_i, Y_i)$ are ordered by their $Y$-values, the corresponding $X_i$'s are called the \emph{induced order statistics} of the vector $Y$, or its \emph{concomitants}. Gordon \cite{gordon83} first proved the following result for finite $n$ which shows that finite i.i.d s.b.p is a form of induced order statistics. Here we state the infinite sequence version, which is a special case of PPY\cite[Lemma 4.4]{pitmanyor}.
\begin{prop}[PPY \cite{pitmanyor}]\label{prop:coupling} Let $(x_1, x_2, \ldots)$ be a fixed positive sequence with finite sum $t = \sum_{i=1}^\infty x_i$. For $i = 1, 2,\ldots$, let $Y_i = \epsilon_i/x_i$ where $\epsilon_i$ are i.i.d standard exponentials. Let $(Y^{\uparrow}(1),Y^{\uparrow}(2), \ldots)$ be the increasing order statistics of the $Y_i$'s, and let $X^\ast(k)$ be the value of the $x_i$ such that $Y_i$ is $Y^{\uparrow}(k)$. Then $(X^\ast(k), k = 1, 2, \ldots)$ is a size-biased permutation of $(x_1, x_2, \ldots)$. In particular, the size-biased permutation of a positive i.i.d sequence $(X_1, \ldots, X_n)$ is distributed as the induced order statistics of the sequence $(Y_i = \epsilon_i/{X_i}, 1 \leq i \leq n)$ for an independent sequence of i.i.d standard exponentials $(\epsilon_1, \ldots, \epsilon_n)$.
\end{prop}
\begin{proof} Note that the $Y_i$'s are independent exponentials with rates $x_i$. Let $\sigma$ be the random permutation such $Y_{\sigma(i)} = Y^\uparrow(i)$. Note that $X^\ast(k) = x_{\sigma(k)}$. Then
$$\mathbb{P}(\sigma(1) = i) = \mathbb{P}(Y_i = \min\{Y_j, j = 1, 2, \ldots\}) = \frac{x_i}{t},$$
thus $X_n^\ast(1) \stackrel{d}{=} x[1]$. In general, for distinct indices $i_1, \ldots, i_k$, by the memoryless property of the exponential distribution,
\begin{align*}
& \, \mathbb{P}(\sigma(k) = i_k | \sigma(1) = i_1, \ldots, \sigma(k) = i_{k-1} ) \\
=& \, \mathbb{P}\left(Y_{i_k} = \min\{Y_{\sigma(j)}, j \geq k \} | \sigma(1) = i_1, \ldots, \sigma(k) = i_{k-1}\right) = \frac{x_{i_k}}{t - \sum_{j=1}^{k-1} x_{i_j}}.
\end{align*}
Induction on $k$ completes the proof.
\end{proof}
This proposition readily supplies simple proofs for joint, marginal and asymptotic distributions of i.i.d s.b.p, as explicitly computed in this section. For instance, the proof of the following nesting property of i.i.d s.b.p, which can be cumbersome, amounts to i.i.d thinning.
\begin{cor}Consider a finite i.i.d s.b.p $(X_n[1], \ldots, X_n[n])$ from a distribution $F$. For $1 \leq m \leq n$, select $m$ integers $a_1 < \ldots < a_m$ by uniform sampling from $\{1, \ldots, n\}$ without replacement. Then the subsequence $\{X_n[a_j], 1 \leq j \leq m\}$ is jointly distributed as a finite i.i.d s.b.p of length ~$m$ from $F$.
\end{cor}
\subsection{Joint and marginal distribution revisited} \label{subsec:ios.joint}
Proposition \ref{prop:coupling} immediately yields the joint distribution of the $X_n[k]$'s, which is a different formula to (\ref{eqn:joint.xnk}).
\begin{prop}\label{lem:joint.x.u}
$(X_n[k], k = 1, \ldots, n)$ is distributed as the first coordinate of the sequence of pairs $((X^{\ast}_n(k), U_n^\downarrow(k)), k =1, \ldots, n)$, where $U_n^\downarrow(1) \geq \ldots \geq U^\downarrow_n(n)$ is a sequence of uniform order statistics, and conditional on $(U^\downarrow_n(k) = u_k, 1 \leq k \leq n)$, the $X^{\ast}_n(k)$ are independent with distribution $(G_{u_k}(\cdot), k = 1, \ldots, n)$, where
\begin{equation} \label{eqn:Gu}
G_u(dx) = \frac{xe^{-\phi^{-1}(u)x}\, F(dx)}{-\phi'(\phi^{-1}(u))}.
\end{equation}
Here $\phi$ is the Laplace transform of $X$, that is, $\phi(y) =~ \int_0^\infty e^{-yx}\, F(dx)$, $\phi'$ its derivative and $\phi^{-1}$ its inverse function.
\end{prop}
\begin{proof} Let $(X_1, \ldots, X_n)$ be $n$ i.i.d draws from $F$, $Y_i = \epsilon_i/X_i$ for $n$ i.i.d standard exponentials $\epsilon_i$, independent of the $X_i$'s. Note that $\{(X_i, Y_i), 1 \leq i \leq n\}$ is an i.i.d sample from the joint distribution $ F(dx)[xe^{-yx}\, dy].$ Thus $Y_i$ has marginal density
\begin{equation}
P(Y_i \in dy) = -\phi'(y)\, dy, \hspace{1em} 0 < y < \infty, \label{1.b}
\end{equation}
and its distribution function is $F_Y = 1 - \phi$. Given $\{Y_i = y_i, 1 \leq i \leq n\}$, the $X_n^\ast(i)$ defined in Proposition \ref{prop:coupling} are independent with conditional distribution $\widetilde{G}(y_i, \cdot)$ where
\begin{equation}
\widetilde{G}(y, dx) = \frac{xe^{-yx}\, F(dx)}{-\phi'(y)}. \label{eqn:Gy}
\end{equation}
Equation (\ref{eqn:Gu}) follows from writing the order statistics as the inverse transforms of ordered uniform variables
\begin{equation}\label{eqn:ynun}
(Y_n^\uparrow(1), \ldots, Y^\uparrow_n(n)) \stackrel{d}{=} (F_Y^{-1}(U^\downarrow_n(n)), \ldots, F_Y^{-1}(U^\downarrow_n(1))) \stackrel{d}{=} (\phi^{-1}(U^\downarrow_n(1)), \ldots, \phi^{-1}(U^\downarrow_n(n)))
\end{equation}
where $(U^\downarrow_n(k), k = 1, \ldots, n)$ is an independent decreasing sequence of uniform order statistics. Note that the minus sign in (\ref{1.b}) results in the reversal of the sequence $U_n$ in the second equality of (\ref{eqn:ynun}).
\end{proof}
\begin{cor} \label{cor:marginal}
For $1 \leq k \leq n$ and $0 < u < 1$, let
\begin{equation}\label{2.a}
f_{n,k}(u) = \frac{\mathbb{P}(U_{n}(k) \in du)}{du} = n\binom{n-1}{k-1}u^{n-k}(1-u)^{k-1}
\end{equation}
be the density of the $k$-th largest of the $n$ uniform order statistics $(U_n(i), i = 1, \ldots, n)$. Then
\begin{equation}\label{2.c}
\frac{\mathbb{P}(X_n[k] \in dx)}{xF(dx)} = \int_0^\infty e^{-xy}f_{n,k}(\phi(y))\, dy.
\end{equation}
\end{cor}
In particular, for the first and last values,
\begin{align*}
\frac{\mathbb{P}(X_n[1] \in dx)}{xF(dx)} &= n\int_0^\infty e^{-xy}\phi(y)^{n-1}\, dy \\
\frac{\mathbb{P}(X_n[n] \in dx)}{xF(dx)} &= n\int_0^\infty e^{-xy}(1-\phi(y))^{n-1}\, dy.
\end{align*}
\begin{ex}\label{ex:gamma1}
Suppose $F$ is $gamma(a,1)$. Then $\phi(y) = (\frac{1}{1+y})^a$, and $\phi^{-1}(u) =~ u^{-1/a} - 1.$
Hence $G_u$ in (\ref{eqn:Gu}) is
$$G_u(dx) = \frac{x}{au^{(a+1)/a}}\, e^{-(u^{-1/a} - 1)x}\, F(dx) = \frac{x^a}{\Gamma(a+1)}u^{-(a+1)/a}e^{-xu^{-1/a}}.$$
That is, $G_u$ is $gamma(a+1, u^{-1/a})$.
\end{ex}
\subsection{A new beta-gamma identity}\label{subsec:ios.gamma}
When $F$ is $gamma(a,\lambda)$, Lemma \ref{lem:joint.x.u} gives the following result, which is a remarkable complement to the Patil-Taillie representation in Proposition \ref{prop:pt}.
\begin{prop}\label{prop:newgb}
Suppose $F$ is $gamma(a,\lambda)$. Then $G_u$ is $gamma(a+1, \lambda u^{-1/a})$, and
\begin{equation}\label{eqn:xrev.g}
(X_n[k], k = 1, \ldots, n) \stackrel{d}{=} ([U^\downarrow_n(k)]^{1/a}\gamma_k, k = 1, \ldots, n)
\end{equation}
where $\gamma_1, \ldots, \gamma_n$ are i.i.d $gamma(a+1, \lambda)$ random variables, independent of the sequence of decreasing uniform order statistics $(U^\downarrow_n(1), \ldots, U^\downarrow_n(n))$. Alternatively, jointly for $k =~ 1, \ldots, n$
\begin{align*}
X_n^{rev}[1] &= \gamma_1 \beta_{an, 1} \\
X_n^{rev}[2] &= \gamma_2 \beta_{an, 1}\beta_{an-a, 1} \\
& \ldots \\
X_n^{rev}[n-1] &= \gamma_{n-1} \beta_{an, 1}\beta_{an-a, 1}\cdots \beta_{2a, 1} \\
X_n^{rev}[n] &= \gamma_{n} \beta_{an, 1}\beta_{an-a, 1}\cdots \beta_{a, 1}.
\end{align*}
where the $\beta_{an - ia,1}$ for $i = 0, \ldots, n-1$ are distributed as $beta(an-ia, 1)$, and they are independent of each other and the $\gamma_k$'s.
\end{prop}
\begin{proof} The distribution $G_u$ is computed in the same way as in Example \ref{ex:gamma1} and (\ref{eqn:xrev.g}) follows readily from Proposition \ref{lem:joint.x.u}.
\end{proof}
A direct comparison of the two different representations in Proposition \ref{prop:pt} and \ref{prop:newgb} creates $n$ distributional identities. For example, the equality $X_n[1] = X^{rev}_n[n]$ shows that the following two means of creating a product of independent random variables produce the same result in law:
\begin{equation}\label{eqn:beta.ini}
\beta_{a+1, (n-1)a} \, \gamma_{an, \lambda} \stackrel{d}{=} \beta_{an, 1} \, \gamma_{a+1, \lambda}
\end{equation}
where $\gamma_{r,\lambda}$ and $\beta_{a,b}$ denote random variables with distributions $gamma(r, \lambda)$ and $beta(a,b)$, respectively. In fact, by comparing the consecutive ratios $\frac{X_n[i+1]}{X_n[i]}$ for $i = 1, \ldots, n-1$, the other $n-1$ identities all reduce to the following single equation after setting $ia - a = b$.
\begin{cor}\label{cor:newgb2} For $a, b, \lambda > 0$,
\begin{equation}\label{eqn:beta.ratio}
\frac{1 - \beta_{a+1,a+b}}{\beta_{a+1,a+b}}\beta_{a+1, b} \stackrel{d}{=} \frac{\gamma_{a+1,\lambda}}{\gamma'_{a+1,\lambda}}\beta_{a+b,1},
\end{equation}
where the random variables $\beta_{a+1,a+b}, \beta_{a+1,b}, \beta_{a+b,1}, \gamma_{a+1,\lambda}, \gamma'_{a+1,\lambda}$ are mutually independent, beta and gamma distributed with parameters indicated by subscripts.
\end{cor}
The validity of this identity can be checked by comparing moments. Conversely, (\ref{eqn:beta.ini}) and (\ref{eqn:beta.ratio}) allow one to go between Proposition \ref{prop:newgb} and the Proposition \ref{prop:pt}.
\section{Asymptotics of the last $u$ fraction of the size-biased permutation}\label{sec:asymp}
In this section we derive Glivenko-Cantelli and Donsker-type theorems for the distribution of the last $u$ fraction of terms in a finite i.i.d s.b.p. These are special cases of more general statements which hold for arbitrary induced order statistics in $d$ dimensions (see Section \ref{sec:history}).
Features pertaining to i.i.d s.b.p are presented in the following lemma, which has an interesting successive sampling interpretation; see Section \ref{subsec:heuristic}.
\begin{lem}\label{lem:ode} Suppose $F$ has support on $[0, \infty)$ and finite mean. For $u \in (0,1)$, define
\begin{equation}\label{eqn:Fu}
F_u(dx) = \frac{e^{-x\phi^{-1}(u)}}{u}\, F(dx)
\end{equation}
and extend the definition to $\{0, 1\}$ by continuity, where $\phi$ is the Laplace transform of $F$ as in Proposition \ref{lem:joint.x.u}. Then $F_u$ is a probability distribution on $[0, \infty)$ for all $u \in [0,1]$, and $G_u$ in (\ref{eqn:Gu}) satisfies
\begin{equation}\label{eqn:Gu.is.sb}
G_u(dx) = xF_u(dx)/ \mu_u
\end{equation}
where $\mu_u = \int x F_u(dx) = \frac{-\phi'(\phi^{-1}(u))}{u}$. Furthermore,
\begin{equation}\label{eqn:integral} \int_0^u G_s(dx)\, ds = F_u(dx) \end{equation}
for all $s \in [0,1]$. In other words, the density
$$f(u,x) = F_u(dx) / F(dx) = u^{-1}e^{-x\phi^{-1}(u)}$$
of $F_u$ with respect to $F$ solves the differential equation
\begin{equation}\label{eqn:evo}
\frac{d}{du}[uf(u,x)] = \frac{-xf(u,x)}{\mu_u} \end{equation}
with boundary condition $f(1,x) \equiv 1$.
\end{lem}
\begin{proof} By direct computation.
\end{proof}
We now state a Glivenko-Cantelli-type theorem which applies to size-biased permutation of finite deterministic sequences. Versions of this result are known in the literature \cite{bnw, holst, gordon83, rosen}, see discussions in Section \ref{sec:history}. We offer an alternative proof using induced order statistics.
\begin{thm}\label{thm:gc} Let $((x_i^n, 1 \leq i \leq n), 1 \leq n)$ be a deterministic triangular array of positive numbers with corresponding c.d.f sequence $(E_n, 1 \leq n)$. Suppose
\begin{equation}
\sup_x|E_n(x) - F(x)| \to 0 \mbox{ as } n \to \infty \label{eqn:sup.condition}
\end{equation}
for some distribution $F$ on $(0, \infty)$. Let $u \in (0, 1]$. Let $E_{n,u}(\cdot)$ be the empirical distribution of the first $\lfloor nu \rfloor$ terms in a size-biased permutation of the sequence $(x_i^n, 1 \leq i \leq n)$. Then for each $\delta \in (0,1)$,
\begin{equation}\label{eqn:gc.main}
\sup_{u \in [\delta, 1]} \sup_\mathbf{I} | E_{n,u}(\mathbf{I}) - F_u(\mathbf{I})| \stackrel{a.s.}{\longrightarrow} 0 \mbox{ as } n \to \infty,
\end{equation}
where $\mathbf{I}$ ranges over all subintervals of $(0, \infty)$.
\end{thm}
\begin{proof} Define $Y^n_i = \epsilon_i/x^n_i$ for $i = 1, \ldots, n$ where $\epsilon_i$'s are i.i.d standard exponentials as in Proposition \ref{prop:coupling}. Let $H_n$ be the empirical distribution function (e.d.f) of the $Y^n_i$'s,
$$H_n(y) := \frac{1}{n}\sum_{i=1}^n\mathbf{1}_{\{Y^n_i < y\}}.$$
Let $J_n$ denote the e.d.f of $(x^n_i, Y^n_i)$. By Proposition \ref{lem:joint.x.u},
\begin{equation}\label{eqn:eemp}
E_{n,u}(\mathbf{I}) = \frac{n}{\lfloor nu \rfloor}\, \frac{1}{n} \sum_{i=1}^n\mathbf{1}_{\{x^n_i \in \mathbf{I}\}}\mathbf{1}_{\{Y^n_i < H_n^{-1}(1-u)\}} = \frac{n}{\lfloor nu \rfloor}J_n(\mathbf{I} \times [0, H_n^{-1}(1-u)]).
\end{equation}
Fix $\delta \in (0, 1)$, and let $u \in [\delta, 1]$. Let $\phi$ be the Laplace transform of $F$ and $J$ the joint law of $(X, \epsilon/X)$, where $X$ is a random variable with distribution $F$, and $\epsilon$ is an independent standard exponential. Note that $\frac{1}{u}J(\mathbf{I} \times [0, \phi^{-1}(u)]) = F_u(\mathbf{I}).$
Thus
\begin{align}
\mathbb{E}_{n, u}(\mathbf{I}) - F_u(\mathbf{I}) = & \left(\frac{n}{\lfloor nu \rfloor}J_n(\mathbf{I} \times [0, H_n^{-1}(1-u)]) - \frac{n}{\lfloor nu \rfloor}J_n(\mathbf{I} \times [0, \phi^{-1}(u)])\right) \nonumber \\
& + \left(\frac{n}{\lfloor nu \rfloor}J_n(\mathbf{I} \times [0, \phi^{-1}(u)]) - \frac{1}{u}J(\mathbf{I} \times [0, \phi^{-1}(u)])\right). \label{eqn:split}
\end{align}
Let us consider the second term. Note that
\begin{align*}
J_n(\mathbf{I} \times [0, \phi^{-1}(u)]) &= \int_0^\infty e^{-t\phi^{-1}(u)}\mathbf{1}_{\{t \in \mathbf{I}\}}\, E_n(dt).
\end{align*}
Since $E_n$ converges to $F$ uniformly and $e^{-t\phi^{-1}(u)}$ is bounded for all $t \in (0, \infty)$ and $u \in [\delta, 1]$,
$$\sup_{u \in [\delta, 1]}\sup_\mathbf{I}\left|\frac{n}{\lfloor nu \rfloor}J_n(\mathbf{I} \times [0, \phi^{-1}(u)]) - \frac{1}{u}J(\mathbf{I} \times [0, \phi^{-1}(u)])\right| \stackrel{a.s.}{\longrightarrow} 0 \mbox{ as } n \to \infty.$$
Let us consider the first term.
Since $J_n$ is continuous in the second variable, it is sufficient to show that
\begin{equation}\label{eqn:hncont}
\sup_{u \in [\delta, 1]}|H_n^{-1}(1-u) - \phi^{-1}(u)| \stackrel{a.s.}{\longrightarrow} 0 \mbox{ as } n \to \infty.
\end{equation}
To achieve this, let $A_n$ denote the `average' measure
$$A_n(y) := \frac{1}{n}\sum_{i=1}^n\mathbb{P}(Y^n_i < y) = 1 - \int_0^\infty e^{-xy}\, dE_n(x).$$
A theorem of Wellner \cite[Theorem 1]{wellner} states that if the sequence of measures $(A_n, n \geq 1)$ is tight, then the Prohorov distance between $H_n$ and $A_n$ converges a.s. to $0$ and $n \to \infty$. In this case, since $E_n$ converges to $F$ uniformly, $A_n$ converges uniformly to $1 - \phi$. Thus $H_n$ converges uniformly to $1-\phi$, and (\ref{eqn:hncont}) follows.
\end{proof}
In particular, when $E_n$ is the e.d.f of $n$ i.i.d picks from $F$, then (\ref{eqn:sup.condition}) is satisfied a.s. by the Glivenko-Cantelli theorem. Thus Theorem \ref{thm:gc} implies that $F_u$ is the limit of the empirical distribution function (e.d.f) of the last $u$ fraction in a finite i.i.d s.b.p.
\begin{cor}\label{cor:emp}For $u \in (0,1]$, let $F_{n,u}(\cdot)$ denote the empirical distribution of the last $\lfloor nu \rfloor$ values of an i.i.d s.b.p. with length $n$. For each $\delta \in (0,1)$, as $n \to \infty$,
\begin{equation}\label{emp.u}
\sup_{u \in [\delta, 1]} \sup_\mathbf{I} | F_{n,u}(\mathbf{I}) - F_u(\mathbf{I})| \stackrel{a.s.}{\longrightarrow} 0,
\end{equation}
where $\mathbf{I}$ ranges over all subintervals of $(0, \infty)$.
\end{cor}
Under the settings in Theorem \ref{thm:gc}, one can also derive a Donsker-type result via entropy methods described in \cite[\S 2]{vaart2}, complementing similar results in the literature \cite[\S 5]{bnw}. In particular, the first term of (\ref{eqn:split}) depends on the flunctuation of empirical uniform order statistics around its limit. Its scaling depends on the rate of convergence of $E_n$ to $F$. The second term of (\ref{eqn:split}) converges in distribution after rescaled by $\sqrt{n}$, by a version of Donsker's theorem, to a Brownian bridge. We omit the details. A similar decomposition applies in Theorem ~\ref{thm:fclt}.
\subsection{A heuristic interpretation} \label{subsec:heuristic}
Since $X_n^{rev}[\lfloor u \, n \rfloor]$ converges in distribution to $G_u$ for $u \in [0, 1]$, Corollary \ref{cor:emp} lends a sampling interpretation to Lemma \ref{lem:ode}. Equation (\ref{eqn:evo}) has the heuristic interpretation as characterizing the evolution of the mass at $x$ over time $u$ in a successive sampling scheme. To be specific, consider a successive sampling scheme on a large population of $N$ individuals, with species size distribution $H$. Scale time such that at time $u$, for $0 \leq u \leq 1$, there are $Nu$ individuals (from various species) remaining to be sampled. Let $H_u$ denote the distribution of species sizes at time $u$, and fix the bin $(x, x + dx)$ of width $dx$ on $(0, \infty)$. Then $NuH_u(dx)$ is the number of individuals whose species size lie in the range $(x, x + dx)$ at time $u$. Thus $\frac{d}{du}NuH_u(dx)$ is the rate of individuals to be sampled from this range of species size at time $u$. The probability of an individual whose species size is in $(x, x+dx)$ being sampled at time $u$ is $\frac{xH_u(dx)}{\int_0^\infty xH_u(dx)}$. As we scaled time such that $u \in [0,1]$, in time $du$ we sample $N \, du$ individuals. Thus
$$ \frac{d}{du}NuH_u(dx) = -N\, \frac{xH_u(dx)}{\int_0^\infty xH_u(dx) }.$$
Let $f(u,x) = H_u(dx) / H(dx)$, then as a function in $u$, the above equation reduces to (\ref{eqn:evo}).
\subsection{Functional central limit theorem}\label{subsec:fclt}
We now state a functional central limit theorem for i.i.d s.b.p. This is a special case of the general result of Davydov and Egorov \cite{egorov89} for induced order statistics.
\begin{thm}[Davydov and Egorov, \cite{egorov89}]\label{thm:fclt} Suppose the first two moments of $F$ are finite. For a distribution $H$, let $\mu(H)$, $\sigma(H)$ denote its mean and standard deviation. For $u \in (0, 1]$, define
\begin{align*}
\xi_n(u) &= \frac{1}{n}\sum_{j=1}^{\lfloor nu \rfloor} X^{rev}_n[j] \\
\eta_n(u) &= \frac{1}{n}\sum_{j=1}^n X_j \mathbf{1}(Y_j < \phi^{-1}(u)) \\
m(u) &= \int_0^u \mu(G_s) \, ds = \mu(F_u).
\end{align*}
Then $\xi_n \to m$, $\eta_n \to m$ uniformly on $[0,1]$, and
$$\sqrt{n}(\xi_n - m) \Rightarrow \alpha, \hspace{2em} \sqrt{n}\, (\eta_n - m) \Rightarrow \eta $$
in the Skorokhod topology, where
$$\alpha(u) = \int_0^u \sigma(G_s) dW(s), \hspace{1em} \eta(u) = \alpha(u) + \int_0^u m(s) \, dV(s).$$
$W$ is a standard Brownian motion, and $V$ is a Brownian bridge, independent of $W$.
\end{thm}
The difference in $\xi_n$ and $\eta_n$ is the fluctuation of empirical uniform order statistics around its limit. Recall the analogue discussed at the end of Theorem \ref{thm:gc}. The proof of Theorem \ref{thm:fclt} can be found in \cite{egorov89}, together with similar statements on functional law of iterated logarithm for the processes $\eta_n, \xi_n$.
\subsection{Historical notes on induced order statistics and successive sampling} \label{sec:history}
Induced order statistics were first introduced by David \cite{david73} and independently by Bhattacharya \cite{bhat74}. Typical applications stem from modeling an indirect ranking procedure, where subjects are ranked based on their $Y$-attributes although the real interest lies in ranking their $X$-attributes, which are difficult to obtain at the moment where the ranking is required.\footnote{One often uses $X$ for the variable to be ordered, and $Y$ for the induced variable, with the idea that $Y$ is to be predicted. Here we use $X$ for the induced order statistics since $X_n[k]$ has been used for the size-biased permutation. The role of $X$ and $Y$ in our case is interchangeable, as evident when one writes $X_iY_i = \epsilon_i$.} For example in cattle selection, $Y$ may represent the genetic makeup, for which the cattle are selected for breeding, and $X$ represents the milk yields of their female offspring. Thus a portion of this literature focuses on comparing distribution of induced order statistics to that of usual order statistics \cite{dayang77, yang77, nagada94, hall94}.
The most general statement on asymptotic distributions is obtained by Davydov and Egorov \cite{egorov89}, who proved functional central limit theorem and functional law of the iterated logarithm for the process $S_{n,u}$ under tight assumptions. Their theorem translates directly into Theorem \ref{thm:fclt} for finite i.i.d s.b.p as discussed in Section \ref{subsec:fclt}.
Various versions of results in Section \ref{sec:asymp}, including Theorem \ref{thm:fclt} are also known in the successive sampling community \cite{holst, rosen, gordon83, bnw, sen}. For example, Bickel, Nair and Wang \cite{bnw} proved Theorem \ref{thm:gc} with convergence in probability when $E_n$ and $F$ have the same discrete support on finitely many values.
\section{Poisson coupling of size-biased permutation and order statistics}\label{sec:ppp}
Comparisons between the distribution of induced order statistics and order statistics of the same sequence have been studied in the literature \cite{dayang77, yang77, nagada94, hall94}. However, finite i.i.d s.b.p has the special feature that there exists an explicit coupling between these two sequences as described in Proposition ~\ref{prop:coupling}. Using this fact, we now derive Theorem \ref{thm}, which gives a Poisson coupling between the \emph{last} $k$ size-biased terms $X_n^{rev}[1], \ldots, X_n^{rev}[k]$ and the $k$ smallest order statistics $X_n^{\uparrow}(1), \ldots X^{\uparrow}_n(k)$ as $n \to \infty$. \footnote{The superscript `rev' indicates that the order statistics in consideration are arranged in increasing order, as opposed to decreasing, which has been the convention of this paper.}
The existence of a Poisson coupling should not be surprising, since it is well-known that the increasing sequence of order statistics $(X_n^{\uparrow}(1), X_n^{\uparrow}(2), \ldots)$ converges to points in a Poisson point process (p.p.p) whose intensity measure depends on the behavior of $F$ near the infimum of its support, which is $0$ in our case. This standard result in order statistics and extreme value theory dates back to R\'{e}nyi \cite{renyi}, and can be found in \cite{evtHaan}.
Our theorem is closely related to a result in PPY \cite[\S 4]{pitmanyor}. When $(X^\downarrow(1), X^\downarrow(2), \ldots)$ are ranked jumps of a subordinator, these authors noted that one can couple the size-biased permutation with the order statistics via the following p.p.p
\begin{equation}\label{eqn:ppois}
N(\cdot) := \sum_{k\geq 1}\mathbf{1}[(X[k],Y^{\uparrow}(k)) \in \cdot] = \sum_{k\geq1}\mathbf{1}[(X^{\downarrow}(k),Y_k) \in \cdot]. \end{equation}
$N(\cdot)$ has measure $ m(dxdy) = xe^{-xy}\, \Lambda(dx) dy. $ The first expression in (\ref{eqn:ppois}) defines a scatter of $(x,y)$ values in the plane listed in increasing $y$ values, and the second represents the same scatter listed in decreasing $x$ values. Since $\sum_{i\geq 1}X^\downarrow(i)$ is a.s. finite, the $x$-marginal of the points in (\ref{eqn:ppois}) has the distribution of the size-biased permutation of the infinite sequence $(X^\downarrow(1), X^\downarrow(2), \ldots)$ since it prescribes the joint distribution of the first $k$ terms $X[1], \ldots, X[k]$ for any finite $k$.
PPY use this p.p.p representation to generalize size-biased permutation to $h$-biased permutation, where the `size' of a point $x$ is replaced by an arbitrary strictly positive function $h(x)$; see \cite[\S 4]{pitmanyor}.
More generally, one obtains a random permutation of $\mathbb{N}$ from ranking points $(x_i, y_i)$ in a Poisson scatter $N(\cdot)$ on $(0, \infty)^2$ according to either their $x$ or $y$ values. For $i = 1, 2, \ldots$, let $(x^\ast_i)$ and $(y^\ast_i)$ denote the induced order statistics of the sequence $(x_i)$ and $(y_i)$ obtained by ranking points by their $y$ and $x$ values in increasing order, respectively. For $j, k = 1, 2, \ldots$, define sequences of integers $(K_j), (J_k)$ such that $x^{\uparrow}(J_k) = x^{\ast}_k$, $y^{\uparrow}(K_j) = y^{\ast}_j$; see Figure \ref{fig:scatter}.
\begin{figure}[hbtp]
\vskip-1.5em
\includegraphics[width = 0.7\textwidth]{./pic1.pdf}
\vskip-4.5em
\caption{A point scatter on the plane. Here $J_1 = 5, J_2 = 3, J_3 = 1, J_4 = 4, J_5 = 2$, and $K_1 = 3, K_2 = 5, K_3 = 2, K_4 = 4, K_5 = 1$. The permutations $J$ and $K$ are inverses. Conditioned on $x^\uparrow(2) = a$ and $y_2^\ast = b$, the number of points lying in the shaded region determines $J_2 - 1$.}
\label{fig:scatter}
\end{figure}
Suppose $N(\cdot)$ has intensity measure $m$ such that for all $x, y \in (0, \infty)$, $m((0, x) \times (0, \infty)) < \infty, m((0, \infty) \times (0, y)) < \infty$. For $j \geq 1$, conditioned on $x(j) = x, y^{\ast}_j = y$,
\begin{equation}\label{eqn:kj}
K_j - 1 \stackrel{d}{=} Poisson\left(m((x, \infty) \times (0, y))\right) + Binomial\left(j-1, \frac{m((0, x) \times (0, y))}{m((0, x) \times (0, \infty))}\right),
\end{equation}
where the two random variables involved are independent. Similarly, for $k \geq 1$, conditioned on $x^\ast_k = x$, $y(k ) = y$,
\begin{equation}\label{eqn:jk}
J_k - 1 \stackrel{d}{=} Poisson\left(m((0,x) \times (y,\infty))\right) + Binomial\left(k-1, \frac{m((0, x) \times (0, y))}{m((0, \infty) \times (y, \infty))}\right),
\end{equation}
where the two random variables involved are independent. When $m$ is a product measure, as is the case of i.i.d s.b.p, it is possible to compute the marginal distribution of $K_j$ and $J_k$ explicitly for given $j, k \geq 1$. We demonstrate such computations in Proposition ~\ref{prop:compute.general}.
Before stating the theorem we need some technical results. The distribution of the last few size-biased picks depends on the behavior of $F$ near $0$, the infimum of its support. We shall consider the case where $F$ has `power law' near $0$, like that of a Gamma distribution.
\begin{lem} \label{lem:approx.G.by.Gamma}
Suppose $F$ is supported on $(0, \infty)$ with Laplace transform $\phi$. Let $u = \phi(y)$, $X_u$ a random variable distributed as $G_u(dx)$ defined in (\ref{eqn:Gu}). For $\lambda, a > 0$,
\begin{equation}
F(x) \sim \frac{\lambda^a x^a}{\Gamma(a + 1)} \mbox{ as } x \to 0, \label{eqn:assump.F}
\end{equation}
if and only if,
\begin{align}
\phi(y) & \sim \lambda^a/y^{a} \hspace{1em} \mbox{ as } y \to \infty \label{eqn:phi.asymp}.
\end{align}
Furthermore, (\ref{eqn:assump.F}) implies
\begin{align}
u^{-1/a}X_u & \stackrel{d}{\longrightarrow} gamma(a+1,\lambda) \hspace{1em} \mbox{ as } u \to 0.
\label{eqn:xu.gamma}
\end{align}
\end{lem}
\begin{proof} The equivalence of (\ref{eqn:assump.F}) and (\ref{eqn:phi.asymp}) follows from a version of Karamata Tauberian Theorem \cite[\S 1.7]{regvar}. Assume (\ref{eqn:assump.F}) and (\ref{eqn:phi.asymp}). We shall prove (\ref{eqn:xu.gamma}) by looking at the Laplace transform of the non-size-biased version $X_u'$, which has distribution $F_u$. For $\theta \geq 0$,
\begin{equation}\label{eqn:laplace.xpu}
E(\exp(-\theta X_u')) = \int_0^\infty u^{-1}\exp(-yx - \theta x)\, F(dx) = u^{-1}\phi(y + \theta) = \frac{\phi(y+\theta)}{\phi(y)}.
\end{equation}
Now as $y \to \infty$ and $u = \phi(y) \to 0$, for each fixed $\eta > 0$, (\ref{eqn:phi.asymp}) implies
$$
E(\exp(-\eta u^{-1/a}X_u')) = \frac{\phi(y + \eta\phi(y)^{-1/a})}{\phi(y)}
\sim \frac{\lambda^a(y+\eta \lambda^{-1} y)^{-a} }{\lambda^ay^{-a}} = \left(\frac{\lambda}{\lambda+\eta}\right)^a.
$$
That is to say
\begin{equation}\label{eqn:xuprime.gamma}
u^{-1/a} X_u' \stackrel{d}{\longrightarrow} gamma(a, \lambda).
