text stringlengths 14 5.77M | meta dict | __index_level_0__ int64 0 9.97k ⌀ |
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Q: Complex number - arctan $$z=-2+2\sqrt{3}i\implies x=-2, y=2\sqrt3$$
$$r=\sqrt{x^2+y^2}=\sqrt{4+12}=4$$
$$\text{Angle}=\arctan\left(\frac{2\sqrt3}{-2}\right)+\pi=\frac{2\pi}{3}=120^\circ$$
1) May I know how $\arctan\left(\dfrac{2\sqrt3}{-2}\right)+\pi$ turns into $\dfrac{2\pi}{3}$?
2) Can I use calculator to do the calculation and how?
Thanks for the kindness!
A: Since $\arctan\left(-\sqrt3\right)=-\frac\pi3$, $\arctan\left(-\sqrt3\right)+\pi=\frac{2\pi}3$.
A: Notice, $\tan(-\theta)=-\tan\theta$
$$\therefore \arctan\left(\frac{2\sqrt3}{-2}\right)+\pi=\arctan\left(-\sqrt{3}\right)+\pi=-\frac{\pi}{3}+\pi=\frac{2\pi}{3}=120^\circ$$
You can use calculator to find amplitude & argument. Do remember $\tan^{-1}\sqrt3=\frac{\pi}{3}$
A: First of all, you need to know that arctan is the inverse function of tangent.
So, if $\tan(x)=y$, it follows that $\arctan(y):=\tan^{-1}(y)=\tan^{-1}(\tan(x))=x$
In this case, we have $\arctan(-\sqrt3)=\tan^{-1}(-\sqrt3)=-\pi/3$
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,249 |
1918 – The second anniversary of the execution of Roger Casement proved a quiet affair.
1918 – About 1,500 camogie, football and hurling matches were played across Ireland as part of a GAA protest against recent restrictions on the playing of these games without a permit.
1996 – A meeting between the Apprentice Boys of Derry and the Bogside Residents Association ended without agreement about the march due to take place on 10 August 1996. A series of meetings between the two groups had been chaired by the local Member of Parliament John Hume, then leader of the Social Democratic and Labour Party (SDLP).
1997 – A Catholic taxi driver survived an attempt to kill him when the gun being used by a Loyalist paramilitary jammed. The attack occurred in the Parkmore estate in Lurgan. The Loyalist Volunteer Force (LVF) later claimed responsibility for the attack.
1997 – A hoax bomb was sent to Sammy Wilson, a Democratic Unionist Party (DUP) councillor, at Belfast City Hall.
1997 – Secretary of State for Northern Ireland, Marjorie (Mo) Mowlam, held her first meeting with President of Sinn Féin (SF), Gerry Adams, since the IRA announced its renewed ceasefire.
1997 – The Irish Times carried a report that John Hume, leader of the Social Democratic and Labour Party (SDLP), was considering accepting the position of President of the Republic of Ireland as an agreed all-party candidate. Hume did not comment on the story.
1997 – The Bogside Residents Group (BRG) gave agreement to the planned Apprentice Boys of Derry (ABD) march in the city on 9 August 1997. This followed the news that the RUC would reroute a number of ABD 'feeder' parades in other Nationalist areas of Northern Ireland.
1999 – Two pipe-bombs were discovered by the RUC in a hedge in Glengormley, Co Antrim. Police made the discovery at 2.45am during a search carried out at the junction between Elmfield Crescent and Elmfield Road in the town. | {
"redpajama_set_name": "RedPajamaC4"
} | 291 |
/****************************************************************************
* wqueue/uwqueue/uwork_queue.c
*
* Copyright (C) 2009-2011, 2014 Gregory Nutt. All rights reserved.
* Author: Gregory Nutt <gnutt@nuttx.org>
*
* Redistribution and use in source and binary forms, with or without
* modification, are permitted provided that the following conditions
* are met:
*
* 1. Redistributions of source code must retain the above copyright
* notice, this list of conditions and the following disclaimer.
* 2. Redistributions in binary form must reproduce the above copyright
* notice, this list of conditions and the following disclaimer in
* the documentation and/or other materials provided with the
* distribution.
* 3. Neither the name NuttX nor the names of its contributors may be
* used to endorse or promote products derived from this software
* without specific prior written permission.
*
* THIS SOFTWARE IS PROVIDED BY THE COPYRIGHT HOLDERS AND CONTRIBUTORS
* "AS IS" AND ANY EXPRESS OR IMPLIED WARRANTIES, INCLUDING, BUT NOT
* LIMITED TO, THE IMPLIED WARRANTIES OF MERCHANTABILITY AND FITNESS
* FOR A PARTICULAR PURPOSE ARE DISCLAIMED. IN NO EVENT SHALL THE
* COPYRIGHT OWNER OR CONTRIBUTORS BE LIABLE FOR ANY DIRECT, INDIRECT,
* INCIDENTAL, SPECIAL, EXEMPLARY, OR CONSEQUENTIAL DAMAGES (INCLUDING,
* BUT NOT LIMITED TO, PROCUREMENT OF SUBSTITUTE GOODS OR SERVICES; LOSS
* OF USE, DATA, OR PROFITS; OR BUSINESS INTERRUPTION) HOWEVER CAUSED
* AND ON ANY THEORY OF LIABILITY, WHETHER IN CONTRACT, STRICT
* LIABILITY, OR TORT (INCLUDING NEGLIGENCE OR OTHERWISE) ARISING IN
* ANY WAY OUT OF THE USE OF THIS SOFTWARE, EVEN IF ADVISED OF THE
* POSSIBILITY OF SUCH DAMAGE.
*
****************************************************************************/
/****************************************************************************
* Included Files
****************************************************************************/
#include <tinyara/config.h>
#include <sys/types.h>
#include <stdint.h>
#include <signal.h>
#include <assert.h>
#include <queue.h>
#include <errno.h>
#include <tinyara/clock.h>
#include <tinyara/wqueue.h>
#include "wqueue.h"
/****************************************************************************
* Pre-processor Definitions
****************************************************************************/
/****************************************************************************
* Private Type Declarations
****************************************************************************/
/****************************************************************************
* Public Variables
****************************************************************************/
/****************************************************************************
* Private Variables
****************************************************************************/
/****************************************************************************
* Private Functions
****************************************************************************/
/****************************************************************************
* Public Functions
****************************************************************************/
/****************************************************************************
* Name: work_queue
*
* Description:
* Queue user-mode work to be performed at a later time. All queued work
* will be performed on the worker thread of of execution (not the caller's).
*
* The work structure is allocated by caller, but completely managed by
* the work queue logic. The caller should never modify the contents of
* the work queue structure; the caller should not call work_queue()
* again until either (1) the previous work has been performed and removed
* from the queue, or (2) work_cancel() has been called to cancel the work
* and remove it from the work queue.
*
* Input parameters:
* qid - The work queue ID (index)
* work - The work structure to queue
* worker - The worker callback to be invoked. The callback will invoked
* on the worker thread of execution.
* arg - The argument that will be passed to the workder callback when
* int is invoked.
* delay - Delay (in clock ticks) from the time queue until the worker
* is invoked. Zero means to perform the work immediately.
*
* Returned Value:
* Zero on success, a negated errno on failure
*
****************************************************************************/
int work_queue(int qid, FAR struct work_s *work, worker_t worker, FAR void *arg, clock_t delay)
{
int ret;
if (qid == USRWORK) {
struct wqueue_s *usrwq = get_usrwork();
ret = work_qqueue(usrwq, work, worker, arg, delay);
if (ret != OK) {
return ret;
}
return work_signal(USRWORK);
} else {
return -EINVAL;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 9,591 |
Q: How to send key events to webkit.WebView() control? I created mini browser for touch panels. I used Python, GTK and WebKit. I want to create touch keyboard built-in that browser. How to send key events to webkit.WebView() control?
My code:
#!/usr/bin/python
import webkit, gtk
window = gtk.Window()
browser = webkit.WebView()
window.add(browser)
window.show()
browser.show()
browser.load_uri("http://google.com")
window.fullscreen()
window.connect("delete-event", gtk.main_quit)
window.set_title("Dashboard")
gtk.main()
Sorry for my bad English...
A: You can do it like this:
event = gtk.gdk.Event(gtk.gdk.KEY_PRESS)
event.keyval = gtk.keysyms.Return
event.time = 0
browser.emit('key-press-event', event)
for example, to send an Enter key press.
A: You will have to use javascript in the code, there is a good example of communicating with webkit on this page:
http://www.aclevername.com/articles/python-webgui/
Read trough it all to understand his logic. Essentially he uses a simple javascript to send data back via titlechange in the browser.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,459 |
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<p><strong>Widerruf Ihrer Einwilligung zur Datenverarbeitung</strong></p>
<p>Nur mit Ihrer ausdrücklichen Einwilligung sind einige Vorgänge der Datenverarbeitung möglich. Ein Widerruf Ihrer bereits erteilten Einwilligung ist jederzeit möglich. Für den Widerruf genügt eine formlose Mitteilung per E-Mail. Die Rechtmäßigkeit der bis zum Widerruf erfolgten Datenverarbeitung bleibt vom Widerruf unberührt.</p>
<p><strong>Recht auf Beschwerde bei der zuständigen Aufsichtsbehörde</strong></p>
<p>Als Betroffener steht Ihnen im Falle eines datenschutzrechtlichen Verstoßes ein Beschwerderecht bei der zuständigen Aufsichtsbehörde zu. Zuständige Aufsichtsbehörde bezüglich datenschutzrechtlicher Fragen ist der Landesdatenschutzbeauftragte des Bundeslandes, in dem sich der Sitz unseres Unternehmens befindet. Der folgende Link stellt eine Liste der Datenschutzbeauftragten sowie deren Kontaktdaten bereit: <a href="https://www.bfdi.bund.de/DE/Infothek/Anschriften_Links/anschriften_links-node.html" target="_blank">https://www.bfdi.bund.de/DE/Infothek/Anschriften_Links/anschriften_links-node.html</a>.</p>
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<p>Ihnen steht das Recht zu, Daten, die wir auf Grundlage Ihrer Einwilligung oder in Erfüllung eines Vertrags automatisiert verarbeiten, an sich oder an Dritte aushändigen zu lassen. Die Bereitstellung erfolgt in einem maschinenlesbaren Format. Sofern Sie die direkte Übertragung der Daten an einen anderen Verantwortlichen verlangen, erfolgt dies nur, soweit es technisch machbar ist.</p>
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<p>Sie haben jederzeit im Rahmen der geltenden gesetzlichen Bestimmungen das Recht auf unentgeltliche Auskunft über Ihre gespeicherten personenbezogenen Daten, Herkunft der Daten, deren Empfänger und den Zweck der Datenverarbeitung und ggf. ein Recht auf Berichtigung, Sperrung oder Löschung dieser Daten. Diesbezüglich und auch zu weiteren Fragen zum Thema personenbezogene Daten können Sie sich jederzeit über die im Impressum aufgeführten Kontaktmöglichkeiten an uns wenden.</p>
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<p>In Server-Log-Dateien erhebt und speichert der Provider der Website automatisch Informationen, die Ihr Browser automatisch an uns übermittelt. Dies sind:</p>
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<li>Besuchte Seite auf unserer Domain</li>
<li>Datum und Uhrzeit der Serveranfrage</li>
<li>Browsertyp und Browserversion</li>
<li>Verwendetes Betriebssystem</li>
<li>Referrer URL</li>
<li>Hostname des zugreifenden Rechners</li>
<li>IP-Adresse</li>
</ul>
<p>Es findet keine Zusammenführung dieser Daten mit anderen Datenquellen statt. Grundlage der Datenverarbeitung bildet Art. 6 Abs. 1 lit. b DSGVO, der die Verarbeitung von Daten zur Erfüllung eines Vertrags oder vorvertraglicher Maßnahmen gestattet.</p>
<p><small>Quelle: Datenschutz-Konfigurator von <a href="http://www.mein-datenschutzbeauftragter.de" target="_blank">mein-datenschutzbeauftragter.de</a></small></p>
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Dreiner Weg 32 <br />
51688 Wipperfürth <br />
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,114 |
Q: AttributeError: 'str' object has no attribute 'datetime': Python This is my code:
# Fetch today's date
Date = datetime.today().strftime('%Y-%m-%d-%H.%M.%S')
# Variable for log file
LogFile = os.getcwd()
print(LogFile)
os.mkdir("Logs12")
f = open("Password_Expiry_Date_Log_"+(Date)+".txt", "w+")
#Date Calculations
Date_Before = Date.datetime(Days_Before)
Days_After = Date.datetime(Days_After)
When I try to initialize the variable 'Date_Before', i get the error AttributeError: 'str' object has no attribute 'datetime'. However, I need date to be in a string format to write in into a text filename. Is there any workaround for this?
Thanks in advance.
A: You could do this:
from datetime import datetime, timedelta
# Fetch today's date
date = datetime.today()
string_date = date.strftime('%Y-%m-%d-%H.%M.%S')
# Variable for log file
log_file = os.getcwd()
print(log_file)
os.mkdir("Logs12")
f = open(f"Password_Expiry_Date_Log_{string_date}.txt", "w+")
f.close()
#Date Calculations
date_before = datetime.today() - timedelta(days=1)
days_after = datetime.today() + timedelta(days=1)
I also updated your string names to conform to PEP8
Edit: I also improved your syntax, please remember that you always need to close your files.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 9,889 |
Q: email verification and forgot password being tag as spam in Laravel 8 I upload my app on siteground. I ask them already a bunch of question regarding this issue and they just gave me links to their tech articles, followed them but still doesn't work. I been searching for an answer for too long. I hope you can suggest a procedure or help me to fix this.
The issue I am having is the email verification and forgot password is being sent to spam section or in some other instance it is blocked due to security reason.
here's the bounce email I am receiving
This message was created automatically by mail delivery software.
A message that you sent could not be delivered to one or more of its
recipients. This is a permanent error. The following address(es) failed:
user@gmail.com
host outbound.mailspamprotection.com [xx.xxx.xxx.xx]
SMTP error from remote mail server after end of data:
550 This message has been reported as spam by other users.
Reporting-MTA: dns; xxxxxxx.siteground.xxxxxx
**Action:** failed
**Final-Recipient:** rfc822;user@gmail.com
**Status:** 5.0.0
**Remote-MTA:** dns; outbound.mailspamprotection.com
**Diagnostic-Code:** smtp; 550 This message has been reported as spam by other users.
MESSAGE HEADER
**Return-path:** <>
**Envelope-to:** serveremail@mysite.com
**Delivery-date:** Fri, 26 Nov 2021 19:16:44 +0000
**Received:** from mailnull by xxxxxxxx.siteground.xxxxxx with local (Exim 4.90-.1)
id 1mqgiC-000Aqa-Ln
for serveremail@mysite.com; Fri, 26 Nov 2021 19:16:44 +0000
**X-Failed-Recipients:** user@gmail.com
**Auto-Submitted:** auto-replied
**From:** Mail Delivery System <Mailer-Daemon@sgp1000.siteground.asia>
**To:** serveremail@mysite.com
**References:** <f1bb46cdbbacd8db694337f14ad1fafc@127.0.0.1>
**Content-Type:** multipart/report; report-type=delivery-status; boundary=1637954204-eximdsn-1244935018
**MIME-Version:** 1.0
**Subject:** Mail delivery failed: returning message to sender
**Message-Id:** <E1mqgiC-000Aqa-Ln@sgp1000.siteground.asia>
**Date:** Fri, 26 Nov 2021 19:16:44 +0000
here's what I have in my env file
MAIL_DRIVER=smtp
MAIL_PORT=465
MAIL_HOST=xxxxxxx.siteground.xxxxxx
MAIL_USERNAME=serveremail@site.com
MAIL_PASSWORD='mypassword'
MAIL_ENCRYPTION=ssl
MAIL_FROM_ADDRESS=serveremail@site.com
MAIL_FROM_NAME='my app name'
I have other notification functions my app that notify/email the admins. It triggers when some certain actions are made like if there's new application etc. those notifications are being sent to inbox and not in spam.
Any suggestions? or links with really working steps that I can follow? Any suggestions will be very much appreciated.
Thank you!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 1,618 |
Q: Rails Validations :format getting skipped on create Rails is skipping the :format validation on create. On create, it's accepting anything at all. Then on update :presence and :format are both working as expected. How can I alter this so it'll :allow_blank on create and check the format too?
validates :mail, :allow_blank => true, :on => :create,
:format => { :with => /^([^@\s]+)@((?:[-a-z0-9]+\.)*[a-z]{2,})$/i },
:length => { :maximum => 60 },
:presence => true, :on => :update
A: Try using separate validates statements for :create, :update, and all life cycle events. Your :on => :update option is essentially overwriting the :on => :create option when you lump them all together.
validates :mail, :format => { :with => /^([^@\s]+)@((?:[-a-z0-9]+\.)*[a-z]{2,})$/i },
:length => { :maximum => 60 }
validates :mail, :allow_blank => true, :on => :create,
validates :mail, :presence => true, :on => :update
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,348 |
Q: cURL login : Invalid length for a Base-64 char array I've searched my problem.
Questions similar to the problem was too much, but I could not get to any conclusion.
I have written the following code
<?php
$aPosts = array("__VIEWSTATE" => "/wEPDwULLTEwMDM5MDE3NDUPZBYEZg8VARJTYWZhcmFmYXJpbi1TZXBlaHJkAgEPZBYEAgUPEA8WAh4LXyFEYXRhQm91bmRnZGRkZAINDxYCHgdWaXNpYmxlaBYCZg9kFgJmDxUBE9io2K7YtNmG2KfZhdmHINmH2KdkZMdboHK0C5vwU/0H8jDKN8JIUGAWK7y6psPERPkF/dYP",
"__EVENTVALIDATION" => "/wEWBwLVs6iwDQKl1bK4CQK1qbSRCwLHlbuRCALGla+RCAKC3IeGDAKqs9XKA+XnuPlm+7BeBMKdG9tnXAq2MVVFKjVmbCbQG4g9jB9u",
"txtUsername" => "",
"txtPassword" => "d41d8cd98f00b204e9800998ecf8427e",
"dplLanguage" => "en",
"btnPublicLogin" => "login",
);
$posts = "";
foreach($aPosts as $key => $value){
$posts .= $key . "=" . $value . "&";
}
$url="http://www.example.com/login.aspx";
$cookie="cookiew.txt";
$rnd = rand(2547896321456987, 9999999999999999);
//$postdata = 'dplLanguage=fa&txtUsername=&txtPassword=d41d8cd98f00b204e9800998ecf8427e&__VIEWSTATE=/wEPDwULLTEwMDM5MDE3NDUPZBYEZg8VARJTYWZhcmFmYXJpbi1TZXBlaHJkAgEPZBYEAgUPEA8WAh4LXyFEYXRhQm91bmRnZGRkZAINDxYCHgdWaXNpYmxlaBYCZg9kFgJmDxUBE9io2K7YtNmG2KfZhdmHINmH2KdkZMdboHK0C5vwU/0H8jDKN8JIUGAWK7y6psPERPkF/dYP&__EVENTVALIDATION=/wEWBwLVs6iwDQKl1bK4CQK1qbSRCwLHlbuRCALGla+RCAKC3IeGDAKqs9XKA+XnuPlm+7BeBMKdG9tnXAq2MVVFKjVmbCbQG4g9jB9u';
$postdata = $posts;
$ch = curl_init();
curl_setopt ($ch, CURLOPT_URL, $url);
curl_setopt ($ch, CURLOPT_SSL_VERIFYPEER, FALSE);
curl_setopt ($ch, CURLOPT_USERAGENT, "Mozilla/5.0 (Windows; U; Windows NT 5.1; en-US; rv:1.8.1.6) Gecko/20070725 Firefox/2.0.0.6");
curl_setopt ($ch, CURLOPT_TIMEOUT, 60);
curl_setopt ($ch, CURLOPT_FOLLOWLOCATION, 1);
curl_setopt ($ch, CURLOPT_RETURNTRANSFER, 0);
curl_setopt ($ch, CURLOPT_COOKIEJAR, $cookie);
$rnd = rand(2547896321456987, 9999999999999999);
curl_setopt ($ch, CURLOPT_REFERER, $url);
curl_setopt ($ch, CURLOPT_POSTFIELDS, $postdata);
curl_setopt ($ch, CURLOPT_POST, 1);
$a = curl_exec ($ch);
$url= 'http://www.example.com/search.aspx';
curl_setopt ($ch, CURLOPT_URL, $url);
$a = curl_exec ($ch);
curl_close($ch);
?>
If I don't send the variables __VIEWSTATE and __EVENTVALIDATION, Page returns to login screen.