\end{equation}
Since $\phi$ is differentiable, (\ref{eqn:laplace.xpu}) implies $E(X'_u) = \phi'(y) / \phi(y)$. Now $\phi$ has an increasing derivative $\phi'$, thus (\ref{eqn:phi.asymp}) implies $\phi'(y) \sim a\lambda^a/y^{a+1}$ as $y \to \infty.$ Therefore,
$$u^{-1/a}E(X_u') = \frac{\phi'(y)}{\phi(y)^{1 + 1/a}} \to \frac{a}{\lambda},$$
which is the mean of a $gamma(a,\lambda)$ random variable. Thus the random variables $u^{-1/a}X'_u$ are uniformly integrable, so for any bounded continuous function $h$, we can compute
$$E(h(u^{-1/a}X_u)) = \frac{E[(u^{-1/a}X_u')h(u^{-1/a}X_u')]}{u^{-1/a}E(X_u')} \to \frac{E[\gamma_{a,\lambda}h(\gamma_{a,\lambda})]}{E(\gamma_{a,\lambda})} = E(h(\gamma_{a+1,\lambda})) $$
where $\gamma_{b,\lambda}$ is a $gamma(b,\lambda)$ random variable. This proves (\ref{eqn:xu.gamma}).
\end{proof}
Now suppose $F$ satisfies (\ref{eqn:assump.F}) for some $\lambda, a > 0$. By standard results in order statistics \cite[Theorem 2.1.1]{evtHaan}, as $n \to \infty$,
\begin{equation}\label{eqn:fidi.order}
\left(n^{1/a} X^{rev}_n(k), 1 \leq k \leq n\right) \stackrel{f.d.d.}{\longrightarrow} \left(\xi^{rev}(k): 1 \leq k < \infty\right),
\end{equation}
where f.d.d. stands for finite dimensional distribution. Here
\begin{equation}\label{eqn:fidi.sbs}
\xi^{rev}(k) = S_{(k)}^{1/a}\Gamma(a+1)^{1/a}/\lambda
\end{equation}
for $S_{(k)} = \epsilon_1 + \ldots + \epsilon_k$, where $\epsilon_i$ are i.i.d standard exponentials.
We now present the analogue of (\ref{eqn:fidi.sbs}) for the last few size-biased picks $X_n^{rev}[1], \ldots, X_n^{rev}[k]$ and the promised Poisson coupling.
\begin{thm} \label{thm}
Suppose that (\ref{eqn:assump.F}) holds for some $\lambda, a > 0$. \\
\textbf{a)} As $n \to \infty$,
\begin{equation}\label{15.a}
\left(n^{1/a} X_n^{rev}[k], 1 \leq k \leq n\right) \stackrel{f.d.d}{\longrightarrow} \left(\xi^{rev}[k]: 1 \leq k < \infty\right),
\end{equation}
with
\begin{equation} \label{15.b}
\xi^{rev}[k] = T_{(k)}^{1/a}\gamma_k / \lambda,
\end{equation}
where $T_{(k)} = \epsilon'_1 + \ldots + \epsilon'_k$ for i.i.d standard exponential r.v. $\epsilon'_i$, $\gamma_k, k = 1, \ldots, n$ are i.i.d $gamma(a+1,1)$, and the $(\epsilon_k')$ and $(\gamma_k)$ are independent.
\vskip12pt \noindent
\textbf{b)} Let $\xi^{rev}(1) < \xi^{rev}(2) < \ldots $ be the increasing order statistics of the $\xi^{rev}[k]$'s. Then the $\xi^{rev}(k)$'s are jointly distributed as (\ref{eqn:fidi.sbs}), for a sequence $(\epsilon_k)$ which is implicitly defined in terms of the $(\epsilon'_k)$ and $(\gamma_k)$ above, and the f.d.d. convergence in (\ref{eqn:fidi.order}) and (\ref{15.a}) hold jointly.
\vskip12pt \noindent
\textbf{c)} For each $n$, let $J_n = (J_{nk}, 1 \leq k \leq n)$ be the permutation of $\{1, \ldots, n\}$ defined by
$$ X^{rev}_n[k] = X_n(J_{nk}). $$
As $n \to \infty$,
\begin{equation}\label{15.g}
\left(J_{nk}, 1 \leq k \leq n\right) \stackrel{f.d.d.}{\longrightarrow} \left(J_k: 1 \leq k < \infty\right),
\end{equation}
where $J_k$ is the random permutation of $\{1, 2, \ldots, \}$, defined by
\begin{equation}\label{15.h}
\xi^{rev}[k] = \xi^{rev}(J_k)
\end{equation}
for $k = 1, 2, \ldots,$ and the f.d.d. convergence in (\ref{eqn:fidi.order}), (\ref{15.a}), (\ref{15.g}) all hold jointly.
\end{thm}
\begin{proof} By Lemma \ref{lem:approx.G.by.Gamma}, it is sufficient to prove the theorem for the case $F$ is $gamma(a, \lambda)$. Part a follows immediately from Proposition \ref{prop:newgb} and law of large numbers, in the same way that part b follows from the proof of (\ref{eqn:fidi.sbs}). For the last part, note that in principle everything has been presented as a function of the variables $T_{(k)}$ and $\gamma_k$. For $a, \lambda > 0$, define
$$\Psi_{a,\lambda}(s) = s^{1/a}\Gamma(a+1)^{1/a}/\lambda.$$
Observe that $\Psi_{a,\lambda}$ has inverse function $\Psi_{a,\lambda}^{-1}(x) = \lambda^a x^a / \Gamma(a+1)$. Rewrite (\ref{eqn:fidi.sbs}) as
\begin{equation}\label{eqn:xik}
\xi^{rev}(k) = \Psi_{a,\lambda}(S_{(k)}),
\end{equation}
then (\ref{15.b}) implies
\begin{equation}\label{16.a}
S_k = \Psi_{a,\lambda}^{-1}(\xi^{rev}[k]) = \lambda^a \xi^{rev}[k]^a/\Gamma(a+1) = T_{(k)}\gamma_k^a / \Gamma(a+1).
\end{equation}
Comparing (\ref{eqn:xik}) and (\ref{16.a}) gives a pairing between $S_k$ and $S_{(k)}$ via $\xi(k)$ and $\xi^{rev}[k]$. Hence we obtain another definition of $J_k$ equivalent to (\ref{15.h}):
$$ S_k = S_{(J_k)}. $$
Let $T_j$ be the $T$ value corresponding to the order statistic $S_{(j)}$ of the sequence $S_k$. That is,
$$T_j = T_{(K_j)}$$
where $(K_j)$ is a random permutation of the positive integers. By (\ref{16.a}), $(J_k)$ is the inverse of $(K_j)$. Together with (\ref{eqn:fidi.order}) and (\ref{15.a}), this implies (\ref{15.g}), completing the proof of part c.
\end{proof}
Since the $T_{(k)}$'s are increasing ordered points of a p.p.p on $(0, \infty)$ with rate 1 and independent of the $\gamma_k$'s, the point process on the positive quadrant $(0, \infty)^2$
\begin{equation}\label{16.b}
N(\cdot) = \sum_k\mathbf{1}[(S_k, T_{(k)}) \in \cdot] = \sum_j\mathbf{1}[(S_{(j)}, T_j) \in \cdot]
\end{equation}
is a p.p.p for the measure $\mu$ with density
\begin{equation}\label{eqn:measure.mu}
\frac{\mu(ds\, dt)}{ds\, dt} = \frac{1}{a} \Gamma(a+1)^{1/a}(s/t)^{1/a}\exp\{-(\Gamma(a+1)s/t)^{1/a}\}
\end{equation}
The random permutation $J_k$ and its inverse $K_j$ are obtained from the p.p.p in (\ref{16.b}) in the $(s,t)$ plane by ranking the points in two different orders according to $s$-value or $t$-value. Since the projection of a Poisson process is Poisson, the $s$ and $t$-marginal of $\mu$ is just Lebesgue measure. Thus, the ranked $s$-values form a p.p.p with rate 1 which determine the ranked $\xi(j)$ by the \emph{deterministic} increasing transformation
$$ \xi^{rev}(k) = \Psi_{a,\lambda}(S_{(k)}).$$
Furthermore,
$$\xi^{rev}[k] = \xi^{rev}(J_k) = \Psi_{a,\lambda}(S_k)$$
comes from the same set of $s$-values listed in order of increasing $t$-values.
Marginal distributions of the random permutation $(J_k)$ and its inverse $(K_j)$ are given in (\ref{eqn:kj}) and (\ref{eqn:jk}). Note that for $k = 1,2, \ldots$, $S_k = T_{(k)}\gamma_k^a/\Gamma(a+1)$ for i.i.d $\gamma_k$ distributed as $gamma(a+1,1)$, independent of the sequence $(T_{(k)})$, and $T_k = \Gamma(a+1)S_{(k)}\tilde{\epsilon}_k^{-a}$ for i.i.d standard exponentials $\epsilon_k$, independent of the sequence $(S_{(k)})$ but not of the $\gamma_k$'s. Thus by conditioning on either $S_{(k)}$ or $T_{(k)}$, one can evaluate (\ref{eqn:kj}) and (\ref{eqn:jk}) explicitly. In particular, by a change of variable $r = \Gamma(a+1)^{1/a}(s/t)^{1/a}$, one can write $\mu$ in product form. This leads to the following.
\begin{prop}\label{prop:compute.general}
For $j \geq 1$, conditioned on $S_{(j)} = s, T_j = \Gamma(a+1)\, s \, r^{-a}$ for some $r > 0$, $K_j-1$ is distributed as
\begin{equation}\label{eqn:marginal.kj}
Poisson(m(s,r)) + Binomial(j-1, p(s,r))
\end{equation}
with
\begin{equation}\label{eqn:ppp.density.kj}
m(s,r) = asr^{-a}\, \int_r^\infty x^{a-1}e^{-x}\, dx,
\end{equation}
and
\begin{equation}\label{eqn:psr}
p(s,r) = as^{2/a-2}r^{a-2}\int_0^r x^{a-1}e^{-x}\, dx,
\end{equation}
where the Poisson and Binomial random variables are independent.
Similarly, for $k \geq 1$, conditioned on $T_{(k)} = t, S_k = t \, r^a / \Gamma(a+1)$ for some $r > 0$, $J_k - 1$ is distributed as
\begin{equation}\label{eqn:marginal.jk}
Poisson(m'(t, r)) + Binomial(k-1, p'(t,r))
\end{equation}
with
\begin{equation}\label{eqn:ppp.density.jk}
m'(t,r) = t \, \left(\frac{r^a + a \, \int_r^\infty x^{a-1}e^{-x}\, dx}{\Gamma(a+1)} - 1\right),
\end{equation}
and
\begin{equation}\label{eqn:ptr}
p'(t,r) = \Gamma(a+1)^{1-2/a}\, at^{2/a-2}r^{a-1/a}\int_0^r x^{a-1}e^{-x}\, dx,
\end{equation}
where the Poisson and Binomial random variables are independent.
\end{prop}
\begin{prop}[Marginal distributions of $K_1$ and $J_1$]\label{prop:compute}
Suppose that (\ref{eqn:assump.F}) holds for some $\lambda > 0$ and $a = 1$. Then the distribution of $K_1$, the $k$ such that $\xi(1) = \xi^{rev}[k]$, is a mixture of geometric distributions, and so is that for $J_1$, the $j$ such that $\xi^{rev}[1] = \xi(j)$. In particular,
\begin{equation}\label{18.k1}
P(K_1 = k) = \int_0^\infty p_rq_r^{k-1}e^{-r}\, dr
\end{equation}
where $p_r = r/(r + e^{-r})$, $q_r = 1 - p_r$, and
\begin{equation}\label{18.j1}
P(J_1 = j) = \int_0^\infty \tilde{p}_r\tilde{q}_r^{j-1}re^{-r}\, dr
\end{equation}
where $\tilde{p}_r = 1/(r+e^{-r})$, $\tilde{q}_r = 1 - \tilde{p}_r$.
\end{prop}
\begin{proof} When $a = 1$, $\int_r^\infty t^{a-1}e^{-t}\, dt = e^{-r}$. Substitute to (\ref{eqn:ppp.density.kj}) and (\ref{eqn:ppp.density.jk}) give
$$m(s,r) = sr^{-1}e^{-r}, \hspace{1em} m'(t,r) = t(r-1+e^{-r}).$$
By a change of variable, (\ref{eqn:measure.mu}) becomes
$$\frac{\mu(ds\, dr)}{ds\, dr} = se^{-r}, \hspace{1em} \frac{\mu(dt\, dr)}{dt\, dr} = tre^{-r}.$$
Thus, \emph{conditioned on $s$ and $r$}, $K_1 - 1$ is distributed as the number of points in a p.p.p with rate $r^{-1}e^{-r}$ before the first point in a p.p.p with rate 1. This is the geometric distributions on $(0, 1, \ldots)$ with parameter $p_r = 1/(1 + r^{-1}e^{-r})$. Since the marginal density of $r$ is $e^{-r}$, integrating out $r$ gives (\ref{18.k1}). The computation for the distribution of $J_1$ is similar.
\end{proof}
One can check that (\ref{18.k1}) and (\ref{18.j1}) sum to 1. We conclude with a `fun' computation. Suppose that (\ref{eqn:assump.F}) holds for some $\lambda > 0$ and $a = 1$. That is, $F$ behaves like an exponential c.d.f near $0$. By Proposition \ref{prop:compute}, $E(J_1) = 9/4$ and $E(K_1) = \infty$. That is, the last size-biased pick is expected to be almost the second smallest order statistic, while the smallest order statistic is expected to be picked infinitely earlier on in a successive sampling scheme(!). The probability that the last species to be picked in a successive sampling scheme is also the one of smallest species size is
\begin{align*}
\lim_{n \to \infty} P(X_n^{rev}[1] = X_n(1)) &= P(\xi^{rev}[1] = \xi(1)) = P(J_1 = 1) = P(K_1 = 1) \\
&= \int_0^\infty \frac{re^{-r}}{r + e^{-r}}\, dr = 1 - \int_0^1 \frac{u}{u - \log u}\, du \approx 0.555229
\end{align*}
\bibliographystyle{plain}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,664 |
Q: Blocked JS deviceorientation devicemotion on Android WebView I've written a JS SDK for Android WebView that collects device orientation and motion. The SDK listens to deviceorientation and devicemotion events on the window like so:
window.addEventListener('devicemotion', (event) => {...})
window.addEventListener('deviceorientation', (event) => {...})
On some devices/integrations, I get no sensors data. I've tried to mimic a "bad" integration, attempting to block the WebView sensors access by adding the following to the app manifest but with no luck. The JS events are still triggered:
<activity
android:screenOrientation="portrait"
android:configChanges="keyboardHidden|orientation|screenSize"
What are other possible ways to block the WebView from triggering the events, besides disabling JS all together?
Update:
Some insights:
the most problematic devices are:
*
*Samsung Galaxy Tab a 10.1 SM-T580
*Samsung Galaxy J5 Prime SM-G570M
Update 2
I have similar issues on IOS on some devices, most problematic is:
*
*Mozilla/5.0 (iPhone; CPU iPhone OS 12_3_1 like Mac OS X)
AppleWebKit/605.1.15 (KHTML, like Gecko) Mobile/15E148
A: Might be, it's not possible to stop only triggering JS events through android, if you want it then it'll disable whole JS.
//disabled
wView.getSettings().setJavaScriptEnabled(false);
//enabled
wView.getSettings().setJavaScriptEnabled(true);
As, you told that on some devices you aren't getting sensors data, for this you need to check your webview decleration code.
Sample code is here:
wView= new WebView(this);
wView.getSettings().setLoadsImagesAutomatically(true);
wView.getSettings().setJavaScriptEnabled(true);
wView.setScrollBarStyle(View.SCROLLBARS_INSIDE_OVERLAY);
wView.setWebViewClient(new WebViewClient() {
@SuppressWarnings("deprecation")
@Override
public void onReceivedError(WebView view, int errorCode, String description, String failingUrl) {
Toast.makeText(MainActivity.this, description, Toast.LENGTH_SHORT).show();
}
@TargetApi(android.os.Build.VERSION_CODES.M)
@Override
public void onReceivedError(WebView view, WebResourceRequest req, WebResourceError rerr) {
// Redirect to deprecated method, so you can use it in all SDK versions
onReceivedError(view, rerr.getErrorCode(), rerr.getDescription().toString(), req.getUrl().toString());
}
});
wView.loadUrl(HOST_URL);
In Manifest
<activity
android:name=".MainActivity"
android:configChanges="orientation|screenSize" />
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 6,562 |
\section{Introduction}
\label{sec.introduction}
Personalized recommendation aims to provide appropriate items for users according to their preferences, where user historical behavior sequence is an informative source for user understanding. \emph{Sequential recommendation}, which takes users' historical behavior sequences as inputs and outputs next predicted items, is widely studied and deployed in practice \cite{kang2018self,sun2019bert4rec,xie2021adversarial}.
With the thriving of pre-training in NLP \cite{devlin2019bert}, there are lots of efforts that bring \emph{pre-training} into sequential recommendation \cite{zeng2021knowledge}. These pre-trained recommendation models usually consider user behavior sequences as token sequences in NLP, using pre-training techniques to improve the sequence modeling ability, which can alleviate the sparsity issues in real-world recommendation systems \cite{xiao2021uprec,xie2020contrastive}.
Recently, in the overwhelming trend of pre-training, how to effectively and efficiently extract useful information from pre-trained models becomes a promising direction. \textbf{Prompt-tuning} \cite{brown2020language,lester2021power} is a representative and powerful method that has remarkable superiority over the generic fine-tuning paradigm especially in few-shot scenarios. It usually inserts hard text templates \cite{brown2020language,gao2021making} or soft continuous embeddings \cite{li2021prefix,qin2021learning} as \textbf{prompts}, and transforms the downstream tasks into similar well-trained pre-training tasks. The advantages of prompt-tuning locate in two aspects:
(1) it bridges the gap between pre-training and downstream objectives, which could better utilize the knowledge in pre-training models. This advantage will be multiplied in cold-start scenarios.
(2) Prompt-tuning only needs to tune a small set of parameters for the prompts and labels, which is more efficient.
Looking back to the sequential recommendation task, the cold-start user issues (including zero-shot and few-shot scenarios) are crucial challenges due to the sparsity of interactions in real-world systems. We attempt to adopt the powerful pre-training with prompt-tuning to address the cold-start recommendation.
\begin{figure}[!hbtp]
\centering
\includegraphics[width=0.99\columnwidth]{figs/example.pdf}
\caption{An example of prompt-tuning in recommendation.}
\label{fig:example}
\end{figure}
However, introducing prompts to recommendation is non-trivial due to the following challenges:
(1) \emph{\textbf{How to transfer the prompt-tuning of NLP into recommendation}}? Differing from words in NLP, the behaviors (items) in recommendation are hard to be directly used to build hard explainable prompts and labels. Moreover, it is also challenging to design an appropriate framework to make full use of pre-training knowledge in personalized recommendation.
(2) \emph{\textbf{How to build appropriate prompts for personalized recommendation}}? Compared with NLP, recommendation further values personalization. Therefore, the proposed prompts should better be personalized so as to more pertinently extract user-related knowledge from the huge pre-training model.
To address these challenges, we propose a novel \textbf{Personalized prompt-based recommendation (PPR)} framework, which first adopts prompt-tuning in sequential recommendation. Specifically, PPR designs the \emph{personalized soft prefix prompt} learned in the proposed prompt generator based on user profiles, which could better extract user-related knowledge from pre-training models for various downstream tasks in few-shot and zero-shot scenarios within a universal framework. We also propose a prompt-oriented contrastive learning (CL) via both prompt-based and behavior-based data augmentations to further enhance the training of prompts.
Compared with the generic fine-tuning, our PPR has the following advantages:
(1) PPR enables the prompt-tuning in recommendation, making full use of pre-trained models for more effective and efficient tuning via personalized prompts.
(2) PPR designs a set of prompt-oriented contrastive learning losses on prompt-enhanced behavior sequences, which enables a more sufficient training for prompts.
(3) PPR is effective, universal, and easy-to-deploy, which could be conveniently adopted for other downstream tasks such as cross-domain recommendation and user profile prediction.
In experiments, we conduct extensive evaluations to verify the effectiveness and universality of our personalized prompt-based recommendation. We conduct CTR prediction evaluations on three large-scale practical datasets in both few-shot and zero-shot scenarios, which confirms that PPR can achieve significant improvements on both scenarios. Ablation study and sparsity analysis are also conducted. Moreover, we also verify PPR's universality on other pre-training models and other challenging downstream tasks (e.g., cross-domain recommendation and user profile prediction), shedding light on the promising applications of personalized prompts in practice. The contributions of this work are concluded as follows:
\begin{itemize}
\item We propose a novel personalized prompt-based recommendation to better extract useful knowledge from pre-training models. To the best of our knowledge, we are the first attempt to bring prompt-tuning in sequential recommendation.
\item We design an universal prompt-based framework with the help of prompt-oriented contrastive learning considering prompt- and behavior- based augmentations, which further improves the model performances.
\item We have verified the effectiveness and universality of PPR on different pre-trained models and downstream tasks, including few-shot recommendation, zero-shot recommendation, cross-domain recommendation, and user profile prediction.
\end{itemize}
\section{Related Works}
\label{sec.related_work}
\noindent
\textbf{Sequential Recommendation.}
Sequential recommendation models mainly leverage users' chronological behavior sequences to learn user preferences.
Recently, various deep neural networks have been employed for sequence-based recommendation. GRU4Rec \cite{hidasi2016session} proposes to use Gated Recurrent Units in the session-based recommendation. Inspired by the success of Transformer and BERT \cite{devlin2019bert}, SASRec \cite{kang2018self} and Bert4Rec \cite{sun2019bert4rec} adopt self-attention mechanisms to model user behavior sequence. The cold-start problem on sequential recommendation also attracts wide attention of researchers. \citet{zheng2021cold} utilizes a meta-learning mechanism to alleviate the cold-start item problem in sequential recommendation. \citet{liu2021augmenting} augments short behavior sequence by reversely predicted items.
\noindent
\textbf{Pre-training in Recommendation.}
Recently, pre-training models have achieved great successes in NLP \cite{devlin2019bert,liu2019roberta} and CV \cite{he2021masked}. It aims to learn prior knowledge from general large-scale datasets to help the specific downstream tasks. After pre-training, models are further fine-tuned on downstream supervised signals to fit the specific task. This \emph{pre-training and fine-tuning} paradigm is widely applied to various tasks \cite{devlin2019bert}.
With the thriving of pre-training, many pre-training models have been proposed in recommendation. BERT4Rec \cite{sun2019bert4rec} adopts masked item prediction in sequential recommendation. $S^3$Rec \cite{zhou2020s3} pre-trains the sequential encoder considering the correlations among item's attribute, item, subsequence, and sequence. CL4Rec \cite{xie2020contrastive} applies CL via item crop, mask, and reorder on sequence modeling. PeterRec\cite{yuan2020parameter} pre-trains a CNN-based model and transfers it to solve cross-domain recommendation problem by adapter technology. UPRec \cite{xiao2021uprec} further highlights user profiles and social relations in pre-training.
Inspired by the successes of pre-training, we propose PPR, which aims to (a) narrow the gap between pre-training models and downstream tasks, and (b) better extract useful knowledge from pre-trained models by replacing fine-tuning with personalized prompt-tuning. We have deploy PPR on different tasks and pre-training models to verify its effectiveness.
\noindent
\textbf{Prompt Tuning.}
Prompt-tuning is first proposed in NLP, and is widely explored and dominating especially in few-shot scenarios) \cite{radford2019language,brown2020language}.
\citet{schick2020exploiting} and \citet{brown2020language} adopt hard prompts that consist of discrete real words. Considering that manually designing the hard prompt is both time-consuming and trivial, other works \cite{gao2021making,jiang2020can,shin2020autoprompt} focus on automatically searching for hard prompts.
In contrast, soft prompt is composed of several continuous learnable embeddings randomly initialized.
Prefix-tuning \cite{li2021prefix} optimizes continuous prompts for generation tasks in NLP.
P-tuning \cite{liu2021gpt} designs prompts to GPT for natural language understanding. PPT \cite{gu2021ppt} further conducts pre-training on the prompts to better initialize the soft prompts. In this work, we first introduce prompts into recommendation. Different from prompts in NLP, we design a personalized prompt for each user.
\section{Methodology}
\label{sec.method}
\subsection{Preliminaries}
\label{sec.preliminaries}
\begin{figure*}[!hbtp]
\centering
\includegraphics[width=0.90\textwidth]{figs/PPR_main.pdf}
\caption{Overall architecture of (a) conventional fine-tuning, and (b) our Personalized prompt-based recommendation.}
\label{fig:overall}
\end{figure*}
\subsubsection{Prompt in NLP}
In NLP pre-training, prompt is often a piece of text inserted into the input text sequence. For example, in sentiment analysis, a prompt ``it is [mask]'' is inserted after the review ``a real joy'' as a natural sentence: ``a real joy, it is [mask]''. In prompt-tuning, the predicted tokens of ``[mask]'' (e.g., great) will be mapped to the sentiment labels (e.g., positive) via a verbalizer. In this case, the original task (e.g., sentiment classification) can be formulated as a masked language model task, which has been fully optimized in pre-training and thus has better performances \cite{radford2019language,brown2020language,shin2020autoprompt}.
The advantages of prompt-tuning are as follows: (1) it can better extract useful knowledge in pre-trained models via similar modeling and objectives fully learned in pre-training based on prompts, especially for few-shot scenarios compared to fine-tuning \cite{gu2021ppt}, and (2) it is more efficient compared to fine-tuning, for most prompt-tuning does not fully update all parameters in pre-trained models.
\subsubsection{Tasks of PPR}
In this work, we propose Personalized prompt-based recommendation to provide a more effective and efficient tuning for cold-start downstream tasks based on pre-training models in sequential recommendation.
The \emph{pre-training and fine-tuning} is a classical widely-used training paradigm, which first trains a pre-trained model on large-scale general dataset, and then fine-tunes the whole pre-trained model on the few-shot supervised information of downstream tasks.
Recently, the \textbf{pre-training and prompt-tuning} paradigm has been widely verified in NLP as introduced in Sec. \ref{sec.related_work}, where the prompts and verbalizer are mainly updated. PPR aims to better extract useful personalized information from pre-trained models via prompt-tuning rather than fine-tuning.
Precisely, the main task that our PPR focuses on is cold-start recommendation (since prompt-tuning functions well especially in few-shot learning), which consists of both \emph{few-shot recommendation} and \emph{zero-shot recommendation}. Moreover, we also deploy PPR for other downstream tasks such as \emph{cross-domain recommendation} and \emph{user profile prediction} in Sec. \ref{sec.explorations} for universality.
\subsubsection{Notions of PPR}
The key notions used in PPR are defined as follows. We denote user and item as $u\in U$ and $v\in V$, where $U$ and $V$ are the overall user set and item set. Each user $u$ has a historical behavior sequence $s_u=\{v_1^{u},v_2^{u},...,v_{|s_u|}^{u}\}$ (of length $|s_u|$) ordered by time. Each user has $m$ basic attributes $A_u=\{a_1^u,a_2^u,...,a_m^u\}$ (i.e., user profiles).
In cold-start recommendation, we split users into a warm user set $U^W$ (for pre-training) and a cold user set $U^C$ (for tuning) by their behavior sequence length.
Then, we pre-train a sequential model $f_{seq}(\{v_1^u,v_2^u,...,v_{|s_u|}^u\}|\Theta)$ to learn user representations by warm users' historical behaviors, where $f_{seq}(.)$ is the sequential model, $\Theta$ is the pre-trained model's parameters. After that, we fine-tune the pre-trained model on cold users' historical behaviors and get the fine-tuned model $f_{seq}(\{v_1^u,v_2^u,...,v_{|s_u|}^u\}|\hat{\Theta})$, where $\hat{\Theta}$ is the fine-tuned model's parameters. Different from fine-tuning, our PPR inserts personalized prompts (i.e., prefix tokens) before cold users' behaviors, noted as $f_{seq}(\{p_1^u,p_2^u,...,p_n^u,v_1^u,v_2^u,...,v_{|s_u|}^u\}|\Theta,\vartheta)$, where $\{p_1^u,p_2^u,..,p_n^u\}$ is the personalized prompt built from user profiles, and $\vartheta$ is the parameters of the prompt generator.
\subsection{Overall Framework}
\label{sec.overall}
The overall framework of PPR is illustrated in Fig. \ref{fig:overall}. For each user, we first build the personalized prompts according to user profiles (user static attributes such as age, gender) via the prompt generator, and insert them into the beginning of the user behavior sequence. The new sequence is then fed into the pre-trained sequential model to get users' behavioral preferences. The user profiles are further added into another network to generate users' attribute preferences, which is an essential supplement for cold-start users widely used in industry. Finally, the user's behavioral and attribute preferences are combined to get the final user representation.
In PPR(light), the pre-training model is fixed during prompt-tuning. To enable a more sufficient training for personalized prompts, we further design a set of contrastive learning losses in PPR, adopting data augmentations on both the prompt generator and sequence modeling parts. PPR has been deployed and verified on various downstream tasks.
\subsection{Pre-training of PPR}
\label{sec.pre-training}
We first introduce the pre-training part of PPR. There are various self-attention based sequential models \cite{kang2018self,sun2019bert4rec,xie2020contrastive} verified to be effective as pre-training models. Following \citet{xie2020contrastive}, we also use the classical SASRec \cite{kang2018self} as PPR's pre-training model. Specifically, SASRec stacks Transformer(·) blocks to encode the historical behavior sequence. For the input behavior sequence $s_u$, we define its $l$-layer's behavior matrix as $\bm{H}_u^l=\{\bm{h}_{u,1}^l,\bm{h}_{u,2}^l, \cdots, \bm{h}^{l}_{u,|s_u|}\}$, where $\bm{h}_{u,i}^l$ is the $i$-th behavior's representation of $u$ at the $l$-th layer. The $(l+1)$-layer's behavior matrix $\bm{H}_u^{l+1}$ is then learned as follows:
\begin{equation}
\begin{aligned}
\bm{H}_u^{l+1}=\mathrm{Transformer}^l(\bm{H}_{u}^{l}), \quad
\bm{u}_o=f_{seq}(s_u|\Theta)=\bm{h}_{u,|s_{u}|}^{L}.
\end{aligned}
\label{eq.transformer}
\end{equation}
$\bm{u}_o$ is the final user representation of $u$ learned in pre-training to predict user's next items, which is generated as the last behavior's representation at the $L$-th layer (i.e., $\bm{h}_{u,|s_{u}|}^{L}$). Here, $L$ is the number of Transformer layers.
Following classical ranking models \cite{rendle2009bpr,kang2018self,xie2020contrastive}, the pre-training model is optimized under the objective $L_o$ as follows:
\begin{equation}
\begin{split}
L_o = -\sum_{(u,v_i) \in S^w_+} \sum_{(u,v_j) \in S^w_-} \log \sigma (\bm{u}_o^\top\bm{v}_i - \bm{u}_o^\top\bm{v}_j), \quad u\in U^w,
\end{split}
\label{eq.simple_loss}
\end{equation}
where $(u,v_i) \in S^w_+$ indicates the positive set where $u$ has clicked $v_i$, and $(u,v_j) \in S^w_-$ indicates the negative set where $v_j$ is randomly sampled negative items. $\sigma (\cdot)$ is the sigmoid function. We optimize the parameters $\Theta$ of the pre-training model via $L_o$ on pre-training dataset of $u \in U^w$ for downstream tasks.
Differing from classical prompt-tuning in NLP that usually adopts the masked language model (MLM) pre-training task \cite{devlin2019bert}, PPR mainly concentrates on the next-item prediction task in pre-training. It is because that the next-item prediction task perfectly fits our downstream tasks (cold-start recommendation and user profile prediction) with PPR, which can stimulate the maximum potential of prompt.
Note that our PPR can be flexibly adopted on different pre-training models. To verify the universality of PPR, we further deploy PPR on CL4Rec \cite{xie2020contrastive}, which also achieves consistent improvements (in Sec. \ref{sec.universality}).
\subsection{Personalized Prompt-tuning}
After pre-training, PPR conducts a personalized prompt-tuning instead of fine-tuning to better learn from pre-trained models.
\subsubsection{Personalized Prompt Generator}
The key of our PPR is generating an effective prompt that helps to narrow the gap between pre-trained models and downstream tasks. However, it is challenging to find appropriate prompts in recommendation, since (1) it is difficult to build hard prompts and labels (i.e., some real tokens) in PPR. Unlike words in NLP, the tokens (i.e., items) in recommendation do not explicit meaningful semantics. (2) Moreover, unlike NLP, recommendation should be personalized, thus the prompts should also be customized for different users. In a sense, each user's recommendation can be viewed as a task, while there are millions of users in a real-world system. It is impossible to manually design personalized prompts for all users.
In PPR, to automatically build personalized prompts for all users, we rely on the essential and informative user profiles. User profiles can be learned from all types of user information, such as user static attributes (e.g., age, gender, location), user cumulative interests, and user behaviors in other domains. In the task of cold-start recommendation, we mainly consider the user attributes $A_u$. Specifically, we concatenate $m$ user profile embeddings as $\bm{x}_u=[\bm{a}_1^u||\bm{a}_2^u|| \cdots ||\bm{a}_m^u]$, where $\bm{a}_i^u$ is the $i$-th user profile embedding. We conduct a Multi-layer perceptron (MLP) to learn the personalized prompt representation $\bm{P}^u=\{\bm{p}_1^u, \cdots, \bm{p}_n^u\}$ containing $n$ tokens as follow:
\begin{equation}
\begin{split}
\bm{P}^u = \mathrm{PPG} (\bm{x}_u|\vartheta)=\bm{W}_2\sigma(\bm{W}_1\bm{x}_u+\bm{b}_1)+\bm{b}_2,
\end{split}
\label{eq.prompt}
\end{equation}
where $\bm{W}_1\in \mathbb{R}^{d_1 \times d'}$ ,$\bm{W}_2\in \mathbb{R}^{d'\times d_2\times n}$, $\bm{b}_1\in \mathbb{R}^{d'}$ and $\bm{b}_2 \in \mathbb{R}^{d_2 \times n}$ are trainable parameters in $\vartheta$. $d_1$, $d'$, and $d_2$ are the embedding sizes of concatenated user profile $\bm{x}_u$, hidden layer, and output prompts respectively. $n$ is the number of prompt tokens.
Inspired by the success of prefix-tuning \cite{li2021prefix}, we adopt the prompt tokens as a prefix and connect them with user behavior sequence as: $\hat{s}_u=\{p_1^u,p_2^u,...,p_n^u,v_1^u,v_2^u,...,v_{|s_u|}^u\}$. This \emph{prompt-enhanced sequence} contains both task-specific and user-specific information. For example, if $A_u=\{$20-year-old, female$\}$, the prompt-enhanced sequence $\hat{s}_u$ can be translated as ``In cold-start recommendation, a user is a 20-year-old female, she likes $v_1$, $v_2$, $\cdots$ '', which provides a more personalized context of user behavior sequences. Through these personalized prompts, the information of various user profiles is naturally fused into user behavioral information and jointly optimized. Hence, the power of pre-trained models (e.g., sequential modeling) can be fully reused to capture user diverse preferences from different sources, which is essential in cold-start scenarios.
\subsubsection{Tuning of PPR}
\label{sec.tuning_PPR}
After generating the personalized prompts, we need to construct the final user representation for recommendation. Precisely, we first input the prompt-enhanced sequence $\hat{s}_u$ to the pre-trained sequential model in Eq. (\ref{eq.transformer}) to get the user behavioral preference $\bm{u}_s$. Next, we directly learn from user profiles via an MLP to get the user attribute preference $\bm{u}_a$. Finally, both $\bm{u}_s$ and $\bm{u}_a$ are combined to get the final user representation $\bm{u}_p$. We have:
\begin{equation}
\begin{split}
\bm{u}_p = \bm{u}_a + \bm{u}_s, \quad
\bm{u}_a = \mathrm{MLP}_a (\bm{x}_u|\phi), \quad
\bm{u}_s = f_{seq}(\hat{s}_u|\Theta,\vartheta).
\end{split}
\label{eq.prompt_user}
\end{equation}
$\Theta$, $\vartheta$, and $\phi$ are parameters of the sequential model, the prompt generator, and the user profile learner for $\bm{u}_a$ respectively.
To tune these parameters, we propose two prompt tuning strategies, PPR(light) and PPR(full), to balance effectiveness and efficiency in the scene of sequential recommendation.
In \textbf{PPR(light)}, we merely update the newly-introduced parameters, i.e., $\vartheta$ of the prompt generator in Eq. (\ref{eq.prompt}) and $\phi$ of the user profile learner in Eq. (\ref{eq.prompt_user}), with other parameters fixed. It is similar to the typical prompt-tuning manner in NLP \cite{li2021prefix}. This is a straightforward and efficient prompt-tuning manner, which completely relies on the pre-trained models in sequence modeling and item representation learning. The tuned parameters of PPR(light) are greatly reduced compared to fine-tuning (note that even the item embeddings are also fixed for efficiency). The tuned personalized prompts works as an inducer, which smartly extracts useful knowledge related to the current user from large-scale pre-trained model.
However, due to the huge gaps between NLP and recommendation tasks, the widely-verified ``light'' prompt-tuning does not always perform satisfactory enough in downstream tasks. It is because that in plenty of NLP downstream tasks such as sentiment analysis, text classification, and sequence labeling, the numbers of predicted labels are limited. On the contrary, the predicted items in recommendation (i.e., labels of verbalizer) are often million-level in practice, which should also be trained sufficiently in tuning.
Hence, we propose another \textbf{PPR(full)} manner for a wider range of tuning, which further tunes the parameters of $\Theta$ (including parameters of item embeddings and sequential model) besides $\vartheta$ and $\phi$.
We follow the pre-training objective in Eq. (\ref{eq.simple_loss}) to build the optimization objective $L_p$ of our prompt-tuning as follows:
\begin{equation}
\begin{split}
L_p = -\sum_{(u,v_i) \in S^c_+} \sum_{(u,v_j) \in S^c_-} \log \sigma (\bm{u}_p^\top\bm{v}_i - \bm{u}_p^\top\bm{v}_j), \quad u\in U^c.
\end{split}
\label{eq.prompt_loss}
\end{equation}
$S^c_+$ and $S^c_-$ are similar positive and negative sample sets in tuning.
\subsection{Prompt-oriented Contrastive Learning}
\label{sec.data_augmentation}
The main challenge of cold-start recommendation is the lack of sufficient tuning instances. Recently, contrastive learning (CL) has shown its power in recommendation \cite{xie2020contrastive,zhou2020s3}. These CL-based models usually conduct self-supervised learning (SSL) as supplements to supervised information via certain data augmentations, which could obtain more effective and robust user representations and alleviate the data sparsity issues. Inspired by those methods, we also adopt CL as auxiliary losses via two types of data augmentations based on elements of our prompt-tuning.
\subsubsection{Prompt-based Augmentation}
In real-world systems, the user basic attributes are usually noisy or even missing, while they are the main source of our personalized prompts, which are essential especially in zero-shot scenarios. Therefore, we design a prompt-based data augmentation to improve the effectiveness and robustness of the prompt generator. Specifically, we conduct a random element-level masking on the feature elements of user profile embeddings $\bm{x}_u$ to obtain $\bm{\bar{x}}_u$ with a certain mask ratio $\gamma_1$. Formally, the augmented prompt-enhanced behavior sequence is noted as $\bar{s}_u^1=\{\bar{p}_1^u,\bar{p}_2^u, ...,\bar{p}_n^u,v_1^u,v_2^u, ...,v_{|s_u|}^u\}$, where $\bm{\bar{P}}^u = \mathrm{PPG} (\bm{\bar{x}}_u|\vartheta)$. This augmentation can also avoid the possible overfitting of our prompt generator on some abnormal profiles.
\subsubsection{Behavior-based Augmentation}
Besides the prompt generator, we also conduct data augmentations on the original user historical behaviors for SSL on prompt-enhanced sequence modeling. Following previous CL-based models \cite{xie2020contrastive,zhou2020s3}, we randomly zero-mask proportional items in user behavior sequence with the mask ratio $\gamma_2$. Intuitively, the augmented prompt-enhanced behavior sequence is noted as $\bar{s}_u^2=\{{p}_1^u,{p}_2^u, ...,{p}_n^u,v_1^u,[mask], ...,v_{|s_u|}^u\}$, which strengthens the sequence modeling ability from another aspect.
\subsubsection{Contrastive Learning}
In prompt-oriented contrastive learning, we hope PPR can distinguish whether two user representations learned from (augmented) prompt-enhanced behavior sequences derive from the same user. To achieve this goal, we need to minimize the differences between the original and augmented sequences of the same users while maximize the gaps between different users' representations.
Specifically, for a batch $B$ with size $N$, we apply the above two augmentations to each user $u$, and get the augmented sequences $\bar{s}_u^1$ and $\bar{s}_u^2$. We regard the original and the corresponding augmented user behavioral representations of $u$ as the positive pair $(\bm{u}_s,\bm{\bar{u}}_s)$. The rest augmented user representations $\bm{\bar{u}}'_s$ of other users $u'$ in the batch form the negative set $S^u_-$ of $u$. We use the cosine similarity $\mathrm{sim}(\cdot,\cdot)$ to measure the similarity. Formally, the loss function of contrastive learning $L_{CL}$ is formulated as:
\begin{equation}
\begin{aligned}
L_{CL}=-\sum_{u \in U^c}\log\frac{\exp(\mathrm{sim}(\bm{u}_s,\bm{\bar{u}}_s)/\tau)}
{\exp(\mathrm{sim}(\bm{u}_s,\bm{\bar{u}}_s)+\sum_{u'\in {S^u_-}}\exp(\mathrm{sim}(\bm{u}_s,\bm{\bar{u}}'_s)}.
\end{aligned}
\end{equation}
Here, the augmented user representation $\bm{\bar{u}}_s$ equals $f_{seq}(\bar{s}_u^1|\Theta,\vartheta)$ or $f_{seq}(\bar{s}_u^2|\Theta,\vartheta)$. $\tau$ is the temperature hyper-parameter.
The prompt-oriented contrastive learning loss is used as an auxiliary task of $L_p$ in prompt-tuning to fully train the personalized prompts. The overall loss $L_{all}$ is defined with the loss weight $\lambda$ as follows:
\begin{equation}
L_{all} = L_p+\lambda L_{CL}.
\end{equation}
\section{Experiments}
In this section, we conduct extensive experiments to answer the following six research questions:
(\textbf{RQ1}): How does PPR perform in few-shot scenarios (Sec. \ref{sec.few-shot})?
(\textbf{RQ2}): Can PPR work well in zero-shot recommendation (Sec. \ref{sec.zero-shot})?
(\textbf{RQ3}): What are the effects of different components in PPR (Sec. \ref{sec.ablation})?
(\textbf{RQ4}): Can PPR also work on different pre-training models (Sec. \ref{sec.universality})?
(\textbf{RQ5}): Can PPR achieve improvements under different sparsity (Sec. \ref{sec.sparsity})?
(\textbf{RQ6}): Is PPR still effective on other downstream recommendation tasks (Sec. \ref{sec.explorations})?
\subsection{Datasets}
\label{sec.dataset}
We evaluate PPR on three real-world open datasets, namely CIKM, QQBrowser, and AliEC\&AliAD. In all datasets, the users are split into warm users and cold-start users according to a threshold of interacted items (users having less than $10$ clicks are regarded as cold-start users). The click instances of warm users are used as the pre-train set, while those of cold-start users are used as the tuning set for downstream tasks. For cold-start recommendation, we randomly split the cold-start users into train ($80\%$) and test ($20\%$) sets in tuning. More details are in Table \ref{tab:dataset}.
\textbf{CIKM}. The CIKM dataset is an E-commerce recommendation dataset released by Alibaba \footnote{https://tianchi.aliyun.com/competition/entrance/231719/introduction}. It has $60$ thousand warm users with $2.1$ million click instances in the pre-train set. Other $21$ thousand cold users with $143$ thousand instances are used for tuning and testing. Each user has $3$ attributes: gender, age, consumption level.
\textbf{QQBrowser}. It is collected from QQ Browser \cite{yuan2020parameter} on news/videos. This dataset has $107$ thousand warm users and $28$ thousand cold users. Each user has $3$ attributes: gender, age, life status.
\textbf{AliEC\&AliAD}. This dataset contains two sub datasets: AliEC for E-commerce and AliAD for advertising \footnote{https://tianchi.aliyun.com/dataset/dataDetail?dataId=56}. AliAD is much sparser than AliEC. For cold-start recommendation and user profile prediction tasks, we evaluate on AliEC. It has nearly $99$ thousand warm users and $6.4$ thousand cold users. AliAD is used for cross-domain recommendation. This dataset has $8$ user attributes for each user.
\begin{table}[!htbp]
\caption{Detailed statistics of three datasets.}
\label{tab:dataset}
\center
\begin{tabular}{l|cccc}
\toprule
Dataset &\# user&\# item&\tabincell{c}{\# pre-training\\ instance}& \tabincell{c}{\# tuning \\ instance}\\
\midrule
\multirow{1}{*}{CIKM}
~ &80,964&87,894&2,103,610&143,726\\
\multirow{1}{*}{QQBrowser}
~ &134,931&97,904&15,359,880&263,572\\
\multirow{1}{*}{AliEC}
~ &104,984&109,938&8,768,915&39,292\\
\bottomrule
\end{tabular}
\end{table}
\begin{table*}[!htbp]
\caption{Results on few-shot recommendation. All improvements are significant over baselines (t-test with p<0.05).}
\label{tab:few-shot}
\center
\begin{tabular}{l|l|ccccccccc}
\toprule
\small
Dataset & Model &AUC& HIT@5& NDCG@5& HIT@10& NDCG@10&HIT@20& NDCG@20&HIT@50& NDCG@50 \\
\midrule
\multirow{6}{*}{CIKM}
~ & BERT4Rec & 0.8482&0.5191&0.4279&0.6235&0.4616&0.7367&0.4902&0.8965&0.5220 \\
~ & CL4Rec & 0.8590&0.5865&0.4942&0.6699&0.5252&0.7612&0.5481&0.8960&0.5750\\
~ &pre-train&0.8630&0.5779 & 0.4906 & 0.6695 & 0.5203 & 0.7660 & 0.5446&0.9039&0.5721\\
~ &fine-tuning&0.8731&0.5886&0.4948&0.6836&0.5255&0.7837&0.5508&0.9159&0.5772\\
\cmidrule{2-11}
~ &PPR(light)&\textbf{0.8780}&0.5895&0.4955&0.6854&0.5265&0.7894&0.5528&\textbf{0.9230}&0.5795\\
~ &PPR(full)&0.8774&\textbf{0.5918}&\textbf{0.4976}&\textbf{0.6889}&\textbf{0.5290}&\textbf{0.7903}&\textbf{0.5547}&0.9211&\textbf{0.5807}\\
\midrule
\midrule
\multirow{6}{*}{QQBrowser}
~ & BERT4Rec &0.9546&0.7782&0.6295&0.8734&0.6606&0.9386&0.6772&0.9858&0.6867 \\
~ & CL4Rec & 0.9554&0.7817&0.6304&0.8779&0.6618&0.9426&0.6783&0.9863&0.6872\\
~ &pre-train&0.9572&0.7842&0.6338&0.8818&0.6657&0.9448&0.6817&0.9873&0.6904\\
~ &fine-tuning&0.9645&0.8142&0.6659&0.9017&0.6944&0.9578&0.7087&0.9919&0.7156\\
\cmidrule{2-11}
~&PPR(light)&0.9640&0.8068&0.6545&0.8982&0.6843&0.9575&0.6994&\textbf{0.9926}&0.7066\\
~&PPR(full)&\textbf{0.9652}&\textbf{0.8169}&\textbf{0.6680}&\textbf{0.9031}&\textbf{0.6960}& \textbf{0.9589}& \textbf{0.7104}&0.9920& \textbf{0.7171}\\
\midrule
\midrule
\multirow{6}{*}{AliEC}
~ & BERT4Rec & 0.8758&0.5543&0.4370&0.6738&0.4757&0.7898&0.505&0.9266&0.5324\\
~ & CL4Rec & 0.8787&0.5878&0.4696&0.6977&0.5052&0.8021&0.5316&0.9248&0.5562\\
~& pre-train& 0.8838&0.5880&0.4710&0.7006&0.5073&0.8110&0.5351&0.9308&0.5592\\
~&fine-tuing&0.8907&0.6058&0.4851&0.7189&0.5217&0.8212&0.5475&0.9371&0.5708\\
\cmidrule{2-11}
~&PPR(light)&\textbf{0.8975}&0.6123&0.4873&\textbf{0.7275}&0.5246&\textbf{0.8363}&0.5521&\textbf{0.9422}&0.5734\\
~&PPR(full)&0.8941&\textbf{0.6126}&\textbf{0.4896}&0.7241&\textbf{0.5256}&0.8284&\textbf{0.5522}&0.9374&\textbf{0.5739}\\
\bottomrule
\end{tabular}
\end{table*}
\subsection{Competitors}
\label{sec.baseline}
In this work, we adopt the representative SASRec \cite{kang2018self} as our base pre-training model, while it is also convenient to deploy our PPR on other pre-training models (e.g., CL4Rec \cite{xie2020contrastive}, see Sec. \ref{sec.universality}). In few-shot recommendation. we compare our PPR with several competitive sequential models as follows:
(1) \textbf{BERT4Rec} \cite{sun2019bert4rec}, which is a classical pre-training recommendation model based on BERT. It uses masked item prediction as its pre-training task.
(2) \textbf{CL4Rec} \cite{xie2020contrastive}, which is a classical sequential recommendation model enhanced by contrastive learning with several sequence-based augmentations.
(3) \textbf{Pre-train} (i.e., SASRec \cite{kang2018self}). We directly conduct the pre-trained model on cold users as a baseline.
(4) \textbf{Fine-tuning}. It tunes all pre-trained model's parameters in tuning. Note that fine-tuning model also jointly considers user profiles and historical behaviors as in Fig. \ref{fig:overall}.
We also implement two PPR versions, namely PPR(light) and PPR(full) as introduced in Sec. \ref{sec.tuning_PPR}. PPR(light) only updates the parameters $\vartheta$ of the prompt generator and $\phi$ of the user profile learner with other parameters fixed, which is more efficient compared to fine-tuning. PPR(full) updates all parameters as fine-tuning.
We should highlight that all models share the same input features (including user profiles and behaviors) and training instances in tuning for fair comparisons (except for pre-training that only uses the pre-train set). Fine-tuning and PPR have the same pre-training model. We also deploy PPR on CL4Rec to verify the universality.
\subsection{Experimental Settings}
\label{sec.experimental_settings}
\noindent
We mainly focus on the few-shot and zero-shot recommendation tasks of cold-start users.
For all cold-start users in test set, their first clicked items are used for zero-shot recommendation (since there is no historical behavior at this time), while the rest click behaviors are used for few-shot recommendation.
We use the classical AUC, top-N hit rate (HIT@N), and Normalized Discounted Cumulative Gain (NDCG@N) as our evaluation metrics. For HIT@N and NDCG@N, we report top $5$, $10$, $20$ and $50$. For each ground truth, we randomly sample $99$ items that the user did not click as negative samples as \cite{zhou2020s3,sun2019bert4rec}.
\subsection{Few-shot Recommendation (RQ1)}
\label{sec.few-shot}
We first evaluate models on the few-shot recommendation. Table \ref{tab:few-shot} shows the results on three datasets. We can find that:
(1) Our personalized prompt-based model outperforms all baselines on all metrics in three datasets (the significance level is $p$<$0.05$ obtained via t-test). It indicates that our personalized prompts can better extract useful information related to the current user from the huge knowledgeable pre-training models, which is beneficial especially for cold-start scenarios. Through the PPR, the powerful sequence modeling ability of the pre-trained model could be smoothly transferred into the few-shot recommendation task.
(2) Comparing among different PPR settings, we observe that PPR(full) generally achieves better performances in few-shot scenarios. Different from tasks in NLP (e.g., sentiment analysis with several labels), the label set of cold-start recommendation is the whole item set (often million-level). It is natural that the tuned item representations of PPR(full) can further improve the performances.
(3) PPR(light) still achieves some SOTA results on HIT@N in AliEC dataset, and generally outperforms fine-tuning in two datasets. It is challenging since PPR(light) only tunes the personalized prompt part with all pre-trained parameters unchanged. Compared to fine-tuning and PPR(full), PPR(light) is more efficient in tuning stage. We can select between the full and light PPR versions according to the priority of effectiveness and efficiency.
PPR models also outperform other classical baselines such as BERT4Rec and CL4Rec.
\begin{table*}[!htbp]
\caption{Results on zero-shot recommendation. The improvements are significant over baselines (t-test with p<0.05).}
\label{tab:zero-shot}
\center
\begin{tabular}{l|l|ccccccccc}
\toprule
\small
Dateset & Model &AUC& HIT@5& NDCG@5& HIT@10&NDCG@10&HIT@20&NDCG@20&HIT@50& NDCG@50 \\
\midrule
\multirow{3}{*}{CIKM}
~&fine-tuning&0.7753&0.2991&0.2061&0.4235&0.2463&0.5946&0.2894&0.8460&0.3394\\
~&PPR(light)&\textbf{0.7845}&0.2998&0.2063&\textbf{0.4337}&0.2493&\textbf{0.6041}&\textbf{0.2923}&\textbf{0.8621}&0.3436\\
~&PPR(full)&0.7825&\textbf{0.3069}&\textbf{0.2100}&0.4329&\textbf{0.2507}&0.5974&0.2920&0.8616&\textbf{0.3446}\\
\midrule
\multirow{3}{*}{QQBrowser}
~ &fine-tuning&0.8743&0.4950&0.3538&0.6427&0.4016&\textbf{0.7822}&0.4368&0.9393&0.4684\\
~&PPR(light)&0.8635&0.4829&0.3418&0.6337&0.3907&0.7770&0.4269&0.9262&0.4569\\
~&PPR(full)&\textbf{0.8757}&\textbf{0.5012}&\textbf{0.3612}&\textbf{0.6443}&\textbf{0.4074}&0.7813&\textbf{0.4421}&\textbf{0.9411}&\textbf{0.4742}\\
\midrule
\multirow{3}{*}{AliEC}
~&fine-tuning&0.8377&0.3905&0.2713&0.5675&\textbf{0.3286}&0.7252&0.3686&0.9006&\textbf{0.4039}\\
~&PPR(light)&0.8349&\textbf{0.4014}&\textbf{0.2743}&0.5575&0.3247&\textbf{0.7337}&\textbf{0.3693}&0.9022&0.4028\\
~&PPR(full)&\textbf{0.8380}&0.3983&0.2708&\textbf{0.5691}&0.3256&0.7283&0.3661&\textbf{0.9037}&0.4013\\
\bottomrule
\end{tabular}
\end{table*}
\begin{table*}[!hbtp]
\caption{Results of joint cold-start recommendation (few-shot+zero-shot). Improvements are significant with p<0.05.}
\label{tab:joint_few_zero}
\center
\begin{tabular}{l|l|ccccccccc}
\toprule
\small
Database & Model &AUC& HIT@5& NDCG@5& HIT@10&NDCG@10&HIT@20&NDCG@20&HIT@50& NDCG@50 \\
\midrule
\multirow{3}{*}{CIKM}
~ &fine-tuning&0.8418&0.5202&0.4329&0.6163&0.4639&0.7257&0.4915&0.8891&0.5240\\
~&PPR(light)&\textbf{0.8606}&\textbf{0.5399}&0.4445&\textbf{0.6427}&0.4777&\textbf{0.7562}&0.5064&\textbf{0.9110}&0.5372\\
~&PPR(full)&0.8585&0.5396&\textbf{0.4476}&0.6409&\textbf{0.4802}&0.7522&\textbf{0.5084}&0.9068&\textbf{0.5392}\\
\midrule
\multirow{3}{*}{QQBrowser}
~ &fine-tuning&0.9593&0.7940&0.6459&0.8838&0.6751&0.9469&0.6912&0.9899&0.6999\\
~&PPR(light)&0.9589&0.7866&0.6350&0.8812&0.6658&0.9471&0.6826&\textbf{0.9904}&0.6914\\
~&PPR(full)&\textbf{0.9602}&\textbf{0.7965}&\textbf{0.6490}&\textbf{0.8863}&\textbf{0.6782}&\textbf{0.9486}&\textbf{0.6941}&0.9902&\textbf{0.7026}\\
\midrule
\multirow{3}{*}{AliEC}
~ &fine-tuning&0.8818&0.5708&0.4480&0.6912&0.4869&0.8057&0.5158&0.9317&0.5411\\
~&PPR(light)&\textbf{0.8904}&\textbf{0.5729}&0.4465&\textbf{0.7032}&0.4886&\textbf{0.8173}&0.5177&\textbf{0.9426}&0.5428\\
~&PPR(full)&0.8817&0.5726&\textbf{0.4519}&0.6934&\textbf{0.4908}&0.8050&\textbf{0.5190}&0.9308&\textbf{0.5443}\\
\bottomrule
\end{tabular}
\end{table*}
\subsection{Zero-shot Recommendation (RQ2)}
\label{sec.zero-shot}
The data sparsity issue is extremely serious in practical systems, where zero-shot users widely exist. Conventional sequential recommendation models cannot handle the zero-shot scenarios, since there is no historical behavior. We attempt to jointly address the zero-shot recommendation with the same PPR framework.
For PPR, we directly input user profiles into the prompt generator as the few-shot recommendation without behavioral inputs. For fine-tuning, only the user profile learner (i.e., the MLP in Eq. (\ref{eq.prompt_user})) is activated to learn from user profiles.
From Table \ref{tab:zero-shot} we can observe that:
(1) Our PPR models still achieve the best performances on most metrics of three datasets, which confirms the effectiveness of PPR in zero-shot user understanding via fully reusing pre-training knowledge. We should highlight that getting improvements in zero-shot scenarios is challenging, since all models share the same sparse user profiles and training instances in tuning. Compared with fine-tuning, PPR can better transfer the sequence modeling ability via prompt, which is the main reason of our improvements.
(2) Currently, the user profiles for prompt construction are not that sufficient in the open datasets, which may limit the modeling ability of prompt-tuning. It can be expected that the improvements will be more significant if enhanced with more user features (e.g., full user profiles or user embeddings learned from other domains).
We have also tested the effectiveness of PPR in cross-domain recommendation and user profile prediction tasks in Sec. \ref{sec.explorations}.
\paragraph{\textbf{Joint Few-shot and Zero-shot Recommendation}}
Table \ref{tab:few-shot} and \ref{tab:zero-shot} demonstrate the effectiveness of our PPR in both few-shot and zero-shot recommendation. Considering the storage efficiency and maintenance cost in online deployment, we further conduct a challenging evaluation, using one set of model parameters to jointly address two cold-start recommendation tasks.
Table \ref{tab:joint_few_zero} shows the results of joint cold-start recommendation. We can find that: PPR models significantly outperform fine-tuning by a larger margin in the joint scenarios. It indicates that the powerful personalized prompts could help to understand few-shot and zero-shot users jointly in practice, which cannot be accomplished well by most sequential models particularly relying on user behaviors.
\begin{figure}[!hbtp]
\centering
\includegraphics[width=0.99\columnwidth]{figs/ablation.pdf}
\caption{Ablation study on (a) CIKM, and (b) AliEC.}
\label{fig:ablation_study}
\end{figure}
\begin{table*}[!hbtp]
\caption{Results of PPR on few-shot recommendation based on CL4Rec. Improvements are significant (t-test with p<0.05).}
\label{tab:few-shot on CL4rec}
\center
\begin{tabular}{l|l|ccccccccc}
\toprule
\small
Dataset & Model &AUC& HIT@5& NDCG@5& HIT@10& NDCG@10&HIT@20& NDCG@20&HIT@50& NDCG@50 \\
\midrule
\multirow{4}{*}{CIKM}
~ &pre-train& 0.8590&0.5865&0.4942&0.6699&0.5252&0.7612&0.5481&0.8960&0.5750\\
~ &fine-tuning&0.8660&0.5948&0.5035&0.6797&0.5309&0.7719&0.5542&0.9034&0.5804\\
\cmidrule{2-11}
~ &PPR(light)&0.8723&0.6013&0.5053&0.6891&0.5338&0.7851&0.5580&0.9106&0.5830\\
~ &PPR(full)&\textbf{0.8769}&\textbf{0.6066}&\textbf{0.5093}&\textbf{0.6982}&\textbf{0.5390}&\textbf{0.7928}&\textbf{0.5628}&\textbf{0.9157}&\textbf{0.5873}\\
\midrule
\midrule
\multirow{4}{*}{QQBrowser}
~ &pre-train& 0.9554&0.7817&0.6304&0.8779&0.6618&0.9426&0.6783&0.9863&0.6872\\
~ &fine-tuning&0.9623&0.8083&0.6593&0.8971&0.6883&0.9548&0.7030&0.9896&0.7101\\
\cmidrule{2-11}
~&PPR(light)&0.9641&0.8077&0.6566&0.8989&0.6863&\textbf{0.9583}&0.7015&\textbf{0.9919}&0.7084\\
~&PPR(full)&\textbf{0.9642}&\textbf{0.8138}&\textbf{0.6652}&\textbf{0.9008}&\textbf{0.6936}&0.9580&\textbf{0.7082}&0.9911&\textbf{0.7150}\\
\midrule
\midrule
\multirow{4}{*}{AliEC}
~& pre-train& 0.8787&0.5878&0.4696&0.6977&0.5052&0.8021&0.5316&0.9248&0.5562\\
~&fine-tuning&0.8912&0.6057&0.4857&0.7226&0.5234&0.8242&0.5491&0.9365&0.5716\\
\cmidrule{2-11}
~&PPR(light)&\textbf{0.8992}&0.6114&0.4857&\textbf{0.7328}&0.5250&\textbf{0.8386}&0.5518&\textbf{0.9441}&0.5729\\
~&PPR(full)&0.8934&\textbf{0.6140}&\textbf{0.4918}&0.7277&\textbf{0.5284}&0.8290&\textbf{0.5542}&0.9362&\textbf{0.5757}\\
\bottomrule
\end{tabular}
\end{table*}
\begin{figure*}[!hbtp]
\centering
\includegraphics[width=0.85\textwidth]{figs/Sparisty.pdf}
\caption{Improvements of PPR models on fine-tuning with different degrees of data sparsity in CIKM dataset.}
\label{fig:Sparisty}
\end{figure*}
\subsection{Ablation Study (RQ3)}
\label{sec.ablation}
In this section, we aim to confirm that the prompt-oriented contrastive learning is essential for PPR. Fig. \ref{fig:ablation_study} shows the results of different ablation versions of PPR(light) and PPR(full) on the few-shot recommendation task with multiple datasets. We can find that:
(1) Generally, the prompt-oriented CL brings consistent improvements on almost all metrics on both datasets, except AUC on PPR(light). It reconfirms the effectiveness of prompt-based SSL in cold-start recommendation.
(2) PPR still outperforms fine-tuning on most metrics even without the prompt-oriented CL. It verifies that our personalized prompt-tuning is truly superior to fine-tuning in transferring pre-trained knowledge to downstream tasks. Besides, we deploy PPR on a CL-based model CL4Rec as the pre-training model and still achieve consistent improvements (Sec. \ref{sec.universality}).
\subsection{Universality of PPR (RQ4)}
\label{sec.universality}
PPR is an effective and universal tuning framework, which can be easily deployed on different pre-training models. In this section, we further adopt PPR with CL4Rec \cite{xie2020contrastive} used as the pre-training model. The results of PPR on few-shot recommendation based on CL4Rec are given in Table \ref{tab:few-shot on CL4rec}, from which we can find that:
(1) PPR models still achieve the state-of-the-art performances on all metrics in three datasets, which proves the universality of PPR. Our personalized prompts could consistently improve cold-start recommendation with different pre-training models.
(2) Generally, PPR(full) consistently outperforms PPR(light) on most metrics. It indicates that fine-tuning item embeddings (i.e., the labels to be predicted) is essential in cold-start recommendation, which differs from the prompt-tuning in NLP. Nevertheless, PPR(light) still performs better than fine-tuning on most metrics. We can flexibly choose different implementations of PPR considering the practical demand on efficiency.
\subsection{Model Analyses on Sparsity (RQ5)}
\label{sec.sparsity}
We further explore the influence of data sparsity on PPR to show its robustness. Specifically, we crop all user behavior sequences of both train and test sets in tuning to represent different degrees of sparsity. We define the k-shot setting where all user behavior sequences are cropped and no longer than $k$, with $k=1,5,7$. Fig. \ref{fig:Sparisty} shows the relative improvements of PPR(light) and PPR(full) on fine-tuning on AUC, HIT, and NDCG in CIKM. We find that:
(1) Both PPR(light) and PPR(full) consistently outperform fine-tuning with different sparsity, which verifies the robustness of PPR on different cold-start scenarios.
(2) The improvements of PPR models increase with the maximum user behavior sequence length decreasing. It indicates that our PPR can perform better on more sparse scenarios compared to fine-tuning. It is intuitive since the personalized prompts are more dominating when the behavioral information is sparser.
\subsection{Explorations on Other Tasks (RQ6)}
\label{sec.explorations}
Besides the main few-shot and zero-shot recommendation tasks, we further explore other promising usages of personalized prompts on more diversified and challenging downstream tasks, including cross-domain recommendation and user profile prediction.
\subsubsection{Cross-domain Recommendation}
Cross-domain recommendation (CDR) aims to transfer useful knowledge from the source domain to help the target domain \cite{hu2018conet}. We focus on the CDR scenario with overlapping users. In this work, the source domain (pre-train set) is AliEC and the target domain (tuning set) is AliAD. We directly use the user embeddings trained on the source domain as our personalized prompts. Similarly, these source-domain user embeddings are also applied as side information for fine-tuning and PPR. We also implement SASRec(target), which is only trained on the target domain without source information. Since there is no overlapping items, we do not report the result of PPR(light). The results are shown in Table \ref{tab:cross-domain}. We can find that PPR significantly outperforms all baselines on all metrics. The only difference between PPR and fine-tuning is the personalized prompt, which brings in impressive improvements. It implies the promising application of adopting PPR on pre-training based CDR models.
\subsubsection{User Profile Prediction}
User profile prediction task aims to predict users' profiles via their behavior sequences.
We conduct an exploration of PPR on this task by predicting users' graduate state in AliEC, which is a binary classification task. Precisely in PPR, the personalized prompts are generated by users' all profiles except the one to be predicted. For classification, we add a profile classifier on the final user representation, and so as the fine-tuning.
The results are shown in Table \ref{tab:user_profile}. We can find that PPR significantly outperforms fine-tuning on ACC, precision, and F1. It is be because that PPR can well prompt the pre-trained sequential model to better extract user-related knowledge for downstream tasks. We also evaluate PPR on other user profile predictions such as gender and age, where PPR(light) performs comparable with fine-tuning involving less parameter tuning. In the future, we will design customized personalized prompts specially for different downstream tasks to further improve the performance of user profile prediction.
\begin{table}[!htbp]
\label{sec.user_profile}
\caption{Results of user profile prediction on AliEC.}
\label{tab:user_profile}
\center
\begin{tabular}{l|l|cccp{0.9cm}<{\centering}}
\toprule
\small
Dataset & Model &ACC&Precision&Recall&F1 \\
\midrule
\multirow{2}{*}{AliEC}
~ &fine-tuning& 0.89&0.33&\textbf{0.82}&0.47 \\
~&PPR(light)&\textbf{0.92}&\textbf{0.42}&0.74&\textbf{0.54}\\
\bottomrule
\end{tabular}
\end{table}
\begin{table*}[!htbp]
\label{sec.cross-domain}
\caption{Results of cross-domain recommendation on AliEC->AliAD. All improvements are significant (t-test with p<0.01).}
\label{tab:cross-domain}
\center
\begin{tabular}{l|l|ccccccccc}
\toprule
\small
Dataset & Model &AUC& HIT@5& NDCG@5& HIT@10&NDCG@10&HIT@20&NDCG@20&HIT@50& NDCG@50 \\
\midrule
\multirow{2}{*}{\tabincell{c}{AliEC \\-> \\AliAD}}
~&SASRec(target)&0.6423&0.2752&0.2106&0.3590&0.2375&0.4594&0.2628&0.6568&0.3017\\
~&fine-tuning&0.7165&0.3362&0.2494&0.4357&0.2813&0.5552&0.3115&0.7323&0.3465\\
~&PPR(full)&\textbf{0.7343}&\textbf{0.3632}&\textbf{0.2799}&\textbf{0.4595}&\textbf{0.3109}&\textbf{0.5761}&\textbf{0.3402}&\textbf{0.7631}&\textbf{0.3772}\\
\bottomrule
\end{tabular}
\end{table*}
\section{Conclusion and Future Work}
In this work, we propose a personalized prompt-based recommendation to improve the tuning in pre-training recommendation. We conduct extensive experiments to verify the effectiveness, robustness, and universality of our PPR on cold-start recommendation tasks, and also explore the potential extensions of PPR on various downstream tasks.