But when I use this vars I see this error:
Invalid length for a Base-64 char array.
A: Try:
foreach($aPosts as $key => $value){
$posts .= $key . "=" . urlencode($value) . "&";
}
$posts = trim($posts, "=");
Hope it helps
A: Replace your foreach by this function :
http_build_query($aPosts);
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 624 |
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Noted Psychoacoustics and Audio Coding Researcher Louis D. Fielder Slated for Heyser Lecture at AES New York 2019
Louis D. Fielder, Psychoacoustics and Audio Coding Pioneer, to Give Heyser Lecture at AES New York 2019
— AES Fellow and former Society president takes the stage of the lauded lecture series to share insights from his vast experience in research and development of cutting-edge industry technologies —
The Audio Engineering Society has announced that Louis D. Fielder, a leading researcher in the fields of psychoacoustics and audio coding, will deliver the Heyser Memorial Lecture at the upcoming AES New York Convention on Thursday, October 17 at 6:30 pm. Offering insights from decades of industry research and development, Fielder will deliver the lecture titled "Psychoacoustics Applied to Dynamic-Range and Nonlinear-Distortion Assessment" as part of the Special Events series taking place during the AES New York Convention, October 16 – 19.
"The psychoacoustics of noise detection, measurements of noise in the digital-audio recording / storage / reproduction chain, and measurements of peak-acoustic pressures in music performances are combined to determine the requirements for noise-free reproduction of music," begins Fielder's abstract for the lecture. "It is found that the required ratio between the maximum reproduction levels and the perceived audibility of noise can be as much as 124 decibels. When more practical circumstances are considered, this requirement is shown to drop to more feasible values." The lecture will continue by examining how auditory masking allows for the assessment of nonlinear distortions in digital-audio conversion systems operating at low-signal levels, along with examples of digital-audio conversion systems. Fielder's lecture will conclude with a discussion of expanded use of masking and present a model for determining non-linear distortions in headphones and low-frequency loudspeakers.
"Louis Fielder's expertise and experience exemplify the stature of Heyser Lecture presenters," shares AES Technical Council Chair Steve Hutt. "As we celebrate the 20th year of the Heyser Lecture series, we are delighted to bring Louis to the stage to share his unique insights."
Fielder's industry experience began in 1976 as an electronic component designer for custom sound reinforcement systems at Paul Veneklasen and Associates. In 1978, he became involved in digital-audio and magnetic recording research at the Ampex Corporation. His interest in applying psychoacoustics to the design and analysis of digital-audio conversion systems soon led to other career opportunities, including his work at Dolby Laboratories (1984 – 2018) on the application of psychoacoustics to the development of audio systems and the development of a number of bit-rate reduction audio coders for music distribution, transmission, and storage applications including AC-1, AC-2, AC-3, Enhanced AC-3, AAC, and Dolby E. Additionally, Fielder managed the Sound Technology Research Department at Dolby Laboratories in San Francisco (2005 – 2009) and has led further research into perceptually derived limits for the performance for digital audio conversion, low-frequency loudspeaker systems, distortion limits for loudspeakers/headphones, loudspeaker-room equalization, and headphone virtualization. Other industry involvement and recognition include his status as a life Fellow of the AES, a recipient of the AES Silver Medal Award, a senior life member of the IEEE, a life member of the SMPTE, and an emeritus member of the Acoustical Society of America. Fielder served on the AES Board of Governors (1990–1992), as AES President (1994 – 1995) and as AES Treasurer (2005 – 2009).
The Richard C. Heyser Memorial Lecture series, hosted by the AES Technical Council at each convention, honors the extensive contributions to the Society made by Heyser, who was widely known for his ability to communicate new and complex technical ideas with great clarity and patience.
Visit aesshow.com to find out how you can attend the AES New York Convention's Heyser Memorial Lecture, as part of the Convention's four full days of pro audio events, exhibition and networking opportunities with audio engineers from around the world.
Event Link: http://www.aes.org/events/147/specialevents/?ID=6819
Posted: Wednesday, August 21, 2019
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Centro Universitário Ítalo Brasileiro (popularly known as UniÍtalo), is a private Brazilian university located in the Santo Amaro district of São Paulo.
UniÍtalo offers 18 undergraduate and 27 graduate degree programs. It also offers university extension courses, online courses and language courses.
History
IEPAC, UniÍtalo's sponsor, was founded by professor and Italian immigrant Pasquale Cascino on January 25, 1949. The university originally only offered typing courses and began offering undergraduate courses in 1972.
In 1994, the institution expanded and moved to the Santo Amaro district of São Paulo. During this period, the university expanded its undergraduate course offerings.
The university began offering graduate courses in 2006. That same year UniÍtalo was formally recognized by the Brazilian Ministry of Education as Centro Universitário Ítalo Brasileiro.
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UniÍtalo offers classes in several time slots, including nontraditional times, such as 5:45 AM (dawn). Other available class times are 8 AM, 8:50 AM, 1:30 PM, 2:20 PM and 7 PM.
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Health Management: Hospital Management / Public Health / PSF
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Universities and colleges in São Paulo (state)
Educational institutions established in 1949
1949 establishments in Brazil
Private universities and colleges in Brazil | {
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{"url":"https:\/\/networkx.org\/documentation\/networkx-2.3\/_modules\/networkx\/algorithms\/connectivity\/cuts.html","text":"Warning\n\nThis documents an unmaintained version of NetworkX. Please upgrade to a maintained version and see the current NetworkX documentation.\n\n# Source code for networkx.algorithms.connectivity.cuts\n\n# -*- coding: utf-8 -*-\n\"\"\"\nFlow based cut algorithms\n\"\"\"\nimport itertools\nimport networkx as nx\n\n# Define the default maximum flow function to use in all flow based\n# cut algorithms.\nfrom networkx.algorithms.flow import edmonds_karp\nfrom networkx.algorithms.flow import build_residual_network\ndefault_flow_func = edmonds_karp\n\nfrom .utils import (build_auxiliary_node_connectivity,\nbuild_auxiliary_edge_connectivity)\n\n__author__ = '\\n'.join(['Jordi Torrents <jtorrents@milnou.net>'])\n\n__all__ = ['minimum_st_node_cut',\n'minimum_node_cut',\n'minimum_st_edge_cut',\n'minimum_edge_cut']\n\n[docs]def minimum_st_edge_cut(G, s, t, flow_func=None, auxiliary=None,\nresidual=None):\n\"\"\"Returns the edges of the cut-set of a minimum (s, t)-cut.\n\nThis function returns the set of edges of minimum cardinality that,\nif removed, would destroy all paths among source and target in G.\nEdge weights are not considered. See :meth:minimum_cut for\ncomputing minimum cuts considering edge weights.\n\nParameters\n----------\nG : NetworkX graph\n\ns : node\nSource node for the flow.\n\nt : node\nSink node for the flow.\n\nauxiliary : NetworkX DiGraph\nAuxiliary digraph to compute flow based node connectivity. It has\nto have a graph attribute called mapping with a dictionary mapping\nnode names in G and in the auxiliary digraph. If provided\nit will be reused instead of recreated. Default value: None.\n\nflow_func : function\nA function for computing the maximum flow among a pair of nodes.\nThe function has to accept at least three parameters: a Digraph,\na source node, and a target node. And return a residual network\nthat follows NetworkX conventions (see :meth:maximum_flow for\ndetails). If flow_func is None, the default maximum flow function\n(:meth:edmonds_karp) is used. See :meth:node_connectivity for\ndetails. The choice of the default function may change from version\nto version and should not be relied on. Default value: None.\n\nresidual : NetworkX DiGraph\nResidual network to compute maximum flow. If provided it will be\nreused instead of recreated. Default value: None.\n\nReturns\n-------\ncutset : set\nSet of edges that, if removed from the graph, will disconnect it.\n\n--------\n:meth:minimum_cut\n:meth:minimum_node_cut\n:meth:minimum_edge_cut\n:meth:stoer_wagner\n:meth:node_connectivity\n:meth:edge_connectivity\n:meth:maximum_flow\n:meth:edmonds_karp\n:meth:preflow_push\n:meth:shortest_augmenting_path\n\nExamples\n--------\nThis function is not imported in the base NetworkX namespace, so you\nhave to explicitly import it from the connectivity package:\n\n>>> from networkx.algorithms.connectivity import minimum_st_edge_cut\n\nWe use in this example the platonic icosahedral graph, which has edge\nconnectivity 5.\n\n>>> G = nx.icosahedral_graph()\n>>> len(minimum_st_edge_cut(G, 0, 6))\n5\n\nIf you need to compute local edge cuts on several pairs of\nnodes in the same graph, it is recommended that you reuse the\ndata structures that NetworkX uses in the computation: the\nauxiliary digraph for edge connectivity, and the residual\nnetwork for the underlying maximum flow computation.\n\nExample of how to compute local edge cuts among all pairs of\nnodes of the platonic icosahedral graph reusing the data\nstructures.\n\n>>> import itertools\n>>> # You also have to explicitly import the function for\n>>> # building the auxiliary digraph from the connectivity package\n>>> from networkx.algorithms.connectivity import (\n... build_auxiliary_edge_connectivity)\n>>> H = build_auxiliary_edge_connectivity(G)\n>>> # And the function for building the residual network from the\n>>> # flow package\n>>> from networkx.algorithms.flow import build_residual_network\n>>> # Note that the auxiliary digraph has an edge attribute named capacity\n>>> R = build_residual_network(H, 'capacity')\n>>> result = dict.fromkeys(G, dict())\n>>> # Reuse the auxiliary digraph and the residual network by passing them\n>>> # as parameters\n>>> for u, v in itertools.combinations(G, 2):\n... k = len(minimum_st_edge_cut(G, u, v, auxiliary=H, residual=R))\n... result[u][v] = k\n>>> all(result[u][v] == 5 for u, v in itertools.combinations(G, 2))\nTrue\n\nYou can also use alternative flow algorithms for computing edge\ncuts. For instance, in dense networks the algorithm\n:meth:shortest_augmenting_path will usually perform better than\nthe default :meth:edmonds_karp which is faster for sparse\nnetworks with highly skewed degree distributions. Alternative flow\nfunctions have to be explicitly imported from the flow package.\n\n>>> from networkx.algorithms.flow import shortest_augmenting_path\n>>> len(minimum_st_edge_cut(G, 0, 6, flow_func=shortest_augmenting_path))\n5\n\n\"\"\"\nif flow_func is None:\nflow_func = default_flow_func\n\nif auxiliary is None:\nH = build_auxiliary_edge_connectivity(G)\nelse:\nH = auxiliary\n\nkwargs = dict(capacity='capacity', flow_func=flow_func, residual=residual)\n\ncut_value, partition = nx.minimum_cut(H, s, t, **kwargs)\nreachable, non_reachable = partition\n# Any edge in the original graph linking the two sets in the\n# partition is part of the edge cutset\ncutset = set()\nfor u, nbrs in ((n, G[n]) for n in reachable):\ncutset.update((u, v) for v in nbrs if v in non_reachable)\n\nreturn cutset\n\n[docs]def minimum_st_node_cut(G, s, t, flow_func=None, auxiliary=None, residual=None):\nr\"\"\"Returns a set of nodes of minimum cardinality that disconnect source\nfrom target in G.\n\nThis function returns the set of nodes of minimum cardinality that,\nif removed, would destroy all paths among source and target in G.\n\nParameters\n----------\nG : NetworkX graph\n\ns : node\nSource node.\n\nt : node\nTarget node.\n\nflow_func : function\nA function for computing the maximum flow among a pair of nodes.\nThe function has to accept at least three parameters: a Digraph,\na source node, and a target node. And return a residual network\nthat follows NetworkX conventions (see :meth:maximum_flow for\ndetails). If flow_func is None, the default maximum flow function\n(:meth:edmonds_karp) is used. See below for details. The choice\nof the default function may change from version to version and\nshould not be relied on. Default value: None.\n\nauxiliary : NetworkX DiGraph\nAuxiliary digraph to compute flow based node connectivity. It has\nto have a graph attribute called mapping with a dictionary mapping\nnode names in G and in the auxiliary digraph. If provided\nit will be reused instead of recreated. Default value: None.\n\nresidual : NetworkX DiGraph\nResidual network to compute maximum flow. If provided it will be\nreused instead of recreated. Default value: None.\n\nReturns\n-------\ncutset : set\nSet of nodes that, if removed, would destroy all paths between\nsource and target in G.\n\nExamples\n--------\nThis function is not imported in the base NetworkX namespace, so you\nhave to explicitly import it from the connectivity package:\n\n>>> from networkx.algorithms.connectivity import minimum_st_node_cut\n\nWe use in this example the platonic icosahedral graph, which has node\nconnectivity 5.\n\n>>> G = nx.icosahedral_graph()\n>>> len(minimum_st_node_cut(G, 0, 6))\n5\n\nIf you need to compute local st cuts between several pairs of\nnodes in the same graph, it is recommended that you reuse the\ndata structures that NetworkX uses in the computation: the\nauxiliary digraph for node connectivity and node cuts, and the\nresidual network for the underlying maximum flow computation.\n\nExample of how to compute local st node cuts reusing the data\nstructures:\n\n>>> # You also have to explicitly import the function for\n>>> # building the auxiliary digraph from the connectivity package\n>>> from networkx.algorithms.connectivity import (\n... build_auxiliary_node_connectivity)\n>>> H = build_auxiliary_node_connectivity(G)\n>>> # And the function for building the residual network from the\n>>> # flow package\n>>> from networkx.algorithms.flow import build_residual_network\n>>> # Note that the auxiliary digraph has an edge attribute named capacity\n>>> R = build_residual_network(H, 'capacity')\n>>> # Reuse the auxiliary digraph and the residual network by passing them\n>>> # as parameters\n>>> len(minimum_st_node_cut(G, 0, 6, auxiliary=H, residual=R))\n5\n\nYou can also use alternative flow algorithms for computing minimum st\nnode cuts. For instance, in dense networks the algorithm\n:meth:shortest_augmenting_path will usually perform better than\nthe default :meth:edmonds_karp which is faster for sparse\nnetworks with highly skewed degree distributions. Alternative flow\nfunctions have to be explicitly imported from the flow package.\n\n>>> from networkx.algorithms.flow import shortest_augmenting_path\n>>> len(minimum_st_node_cut(G, 0, 6, flow_func=shortest_augmenting_path))\n5\n\nNotes\n-----\nThis is a flow based implementation of minimum node cut. The algorithm\nis based in solving a number of maximum flow computations to determine\nthe capacity of the minimum cut on an auxiliary directed network that\ncorresponds to the minimum node cut of G. It handles both directed\nand undirected graphs. This implementation is based on algorithm 11\nin [1]_.\n\n--------\n:meth:minimum_node_cut\n:meth:minimum_edge_cut\n:meth:stoer_wagner\n:meth:node_connectivity\n:meth:edge_connectivity\n:meth:maximum_flow\n:meth:edmonds_karp\n:meth:preflow_push\n:meth:shortest_augmenting_path\n\nReferences\n----------\n.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.\nhttp:\/\/www.cse.msu.edu\/~cse835\/Papers\/Graph_connectivity_revised.pdf\n\n\"\"\"\nif auxiliary is None:\nH = build_auxiliary_node_connectivity(G)\nelse:\nH = auxiliary\n\nmapping = H.graph.get('mapping', None)\nif mapping is None:\nraise nx.NetworkXError('Invalid auxiliary digraph.')\nif G.has_edge(s, t) or G.has_edge(t, s):\nreturn []\nkwargs = dict(flow_func=flow_func, residual=residual, auxiliary=H)\n\n# The edge cut in the auxiliary digraph corresponds to the node cut in the\n# original graph.\nedge_cut = minimum_st_edge_cut(H, '%sB' % mapping[s], '%sA' % mapping[t],\n**kwargs)\n# Each node in the original graph maps to two nodes of the auxiliary graph\nnode_cut = set(H.nodes[node]['id'] for edge in edge_cut for node in edge)\nreturn node_cut - set([s, t])\n\n[docs]def minimum_node_cut(G, s=None, t=None, flow_func=None):\nr\"\"\"Returns a set of nodes of minimum cardinality that disconnects G.\n\nIf source and target nodes are provided, this function returns the\nset of nodes of minimum cardinality that, if removed, would destroy\nall paths among source and target in G. If not, it returns a set\nof nodes of minimum cardinality that disconnects G.\n\nParameters\n----------\nG : NetworkX graph\n\ns : node\nSource node. Optional. Default value: None.\n\nt : node\nTarget node. Optional. Default value: None.\n\nflow_func : function\nA function for computing the maximum flow among a pair of nodes.\nThe function has to accept at least three parameters: a Digraph,\na source node, and a target node. And return a residual network\nthat follows NetworkX conventions (see :meth:maximum_flow for\ndetails). If flow_func is None, the default maximum flow function\n(:meth:edmonds_karp) is used. See below for details. The\nchoice of the default function may change from version\nto version and should not be relied on. Default value: None.\n\nReturns\n-------\ncutset : set\nSet of nodes that, if removed, would disconnect G. If source\nand target nodes are provided, the set contains the nodes that\nif removed, would destroy all paths between source and target.\n\nExamples\n--------\n>>> # Platonic icosahedral graph has node connectivity 5\n>>> G = nx.icosahedral_graph()\n>>> node_cut = nx.minimum_node_cut(G)\n>>> len(node_cut)\n5\n\nYou can use alternative flow algorithms for the underlying maximum\nflow computation. In dense networks the algorithm\n:meth:shortest_augmenting_path will usually perform better\nthan the default :meth:edmonds_karp, which is faster for\nsparse networks with highly skewed degree distributions. Alternative\nflow functions have to be explicitly imported from the flow package.\n\n>>> from networkx.algorithms.flow import shortest_augmenting_path\n>>> node_cut == nx.minimum_node_cut(G, flow_func=shortest_augmenting_path)\nTrue\n\nIf you specify a pair of nodes (source and target) as parameters,\nthis function returns a local st node cut.\n\n>>> len(nx.minimum_node_cut(G, 3, 7))\n5\n\nIf you need to perform several local st cuts among different\npairs of nodes on the same graph, it is recommended that you reuse\nthe data structures used in the maximum flow computations. See\n:meth:minimum_st_node_cut for details.\n\nNotes\n-----\nThis is a flow based implementation of minimum node cut. The algorithm\nis based in solving a number of maximum flow computations to determine\nthe capacity of the minimum cut on an auxiliary directed network that\ncorresponds to the minimum node cut of G. It handles both directed\nand undirected graphs. This implementation is based on algorithm 11\nin [1]_.\n\n--------\n:meth:minimum_st_node_cut\n:meth:minimum_cut\n:meth:minimum_edge_cut\n:meth:stoer_wagner\n:meth:node_connectivity\n:meth:edge_connectivity\n:meth:maximum_flow\n:meth:edmonds_karp\n:meth:preflow_push\n:meth:shortest_augmenting_path\n\nReferences\n----------\n.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.\nhttp:\/\/www.cse.msu.edu\/~cse835\/Papers\/Graph_connectivity_revised.pdf\n\n\"\"\"\nif (s is not None and t is None) or (s is None and t is not None):\nraise nx.NetworkXError('Both source and target must be specified.')\n\n# Local minimum node cut.\nif s is not None and t is not None:\nif s not in G:\nraise nx.NetworkXError('node %s not in graph' % s)\nif t not in G:\nraise nx.NetworkXError('node %s not in graph' % t)\nreturn minimum_st_node_cut(G, s, t, flow_func=flow_func)\n\n# Global minimum node cut.