In the future, we attempt to verify our PPR on more recommendation tasks with customized prompt-tuning manners or pre-training tasks. We will also explore the effectiveness of PPR on industrial extremely large-scale recommendation datasets.
\bibliographystyle{ACM-Reference-Format}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,337 |
Sorry. But how many chocolate oranges do you melt and add to the filling? I am assuming. I get just 1 and the other used for decoration.
Hoping to make it this afternoon with my son.
Hi Joey – I add both chocolate oranges to the cheesecake mixture and decorate with another. Hope you and your son enjoying making the cheesecake!
I only used 1 an a half will it still set?
Hi Natasha – you can add as much or as little chocolate as orange as you want. Please note adding less will change the flavour of the cheesecake but it will set even if you used just one and a half chocolate oranges.
Hey, what size tin did you use for this?
Hi Sophie – I use a 23cm springform tin for all cheesecake recipes I share. I hope you enjoy this cheesecake!
Hi Geni – thanks for stopping by my blog! 🙂 Any kind of cream that can be whipped will work in this recipe. I hope that answers you question, let me know if you have any other queries. Have a great Christmas and enjoy the cheesecake!
Hi Geni – just follow the recipe method and it work out just fine. Let me know how you get on!
Hi Barbara – I haven't ever attempted to freeze cheesecake before so I wouldn't be able to confidently say whether it would be ok to do. I reckon if you leave the cheesecake in the tin and cover it completely with clingfilm it will be fine to freeze for up to a month. When you're thinking of serving it, allow the cheesecake to thaw overnight in the fridge before slicing and it will need to be eaten within 3 days. Hope that helps and that you enjoy this recipe!
I made your cheesecake & my family loved it. Though I seemed to have a slight problem , I followed your recipe exactly to the letter but didn't seem to get the colour you have. Mine was slightly whiter than chocolatey coloured .
What did I do wrong ? Was the choc maybe too cool or mix ? as mix seemed to have all bits of choc.
Hi Laura, thanks for stopping by my blog! I'm super pleased you and your family enjoyed this cheesecake, that's fantastic to hear. I'm not sure what could've been the reason the cheesecake was not as chocolatey in colour as mine was, I can only think that recently they changed the size of the chocolate oranges here so it be the fact because they are smaller and cheesecake had less chocolate added to it. Also, when I say to leave the chocolate to cool slightly it should still be warmish before adding to the other cheesecake ingredients, if it's cooled then lumps will form when it's combined with the remaining ingredients. I will update the blog post and make that clear on my recipe and method.
Disappointed as chocolate didn't melt properly either over pan or in microwave – was it because I used a Milk chocolate instead of a dark chocolate, that the cheesecake is no full of lumps.
Hi Lois – I can't understand why the chocolate couldn't have melted properly. Milk chocolate should've melted just as well as dark chocolate does, I used milk chocolate in this cheesecake and I didn't seem to have a problem melting it over a bain-marie or in the microwave.
As I explained in the recipe and method to avoid lumps you need to make sure your cream cheese is softened (so it should really be left out for at least 30 mins or up to an hour at room temperature), also the melted chocolate should still be slightly warm when you add it to the cheesecake filling otherwise lumps will form and it won't completely incorporate. I hope that answers your query.
Great recipe – thanks for sharing!
Hi Catherine – I've personally never found this cheesecake to be sour on the occasions I have made it. Perhaps if you found it too sour for your liking you could add a touch extra icing sugar, but try not to add too much otherwise it affect the consistency of the cheesecake.
Thanks for the fantastic review, Gemma!
Hi, I'm really pleased that you like the look this cheesecake! You can substitute the orange chocolate digestives for regular chocolate digestives or even plain digestives without the chocolate topping. Hope you enjoy the recipe!
Hi Cheryl, thank you your comment. I'm really pleased that you're enjoying this cheesecake so much, it's definitely one of my favourite recipes!
How much cream goes into this filling? I can't work out if some of the 300ml is used to decorate or it all goes into the filling.
Lovely recipe that I'd like to try. Which cream cheese do you use?
Thank you! You can use regular full-fat cream cheese or mascarpone (this has a slighter higher fat content so will be richer).
Hi Lisa – the texture the cheesecake will be different because there's no gelatine and it's not a baked cheesecake. I have tried making this cheesecake with mascarpone, however this has a higher fat content compared to full-fat cream cheese, so bear in mind that the cheesecake would be slightly more decadent and the flavour will be near enough the same. | {
"redpajama_set_name": "RedPajamaC4"
} | 1,253 |
La batalla de Pankalia fue una batalla que enfrentó en 978 o 979 al ejército leal al emperador bizantino Basilio II, comandado por Bardas Focas el joven, con las fuerzas del general rebelde Bardas Esclero.
Las fuentes son contradictorias con respecto a la batalla: mientras que los historiadores clásicos seguían el relato de Juan Escilitzes según el cual la batalla fue una victoria decisiva de los lealistas en marzo de 979, los historiadores modernos dan más peso a la crónica de León el diácono que lo describe como una victoria rebelde en junio de 978.
Fuentes y reconstrucción de acontecimientos
Las fuentes sobre la revuelta de Bardas Esclero difieren en el orden y ubicación de las batallas que acabaron con su rebelión. Una de ellas tuvo lugar en la llanura de Pankalia (Παγκάλεια), nordeste de Amorium. La historia de Juan Escilitzes, escrita a finales del , cuenta que Esclero ganó una primera batalla cerca de Amorium, así como una segunda en Basilika Therma (actual Sarıkaya) y que fue en un tercer encuentro en Pankalia que Focas triunfó. Según Escilitzes, Focas había recibido jinetes georgianos de Tao como refuerzos gracias a su amistad con su gobernante, David, pero su ejército otra vez empezó a ceder. Focas cargó directamente contra Esclero y en combate personal hirió e hizo huir al general rebelde.
El autor de finales del León el Diácono describe en cambio un primer encuentro con derrota para el ejército de Focas en Pankalia y una victoria deceisva en una segunda batalla cuya ubicación no recoge.
Otros autores proporcionan más detalles: Miguel Psellos también recoge el duelo de los dos generales en la batalla decisiva, mientras el historiador árabe cristiano de comienzos del Yahya de Antioquía alude a dos batallas y da las fechas de 19 de junio de 978 y 24 de marzo de 979 respectivamente. Lo que es unánimemente narrado en todas las fuentes es que después de su derrota, Esclero huyó a refugiarse con su aliado árabe, el emir hamdánida Abu Taghlib y luego en la corte búyida en Bagdad, donde permaneció siete años.
Basándose en fuentes georgianas, P. M. Tarchnichvili sugirió en 1964 que la victoria de Focas tuvo lugar en un sitio llamado «Sarvenis» en georgiano, que identificó como Aquae Saravenae, al norte de Caesarea (actual Kayseri), y que la tercera batalla según Escilitzes (que erróneamente coloca Pankalia cerca del río Halys, el cual corresponde al sitio de Kırşehir) es un mezcla ficticia de las dos batallas. Esta visión es compartida por John Forsyth en su edición crítica de la crónica de Yahya en inglés de 1977. Según Catherine Holmes, el relato de Escilitzes, a pesar de ser «sin duda embellecido», probablemente tenía alguna fuente real dado su nivel de detalle siendo el «asegurar alguna de las opciones imposible» salvo que la batalla decisiva tuvo lugar en marzo de 979.
Académicos anteriores como George Finlay y Georg Ostrogorsky, generalmente habían tomado como veraz la crónica de Juan Escilitzes, con la batalla final decisiva entre Esclero y Focas en Pankalia en marzo de 979 a veces etiquetada de «segunda» batalla de Pankalia, dependiendo de si la primera batalla cerca de Amorium era también considerado en el mismo lugar. Los académicos modernos, por otro lado, han adoptado una reconstrucción diferente con una primera batalla que tiene lugar en Pankaleia en junio de 978, una segunda batalla en Basilika Therma en Carsiano en invierno/otoño y una tercera y decisiva batalla que tuvo lugar en Sarvenis en marzo de 979.
Referencias
Bibliografía
Pankalia
Pankalia
Georgia en el siglo X
Pankalia | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 4,147 |
"EmpressAK is an amalgamation of righteousness, sagacity, recklessness, wit and a dash of spice; the perfect dichotomy. In fact, everyone has a bit of EmpressAK (balance) in them, some not cognizant of it, others afraid to truly express it. The aim of this site is to provide a platform where anyone can weigh in on a plethora of topics from cultural, social, personal. These topics are packaged in different forms; thought provoking to humorous. Let's build, laugh, learn!
This cute sista is Latrice. She is a blogger on WordPresss and goes by the name Latti Nerd Gangsta. She has a blog,Facebook page and YouTube channel. Be sure to check out her websites and show the sista some support. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,348 |
<?php
/**
*
* @author Vee W.
* @license http://opensource.org/licenses/MIT
*
*/
namespace Blog;
class Model_BlogComment extends \Orm\Model
{
protected static $_table_name = 'blog_comment';
protected static $_primary_key = array('comment_id');
// relations
protected static $_belongs_to = array(
'blog' => array(
'model_to' => '\Blog\Model_Blog',
'key_from' => 'post_id',
'key_to' => 'post_id',
)
);
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,270 |
"""Unit tests for the NetApp-specific ssc module."""
import BaseHTTPServer
import copy
import httplib
from lxml import etree
from mox3 import mox
import six
from cinder import exception
from cinder import test
from cinder.volume.drivers.netapp.dataontap.client import api
from cinder.volume.drivers.netapp.dataontap import ssc_cmode
class FakeHTTPRequestHandler(BaseHTTPServer.BaseHTTPRequestHandler):
"""HTTP handler that doesn't spam the log."""
def log_message(self, format, *args):
pass
class FakeHttplibSocket(object):
"""A fake socket implementation for httplib.HTTPResponse."""
def __init__(self, value):
self._rbuffer = six.StringIO(value)
self._wbuffer = six.StringIO('')
oldclose = self._wbuffer.close
def newclose():
self.result = self._wbuffer.getvalue()
oldclose()
self._wbuffer.close = newclose
def makefile(self, mode, _other):
"""Returns the socket's internal buffer"""
if mode == 'r' or mode == 'rb':
return self._rbuffer
if mode == 'w' or mode == 'wb':
return self._wbuffer
RESPONSE_PREFIX_DIRECT_CMODE = """<?xml version='1.0' encoding='UTF-8' ?>
<!DOCTYPE netapp SYSTEM 'file:/etc/netapp_gx.dtd'>"""
RESPONSE_PREFIX_DIRECT = """
<netapp version='1.15' xmlns='http://www.netapp.com/filer/admin'>"""
RESPONSE_SUFFIX_DIRECT = """</netapp>"""
class FakeDirectCMODEServerHandler(FakeHTTPRequestHandler):
"""HTTP handler that fakes enough stuff to allow the driver to run."""
def do_GET(s):
"""Respond to a GET request."""
if '/servlets/netapp.servlets.admin.XMLrequest_filer' not in s.path:
s.send_response(404)
s.end_headers
return
s.send_response(200)
s.send_header("Content-Type", "text/xml; charset=utf-8")
s.end_headers()
out = s.wfile
out.write('<netapp version="1.15">'
'<results reason="Not supported method type"'
' status="failed" errno="Not_Allowed"/></netapp>')
def do_POST(s):
"""Respond to a POST request."""
if '/servlets/netapp.servlets.admin.XMLrequest_filer' not in s.path:
s.send_response(404)
s.end_headers
return
request_xml = s.rfile.read(int(s.headers['Content-Length']))
root = etree.fromstring(request_xml)
body = [x for x in root.iterchildren()]
request = body[0]
tag = request.tag
localname = etree.QName(tag).localname or tag
if 'volume-get-iter' == localname:
body = """<results status="passed"><attributes-list>
<volume-attributes>
<volume-id-attributes>
<name>iscsi</name>
<owning-vserver-name>Openstack</owning-vserver-name>
<containing-aggregate-name>aggr0
</containing-aggregate-name>
<junction-path>/iscsi</junction-path>
<type>rw</type>
</volume-id-attributes>
<volume-space-attributes>
<size-available>214748364</size-available>
<size-total>224748364</size-total>
<space-guarantee-enabled>enabled</space-guarantee-enabled>
<space-guarantee>file</space-guarantee>
</volume-space-attributes>
<volume-state-attributes>
<is-cluster-volume>true
</is-cluster-volume>
<is-vserver-root>false</is-vserver-root>
<state>online</state>
<is-inconsistent>false</is-inconsistent>
<is-invalid>false</is-invalid>
<is-junction-active>true</is-junction-active>
</volume-state-attributes>
</volume-attributes>
<volume-attributes>
<volume-id-attributes>
<name>nfsvol</name>
<owning-vserver-name>Openstack
</owning-vserver-name>
<containing-aggregate-name>aggr0
</containing-aggregate-name>
<junction-path>/nfs</junction-path>
<type>rw</type>
</volume-id-attributes>
<volume-space-attributes>
<size-available>14748364</size-available>
<size-total>24748364</size-total>
<space-guarantee-enabled>enabled
</space-guarantee-enabled>
<space-guarantee>volume</space-guarantee>
</volume-space-attributes>
<volume-state-attributes>
<is-cluster-volume>true
</is-cluster-volume>
<is-vserver-root>false</is-vserver-root>
<state>online</state>
<is-inconsistent>false</is-inconsistent>
<is-invalid>false</is-invalid>
<is-junction-active>true</is-junction-active>
</volume-state-attributes>
</volume-attributes>
<volume-attributes>
<volume-id-attributes>
<name>nfsvol2</name>
<owning-vserver-name>Openstack
</owning-vserver-name>
<containing-aggregate-name>aggr0
</containing-aggregate-name>
<junction-path>/nfs2</junction-path>
<type>rw</type>
</volume-id-attributes>
<volume-space-attributes>
<size-available>14748364</size-available>
<size-total>24748364</size-total>
<space-guarantee-enabled>enabled
</space-guarantee-enabled>
<space-guarantee>volume</space-guarantee>
</volume-space-attributes>
<volume-state-attributes>
<is-cluster-volume>true
</is-cluster-volume>
<is-vserver-root>false</is-vserver-root>
<state>online</state>
<is-inconsistent>true</is-inconsistent>
<is-invalid>true</is-invalid>
<is-junction-active>true</is-junction-active>
</volume-state-attributes>
</volume-attributes>
<volume-attributes>
<volume-id-attributes>
<name>nfsvol3</name>
<owning-vserver-name>Openstack
</owning-vserver-name>
<containing-aggregate-name>aggr0
</containing-aggregate-name>
<junction-path>/nfs3</junction-path>
<type>rw</type>
</volume-id-attributes>
<volume-space-attributes>
<space-guarantee-enabled>enabled
</space-guarantee-enabled>
<space-guarantee>volume
</space-guarantee>
</volume-space-attributes>
<volume-state-attributes>
<is-cluster-volume>true
</is-cluster-volume>
<is-vserver-root>false</is-vserver-root>
<state>online</state>
<is-inconsistent>false</is-inconsistent>
<is-invalid>false</is-invalid>
<is-junction-active>true</is-junction-active>
</volume-state-attributes>
</volume-attributes>
</attributes-list>
<num-records>4</num-records></results>"""
elif 'aggr-options-list-info' == localname:
body = """<results status="passed">
<options>
<aggr-option-info>
<name>ha_policy</name>
<value>cfo</value>
</aggr-option-info>
<aggr-option-info>
<name>raidtype</name>
<value>raid_dp</value>
</aggr-option-info>
</options>
</results>"""
elif 'sis-get-iter' == localname:
body = """<results status="passed">
<attributes-list>
<sis-status-info>
<path>/vol/iscsi</path>
<is-compression-enabled>
true
</is-compression-enabled>
<state>enabled</state>
</sis-status-info>
</attributes-list>
</results>"""
elif 'storage-disk-get-iter' == localname:
body = """<results status="passed">
<attributes-list>
<storage-disk-info>
<disk-raid-info>
<effective-disk-type>SATA</effective-disk-type>
</disk-raid-info>
</storage-disk-info>
</attributes-list>
</results>"""
else:
# Unknown API
s.send_response(500)
s.end_headers
return
s.send_response(200)
s.send_header("Content-Type", "text/xml; charset=utf-8")
s.end_headers()
s.wfile.write(RESPONSE_PREFIX_DIRECT_CMODE)
s.wfile.write(RESPONSE_PREFIX_DIRECT)
s.wfile.write(body)
s.wfile.write(RESPONSE_SUFFIX_DIRECT)
class FakeDirectCmodeHTTPConnection(object):
"""A fake httplib.HTTPConnection for netapp tests.
Requests made via this connection actually get translated and routed into
the fake direct handler above, we then turn the response into
the httplib.HTTPResponse that the caller expects.
"""
def __init__(self, host, timeout=None):
self.host = host
def request(self, method, path, data=None, headers=None):
if not headers:
headers = {}
req_str = '%s %s HTTP/1.1\r\n' % (method, path)
for key, value in headers.iteritems():
req_str += "%s: %s\r\n" % (key, value)
if data:
req_str += '\r\n%s' % data
# NOTE(vish): normally the http transport normailizes from unicode
sock = FakeHttplibSocket(req_str.decode("latin-1").encode("utf-8"))
# NOTE(vish): stop the server from trying to look up address from
# the fake socket
FakeDirectCMODEServerHandler.address_string = lambda x: '127.0.0.1'
self.app = FakeDirectCMODEServerHandler(sock, '127.0.0.1:80', None)
self.sock = FakeHttplibSocket(sock.result)
self.http_response = httplib.HTTPResponse(self.sock)
def set_debuglevel(self, level):
pass
def getresponse(self):
self.http_response.begin()
return self.http_response
def getresponsebody(self):
return self.sock.result
def createNetAppVolume(**kwargs):
vol = ssc_cmode.NetAppVolume(kwargs['name'], kwargs['vs'])
vol.state['vserver_root'] = kwargs.get('vs_root')
vol.state['status'] = kwargs.get('status')
vol.state['junction_active'] = kwargs.get('junc_active')
vol.space['size_avl_bytes'] = kwargs.get('avl_byt')
vol.space['size_total_bytes'] = kwargs.get('total_byt')
vol.space['space-guarantee-enabled'] = kwargs.get('sg_enabled')
vol.space['space-guarantee'] = kwargs.get('sg')
vol.space['thin_provisioned'] = kwargs.get('thin')
vol.mirror['mirrored'] = kwargs.get('mirrored')
vol.qos['qos_policy_group'] = kwargs.get('qos')
vol.aggr['name'] = kwargs.get('aggr_name')
vol.aggr['junction'] = kwargs.get('junction')
vol.sis['dedup'] = kwargs.get('dedup')
vol.sis['compression'] = kwargs.get('compression')
vol.aggr['raid_type'] = kwargs.get('raid')
vol.aggr['ha_policy'] = kwargs.get('ha')
vol.aggr['disk_type'] = kwargs.get('disk')
return vol
class SscUtilsTestCase(test.TestCase):
"""Test ssc utis."""
vol1 = createNetAppVolume(name='vola', vs='openstack',
vs_root=False, status='online', junc_active=True,
avl_byt='1000', total_byt='1500',
sg_enabled=False,
sg='file', thin=False, mirrored=False,
qos=None, aggr_name='aggr1', junction='/vola',
dedup=False, compression=False,
raid='raiddp', ha='cfo', disk='SSD')
vol2 = createNetAppVolume(name='volb', vs='openstack',
vs_root=False, status='online', junc_active=True,
avl_byt='2000', total_byt='2500',
sg_enabled=True,
sg='file', thin=True, mirrored=False,
qos=None, aggr_name='aggr2', junction='/volb',
dedup=True, compression=False,
raid='raid4', ha='cfo', disk='SSD')
vol3 = createNetAppVolume(name='volc', vs='openstack',
vs_root=False, status='online', junc_active=True,
avl_byt='3000', total_byt='3500',
sg_enabled=True,
sg='volume', thin=True, mirrored=False,
qos=None, aggr_name='aggr1', junction='/volc',
dedup=True, compression=True,
raid='raiddp', ha='cfo', disk='SAS')
vol4 = createNetAppVolume(name='vold', vs='openstack',
vs_root=False, status='online', junc_active=True,
avl_byt='4000', total_byt='4500',
sg_enabled=False,
sg='none', thin=False, mirrored=False,
qos=None, aggr_name='aggr1', junction='/vold',
dedup=False, compression=False,
raid='raiddp', ha='cfo', disk='SSD')
vol5 = createNetAppVolume(name='vole', vs='openstack',
vs_root=False, status='online', junc_active=True,
avl_byt='5000', total_byt='5500',
sg_enabled=True,
sg='none', thin=False, mirrored=True,
qos=None, aggr_name='aggr2', junction='/vole',
dedup=True, compression=False,
raid='raid4', ha='cfo', disk='SAS')
def setUp(self):
super(SscUtilsTestCase, self).setUp()
self.stubs.Set(httplib, 'HTTPConnection',
FakeDirectCmodeHTTPConnection)
def test_cl_vols_ssc_all(self):
"""Test cluster ssc for all vols."""
na_server = api.NaServer('127.0.0.1')
vserver = 'openstack'
test_vols = set([copy.deepcopy(self.vol1),
copy.deepcopy(self.vol2), copy.deepcopy(self.vol3)])
sis = {'vola': {'dedup': False, 'compression': False},
'volb': {'dedup': True, 'compression': False}}
mirrored = {'vola': [{'dest_loc': 'openstack1:vol1',
'rel_type': 'data_protection',
'mirr_state': 'broken'},
{'dest_loc': 'openstack2:vol2',
'rel_type': 'data_protection',
'mirr_state': 'snapmirrored'}],
'volb': [{'dest_loc': 'openstack1:vol2',
'rel_type': 'data_protection',
'mirr_state': 'broken'}]}
self.mox.StubOutWithMock(ssc_cmode, 'query_cluster_vols_for_ssc')
self.mox.StubOutWithMock(ssc_cmode, 'get_sis_vol_dict')
self.mox.StubOutWithMock(ssc_cmode, 'get_snapmirror_vol_dict')
self.mox.StubOutWithMock(ssc_cmode, 'query_aggr_options')
self.mox.StubOutWithMock(ssc_cmode, 'query_aggr_storage_disk')
ssc_cmode.query_cluster_vols_for_ssc(
na_server, vserver, None).AndReturn(test_vols)
ssc_cmode.get_sis_vol_dict(na_server, vserver, None).AndReturn(sis)
ssc_cmode.get_snapmirror_vol_dict(na_server, vserver, None).AndReturn(
mirrored)
raiddp = {'ha_policy': 'cfo', 'raid_type': 'raiddp'}
ssc_cmode.query_aggr_options(
na_server, mox.IgnoreArg()).AndReturn(raiddp)
ssc_cmode.query_aggr_storage_disk(
na_server, mox.IgnoreArg()).AndReturn('SSD')
raid4 = {'ha_policy': 'cfo', 'raid_type': 'raid4'}
ssc_cmode.query_aggr_options(
na_server, mox.IgnoreArg()).AndReturn(raid4)
ssc_cmode.query_aggr_storage_disk(
na_server, mox.IgnoreArg()).AndReturn('SAS')
self.mox.ReplayAll()
res_vols = ssc_cmode.get_cluster_vols_with_ssc(
na_server, vserver, volume=None)
self.mox.VerifyAll()
for vol in res_vols:
if vol.id['name'] == 'volc':
self.assertEqual(vol.sis['compression'], False)
self.assertEqual(vol.sis['dedup'], False)
else:
pass
def test_cl_vols_ssc_single(self):
"""Test cluster ssc for single vol."""
na_server = api.NaServer('127.0.0.1')
vserver = 'openstack'
test_vols = set([copy.deepcopy(self.vol1)])
sis = {'vola': {'dedup': False, 'compression': False}}
mirrored = {'vola': [{'dest_loc': 'openstack1:vol1',
'rel_type': 'data_protection',
'mirr_state': 'broken'},
{'dest_loc': 'openstack2:vol2',
'rel_type': 'data_protection',
'mirr_state': 'snapmirrored'}]}
self.mox.StubOutWithMock(ssc_cmode, 'query_cluster_vols_for_ssc')
self.mox.StubOutWithMock(ssc_cmode, 'get_sis_vol_dict')
self.mox.StubOutWithMock(ssc_cmode, 'get_snapmirror_vol_dict')
self.mox.StubOutWithMock(ssc_cmode, 'query_aggr_options')
self.mox.StubOutWithMock(ssc_cmode, 'query_aggr_storage_disk')
ssc_cmode.query_cluster_vols_for_ssc(
na_server, vserver, 'vola').AndReturn(test_vols)
ssc_cmode.get_sis_vol_dict(
na_server, vserver, 'vola').AndReturn(sis)
ssc_cmode.get_snapmirror_vol_dict(
na_server, vserver, 'vola').AndReturn(mirrored)
raiddp = {'ha_policy': 'cfo', 'raid_type': 'raiddp'}
ssc_cmode.query_aggr_options(
na_server, 'aggr1').AndReturn(raiddp)
ssc_cmode.query_aggr_storage_disk(na_server, 'aggr1').AndReturn('SSD')
self.mox.ReplayAll()
res_vols = ssc_cmode.get_cluster_vols_with_ssc(
na_server, vserver, volume='vola')
self.mox.VerifyAll()
self.assertEqual(len(res_vols), 1)
def test_get_cluster_ssc(self):
"""Test get cluster ssc map."""
na_server = api.NaServer('127.0.0.1')
vserver = 'openstack'
test_vols = set(
[self.vol1, self.vol2, self.vol3, self.vol4, self.vol5])
self.mox.StubOutWithMock(ssc_cmode, 'get_cluster_vols_with_ssc')
ssc_cmode.get_cluster_vols_with_ssc(
na_server, vserver).AndReturn(test_vols)
self.mox.ReplayAll()
res_map = ssc_cmode.get_cluster_ssc(na_server, vserver)
self.mox.VerifyAll()
self.assertEqual(len(res_map['mirrored']), 1)
self.assertEqual(len(res_map['dedup']), 3)
self.assertEqual(len(res_map['compression']), 1)
self.assertEqual(len(res_map['thin']), 2)
self.assertEqual(len(res_map['all']), 5)
def test_vols_for_boolean_specs(self):
"""Test ssc for boolean specs."""
test_vols = set(
[self.vol1, self.vol2, self.vol3, self.vol4, self.vol5])
ssc_map = {'mirrored': set([self.vol1]),
'dedup': set([self.vol1, self.vol2, self.vol3]),
'compression': set([self.vol3, self.vol4]),
'thin': set([self.vol5, self.vol2]), 'all': test_vols}
test_map = {'mirrored': ('netapp_mirrored', 'netapp_unmirrored'),
'dedup': ('netapp_dedup', 'netapp_nodedup'),
'compression': ('netapp_compression',
'netapp_nocompression'),
'thin': ('netapp_thin_provisioned',
'netapp_thick_provisioned')}
for type in test_map.keys():
# type
extra_specs = {test_map[type][0]: 'true'}
res = ssc_cmode.get_volumes_for_specs(ssc_map, extra_specs)
self.assertEqual(len(res), len(ssc_map[type]))
# opposite type
extra_specs = {test_map[type][1]: 'true'}
res = ssc_cmode.get_volumes_for_specs(ssc_map, extra_specs)
self.assertEqual(len(res), len(ssc_map['all'] - ssc_map[type]))
# both types
extra_specs =\
{test_map[type][0]: 'true', test_map[type][1]: 'true'}
res = ssc_cmode.get_volumes_for_specs(ssc_map, extra_specs)
self.assertEqual(len(res), len(ssc_map['all']))
def test_vols_for_optional_specs(self):
"""Test ssc for optional specs."""
test_vols =\
set([self.vol1, self.vol2, self.vol3, self.vol4, self.vol5])
ssc_map = {'mirrored': set([self.vol1]),
'dedup': set([self.vol1, self.vol2, self.vol3]),
'compression': set([self.vol3, self.vol4]),
'thin': set([self.vol5, self.vol2]), 'all': test_vols}
extra_specs =\
{'netapp_dedup': 'true',
'netapp:raid_type': 'raid4', 'netapp:disk_type': 'SSD'}
res = ssc_cmode.get_volumes_for_specs(ssc_map, extra_specs)
self.assertEqual(len(res), 1)
def test_query_cl_vols_for_ssc(self):
na_server = api.NaServer('127.0.0.1')
na_server.set_api_version(1, 15)
vols = ssc_cmode.query_cluster_vols_for_ssc(na_server, 'Openstack')
self.assertEqual(len(vols), 2)
for vol in vols:
if vol.id['name'] != 'iscsi' or vol.id['name'] != 'nfsvol':
pass
else:
raise exception.InvalidVolume('Invalid volume returned.')
def test_query_aggr_options(self):
na_server = api.NaServer('127.0.0.1')
aggr_attribs = ssc_cmode.query_aggr_options(na_server, 'aggr0')
if aggr_attribs:
self.assertEqual(aggr_attribs['ha_policy'], 'cfo')
self.assertEqual(aggr_attribs['raid_type'], 'raid_dp')
else:
raise exception.InvalidParameterValue("Incorrect aggr options")
def test_query_aggr_storage_disk(self):
na_server = api.NaServer('127.0.0.1')
eff_disk_type = ssc_cmode.query_aggr_storage_disk(na_server, 'aggr0')
self.assertEqual(eff_disk_type, 'SATA')
| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,488 |
\section{Introduction and Conclusions}
\label{sec:Introduction}
The Sachdev-Ye-Kitaev (SYK) model \cite{Sachdev:1992fk,KitaevTalks} has received much attention recently \cite{Polchinski:2016xgd,Maldacena:2016hyu,Jevicki:2016bwu,Fu:2016yrv,You:2016ldz,Jevicki:2016ito}. It is a simple model with solvable aspects which exhibits interesting connections to quantum chaos \cite{Shenker:2013pqa, Shenker:2013yza,Shenker:2014cwa,Maldacena:2015waa,Reynolds:2016pmi}, black hole physics and quantum gravity in 1+1 dimensions
\cite{Almheiri:2014cka,Sachdev:2015efa,Maldacena:2016upp,Jensen:2016pah,Sekino:2008he,Engelsoy:2016xyb,Cvetic:2016eiv}. Understanding these relations better in this model, and potential extensions of it, is an active and exciting research direction that promises to improve our knowledge of holography.
The SYK model is a quantum mechanical model involving Majorana fermions interacting with non-local random couplings. Much of the interesting physics of the model, including low energy near-conformal symmetry\footnote{The relation to $AdS_2$ was first pointed out in \cite{PhysRevLett.105.151602}.}and maximal scrambling, is not manifestly related to the specific construction in a transparent manner. To gain a better understanding of these features and their origin, it is therefore important to explore generalizations of the model. There has already been some work in this direction\footnote{In \cite{Gu:2016oyy}, a 1+1 translationally invariant model (on average), and higher dimensional extensions, were proposed. In \cite{Gross:2016kjj}, the authors propose a generalization by introducing an extra flavour index for fermions and considering complicated interactions between them. These models have neither hopping term nor a low momentum filter. Hence, we believe our models are qualitatively different from theirs. Another generalization including hopping term (but not low momentum filter) was introduced in \cite{PhysRevB.59.5341}. However the details of the interaction are different from ours.}.
Our approach to exploring the generalization to the SYK model is motivated by holography. In this work we consider two models. In the first one, we consider a 1+1d generalization in which we add locality in the extra dimension and implement random couplings as a function of the momentum. In the second model, we consider probe fermions interacting with a core of SYK degrees of freedom.
The first extension consists of a chain of SYK models, with Majorana fermions at each site with nearest hopping. This adds local physics in the added spatial direction. The fermions interact via random couplings, as in the original SYK model, but only at low enough momenta. This is achieved by first passing the Majorana fermions via a low-pass filter, and then coupling these via an SYK random interaction. The motivation for this is to have a model which is an ordinary relativistic field theory above some scale (and below a UV cut-off), which may model an asymptotically $AdS_3$ space, and some complicated IR dynamics encoding an object in the interior of $AdS_3$. Besides ensuring that the high momenta modes are filtered out, in this work we also focus on {\it chiral} filters, thus only one chiral half of the fermions participate in the interactions.
The second extension considers a core of SYK fermions to which a probe fermion is coupled to. In this approach we interpret the SYK fermions as describing the interior of the black hole, and the probe fermions as a single trace operator outside of it.
The results we obtain for the first class of models demonstrate, depending on the precise way the low pass filter is implemented, a rich variety of IR theories generalizing the SYK family of models. At high momentum, the model asymptotes to a free 1+1 dimensional field theory. As we decrease the momentum, the modes interact more strongly, until the new scaling regime is approached. In this scaling regime the fermions acquire an anomalous dimension, with different scaling for the space and time coordinate. In other words, we get a general hyperscaling at low energies. The dynamical critical exponent $z$ depends on the type of low-pass filter we use, and we discuss the range of sensible possibilities that arise. We also solve an example of the second class of models and discuss the new scaling dimensions appearing for these fermions.
One could consider our models as a particular class of disordered large $N$ theories at strong coupling. Applications of holography to such theories have been explored in \cite{Aharony:2015aea}. Furthermore, that inherent randomness in the disordered theory might be crucial for understanding black hole physics has recently been pointed out \cite{Balasubramanian:2014gla}. Interesting connections between 1+1d theories and black holes in gravity were also previously explored \cite{Guica:2008mu,Berkooz:2006wc,Berkooz:2014uwa}.