\n# Analog to the algorithm 11 for global node connectivity in [1].\nif G.is_directed():\nif not nx.is_weakly_connected(G):\nraise nx.NetworkXError('Input graph is not connected')\niter_func = itertools.permutations\n\ndef neighbors(v):\nreturn itertools.chain.from_iterable([G.predecessors(v),\nG.successors(v)])\nelse:\nif not nx.is_connected(G):\nraise nx.NetworkXError('Input graph is not connected')\niter_func = itertools.combinations\nneighbors = G.neighbors\n\n# Reuse the auxiliary digraph and the residual network.\nH = build_auxiliary_node_connectivity(G)\nR = build_residual_network(H, 'capacity')\nkwargs = dict(flow_func=flow_func, auxiliary=H, residual=R)\n\n# Choose a node with minimum degree.\nv = min(G, key=G.degree)\n# Initial node cutset is all neighbors of the node with minimum degree.\nmin_cut = set(G[v])\n# Compute st node cuts between v and all its non-neighbors nodes in G.\nfor w in set(G) - set(neighbors(v)) - set([v]):\nthis_cut = minimum_st_node_cut(G, v, w, **kwargs)\nif len(min_cut) >= len(this_cut):\nmin_cut = this_cut\n# Also for non adjacent pairs of neighbors of v.\nfor x, y in iter_func(neighbors(v), 2):\nif y in G[x]:\ncontinue\nthis_cut = minimum_st_node_cut(G, x, y, **kwargs)\nif len(min_cut) >= len(this_cut):\nmin_cut = this_cut\n\nreturn min_cut\n\n[docs]def minimum_edge_cut(G, s=None, t=None, flow_func=None):\nr\"\"\"Returns a set of edges of minimum cardinality that disconnects G.\n\nIf source and target nodes are provided, this function returns the\nset of edges of minimum cardinality that, if removed, would break\nall paths among source and target in G. If not, it returns a set of\nedges of minimum cardinality that disconnects G.\n\nParameters\n----------\nG : NetworkX graph\n\ns : node\nSource node. Optional. Default value: None.\n\nt : node\nTarget node. Optional. Default value: None.\n\nflow_func : function\nA function for computing the maximum flow among a pair of nodes.\nThe function has to accept at least three parameters: a Digraph,\na source node, and a target node. And return a residual network\nthat follows NetworkX conventions (see :meth:maximum_flow for\ndetails). If flow_func is None, the default maximum flow function\n(:meth:edmonds_karp) is used. See below for details. The\nchoice of the default function may change from version\nto version and should not be relied on. Default value: None.\n\nReturns\n-------\ncutset : set\nSet of edges that, if removed, would disconnect G. If source\nand target nodes are provided, the set contains the edges that\nif removed, would destroy all paths between source and target.\n\nExamples\n--------\n>>> # Platonic icosahedral graph has edge connectivity 5\n>>> G = nx.icosahedral_graph()\n>>> len(nx.minimum_edge_cut(G))\n5\n\nYou can use alternative flow algorithms for the underlying\nmaximum flow computation. In dense networks the algorithm\n:meth:shortest_augmenting_path will usually perform better\nthan the default :meth:edmonds_karp, which is faster for\nsparse networks with highly skewed degree distributions.\nAlternative flow functions have to be explicitly imported\nfrom the flow package.\n\n>>> from networkx.algorithms.flow import shortest_augmenting_path\n>>> len(nx.minimum_edge_cut(G, flow_func=shortest_augmenting_path))\n5\n\nIf you specify a pair of nodes (source and target) as parameters,\nthis function returns the value of local edge connectivity.\n\n>>> nx.edge_connectivity(G, 3, 7)\n5\n\nIf you need to perform several local computations among different\npairs of nodes on the same graph, it is recommended that you reuse\nthe data structures used in the maximum flow computations. See\n:meth:local_edge_connectivity for details.\n\nNotes\n-----\nThis is a flow based implementation of minimum edge cut. For\nundirected graphs the algorithm works by finding a 'small' dominating\nset of nodes of G (see algorithm 7 in [1]_) and computing the maximum\nflow between an arbitrary node in the dominating set and the rest of\nnodes in it. This is an implementation of algorithm 6 in [1]_. For\ndirected graphs, the algorithm does n calls to the max flow function.\nThe function raises an error if the directed graph is not weakly\nconnected and returns an empty set if it is weakly connected.\nIt is an implementation of algorithm 8 in [1]_.\n\n--------\n:meth:minimum_st_edge_cut\n:meth:minimum_node_cut\n:meth:stoer_wagner\n:meth:node_connectivity\n:meth:edge_connectivity\n:meth:maximum_flow\n:meth:edmonds_karp\n:meth:preflow_push\n:meth:shortest_augmenting_path\n\nReferences\n----------\n.. [1] Abdol-Hossein Esfahanian. Connectivity Algorithms.\nhttp:\/\/www.cse.msu.edu\/~cse835\/Papers\/Graph_connectivity_revised.pdf\n\n\"\"\"\nif (s is not None and t is None) or (s is None and t is not None):\nraise nx.NetworkXError('Both source and target must be specified.')\n\n# reuse auxiliary digraph and residual network\nH = build_auxiliary_edge_connectivity(G)\nR = build_residual_network(H, 'capacity')\nkwargs = dict(flow_func=flow_func, residual=R, auxiliary=H)\n\n# Local minimum edge cut if s and t are not None\nif s is not None and t is not None:\nif s not in G:\nraise nx.NetworkXError('node %s not in graph' % s)\nif t not in G:\nraise nx.NetworkXError('node %s not in graph' % t)\nreturn minimum_st_edge_cut(H, s, t, **kwargs)\n\n# Global minimum edge cut\n# Analog to the algorithm for global edge connectivity\nif G.is_directed():\n# Based on algorithm 8 in [1]\nif not nx.is_weakly_connected(G):\nraise nx.NetworkXError('Input graph is not connected')\n\n# Initial cutset is all edges of a node with minimum degree\nnode = min(G, key=G.degree)\nmin_cut = set(G.edges(node))\nnodes = list(G)\nn = len(nodes)\nfor i in range(n):\ntry:\nthis_cut = minimum_st_edge_cut(H, nodes[i], nodes[i + 1], **kwargs)\nif len(this_cut) <= len(min_cut):\nmin_cut = this_cut\nexcept IndexError: # Last node!\nthis_cut = minimum_st_edge_cut(H, nodes[i], nodes[0], **kwargs)\nif len(this_cut) <= len(min_cut):\nmin_cut = this_cut\n\nreturn min_cut\n\nelse: # undirected\n# Based on algorithm 6 in [1]\nif not nx.is_connected(G):\nraise nx.NetworkXError('Input graph is not connected')\n\n# Initial cutset is all edges of a node with minimum degree\nnode = min(G, key=G.degree)\nmin_cut = set(G.edges(node))\n# A dominating set is \\lambda-covering\n# We need a dominating set with at least two nodes\nfor node in G:\nD = nx.dominating_set(G, start_with=node)\nv = D.pop()\nif D:\nbreak\nelse:\n# in complete graphs the dominating set will always be of one node\n# thus we return min_cut, which now contains the edges of a node\n# with minimum degree\nreturn min_cut\nfor w in D:\nthis_cut = minimum_st_edge_cut(H, v, w, **kwargs)\nif len(this_cut) <= len(min_cut):\nmin_cut = this_cut\n\nreturn min_cut","date":"2023-01-29 02:37:49","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.4587076008319855, \"perplexity\": 7675.208569149221}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 20, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2023-06\/segments\/1674764499697.75\/warc\/CC-MAIN-20230129012420-20230129042420-00054.warc.gz\"}"} | null | null |
\section{Introduction}\label{sec:intro}
Astrophysical observations suggest that ordinary luminous and baryonic matter contributes only about 5\% to the total energy density of the Universe, with the rest due to dark matter (DM) at $\sim 25\%$, and dark energy (DE) at $\sim 70\%$.
Despite the overwhelming cosmological evidence for the existence of DM, and the considerable effort of the scientific community over several decades, there is as of yet no definitive evidence for DM in terrestrial experiments.
Currently all the evidence for DM comes from observations carried out over distances greater than or comparable to galactic scales \cite{Bertone2005}.
In general, in order to perform a direct DM detection experiment, these vast 10\,kpc ($\sim10^{13}\,$m) distances must be extrapolated down to scales that are accessible in laboratory settings ($\sim$\,1\,m).
This extrapolation leads to a variety of plausible theoretical possibilities for DM models, ranging from elementary particles to black holes.
Considering the broad variety of models and the associated assortment of non-gravitational interactions of DM with ordinary matter, it is important to constrain DM models by creatively reanalyzing archival data~\cite{BudkerPT2015}.
Compared to investments into dedicated experiments, this is a relatively low-cost strategy with potential for important discovery.
Here we develop a method based on Bayesian statistics for a time-domain DM search using data accumulated by networks of precision measurements devices.
The field of low-energy precision measurements has proven to be an important area for probing fundamental laws and searching for new physics, that is often complementary to collider experiments~\cite{AtomicReview2017}.
The idea of using a distributed network of precision measurement devices to search for DM and other exotic physics signatures is one promising approach~\cite{Pospelov2013,Pustelny2013,DereviankoDM2014,GPSDM2017,Delva2017}.
The particular network considered here is the Global Positioning System (GPS), a satellite constellation of nominally 32 satellites in medium-Earth orbit (altitude $\sim20,000$\,km) housing atomic clocks, as well as a large number of atomic clocks on ground-based receiver stations.
Following the proposal of Ref.~\cite{DereviankoDM2014}, we use the GPS constellation as a $\sim50,000\,$km aperture sensor array, analyzing the satellite and terrestrial atomic clock data for transient signatures of exotic physics, such as DM and DE.
High-quality timing data from the GPS network exists for the past 18 years, and is made freely available by, e.g., the Jet Propulsion Laboratory (JPL), NASA \cite{[{Data freely available online:~}][]JPLigsac,MurphyJPL2015}. This dataset is routinely augmented with more recent data.
The global scale of the GPS network offers a unique opportunity to search for spatially-extended DM objects (or ``clumps''), such as topological defects (TDs)~\cite{Vilenkin1985}, which are otherwise not detectable by most ongoing and planned DM searches.
The large number of clocks and the very large aperture of the network increase both the chance of an interaction and the sensitivity of the search, since we seek the correlated propagation of new physics signals throughout the entire network.
The large network diameter also increases the overall interaction time.
Therefore, by analyzing the GPS timing data, one can perform a sensitive search for transient signals of exotic physics, and if no sought signals are found, stringent limits on the relevant interaction strengths can be placed.
Recently, our GPS.DM collaboration carried out an initial analysis~\cite{GPSDM2017} of the archival GPS data, looking for signatures of a particular type of TDs (domain walls, quasi-2D cosmic structures).
While no such signatures were found, we placed limits on certain DM couplings to atoms that are many orders of magnitude more stringent than the previous constraints.
Here, we present a search method based on Bayesian statistics. We demonstrate that compared to our initial search, the Bayesian approach greatly increases the search sensitivity. This approach also broadens the discovery reach to more general DM models and to lower DM field masses.
Our approach is not limited in scope to the GPS network, but applies equally to other networks of precision measurement devices.
In principle, timing data from any other atomic clocks as well as data from other precision measurement devices can be included in the analysis.
In particular, there are similarities to another experiment, the Global Network of Optical Magnetometers for Exotic physics (GNOME), which employs a geographically-distributed Earth-based network of magnetometers to search for transient signatures of exotic physics, including topological defect DM \cite{Pospelov2013,Pustelny2013}.
Techniques described in this work may prove useful for such experiments.
Beyond ``clumpy'' DM models, one can use networks to search for other types of DM, such as non-self-interacting virialized ultralight fields (VULFs), that lead to signals that oscillate at the DM Compton frequency.
Such a search would rely on a multi-node spatio-temporal correlation function \cite{DereviankoVULF2016}.
One may also search for both transient and oscillating effects due to ultralight DM (and other exotic physics) with laser interferometers and gravitational wave detectors
\cite{
Arvanitaki2014,Arvanitaki2015,StadnikLaser2015,*StadnikLasInf2015,StadnikDMalpha2015,Arvanitaki2016,Hall2016,Yang2016},
by directly exploiting the scalar--photon coupling
\cite{Sikivie1983,
ADMX2010
CAST2015,*CAST201
},
atomic spectroscopy \cite{Tilburg2015,Hees2016,Kalaydzhyan2017} and
noise statistics \cite{RobertsAsymm2018,Kalaydzhyan2018},
electric dipole moment searches
\cite{Graham2013,Budker2014,RobertsCosPRD2014,*RobertsCosmic2014},
and even pulsar timing \cite{StadnikDefects2014}.
The structure of this paper is as follows.
Section~\ref{sec:theory} reviews the background theory for topological defect DM, the DM-induced transient variations of fundamental constants, and how atomic clocks can be used to search for DM signatures.
Section \ref{sec:noise} discusses aspects of the GPS network relevant to our search.
In Section \ref{sec:SearchMethod} we describe the Bayesian statistics method for the data analysis and the search, and in Section \ref{sec:TestMethod} we use this method with simulated data to demonstrate its efficacy.
Finally, in Section \ref{sec:results} we present the projected sensitivity and the discovery reach of the search.
This paper has three appendices, which include the derivation of the velocity distributions for macroscopic DM objects, a brief characterization of the noise properties of the clock data relevant to our search, and the expected signals for a few specific DM models.
The supplementary information~\cite{[{See the Supplementary Information ``GPS satellite clock noise characteristics'' in the {\em ancillary files} section of this paper's arXiv page}][]Supplement}
presents a detailed analysis of noise characteristics such as Allan variance, power spectrum, and autocorrelation for individual GPS satellite clocks.
Since the intended audience includes both atomic and particle physics communities, we restore $\hbar$ and $c$ in the formulas. We use the rationalized Heaviside-Lorentz units for electromagnetism.
\section{Theory}\label{sec:theory}
\subsection{Ultralight dark matter and topological defects}\label{sec:TDs}
Despite the extensive searches, both laboratory direct detection and high-energy collider experiments have so far failed to yield convincing evidence for the existence of weakly interacting massive particles (WIMPs) with masses $\sim$\,10\,--\,$10^4\un{GeV}$, see, e.g.,
Refs.~\cite{AgneseSCDMS2014,
PandaX2016,
LUXjan2017,
XENON100-2016,XENON-1T-2017}.
While WIMPs are theoretically well-motivated, they are by no means the only DM candidate.
The null WIMP searches have partially motivated searches for ultralight bosonic DM, such as axions \cite{Peccei1977a,*Peccei1977b,Srednicki1981,Sikivie1983,Preskill1983}.
While direct DM searches with particle detectors rely on measuring energy deposition by individual DM particles, precision measurement techniques are well suited for detecting candidates that act as coherent entities on the scale of individual devices or their networks.
In other words, precision measurement devices can be used for detecting ultralight DM and this approach probes the mass region that is
complementary to particle detectors.
Ultralight fields may form coherent (on certain time-scales) oscillating fields, or they may form stable macroscopic-scale objects
\cite{
Sikivie1982,
Press1989,
Vilenkin1994,
Battye1999,
Durrer2002,
Friedland2003,
Avelino2008%
}. The formation of macroscopic ``clumpy'' DM objects requires self-interactions in the dark sector.
An example of macroscopic DM are topological defects, which may have various dimensionalities: monopoles (0D), strings (1D), and domain walls (2D). Depending on their cosmological fluid equation of state, these objects can contribute to both DM and DE.
The interactions of light scalar fields with standard model (SM) fields can be phenomenologically parameterized as a sum of effective interaction Lagrangians (portals) \cite{DereviankoDM2014}
\begin{equation}\label{eq:effL}
\ensuremath{{\cal L}}_{\rm int} = \ensuremath{{\cal L}}^{\rm PS} + \ensuremath{{\cal L}}^{\rm S^1} + \ensuremath{{\cal L}}^{\rm S^2} + \; \ldots \; ,
\end{equation}
where $\ensuremath{{\cal L}}^{\rm PS}$ represents the pseudoscalar (axionic) portal, and $\ensuremath{{\cal L}}^{\rm S^1}$ and $\ensuremath{{\cal L}}^{\rm S^2}$ are the linear and quadratic scalar portals, respectively.
The linear and quadratic scalar portals can lead to changes in the effective values of certain fundamental constants and thus cause shifts in atomic transition frequencies. Atomic clocks in particular are sensitive probes of varying fundamental constants.
The axionic portal leads to interactions that mimic spin-dependent shifts due to fictitious magnetic fields, and thus are well suited for magnetometry searches \cite{Pospelov2013,*Pustelny2013,KimballQball2017}.
We also note that there are stringent limits on the interaction strength for the linear scalar interaction coming from astrophysics and gravitational experiments (see, e.g., \cite{Raffelt1999,Bertotti2003}).
However, the constraints on the quadratic portal are substantially weaker \cite{Olive2008}.
For concreteness, here we will focus on the quadratic scalar portal.
While we refer to specific models, namely topological defect DM with quadratic scalar couplings, it is important to note that the search technique is not limited in scope to this possibility.
Any large (on laboratory scales), ``clumpy'' object that interacts with standard model particles is detectable using this scheme.
Examples of such other models include $Q$-balls~\cite{Coleman1985,Kusenko2001,Lee1989}, solitons~\cite{Marsh2015,Schive2014}, axion stars~\cite{Hogan1988,Kolb1993}, and other stable objects formed due to self-interactions in the DM sector.
\subsection{Searching for dark matter with atomic clocks}
Since the microscopic nature of DM is unknown, we take a phenomenological approach for the non-gravitational interactions with ordinary matter~(see, e.g., \cite{DereviankoDM2014}).
Explicitly for the quadratic scalar portal, we have
\begin{equation}
\label{eq:scalarPortal}
-\ensuremath{{\cal L}}^{\rm S^2}= \phi^2\left(
{\Gamma_f}{m_f c^2 \bar \psi_f \psi_f }
+\Gamma_\alpha\frac{F_{\mu\nu}^2 }{4}
+ \,\ldots \right),
\end{equation}
where $\phi$ is the scalar DM field (measured in units of energy), $m_f$ are the fermion masses,
$\psi_f$ and $F_{\mu\nu}$
are the SM fermion fields and the electromagnetic Faraday tensor, respectively,
and $\Gamma$ are coupling constants that quantify the strength of the DM--SM interaction.
There is an implicit sum over the SM fermions $f$ in the above equation.
The above Lagrangian leads to the effective redefinition of fundamental masses and coupling constants,
\begin{align}
\label{eq:varalpha}
\alpha^{\rm eff}(\v{r},t) &= \left[1+ \Gamma_{\alpha}\,{\phi^2(\v{r},t)}\right] {\alpha },\\
m_{f}^{\rm eff}(\v{r},t) &= \left[1+ \Gamma_{f}\,{\phi^2(\v{r},t)}\right] {m_{f}},
\label{eq:varqcd}
\end{align}
where $\alpha\approx1/137$ is the electromagnetic fine-structure constant
and $m_f$ are the fermion (electron $m_e$ and light quark $m_q\equiv[m_u+m_d]/2$) masses.
The coupling constants $\Gamma$ have units of $[{\rm Energy}]^{-2}$ and to aid the comparison with previous literature we also define the effective energy scales $\Lambda_X \equiv 1/\sqrt{\abs{\Gamma_X}}$ with $X=\alpha,\,m_e,\,m_q$.
Considering TDs, the DM field $\phi^2\to0$ outside the defect, hence the effective couplings are only realized inside the defect.\footnote{Strictly speaking, this condition requires an auxiliary DM field, see Ref.~\cite{DereviankoDM2014} for the mechanism. }
The field amplitude inside the defect, $A$, can be linked to the average energy density inside the defect as $\rho_{\rm inside}= A^2/(\h c \, d^2)$, where $d$ is the spatial extent or width of the defect.
In TD models, the width $d$ is set naturally by the field Compton wavelength, $d=\hbar/(m_\phi c)$, where $m_\phi$ is the mass of the DM field particles; in general, we treat $d$ as a free observational parameter.
Further, in the assumption that these objects saturate the local DM energy density, one can link $A$ and $d$ to the local DM energy density $\rho_{\rm DM}$,
\begin{equation}\label{eq:A2}
A^2 = (\hbar c) \, \rho_{\rm DM} v_g \ensuremath{{\cal T}} d ,
\end{equation}
where
$\ensuremath{{\cal T}}$
is the average time between close encounters of the DM objects with a point-like device, and the galactic velocity $v_g=\braket{v}\sim300\un{km}\un{s}^{-1} \sim 10^{-3} c$ is the average relative velocity of DM objects that cross paths with the Earth.
From Eqs.~(\ref{eq:varalpha}) -- (\ref{eq:varqcd}), we may relate the observable DM-induced atomic frequency shift to the transient variation of fundamental constants (and thus to the DM field parameters).
The fractional shift in the frequency $\omega_0$ of a particular clock transition can be expressed as
\begin{equation}
\label{eq:variation}
\frac{\delta \omega(\v{r}, t)}{ \omega_0}
= \sum_X \k_X\Gamma_X{ \phi^2(\v{r}, t)}{} \equiv \ensuremath{\Gamma_{\rm eff}}\,\phi^2(\v{r}, t),
\end{equation}
where $X$ runs over relevant fundamental constants, and $\k_X$ are dimensionless sensitivity coefficients.
For convenience, we introduced the effective constant, $\Gamma_{\rm eff} \equiv \sum_X \kappa_X \Gamma_X$, which depends on the specific clock.
The dimensionless sensitivity coefficients $\k_X$ are known from atomic and nuclear structure calculations.
For example, considering only the variation in the fine-structure constant $\alpha$ and ignoring relativistic effects, the optical and microwave transitions frequencies scale as
$\omega^{\rm opt}_c \propto \alpha^2$, and
$\omega^{\rm mw}_c \propto \alpha^4$, respectively.
Relativistic atomic-structure effects add small corrections to these scalings~\cite{Dzuba2003,*Angstmann2004}.
For the microwave Rb, Cs, and H clocks of the GPS network, the effective coupling constants read (using computations~\cite{Dzuba2003,*Angstmann2004,Flambaum2006a,*Dinh2009})
\begin{align}
\label{eq:Crb}
\ensuremath{\Gamma_{\rm eff}}({\rm ^{87}Rb})&={4.34}\,{\Gamma_\a}-{0.069}\,{\Gamma_{m_q}}+{2}\,{\Gamma_{m_e}},
\\
\label{eq:Ccs}
\ensuremath{\Gamma_{\rm eff}}({\rm ^{133}Cs})&={4.83}\,{\Gamma_\a}-{0.048}\,{\Gamma_{m_q}}+{2}\,{\Gamma_{m_e}},
\\
\label{eq:Ch}
\ensuremath{\Gamma_{\rm eff}}({\rm ^{1}H})&={4}\,{\Gamma_\alpha}-{0.150}\,{\Gamma_{m_q}}+{2}\,{\Gamma_{m_e}}.
\end{align}
The values of $\k_{m_q}$ come from a combination of shifts in the nuclear magnetic moment and in the nuclear size~\cite{Flambaum2006,
[{For Cs, the contributions from these two effects are roughly equal in magnitude and opposite in sign ($\k_\mu+\k_{\rm hq}=0.009-0.007$ \cite{Dinh2009}), and this part of $\k_q$ for Cs is therefore sensitive to uncertainties in the nuclear structure calculations}][]bibtexnote}, and from the variation in the proton mass with $\delta m_p/m_p = 0.05\,\delta m_q/m_q$ \cite{Flambaum2004}.
Although each clock type is sensitive to a combination of three coupling constants,
by combining results for three types of clocks within the network one can unfold individual coupling constants $\Gamma_X$ or, equivalently, individual energy scales $\Lambda_X$.
Until recently, the existing constraints came from observations of supernova emission~\cite{Olive2008}:
$\Lambda_{m_e,\a}\gtrsim3\un{TeV}$, and
$\Lambda_{m_p}\gtrsim15\un{TeV}$.
More stringent constraints for certain regions of the $(d,\ensuremath{{\cal T}})$ parameter space have recently been placed on $\Lambda_{\a}$ using a laboratory optical Sr clock by the Toru\'n group~\cite{Wcislo2016}.
Using 16 years of archival GPS data, our GPS.DM collaboration constrained $\Lambda_{\a}, \Lambda_{m _e}$, and $\Lambda_{m_q}$~\cite{GPSDM2017};
that initial search focused on domain walls.
These newly-established constraints reach the $\sim 10^7\un{TeV}$ level depending on the size of the objects and the frequency of their encounters with the Earth.
With the model-specific theoretical background established, now we proceed to developing a method for a sensitive search for macroscopic DM objects.
We will demonstrate that compared to our initial search~\cite{GPSDM2017}, the developed method improves the sensitivity by several orders of magnitude, and also substantially increases the range of probed DM field masses.
It is also sufficiently general to enable mining for signatures of all the prototypical topological defects: monopoles, strings, and walls.
The method is Bayesian in nature and we start with describing known DM halo properties, velocity distribution and directionality, that serve as priors to the search.
\subsection{Priors on velocity distribution and event rate}
\label{Sec:Priors}
\begin{figure}
\includegraphics[width=0.3\textwidth]{./perp}
\caption{\small Geometry of a domain wall encounter with the Earth.
Here, $d$ is the domain wall width, $\v{v}$ is the relative velocity of the encounter with component $\v{v}_\perp$ perpendicular to the wall surface, and $\eta$ is the angle between $\v{v}$ and $\v{v}_\perp$.
The incident direction of the wall, $\v{\hat n}$, is defined as pointing away from the Earth center, so that $\v{\hat n} = -\v{v}/\abs{\v{v}}$ and $\v{\hat n}_\perp \equiv -\v{v}_\perp/\abs{\v{v}_\perp}$.
}\label{fig:vperp}
\end{figure}
\begin{figure*}
\includegraphics[width=0.30\textwidth]{./v-scalar}~~~
\includegraphics[width=0.30\textwidth]{./angles}~~~
\includegraphics[width=0.30\textwidth]{./tau}
\caption{\small Probability densities for the DM scalar velocity (left), incident angles (middle), and transit time for the GPS constellation (right).
The forward-facing angle $\psi=\pi$ points in the direction of the galactic motion of the Solar system, towards the Cygnus constellation.
The blue curves show the standard halo model distributions (relevant for monopole-like DM objects), and the red curves show the distributions for velocities perpendicular to the wall (relevant for domain walls and strings).
}
\label{fig:distros}
\end{figure*}
We form our search priors based on the standard halo model (SHM), see, e.g., Ref.~\cite{Bovy:2012tw}.
Within the SHM framework, the velocity distribution of DM objects in the galactic rest frame is isotropic and quasi-Maxwellian; further details are given in Appendix~\ref{sec:vel-distro}.
The Milky Way rotates through the DM halo, with the Sun moving at $v\simeq 220\un{km}\un{s}^{-1}$ in the direction towards the Cygnus constellation.
This defines the most probable incident direction for a collision with a DM object; in fact, more than 90\% of events are expected to come from the forward-facing hemisphere, as shown below.
We define the unit-vector, $\v{\hat n}_g$, that points from the Earth center along this direction.
Further, as shown in Fig.~\ref{fig:vperp}, we define the incident direction of the DM object, $\v{\hat n}$, to be pointing away from the Earth center, so that $\v{\hat n} = -\v{v}/\abs{\v{v}}$, where $\v{v}$ is the velocity vector of the DM object.
The angle of incidence $\psi$ is defined as $\v{\hat n} \cdot \v{\hat n}_g=-\cos\psi$. According to this definition, the forward-facing angle $\psi=\pi$ points in the direction of the galactic motion of the Solar system, towards the Cygnus constellation.
We consider three topological defect ``templates'': domain walls, strings, and monopoles.
We assume that over the length scales of the GPS constellation a string/wall can be modeled to be straight/flat.
A domain wall that crosses the GPS constellation incident with a velocity $\v{v}$ that is at an angle $\eta$ to the vector normal to the wall, would be indistinguishable from a wall (of the same width) incident with a (slower) velocity
\begin{equation}\label{eq:eta}
v_\perp=v\cos\eta
\end{equation}
that is normal to the wall, see Fig.~\ref{fig:vperp}.
We will refer to the $\v{v}_\perp$ component of the relative velocity $\v{v}$ as the ``perpendicular'' velocity, and define $\v{\hat n}_\perp \equiv -\v{v}_\perp/\abs{\v{v}_\perp}$.
The same argument applies to strings (for strings, $\v{v}_\perp$ is defined to lie in the plane containing $\v{v}$ and the string symmetry axis).
Therefore, in these cases, the more relevant quantity is the distribution of the perpendicular velocities, $f_{v_\perp}$; this distribution is derived in Appendix~\ref{sec:vel-distro}.
Since we are focusing on macroscopic DM objects, it is also instructive to consider the distribution of transit durations.
The transit duration, $\t=d/v$, is defined as the time it takes a DM object of width $d$ to sweep through a point in space (or a single device).
Similarly we can consider $\t_{\rm GPS}=D_{\rm GPS}/v$, the time for the center of the DM object to pass the entire GPS constellation.
Our derived speed, incident angle, and transit duration distributions are shown for monopole- and wall-like objects in Fig.~\ref{fig:distros}.
We treat the expected event rate, $1/\ensuremath{{\cal T}}$ [see Eq.~(\ref{eq:A2})], as a free parameter.
This parameter can be linked to the number density of DM objects in the galaxy.
For monopole-like objects (including non-topological solitons, Q-balls, bubbles etc.), the relevant quantity is the volume number density, while for strings and domain walls it is the areal and linear number densities, respectively.
Thereby, $\ensuremath{{\cal T}}$ can be related to the energy density inside the DM object as
\begin{equation}\label{eq:T}
\ensuremath{{\cal T}} = \frac{\rho_{\rm inside}}{\rho_{\rm TDM}} \frac{d}{v_g},
\end{equation}
where $\rho_{\rm TDM}$ is the galactic energy density of the considered DM objects.
In the assumption that these objects saturate the local DM density, we have $\rho_{\rm TDM} = \rho_{\rm DM}$.
Direct measurements of the local DM density give $0.3\pm0.1\un{GeV}\un{cm}^{-3}$~\cite{Bovy:2012tw}; we take $\rho_{\rm DM}\approx 0.4\un{GeV}\un{cm}^{-3}$ for definitiveness and to be consistent with recent literature.
Note that Eq.~(\ref{eq:T}) is model independent and applies to any DM object of characteristic size $d$.
\section{GPS Architecture and Clock Solutions}\label{sec:noise}
A detailed description of modern GPS data acquisition and processing techniques and their application in precision geodesy can be found in Ref.~\cite{[][{; see also references therein.}]Blewitt2015307}. Details relevant to DM searches with GPS constellation
are given in Ref.~\cite{GPSDM2017}. Here, we briefly review the main aspects of GPS and introduce relevant concepts and terminology.
GPS works by broadcasting microwave signals from nominally 32 satellites in medium-Earth orbit (altitude $\sim$\,$20,000$\,km).
The transmissions are driven by an atomic clock (either based on Rb or Cs atoms) on board each satellite.
It is namely the carrier phase of these microwave signals that is measured by the specialized GPS receivers and is used in deriving the GPS clock solutions.
Typically, each satellite houses four atomic clocks, only one of which is broadcasting at any given time.
Clock swaps are marked in databases supporting the archival GPS dataset.
There are also a large number of ground-based receiver stations, several of which employ highly-stable H-maser clocks.
The more recent satellites predominantly employ Rb clocks as it has become clear that unpredictable variations in clock phases for the Cs-clock satellites are significantly worse than for Rb.
As of early September 2017, there were 30 Rb satellites and only two Cs satellites in orbit.
The GPS satellites are grouped into several generations, called blocks: II, IIA, IIR, and IIF~\cite{GPS.gov}, and each satellite is assigned a unique identifier known as the Space Vehicle Number (SVN).
Each subsequent block was built with significant improvements, and the effect of these improvements can be seen in the noise characteristics of the satellite clocks, as discussed below.
Block III satellites are currently under development, and are to be launched from mid-2018.
Table~\ref{tab:GPSsats} presents a summary, including the number of days worth of data that is available for each satellite block in the archival data set.
Further, the network can be extended to incorporate the network of Earth-based receiver clocks, as well as clocks from other Global Navigation Satellite Systems, such as the European Galileo, Russian GLONASS, and Chinese BeiDou, and networks of laboratory clocks~\cite{Hees2016,Wcislo2016,Riehle2017,Delva2017}.
\begin{table}%
\centering%
\caption{\small Summary of GPS satellite clocks. The currently employed data set, ranging from 5 May 2000 to 2 September 2017, consists of a total of 186,\,700 clock-days.}%
\begin{ruledtabular}%
\begin{tabular}{llrrr}%
\multicolumn{1}{c}{Block}&\multicolumn{1}{l}{Years active}&\multicolumn{2}{c}{Days in data}&\multicolumn{1}{l}{In orbit\tablenotemark[1]} \\
\cline{3-4}
\multicolumn{2}{c}{}&\multicolumn{1}{c}{Cs}&\multicolumn{1}{c}{Rb}&\multicolumn{1}{c}{} \\
\hline%
I&1978--1995&0 &0 &0\\
II&1989--2007&5304&1181&0\\
IIA&1990--2017&39557&33319&1\\
IIR&1997--&0&92287&19\\
IIF&2010--&2541&12511&12\\
\end{tabular}%
\end{ruledtabular}%
\label{tab:GPSsats}%
\tablenotetext[1]{As of September 2017. Also as of that date, only two satellites (both block IIF) use Cs clocks.}
\end{table}%
Here we analyze data generated by the Jet Propulsion Laboratory (JPL)~\cite{JPLigsac}, in which clock time-series are given at $\tau_0=30\,{\rm s}$ intervals (epochs).
The data are clock biases, that is the difference in the time readings (clock phases) between the given clock and a reference clock.
The same reference clock is used for the entire GPS network for each day.
The biases are generated using data from a global network of $\sim100$ GPS station receivers \cite{IGSnetwork} by a mature analysis system that is used routinely for purposes of centimeter-level satellite orbit determination, and millimeter-level positioning for scientific purposes, such as plate tectonics, Earth rotation, and geodynamics.
We also note that while the currently available clock time-series are sampled every 30 seconds, the raw GPS data is sampled every second for some stations.
It is therefore possible to re-process the GPS data to generate higher-rate 1\un{s} clock solutions.
This work is currently underway in our group.
Notice that a fiducial DM object sweep through the entire constellation takes about 170 seconds, thereby it lasts for just 6 epochs for the currently available 30\un{s} sampling intervals. Clearly, the resolution would improve for the 1\un{s} data.
In the initial GPS data processing (performed by JPL~\cite{MurphyJPL2015}), there is effectively no restriction on the allowed behavior of the clocks from one epoch to the next.
Crucially, if a clock were to have a real transient that far exceeded engineering expectations, the data over that time window would not have been removed as outliers.
The clock biases from JPL~\cite{JPLigsac} also come with a ``formal error'', $\sigma_F$.
The formal error, typically on the order of $\sigma_F\sim0.02-0.03\,{\rm ns}$, quantifies uncertainty in the determination of the clock bias, and does not directly incorporate the intrinsic clock noise or slowly varying biases due to correlated orbit errors and higher-order general relativistic effects ($\sim0.1\un{ns}$).
Only the most recent satellite clocks (Rb clocks on board the block IIF satellites) have observed temporal variations from one epoch to the next that are at a similar level as the formal error, indicating that temporal variations in older clocks are actually due to clock behavior rather than estimation error.
In fact, the observed variances in the data from the most modern Rb IIF satellite clocks are significantly better than suggested by the formal error, see Appendix~\ref{sec:clocknoise}.
Due to frequency drifts and other long time-scale ($\sim$ hours) effects, it is typical for a second-order polynomial ($y_2$) to be subtracted from the raw GPS time series data before the analysis~\cite{Blewitt2015307}.
One may form $y_2(j)$ for each clock, for each individual day, using a weighted least-squares approach, taking the weights as the inverse of the formal error.
Then the polynomial-reduced data (residuals) are defined
$d_j^{(0)} = x_j-y_2(j)$,
where $\{x_j\}$ are the raw time-series data, and $j$ denotes the same-time (epoch).
This procedure is useful for visualizing the data, however, it is not necessary for our analysis.
Unless noted otherwise, we take $d_j^{(0)} \equiv x_j$ in this paper.
The relative phase of an atomic clock (bias) $d_j^{(0)}$ is a non-stationary time series, dominated by random walk noise.
To perform the analysis, we must first ``whiten'' the data.
To this end, we employ (depending on the clock type, as discussed below) either a first- or second-order differencing, and define
\begin{align}
\label{eq:difference}
d_j^{(1)} &\equiv d_{j}^{(0)}-d_{j-1}^{(0)}, \\
d_j^{(2)} &\equiv d_{j}^{(0)}-2d_{j-1}^{(0)}+d_{j-2}^{(0)}.
\end{align}
In general, first-order differencing is sufficient for Rb clocks, while second-order differencing is required for Cs clocks.
Since $d_j^{(1)}$ is proportional to the discreet derivative of the clock biases, we refer to it as a pseudo-frequency.
Further discussion of the clock noise characteristics is presented in Appendix~\ref{sec:clocknoise}; see also the Supplementary Information, where we quantify the noise characteristics of each satellite clock individually.
\section{Bayesian search for DM events}\label{sec:SearchMethod}
\subsection{Likelihoods and odds ratio}\label{sec:likelihoods}
A DM-induced perturbation in the device data would be indistinguishable from a perturbation caused by other external non-DM factors or random statistical processes.
The key then, is to rely on the correlated propagation of clock ``glitches'' across a network caused by the sweep of a DM object through the network.
Based on the standard halo model (see Sec.~\ref{Sec:Priors}), DM objects are expected to travel relative to the Earth with galactic-scale speeds, $v_g\sim300\un{km}\un{s}^{-1}$, incident from a certain direction.
Thus the speed and the directionality of the sweeps serve as DM signatures.
There are similarities between the method we describe and those employed in gravitational wave detection by the LIGO collaboration, see, e.g., Refs.~\cite{Anderson2001,Allen2012}.
Consider a candidate model, denoted $M$, that predicts a DM signal across the network (for example, the passing of a domain wall).
In order to determine whether such a model is supported by the data $D$, we employ a Bayesian technique, see, e.g., Ref.~\cite{GregoryBayesian2005}.
In Bayesian statistics, model selection is based on forming the odds ratio of two probabilities (likelihoods)
\begin{equation}
O_{M,\bar{M}}(j_0)=\frac{p(D_{j_0}|M,I)}{p(D_{j_0}|\bar M,I)}.
\label{eq:odds}
\end{equation}
Here, $\bar M$ denotes the proposition that no signal is present in the data, i.e. the data is purely random,
and $I$ encodes the knowledge of the SHM priors discussed in Sec.~\ref{Sec:Priors}.
The data stream $D$ is sampled at intervals of $\t_0$ ($\t_0=30\un{s}$ for our current GPS data set).
Since we search for transient events of finite duration, the odds ratio is tested in a time window of length $J_W$ points, centered at epoch $j_0$.
The value of $J_W$ is determined by the maximum duration of the transient signals to be tested.
In the analysis, we scan over $j_0$ for a fixed value of $J_W$, so the odds ratio is an explicit function of $j_0$.