The outline of our paper is as follows. In section 2 we introduce the model of interacting fermions on a discrete lattice and the associated low-pass filters. We solve for the two point function in section 3 for scaling filters, exhibit different deep IR scaling dimensions, and discuss the continuum limit. In section 4 we discuss the probe fermion models and solve one such example. In appendix A, we derive the Schwinger-Dyson equations using the replica formulation and in appendix B we discuss gaussian filters.
While we focus below on simple 1+1 dimensional chains, it is possible to extend the model to several spatial directions and various interesting lattice structures in those directions. More generally, we view this work as a preliminary study of a large set of models generalizing the SYK construction both in the UV and the IR. It would be interesting to further study those models, compare and contrast their features with those of the original SYK model. In particular, studying the 4-point function would allow us to probe the chaotic behaviour of the system and the spatial spread of chaos, as manifested by the butterfly velocity. It might also be interesting to compute entanglement entropy (perhaps numerically as in \cite{Fu:2016yrv}). More ambitiously, one can hope that subtle issues like the information paradox \cite{Mathur:2010kx,Almheiri:2012rt} might be clearer if one has solvable models capturing the relevant physics of higher dimensional versions of the AdS/CFT correspondence.
\section{The 1+1D "low pass" SYK model}
\subsection{Definition of the Model}
Consider an extension of the SYK model involving a one-dimensional lattice with $L$ sites having $N$ Majorana fermions $\chi^{i,a}$ on each site $i$ with an
$\mathrm{SO}(N)$ index $a$. Its euclidean space lagrangian is
\begin{equation}
\mathcal{L}_E=\sum_{i,a}\left\{ \frac{1}{2}\chi^{i,a}\partial_{\tau}\chi^{i,a} - i\alpha[\chi^{i,a},\chi^{i+1,a}] \right\} +\sum_{i,abcd}J_{i,abcd}\eta^{i,a}\eta^{i,b}\eta^{i,c}\eta^{i,d}\,.
\label{trans-inv}
\end{equation}
We can either take the lattice to be a discretized circle or a discretized infinite line. The free theory involves a hopping term with bare parameter $\alpha$. The interaction term involves random couplings satisfying the disorder average
\begin{equation}
\langle J_{iabcd}J_{jabcd}\rangle=\frac{3!J^{2}}{N^3} \delta_{ij} \hspace{10mm} \mbox{no sum over }a,b,c,d
\label{dis-av}
\end{equation}
and \textit{low pass} fermions $\eta^{i,a} $ defined by a filter function $F$
\begin{equation}
\eta^{i,a}=\sum_{j} {\tilde F}(i-j)\chi^{j,a}\,.
\label{lowp}
\end{equation}
We will be interested in two filters : the standard {\it gaussian} filter
\begin{equation}
{\tilde F}(i-j)={\tilde A} \, \exp({-\hat{D}^{2}\frac{(i-j)^{2}}{L^{2}}})\,,
\end{equation}
where $\hat D$ sets the scale of the filter, and the {\it "scaling"} filters
\begin{equation}
{\tilde F}(i-j) \sim {{\tilde A}\over |i-j|^{\gamma\prime}}\label{filtpos}
\end{equation}
for a large enough range of $|i-j|$ and an appropriate range of $\gamma^\prime$. Depending on the latter, we may need to soften the filter at short distances or provide a sharper cut-off at large distances. We will assume the filter behavior is as in \eqref{filtpos} for a large enough range of lattice site separations, and discuss potential UV and IR modifications when we need it.
As mentioned in the introduction, some of the features of this model attempt to resemble the physics of $AdS_3$ black holes :
\begin{itemize}
\item At high momenta, the $\chi$ fermions decouple from the random interaction and become free -- this is the analogue of the region of $AdS_3$ far from the black hole. In the intermediate regime, where momentum is larger than the scale of the low pass filter but still smaller then the inverse lattice spacing, the model describes the simplest conformal field theory -- N species of Majorana fermions -- and, at least kinematically, we can think about it as $AdS_3$. We could of course also complicate the theory further in that regime, but in this work we keep the UV theory free (in section 5 we will discuss another interpretation where the free modes are modes outside the horizon of single trace operators).
\item At low momenta, the $\chi$ quanta become strongly interacting, in the appropriate SYK limit $J\rightarrow\infty$ -- this is akin to low momentum modes of the dual field theory forming a plasma, encoded by a dual black hole.
\end{itemize}
The model is not quite SYK -- other than the zero modes, all other modes are gapped in a specific pattern (if we think about them as quantum mechanics). The number of interacting fermions first increases, as a function of their momentum, as more and more modes participate in the interaction. It then decreases when the cut-off of the filter is reached. This will bring about different IR scaling behaviors, depending on the shape of the smearing function. We will see below that we obtain a large set of models with distinct scaling behaviors for the time coordinate and for the spatial coordinate.
The specific model that we will discuss is a chiral theory. As is standard with lattice fermions, the free model flows in the IR to a non-chiral theory of Majorana fermions. However, the low pass filter that we defined above keeps the low momentum modes of only one chirality of the fermions. The low-pass fermion for the other chirality is defined as (for the gaussian filter as an example)
\begin{equation}
\eta^{R,i,a}={\hat A}\sum_{j}e^{-\hat{D}^{2}\frac{(i-j)^{2}}{L^{2}}}(-1)^{i-j}\chi^{j,a}
\end{equation}
The present model includes only interactions of the low $\eta^L$ and not of low momenta $\eta^R$. We will refer to it as the chiral model -- similar models where interactions involve both left and right can be made non-chiral but here we will discuss only the former.
\subsubsection{Possible generalizations}
Several interesting generalizations are possible for the model discussed above.
\begin{itemize}
\item One possible generalization is the non-chiral model just mentioned. In this generalization, there will be a random interaction for the right movers $\eta^{R,i,a}$ as well. This theory will preserve parity symmetry (discussed in some detail in section \ref{sec:FreeTheory}). We can also add a direct coupling between the left and right moving sectors of the form $J_{i,abcd}\eta^{L,i,a}\eta^{L,i,b}\eta^{R,i,c}\eta^{R,i,d}$.
\item The random interactions in \eqref{trans-inv} involve 4 Majorana fermions. More generally, as was done in the 0+1 dimensional SYK model \cite{Maldacena:2016upp}, one can have a random interaction involving an arbitrary number $q$ of fermions. In that case, the theory is exactly solvable for $q=2$ and has important implications for large values of $q$ that allow to obtain an analytic understanding of the entire flow. We expect the same to hold here.
\item
The model \eqref{trans-inv} is not translationally invariant because the interactions $J_{i,abcd}$ depend on the lattice site $i$, although correlation functions are translationally invariant after the disorder average. We can modify the model above to accommodate strict translational invariance replacing the interaction term by, for example,
\begin{equation}
\mathcal{L}_{\text{int}} = \sum_i \sum_{a,b,c,d}\sum_{d_1,d_2,d_3} J_{abcdd_1d_2d_3} \eta^{i,a}\eta^{i+d_1,b}\eta^{i+d_2,c}\eta^{i+d_3,d}
\end{equation}
This model is not easily solvable using the tools we discuss below, but we hope to return to it in the near future.
\end{itemize}
\subsection{The free theory}\label{sec:FreeTheory}
Using euclidean conventions (with $Z=e^{-S_E},\ S_E=\int d\tau {\mathcal L}_E, \ d\tau=idt$), the lagrangian is
\begin{equation}\label{EuclideanFree}
{\mathcal L}^{\text{free}}_E=\sum_{i,a} \left\lbrace {1 \over { 2}} \chi^{i,a}\partial_\tau \chi^{i,a} - i\alpha [\chi^{i,a},\chi^{i+1,a}] \right\rbrace \, .
\end{equation}
To describe a periodic lattice, we identify $\chi^{i,a}=\chi^{i+L,a}$. Hence, as operators, the commutation relations are
\begin{equation}
\{\chi^{i,a},\chi^{j,b}\}=\delta^{ab}\sum_{p\in \mathbb{Z}}\delta^{i(j+pL)}\,.
\end{equation}
The momentum space fermions $\chi_k^a$ are defined, for integer $k$, as
\begin{equation}
\chi_k^a = \frac{1}{\sqrt{L}}\sum_j e^{2\pi i {jk\over L}}\, \chi^{j,a}\, \quad \text{with} \quad \chi^{j,a}= \frac{1}{\sqrt{L}} \sum_k e^{-2\pi i {jk\over L}}\,\chi_k^a\,.
\label{mfermion}
\end{equation}
The conventions will be that momentum index is down and position index is up.
We assume that L is even. Since $\chi_k^a=\chi_{k+L}^a$, we can either take $k$
to be an arbitrary integer with this periodicity, or we can restrict ourselves to the range
\begin{equation}
k= -L/2+1,...L/2
\end{equation}
We will use both these descriptions. As operators, the momentum space fermions satisfy the commutation relations
\begin{equation}\label{anticom}
\{\chi_k^a,\chi_{k'}^b\} = \delta^{ab}(\delta_{k+k',0}+\delta_{k+k',L} )\,,
\end{equation}
where the second contribution is only non-zero when $k=k'=L/2$. The free theory \eqref{EuclideanFree} in momentum space modes becomes\footnote{We have dropped some non-essential constant pieces in obtaining this expression.}
\begin{equation}\label{EuclideanFreeMomentum}
{\mathcal L}^{\text{free}}_E = \sum_a \left\lbrace \sum_{k=1}^{{L \over 2}-1} \chi^a_{-k} ( \partial_\tau + E_k) \chi^a_k + { \chi^a_0 \partial_\tau \chi^a_0 + \chi^a_{L \over 2} \partial_\tau \chi^a_{L \over 2} \over 2} \right\rbrace
\end{equation}
where
\begin{equation}\label{Energy}
E_k \equiv 4 \alpha \left| \sin\left({2\pi k \over L}\right) \right|\,.
\end{equation}
Since $(\chi^a_k)^\dagger = \chi_{-k}^a$, the fermions $\chi^{a}_k$ for $1 \le k < {L \over 2}$ can be thought of as complex fermions, whereas $\chi^a_0 = {\chi^a_0}^\dagger,\chi^a_{L \over 2}={\chi^a_{L \over 2}}^\dagger$ are $2N$ Majorana fermions.
\subsubsection*{Linearized theory}
The free theory \eqref{EuclideanFree} is non-chiral since it is invariant under the parity transformation
\begin{equation}
\chi^{i,a}\to (-1)^i \chi^{L-i,a}
\end{equation}
Its action in momentum space is
\begin{eqnarray}\nonumber
\chi^a_k \leftrightarrow \chi^a_{ {L \over 2} - k} \,, \,\,\chi^a_{-k} & \leftrightarrow & \chi^a_{- ({L \over 2} - k)}\hspace{10mm} \mbox{ for } 1 \le k < {L \over 2} \\
\chi^a_0 &\leftrightarrow & \chi^a_{L \over 2}
\end{eqnarray}
If we define\footnote{We will take $L$ not divisible by $4$ for simplicity. $[x]$ below denotes the integer part of $x$.}
\begin{equation}\label{LRfermions}
\chi^{L,a}_k \equiv \chi_k^a,\ \ \ \chi^{R,a}_{-k} \equiv \chi_{{L \over 2}-k}^a \mbox{ for } 1 \le k \le [{L \over 4}] \,,
\end{equation}
parity maps left $\chi^{L,a}_k$ to right $\chi^{R,a}_{-k}$ fermions. Using these degrees of freedom, the action \eqref{EuclideanFreeMomentum} becomes
\begin{equation}\label{FreeMomentumDiscrete}
{\mathcal L}^{\text{free}}_E = \sum_a \left\lbrace \sum_{k=1}^{ [{L \over 4}]} \left[ \chi^{L,a}_{-k} ( \partial_\tau + E_k) \chi^{L,a}_k + \chi^{R,a}_{k} ( \partial_\tau + E_k) \chi^{R,a}_{-k} \right] + { \chi^a_0 \partial_\tau \chi^a_0 + \chi^a_{L \over 2} \partial_\tau \chi^a_{L \over 2} \over 2} \right\rbrace\,.
\end{equation}
This form of the action will is more useful when taking the continuum limit $L\to \infty$ and linearising the dispersion relation.
\subsubsection*{States of the free theory}
The ground states of the free theory $ |\beta \rangle $ satisfy
\begin{equation}
\chi_k^a | \beta\rangle = 0 \mbox{ for } 1 \le k < L/2
\end{equation}
since $\chi_{k}^a$ are annihilation operators for $k>0$ and creation operators for $k<0$. Hence, they form $2^{N}$ dimensional representations of $\chi^a_0, \chi^a_{L \over 2}$. These states can equivalently be described in the left and right representation as
\begin{equation}
\chi_k^{L,a}|\beta\rangle=\chi_{-k}^{R,a}|\beta\rangle=0,\ \ \ k>0
\end{equation}
Excited states are of the form
\begin{equation}
\chi^a_{-k} | \beta \rangle \mbox{ for } 1 \le k < L/2
\end{equation}
They carry energy $E_k$ as in \eqref{Energy}.
\subsubsection*{Momentum space correlators}
We now compute the time ordered propagator for the momentum space fermions in any of the vacuum states
\begin{equation}
G^{ab}_{k,k'}(\tau) \equiv \langle T\left[ \chi^a_k(\tau) \chi^b_{-k}(0) \right] \rangle\,.
\end{equation}
Using standard methods, we obtain
\begin{eqnarray}
\langle T\left[ \chi^a_k(\tau) \chi^b_{-k}(0) \right] \rangle &=& \theta(\tau) e^{-E_k \tau} \delta_{ab} \mbox{ for } 1 \le k <{ L \over 2}\,, \\
\langle T\left[ \chi^a_{-k}(\tau) \chi^b_{k}(0) \right] \rangle &=& -\theta(-\tau) e^{-E_k |\tau|} \delta_{ab} \mbox{ for } 1 \le k <{ L \over 2}\,, \\
\langle T\left[ \chi^a_0(\tau) \chi^b_{0}(0) \right] \rangle &=& \langle T\left[ \chi^a_{L \over 2}(\tau) \chi^b_{L \over 2}(0) \right] = \frac{1}{2}\mbox{sgn}(\tau)\,. e^{-\epsilon |\tau|} \delta_{ab}
\end{eqnarray}
Here $\epsilon$ is small and positive and constitutes some effective "$\epsilon$ prescription".Correlators of left and right fermions in equal
\begin{equation}
\langle T\left[ \chi^{L,a}_k(\tau) \chi^{L,b}_{-k}(0) \right] \rangle = \langle T\left[ \chi^{R,a}_{-k}(\tau) \chi^{R,b}_{k}(0) \right] \rangle= \theta(\tau) e^{-E_k \tau} \delta_{ab}\,.
\end{equation}
It is possible to assemble all these propagators in a more compact notation
\begin{equation}
G^{ab}_{k,k'}(\tau) = \langle T\left[ \chi^a_k(\tau) \chi^b_{k'}(0) \right] \rangle = \delta_{k+k'=0,L} \delta_{ab}
\left[
\theta(\tau) e^{-E_k \tau} H(k) - \theta(-\tau) e^{-E_{k'} |\tau|} H(k') \right]
\end{equation}
introducing the function
\begin{equation}
H(k) = \begin{cases}
1 &, \mbox{ if } 1 \le k <{ L \over 2}\\
0 &, \mbox{ if } -{L \over 2} < k \le 1 \\
{1 \over 2} &, \mbox{ if } k =0,{L \over 2} \\
\end{cases}
\end{equation}
Here $E_0 = E_{L/2} = \epsilon$ which is a regularization prescription. We have also used
$E_k = E_{-k} = 4 \alpha |\sin({2 \pi k \over L})|$.
It is also possible to write the correlators in frequency space $(\omega)$
\begin{equation}
G^{ab}_{k,k'}(\omega) \equiv \int_{-\infty}^\infty d\tau e^{i\omega \tau} G^{ab}_{k,k'}(\tau) = \delta_{k+k'=0,L} \delta_{ab}
\left[
{H(k) \over -i \omega + E_k } + { H(k') \over -i \omega - E_{k'} } \right] \,.
\end{equation}
\subsubsection*{Continuum limit of the free theory}
We describe the continuum of the free theory here, since it would be relevant when we solve the interacting theory below. We consider the theory on a circle of size $R$. Hence, we define the continuum limit as $L \to \infty$ keeping the coordinate $x = {i \over L} R$ fixed. The physical momentum modes are $p = {2 \pi k \over R}$ with $k =0,\pm 1,\dots$ and the UV cutoff is $\Lambda \ll {L \over R}$.
Linearizing the energy spectrum in \eqref{Energy} for small $k$, one gets $E_p = E_k \approx {4 \alpha R \over L} p$. We will choose the bare parameter $\alpha = {L \over 4R}\equiv {\Lambda_0 \over 4}$ from now onwards to obtain a relativistic theory, but we could have accommodated other values. Since we will be interested in chiral models, we give here the chiral sector of the free theory \eqref{EuclideanFreeMomentum} in the above limit
\begin{equation}
{\mathcal L}^{\text{free}}_E = \sum_a \left\lbrace \sum_{p={2 \pi \over R}}^{\Lambda} \chi^a_{-p} ( \partial_\tau + p) \chi^a_p + {1 \over 2} \chi^a_0 \partial_\tau \chi^a_0 \right\rbrace\,.
\label{freeR}
\end{equation}
\subsection{The interacting theory}
The interacting theory in momentum space equals
\begin{equation}
\begin{aligned}
\mathcal{L}_E &=\mathcal{L}^{\text{free}}_E + \mathcal{L}^{\text{int}}_E = \sum_{k}\chi_{-k}^{a}(\partial_{\tau }+E_{k})\chi_{k}^{a} \\
& + \frac{1}{L^2}\sum_{a,b,c,d,k_{1},k_{2},k_{3},k_{4}}\eta_{k_{1}}^{a}\eta_{k_{2}}^{b}\eta_{k_{3}}^{c}\eta_{k_{4}}^{d}
\left(\sum_{j}e^{-2\pi i{(k_{1}+k_{2}+k_{3}+k_{4})j\over L}}J_{j,abcd}\right)
\end{aligned}
\label{model}
\end{equation}
where the low pass momentum fermions $\eta^a_k$ are defined as in \eqref{mfermion}. These are related to the physical fermions $\chi_k^a$ through the low pass filter in momentum space $F(k)$ by
\begin{equation}
\eta^i_k = A\,F(k) \chi^i_k\,.
\end{equation}
where $A$ is a normalization constant and $F(k)$ is the Fourier transform of $\tilde F$ introduced before. This allows to write the interaction lagrangian in terms of the physical fermions $\chi_k^a$
\begin{equation}
{\mathcal L}^{\text{int}}_E = {A^4\over L^2}\sum_{k_{1,2,3,4}} \sum_{a,b,c,d} \left(\prod_{h=1}^4 F(k_h)\right) \chi_{k_1}^a \chi_{k_2}^b \chi_{k_3}^c \chi_{k_4}^d
\biggl(\sum_j J_{j,abcd}\,e^{-i2\pi {j\sum_{l=1}^4 k_l \over L}}\biggr)
\label{Lintchimomenta}
\end{equation}
Remember that our convention for the impurity average will be
\begin{equation}
E[J_{iabcd} J_{jabcd}] = {3!J^2 \over N^3} \delta_{ij}\,,
\label{impurity}
\end{equation}
and we will mainly be interested in scaling filters of the form
\begin{equation}
F(k)= |k|^{-\gamma}\,,
\end{equation}
or in gaussian filters
\begin{equation}
F(k)= e^{-\pi^2k^2\over {\hat D}^2}\,.
\end{equation}
\section{Solution of the model}
In this section we find a saddle point solution to our model \eqref{model} with a scaling filter, generalizing the SYK model one. Gaussian filters are discussed in appendix \ref{sgaussian}. We use the saddle point equations to calculate the two-point functions
\begin{equation}
G^{ab}_{k',k}(\omega)=\langle [\chi^a_{k'} \chi^b_{k}]\rangle(\omega)\,,
\end{equation}
in an scaling regime at low energies. We will find a rich variety of behaviour for these correlators.
\subsection{Recalling SYK}
We start by briefly recalling the results in the original 0+1 dimensional SYK model
\begin{equation}
H_{SYK}= \sum_{abcd} J_{abcd} \chi^a\chi^b\chi^c\chi^d\,.
\end{equation}
where $J_{abcd}$ is the random coupling. The two-point function for the free theory is
\begin{equation}
G_{\text{free}}(\tau)={1\over 2}\mbox{sgn}(\tau),\ \ \ \ G_{\text{free}}(w)=\int dt e^{iwt} G_{\text{free}}(\tau) = -{1\over iw}\, .
\end{equation}
In the interacting theory the connected 2 point functions are \cite{KitaevTalks,Polchinski:2016xgd,Maldacena:2016hyu}
\begin{equation}
G_c(\tau) \sim{ 1\over |\tau|^{2\Delta} } \mbox{sgn}(\tau)~~~~~~~\Delta=\frac{1}{4}
\end{equation}
where to transform to frequency space the following Fourier transform formula can be used
\begin{equation}
\int_{-\infty}^\infty d\tau e^{iw\tau} { \mbox{sgn}(\tau) \over |\tau|^{2\Delta}}= i\, 2^{1-2\Delta}\sqrt{\pi}{\Gamma(1-\Delta)\over \Gamma({1\over 2}+\Delta )}|w|^{2\Delta-1}\mbox{sgn}(w)
\label{MS}
\end{equation}
\subsection{Single k and collective equation}
Given the interaction Lagrangian \eqref{Lintchimomenta}, the Schwinger-Dyson (SD) equations are given by
\begin{eqnarray}
\label{SDEqn1init}
\Sigma^{aa^\prime}_{k_1k_1^\prime}(t) &=& {J^2A^8\over L^3} \delta^{aa^\prime}F(k_1)F(k_1^\prime) \sum_{k_2,k_3,k_4,k_2^\prime,k_3^\prime,k_4^\prime} \delta_{\sum_{i=1}^4 (k_i + k'_i)=0}\,
\prod_{i=2}^4 [F(k_i)F(k'_i)G_{k_ik'_i}\ \ \ \ \\
\label{SDEqn2init}
( G^{aa'}_{kk'})^{-1} &=& (G^{(0)aa'}_{kk'})^{-1} - \Sigma^{aa'}_{kk'}(\omega)\,.
\end{eqnarray}
Here $G^{(0)aa'}_{kk'}$ is the free two-point function. In going from the interaction lagrangian to the SD equation we carried out the disorder average. We also assumed that the filter, in momentum space, cuts off the interaction before reaching momentum $\sim {\cal O}(L)$, such that the other chirality (in our conventions) does not participate in the interaction (i.e., we are in the chiral model). Practically, this enforces strict momentum conservation, rather than up to multiples of L.
We will assume that after disorder averaging both the $\mathrm{SO}(N)$ symmetry and the $\mathbb{Z}_L$ lattice translations are preserved\footnote{Verifying this requires an analysis of stability, which we will not do here.}. We implement this by defining
\begin{equation}
G^{ab}_{k',k}(w)=\delta^{ab}\delta_{k+k'=0} G_k(w).
\label{diagG}
\end{equation}
Under this assumption the SD equation \eqref{SDEqn1init} forces the self-energy $\Sigma(\tau)$ to be diagonal too. Thus it is natural to define
\begin{equation}
\Sigma^{ab}_{k_1k_1^\prime}(\tau)=\delta^{ab}\Sigma_{k_1}(\tau)\delta_{k_1+k_1^\prime=0}\,.
\end{equation}
The SD equations \eqref{SDEqn1init}-\eqref{SDEqn2init} become
\begin{eqnarray}\label{SDEqn1fin}
\frac{1}{G_k(\omega)} &=& {1\over G_{k}^{(0)}(\omega) }-\Sigma_k(\omega)\,, \\
\label{SDEqn2fin}
\Sigma_k(\tau) &=& {A^8J^2\over L^3} \,F(k)^2\prod_{h=1}^{3} \biggl( \sum_{k_h} F(k_h)^2 \,G_{k_h}(\tau)\biggr)\,,
\end{eqnarray}
where we also assumed $F(k)$ is an even function of k. Let us introduce new quantities
\begin{equation}\label{SDdef}
{\tilde G}_k(\tau) \equiv F(k)^2 G_k(\tau)\,,\, \, \, {\tilde G}(\tau) \equiv \sum_k {\tilde G}_k(\tau) \,,\,\,\,
{\tilde \Sigma}(\tau) \equiv F(k)^{-2} \Sigma_k(\tau)\,,
\end{equation}
where we already dropped the momentum subscript in ${\tilde{\Sigma}}$ since this is independent of $k$. The SD equations simplify to
\begin{eqnarray}
\tilde{\Sigma}(\tau) & = & {A^8J^{2} \over L^3} \, \tilde{G}(\tau)^{3} \label{SDEqn1}\\
\tilde{G}(\omega) & = & \sum_{k=-{L/2}+1}^{\frac{L}{2}}\frac{1}{(F^2(k)G_{k}^{(0)})^{-1}(w)-\tilde{\Sigma}(\omega)} \label{SDEqn2}\\
{1\over G_k(w) } &=& {1\over G_{k}^{(0)}(\omega)} - F(k)^2 {\tilde\Sigma}(\omega) \label{2ptG}
\end{eqnarray}
In Appendix \ref{replica} we re-derive this set of equations using the replica method.
We will refer to ${\tilde G}$ and $\tilde \Sigma$ as the collective quantities and to \eqref{SDEqn1} and \eqref{SDEqn2} as the collective equations. Solving them requires the evaluation of the sum \eqref{SDEqn2} to write ${\tilde G}(\omega)$ as a function of ${\tilde \Sigma}(\omega)$. \eqref{SDEqn1} and \eqref{SDEqn2} are then two equations for two functions, one in time and one in frequency, which we can hope to solve. For some choices of $F(k)$ this can be done analytically (in the scaling regime at least), and in other cases we can try and evaluate them numerically. In any case, their basic complexity is not much worse than the original ones in the SYK model. Once we know their solution, we can plug $\tilde \Sigma(\omega)$ into the last equation to compute the momentum 2 point propagator.
It may be instructive to revisit part of our holographic motivation to consider these models at this stage. In terms of a possible bulk interpretation, start by examining the propagator at large $k$. In our case, this is approximately the free propagator with a small correction from $\tilde\Sigma(\omega)$ because $F(k)$ cuts off the coupling with $\tilde\Sigma(\omega)$ at high momenta. In the bulk interpretation, these should correspond to the UV modes near the boundary of $AdS$, interacting with some object living in the bulk interior. As the interaction increases, the momentum modes feel more and more this IR object. In this interpretation, the collective quantities encode the dynamics of whatever macroscopic object we have in the bulk, comprised of the strongly coupled dynamics of the low energy (below the filter scale) modes. They are SYK in nature, although we will see that different filters can give us different scaling theories. The presence of this IR object in the bulk feeds into the high momentum modes like a semi-classical object, correcting their propagators.
\subsection{Solving the collective equations in the continuum}
\label{Rcontinuum}
Since evaluating the sum \eqref{SDEqn2} is complicated, and we are interested in the continuum theory any way, we start discussing the latter here. We think of our model as defined on a circle of fixed size $R$ (which we will think of as large), so that the lattice spacing is $R/L$ and take $L\to\infty$ at the end of the computation. The coordinate around the circle $x=\frac{i}{L}\,R$ will be kept fixed. We will require our filter to cut-off momenta at scales much larger than $1/R$ and much smaller than $L/R$, and kept fixed in the limit $L\to\infty$. In this scaling, many momentum modes participate in the collective dynamics, but we still have effectively a continuum theory for the momentum modes above the filter and below $1/L$.
The $L\to \infty$ limit allows us to approximate \eqref{SDEqn2} by the integral
\begin{equation}
\begin{aligned}
{\tilde G}(\omega)&=\biggl(\int_{-\infty}^{0} dk +\int_{0}^{\infty} dk\biggr) {1\over \bigl( F^2(k)G_{k}^{(0)}(\omega)\bigr)^{-1}-{\tilde\Sigma}(\omega) }\\
&=\int_{-\infty}^{0} {dk\over \bigl( F^{-2}(k)(-i\omega-E_k\bigr)-{\tilde\Sigma}(\omega) } +
\int_{0}^{\infty} {dk\over \bigl( F^{-2}(k)(-i\omega+E_k\bigr)-{\tilde\Sigma}(\omega) }\,.
\end{aligned}
\end{equation}
Since ${\tilde G}(\omega)$ and ${\tilde\Sigma}(\omega)$ are both purely imaginary, we obtain
\begin{equation}\label{tildeGexp}
{\tilde G}(\omega)=2i\, {\it{\text{Im}}} \int_{0}^{\infty} {dk \over F(k)^{-2}(-i\omega+E_k)-{\tilde\Sigma}(\omega) }\,.
\end{equation}
We have assumed that the function $F^{-2}(k)$ increases rapidly enough for large $k$ such that the integral converges and we can replace the cut-off $L$ by infinity.
The discussion above about the scale of the filter is a little subtle since will be interested in scaling filters of the form
$F(k)\sim |k|^{-\gamma}$ for a range of $\gamma$. These filters need to be cut off at small $k$ and/or at large $k$, depending on $\gamma$. In position space the filter goes like $(x-y)^{\gamma-1}$. We will approach the issue of the cut-off by examining the integral after the fact.
\begin{itemize}
\item If the integral converges at large $k$ then we don't need to introduce an additional cut-off in equation \eqref{tildeGexp}, or more precisely, introducing such a cut-off $\Lambda$ will change the results by some negative power of $\Lambda$. However, we may still keep this cut-off, as in \eqref{2ptG}, if we want to.
\item The integral \eqref{tildeGexp} itself has no problem in low momenta, for finite ${\tilde \Sigma}$, since $F^{-2}(0)=0$. This means that the interaction will effectively mix the low momenta degrees of freedom and set them at some scale defined by the filter. Phrased in another way, one might worry that the process of going from the sum to the integral is not correct. If we were to do the exact sum, then momentum modes around $k\sim 1$ (physical momenta of order $1/R$) would give contributions that scale like $1/{\tilde \Sigma}$ in the limit of large interaction strength. We will see that for the filters that we present below, this is much smaller than the total integral contribution and hence we don't need to worry about anomalous contributions of some global modes.
\end{itemize}
For the scaling low pass filter defined by $F^{-1}(k) = |k|^\gamma$, the integral \eqref{tildeGexp} becomes
\begin{equation}
{\tilde G}(\omega)=2i\, {\it{\text{Im}}} \int_0^\infty {dk\over k^{2\gamma}(-i\omega+ 2\pi\,k/R)-{\tilde\Sigma}(\omega)}
\end{equation}
where we already linearised the spectrum as in \eqref{freeR}. Working at fixed low frequency and large $\tilde\Sigma(\omega)$, we can eventually neglect $\omega$ with respect to $k$. We will assume this for now and check for self consistency at the end of the computation.
Using the variable $k = \left[-R\,\tilde{\Sigma}(\omega)/(2\pi) \right]^{1 \over 2 \gamma +1} \tilde k$, the above integral becomes
\begin{equation}
{\tilde G}(\omega)=2i\, {\text{Im}}\left[c_\gamma \left(\frac{R}{2\pi}\right)^{{1 \over 2 \gamma + 1}} (-\tilde \Sigma(\omega))^{-{2 \gamma \over 2 \gamma +1}}\right]\,,
\label{GSigma}
\end{equation}
where $c_\gamma = \int_{0}^{\infty} {d \tilde k \over k^{2\gamma+1}+1} = {\pi \over 2 \gamma +1} \csc({\pi \over 2 \gamma + 1})$ \footnote{The actual integral we get has the contour from $0$ to $\infty$ at an angle $\theta = -{\pi i \over 2(2 \gamma +1)}$ in the complex plane. Since there are no poles and the contour at $\infty$ vanishes for $\gamma >0 $, we can rotate it back to real axis. One can then use the Formulae $\int_0^\infty {dx \over 1 + x^\nu} = {\pi \over \nu} \csc({\pi \over \nu}) \mbox{ for } \mbox{Re}(\nu) > 1$.}. The resulting collective equations are solved by
\begin{eqnarray}
\tilde{\Sigma}(\omega) &= &i\,\mbox{sgn}(\omega)\,\frac{2\pi \kappa^3}{R\,Z}\,{\cal K}(\gamma)\,|\omega|^{6\Delta -1} \quad \text{with} \quad {\cal K}(\gamma)\equiv 2^{1 - 6 \Delta} \sqrt{\pi} {\Gamma(1 - 3 \Delta) \over \Gamma({1 \over 2}+3 \Delta )}\,, \label{sigma-sol} \\
\tilde G(\tau) &=& Z^{\frac{2\gamma}{2\gamma+1}}\,\frac{R}{2\pi}\frac{\kappa}{|\tau|^{2 \Delta }}\, \mbox{sgn}(\tau)
\end{eqnarray}
provided
\begin{equation}
\Delta = \frac{1+4 \gamma }{2 (1+8 \gamma )}\,,
\label{Delta}
\end{equation}
$\kappa(\gamma)$ satisfies the constraint
\begin{equation}
\kappa^{\frac{1+8\gamma}{1+2\gamma}}\, 2^{1-2\Delta} \sqrt{\pi} {\Gamma(1 - \Delta) \over \Gamma({1 \over 2}+ \Delta )} = 2 c_\gamma \,\sin({\pi \gamma\over 2 \gamma + 1})\,{\cal K}(\gamma)^{-2\gamma/(2\gamma+1)}\,,
\end{equation}
and $Z$ is identified as
\begin{equation}\label{defZ}
Z = \left(\frac{A^8 J^2\,R^4}{L^3\,(2\pi)^4}\right)^{-\frac{1+2\gamma}{1+8\gamma}}\,.
\end{equation}
For completeness, we also write the solution to the collective equation in frequency space
\begin{equation}
\tilde{G}(\omega) = i\,\mbox{sgn}(\omega)\,\frac{\kappa\,R}{2\pi} \, Z^{\frac{2\gamma}{2\gamma+1}}\,2^{1-2\Delta} \sqrt{\pi} {\Gamma(1 - \Delta) \over \Gamma({1 \over 2}+ \Delta )}\,|\omega|^{2\Delta -1}\,,
\end{equation}
with $\Delta$ given in \eqref{Delta}.
\subsubsection{IR contribution}
Here, we check the consistency of our approximations and include further comments on the possible modifications that our scaling filters may require. We will be interested in working in the regime $|\omega|\,R\gg 1$ and fixed. There are three reasons to examine the IR more closely.
First, to check that physical momentum modes close to $1/R$ do not change the conclusions above, we require the magnitude $|{\tilde G}(\omega)|$ from
\eqref{GSigma}, which captures the contribution from higher momentum modes within the filter, to be larger than the $|{\tilde\Sigma}(\omega)|^{-1}$ contribution from the lower modes. This amounts to working in the range
\begin{equation}
1\ll R\,|{\tilde\Sigma}(\omega)| \quad \Rightarrow \quad \biggl( {A^8\,R^2\,J^2\over L^3}\biggr)\, (R|\omega|)^2 \gg 1
\end{equation}
This condition is compatible with $|\omega|\,R\gg 1$ fixed for large enough $J$.
Furthermore, our analysis neglected the $-i\omega$ term in $\tilde{G}(\omega)$, while working in some momentum range satisfying $k^{2\gamma+1}\sim {\tilde{\Sigma}}$. Consistency of both conditions is equivalent to
\begin{equation}
\frac{k}{R\,|\omega|} \gg \frac{k^{2\gamma + 1}}{R\,|\tilde{\Sigma}(\omega)|} \sim 1
\end{equation}
which can also be satisfied in our desired regime. This is again the statement that many momentum modes, much above physical momenta $1/R$ participate in the collective quantities.
Second, the discussion above is valid at intermediate frequencies, as long as we stay $\omega \gg 1/R$, at which point we see that momentum is quantized. It could still be that there is an almost continuum spectrum of energies originating from large $N$, but in any case, we expect that this limit to be governed by a different limit of SD equations.
Third, a filter of the form $F\sim 1/|k|^\gamma$ is non trivial in position space. Its Fourier transform is ${\tilde F}=x^{\gamma-1}$. We can work with $\gamma<1$ to obtain reasonable behavior in position space, or we can modify the solution to decay further at some long distance, for example by considering
\begin{equation}
F(k)=k^{-\gamma}e^{-D^2/k^2}\, .
\end{equation}
We will then need to choose $D$ to be small enough. Under this assumption, the discussion above remains the same. We could analogously add a hard UV cut-off to improve the UV convergence.
\subsection{Going to the infinite line}\label{sec:NonCompactSoln}
At finite radius $R$, we can always rescale $A$ and $J$ keeping $Z$ fixed, so that the model provides finite, L-independent, results. This is the standard RG approach of keeping the IR fixed and running the UV appropriately. In the following, we investigate whether we can achieve the same finiteness in the non-compact limit $R\to \infty$.
In the non-compact limit, finite physical momenta are labelled by $p=\frac{2\pi k}{R}$. Despite the rescaling of momenta, the diagonal contribution to the 2 point function in \eqref{diagG} remains finite
\begin{equation}
G^{\infty}(p) = G_{k=pR/2\pi}\,.
\end{equation}
To investigate the existence of a finite interacting theory in the deep IR in the non-compact limit, we will require both the filter function and the two point function \eqref{2ptG} to be finite.
The first condition requires to write the filtering function $A\,F(k)= \frac{A}{|k|^\gamma}$ relating the interacting fermions $\eta$ with the physical fermions $\chi$ in terms of the physical momentum $p$ as $A_\infty\,F(p) \equiv \frac{A_\infty}{|p|^\gamma}$. Hence, we learn $A\sim A_\infty\,R^\gamma$, with $A_\infty$ fixed.
The second condition is studied by replacing \eqref{sigma-sol} into \eqref{2ptG}
\begin{equation}
G^{\infty}(p) = {1 \over [G^{(0)}_{k}(\omega)]^{-1} - i\, \mbox{sgn}(\omega) |\omega|^{6 \Delta -1}\,\kappa^3\,\frac{(2\pi)^{2\gamma+1}{\cal K}(\gamma)}{R^{2\gamma+1}\,Z\,p^{2\gamma}}}
\label{finiteGR}
\end{equation}
Requiring the interaction to be finite is equivalent to keeping
\begin{equation}
\frac{ J^2}{\Lambda_0^3} \ \ \mbox{fixed}\,.
\label{nonc}
\end{equation}
where recall that $\Lambda_0 = {L \over R}$ is the inverse lattice spacing.
\subsubsection{Hyperscaling}
The finite 2 point function \eqref{finiteGR} has an hyperscaling symmetry in the deep IR $\omega\to \lambda^{-z}\omega$ and $p\to \lambda^{-1}\,p$\footnote{The $z$ exponent is usually defined in real space by the scaling relations $t\to \lambda^z\,t$ and $x\to \lambda\,x$.}
with dynamical scaling exponent $z$:
\begin{equation}
z = \frac{2\gamma+1}{6\Delta - 1} = {1 \over 2} + 4 \gamma\,.
\end{equation}
This covers the range $z>1/2$. In particular, it includes unitary theories, i.e. those with $z\geq 1$.
The range of $z$'s slightly changes when we consider random couplings of $q$ fermions ($q=4$ in our previous analysis). Then $\Delta_q \equiv {1 + 4 \gamma \over 2(1+2 q \gamma)} $ and the dynamical exponent becomes
\begin{equation}
z_q = \frac{2\gamma+1}{6\Delta_q - 1} = {1 + 2 q \gamma \over q-2 } \,.
\end{equation}
This opens further possibilities for the range of $z$.
\subsection{Continuum non-compact limit}
\label{Raction}
In this subsection we discuss again the continuum model formulated on an infinite line, but from the action perspective. We will recover the condition \eqref{nonc} and in the process we will give the continuum version of the impurity average \eqref{impurity}.
First, let us write the continuum $L\to \infty$ limit of our interacting theory on a circle of size $R$. Using \eqref{freeR}) and \eqref{Lintchimomenta},
\begin{equation*}
\mathcal{L} = \sum_p \, \chi^{a}_{-p}(\partial_{\tau }+ p )\chi^{a}_p + {(2\pi)^{4\gamma} A^4 \over L^2 R^{4 \gamma} } \sum_{a_i, p_i} \prod_{i=1}^4 [F(p_i) \chi^a_{p_i}] \int {Ldx \over R} e^{- i x \sum_{j=1}^4 p_j}
J_{a_1a_2a_3a_4}(x)\,.
\end{equation*}
Here the sum over momentum $p$ runs from the IR cutoff $\Lambda_{\text{IR}} \sim {1 \over R}$ to the UV
cutoff $\Lambda $. We also defined $F(p) \equiv {1 \over |p|^\gamma}$ as in our discussion in subsection \ref{sec:NonCompactSoln}.
Rewriting the original filter parameter $A$ in terms of the fixed $A_{\infty} = A ({2 \pi \over R})^{\gamma}$, as in subsection \ref{sec:NonCompactSoln}, the interaction lagrangian becomes
\begin{equation}
{\mathcal L}_{\text{int}} = { A_\infty^4 \over L R } \sum_{a_i, p_i} \prod_{i=1}^4 [F(p_i) \chi^{a_i}_{p_i}] \int dx e^{- i x \sum_{j=1}^4 p_j}\,
J_{a_1a_2a_3a_4}(x)\,.
\end{equation}
Lastly, we take the non-compact limit $R \to \infty$. Sums over momentum $\sum_p$ are proportional to $R \int dp$. To keep a finite kinetic term, we need to rescale the physical fermions as $\chi^a_p \sim {\chi^a(p) \over \sqrt R}$, leading to a non-compact lagrangian
\begin{equation*}
\int dp \chi^{a}(-p)(\partial_{\tau }+ p )\chi^{a}(p) +{ A_\infty^4 R \over L } \sum_{a_i} \int \prod_{i=1}^4 [dp_i F(p_i) \chi^a(p_i)] \int dx e^{- i x \sum_{j=1}^4 p_j}\,J_{a_1a_2a_3a_4}(x)\,.
\end{equation*}
Since the non-compact version of the impurity average \eqref{impurity}
\begin{equation}\label{continuumdisorder}
E[J_{a_1a_2a_3a_4}(x)\,J_{b_1b_2b_3b_4}(y)] = \frac{3! J^2}{N^3} \frac{R}{L} \delta (x-y)\,,
\end{equation}
includes an additional $R \over L$ factor from the continuum limit of the discrete Kronecker delta $\delta_{ij}$, we can write the continuum version of the SD equation as
\begin{eqnarray} \nonumber
\Sigma^{a_1a^\prime_1}(p_1,p^\prime_1,\tau) &=&\delta^{a_1a^\prime_1}{ J^2 A_\infty^8 R^3\over L^3} F(p_1) F(p^\prime_1) \sum_{a_i} \prod_{i=2}^4[\int dp_idp^\prime_i F(p_i) F(p^\prime_i) G^{a_ia_i}(p_i,p^\prime_i,\tau) ] \cdot \\
&& \hspace{10mm}\cdot \int dx\, e^{-i x \sum_i(p_i + p'_i)}\,,
\end{eqnarray}
where the last term will implement conservation of momentum $\delta(\sum_i(p_i + p^\prime_i))$. It is now clear that to keep a non-trivial interaction in the non-compact limit we must work with
\begin{equation}
{J^2 R^3 \over L^3} = {J^2 \over \Lambda_0^3} \ \ \ \mbox{ fixed}
\end{equation}
Hence we reproduce our previous claim \eqref{nonc} provided we take the disorder average in the continuum non-compact limit to be as in eq(\ref{continuumdisorder}).
\section{A probe model}
In the previous class of models, we were interpreting the high momentum modes of the physical fermions $\chi^a$ as living outside of a black hole in some putative bulk, while the strongly interacting low momentum modes of $\chi^a$, i.e. the $\eta^a$ degrees of freedom, built the putative black hole. In this section, we explore a second class of models with a similar holographic motivation.
Consider models consisting of two types of fermions : $\eta^a,\ a=1..N$ interacting via an SYK model or its 1+1 extension described in previous sections and a single degree of freedom (or maybe a few) $\rho$ acting as a probe. We envision a situation in which the $\eta^a$ fermions describe the degrees of freedom of a black hole (in some approximate sense), while the $\rho$'s encode the analogue of single trace operators in the AdS/CFT correspondence
More specifically, we will take $\rho$ to be a 1+1 system (but we can take them in any dimension), so that they become $\rho^i$ where $i$ is the spatial index. It can either be fields that go to a free fermion in the continuum, or we can maybe take them to be some generalized free fields, in which case we can hope to find a field of arbitrary dimension.
\subsection{A 2D probe model with SYK kernel}
In this subsection we introduce, and solve in some regime, one such model. Let $\eta^{a}$, $a=1,..N$ be the $0+1$d SYK Majorana fermions and $\rho^{i}$ be Majorana fermions on a periodic lattice of length $L$ ($i=0,..L-1$ is the spatial index and $\rho^0\equiv \rho^L$). Following previous sections, we will find it useful to have a filter for the $\rho$ fermions.
We take our model to have the action $S = S_{\eta} + S_{\rho} + S_{\rho,\eta}$ where
\begin{eqnarray}
S_{\eta} &=&\int d\tau [ {1 \over 2} \eta^a \partial_\tau \eta^a + \sum_{a,b,c,d} J_{abcd}\eta^{a}\eta^{b}\eta^{c} \eta^{d} ]\\
S_{\rho} &=& \sum_i \int d\tau \{ {1 \over 2} \rho^i \partial_\tau \rho^i - i \alpha [\rho^i , \rho^{i+1} ] \} = \sum_k \int d\tau \{ {1 \over 2} \rho_{-k} (\partial_\tau +E_k) \rho_k \} \\
S_{\rho ,\eta} &=& \int d\tau \sum_{i_{1}..i_{k}}\hat{J}_{i_{1}..i_{k}a_1..a_m}\bar \rho^{i_{1}}\bar \rho^{i_{2}}..\bar \rho^{i_{k}}\eta^{a_{1}}..\eta^{a_{m}} \\
&=& {A^k \over L^{k \over 2}} \int d\tau \sum_{i_1,..i_k, k_1,..k_k,a_1..a_m} [\prod_{j=1}^k F(k_j) \rho_{k_j} ]
e^{-{2\pi i \over L} \sum_{j=1}^k i_j k_j } \ \eta^{a_{1}}..\eta^{a_{m}} \hat J_{i_{1}..i_{k}a_1..a_m}
\end{eqnarray}
$\rho_k$ stands for the Fourier transform of the $\rho^i$ lattice fermions, whereas $\bar \rho_i$ fermions are the corresponding low pass fermions $\bar \rho_i \equiv {1 \over \sqrt L} \sum_k F(k) \rho_k e^{-2 \pi i k \over L } $ interacting with the SYK fermions $\eta^a$. The $\hat{J}_{i_{1}i_{k}a_1..a_m}$ are taken to be random variables with impurity average
\[
E[ \hat{J}_{i_{1}..i_{k}a_1..a_m}\hat{J}_{i_{1}..i_{k}a_1..a_m}] =\frac{k! m!\hat{J}^{2}}{N^{m}}
\]
and $E[ J_{abcd} J_{abcd}] = {m! J^2 \over N^3}$ as for SYK fermions.
The $N$ scaling was chosen so that the $\rho^i$ fermions behave like probes, i.e. their propagator will be corrected by the interactions whereas the $\eta^a$ propagators will remain unmodified at leading order. More precisely, there are two leading 1-loop diagrams contributing to the 1PI self-energy
of the $\eta^a$ propagator, as indicated in figure \ref{diag1}. Diagram (A) scales like $E[J^{2}_{..}]N^{3}\sim{\cal O}(N^{0})$ as in SYK. Diagram (B) is subleading since it scales like $E[\hat{J}^{2}_{..}]N^{m-1}\sim{\cal O}(N^{-1})$. Hence, the $\eta^a$ propagators which we denote by $G(\tau)$ are indeed unmodified at leading order.
\begin{figure}
\begin{subfigure}{.5\textwidth}
\vspace{-50pt}
\includegraphics[scale=0.5]{fig1.pdf}\\
\vspace{-300pt}
\end{subfigure}
\begin{subfigure}{.5\textwidth}
\vspace{-50pt}
\includegraphics[scale=0.45]{fig2.pdf}\\
\vspace{-250pt}
\end{subfigure}
\caption{Diagrams contributing to the $\eta$ propagator. }
\label{diag1}
\end{figure}
Next we will solve for the $\rho$ propagator which we will denote by $\cal G$. The leading 1-loop diagram (denoted by $\cal S$) contributing to its 1PI self-energy is given in figure \ref{diag2}.
\begin{figure}\label{diag2}
\centering
\includegraphics[scale=0.45]{fig3.pdf}
\vspace{-250pt}
\caption{Diagram contributing to the $\rho$ propagator. }
\label{diag2}
\end{figure}
Since this diagram scales like $\hat{J}^{2}_{..}N^{m}\sim{\cal O}(N^{0})$, it gives rise to a non-trivial correction. In fact we find the SD equations for $\rho$ to be
\begin{eqnarray} \nonumber
{\cal S}_{k_1,k_1'}(\tau) &=& {A^{2k} \hat J^2 \over L^k} G(\tau) ^m F(k_1) F(k_1') \times \\
&& \sum_{i_1,..i_k,k_2..k_k,k_2'..k_k'} \prod_{j=2}^{k} [F(k_j)F(k_j'){\cal G}_{k_jk_j'}(\tau)] e^{-{2\pi i \over L} \sum_{j=1}^k i_j (k_j+k'_j) } \\
{\cal G}_{k,k'}^{-1}(\omega) &=& {{\cal G}^{(0)}}_{k,k'}^{-1}(\omega) - {\cal S}_{k,k'}(\omega)
\end{eqnarray}
It is clear that the SD equations force self energy ${\cal}S$ and hence the propagator ${\cal G}$ to be diagonal in momentum space. We will assume that $F(k)$ is an even function. Let us define
\begin{equation}
{\cal S}_{k,k'}(\tau) \equiv \delta_{k+k'=0} {\cal S}_k(\tau) \hspace{10mm} {\cal G}_{k,k'}(\tau) \equiv \delta_{k+k'=0} {\cal G}_k(\tau)
\end{equation}
The SD equations then become
\begin{eqnarray}
{\cal G}_{k}^{-1}(\omega) &=& {{\cal G}^{(0)}}_{k}^{-1}(\omega) - {\cal S}_k(\omega) \\
{\cal S}_k(\tau) &=& A^{2k} \hat J^2 F(k)^2 G(\tau)^m [\sum_{k'} F(k') {\cal G}_{k'}(\tau)]^{k-1}
\end{eqnarray}
We finally write a collective SD equation by defining
\begin{equation}
{\cal S}_{k}(\tau) \equiv F(k)^2 \tilde {\cal S}(\tau) \hspace{10mm} \tilde {\cal G}(\tau) \equiv \sum_k F(k)^{2} {\cal G}_k(\tau)
\end{equation}
We have dropped the subscript $k$ on $\tilde {\cal S}$ because the SD equations force it to be independent of $k$. The final SD equations become
\begin{eqnarray}
\tilde {\cal G}(\omega) &=& \sum_k {1 \over F(k)^{-2} {{\cal G}^{(0)}}_{k}^{-1}(\omega) - \tilde {\cal S}(\omega) }\\
\tilde {\cal S}(\tau) &=& A^{2k} \hat J^2 G(\tau)^m \tilde {\cal G}(\tau)^{k-1} \\
{\cal G}_{k}(\omega) &=& {1 \over {{\cal G}^{(0)}}_{k}^{-1}(\omega) - F^2(k) \tilde {\cal S}(\omega) }
\end{eqnarray}
These equations can be solved using the available SYK solutions in the conformal window $G(\tau) \sim {1 \over \tau^{1 \over 2}}$ (see \cite{Maldacena:2016hyu}), and the same strategy we followed in section 3. Rather than presenting the solution for arbitrary $k$ and $m$, we focus on the $k=m=2$ case.
The scaling of the collective probe propagator and self-energy is
\begin{eqnarray}
{\tilde{\cal G}}(\tau)\sim |\tau|^{-\Delta_1},\ \ {\tilde{\cal G}}\sim |\omega|^{\Delta_1-1}\\
{\tilde{\cal S}}(\tau)\sim |\tau|^{-\Delta_2},\ \ {\tilde{\cal S}}\sim |\omega|^{\Delta_2-1}\\
\end{eqnarray}
with
\begin{equation}
\Delta_2-1=\frac{1+2\gamma}{1+4\gamma},\ \ \ \Delta_1-1=- \frac{2\gamma}{1+4\gamma}\,.
\end{equation}
Notice these are consistent with the assumption $|\omega|\ll {\tilde{\cal S}}$ holding in the deep scaling IR regime.
\subsection*{Acknowledgements}
We would like to thank S. Ross for collaboration at early stages of this project. The work of MB is supported by an ISF center of excellence grant (1989/14). PN gratefully acknowledges the support from International Centre for Theoretical Sciences (ICTS), India. The work of MR is supported by a Discovery grant from NSERC. The work of JS is supported by the Science and Technology Facilities Council (STFC) [grant number ST/L000458/1].
\begin{appendix}
\section{Equations via the replica method}
\label{replica}
We now rederive the Schwinger-Dyson equations for our model using
replica methods. In this framework those equations represent the saddle point approximation to the effective action of the model.
In order to compute $S^{(n)}$, the n'th Renyi entropy, we construct $n$ replicas
of our model, labelled by $\alpha=1,\dots n$ The Euclidean action for the replicated theory is
\begin{eqnarray}
S^{(n)}&=&\int d\tau\sum_{\alpha} \left[ \frac{1}{2}\sum_{i,a}\chi_{\alpha}^{i,a}(\tau)\partial_{\tau}\chi_{\alpha}^{i,a}(\tau) - i\alpha\sum_{i,a}[\chi_{\alpha}^{i,a}(\tau),\chi_{\alpha}^{i+1,a}(\tau)] \right. \nonumber \\
&&~~~~~~~~~~~~~~~~~~~~~~~~~~\left. + \sum_{i,abcd}J_{i,abcd} \, \eta_{\alpha}^{i,a}(\tau)\eta_{\alpha}^{i,b}(\tau)\eta_{\alpha}^{i,c}(\tau)\eta_{\alpha}^{i,d}(\tau)\right]
\end{eqnarray}
We now perform the disorder average, recalling that we have independent disorder variables at each site, we get \begin{eqnarray}\nonumber
S^{(n)}&=&\int d\tau\sum_{\alpha,a,i}\left[\frac{i}{2} \chi_{\alpha}^{i,a}(\tau)\partial_{\tau}\chi_{\alpha}^{i,a}(\tau)-i\alpha [\chi_{\alpha}^{i,a}(\tau),\chi_{\alpha}^{i+1,a}(\tau)]\right]\\
&&~~~~~~~~~~ - \frac{4 J^{2}L^3}{N^{3}}\sum_{\alpha,\beta}\int d\tau\int d\tau' \,\sum_i\left[\sum_{a}\eta_{\alpha}^{i,a}(\tau)\eta_{\beta}^{i,a}(\tau')\right]^{4}
\label{ReplicaStart}
\end{eqnarray}
where we use the convention $E(J_{....},J_{....}) \sim \frac{J^2 L^3}{3! N^3}$ for each randomly distributed variable.
One can now perform a Hubbard-Stratonovich transformation by introducing the real decoupling field $Q^{i}_{\alpha\beta}(\tau,\tau')$, symmetric in replica indices, for each site
\begin{eqnarray}
S^{(n)}&=&\int d\tau\sum_{\alpha,a,i}\left[\frac{1}{2} \chi_{\alpha}^{i,a}(\tau)\partial_{\tau}\chi_{\alpha}^{i,a}(\tau)-i\alpha [\chi_{\alpha}^{i,a}(\tau),\chi_{\alpha}^{i+1,a}(\tau)]\right]\\
\nonumber
& & \ \ +\sum_{\alpha,\beta}\int d\tau \int d\tau'\sum_i\left[ \frac{N}{4 L J^2} Q^i_{\alpha\beta}(\tau,\tau')^{2}- \frac{2 L }{ N}Q^i_{\alpha\beta}(\tau,\tau') \left( \sum_{a}\eta_{\alpha}^{i,a}\eta_{\beta}^{i,a}\right)^{2}\right]
\end{eqnarray}
Finally we introduce another set of decoupling fields $P^i_{\alpha \beta}$, also real and symmetric in replica indices, to obtain
\begin{eqnarray}
S^{(n)}&=&\int d\tau\sum_{\alpha,a,i}\left[\frac{1}{2} \chi_{\alpha}^{i,a}(\tau)\partial_{\tau}\chi_{\alpha}^{i,a}(\tau)-i\alpha [\chi_{\alpha}^{i,a}(\tau),\chi_{\alpha}^{i+1,a}(\tau)]\right] \nonumber\\
\nonumber
& & \ \ +\frac{N}{L} \sum_{\alpha,\beta}\int d\tau \int d\tau' \sum_i\left[ \frac{Q^i_{\alpha\beta}(\tau,\tau')^{2}}{4 J^2} +\frac{Q^i_{\alpha\beta}(\tau,\tau') P^i_{\alpha\beta}(\tau,\tau')^2}{2} \right.\\
&& \left.\hspace{40mm}- Q^i_{\alpha\beta}(\tau,\tau') P^i_{\alpha \beta}(\tau',\tau) \left(\frac{L}{N} \sum_{a}\eta_{\alpha}^{i,a}\eta_{\beta}^{i,a} \right) \right]
\end{eqnarray}
This is simply the sum over the replicated action obtained for each site separately, for a direct comparison see for example the discussion in \cite{Sachdev:2015efa,Fu:2016yrv}. Note that the saddle point equations set
\begin{eqnarray}
P^i_{\alpha \beta}(\tau,\tau')&=&\, \frac{L}{N }\sum_a \langle \eta^i_\alpha (\tau)\eta^i_\beta(\tau') \rangle \nonumber \\
Q^{i}_{\alpha \beta}(\tau,\tau')&=& {J^2} P^i_{\alpha,\beta}(\tau,\tau')^2
\end{eqnarray}
We now assume that replica symmetry is not broken, so that $P^i_{\alpha\beta}=P^i \delta_{\alpha\beta}$ and $Q^i_{\alpha\beta}=Q^i
\delta_{\alpha\beta}$. Similarly we assume that upon disorder averaging the $SO(N)$ symmetry is restored. Therefore we can drop the fermion $SO(N)$ index and refer to a single fermion. We further can go to a single replica, obtaining the action
\begin{eqnarray}
S &=&N \int d\tau\sum_{i}\left[\frac{1}{2} \chi^{i}(\tau)
\partial_{\tau}\chi^{i}(\tau)-i\alpha
[\chi^{i}(\tau),\chi^{i+1}(\tau)]\right]
\\
\nonumber
& & \ \ +\frac{N}{L} \int d\tau \int d\tau' \, \sum_i\left[
\frac{Q^i(\tau,\tau')^{2}}{4 J^2} +\frac{Q^i(\tau,\tau') P^i(\tau,\tau')^2}{2}
- L \, Q^i(\tau,\tau') P^i(\tau',\tau) \eta^{i}(\tau)\eta^{i}(\tau')
\right]
\end{eqnarray}
We are now ready to integrate out the fermions $\chi^{i}$. Define the mass shifts $\tilde \Sigma^i= Q^i P^i$, making the same self-consistent assumption as above, namely that the mass shifts are on-site only (i.e. they are all equal in momentum space), so that the saddle point solution satisfies $\tilde \Sigma_k=\tilde \Sigma$, i,.e the same value for each Fourier mode $k$. With this assumption we can now Fourier transform the action and integrate out the fermions:
\begin{eqnarray}
S&=& N \, \sum_{\omega,k} \log \text{Pf} \left[ \partial_{\tau} - E_k - \tilde \Sigma(\tau,\tau') F(k)^2\right] \\
\nonumber
& & ~~~~ +{N} \sum_{k}\int d\tau \int d\tau' \, \left[ \frac{Q_k(\tau,\tau') Q_{-k}(\tau,\tau')}{4 J^2} + \frac{P_{k}(\tau,\tau') \, \tilde \Sigma (\tau,\tau') }{2} \right]
\end{eqnarray}
where we have used that
in momentum space $\eta_k=F(k) \chi_k$. We can further use the identity $Q^i= J^2 (P^{i})^2$, and denote $\tilde G^i=\frac{P^i}{L}$ to obtain
\begin{eqnarray}
S&=& N \, \sum_k \log \text{Pf} \left[ \partial_\tau - E_k - \tilde \Sigma(\tau,\tau') F(k)^2\right] \\
\nonumber
& & ~~~~~~~~ +{N}\sum_{k}\int d\tau \int d\tau' \, \left[ \frac{J^2 (\tilde G(\tau,\tau')^2)_k (\tilde G(\tau,\tau')^2)_{-k}}{4 L^4 } + \frac{ \tilde G_{k}(\tau,\tau') \, \tilde \Sigma (\tau,\tau')}{L}\right]
\end{eqnarray}
where $(\tilde G(\tau,\tau')^2)_k$ denotes the Fourier transform of $\tilde G^i(\tau,\tau')^2$.
We can now obtain the saddle point equations following from the action. Varying with respect to $\tilde \Sigma$ gives (in frequency space)
\begin{eqnarray}
\tilde G_k(\omega) = {1 \over -i \omega -E_k -\tilde \Sigma(\omega) F(k)^2}
\end{eqnarray}
whereas varying with respect to $G(\tau)=\sum_k\tilde G_k(\tau)$ gives
\begin{equation}
\tilde \Sigma(\tau)= \frac{ J^2}{L^3} \tilde G(\tau)^3 \end{equation}
To exhibit the dependence on the normalization $A$, we redefine $F(k)\rightarrow A F(k)$ and rescale $\tilde{\Sigma}\rightarrow A^{-2} \tilde \Sigma$ and $\tilde G\rightarrow A^{2} \tilde G$. This yields the same equations as those derived in subsections \ref{Rcontinuum} and \ref{Raction}, where the normalization constant $A$ is shown explicitly.
\section{Gaussian low pass filter}
\label{sgaussian}
In this Appendix, we consider the gaussian low pass filter. Although we will not be able to solve the SD equations exactly (even in the deep IR), we will determine the scaling of the 2 point function in frequency space $()\omega)$. Recall the gaussian low pass filter is defined by the function
\begin{equation}
\label{gaussian}
F(k)=e^{-\pi^2 k^2\over R^2 D^2}\,,
\end{equation}
where $D = {\hat D \over R}$ is the physical scale of the filter as can be seen by taking the non-compact limit $R\to \infty$ keeping the physical momentum $p\sim \frac{k}{R}$ fixed.
If we were to consider an step function filter, one would expect to obtain similar physics to the SYK model for the modes passing the filter, while decoupling those being filtered out. What we show below is that the gaussian filter model provides logarithmic corrections to the SYK scaling behaviour for very long times.
To solve this model in the same regime as we discussed the solution for the power law filter, we need to consider the integral \eqref{tildeGexp} with $F(k)$ given by \eqref{gaussian}. This looks like (dropping the $-i\omega$ term compared to $\Sigma(\omega)$ in the deep IR)
\begin{equation}
\tilde G(\omega) = {2 i} \mbox{ Im} \int_0^\infty {d \tilde k \over e^{\pi^2 \tilde k^2} \tilde k - {\tilde \Sigma(\omega) \over R D } } = 2i\,\frac{|{\tilde\Sigma(\omega)| }}{RD}\,\int_0^\infty {d \tilde k \over \tilde k^2\,e^{2\pi^2 \tilde k^2} + \frac{|{\tilde\Sigma(\omega)}|^2}{R^2\,D^2} }
\end{equation}
where we used that $\tilde \Sigma(\omega)$ is purely imaginary. Although we could not solve the above integral exactly, we can estimate it for large values of
$\frac{|{\tilde\Sigma(\omega)|}}{RD}$. In this regime, the integral cuts off when the two terms become comparable. This occurs around $k \sim \sqrt{\log
\frac{|\tilde\Sigma(\omega)|}{RD}}$. Thus we get the following estimate
\begin{equation}\label{gaussianSD2}
|\tilde{G}(\omega)| \sim \frac{\sqrt{\log \frac{|\tilde\Sigma(\omega)|}{RD}}}{\frac{|\tilde\Sigma(\omega)|}{RD}}\,,
\end{equation}
The other SD equation \eqref{SDEqn1} becomes
\begin{equation}\label{gaussianSD1}
{\Sigma(\tau) \over R D} = {A^8 J^2 \over R D L^3} G(\tau)^3
\end{equation}
Let us now assume a simple ansatz $\tilde \Sigma(\omega) \sim (\omega)^\alpha |\log \omega|^\beta$. Since we will work in small $\omega$ and large time $t$ we can use the following approximation when performing the Fourier transform
\begin{eqnarray}\nonumber
\int d\omega e^{i \omega t} (\omega)^\alpha |\log \omega|^\beta &\sim& {(\log t)^\beta \over t^{1 +\alpha }} \left[\int_{-\infty}^\infty d\hat \omega e^{i \hat \omega} \hat \omega^\alpha \left( 1 - {\beta |\log \hat \omega | \over(\log t)^\beta } + {\cal O}(\log t)^{-2\beta} \right) \right] \\
&=& {(\log t)^\beta \over t^{1 +\alpha }} \left( 1 + + {\cal O}(\log t)^{-\beta} \right)
\end{eqnarray}
The situation is thus very similar to the original SYK model, except for the extra $\log$ pieces. Defining ${\cal J}^2 \equiv {A^8 J^2 \over R D L^3}$, one can check that the SD equations \eqref{gaussianSD1} and \eqref{gaussianSD2} are solved by
\begin{equation}
|\tilde{G}(\omega)|\sim {\cal J}^{-1\over 2}\,\frac{|\log (\omega/{\cal J}) |^{\frac{1}{8}}}{|\omega/{\cal J}|^{\frac{1}{2}}}\,, \quad\quad |\tilde{\Sigma}(\omega)|\sim {\cal J}^{\frac{1}{2}}|\omega/{\cal J}|^{\frac{1}{2}}|\log (\omega/{\cal J})|^{\frac{3}{8}}\,.
\end{equation}
or in euclidean time $\tau$
\begin{equation}
|\tilde{G}(\tau)| \sim {\cal J}^{-1/2}\,\frac{\log |{\cal J}\tau|^{\frac{1}{8}}}{|{\cal J}\tau|^{1/2}}\,,\quad \quad |\tilde{\Sigma}(\tau)| \sim {\cal J}^{1/2}\,\frac{ \log |{\cal J}\tau|^{\frac{3}{8}}}{|{\cal J}\tau |^{\frac{3}{2}}}\,.
\end{equation}
We see that the resulting theory has $\log (\omega)$ enhancement compared to SYK in the free energy.
\end{appendix}
\bibliographystyle{JHEP}
| {
"redpajama_set_name": "RedPajamaArXiv"
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Courses 2018-2019
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AR323: Destroying Culture: Iconoclasm from Antiquity to Today
Students in this humanities theme lab work together to create a digital map and timeline that traces the history of iconoclasm and cultural destruction from antiquity to the present. They explore the religious and political contexts linked to the production, protection, and destruction of material culture by examining specific case studies over a wide geographic and historic span. Students are encouraged to question the forces behind different instances of destruction as well as the meaning they hold for us today. Assessment consists of reading responses, short writing assignments, and a group project.
AR425: Intimate Things
Four credit hours. Harkett
Focusing on eighteenth- and nineteenth-century Europe and America, this seminar explores relationships linking objects, intimate experience and memory. We will ask: How did everyday practices of keeping, wearing, touching, and viewing things shape personal identities, connect people together, and enact stories about the past and the present? How, for example, did miniature paintings mediate relationships between lovers and among friends and family? How did albums and private museums collect and represent the past? How did death masks and hair jewelry help people come to terms with loss? Students will address these questions by reading relevant texts and developing a semester-long research project. Presence of the Past theme course.
AY298: Cultural Accounting of Business and Work
Four credit hours. Menair
An intellectual opportunity to examine business and work as part of culture. We focus on the motives and methods of business with readings from Veblen, Weber, Marx, and Graeber as well as contemporary business textbooks. Students will reflect on people's lived experiences of markets and work, the culture of modern individualism, and the precarity of work in the 21st century.