The likelihoods $p(D_{j_0}|M,I)$ entering Eq.~(\ref{eq:odds}) are described by the Gaussian multi-variate distributions marginalized over model parameters,
\begin{widetext}
\begin{align}
\label{eq:m-likelihood}
p(D_{j_0}|M,I)&=
K\int\ensuremath{{\rm d}}^3v \; p(\v{v}|M,I)
\int\ensuremath{{\rm d}} h\; p(h|M)\;
\int\ensuremath{{\rm d}} x\; p(x|M)\;
\int_{ (j_{0}-1)\tau_0}^{ j_0\tau_0}\frac{1}{\t_0}\,\ensuremath{{\rm d}} t_0\;
\exp
\left(
-\frac{1}{2}\chi^2(s)
\right)\,.
\end{align}
\end{widetext}
Here,
$\v{v}$ is the velocity of the incident DM object in the Earth-centered inertial (ECI) frame,
$h$ quantifies the amplitude of the DM signal,
$x$ stands for the remaining model-specific parameters of the DM object,
and $K$ is a normalization factor.
Further, $t_0$ is the moment of time at which the DM object passes by the center of the Earth.
It is assumed to occur in the time interval $\left( (j_0-1)\t_0, j_0 \t_0 \right]$, and we marginalize over $t_0$ in the last integral (\ref{eq:m-likelihood}).
Note that compared to our initial search~\cite{GPSDM2017}, the single-device sweep time ($d/v$) may last longer than $\t_0$.
The data and the model-prescribed DM signal are combined in the argument of the exponential,
\begin{equation}\label{eq:chis}
\chi^2(s)
=\sum_{ab}^{\ensuremath{N_{\rm clk}} }\sum_{jl}^{J_W}\left[{d^a_j}-{s^a_j}\right]{({E^{-1}})^{ab}_{jl}}\left[{d^b_l}-{s^b_l}\right],
\end{equation}
where $E$ is the covariance matrix discussed in the following section.
Here and below we use the ``upstairs'' indices to label devices, and the ``downstairs'' indices to denote epochs (sampling times).
The indices $a$ and $b$ run over all $\ensuremath{N_{\rm clk}} $ devices in the network, and the indices $j$ and $l$ run over the $J_W$ data points in the time window. The device data $d$ notation is generic and it can stand for the singly-- or doubly--differenced clock bias data, Eq.~(\ref{eq:difference}).
Finally, $s^a_j=s^a_j(M,t_0,\v{v},h,x)$ is the model-prescribed DM signal in device $a$ at epoch $j$, discussed in Sec.~\ref{sec:signals}.
Continuing with the discussion of factors entering the likelihood, $p(\v{v}|M,I)$ is the (normalized) probability density for the velocity distribution of DM objects in the ECI frame.
In the case of monopoles, for example, it is reasonable to take this to be given by the SHM.
Likewise, the function $p(h|M)$ is the normalized probability density for the DM signal amplitude in the time series, and is described by a flat prior.\footnote{We note that the normalization for the $h$ prior is arbitrary. For our purposes it is not important, since we do not rely on the actual value of the likelihoods function but rather define some threshold, above which false-positives are sufficiently rare, as discussed in the following sections.}
To calculate the likelihoods, we perform the integral over $h$ analytically (possible because $s$ is linear in $h$, see below), and use a randomized Monte-Carlo integration for the other parameters.
Finally, the likelihood that no signal is present in the data
is given simply by
\begin{equation}
\label{eq:null-likelihood}
p(D_{j_0}|\bar{M},I)=K
\exp
\left(
-\frac{1}{2}\chi^2(0)
\right),
\end{equation}
where the DM signal is set to zero.
The window size $J_W$ is kept the same as in the $p(D_{j_0}|M,I)$ computations.
The likelihood functions (\ref{eq:m-likelihood}) and (\ref{eq:null-likelihood}) are calculated for every available epoch $j_0$, and the odds ratios~(\ref{eq:odds}) are formed. Large spikes in the odds ratio as a function of $j_0$ can indicate potential DM events.
\subsection{Correlations and covariance}
\label{sec:covariance}
The covariance matrix entering Eq.~(\ref{eq:chis}) is defined as
\begin{equation}\label{eq:Eijkl}
E^{ab}_{jl}\equiv\braket{d^a_jd^b_l},
\end{equation}
where $\braket{\cdots}$ denotes averaging.
To compute its elements, one requires a stationary time series, for which (depending on the clock type) we use either the first- or second-order differenced data (\ref{eq:difference}).
Note that for pure uncorrelated white noise, the covariance matrix is completely diagonal, with elements given by the variances.
In this case, the matrix inversion required for computing
$\chi^2(s)$ (\ref{eq:chis}) is trivial.
Realistic device noise is, however, correlated.
Specific to the GPS clocks, additional short-range anti-correlation for individual clocks is introduced by the propagation of the formal error (which is roughly white noise in $d^{(0)}$) through the differencing procedure (\ref{eq:difference}).
Moreover, the underlying clock biases $d^{(0)}$ are the differences between the phases of the given clock and a reference clock.
Since the reference clock is common to all clocks, biases and the differenced data $d^{(1)}$ and $d^{(2)}$ are correlated between different clocks.
It is convenient to split the covariance matrix into two contributions, $E=A+B$, where
\begin{align}\label{eq:E=A+B}
A^{ab}_{jl} &\equiv E^{ab}_{jl}\delta^{ab},\\
B^{ab}_{jl} &\equiv E^{ab}_{jl}(1-\delta^{ab}).
\end{align}
The first term, $A$, represents the correlation between data points for a single clock, i.e., auto-correlation.
The $B$ contribution describes the correlations between different clocks, and is referred to herein as the cross-correlation.
The autocorrelation part of the covariance matrix is block diagonal, built from $\ensuremath{N_{\rm clk}}$ independent symmetric $J_W \times J_W$ matrices.
The elements of $A$ depend only on the distance from the diagonal, and can be related to the autocorrelation function $A^a(\Delta t)$ as
$A^{aa}_{jl} = {(\s^a)}^2 A^a(\Delta t_{jl}),$
where $\Delta t_{jl} = \abs{j-l}\tau_0$ is the lag and $\s^a$ is the standard deviation (see Appendix~\ref{sec:clocknoise}).
Each clock in the network is referenced against a common reference clock.
This adds a common noise component to all the data streams, and is main source of cross-correlations.
Therefore, the time series for each clock can be decomposed as
\begin{equation}\label{eq:dec}
d^a_j = e^a_j + c_j,
\end{equation}
where $c_j$ is the component due to the shared reference clock, and $e^a_j$ is the component unique to clock $a$ ($\braket{e^a_je^b_l}=0$ for $a\neq b$).
Then, it is clear that $B$ depends only on the reference clock, and is independent of $a$, $b$:
\begin{equation}
B^{a,b\neq a}_{jl} = \braket{d^a_jd^b_l} = \braket{c_jc_l} \equiv b_{jl}.
\end{equation}
To calculate the likelihoods, we need to invert the covariance matrix.
First, we note that the Earth-based H-maser clocks used as reference in the JPL data processing are typically much quieter than the satellite clocks.
Therefore, the cross-correlation contribution $B$ is typically smaller than $A$, so $B$ can be treated perturbatively.
Further, we may neglect the even smaller terms $B^{ab}_{jl}$ with $j\neq l$, defining $b_0\equiv B^{ab}_{jj}$.
Thus, we express the inverse of the covariance matrix as
${E}^{-1} = H + W,$
where
\begin{align}
H^{aa}_{jl} &= ({A}^{-1})^{aa}_{jl} , \\
W^{ab}_{jj} &\approx \frac{-b_0}{({\s^a\s^b})^2}(1-\delta^{ab}). \label{eq:EHW}
\end{align}
Then, Eq.~(\ref{eq:chis}) can be expressed (with $\eta \equiv d-s$) as
\begin{multline}%
\chi^2(s)=
\sum_{a}^{\ensuremath{N_{\rm clk}} }\sum_{jl}^{J_W}\eta^a_j\,{H}_{jl}^{aa}\,\eta^a_l
-
\sum_{a \neq b}^{\ensuremath{N_{\rm clk}} } \sum_{j}^{J_W} \frac{b_0\, \eta^a_j\,\eta^b_j }{({\s^a\s^b})^2}.
\label{eq:chis2}
\end{multline}
The described approximation holds when the clock noises far exceed that of the reference clock.
This approximation breaks down if the network includes clocks with noise levels similar to that of the reference clock.
For example, when including multiple station, Rb-IIF, or laboratory clocks, Eq.~(\ref{eq:EHW}) is no longer valid.
In this case, we define the weighted mean of all (other) clocks
\[\bar d^{\bar a}_j = \frac{\sum_{b\neq a} d^b_j \, (\s^b)^{-2}}{\sum_{b\neq a} {(\s^b})^{-2}} \approx c_j \pm \sigma/\sqrt{\ensuremath{N_{\rm clk}}} ,\]
which is subtracted from each time series [Eq.~(\ref{eq:dec})]:
\begin{equation}\label{eq:WM}
d^a_j-\bar d^{\bar a}_j \approx e^a_j \pm \sigma/\sqrt{\ensuremath{N_{\rm clk}}}.
\end{equation}
Here, $\s$ is the typical standard deviation of the clock data.
Each new data stream still contains a common component, $\sim$\,$\sigma/\sqrt{\ensuremath{N_{\rm clk}}}$, however this is small enough so that the above approximation (\ref{eq:EHW}) holds true.
In these cases, the same procedure must be applied also to the expected signals $s^a_j \to s^a_j-\bar s^{\bar a}_j $.
\subsection{Transient dark matter signals}\label{sec:signals}
The likelihood function in Eq.~(\ref{eq:m-likelihood}) requires a model-prescribed DM signal, ${s^a_j}$, for the data streams to be compared against.
The DM signal depends on the assumed coupling strength to the device, and on the kinematics and spatial structure of the DM object (monopole, domain wall, etc.). To quantify the transient signal we need to specify the collision geometry.
We work in the ECI (Earth-centered inertial) J2000 frame, which has its origin (denoted ECI0) at the center of mass of the Earth, and $z$-axis aligned with Earth's spin axis.
The $x$-axis is aligned with the mean equinox at 12:00 Terrestrial Time on 1 January 2000.
The important aspect is that the ECI frame orientation remains fixed in the galactic rest frame, i.e., it does not rotate with the Earth.
Here, we consider three generic and geometrically unique templates: walls, strings, and monopoles.
Albeit more complex geometries are plausible, such as walls closing on themselves forming cosmic bubbles, the presentation below is sufficient for extending the formalism to such more complex object geometries.
While the field profile inside the DM object can be arbitrary, we focus on Gaussian profiles.
Beyond qualitative arguments, the reasons for Gaussian-profiled objects can be also supported by Bayesian logic.
Indeed, application of the maximum entropy principle to a distribution with the mean and variance (determined by the defect size $d$ in our case) as the only given information yields the Gaussian distribution~\cite{GregoryBayesian2005}.
In any case, the presented formalism can be applied to arbitrarily-shaped DM object profiles.
We assume that the linear trajectory and velocity of the DM object are not affected by the gravitational pull of the Earth or the portal couplings to the Earth constituents, and that the shape of the DM object is preserved through the encounter.
Another assumption is that the DM encounters are well separated, i.e.\ DM objects do not overlap and at most one of them interacts with the entire network at any given time.
Finally, we assume that objects lacking spherical symmetry do not rotate.
Consider an event in which the center of a DM object moving with velocity \v{v} crosses the plane that is perpendicular to $\v{v}$ and contains ECI0 at time $t_0$, as shown in Fig.~\ref{fig:monopole}.
The accumulated time bias between a clock thats frequency is perturbed by $\delta\omega$ and an unaffected clock ($\omega_0$) is given by $\int_{-\infty}^t\frac{\delta\omega(t')}{\omega_0}\ensuremath{{\rm d}} t'$.
Therefore, at time $t$, the DM-induced clock phase bias in clock $a$ reads
\begin{equation}
\label{eq:generalsignal}
{s^a}^{(0)}(t) = \int\limits_{-\infty}^t
\left[
h^a \varphi_M^2(t^a,{\rho^a},t') - h^R \varphi_M^2(t^R,{\rho^R},t')
\right] \ensuremath{{\rm d}} t' ,
\end{equation}
where $\varphi_M^2$ is the normalized profile\footnote{The DM ``profile'' $\varphi$ differs from the field $\phi$ [Eq.~(\ref{eq:scalarPortal})] only by normalization, and is defined for convenience; see Appendix~\ref{sec:specific-signals}.} of the DM object (for specific model $M$),
${\rho^a}$ is the impact parameter,
$h \propto A^2\,\Gamma_\mathrm{eff}$
is a clock-specific constant that determines the magnitude of the signal in the data (see Appendix~\ref{sec:specific-signals}), and $t^a$ ($t^R$) is the time of encounter for clock $a$ (reference clock).
The time of encounter is defined as the moment the DM object passes by clock $a$.
More precisely, it is the time at which the center of the DM object (central plane for walls, or central axis for strings) crosses the plane that is perpendicular to \v{v} and contains the given clock:
\begin{equation}\label{eq:tiR}
t^a = t_0-\frac{{\v{r}^a}\cdot\v{\hat n}}{v},
\end{equation}
where $\v{\hat n}$ is the unit vector that points from ECI0 parallel to the incident direction of the DM object
($\v{v}=-v\v{\hat n}$, see Fig.~\ref{fig:monopole}), and ${\v{r}^a}$ is coordinate of clock $a$ in the ECI frame.
The satellite and ground station positions $\v{r}^a$ are a part of the JPL GPS dataset, and are known with $\sim\mathrm{cm}$ and $\sim\mathrm{mm}$ accuracies, respectively.
Note that the impact parameters are zero for domain walls, but are, in general, non-zero for strings and monopoles; see Appendix~\ref{sec:specific-signals}.
\begin{figure}
\includegraphics[width=0.35\textwidth]{./monopole}
\caption{\small Example geometry for a monopole crossing.}
\label{fig:monopole}
\end{figure}
For domain walls and strings, we use $\v{v}_\perp$ and $\v{\hat n}_\perp$, see Fig.~\ref{fig:vperp} and the discussion around Eq.~(\ref{eq:eta}).
The discreet matrix ${s^a_j}$ is generated by integrating to the specific values of $t$ that correspond to the GPS epochs (data sampling times).
Then we form either the first- or second-order differenced DM signals as in Eq.~(\ref{eq:difference}), with $d\to s$.
The particular form of $\varphi^2_M$ depends on the spatial structure and the kinematics of the DM object.
In Appendix~\ref{sec:specific-signals}, we present explicit signals for domain walls, monopoles, and strings, and link $\varphi_M$ and $h^{a}$ to the field parameters for these templates.
In our discussion of DM signals, we neglected the Earth orbital motion about the Sun at $\sim 30 \un{km/s}$, orbital velocities of satellites about the Earth ($\sim 4 \un{km/s}$), and the ground station rotational velocities ($\sim 0.5 \un{km/s}$).
While these velocities are much smaller than the galactic velocities, the motional effects can become important for large-scale or slowly-moving objects.
For example, the motional effects become relevant if the overall duration of an encounter is comparable to the 12-hour satellite orbit.
The modification of the DM signal templates to account for clock motion is straightforward, as the satellite and ground station positions are known.
We leave this generalization for future work.
\subsection{Mixed networks}
There are several different clock types (Cs, Rb, H-maser) in the GPS network.
As our search is expanded to include other laboratory clocks (and other high-precisions sensors) the diversity will increase further.
Each clock species may respond differently to the interaction with the DM field, see Eq.~(\ref{eq:variation}).
Therefore, we cannot assume $h$ to be uniform across the network.
There are several approaches for inhomogeneous networks.
One approach, as per Ref.~\cite{GPSDM2017}, is to consider separately the homogeneous sub-networks (e.g., consider only the Rb clocks).
The major drawbacks of this approach is that we lose the benefit of the highly-stable H-maser reference clocks (none of the GPS satellite clocks have H-masers), and that we also limit the total number of clocks that are considered at any given time.
The simplest approach is to assume that one of the couplings in Eq.~(\ref{eq:variation}) dominates, and carry out the analysis separately for each case.
For example, we may assume that $|\Gamma_\alpha|\gg |\Gamma_{m_e}|, |\Gamma_{m_q}|$ in Eq.~(\ref{eq:scalarPortal}).
The drawback of this approach is that it does not account for the possibility that several couplings may produce effects that are of a similar magnitude.
Furthermore, a Bayesian-like approach is to introduce additional marginalization parameters for each extra free parameter in place of $h$.
The number of such free parameters is equal to the smaller of either the number of distinct clock species in the network, or the number of distinct couplings we consider.
For example, considering a network of Rb, Cs, and H clocks (as per GPS), we can substitute
$\int\ensuremath{{\rm d}} h\to \int\ensuremath{{\rm d}} h_{\rm Rb}\int\ensuremath{{\rm d}} h_{\rm Cs}\int\ensuremath{{\rm d}} h_{\rm H}$ in Eq.~(\ref{eq:m-likelihood}).
\subsection{Directional signatures}\label{sec:directional}
A possible scenario is that a large number of small events are flagged by the Bayesian search (by ``small'' we mean the magnitude of the signal in the data compared to the clock noise, or the small magnitude of the spikes in the odds ratio).
Of course, such events may be simply due to random statistical fluctuations, or other conventional-physics non-DM perturbations.
Here we consider signatures unique to DM (or other galactic sources) allowing us to exclude non-DM signals.
While these signatures are included in the Bayesian approach through the priors (e.g., the likelihood are suppressed for velocities outside the SHM range through the prescribed velocity distribution prior), we could also examine inferred values of collision geometry parameters through the Bayesian parameter estimation, as discussed in Sec.~\ref{sec:parameters}.
Being able to resolve the event velocity magnitude and directionality is a powerful feature of geographically distributed networks.
First of all, the distributed nature of the network offers the direct sensitivity to the magnitude of DM object velocities.
If the observed incident velocity falls too far outside of the bounds allowed by the standard halo model, then a DM origin can be excluded.
There is also sensitivity to the directionality of the DM object velocity.
The most probable incident direction is from the average forward direction of the Sun's motion through the galaxy (roughly from the direction of the Cygnus constellation), see Fig.~\ref{fig:distros}.
We are only aware of one external systematic effect that has propagation speeds comparable to $v_g$, which is the solar wind~\cite{SolarWindBook}.
This effect, however, can be vetoed out on the basis of distinct directionality from the Sun, and by the fact that the solar wind does not affect the satellites in the Earth's shadow.
In addition to individual event signatures, one can also focus on the overall event statistics, provided the event rates are sufficiently high on the yearly basis \cite{RobertsAsymm2018}.
For example, due to the $\sim10\%$ annual variation in the relative velocities of the Earth and Sun in the galaxy~\cite{Freese2013}, one would expect to observe the annual modulation in the event rate.
This approach parallels the method employed in WIMP searches, e.g.,~Refs.~\cite{CoGeNT2011,Bernabei2013}.
Unlike WIMP searches, where the event rate may depend strongly on the DM velocity~\cite{RobertsDAMA2016,*RobertsAdiabatic2016} (due to energy dependence of the cross section), here the rate is linear in $v$.
Also unlike (most) WIMP searches, the distributed network approach is additionally sensitive to the annual modulation in the average incident velocity {\em direction}, which varies by $\sim$\,$20{^\circ}$, as shown in Fig.~\ref{fig:AngleVariation}.
(A WIMP-detection scheme that does have directional sensitivity is presented in Ref.~\cite{Rajendran2017}.)
\begin{figure}
\includegraphics[width=0.40\textwidth]{./ngal}
\caption{\small Annual variation in the direction of the Earth's galactic motion (ECI frame, $\theta\in[0,\pi]$ is the polar angle), which is the most probable incident DM direction. The central point, $\v{\hat n}_g$, is the average direction, corresponding to the direction of the Sun's velocity through the galaxy.}
\label{fig:AngleVariation}
\end{figure}
\section{Benchmarking the method}\label{sec:TestMethod}
\subsection{Simulating realistic clock time series}
\label{sec:TestMethod-randts}
\begin{figure*}
\includegraphics[height=0.26\textwidth]{./real}~~~~
\includegraphics[height=0.26\textwidth]{./sim}
\caption{\small Comparison of (polynomial-reduced) real GPS clock data for a few satellites from 21 June 2015 UTC (left) with simulated data for the corresponding SVNs (right). Each time-series is shifted by a constant offset for clarity. The curve labels encode the clock type, GPS block, and SVN.
}
\label{fig:simulated}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.435\textwidth]{./ACF-TestSim}~~~~~
\includegraphics[width=0.435\textwidth]{./AVAR-TestSim}
\caption{\small Comparison of the autocorrelation functions (left) and Allan variances (right) for the real data to those for the simulated data. Clocks are the same as in Fig.~\ref{fig:simulated}. The solid lines are from the real data, and the dotted lines are from the simulated data; they are practically indistinguishable.
}
\label{fig:simulatedAVARACF}
\end{figure*}
We generate simulated time series data that have the same noise characteristics as the real clock data for each individual GPS satellite.
This is achieved by ``coloring'' pseudo-random white noise with the known power spectral densities for each clock~\cite{AlexThesis2016}.
We calculate the power spectral densities for each specific SVN using the clock data provided by JPL, as the clock performance may degrade over time, and the clocks can perform differently when in orbit than when tested in a laboratory environment. We also simulate cross-correlations (correlations between different clocks).
This is achieved by simulating a reference clock, which adds a common noise stream to all the clocks in the network.
In Fig.~\ref{fig:simulated}, we plot several arbitrarily selected real JPL clock solutions, $d^{(0)}$, alongside the simulated clock solutions for the corresponding SVNs (denoted $z^{(0)}$) to demonstrate the quality of simulated data.
The standard deviations of the simulated data (after first- or second-order differencing) match exactly those of the real data for the given SVNs.
Further, the longer-scale noise characteristics also match -- in Fig.~\ref{fig:simulatedAVARACF}, we plot the autocorrelation functions and Allan variances for both the simulated and real data for the same clocks.
These figures demonstrate that the simulated clock data do indeed have the same noise characteristics as the real data.
Having generated simulated time series, we can test our Bayesian search code in a number of distinct ways:
\begin{enumerate}
\item
To gauge the prevalence of statistical false-positives, we run the code for the event-free simulated data.
\item
We inject DM event signals into the simulated data streams to gauge the efficacy of our technique to pick out true-positive events.
\item
We inject ``bad'' events (i.e., signals that are not properly correlated) into the simulated data as a test of the robustness of the search technique.
\item
We use parameter estimation to extract the observed parameters of the injected DM event, and compare the results to those used to generate the injected DM signal as a test of the method accuracy and efficacy.
\end{enumerate}
\subsection{Prevalence of statistical false-positives}
\begin{figure}
\includegraphics[width=0.475\textwidth]{./fp}
\caption{\small
Rate of statistical false positives as a function of the odds ratio threshold, $O_\mathrm{thresh}$, for thin domain walls.
The rate of false positives from simulated GPS networks typical for the given years:
2000 (1 Rb-II, 7 Rb-IIA, 3 Rb-IIR, 5 Cs-II, 11 Cs-IIA),
2005 (1 Rb-II, 8 Rb-IIA, 12 Rb-IIR, 1 Cs-II, 8 Cs-IIA),
2010 (5 Rb-IIA, 19 Rb-IIR, 2 Rb-IIF, 5 Cs-IIA, 2 Earth-based H-masers),
2017 (19 Rb-IIR, 10 Rb-IIF, 5 Earth-based H-masers),
and a possible future network (30 Rb-IIF--style satellites, 20 Earth-based H-masers).
Each curve corresponds to 4 years of 30\,s sampled simulated data.
}
\label{fig:FalsePos}
\end{figure}
We wish to define a threshold for the odds ratio, $O_{M,\bar{M}}(j_0)$, Eq.(\ref{eq:odds}).
If the spike in the odds ratio is larger than this threshold, such an event can be investigated as a potential DM event.
In order to do this, we need to calibrate the rate of statistical false positives.
To this end, we ran multiple simulations of various GPS clock network configurations, and computed the odds ratio (\ref{eq:odds}) for each epoch.
For each combination of clocks, we considered 2048 realizations of 2048 30\un{s}-epochs, amounting to approximately 4 years of simulated data for each simulation.
A plot of the rate of false positives as a function of the threshold is presented in Fig.~\ref{fig:FalsePos}.
This plot is for the specific DM model of ``thin'' ($d \ll 10^4\un{km}$, see Appendix~\ref{sec:specific-signals}) domain walls.
Note that for this exercise, a false positive is counted whenever an epoch has a value for the odds ratio above the given threshold.
This is a conservative definition, since the ``width'' of the odds-ratio spike (due to the imperfect resolution) may lead to the same false-positive event appearing in more than one neighboring epoch.
By our definition, this will be counted several times.
We can also drastically reduce the number of false positives that occur by introducing a minimum value (magnitude), $h_{\rm min}$, for the integral over signal magnitudes (\ref{eq:m-likelihood}).
Of course, this also means we can only detect positive events with $|h|>h_{\rm min}$.
We can then perform the analysis in several sweeps, systematically reducing $h_{\rm min}$ each time until signals of a given magnitude can no longer be excluded.
\subsection{Detecting injected DM events}
\begin{figure}
\includegraphics[width=0.425\textwidth]{./testOdds}
\caption{\small Bayesian detection of an injected thick domain wall ($d = 10^4\un{km}$) signal. The wall sweeps the GPS network of 32 satellite clocks (with $\s=0.01\un{ns}$) at time $t_0 = 0$.
For this simulation, $h = 0.02\un{ns}$.
Bottom panel: simulated clock biases shown for the first 8 clocks (including the injected thin-wall signal). Each time-series is shifted by a constant offset for clarity.
Top panel: the corresponding odds ratio using the same time scale.
}
\label{fig:O}
\end{figure}
To determine the sensitivity of the method, we must know the probability of positively detecting DM events of a given magnitude.
To this end, we generate clock data per Sec.~\ref{sec:TestMethod-randts}, inject randomized DM signals into the data streams and compute the odds ratios. In Fig.~\ref{fig:O}, we present one such simulation as an example.
Here we show the first 8 (of 30) simulated time series' for a 1.5\,hr window.
In this example, for simplicity, the clock noise was taken to be white (in $d^{(1)}$).
Then we injected a single ``thick'' domain-wall event for a wall of size $d=10^{4}\, \mathrm{km}$; the velocity and incident direction were chosen randomly.
The odds ratio was calculated for each epoch.
The spike in the odds ratio at the event is apparent, while the event is not discernible by eye in the data streams.
Note that the search routine is isolated from the simulation -- it is not made aware of the event time, speed, direction, magnitude, wall width (or if there was an event at all).
Figure~\ref{fig:TruePos} shows the fraction of injected thin domain wall events that are correctly identified, as a function of the signal magnitude.
The velocity and incident direction for each wall was chosen randomly (according to the SHM distributions, Fig.~\ref{fig:distros}), and we assumed all clocks were affected by the DM in the same way (i.e., all clocks have the same $\Gamma_{\rm eff}$).
For this analysis, the odds threshold was set to allow fewer than 10 false positive events per year ($O_\mathrm{thresh} \sim 10^3$, see Fig.~\ref{fig:FalsePos}).
Note, for 30\,s data, there are over $10^6$ epochs in a year.
We count an event as ``found'' if there was a spike in the odds ratio above the determined threshold that appears within $\pm1$ epoch from the injected incident time $t_0$.
(Of course, occurrences where a single event leads to an odds-ratio spike for more than one epoch are not double-counted, only one event is injected per trial, and it is either found or not.)
Also shown in Fig.~\ref{fig:tp-odds} is the average of the log odds ratio for each of these simulated networks as a function of the magnitude of the injected domain wall signal.
The large ``gap'' in the sensitivity that occurs around 2010 is due to the introduction of the Rb-IIF satellite clocks, which are substantially more stable than the older generation satellite clocks; see Appendix~\ref{sec:clocknoise}.
In Fig.~\ref{fig:TruePos-white}, we show the same true- and false-positive test results, but for networks of a varying number of identical pure white frequency noise devices ($d^{(1)}$ equivalent).
This is to demonstrate the general efficacy of the method, without specific reference to the properties of the GPS data.
\begin{figure}
\includegraphics[width=0.475\textwidth]{./tp}
\caption{\small
Efficacy of the method for detecting injected thin-wall DM signals, for the same simulated networks as in Fig.~\ref{fig:FalsePos}. Each point represents 128 trials, each curve has $\sim60$ points.
Top panel shows the fraction of injected thin-wall events that were correctly identified, as a function of the event magnitude $h$.
The odds threshold was set to allow fewer than $10$ false positives per year (see Fig.~\ref{fig:FalsePos}).
Bottom panel shows the average log-odds ratio as a function of $h$ on the same scale.
}
\label{fig:TruePos}\label{fig:tp-odds}
\end{figure}
Note that the ground receiver clocks contribute only minimally, even though they are significantly more precise than the GPS satellite clocks.
That is because our current data is sampled only every 30\,s, which is about the time it would take for a DM object to cross the Earth, meaning many of the Earth-bound clocks will be affected by the DM during the same data acquisition interval.
As discussed in Sec.~\ref{sec:noise}, it is possible to re-process the existing raw GPS data to produce 1\,s sampled time series.
In addition to the statistical improvement from the larger data set, this would also further allow us to take full advantage of the highly-stable Earth-based receiver and laboratory clocks.
Of course, this advantage comes at the cost of significantly increased computation time, which scales (roughly) quadratically with the number of data points $J_{W}$ due to the correlations, see Eq.~(\ref{eq:chis}).
\begin{figure}
\includegraphics[width=0.475\textwidth]{./tp-fp-white}
\caption{\small
Monte-Carlo simulations for thin domain walls, using a network of pure white-noise (in $d^{(1)}$) devices.
The green, red, and blue curves are for a network of 20, 30, and 50 identical devices, respectively.
Top panel shows the fraction of events that were correctly identified, as a function of the injected event magnitude $h$ (scaled by $\s$, the standard deviation of the data noise).
This is done requiring an odds threshold such that there are fewer than 10 false positives per year (solid lines), or 1 false positive per day (dotted lines).
Bottom left panel shows the average log-odds ratio as a function of $h/\s$ on the same scale.
Bottom right panel shows the yearly rate of false positives as a function of the odds threshold, $O_{\rm thresh}$.
}
\label{fig:TruePos-white}
\end{figure}
We also check the ``robustness'' of the method, to ensure incorrectly correlated events (that may exist in the data due to Earth-sourced or other non-galactic perturbations) are not flagged as potential events.
To do this, we inject a single perturbation of a specified magnitude into each satellite data stream at a random epoch, all within the same 5 minute time window.
This simulates a domain wall crossing, except in the important fact that the network perturbations are not correctly correlated between different satellites.
Injecting a large $2\sigma$ perturbation of this kind ($\sigma$ is the typical standard deviation of the clock noise) into the simulated data streams for a 30-clock network, fewer than 1\% present odds ratios anywhere within the 5 minute window that are above the threshold.
\subsection{Parameter estimation}\label{sec:parameters}
\begin{figure*}
\includegraphics[width=0.32\textwidth]{./hist-t0}~
\includegraphics[width=0.32\textwidth]{./hist-v}~
\includegraphics[width=0.32\textwidth]{./hist-theta}
\caption{\small Example normalized histograms for the difference between the injected event parameters and the best-fit values extracted from the Bayesian analysis. Results for 2048 randomized simulations of a 25 satellite clock homogeneous network. Each trial has a single $\sim1\s$ thin wall event injected with $v\simeq300\un{km}\un{s}^{-1}$.
{\em Left}: for the incident arrival time, $t_0$, {\em middle}: for the scalar speed, $v$, and {\em right}: for the incident polar angle, $\theta$.
We have resolution of better than $\sim\pm0.1\pi$ radians for the incident angle, and $\sim\pm10\un{s}$ for the incident time (note that this is with 30\un{s} sampled data).}
\label{fig:params}
\end{figure*}
\begin{figure*}
\includegraphics[width=0.32\textwidth]{./hist-v-fp}~
\includegraphics[width=0.32\textwidth]{./hist-theta-fp}~
\includegraphics[width=0.32\textwidth]{./hist-phi-fp}
\caption{\small Normalized histograms for the distribution of the best-fit values extracted from the false positives of the Bayesian analysis, {\em left}: for the scalar speed $v$, {\em middle}: for the polar angle $\theta$, and {\em right}: the azimuthal angle $\phi$.
(The hump in the $\theta$ histogram is due to the solid angle volume element $\sin\theta$.)
Here, a low threshold ($O_{\rm thresh}=10$) was chosen to increase the statistics; when increasing $O_{\rm thresh}$, the shape of the histograms remains constant (it is prohibitively computationally intensive to run enough simulations to form false positive histograms for large $O_{\rm thresh}$, see Fig.~\ref{fig:FalsePos}).
For these simulations, the priors were excluded (i.e., flat priors were assumed).
}
\label{fig:fp-params}
\end{figure*}
When a spike in the odds ratio is above the pre-determined threshold value, we can investigate this region of data as a potential event.
For example, by finding the set of ``best-fit'' parameters that maximize the un-marginalized likelihood,
we can estimate the properties of the possible event (e.g., the time of arrival, size of the object, coupling strength etc.).
In Fig.~\ref{fig:params}, we show histograms of the parameter estimation for a number of simulated trials where event signals were randomly injected into simulated data.
Shown in the plots is the difference between the injected value and the extracted best-fit value for the crossing time $t_0$, speed $v$, and incident polar angle $\theta$, for simulated domain wall crossings.
These parameters are representative of the spatial and temporal resolution of the method.
Note that Fig.~\ref{fig:params} was generated for
30\,s sampled data -- using the re-processed 1\,s data (as discussed above) will lead to a substantially improved resolution in the arrival time and velocity.
We also perform the parameter estimation for the false positive trials, where the analysis is performed on simulated event-free data.
The resultant histograms are presented in Fig.~\ref{fig:fp-params}.
In this case, when neglecting the priors, the histograms are flat, with a slight bias of more false-positives towards higher velocities.
When including the priors, the distribution of extracted parameters from the false-positives match the priors, as expected.
This means that there is a potential to search for events even below the ``false positive floor''.
Reducing the odds ratio threshold will allow us to detect much smaller DM events, but will also lead to a larger number of false positives.
The true positive results, however, are expected to follow the distribution of velocities and incident directions predicted by the standard halo model.
This is relevant for the part of the parameter space with $\ensuremath{{\cal T}}\ll1\un{year}$.
There would also be expected annual modulations in the event rate, average event speed, and most-common incident direction, see Sec.~\ref{sec:directional}.
In this case, the analysis would need to be performed without the priors (i.e., assuming flat priors) to avoid biasing the false positives.
\section{Search sensitivity and discovery reach}\label{sec:results}
Combining Eqs.~(\ref{eq:A2}), (\ref{eq:variation}), and (\ref{eq:generalsignal}), we find the maximum signal amplitude observable in a given clock ($a$, with reference clock $R$) for a domain wall crossing to be
\begin{equation}\label{eq:s1max}
s^{(1)}_{\rm max} \simeq (\hbar c) \rho_{\rm DM} \sqrt{\pi} d \tilde\tau v_g \mathcal{T} \left[ \Gamma_{\rm eff}^a - \Gamma_{\rm eff}^R\exp\left(-\frac{L^2}{d^2}\right) \right],
\end{equation}
where
the interaction duration is given $\tilde\tau = d/v$ for $d/v<\tau_0$ and $\tilde\tau = \tau_0$ otherwise ($\tau_0=30\,$s is the time period between data sample points for GPS),
and $L\sim10^4\,{\rm km}$ is the distance between the clock and the reference clock.
\begin{figure*}
\includegraphics[width=0.47\textwidth]{./projected-d}~~~
\includegraphics[width=0.47\textwidth]{./projected-T}
\caption{\small Projected discovery reach for topological defect dark matter, along with existing constraints for comparison. The red shaded region are the limits (on domain walls) from the initial GPS.DM search using the Rb GPS network~\cite{GPSDM2017}, the shaded orange regions are limits set by optical Sr clock~\cite{Wcislo2016} and from astrophysics observations~\cite{Olive2008}; these apply for walls, strings, and monopoles.
The curves represent the projected sensitivities for our method, with the red, green, and blue colors for domain walls, strings, and monopole-like dark matter, respectively.
For monopoles and strings, we require that at least 3 clocks are affected in the DM crossing, which causes the sharp cut-off for low $d$.
The solid lines are the projections for the global network of GPS microwave clocks, and the dashed lines are the reach for the case when a single optical clock can be incorporated into the analysis.
The sensitivity is slightly lower for large $\ensuremath{{\cal T}}$, since we rely on the older GPS clocks.}
\label{fig:proj}
\end{figure*}
The subtraction of two terms in square brackets in Eq.~(\ref{eq:s1max}) is due to the fact that when the maximum of DM field affects the clock, the reference clock is affected by its exponentially-suppressed tail.
When employing a network of identical clocks, this term leads to a fast decline in sensitivity for large $d$.
This is because the clock and reference clock are affected in the same way, so no bias is built up between them.
In contrast, when employing clocks with significantly different effective couplings $\Gamma_{\rm eff}$ (particularly, combining microwave and optical clocks) this suppression is not realized.
Statistically, the minimum detectable signal is proportional to
\[
s^{(1)}_{\rm min} \propto \frac{\s_y(\t_0)\t_0}{\sqrt{N_{\rm clk} N_{\rm pts}}},
\]
where $N_{\rm clk}$ is the number of affected clocks, and $N_{\rm pts} \sim {d}/{v_g\t_0}$ is the number of data samples per clock for which the DM-induced signal is appreciable, and $\s_y(\t_{0})$ is the Allan deviation.
The proportionality constant depends on the efficacy of the search technique, and on $O_{\rm thresh}$, the odds ratio threshold required to eliminate false positives.
Therefore, we should have sensitivity to the region
\begin{equation} \label{eq:GammaX}
\Gamma_X \gtrsim \varepsilon \, \frac{\sigma_y(\tau_0)\tau_0 \, (\k_X^a-\k_X^Re^{-L^2/d^2})^{-1}}{\h c \rho_{\rm DM}\,\sqrt{N_{\rm clk} N_{\rm pts}}\,\tilde\t v_g\, d \, \ensuremath{{\cal T}} },
\end{equation}
where the factor $\varepsilon\sim O(1)$ is the efficiency factor determined from the simulations, and depends on $O_{\rm thresh}$.
From the results presented in Fig.~\ref{fig:tp-odds}, for a 90\% detection confidence level, and when requiring fewer than ten false-positives per year, we have $\varepsilon\approx5$ for the existing GPS data.
Future improvements in the search method should allow us to further decrease $\varepsilon$.
The $\varepsilon$ factor depends only fairly weakly on the search parameters.
For example, as shown in Figs.~\ref{fig:FalsePos} and Fig.~\ref{fig:TruePos-white}, increasing the odds threshold by a factor of 10 decreases the number of false positives by a factor of 10, while only increasing $s_{\rm min}$, the smallest detectable signal magnitude, by $\sim10\%$.
Therefore, we may estimate that for a 90\% detection confidence level, and when requiring fewer than one false-positive every 10 years, $\varepsilon\approx6$.
The average time between consecutive encounters with a DM object, $\ensuremath{{\cal T}}$, is considered a free parameter in our model (set by the number density of the DM objects).
The dependence of Eq.~(\ref{eq:GammaX}) on $\ensuremath{{\cal T}}$ comes via the DM field amplitude (\ref{eq:A2}), and the requirement to not oversaturate the galactic DM density; the higher the number density of objects, the lower the field amplitude must be per object to compensate.