CL138: Heroes of the World
Four credit hours. O'Neill
The Greeks, the Romans, the Irish: peoples around the globe have produced their own unique heroes appropriate to the needs and desires of their particular cultures. Nevertheless, these heroes share a variety of traits and experiences. We will examine the similarities and differences of the heroes of Ireland, Greece, Rome, and other cultures and explore why we still crave heroes and how that craving has shaped our present.
CL398: Athenian Democracy as Reality and Idea
Four credit hours. Welser
The rise of democracy in ancient Athens had radical consequences not only for Athens itself, but for the entire Greek world and the whole subsequent course of human history. In this seminar, we will explore what democracy meant to the Athenians and how they sought to realize its ideals. We will examine some of the varied presentations of Athenian democracy in later political thought and evaluate the extent to which democracy can be held responsible for the Athenians' triumphs and failures. In so doing, we will seek to sharpen our modern understanding of democracy and assess conventional claims concerning democracy's strengths and weaknesses.
EA263: Buddhism across East Asia
Four credit hours. Orzech
Introduces students to the histories, texts, material culture, and practices of Buddhism in East Asian cultural settings. The spring 2019 offering will focus on Chan/Son/Zen traditions in China, Korea, and Japan. Is there really such a thing as Zen? To answer this question we will do intensive reading of key primary texts (such as the Platform Sutra) and important historical and critical secondary works.
EA355: Aging and Public Policy in East Asia
Four credit hours. Zhang
Students will combine ethnographic studies with demographic data to compare and analyze how East Asian countries cope with challenges of rapid population aging and to explore public policy shifts regarding state and private responsibility for the wellbeing of the elderly. Utilizing interactive data from the United Nation Population Division to compare and project aging trends including fertility rates, life expectancy, median age, and dependence ratio in East Asia. Students will also make two field trips to local eldercare facilities to gain comparative insight on the challenges of aging and eldercare provision in Maine, one of the grayest states in the United States.
EN120: Inventing Nature in New England
Four credit hours. Gibson
This humanities lab course will combine field trips around Maine with work in the Colby Museum and the rare book room. We'll read some of the classics of New England nature writing, make our own "field journals" on Mayflower Hill, and think about how our ideas of and relationships to the natural world are shaped by our knowledge, our technology, and our historical situation. We'll read prose and poetry, from Emerson to Maine writer Sarah Orne Jewett's short stories, to modern poetry broadsides in our library's collection. When spring finally comes we'll make a field trip to the Maine coast to see for ourselves the world described in Celia Thaxter's The Isle of Shoals. We will keep journals and write and revise both research essays and journalistic essays.
EN264: Comparative Studies: Emily Dickinson and English poetry
Four credit hours. Sagaser
This course compares poems by 19th c. American poet Emily Dickinson with poems by writers she admired and read intensely, from Shakespeare and Milton to Keats, the Brontës and E. B. Browning. Students will gain analytical skills and creative strategies for engaging with poetry as they discover poetry's power to bring thoughts and voices from faraway centuries and continents into the minds and memories of newly present readers and thinkers. They will explore some additional contexts for for Dickinson's reading and writing, including her education, material conditions, and the Civil War.
EN298: Fake News and the Rise of the Novel
Four credit hours. Reed
Examines the early novel as a site of conjecture and debate about fake news, truth claims, and credibility. We will read four novels spanning almost a century, and let those novels teach us how to distinguish truth from lies, and where such distinctions are most needed.
EN398: Life in Times of Extinction
Four credit hours. Walker
We are living through an event known as the Sixth Extinction. Human impacts on the environment are causing the largest extinction in the last 65 million years. At the same time, humans are discovering and celebrating life in all its biodiversity. Photographs, films, ethological narratives, and biological databases attesting to human interest in newly discovered, and newly endangered, species proliferate. To address this incongruity, this humanities lab will explore a recent strain of scholarship in the environmental humanities that asks how extinction comes to matter to us culturally, ethically, and evolutionarily.
EN493: Seminar: Poetry and Cognition
Long before psychology and neuroscience were fields of study, poets experimented with language and cognition, discovering ways to engage attention and amplify memory. It makes sense therefore to ask what insights poetry and cognitive science might offer each other now. We'll invite to our table poetry from the Renaissance to the present along with readings from cognitive psychology, neuroscience, linguistics and and philosophy of mind. In connection with this year's Humanities Center theme, we will focus in particular on poetry as a non-electronic yet mighty (because cognition-savvy) technology for bringing together minds and voices not living in the same shares of spacetime.
ES276: Global Change Ecology
Four credit hours. Bruesewitz
Provides an interdisciplinary introduction to the principles of climate, ecosystems, and biogeochemistry needed to understand human impacts on the natural environment. Students will study the impacts of climate warming, our changing atmosphere, land-use change, altered hydrologic and nutrient cycles, and other global changes. We will examine key elements of global ecosystem function and investigate how human activities have altered global ecosystems since the Industrial Revolution. We will critically assess scientific evidence for anthropogenic changes, and consider both impacts and solutions to the challenges of global changes. Relies heavily on reading of primary scientific literature and group participation and discussion. Prerequisite: Environmental Studies 118 and one college-level science course.
FR361: Creolization, Culture, and Society in the Indian Ocean Islands
Four credit hours. Mauguiere
Explores issues of race, gender, identity, diversity, cultural contact, and conflict in Indian Ocean island cultures and literature written in French through selected writings from Mauritius, Madagascar, Reunion, the Seychelles, and the Comoros. We will examine the complex social, cultural and historical context of the region with an interdisciplinary perspective. Topics include slavery, "marronage", cultural hybridity, "métissage," "coolitude," and the development of colonial and postcolonial identities and subjectivities. Students will develop their presentation and writing skills through the production of critical essays and research projects.
GE262:Earth's Climate: Past, Present, and Future
Four credit hours. Koffman
Those who study Earth's climate history see the presence of the past all around us. In this course we will examine the physical, chemical, and biological interactions that define Earth's climate. From this foundation, we will explore the mechanisms that shape environmental evolution across a range of timescales, including the role of humans. By collecting and analyzing a lake sediment core, we will develop our own record of past climate change in Maine (last year's core spanned the past 12,000 years!).
GO238: Politics of War Crime Tribunals
Four credit hours. Rodman
Examines the politics of establishing tribunals to hold individuals criminally accountable for genocide and other atrocity crimes, from the Nuremberg and Tokyo trials after World War II through the International Criminal Court. Central questions involve the nature of post-conflict justice, the degree to which international legal bodies are insulated from or influenced by politics, and the impact of prosecution on transitions from war and dictatorship to peace and democracy. Academic and legal analysis combined with simulated court proceedings. Areas of application include South Africa's Truth and Reconciliation Commission, the Milosevic trial, the Pinochet extradition hearing, and issues surrounding Guantanamo and Abu Ghraib.
GS/AY316: Religion and Social Change in Contemporary Africa
Four credit hours. Halvorson
This seminar will build students' awareness of the religious diversity of contemporary African societies using selected studies from Madagascar, Tanzania, Mali, Mozambique, and other sites. Students will learn to identify the relationship of African religions with diverse, transforming views on biomedicine and healing, urbanization, gender relations, political protest, development and humanitarianism, and the colonial legacy. Emphasis will be placed on theoretical approaches that analyze the role of African religions in dynamic processes of political, economic, and cultural transformation.
GS/AY455: Intervention: The Ethics and Politics of Humanitarianism
What does it mean to seek to relieve suffering on a global scale? How could such an impulse be political? Students will have the opportunity to critically analyze and understand humanitarian action in global perspective. We will investigate the principles and history of humanitarianism and consider their application on a global scale by a range of humanitarian actors, such as NGOs and states. We will investigate the politics and ethics of philanthropy, volunteerism, and humanitarian-military intervention and will discuss and debate the intersections and divergences between humanitarianism, human rights, and development.
HI232: American Women's History, 1870 to the Present
Four credit hours. Leonard
An exploration of critical topics in the history of women in America from Reconstruction to the present, including the struggle for suffrage, black women in the aftermath of slavery, women and the labor movement, the impact on women of two world wars, birth control and reproductive freedom, women's liberation, the feminization of poverty, and the backlash against feminism.
HI255: Slavery, Diaspora, and Revolution: Global Histories of Southeast Asia
Four credit hours. VanderMeer
Southeast Asia is one of the most dynamic economic and cultural regions in the world and central to President Obama's pivot to Asia. As the geographical name implies, the region is seemingly locked between two powerful neighbors, India and China, which means that it has received relatively little attention from scholars and contemporary observers alike. This is a shame, as Southeast Asia – consisting of the modern states of Brunei, Cambodia, Indonesia, Laos, Malaysia, Myanmar, Philippines, Singapore, Thailand, Timor Leste, and Vietnam – has been a crossroads for people, cultures, languages, flora, and fauna for millennia, making it one of the most diverse, surprising, and fascinating regions in the world. For instance, how do we explain that the world's largest Buddhist temple can be found in the world's largest Islamic state?
HI297: New Perspectives on the American Revolution
Four credit hours. Reardon
Patriotic narratives associated with the birth of the republic are deeply engrained within the American political identity. Recently, the hit Broadway musical Hamilton brought the production's namesake and the familiar cast of Founding Fathers back to the center stage of American pop culture. The contributions of political elites of course merit popular and scholarly attention, but should we also consider the experiences, perspectives, and contributions of those outside centers of formal political power? This course will ask that students examine the ways African-Americans, Native Americans, women, loyalists, common farmers, and urban artisans experienced and contributed to the Revolutionary Era.
HI320: Seminar: Joan of Arc: History, Legend and Film
Four credit hours. Taylor
Yes, here we are still talking about her. Joan has inspired tens of thousands of works in every genre since her short lifetime (1412-31). As Marina Warner states: "A story lives in relation to its tellers and receivers, it continues because people want to hear it again, and it changes according to their needs and tastes." The most important questions surrounding Joan concern the "whys." Why did experienced nobles and mercenaries follow a teenaged peasant girl to war? Why were the English so afraid of her that they preordained her execution? Why did other clerics in English-controlled areas suffer exile, imprisonment and threats of death rather than take part in her trial? Why did clothing and gender play such an important role in her trial? Why, in 1920, after 500 years, was she canonized? Was Joan a saint or a heretic? Effective military leader/strategist or a figurehead? Woman, Amazon, or androgyne? Patriot, revolutionary or anti-imperialist? Why has Joan become a figure for all times?
HI377: Crossroads of the World: The History of Modern Southeast Asia
This class explores the fascinating multi-cultural history of Southeast Asia – "Crossroads of the World" – from the 18th century till the present. While the term "Southeast Asia" is fairly recent (post-1945), the region has been a thoroughfare for people, flora, fauna, and commodities between East Asia, South Asia, Africa, and the Pacific for millennia. Subsequently, it is one of the most ethnically and culturally diverse regions in the world today. It is home to over 600 million people, has two of the largest cities in the world (Manila and Jakarta), the 2nd largest port in the world (Singapore), the greatest linguistic diversity, the most countries with a majority Buddhist population, the country with the largest Muslim population, the country with the third largest Catholic population, the largest community of overseas Chinese, and does not only have the fourth most populous country, but as many as four countries with larger populations than the United Kingdom. The geography and natural environment of Southeast Asia – such as the contrasts between the rugged uplands and fertile lowlands, the great river systems of the Indo-Chinese peninsula, the most (active!) volcanoes as well as 25,000 plus islands of the Malay Archipelago – has profoundly shaped its history. Taken together, Southeast Asia was cosmopolitan before there even was such a thing.
HI398: South African Women's Memoir
Four credit hours. Duff
Critical thinking about the entanglement of the past and the present, focusing on a selection of memoirs written by South African women during the segregationist, apartheid, and post-apartheid periods. Memoir was a powerful tool for these women, allowing them to set the (historical) record straight and to describe and rewrite familiar histories from radically new points of view. Provides students with a thorough introduction to modern South African history; they will learn about the politics, uses, and limitations of memoir as genre and will explore the multiple uses of the past in the present.
MU234: From Rockabilly to Grunge: A History of Rock 'n' Roll
Four credit hours. A. Zelensky
A survey of rock music, from its roots in country and blues to the alternative rock scene of the 1990s. Rock music will be considered in relation to race, sex, gender, drugs, technology, marketing, and politics to better understand its powerful position in constructing, challenging, and reinforcing various positions of identity. Students will learn to discuss the musical characteristics of a work, identify its genre and era of composition, and contextualize it within a broader framework of American culture and politics.
PS259: Lifespan Development
Four credit hours. Arterberry
A study of human development across the lifespan with emphasis on the general characteristics of development from birth to death. Various theories will be explored to explain developmental processes. Topics include perceptual, cognitive, social, and identity development; the role of families, communities, and culture in development; and death and dying. Students have the option to participate in civic engagement activities in the local community. This applied work helps students explore how to apply the findings of research or tenets of theory to real-world contexts.
RE182: Jews, Judaism, and the Modern World
Four credit hours. Freidenreich
A survey of the social, cultural, intellectual, and political history of the Jews of Europe, the United States, and Israel/Palestine from the 17th century to the present. Traces the emergence of contemporary Judaism in its various manifestations. In addition to developing basic familiarity with the subject matter, students will learn how to interpret specific ideas, movements, biographies, and works of cultural production within the framework of broader dynamics associated with Jewish life in modern times.
RE217:Religion in the Americas
Four credit hours. Harper
Examines religion and culture in the Americas, beginning with Native American religions and European-Indian contact and moving forward to contemporary movements and phenomena. Topics will include slavery and religion, politics and religion, evangelical Christianity, Judaism and Islam in the United States, "cults" and alternative spiritualities, and religion in/as popular culture. While the United States will serve as the primary focus, we will consider issues of cultural exchange across national boundaries in the Western Hemisphere, especially Mexico, Canada, and Caribbean countries. Prerequisite: Sophomore or higher standing.
RE298: The Jewish Jesus
Four credit hours. Emanuel
If Jesus is the epicenter of modern Christianity, does it make sense to situate him historically in a Jewish context? Could there be a difference between the contemporary Christian "Jesus of faith" and the "Jesus of history?" How have some persons argued that Jesus is best understood historically as Jewish, and others as Aryan? This course engages these questions and offers extended study into the historical, cultural, and theological contexts from which Jesus—and those who wrote about him—emerged. It also introduces students to the various approaches scholars use to guide these investigations.
RE298B: American Spirituality and the Environment
Examines historical and contemporary connections between spirituality and environmentalism in American culture. From early Quakers to mid-19th-century Romantics to contemporary Buddhists, we explore how individuals and groups in the United States have conceived of the relationship between environmentally responsible living, spiritual discipline, and social witness. While the course will span geographic regions, special attention is paid to movements and figures centered in Maine.
SP234: Diversity and Racism in Contemporary Spain
Four credit hours. Allbritton
This course focuses on the cultures and communities that make up contemporary Spain, with particular emphasis on the modern waves of immigration that have radically changed the country. Covering the latter years of the dictatorship and into the democracy (from 1970 forward), we examine how regionalism, multiculturalism, and diversity have been represented across a range of media and literature in Spain. Topics may include Latin American, African and Asian migration and diasporas, sex and sexuality, racial politics, and linguistic and cultural difference in Spain.
SP:398 In the Shadow of Medieval Spain
Four credit hours. Savo
This course examines some well-known medieval literary depictions of Iberian society by Christian, Jewish and Muslim authors, considering the ways in which each literary text portrays, critiques, and/or fabricates a social landscape. These readings are juxtaposed with an exploration of how nostalgia for an absent medieval past is used as a literary topos in modern narrative and poetry. Students will interrogate dichotomies of tolerance and persecution, exile and belonging, originals and translations, while exploring how our modern interpretive frameworks shape the construction of knowledge about the past.
ST120: Information Before and After Google: Impacts and Technologies
Four credit hours. Kugelmeyer
Explores the nature of information and how technology has changed our experience and understanding of it over the past 75 years. Emphasizes the relationship between information and technology and explores the impact of information technologies on societies, organizations, and people. Participants explore how people understand and evaluate information and in what contexts information is valued and why. Students will develop and improve their understanding, critical thought processes, and analytic skills around a range of information technologies. Class format is discussion based, and the focus is on developing scholarly writing skills.
ST132: The Presence of the Past
One credit hour. Cook, Jiang, Walker, van der Meer
How does the past shape our contemporary moment? How does the present inform what we know and feel about the past? To address these questions, this course will explore how our relation to the past is shaped by politics, art, science, and culture. Students will attend public lectures by visiting scholars and Colby faculty. These lectures will examine the political stakes of negotiating between the past and present from a range of disciplinary perspectives. Students will engage in focused discussion and short reflection papers. Nongraded.
HI297: Revolutionary Culture in Contemporary China
Three credit hours. Parker
A study of the Cultural Revolution, investigating how present discourses of revolutionary heritage and nationalism shape and define its history. Combines historicizing interpretations with original documents: photojournalism with poster art, present day news with revolutionary speeches, films with their revolutionary predecessors, memoirs with diaries. Placing culture at the center of historiography, we bring into focus the competing epistemologies of the Cultural Revolution itself — its anti-Party, grassroots and anarchic visions — to grapple with how the Chinese Communist Party deploys competing versions of its own historical legitimacy.
TD261: Topics in Performance: Activist Storytelling Workshop
Three credit hours. Weinblatt
Students will create original story-based performance pieces inspired by their own passion — issues such as the environment, race, poverty, reproductive justice, freedom of speech, LGBTQ+ rights, disability, diversity, access to education, etc. Students will explore a variety of writing and performance styles and techniques to engage in creative process and generate material. Culminates in a showcase presentation of solo and small group pieces at Colby and at a professional performance venue in Portland, which will require additional travel and rehearsal time the final week of Jan Plan. No previous writing or performance experience necessary. Prerequisite: Permission of the instructor.
TD361: Advanced Topics in Performance: Presence/Past
Three credit hours.
Directed by a collaborative team of guest artists rooted in visual art, theater, and dance, students will collaborate to create a multi-arts, immersive performance to be installed and performed on tour in Boston. Through both practiced and cutting edge methods, the process examines the tenuous state of communication in our technologically-mediated culture. Artists will examine the relationship between personal and collective histories translated through memory. Interested students studying abroad in either the fall or spring semesters should contact Professor Annie Kloppenberg. Prerequisite: Theater and Dance 164 or audition.
AR285 History of Photography
Introduction to the major aesthetic and cultural debates surrounding photography, from the announcement of its invention in 1839 through the postmodern era (ca. 1990). Investigates aesthetic styles and the ways they respond to the question of whether a mechanical medium can produce art. Considers documentary and ethnographic uses of photographs and asks how they construct ideas about "the real." Primary focus is on the Anglo-American tradition. Essay assignments, oral presentations, and discussion emphasize visual analysis skills and the ability to read images in their aesthetic and cultural contexts. Prerequisite: Art major or minor, or sophomore or higher standing.
AY232: Oral History Ethnographic Research Lab: Waterville Main Street
Two credit hours. Tate
In this ethnographic research lab, students will explore the theory and practice of oral history. They will read from a range of sources about the challenges of producing oral history, and they will conduct both archival research and produce oral histories examining the history of Waterville Main Street using Colby's Special Collections and with Waterville residents. Drawing on Digital Maine's previous projects (including American Studies 221, "Mapping Waterville"), the class will produce a collective project presenting oral histories of Waterville Main Street.
AY344: Black Radical Imaginations
Four credit hours. Bhimull
A seminar about the complex history of black radical imagination. Explores how black people have long used imagination as a strategy for survival, resistance, emancipation, liberation, and to create worlds of joy and love. It is concerned with black intellectual activism in the African diaspora and examines a range of cultural movements against racialized forms of oppression, including black surrealism and Afrofuturism.
EC470 Seminar: The City in Economic History
Four credit hours. Siodla
Since its founding, the United States has steadily become urbanized. What economic forces have caused people to move to cities? Can history explain today's urban locations and spatial patterns? Focusing primarily on U.S. urban growth since 1800, students will read, present, and discuss academic articles on topics such as suburbanization, zoning, local infrastructure investment, urban quality of life, housing, and racial and economic inequality. Students will build the economic models and tools necessary to complete an original empirical research paper in urban economic history.
EN397: Poetry Remixes
Four credit hours. L. Ardam
Remixing, re-visioning, rewriting, appropriation, quotation, and recycling are key methods and concerns for many 20th- and 21st-century poets. This humanities lab will study 100 years of poetic remixing in units on gender, race and identity, and culture. We will work with Special Collections and the Colby Museum, including a project on the found language poetry of Bern Porter. We will ask questions such as: How and why do poets engage other art and cultural forms? How does remixing shape our understanding of history and politics? What does our poetic engagement with the past tell us about how we view our political moment?
GS245: Memory and Politics
Four credit hours. Yoder
This writing-intensive course invites students to consider how governments and other actors frame the past, for what purposes, and with what effects. The focus is on post-1945 Europe, however students are welcome to examine non-European cases in their own work. Through a variety of writing exercises, students will engage with discipline- and culture-specific debates about whether and how a society should address its past, particularly after periods of violence and authoritarian or totalitarian rule.
HI338 History in Reverse: Backwards through the Records from Now to Then
Four credit hours. H. Leonard
Professional historians are often drawn to the field by their interest in or concern about current affairs, whose historical roots they seek to understand. Similarly, we will begin by focusing collectively on a contemporary issue, problem, or development (such as the presidential candidacy of Hillary Clinton or the collapse of the paper industry in central Maine), and then trace backwards through the relevant historical records for evidence of causation and contingency. Students will then choose a topic of interest and repeat the process, developing skills in effective research, clear and precise writing, critical source analysis, and oral presentation.
LT297 Seneca's Medea
In this class, we shall read selected passages from Seneca's Medea. Seneca's version of the story of Medea's terrible vengeance on the guilty and innocent alike warns us that injustice begets injustice, and asks how divine power can permit evil to triumph. The play draws on contemporary dilemmas of Imperial Rome (Seneca's "Present") but explores them in the "safe" context of a Greek tragic "Past."
RU242: Back to the Future: Recent Russian Cinema
Four credit hours. Monsastireva-Ansdell
What role does Russia's "most important art" play in shaping the nation's myths, its present and future? How does it legitimate or subvert the official notions of usable and unusable past? What has caused the shift from the rigorous interrogation of the Soviet past in the 1990s to the revival and reintegration of Soviet-era policies, practices, and values in the 21st century? How does the Kremlin use the cinema as an ideological tool against the West and Hollywood? What are the artistic and discursive strategies of the Putin-era independent filmmakers, variously referred to as "the New Wave" or "New Quiets," in an increasingly homogeneous and controlled public sphere (Wilmes)? Examins a variety of genres (drama, the war film, comedy, fantasy, criminal thriller, historical epic, the musical), as well as a range of social issues and changing representations of social structures, ethnic relations, and gender roles in contemporary Russia.
WP115G First-Year Writing: Rich and Poor in American Novels
Four credit hours. W1. Harrington
This humanities lab invites students to explore 19th-21st century American novels through the lens of class extremes, with a special focus on homes and material domestic culture. Through a close study of four novels centered on dwelling spaces, from mansions to migrant camps to squats, students will investigate how narrative and artistic production construct and reiterate characterizations of "rich" and "poor," reflecting critically on their own notions of class in today's era of income inequality. Lab components include musical research, a trip to the Victoria Mansion in Portland, a Colby Museum writing assignment, a reflective blog, a curated exhibit in Miller Library, and group presentations on material culture.
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{"url":"http:\/\/www.statemaster.com\/encyclopedia\/Quantum-computer","text":"FACTOID # 16: In the 2000 Presidential Election, Texas gave Ralph Nader the 3rd highest popular vote count of any US state.\n\n Home Encyclopedia Statistics States A-Z Flags Maps FAQ About\n\n WHAT'S NEW\n\nSEARCH ALL\n\nSearch encyclopedia, statistics and forums:\n\n(* = Graphable)\n\nEncyclopedia >\u00a0Quantum computer\nThe Bloch sphere is a representation of a qubit, the fundamental building block of quantum computers.\n\nAlthough quantum computing is still in its infancy, experiments have been carried out in which quantum computational operations were executed on a very small number of qubits. Research in both theoretical and practical areas continues at a frantic pace, and many national government and military funding agencies support quantum computing research to develop quantum computers for both civilian and national security purposes, such as cryptanalysis.[2] (See Timeline of quantum computing for details on current and past progress.) To meet Wikipedias quality standards and make it more accessible, this article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... Cryptanalysis (from the Greek krypt\u00c3\u00b3s, hidden, and anal\u00c3\u00bdein, to loosen or to untie) is the study of methods for obtaining the meaning of encrypted information, without access to the secret information which is normally required to do so. ... Timeline of quantum computers \/\/ 1970 - Stephen Wiesner invents conjugate coding. ...\n\nIf large-scale quantum computers can be built, they will be able to solve certain problems exponentially faster than any of our current classical computers (for example Shor's algorithm). Quantum computers are different from other computers such as DNA computers and traditional computers based on transistors. Some computing architectures such as optical computers may use classical superposition of electromagnetic waves, but without some specifically quantum mechanical resources such as entanglement, they have less potential for computational speed-up than quantum computers. Shors algorithm is a quantum algorithm for factoring an integer N in O((log N)3) time and O(log N) space, named after Peter Shor. ... The tower of a personal computer. ... DNA computing is a form of computing which uses DNA and molecular biology, instead of the traditional silicon-based computer technologies. ... Assorted discrete transistors A transistor is a semiconductor device, commonly used as an amplifier or an electrically controlled switch. ... An optical computer is a computer that uses light instead of electricity (i. ... It has been suggested that Quantum coherence be merged into this article or section. ...\n\n## The basis of quantum computing GA_googleFillSlot(\"encyclopedia_square\");\n\nA classical computer has a memory made up of bits, where each bit holds either a one or a zero. A quantum computer maintains a sequence of qubits. A single qubit can hold a one, a zero, or, crucially, a quantum superposition of these, allowing for an infinite number of states. A quantum computer operates by manipulating those qubits with (possibly a suite of) quantum logic gates. This article is about the unit of information. ... To meet Wikipedias quality standards and make it more accessible, this article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... Quantum superposition is the application of the superposition principle to quantum mechanics. ... A quantum gate or quantum logic gate is a rudimentary quantum circuit operating on a small number of qubits. ...\n\nAn example of an implementation of qubits for a quantum computer could start with the use of particles with two spin states: \"up\" and \"down\" (typically written $|0rangle$ and $|1rangle$). But in fact any system possessing an observable quantity A which is conserved under time evolution and such that A has at least two discrete and sufficiently spaced consecutive eigenvalues, is a suitable candidate for implementing a qubit. That's because any such system can be mapped onto an effective spin-1\/2. In physics, spin refers to the angular momentum intrinsic to a body, as opposed to orbital angular momentum, which is the motion of its center of mass about an external point. ... In physics, particularly in quantum physics, a system observable is a property of the system state that can be determined by some sequence of physical operations. ... In mathematics, a number is called an eigenvalue of a matrix if there exists a nonzero vector such that the matrix times the vector is equal to the same vector multiplied by the eigenvalue. ... In quantum mechanics, spin is an intrinsic property of all elementary particles. ...\n\n## Bits vs. qubits\n\nConsider first a classical computer that operates on a 3-bit register. At any given time, the bits in the register are in a definite state, such as 101. In a quantum computer, however, the qubits can be in a superposition of all the classically allowed states. In fact, the register is described by a wavefunction: In computer architecture, a processor register is a small amount of very fast computer memory used to speed the execution of computer programs by providing quick access to frequently used values\u00e2\u20ac\u201dtypically, these values are involved in multiple expression evaluations occurring within a small region on the program. ... This article discusses the concept of a wavefunction as it relates to quantum mechanics. ...\n\n$|psi rangle = a,|000rangle + b,|001rangle + c,|010rangle + d,|011rangle + e,|100rangle + f,|101rangle + g,|110rangle + h,|111rangle$\nQubits are made up of controlled particles and the means of control (e.g. devices that trap particles and switch them from one state to another).[3]\n\nwhere the coefficients a, b, c,..., h are complex numbers whose amplitudes squared are the probabilities to measure the qubits in each state- for example, | c | 2 is the probability to measure the register in the state 010. It is important that these numbers are complex, because the phases of the numbers can constructively and destructively interfere with one another; this is an important feature for quantum algorithms.[4] Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... In mathematics, a complex number is a number of the form where a and b are real numbers, and i is the imaginary unit, with the property i 2 = \u00e2\u02c6\u20191. ... This article is about a portion of a periodic process. ...\n\nRecording the state of a quantum register requires an exponential number of complex numbers (the 3-qubit register above requires 23 = 8 complex numbers). The number of classical bits required even to estimate the complex numbers of some quantum state grows exponentially with the number of qubits. For a 300-qubit quantum register, somewhere on the order of 1090 classical registers are required, more than there are atoms in the observable universe.[5] A quantum register is the quantum computing analogue of a classical processor register. ... See universe for a general discussion of the universe. ...\n\n## Initialization, execution and termination\n\nIn our example, the contents of the qubit registers can be thought of as an 8-dimensional complex vector. An algorithm for a quantum computer must initialize this vector in some specified form (dependent on the design of the quantum computer). In each step of the algorithm, that vector is modified by multiplying it by a unitary matrix. The matrix is determined by the physics of the device. The unitary character of the matrix ensures the matrix is invertible (so each step is reversible). In linear algebra, a coordinate vector is an explicit representation of a vector in an abstract vector space as an ordered list of numbers or, equivalently, as an element of the coordinate space Fn. ... In mathematics, a unitary matrix is an n by n complex matrix U satisfying the condition where In is the identity matrix and U* is the conjugate transpose (also called the Hermitian adjoint) of U. Note this condition says that a matrix U is unitary if it has an inverse... The term reversible computing refers to any computational process that is (at least to some close approximation) reversible, i. ...\n\nUpon termination of the algorithm, the 8-dimensional complex vector stored in the register must be somehow read off from the qubit register by a quantum measurement. However, by the laws of quantum mechanics, that measurement will yield a random 3-bit string (and it will destroy the stored state as well). This random string can be used in computing the value of a function because (by design) the probability distribution of the measured output bitstring is skewed in favor of the correct value of the function. By repeated runs of the quantum computer and measurement of the output, the correct value can be determined, to a high probability, by majority polling of the outputs. In brief, quantum computations are probabilistic; see quantum circuit for a more precise formulation. The framework of quantum mechanics requires a careful definition of measurement, and a thorough discussion of its practical and philosophical implications. ... Random redirects here. ... In mathematics and statistics, a probability distribution is a function of the probabilities of a mutually exclusive and exhaustive set of events. ... In quantum mechanics, a quantum circuit is a specific model for a quantum computational device. ...\n\nFor more details on the sequences of operations used for various algorithms, see universal quantum computer, Shor's algorithm, Grover's algorithm, Deutsch-Jozsa algorithm, quantum Fourier transform, quantum gate, quantum adiabatic algorithm and quantum error correction. Also refer to the growing field of quantum programming. The universal quantum computer or universal quantum Turing machine (UQTM) is a theoretical machine that combines both Church-Turing and quantum principles. ... Shors algorithm is a quantum algorithm for factoring an integer N in O((log N)3) time and O(log N) space, named after Peter Shor. ... Grovers algorithm is a quantum algorithm for searching an unsorted database with N entries in O(N1\/2) time and using O(logN) storage space (see big O notation). ... The Deutsch-Jozsa algorithm is a quantum algorithm, proposed by David Deutsch and Richard Jozsa in 1992. ... The quantum Fourier transform is the discrete Fourier transform with a particular decomposition into a product of simpler unitary matrices. ... A quantum gate or quantum logic gate is a rudimentary quantum circuit operating on a small number of qubits. ... Quantum error correction is for use in quantum computing to protect quantum information from errors due to decoherence and other quantum noise. ... It has been suggested that Quantum programming language be merged into this article or section. ...