In order to determine the maximum $\ensuremath{{\cal T}}$ that one can have sensitivity to, we assume the sequence of DM events can be modeled as a Poissonian process.
For example, if we expect one DM object to cross the Earth every period of $\ensuremath{{\cal T}}$ on average, then in order to be $\sim90\%$ confident that an event would have occurred in the observation time $T_{\rm obs}$, we must require $T_{\rm obs} \gtrsim 2.3 \ensuremath{{\cal T}}$.
We present the projected sensitivity of our search in Fig.~\ref{fig:proj}, along with the existing constraints.
To be consistent with existing literature, we present the sensitivity in terms of the effective energy scales,
$\Lambda_X \equiv 1/\sqrt{\abs{\Gamma_X}}$.
Specifically, we show the projections for $\Lambda_\alpha$; the projections for $\Lambda_{m_e}$ and $\Lambda_{m_q}$ are essentially the same, the only difference arising from the different sensitivity coefficients $\kappa_X$, see Eq.~(\ref{eq:Crb}).
The reduction in sensitivity above $d\simeq10^4\un{km}$ for the homogeneous clock networks is due to the fact that large DM objects will interact with the clock and reference clock at the same time, see Eq.~(\ref{eq:s1max}).
That is, above this value, we are only sensitive to the gradient in the DM field when using a homogeneous network.
The limits from our previous work \cite{GPSDM2017} have a sharp cut-off above this value, since in that work, we required that the DM signal would be present for just a single data point (see Appendix~\ref{sec:specific-signals}).
The Bayesian method presented in this work does not suffer this constraint.
Performing the simulations for strings and monopoles is substantially more computationally demanding, due to the number of extra free parameters that must be marginalized over (see Appendix~\ref{sec:specific-signals}).
However, the sensitivity can be approximated by analogy with the domain wall case.
For $d\gg R_{\rm GPS}$, the monopole and domain wall cases are essentially the same.
For $d < R_{\rm GPS}$, the sensitivity of the search can be estimated by noting the typical number of clocks that would be affected in a monopole crossing,
$N_{\rm eff} \simeq \left\lceil N_{\rm clk} {d^2}/{R_{\rm GPS}^2}\right\rceil$.
A similar equation exists for strings, $N_{\rm eff} \propto N_{\rm clk} {d}/{R_{\rm GPS}}$.
For strings and monopoles, we required that at least 3 satellite clocks are affected during the DM sweep, $N_{\rm eff}\geq3$, which leads to a sharp drop in sensitivity for small $d$, as shown in Fig.~\ref{fig:proj}.
\section*{Conclusion}
We have described a method to use data from a distributed global network of precision measurement devices to search for transient signals that may be associated with sweeps by macroscopic-scale dark matter.
In particular, we considered the network of microwave atomic clocks on board the GPS satellites and ground stations, for which nearly two decades of archival data is available.
The method was demonstrated using simulated atomic clock data, and the prospects and discovery reach for topological defect dark matter was presented.
This approach can be extended in a straightforward fashion to other networks of high-precision measurement devices.
\acknowledgements
This work was supported by the U.S. National Science Foundation grant PHY-1506424.
We thank Chris Pankow, Derek Jackson Kimball, and Tigran Kalaydzhyan for discussions.
BMR is grateful to the CIERA institute and Northwestern University for hospitality during the CIERA Data Analysis Workshop,
and to the Perimeter Institute for Theoretical physics for support to attend the New Directions in Dark Matter and Neutrino Physics workshop and acknowledges the many helpful discussions that took place there.
\begin{widetext}
\section*{Appendix}
\end{widetext}
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} | 8,574 |
{"url":"https:\/\/answers.opencv.org\/users\/40641\/asa\/?sort=recent","text":"2017-10-24 16:31:25 -0500 asked a question Assertion Failure in SVM Predcit (OpenCV 3.0) Assertion Failure in SVM Predcit (OpenCV 3.0) Hi I was trying out the Bag of Words to classify images. I tried a simple 2016-07-14 10:02:21 -0500 commented answer How to determine angular rotation in appropriate direction The ROI can have additional printed info (say characters) other than the code pattern. The code bars have a specific height and width, so I want to be as accurate as possible to measure the bars before further processing. hence I want to correct for the tilting \/ rotation. I extract contours and then extract rectangles and filter them based on size specification. The approximation of rectangles are more accurate when the image is not tilted than when it is, hence the angle correction. Thanks 2016-07-14 09:57:24 -0500 commented answer How to determine angular rotation in appropriate direction Thanks! It works great. There is another thing, the ultimate focus of the scanning system is to read and decode a certain type barcode like pattern on the paper. The paper size can be huge, so after scanning the system extracts a smaller sub image (ROI) where the barcode should be. The position of the code on each paper is fairly static (it's usually near the top left corner). So, I will be getting only the ROI ( apprximately 500px x 1000px). Now the same issue as above. I can see some ROIs are roated within +-5 degrees. I mean the code bars look tilted. How can I rotate the ROI? I am thinking of using corner detectors and use your idea to rotate the tilted images back to zero degrees. If this is correct how do I get the reference corner points (correctedCorners) ? 2016-07-14 09:47:21 -0500 received badge \u25cf\u00a0Scholar (source) 2016-07-14 09:47:09 -0500 received badge \u25cf\u00a0Supporter (source) 2016-07-13 17:47:27 -0500 asked a question How to determine angular rotation in appropriate direction Hi, I have this image scanning setup where I am scanning printed papers under a camera. I have this prior information that, when the paper passes under the camera it can rotate upto maximum 5 degrees in either clock wise or counter clock wise direction. I want to determine the value of the rotation angle in correct direction and then rotate the image back to make it zero degrees. My question is how can I determine the amount of rotation and the correct direction? thanks 2016-07-11 09:40:14 -0500 asked a question estimate motion with opencv:contrib:reg class Hi, I am estimating Affine motion between two images using the opencv::contirb::reg class. There are some issues I am facing. The images have to of the same size. The calculate() function inside the \"mappergradaffine\" class has a CV_DbgAssert(img1.size() == image2.size()); statement. The inversewarp() function inside the mapaffine class uses the OpenCV function remap(). Now for my purpose even the bordering areas of the image are important. But BORDER_TRANSPARENT is used in remap() which means pixels that cannot be interpolated will not be touched. So after warping I get artifacts along the image borders I tried this workaround: I take a slightly larger reference image and a smaller current image. I use the matchTemplate() function to find the best matching part from the reference image to the current image. Due to time constraint, I resize both the images to 1\/16 size. Then I get the reference image same size as the current image and pass them to mapper class. I estimate affine motion between them and then use the affine matrix to warp the larger golden image. Now, this seems to work, but sometimes the matchTemplate()'s output isn't actually the part that should match with the current image. Also I am wondering as I am estimating affine motion between two slightly smaller images and then applying this affine motion to a slightly larger image, is the affine motion matrix correct for the larger image? (the larger reference image has roughly 300px more on all sides ) Can anyone suggest any good ideas? Thanks 2016-04-27 08:37:26 -0500 received badge \u25cf\u00a0Enthusiast 2016-04-26 15:03:06 -0500 commented question Improve Runtime of a Function Thanks! I'll give them a try 2016-04-26 13:57:12 -0500 commented question Improve Runtime of a Function Got it. So what should I use if I want to measure execution time of functions that involve io operations say imread() and\/or inwrite() ? I have functions that may or may not have io operations and I need to find execution time. 2016-04-26 13:45:54 -0500 commented question Improve Runtime of a Function I use cv::getTickCount() Update: I figured out the problem. The function was being called inside another function and there was a conditional cv::imwrite() in the function. That's why I was getting the problem. The conditional part comes from another section of the program. I've fixed the part and it's working ok. Thanks everyone! 2016-04-26 13:15:52 -0500 asked a question Improve Runtime of a Function Hi, I am using a function to find difference between 2 images. It takes two 8-bit grayscale images, converts them to CV_32FC1, does a subtraction. Here is the function I am using: cv::Mat calculateDiff(const cv::Mat &image_one, const cv::Mat &image_two) { cv::Mat im1, im2, im_dest; image_one.convertTo(im1,CV_32FC1); image_two.convertTo(im2,CV_32FC1); im1 \/= 2.f; im1 += 128.f; im2 \/= 2.f; cv::subtract(im1, im2, im_dest); im_dest.convertTo(im_dest, CV_8UC1); return im_dest; } I have measured the run-time of each major step individually convertTo steps divide by 2, add 128 subtract() function convertTo() When I call this function with images of sizes: 9000 x 6000. I get a run-time of about 900 msec, but each individual step takes a lot less time. Here's one example: Step 1 time: 64 msec Step 2 time: 76 msec Step 3 time: 51 msec Step 4 time: 22 msec When I called the function: I get the function's runtime: 905 msec The function call looks like this: cv::Mat diff_image; diff_image = calculate_diff(input_one, input_two); I measure the runtime using cv::getTickCount() and cv::getTickFrequency() Why is the function's runtime so large where individual step do no take that long? How to improve the runtime? Kindly Help Thanks! 2016-04-07 17:07:41 -0500 commented answer Help Needed with OpenCV reg: Modifying the map there is an efficient way of doing it. Use the scale() function mapAff->scale(alpha)\/\/multiplies the shift vector by a factor somehow I missed this simple function! 2016-04-06 12:10:05 -0500 answered a question Help Needed with OpenCV reg: Modifying the map I was able to solve the issue like this: After MapAffine* mapAff = dynamic_cast(mapPtr.get); I create a MapAffine object using the parameterised constructor where I multiply the shift component by the integer factor: MapAffine mapAff2 = cv::reg::MapAffine(ampAff->getLinTr(), alpha * mapAff->getShift()); \/\/ alpha is the integer factor Then I call inversewarp() using mapAff2: mapAff2.inversewarp(source, destination); If there's a more efficient way of doing it please let me know Thanks 2016-04-04 11:06:34 -0500 asked a question Help Needed with OpenCV reg: Modifying the map Hi, I am working with OpenCV image registration library \"reg\" under \"opencv-contrib\". I am using the MapAffine class to estimate affine motion. I need to modify the shift vector element (multiply it by a constant factor). I can get the linear transformation matrix and shift vector using getLinTr() and getShift(). Before doing the warping (using inverseWarp() ) I want to multiply the shift vector by a constant. This is what I have done so far: (Following this tutorial (https:\/\/github.com\/Itseez\/opencv_cont...) Ptr mapPtr; MapperGradAffine mapper; MapperPyramid mapPyr(mapper); Ptr mapPtr; mapPyr.calculate(image1, image2, mapPtr); MapAffine* mapAff = dynamic_cast(mapPtr.get()); Then doing the warping: mapAff->inversewarp(image2, destination); Now I want to modify the shift vector prior to doing the above step. I have tried to modify the shift part using opencv Mat obejcts: cv::Mat lin_tr = Mat(mapAff->getLintr); \/\/getting the linear part cv::Mat shift = Mat(mapAff->getShift()); \/\/getting the translation cv::Mat* aff_mat; cv::hconcat(lin_tr, 2 * shift, *aff_mat); Now the affine matrix is in a Mat object. My question is how can I recast it to MapAffine so that I can use the inversewarp() function. Or is there another way to modify the mapAffine reference directly? 2016-03-24 12:09:17 -0500 commented question How to Change image intensity range Thanks, Ok so I am working with printed images. I have a set of images and I am checking the cover side of each image. The first image is the template image. the rest are checked against the template. For each subsequent image there is some distortion \/ motion present. So I am doing the registration. I am experimenting with estimateRigidTransform() and fidTransformECC to see alignment performance. I need to check if there is missing print and\/or extra print comparing the aligned image against the template. 2016-03-24 11:54:37 -0500 received badge \u25cf\u00a0Editor (source) 2016-03-24 11:54:24 -0500 commented question How to Change image intensity range Hi berak, In the code example shown here in the showDifference function images are being converted to 32F. Should that be the approach?I want to check two things: missing object and extra object in the aligned image. 2016-03-24 11:45:02 -0500 asked a question How to Change image intensity range Hi, I am working with image registration. I have gray-scale images (CV_8UC1). I have done the registration part. Now I want to check the alignment accuracy. I want to convert the intensity range to [-127 to 128] from [0 to 255]. How can I do that? What I am doing is: subtract the aligned image from template divide the result by 2 add 128 to result Is this correct? Do I need to convert the images from 8U ? 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Patent application title: Methods for Reducing Sample Size of Clinical Trials
Inventors: Richard C. Straube (Bloomsbury, NJ, US) Ashleigh Palmer (Lebanon, NJ, US)
Methods for Reducing Sample Size of Clinical Trials - Patent application <?php require_once('/home/patents/php/mtc.config.php'); require_once('/home/patents/php/mtc.class.php'); $MTC = new MTC(); $MTC->init(); ?>
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Inventors: Richard C. Straube Ashleigh Palmer
Agents: GREENBERG TRAURIG, LLP
Assignees: Critical Biologics Corporation
Origin: BOSTON, MA US
Methods for reducing sample size of clinical trials are disclosed herein. In an embodiment, a method for calculating a sample size for a clinical trial includes choosing values for power, level of significance and size of treatment effect sought for a particular event; selecting a subgroup of people for the clinical trial from a selection of subgroups, the subgroup having a higher value for an event rate than the remaining subgroups; and calculating the sample size for the clinical trial using the values for power, level of significance, size of treatment effect sought and the subgroup event rate, wherein the people of the subgroup have a lower gelsolin concentration than a predetermined baseline value of gelsolin.
1. A method for stratifying patients into subgroups comprising:collecting a body fluid sample from each of the patients;analyzing each of the body fluid samples to determine a concentration of gelsolin;comparing the concentration of gelsolin in each of the body fluid samples to a pre-determined baseline value of gelsolin; andstratifying each of the patients into a subgroup based on the comparison,wherein at least one of the subgroups includes patients that have a lower gelsolin concentration than the pre-determined baseline value, resulting in the patients of the subgroup being more likely to experience a poor outcome.
2. The method of claim 1 wherein the body fluid sample is selected from the group consisting of blood, lymph, saliva, urine and cerebrospinal fluid.
3. The method of claim 2 wherein the body fluid sample is blood.
4. The method of claim 1 wherein the gelsolin in the body fluid sample is plasma gelsolin.
5. The method of claim 1 wherein the pre-determined baseline value of gelsolin is about 150 mg/L.
7. The method of claim 1 wherein the poor outcome is selected from the group consisting of atherosclerosis, lesions, rheumatoid arthritis score, sepsis, sepsis syndrome, acute respiratory failure, acute lung injury, acute respiratory distress syndrome (ARDS), shock, acute kidney failure, disseminated intravascular coagulation, neutropenia, anemia, increase in length of hospitalization, increase in time on mechanical ventilation, death, overall survival rates, disease-free survival rates, treatment-related morbidity, pro-inflammatory cytokine elevation and elevation of bacterial pro-inflammatory mediators.
8. The method of claim 1 further comprising:analyzing each of the body fluid samples to detect if one of F-actin or actin-gelsolin complexes are present; andstratifying each of the patients into a subgroup based on the detection,wherein at least one of the subgroups includes patients having F-actin or actin-gelsolin complexes, resulting in the patients of the subgroup being more likely to experience a poor outcome.
9. The method of claim 8 wherein the concentration of gelsolin is determined by one of measuring actin filament nucleating activity or measuring filament-severing activity.
10. The method of claim 8 wherein the detection of actin-gelsolin complexes is determined by extracting actin-gelsolin complexes using sepharose beads linked to monoclonal anti-gelsolin antibodies.
11. A method for calculating a sample size for a clinical trial comprising:choosing values for power, level of significance and size of treatment effect sought for a particular event;selecting a subgroup of people for the clinical trial from a selection of subgroups, the subgroup having a higher value for an event rate than the remaining subgroups; andcalculating the sample size for the clinical trial using the values for power, level of significance, size of treatment effect sought and the subgroup event rate,wherein the people of the subgroup have a lower gelsolin concentration than a pre-determined baseline value of gelsolin.
12. The method of claim 11 wherein the pre-determined baseline value of gelsolin is about 150 mg/L.
14. The method of claim 11 wherein the event is selected from the group consisting of atherosclerosis, lesions, rheumatoid arthritis score, sepsis, sepsis syndrome, acute respiratory failure, acute lung injury, acute respiratory distress syndrome (ARDS), shock, acute kidney failure, disseminated intravascular coagulation, neutropenia, anemia, increase in length of hospitalization, increase in time on mechanical ventilation, death, overall survival rates, disease-free survival rates, treatment-related morbidity, pro-inflammatory cytokine elevation and elevation of bacterial pro-inflammatory mediators.
15. The method of claim 11 wherein a blood sample is collected from each of the people of the subgroup, and wherein each of the people of the subgroup have F-actin or actin-gelsolin complexes present in the sample.
16. A method for calculating a sample size for a clinical trial comprising:choosing values for power, level of significance and size of treatment effect sought for a particular event;selecting a subgroup of people for the clinical trial from a selection of subgroups, the subgroup having a higher value for an event rate than the remaining subgroups; andcalculating the sample size for the clinical trial using the values for power, level of significance, size of treatment effect sought and the subgroup event rate,wherein a body fluid sample collected from each of the people of the subgroup have detectable levels of actin-gelsolin complexes.
18. The method of claim 16 wherein the body fluid sample collected from each of the people of the subgroup further includes a lower gelsolin concentration than a pre-determined baseline value of gelsolin.
20. The method of claim 18 wherein the predetermined baseline value of gelsolin is about 100 mg/L.
[0001]This application claims the benefit of and priority to U.S. Provisional Application Ser. No. 61/091,580, filed Aug. 25, 2008, the entirety of this application is hereby incorporated herein by reference.
[0002]The embodiments disclosed herein relate to methods for reducing sample size of clinical trials, and more particularly to pre-screening patients to select a placebo group of patients that are more likely to experience a poor outcome.
[0003]The calculation of the sample size needed for a clinical trial is based on both conventional parameters, such as power, level of significance, and the size of treatment effect sought, as well as on estimated parameters, such as the underlying expected outcome (also referred to as the "event rate", such as death, need for bypass, etc.) which is measured by the rate in the placebo group. Since sample size calculation for clinical trials is based on the normal deviate curve, increasing the expected event rate, as seen in the placebo group, will lead to a decrease in the sample size needed in order to show a small statistically significant improvement with the experimental therapy. Therefore, if the expected event rate could be increased by pre-screening patients and choosing only those patients that are found to have an increased chance of having a poor event rate, the sample size needed for a particular clinical trial will be less than the sample size needed if patients were not pre-screened.
[0004]Methods for reducing sample size of clinical trials are disclosed herein.
[0005]According to aspects illustrated herein, there is provided a method for stratifying patients into subgroups that includes collecting a body fluid sample from each of the patients; analyzing each of the body fluid samples to determine a concentration of gelsolin; comparing the concentration of gelsolin in each of the body fluid samples to a pre-determined baseline value of gelsolin; and stratifying each of the patients into a subgroup based on the comparison, wherein at least one of the subgroups includes patients that have a lower gelsolin concentration than the pre-determined baseline value, resulting in the patients of the subgroup being more likely to experience a poor outcome. The method may further include analyzing each of the body fluid samples to detect if one of F-actin or actin-gelsolin complexes are present; and stratifying each of the patients into a subgroup based on the detection, wherein at least one of the subgroups includes patients having F-actin or actin-gelsolin complexes, resulting in the patients of the subgroup being more likely to experience a poor outcome.
[0006]According to aspects illustrated herein, there is provided a method for calculating a sample size for a clinical trial that includes choosing values for power, level of significance and size of treatment effect sought for a particular event; selecting a subgroup of people for the clinical trial from a selection of subgroups, the subgroup having a higher value for an event rate than the remaining subgroups; and calculating the sample size for the clinical trial using the values for power, level of significance, size of treatment effect sought and the subgroup event rate, wherein the people of the subgroup have a lower gelsolin concentration than a pre-determined baseline value of gelsolin.