\n\n## The power of quantum computers\n\nA way out of this dilemma would be to use some kind of quantum cryptography. The are also are some digital signature schemes that are believed to be secure against quantum computers. See for instance Lamport signatures. Quantum cryptography is an approach based on quantum physics for secure communications. ... A digital signature or digital signature scheme is a type of asymmetric cryptography used to simulate the security properties of a signature in digital, rather than written, form. ... In cryptography, a Lamport signature or Lamport one-time signature scheme is a method for constructing a digital signature. ...\n\nThis dramatic advantage of quantum computers has only been discovered for these problems so far: factoring, discrete logarithm. However, there is no proof that the advantage is real: an equally fast classical algorithm may still be discovered. There is one other problem where quantum computers have a smaller, though significant (quadratic) advantage. It is quantum database search, and can be solved by Grover's algorithm. In this case the advantage is provable. This establishes beyond doubt that (ideal) quantum computers are superior to classical computers for at least one problem. 2007 is a common year starting on Monday of the Gregorian calendar. ... Grovers algorithm is a quantum algorithm for searching an unsorted database with N entries in O(N1\/2) time and using O(logN) storage space (see big O notation). ...\n\nConsider a problem that has these four properties:\n\n1. The only way to solve it is to guess answers repeatedly and check them,\n2. There are n possible answers to check,\n3. Every possible answer takes the same amount of time to check, and\n4. There are no clues about which answers might be better: generating possibilities randomly is just as good as checking them in some special order.\n\nAn example of this is a password cracker that attempts to guess the password for an encrypted file (assuming that the password has a maximum possible length). Password cracking is the process of recovering secret passwords from data that has been stored in or transmitted by a computer system, typically, by repeatedly verifying guesses for the password. ... Encrypt redirects here. ...\n\nFor problems with all four properties, the time for a quantum computer to solve this will be proportional to the square root of n (it would take an average of (n\u00a0+\u00a01)\/2 guesses to find the answer using a classical computer.) That can be a very large speedup, reducing some problems from years to seconds. It can be used to attack symmetric ciphers such as Triple DES and AES by attempting to guess the secret key. Regardless of whether any of these problems can be shown to have an advantage on a quantum computer, they nonetheless will always have the advantage of being an excellent tool for studying quantum mechanical interactions, which of itself is an enormous value to the scientific community. A symmetric-key algorithm is an algorithm for cryptography that uses the same cryptographic key to encrypt and decrypt the message. ... In cryptography, Triple DES (also 3DES) is a block cipher formed from the Data Encryption Standard (DES) cipher. ... In cryptography, the Advanced Encryption Standard (AES), also known as Rijndael, is a block cipher adopted as an encryption standard by the U.S. government. ...\n\nThere are currently no other practical problems known where quantum computers give a large speedup over classical computers. Research is continuing, and more problems may yet be found.\n\n## Problems and practicality issues\n\nThere are a number of practical difficulties in building a quantum computer, and thus far quantum computers have only solved trivial problems. David DiVincenzo, of IBM, listed the following requirements for a practical quantum computer:[7]\n\n\u2022 scalable physically to increase the number of qubits\n\u2022 qubits can be initialized to arbitrary values\n\u2022 quantum gates faster than decoherence time\n\u2022 universal gate set\n\u2022 qubits can be read easily\n\nTo summarize the problem from the perspective of an engineer, one needs to solve the challenge of building a system which is isolated from everything except the measurement and manipulation mechanism. Furthermore, one needs to be able to turn off the coupling of the qubits to the measurement so as to not decohere the qubits while performing operations on them. Quantum decoherence is the general term for the consequences of irreversible quantum entanglement. ...\n\n### Quantum decoherence\n\nOne major problem is keeping the components of the computer in a coherent state, as the slightest interaction with the external world would cause the system to decohere. This effect causes the unitary character (and more specifically, the invertibility) of quantum computational steps to be violated. Decoherence times for candidate systems, in particular the transverse relaxation time T2 (terminology used in NMR and MRI technology, also called the dephasing time), typically range between nanoseconds and seconds at low temperature.[8] The issue for optical approaches are more difficult as these timescales are orders of magnitude lower and an often cited approach to overcome it uses an optical pulse shaping approach. Error rates are typically proportional to the ratio of operating time to decoherence time, hence any operation must be completed much more quickly than the decoherence time. In quantum mechanics, quantum decoherence is the mechanism by which quantum systems interact with their environments to exhibit probabilistically additive behavior - a feature of classical physics - and give the appearance of wavefunction collapse. ... NMR redirects here. ... The mri are a fictional alien species in the Faded Sun Trilogy of C.J. Cherryh. ... Debabrata Goswami, is an Indian spectroscopist, winner of the Wellcome Trust Senior Research Fellow Award (2004), Swarnajayanti Award (2004), presently Associate Professor of Chemistry at Indian Institute of Technology, Kanpur (IITK). ...\n\nIf the error rate is small enough, it is thought to be possible to use quantum error correction, which corrects errors due to decoherence, thereby allowing the total calculation time to be longer than the decoherence time. An often cited (but rather arbitrary) figure for required error rate in each gate is 10\u22124. This implies that each gate must be able to perform its task 10,000 times faster than the decoherence time of the system.\n\nMeeting this scalability condition is possible for a wide range of systems. However, the use of error correction brings with it the cost of a greatly increased number of required qubits. The number required to factor integers using Shor's algorithm is still polynomial, and thought to be between L4 and L6, where L is the number of bits in the number to be factored. For a 1000-bit number, this implies a need for 1012 to 1018.[9]\n\nA very different approach to the stability-decoherence problem is to create a topological quantum computer with anyons, quasi-particles used as threads and relying on knot theory to form stable logic gates.[10] A topological quantum computer is a theoretical quantum computer that uses quasiparticles called anyons where their world lines form threads that cross over one another to form braids in a two-dimensional world. ... In mathematics and physics, an anyon is a type of projective representation of a Lie group. ... Trefoil knot, the simplest non-trivial knot. ...\n\n### Candidates\n\nThere are a number of quantum computing candidates, among those:\n\n1. Superconductor-based quantum computers (including SQUID-based quantum computers)\n2. Trapped ion quantum computer\n3. Electrons on helium quantum computers\n4. \"Nuclear magnetic resonance on molecules in solution\"-based\n5. \"Quantum dot on surface\"-based (e.g. the Loss-DiVincenzo quantum computer)\n6. \"Cavity quantum electrodynamics\" (CQED)-based\n7. \"Molecular magnet\"-based\n8. Fullerene-based ESR quantum computer\n9. Solid state NMR Kane quantum computers\n10. Optic-based quantum computers (Quantum optics)\n11. Topological quantum computer\n12. Spin-based quantum computer\n14. Diamond-based quantum computer[12]\n15. Bose\u2013Einstein condensate-based quantum computer[13]\n\nThe large number of candidates shows explicitly that the topic, in spite of rapid progress, is still in its infancy. But at the same time there is also a vast amount of flexibility. Superconductivity is a phenomenon occurring in certain materials at low temperatures, characterised by the complete absence of electrical resistance and the damping of the interior magnetic field (the Meissner effect. ... For other uses, see Squid (disambiguation). ... A Trapped ion quantum computer is a type of quantum computer. ... NMR redirects here. ... 3D (left and center) and 2D (right) representations of the terpenoid molecule atisane. ... Making a saline water solution by dissolving table salt (NaCl) in water This article is about chemical solutions. ... A quantum dot is a semiconductor nanostructure that confines the motion of conduction band electrons, valence band holes, or excitons (bound pairs of conduction band electrons and valence band holes) in all three spatial directions. ... A double quantum dot. ... Molecular magnets are systems where a permanent magnetization and magnetic hysteresis can be achieved (although usually at extremely low temperatures) not through a three-dimensional magnetic ordering, but as a purely one-molecule phenomenon. ... The Icosahedral Fullerene C540 C60 and C-60 redirect here. ... Electron paramagnetic resonance (EPR) or electron spin resonance (ESR) spectroscopy is a technique for studying chemical species that have one or more unpaired electrons, such as organic and inorganic free radicals or inorganic complexes possessing a transition metal ion. ... The Kane quantum computer is a proposal for a scalable quantum computer proposed by Bruce Kane in 19981, then at the University of New South Wales. ... Quantum optics is a field of research in physics, dealing with the application of quantum mechanics to phenomena involving light and its interactions with matter. ... A topological quantum computer is a theoretical quantum computer that uses quasiparticles called anyons where their world lines form threads that cross over one another to form braids in a two-dimensional world. ... Unsolved problems in physics: Is it possible to construct a practical electronic device that operates on the spin of the electron, rather than its charge? Spintronics (a neologism for spin-based electronics), also known as magnetoelectronics, is an emergent technology which exploits the quantum spin states of electrons as well... A Bose\u00e2\u20ac\u201cEinstein condensate (BEC) is a state of matter formed by a system of bosons confined in an external potential and cooled to temperatures very near to absolute zero (0 kelvin or \u00e2\u02c6\u2019273. ...\n\nIn 2005, researchers at the University of Michigan built a semiconductor chip which functioned as an ion trap. Such devices, produced by standard lithography techniques, may point the way to scalable quantum computing tools.[14] An improved version was made in 2006. The University of Michigan, Ann Arbor (U of M, UM or simply Michigan) is a coeducational public research university in the state of Michigan, and one of the foremost universities in the United States. ... Integrated circuit showing memory blocks, logic and input\/output pads around the periphery A monolithic integrated circuit (also known as IC, microchip, silicon chip, computer chip or chip) is a miniaturized electronic circuit (consisting mainly of semiconductor devices, as well as passive components) which has been manufactured in the surface... An ion trap is a combination of electric or magnetic fields that captures ions in a region of a vacuum system or tube. ... Lithography stone and mirror-image print of a map of Munich. ...\n\nD-Wave Systems Inc. claims to be the world\u2019s first \u2014 and only \u2014 provider of quantum computing systems designed to run commercial applications. On 13 February, 2007 they ran an initial demonstration of their Orion quantum computing system, which is built around a 16-qubit superconducting adiabatic quantum computer processor.[15] However, since D-Wave Systems has not released the full details of Orion to the scientific community, many experts in the field have expressed skepticism.[16] D-Wave Systems, Inc. ... D-Wave Systems, Inc. ... To meet Wikipedias quality standards and make it more accessible, this article needs a better explanation of technical details or more context regarding applications or importance to make it more accessible to a general audience, or at least to technical readers outside this specialty. ... Superconductivity is a phenomenon occurring in certain materials at low temperatures, characterised by the complete absence of electrical resistance and the damping of the interior magnetic field (the Meissner effect. ... In quantum mechanics, an adiabatic process is an infinitely slow change in the Hamiltonian of a system. ...\n\n## Quantum computing in computational complexity theory\n\nThe suspected relationship of BQP to other problem spaces[17]\n\nThis section surveys what is currently known mathematically about the power of quantum computers. It describes the known results from computational complexity theory and the theory of computation dealing with quantum computers. Image File history File links BQP_complexity_class_diagram. ... Image File history File links BQP_complexity_class_diagram. ... As a branch of the theory of computation in computer science, computational complexity theory investigates the problems related to the amounts of resources required for the execution of algorithms (e. ... The theory of computation is the branch of computer science that deals with whether and how efficiently problems can be solved on a computer. ...\n\nThe class of problems that can be efficiently solved by quantum computers is called BQP, for \"bounded error, quantum, polynomial time\". Quantum computers only run probabilistic algorithms, so BQP on quantum computers is the counterpart of BPP on classical computers. It is defined as the set of problems solvable with a polynomial-time algorithm, whose probability of error is bounded away from one quarter.[18] A quantum computer is said to \"solve\" a problem if, for every instance, its answer will be right with high probability. If that solution runs in polynomial time, then that problem is in BQP. BQP, in computational complexity theory, stands for Bounded error, Quantum, Polynomial time. It denotes the class of problems solvable by a quantum computer in polynomial time, with an error probability of at most 1\/4 for all instances. ... A randomized algorithm is an algorithm which is allowed to flip a truly random coin. ... This article is about the complexity class. ...\n\nBQP is suspected to be disjoint from NP-complete and a strict superset of P, but that is not known. Both integer factorization and discrete log are in BQP. Both of these problems are NP problems suspected to be outside BPP, and hence outside P. Both are suspected to not be NP-complete. There is a common misconception that quantum computers can solve NP-complete problems in polynomial time. That is not known to be true, and is generally suspected to be false. In complexity theory, the NP-complete problems are the most difficult problems in NP, in the sense that they are the ones most likely not to be in P. The reason is that if you could find a way to solve an NP-complete problem quickly, then you could use... In computational complexity theory, P is the complexity class containing decision problems which can be solved by a deterministic Turing machine using a polynomial amount of computation time, or polynomial time. ... Prime decomposition redirects here. ... In abstract algebra and its applications, the discrete logarithms are defined in group theory in analogy to ordinary logarithms. ...\n\nAn operator for a quantum computer can be thought of as changing a vector by multiplying it with a particular matrix. Multiplication by a matrix is a linear operation. Daniel S. Abrams and Seth Lloyd have shown that if a quantum computer could be designed with nonlinear operators, then it could solve NP-complete problems in polynomial time. It could even do so for #P-complete problems. They do not believe that such a machine is possible. In mathematics, a linear transformation (also called linear map or linear operator) is a function between two vector spaces that preserves the operations of vector addition and scalar multiplication. ... Seth Lloyd is a Professor of Mechanical Engineering at MIT. His research area is the interplay of information with complex systems, especially quantum systems. ... The title given to this article is incorrect due to technical limitations. ...\n\nAlthough quantum computers may be faster than classical computers, those described above can't solve any problems that classical computers can't solve, given enough time and memory (albeit possibly an amount that could never practically be brought to bear). A Turing machine can simulate these quantum computers, so such a quantum computer could never solve an undecidable problem like the halting problem. The existence of \"standard\" quantum computers does not disprove the Church-Turing thesis.[19] For the test of artificial intelligence, see Turing test. ... Undecidable has more than one meaning: In mathematical logic: A decision problem is undecidable if there is no known algorithm that decides it. ... In computability theory the halting problem is a decision problem which can be stated as follows: Given a description of a program and a finite input, decide whether the program finishes running or will run forever, given that input. ... In computability theory the Church-Turing thesis, Churchs thesis, Churchs conjecture or Turings thesis, named after Alonzo Church and Alan Turing, is a hypothesis about the nature of mechanical calculation devices, such as electronic computers. ...\n\nVery recently, many researchers have begun to investigate the possibility of using quantum mechanics for hypercomputation - that is, solving undecidable problems. Such claims have been met with considerable skepticism as to whether it is even theoretically possible; see the hypercomputation article for more details. Hypercomputation refers to various proposed methods for the computation of non-Turing-computable functions. ... Hypercomputation refers to various proposed methods for the computation of non-Turing-computable functions. ...\n\n## Quantum computers in fiction\n\nThe TV show Code Lyoko features a quantum supercomputer. Code Lyoko is a French animated television series featuring both conventional animation and CGI animation. ...\n\nQuantum computing technology is used in the alternate reality game for the Nine Inch Nails album Year Zero to transmit data to the present from the year 2022. NIN redirects here. ... Year Zero (also known as Halo 24) is the sixth Nine Inch Nails studio album, released on April 16, 2007 in Europe, April 17 in the United States, and April 25, 2007 in Japan. ...\n\nIn the 2007 movie Transformers, it is stated that the Decepticons use quantum computing to break the U.S. Army's data signal, therefore being able to access secure files. It is stated that even with the most powerful supercomputer, it would take 22 years to do. For the 1986 animated film, see The Transformers: The Movie. ... The Decepticons (also known as Destrons in Japan) are the enemies of the Autobots, and the villains in the Transformers toyline and related spin-off comics and cartoons. ...\n\nThe Michael Crichton novel Timeline also features use of a quantum computer, especially with regards to deconstructing and reconstructing objects being transported across time (or, more accurately, across the multiverse). Timeline is a science fiction novel by Michael Crichton that was published in November 1999. ...\n\nThe Robert J. Sawyer trilogy The Neanderthal Parallax involves a quantum computer which unintentionally opens a gateway between different versions of Earth. Robert J. Sawyer is a Canadian science fiction writer, dubbed the dean of Canadian science fiction by the Ottawa Citizen in 1999. ... The Neanderthal Parallax is a trilogy of novels by Robert J. Sawyer. ...\n\nQuantum computers and mechanics are also used in the Noein anime series extensively. Original\u00a0run 2005-10-12 \u00e2\u20ac\u201c 2006-03-29 No. ...\n\nIn the MMORPG EvE Online the Caldari State makes use of Quantum computers\n\nIn Dan Brown's novel Digital Fortress, the NSA's cypher breaking computer TRANSLTR uses quantum computing to break encryption keys of encrypted E-mails. This article is about the writer. ... Digital Fortress is a novel by American author Dan Brown and published in 1998 by St. ...\n\nA quantum bus is a device which can be used to store or transfer information between independent qubits in a quantum computer, or combine two qubits into a superposition. ... Timeline of quantum computers \/\/ 1970 - Stephen Wiesner invents conjugate coding. ...\n\n## Notes\n\n1. ^ \"Quantum Computing with Molecules\" article in Scientific American by Neil Gershenfeld and Isaac L. Chuang - a generally accessible overview of quantum computing and so on.\n2. ^ Quantum Information Science and Technology Roadmap for a sense of where the research is heading.\n3. ^ Waldner, Jean-Baptiste (2007). Nanocomputers and Swarm Intelligence. ISTE, p157. ISBN 2746215160.\n4. ^ DiVincenzo, David (October 1995). \"Quantum Computation\". Science 270 (5234): 255-261. Retrieved on 2007-04-25.\n5. ^ Note that the coefficients are not all independent, since the probabilities must sum to 1.\n6. ^ http:\/\/modular.fas.harvard.edu\/edu\/Fall2001\/124\/misc\/arjen_lenstra_factoring.pdf\n7. ^ David P. DiVincenzo, IBM (2000-04-13). The Physical Implementation of Quantum Computation. Retrieved on 2006-11-17.\n8. ^ David P. DiVincenzo, IBM (1995-10-13). Quantum Computation. Retrieved on 2006-11-17.\n9. ^ M. I. Dyakonov, Universit\u00e9 Montpellier (2006-10-14). Is Fault-Tolerant Quantum Computation Really Possible?. Retrieved on 2007-02-16.\n10. ^ Freedman, Michael; Alexei Kitaev, Michael Larsen, Zhenghan Wang (2002-10-20). \"Topological Quantum Computation\". Bulletin of the American Mathematical Society 40 (1): 31\u201438. Retrieved on 2007-05-07.\n11. ^ William M Kaminsky, MIT (Date Unknown). Scalable Superconducting Architecture for Adiabatic Quantum Computation. Retrieved on 2007-02-19.\n12. ^ Wolfgang Gruener, TG Daily (2007-06-01). Research indicates diamonds could be key to quantum storage. Retrieved on 2007-06-04.\n13. ^ Rene Millman, IT PRO (2007-08-03). Trapped atoms could advance quantum computing. Retrieved on 2007-07-26.\n14. ^ Ann Arbor (2005-12-12). U-M develops scalable and mass-producible quantum computer chip. Retrieved on 2006-11-17.\n15. ^ Comment on D-Wave by David Deutsch\n16. ^ Jason Pontin (2007). A Giant Leap Forward in Computing? Maybe Not. The New York Times Company. Retrieved on 2007-04-08.\n17. ^ Michael Nielsen and Isaac Chuang (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 0-521-63503-9.\n18. ^ (Nielsen & Chuang 2000)\n19. ^ Nielsen, Michael and Isaac Chuang (2000), p.\u00a0?\n\n## References\n\n\u2022 DiVincenzo, David P. (2000). \"The Physical Implementation of Quantum Computation\". Experimental Proposals for Quantum Computation. arXiv:quant-ph\/0002077.\n\u2022 DiVincenzo, David P. (1995). \"Quantum Computation\". Science 270 (5234): 255\u2013261.\u00a0 Table 1 lists switching and dephasing times for various systems.\n\u2022 Feynman, Richard (1982). \"Simulating physics with computers\". International Journal of Theoretical Physics 21: 467.\n\u2022 Jaeger, Gregg (2006). Quantum Information: An Overview. Berlin: Springer. ISBN 0-387-35725-4.\u00a0\"\n\u2022 Nielsen, Michael and Isaac Chuang (2000). Quantum Computation and Quantum Information. Cambridge: Cambridge University Press. ISBN 0-521-63503-9.\n\u2022 Singer, Stephanie Frank (2005). Linearity, Symmetry, and Prediction in the Hydrogen Atom. New York: Springer. ISBN 0-387-24637-1.\n\u2022 http:\/\/jquantum.sourceforge.net\/jQuantumApplet.html Java quantum computer simulator.\n\nResults from FactBites:\n\n Quantum computer - Wikipedia, the free encyclopedia (3339\u00a0words) In a classical (or conventional) computer, the amount of data is measured by bits; in a quantum computer, it is measured by qubits. The basic principle of quantum computation is that the quantum properties of particles can be used to represent and structure data, and that quantum mechanisms can be devised and built to perform operations with this data. Quantum computers are different from classical computers such as DNA computers and computers based on transistors, even though these may ultimately use some kind of quantum mechanical effect (for example covalent bonds).\n Reference.com\/Encyclopedia\/Quantum computer (2819\u00a0words) In a classical (or conventional) computer, data are measured by bits; in a quantum computer the data are measured by qubits. In quantum mechanics, the state of a physical system (such as an electron or a photon) is described by an element of a mathematical object called a Hilbert space. Qubits for a quantum computer can be implemented using particles with two spin states: \"up\" and \"down\"; in fact any system, possessing an observable quantity A which is conserved under time evolution and such that A has at least two discrete and sufficiently spaced consecutive eigenvalues, is a suitable candidate for implementing a qubit.\nMore results at FactBites\u00a0\u00bb\n\nShare your thoughts, questions and commentary here","date":"2019-07-22 20:36:44","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 0, \"img_math\": 3, \"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\": 0.5800936818122864, \"perplexity\": 901.0822413561942}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-30\/segments\/1563195528220.95\/warc\/CC-MAIN-20190722201122-20190722223122-00379.warc.gz\"}"} | null | null |
{"url":"https:\/\/zbmath.org\/?q=an:0676.46031","text":"# zbMATH \u2014 the first resource for mathematics\n\nWulff theorem and best constant in Sobolev inequality. (English) Zbl\u00a00676.46031\nLet $$f: {\\mathbb{R}}^ 2\\to {\\mathbb{R}}$$ be a positively homogeneous function of degree one, lower semi-continuous, with $$f(x)>0$$ if $$x\\neq 0$$. For each such function one defines a convex set of $${\\mathbb{R}}^ 2$$, $$W_ f$$, $W_ f=\\{x^*\\in {\\mathbb{R}}^ 2:\\quad f^*(x^*)\\leq 0\\} = \\{x^*\\in {\\mathbb{R}}^ 2:\\quad f^ 0(x^*)\\leq 1\\},$ where $$f^*$$, resp. $$f^ 0$$, is the Legendre transform, resp. the polar transform, of f. Let $$(u,v)\\in W^{1,1}_{per}(a,b)\\times W^{1,1}_{per}(a,b)$$ with $$u^{'2}+v^{'2}\\neq 0$$ a.e. in $$(a,b)$$. Let $F(u,v) = \\int^{b}_{a}f(v'(\\theta),-u'(\\theta))d\\theta,\\quad m(u,v) = \\int^{b}_{a}(v'(\\theta)u(\\theta)-u'(\\theta)v(\\theta))d\\theta.$ Then the following inequality holds $(*)\\quad F^ 2(u,v)-4| W_ f| m(u,v)\\geq 0,$ where $$| W_ f|$$ is the Lebesgue measure of $$W_ f$$. Equality holds if and only if $$(u,v)$$ is a parametrization of $$\\partial W_ f.$$\nInequality (*) is a generalized isoperimetric inequality. Indeed, if $$f$$ is the Euclidean norm and $$(u,v)$$ a parametric representation of the boundary $$\\partial A$$ of a region $$A$$, then $$F(u,v)=\\ell (\\partial A)$$ and $$m(u,v)=| A|$$, where $$\\ell (\\partial A)$$ is the length of $$\\partial A$$ and $$| A|$$ the area of A. In that case $$W_ f$$ is the unit Euclidean disk and $$| W_ f| =\\pi$$. The proof of (*) is a consequence of a generalized Wirtinger inequality $\\inf \\{\\int^{1}_{-1}(f(v',-u'))^ 2 d\\theta \/\\int^{1}_{-1}(f^ 0(u,v))^ 2 d\\theta:\\quad (u,v)\\in {\\mathcal M}\\}=(| W_ f|)^ 2,$ where ${\\mathcal M}=\\{(u,v)\\in H^ 1(-1,1)\\times H^ 1(-1,1);\\quad u(- 1)=u(1),$ $v(-1)=v(1),\\quad \\int^{1}_{-1}f^ 0(u,v)\\partial f^ 0\/\\partial u(u,v)d\\theta =\\int^{1}_{-1}f^ 0(u,v)\\partial f^ 0\/\\partial v(u,v)d\\theta =0\\}.$\nReviewer:\u00a0B.Dacorogna\n\n##### MSC:\n 46E35 Sobolev spaces and other spaces of \u201csmooth\u201d functions, embedding theorems, trace theorems 49J27 Existence theories for problems in abstract spaces 46E39 Sobolev (and similar kinds of) spaces of functions of discrete variables","date":"2021-04-17 18:35:41","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\": 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\": 0.9684929847717285, \"perplexity\": 164.89731818321988}, \"config\": {\"markdown_headings\": true, \"markdown_code\": false, \"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-2021-17\/segments\/1618038461619.53\/warc\/CC-MAIN-20210417162353-20210417192353-00094.warc.gz\"}"} | null | null |
<?php namespace Arcanesoft\Blog\ViewComposers\Admin\Dashboard;
use Arcanesoft\Blog\ViewComposers\AbstractComposer;
use Illuminate\Contracts\View\View;
/**
* Class PostsCountComposer
*
* @package Arcanesoft\Blog\ViewComposers\Dashboard
* @author ARCANEDEV <arcanedev.maroc@gmail.com>
*/
class PostsCountComposer extends AbstractComposer
{
/* -----------------------------------------------------------------
| Constants
| -----------------------------------------------------------------
*/
const VIEW = 'blog::admin._composers.dashboard.posts-total-box';
/* -----------------------------------------------------------------
| Main Methods
| -----------------------------------------------------------------
*/
/**
* Compose the view.
*
* @param \Illuminate\Contracts\View\View $view
*/
public function compose(View $view)
{
$view->with('postsCount', $this->cachedPosts()->count());
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,384 |
Q: Send CAN Messages to Multiple Specified Device IDs in Socketcan / Python-can similar to MCP_CAN Arduino library? Is there an equivalent in Python Socketcan / python-can that allows CAN messages to be sent to a specific destination device ID like how it's done on Arduino devices? The Arduino CAN bus boards use a MCP_CAN Library which allows for defining ID filter mode of the protocol controller in the command sendMsgBuf(ID, EXT, DLC, DATA). I have multiple devices connected which I can send specific CAN messages with ID field set to... CAN.sendMsgBuf(0x01...), CAN.sendMsgBuf(0x02...), CAN.sendMsgBuf(0x03...). However, when on a Linux board running socketcan, there's no equivalent Python-can with the cansend command.
Python-Can / Linux
import can
def send_one()
bus = can.interface.Bus(bustype='socketcan', channel='can0', bitrate=500000)
msg = can.Message(arbitration_id=0xc0ffee,
data=[0, 25, 0, 1, 3, 1, 4, 1],
is_extended_id=False)
try:
bus.send(msg)
print("Message sent on {}".format(bus.channel_info))
except can.CanError:
print("Message NOT sent")
if __name__ == '__main__':
send_one()
MCP_CAN Library for Arduino
//sendMsgBuf(ID, DLC, DATA)
...sendMsgBuf(0x01, DLC, DATA) //Send to Device ID 0x01
...sendMsgBuf(0x02, DLC, DATA) //Send to Device ID 0x02
...sendMsgBuf(0x02, DLC, DATA) //Send to Device ID 0x03
In a standard socketcan example, the can.Message() command sets the arbitration_id for the transmitting device, but it doesn't define the can_id of the device that I'm trying to send it to when I have multiple devices connected to the canbus which are waiting for messages. How would I send CAN messages to let's say can device 0x01, device 0x02, and device 0x03?
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,358 |
78 Views | 1
The Dallas Museum of Art's iconic new exhibition is all about the House of Dior
by Ilia Sybil Sdralli
Christian Dior was the pioneering French couturier who revolutionized fashion and created his fashion powerhouse set to be at the forefront of style and innovation. His work has been widely explored and studies and the subject of several impressive museum retrospectives in all over the world. He is the couturier associated with the iconic New Look and the re-introduction of strong feminity straight after the reserved fashions of the Second World War period-and perceived as one of the most groundbreaking of all time.
This summer, the Dallas Museum of Art is hosting perhaps its most ambitions exhibition-and it's all dedicated to the work of Mr. Dior and the talented designers that followed at the helm of his historical house. The exhibition, "Dior: From Paris to the World," was initially a great success when started in Paris and then transported to the Denver Art Museum, the only other American venue to host it. "The House of Dior has been a legendary force in fashion and visual culture for decades and continues to be an important influence that blurs the lines between fashion and art," said Dr. Agustín Arteaga, the DMA's Eugene McDermott Director.
With the addition of thirty additional dresses exclusively for the DMA, the Dallas exhibition is perhaps the most ambitious exhibition of all. It is curated by the museum's decorative arts curator, Sarah Schleuning and designed by architect Shohei Shigematsu who ambitiously transformed the inside of the museum's central space into a stage resembling a fashion catwalk-and sometimes even a 'fashion cathedral'. In fact, as chief curator Sarah Schleuning described it: "The exhibition takes audiences through more than seven decades of innovation, bringing together the most exciting, dynamic, and pivotal pieces."
The Dallas exhibition is a massive tale about not only monsieur Dior but the overall history of the House of Dior and the designers that shaped its vision. Nearly 200 dresses –some 500 exhibits in total are being showcased by no less than seven prominent designers that worked for Dior covering some 70 years of history. The curatorial approach was one of an open dialogue between the past and the future by presenting the work of chief designers Yves St. Laurent, John Galliano and today's design head Maria Grazia Chiuri. Sarah Schleuning called her work as an ambition to "tell the story of this celebrated haute couture house and to create an enchanting experience for visitors "aiming to "give them a new appreciation for the artistry of fashion and the legacy of Dior."
Dallas has a special, strong connection with Christian Dior. It is during Dior's first trip to the US in 1947 that the designer visited Dallas to be honored with the Neiman Marcus Award for his fashion service from Stanley Marcus, the founder of Neiman Marcus himself. In fact, he was one of the early supporters of Dior's mastery and one that campaigned for Dior to be widely recognized as a pioneer couturier-in a historical time where his current work was received with some criticism. Some 70 years later, Dallas is in fact paying tribute to the genius of Christian Dior and his legacy yet another time.
Dior: From Paris to the World is organized by the Denver Art Museum in association with the Dallas Museum of Art. The exhibition in Dallas is presented by PNC with leadership support by Nancy C. and Richard R. Rogers.
BASIC Spotlight Cars & Fashion @ LA Fashion Week
New Issue! BASIC #Spectacle Zara Larsson | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,326 |
John G. Stephenson (1809 in County Armagh, Ireland - 1893 in New Rochelle, N.Y.), an American coachbuilder, invented and patented the first streetcar to run on rails in the United States. Stephenson also designed the New York and Harlem Railroad which was formally opened on 26 November 1832. Twelve days later a horse-drawn streetcar built at Stephenson's works and named John Mason after the president of the railroad company, started the public service. Stephenson is therefore remembered as the creator of the tramway. Stephenson was the great-grandfather of Alan Stephenson Boyd, the first United States Secretary of Transportation.
Life
John Stephenson emigrated to the United States from Ireland with his parents, James and Grace Stephenson, when he was two years old. After attending public schools in New York City, he completed his education at the Wesleyan University in Middletown, Connecticut. At the age of 19, he became an apprentice to Abram Brower, the pioneer of the Broadway stage lines.
Stephenson died at his summer home in New Rochelle, New York in 1893.
First streetcar
In May 1831, Stephenson started his own business, the John Stephenson Company, on 667 Broadway where he built omnibus cars for Brower until a fire destroyed his shop in March 1832. He immediately moved to a new site on Elizabeth Street near Bleecker where he continued to build omnibuses which proved to be a huge success on the streets of New York.
However, soon afterwards he received an order from John Mason, a successful merchant and banker, to build a horse car for the New York and Harlem Railroad which had just been granted a charter authorizing a route from Fourth Avenue and the Bowery north to the Harlem River. The first stretch was opened from Prince to 14th Street on November 26, 1832, with a procession of the four cars developed for the company. Stephenson's car, named "John Mason" or simply the "Mason" after the company's president, was in the lead with the mayor and other dignitaries. He had modeled it after the English four-wheeled passenger railroad car but dropped the body down over the wheels for easier access. Four horses pulled the car and it carried up to 30 passengers in its three compartments. It was Stephenson's design which was finally adopted. In April 1833, he obtained a U.S. patent for it.
Failures and successes
Orders for his design came in not only received from New York and other U.S. cities but also from Cuba. In 1836, business prospered even more rapidly after Ross Winans developed his eight-wheeled vehicles. Stephenson built a larger factory at Fourth Avenue and 129th Street. At first business prospered and he received an increasing number of orders, especially for railway cars. He was doing good business when the panic of 1837 struck the country, causing him years of distress as the bonds he had accepted in lieu of cash for orders became worthless. In 1842, his business finally failed and he lost all his property. He was only able to pay his creditors 50 cents on the dollar.
Undeterred, Stephenson found a new site on West 27th Street, where in 1843 he started to develop a business which eventually covered 16 city lots. Streetcars continued to gain popularity, allowing Stephenson to prosper for the remainder of his life. It was not long before he had fully reimbursed all his creditors and became known as Honest John Stephenson. From 1852 he put all his efforts into building streetcars of various types as their popularity extended to cities throughout the world including, for example, Port Elizabeth, South Africa, Bombay and Caracas. For many years, he was the world's largest builder of streetcars. By the time of his death in 1893, his factory had 500 employees and was producing some 25 cars a week.
As time went by, Stephenson introduced a number of improvements to his streetcars. Perhaps the most important was to reduce the weight from 6,800 pounds to just 3,500 pounds, allowing just two rather than four horses to pull the vehicle. He achieved this by using hickory or ash instead of oak and adding larger windows rather than wood. He placed seats along the sides of the vehicle and used a single rear entrance rather than doors along the side.
He also devised many other improvements in rail car design and was successful in filing at least 11 patents in his own name. It is estimated that the John Stephenson Company made some 25,000 cars in the period 1876–1891 alone and an untold number over the life of the company.
Further reading
Burrows, Edwin G.; Wallace, Mike, Gotham: a history of New York City to 1898. New York: Oxford University Press, 1999, 1383p,
Carman, Harry James: The street surface railway franchises of New York City. New York, Columbia University; 1919, 259p.
Hornung, Clarence Pearson: Wheels across America: a pictorial cavalcade illustrating the early development of vehicular transportation. New York, A.S. Barnes, 1959, 341p.
Kennedy, William Sloane: Wonders and curiosities of the railway; or, Stories of the locomotive in every land. Chicago, S.C. Griggs and Company, 1884, 254p.
McShane, Clay; Tarr, Joel A: The horse in the city: living machines in the nineteenth century. Baltimore: The Johns Hopkins University Press, 2007, 242p.
White, John H.: Horsecars, cable cars, and omnibuses: all 107 photographs from the John Stephenson Company album, 1888. New York: Dover Publications, 1974, 54 leaves of plates.
References
19th-century American inventors
American railroad mechanical engineers
American railroad pioneers
Businesspeople from New Rochelle, New York
Wesleyan University alumni
1809 births
1893 deaths
Irish emigrants to the United States (before 1923)
People from County Armagh
19th-century American businesspeople | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 1,562 |
set -v on
export CGO_ENABLED=0
if which govendor 2>/dev/null; then
echo 'govendor exist'
else
go get -u -v github.com/kardianos/govendor
echo 'govendor does not exist'
fi
export GOOS=linux GOARCH=amd64
govendor build -o "${PWD##*/}-${GOOS}-${GOARCH}"
export GOOS=windows GOARCH=amd64
govendor build -o "${PWD##*/}-${GOOS}-${GOARCH}.exe"
export GOOS=darwin GOARCH=amd64
go build -o "${PWD##*/}-${GOOS}-${GOARCH}" | {
"redpajama_set_name": "RedPajamaGithub"
} | 2,935 |
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