[0007]According to aspects illustrated herein, there is provided a method for calculating a sample size for a clinical trial that includes choosing values for power, level of significance and size of treatment effect sought for a particular event; selecting a subgroup of people for the clinical trial from a selection of subgroups, the subgroup having a higher value for an event rate than the remaining subgroups; and calculating the sample size for the clinical trial using the values for power, level of significance, size of treatment effect sought and the subgroup event rate, wherein a body fluid sample collected from each of the people of the subgroup have detectable levels of actin-gelsolin complexes.
[0008]The presently disclosed embodiments will be further explained with reference to the attached drawings, wherein like structures are referred to by like numerals throughout the several views. The drawings shown are not necessarily to scale, with emphasis instead generally being placed upon illustrating the principles of the presently disclosed embodiments.
[0009]FIG. 1 is a graph showing the relationship between the placebo event rate and the sample size based on a continuous outcome endpoint. Three different curves are plotted based on the treatment effect sought (i.e., a 25% reduction, a 50% reduction or a 75% reduction).
[0010]FIG. 2 is a graph showing the relationship between the placebo event rate and the sample size based on a dichotomous outcome endpoint. Three different curves are plotted based on the treatment effect sought (i.e., a 25% reduction, a 50% reduction or a 75% reduction).
[0011]FIG. 3 is a flowchart showing the method steps for stratifying patients into subgroups of the presently disclosed embodiments. The method is based on collecting a body fluid sample from each of the patients and comparing a concentration of gelsolin in the body fluid sample to a pre-determined baseline value of gelsolin.
[0012]FIG. 4A shows a schematic illustration of a prior art method for determining the sample size needed for a clinical trial based on both conventional parameters, such as power, level of significance, and the size of treatment effect sought, as well as on estimated parameters, such as the underlying expected event rate as measured in the placebo group.
[0013]FIG. 4B shows a schematic illustration of a method for determining the sample size needed for a clinical trial of the presently disclosed embodiments. The method is based on pre-screening patients to design a placebo group that has a high rate of a poor outcome.
[0014]While the above-identified drawings set forth presently disclosed embodiments, other embodiments are also contemplated, as noted in the discussion. This disclosure presents illustrative embodiments by way of representation and not limitation. Numerous other modifications and embodiments can be devised by those skilled in the art which fall within the scope and spirit of the principles of the presently disclosed embodiments.
[0015]Throughout this Application, various publications are referenced The disclosures of these publications are hereby incorporated by reference into this Application in order to more fully describe the state of the art as of the date of the invention described and claimed herein.
[0016]Sample size should be planned carefully to ensure that the research time, patient effort and support costs invested in any clinical trial are not wasted. Ideally, clinical trials should be large enough to detect reliably the smallest possible differences in the primary outcome with treatment that are considered clinically worthwhile. It is not uncommon for studies to be underpowered, failing to detect even large treatment effects because of inadequate sample size. Also, it may be considered unethical to recruit patients into a study that does not have a large enough sample size for the trial to deliver meaningful information on the tested intervention.
[0017]As used herein, the term "clinical trial" refers to a research study designed to the safety and/or effectiveness of drugs, devices, treatments, or preventive measures in humans or animals. Some clinical trials may last weeks (an acute trial), while others may last months or years (a long-term trial). In an embodiment, the methods disclosed herein are used for calculating a sample size needed to assess the short-term clinical effectiveness of a drug. In an embodiment, the methods disclosed herein are used for calculating a sample size needed to assess the long-term clinical effectiveness of a drug. The methods disclosed herein may be used to calculate the sample size needed for conducting a clinical trial for a disease or disorder selected from the following diseases and disorders, but not limited to, Multiple Sclerosis (MS), Alzheimer's , Rheumatoid Arthritis (RA), Huntington's Chorea (HD), Acute Kidney Failure, Sepsis Syndrome, Acute Respiratory Failure, Acute Lung Injury, Acute Respiratory Distress Syndrome (ARDS), Chronic Kidney Disease, Plasmodium falciparum Malaria, Accelerated Atherosclerosis.
[0018]As used herein, the term "sample size" refers to the number of participants in a clinical trial.
[0019]As used herein, the terms "placebo group", "placebo arm", "control group" and "control arm" refer to the participants in a clinical trial that are given a substance or mock therapy made to look like some form of experimental treatment that has no therapeutic or medicinal qualities.
[0020]As used herein, the terms "interventional group", "interventional arm", "experimental group" and "experimental arm" refer to the participants in a clinical trial that receive the drug, device, treatment, or intervention under study.
[0021]As used herein, the terms "event" and "outcome" refer to the ultimate result of a clinical trial given to patients. Examples of patient-oriented outcomes include, but are not limited to, atherosclerosis, lesions, rheumatoid arthritis score, sepsis, sepsis syndrome, acute respiratory failure, acute lung injury, acute respiratory distress syndrome (ARDS), shock, acute kidney failure, disseminated intravascular coagulation, neutropenia, anemia, increase in length of hospitalization, increase in time on mechanical ventilation, death, overall survival rates, disease-free survival rates, treatment-related morbidity, pro-inflammatory cytokine elevation and elevation of bacterial pro-inflammatory mediators.
[0022]As used herein, the term "event rate" refers to the proportion of patients in a group in whom an event is observed.
[0023]As used herein, the term "gelsolin" encompasses cytoplasmic gelsolin, plasma gelsolin, as well as fragments thereof. A "fragment" is meant to include any portion of a gelsolin molecule.
[0024]The minimum information needed to calculate sample size for a randomized controlled clinical trial in which a specific event is being counted includes the power, the level of significance, the underlying event rate in the population under investigation and the size of the treatment effect sought. The power of a study (denoted by 1-β) is the ability of the study to detect a true difference in outcome between the placebo group and the intervention group. The value for power is usually chosen to be 80%. By definition, a study power set at 80% accepts a likelihood of one in five (that is, 20%) of having the difference between two treatment groups not be statistically significant when one really exists. Thus, the power for large trials is occasionally set at 90% to reduce to 10% the possibility of a so-called "false-negative" result. The chosen level of significance (denoted by a) sets the likelihood of detecting a treatment effect when no effect exists (leading to a so-called "false-positive" result) and defines the threshold "P value". Results with a P value above the threshold lead to the conclusion that an observed difference may be due to chance alone, while those with a P value below the threshold lead to rejecting chance and concluding that the intervention has a real effect. The level of significance is most commonly set at 5% (that is, P=0.05) or 1% (P=0.01). Two-side statistical tests are most often specified for clinical trials to test both for better and worse outcomes in the intervention group when compared to the placebo group.
[0025]Unlike the statistical power and level of significance, which are generally chosen a-priori by researchers based upon convention, the underlying expected event rate (as reflected in the placebo group rate) is typically established by other means, usually from previous studies, including observational cohorts. Usually, investigators start by estimating the event rate in the placebo group. However, estimation of the event rate has mystical overtones, sometimes scant data leads to unreliable estimates. The data often provides the best information available, but may over- or under-estimate event rates, as the data can be from a different time or place, and thus subject to changing and differing background practices. Additionally, trial participants are often "healthy volunteers", or at least people with stable conditions without other comorbidities, which may further erode the study event rate compared with observed rates in the population. Great care is required in specifying the event rate and, even then, during ongoing trials it may be necessary to adjust the sample size, especially if the overall event rate proves to be unexpectedly low. The effect of treatment in a trial can be expressed as an absolute difference. That is, the difference between the rate of the event in the placebo group and the rate of the event in the intervention group, or as a relative reduction, that is, the proportional change in the event rate with treatment. If the event rate in the placebo group is 6.3% and the event rate in the intervention group is 4.2%, the absolute difference is 2.1%; the relative reduction with intervention is 2.1%/6.3%, or 33%.
[0026]Actin is the major protein within many cell types and because of the mass of muscle in humans, may be the most abundant body protein. Cell injury could therefore expose large amounts of actin to the extracellular space, where the ionic conditions favor polymerization of actin into filaments (F-actin), which, in solution, can be many microns in length. Two plasma proteins bind F-actin with high affinity: plasma gelsolin and the vitamin D-binding protein (DVP).
[0027]Plasma gelsolin (pGSN) is an about 83 to an about 85 kilo Dalton (kDa) secretory form of cellular gelsolin that circulates in high concentrations in the plasma of all people. The protein is molecularly identical with the cellular form with the addition of a 25 amino acid secretory extension. pGSN consists of six-similar domains that have very different properties. The properties of three of the domains have been discovered: gelsolin segment 1 (G1) and gelsolin segment 4 (G4) binds monomeric actin and gelsolin segment 2 (G2) binds F-actin and tropomyosin. When pGSN is added to F-actin, pGSN severs the filament in a nonproteolytic manner and remains bound to one end of the newly formed filament. If free pGSN molecules are present, the free molecules will sever the filament successively until only 2:1 actin-gelsolin complexes are present, thereby rapidly depolymerizing the filament. Free and complexed (to actin) pGSN molecules differ in functional properties. Although free pGSN can sever filaments, actin-gelsolin complexes cannot. Besides binding to actin and tropomyosin, pGSN appears to importantly bind a series of lipid mediators of inflammation including lysophosphatidic acid, diadenosine phosphate lipopolysaccharide (LPS; endotoxin), lipoteichoic acid (LTA; gram positive bacterial cell wall active lipoprotein), Aβ peptide (a peptide implicated in the pathogenesis of Alzheimer's disease), platelet-activating factor and possibly others. Without the action of the plasma depolymerizing proteins (pGSN and DBP), actin filaments could reach lengths of several microns and might affect blood flow through the microcirculation or cell migration through the extracellular space. The demonstration that long actin filaments can affect fibrin clot formation by inhibiting the lateral association of fibrils into bundles and the abrogation of this effect by shortening the actin filaments with pGSN give some support for the hypothesis that long actin filaments might interfere with normal physiologic processes. Increasing evidence suggests that pGSN, through its ability to bind to the potent early lipid mediators that initiate the inflammatory cascade, also plays a critical role in localizing the inflammatory process to the site of injury and preventing inappropriate systemic inflammation from occurring. It appears that local tissue injury causes the cytoskeleton of damaged cells to be exposed and local, extracellular pGSN avidly binds to the actin causing a local depletion of pGSN. Both exogenous mediators (LPS, LTA) and early endogenous lipid mediators (PAF, LPA, and potential others) are no longer inactivated by binding to the pGSN and can initiate local, beneficial inflammation. Mediators that diffused from the local site of injury to the systemic circulation are rapidly inactivated by binding to the huge, systemic reservoir of pGSN. However, in the uncommon situation in which the extent of tissue is massive, critical depletion of circulating pGSN allows these mediators to "escape" the local site unimpeded and start the process that can lead to inappropriate, potentially life-threatening systemic inflammation such as the sepsis syndrome.
[0028]Studies in humans have shown that pGSN levels fall in major inflammatory states including, but not limited to, acute malaria, acute lung injury (ALI), sepsis, critically ill post-surgical patients, major trauma, acute liver injury, and hematopoietic stem cell transplantation. In these studies the degree of decrease in pGSN levels correlates with the extent of tissue injury and the rate of development of disease specific complications such as death in ICU patients and septic patients, the development of idiopathic pneumonia syndrome in stem cell transplant recipients, prolonged mechanical ventilation and ICU stays. Decreases in pGSN levels have also been shown in patients having Alzheimer's , Multiple Sclerosis, Plasmodium falciparum Malaria and Huntington's chorea. Several studies have shown a similar decrease in pGSN levels in animal models including hyperoxia in mice, LPS challenge, oleic acid-induced ALI, cecal ligation/puncture model of sepsis, blast-induced lung injury, and burns. In patients with serious, systemic inflammation, critically depleted pGSN levels have been epidemiologically linked to poor outcome, including the development of the sepsis syndrome and death. pGSN levels differentiate otherwise identically appearing patients that will have higher rates of poor outcomes. Poor outcomes of interest for a clinical trial includes both clinically important events including, but not limited to, the development of sepsis, sepsis syndrome, acute respiratory failure, shock, acute kidney failure, disseminated intravascular coagulation, severe neutropenia, severe anemia and death, as well as biochemical abnormalities including, but not limited to, pro-inflammatory cytokine elevation, elevation of bacterial pro-inflammatory mediators including endotoxin (lipopolysaccharide; LPS) and lysophosphatidic acid (LPA).
[0029]By measuring the concentration of gelsolin in body fluid samples taken from patients and/or detecting the presence/absence of F-actin or actin-gelsolin complexes, patients may be stratified into subgroups that are more likely to experience a poor outcome. These measurements may be used as entry criteria for clinical trials to select a patient population in which base outcome event rate (as measured by the rate in the placebo group) will have a higher event rate than it otherwise would have (for example, by the conventional method of estimating the event rate). By increasing the expected event rate, the clinical trial can be completed with fewer patients. This reduction in sample size can be seen regardless of the therapy being examined. The screening of patients for clinical trials by measuring gelsolin levels and/or detecting the presence/absence of F-actin or actin-gelsolin complexes, may be used to identify a subgroup of patients who will experience several-fold higher poor outcome rates than the general patient population.
[0030]Clinical trials focus on improving the outcome of patients and use a variety of measures of poor outcome endpoints. Poor outcomes of interest for a clinical trial include, but are not limited to, both clinically important events such as the development of sepsis, the development sepsis syndrome, acute respiratory failure, acute lung injury, acute respiratory distress syndrome (ARDS), shock, acute kidney failure, disseminated intravascular coagulation, neutropenia, anemia, increase in length of hospitalization, increase in time on mechanical ventilation and death, as well as biochemical abnormalities including, but not limited to, pro-inflammatory cytokine elevation and elevation of bacterial pro-inflammatory mediators including endotoxin (lipopolysaccharide; LPS) and lysophosphatidic acid (LPA). The outcome endpoint for a clinical trial is based either on a dichotomous outcome (these are `yes` or `no` outcomes, such as for example, dead or alive, development of acute respiratory distress syndrome (ARDS) or no development, stroke or no stroke, etc) or a continuous outcome (a variable that can theoretically take an unlimited number of values, such as number of hours in the Intensive Care Unit, hours on mechanical ventilation, etc.). The underlying concept of increasing the event rate(s) or event duration resulting in lower patient numbers needed for statistical demonstration of benefit for these two types of endpoints are identical, although the mathematical calculations are different and are presented separately. The statistical literature has extensive descriptions for modifying the basic equations given below to account for specific theoretical conditions. Although use of these modifications will give slight numeric differences in the final calculations, the same fundamental results apply. Pre-screening patients based on gelsolin levels or gelsolin levels and the presence/absence of F-actin or actin-gelsolin complexes can select a patient population with increased expected event rates/durations and this increase allows the clinical trial to be completed with fewer patients. This reduction in sample size follows a nonlinear, chi distribution, so even relatively small changes in the expected outcome rates as measured in the placebo group can have substantial impact on the number of patients needed to show a statistically significant difference in outcomes with the same relative intervention improvement.
[0031]As shown mathematically below, for a clinical trial with a fixed statistical threshold, power, and minimally important intervention effect, the greater the expected event rate as measured in the placebo group, the smaller the sample size needed. This relative decrease in sample size with pre-screening patients with gelsolin and/or actin testing is more important the lower the initial event rate in the placebo group. This is important since most critical care trials involve endpoints that occur in an about 15 to an about 30% range. In such situations, an increase in an event rate of an absolute value of about 5 to about 20%, may have a major impact in the sample size needed to reach statistical significance. This in turn has profound impacts on the cost, feasibility, duration and size of the infrastructure necessary to perform the clinical trial.
Determining Sample Size Needed for a Clinical Trial Based on a Continuous Outcome Endpoint
[0032]Methods to calculate sample size needed to show a statistical difference between treatment groups (placebo group vs. intervention group) are well described and accepted based on estimated differences in the values for a primary outcome endpoint. Assuming that the true difference in the values between the intervention and placebo values is represented by "δ" and the experimentally measured mean difference is D= Xplacebo- Xintervention, where Xplacebo and Xintervention are the mean values for the placebo and intervention endpoint, respectively. Assuming a normal distribution of the difference around the true population differences, the variance of this difference is σD giving a standard deviation of σD/ {square root over (n)}. In order to be statistically significant, D must exceed Z.sub.ασD/ {square root over (n)}, where Z.sub.α is the normal deviate corresponding to the significance level α. The distribution of the power function (1-β) is also a normal deviate that can be symbolized by Z.sub.β. D has a mean of ∂ with a standard deviation of σD/ {square root over (n)} and hence the quantity ( D-∂)/(σD/ n) is normally distributed. For a two-tailed test, the probability that D exceeds -β (power-1) is given by equation 1:
( D-∂)/(σD/ n)=-Z.sub.β.sub.(1) (1)
Solving for n, the sample size can be calculated by equation 2:
n=(Z.sub.α+Z.sub.β)ZσD2/∂.sup- .2 (2)
[0033]The implication of equation 2 is perhaps best seen by plotting the sample size result for a continuous outcome endpoint against the placebo event rate for a fixed drug effect rate. As shown in FIG. 1, the sample size is plotted for a hypothetical clinical trial in which the α is set at 0.05, the power (1-β) set at 0.90 and the drug effect is set at 25%, 50% or 75% reduction in days in the hospital with a standard deviation for both groups of 2 days. As shown, as the drug effect rate increases the sample size decreases. For each drug effect rate, increasing the event rate in the placebo group decreases the size of the clinical trial to show a statistically significant improvement. Therefore, by selecting a patient population in which the baseline outcome, as measured by the rate in the placebo group, is worse (e.g., increased hospital days, death), the sample size needed is smaller. This indicates that by choosing patients having baseline gelsolin levels that are below a baseline value, the size of the clinical trial may be reduced. Therefore, using a baseline gelsolin level as part of the admission criteria for the placebo group results in a smaller clinical trial.
Determining Sample Size Needed for a Clinical Trial Based on a Dichotomous Outcome Endpoint
[0034]With exactly the same sample results but slightly different underlying derivation, the sample size of a dichotomous outcome endpoint clinical trial yields a result that increasing the event rate in the placebo group of the clinical trial decreases the total sample size of the study. The standard calculation of normal approximation of the binomial equation is shown in equation 3:
n=(Z.sub.α+Z.sub.β)Z(p1q1+p2q2)/(p.sub- .2-p1)2 (3)
Where p1 is the proportion in the placebo group having the outcome, p2 is the proportion in the intervention group, Z.sub.α is the normal deviate corresponding to the significance level to be used in the test, P' is the of declaring a significant result, β=2(1-P'), Z.sub.β is the normal deviate corresponding to two-tail probability β, q1 is equal to (1-p1) and q2 is equal to (1-p2).
[0035]As shown in FIG. 2, the sample size is plotted for a hypothetical clinical trial in which the α is set at 0.05, the power (1-β) set at 0.90 and the drug effect is set at 25%, 50% or 75%. Results indicate that regardless of the drug effect, increasing the event rate in the placebo group decreases the size of the clinical trial needed to show a statistically significant improvement.
[0036]According to aspects illustrated herein, a method is provided for stratifying patients into subgroups to create a subgroup of patients that are more likely to experience a poor outcome (for example, death, increased hospital stay, sepsis). The method is based on collecting a body fluid sample from each of the patients and comparing a concentration of gelsolin in each of the samples to a pre-determined baseline value of gelsolin. The subgroup of patients that are more likely to experience a poor outcome have a lower gelsolin concentration than the pre-determined baseline value. The level of the gelsolin for the patient may be obtained by any art recognized method. Typically, the level is determined by measuring the level of the marker in a body fluid, for example, blood, lymph, saliva, urine, cerebrospinal fluid and the like. If the body fluid sample is blood, the blood can be separated into blood cells and plasma. The plasma may further be separated into serum. The level can be determined by ELISA, or immunoassays or other conventional techniques for determining the presence of the marker. Conventional methods include sending a sample(s) of a patient's body fluid to a commercial laboratory for measurement.
[0037]The invention also involves comparing the level of gelsolin for the patient with a pre-determined value. The pre-determined value can take a variety of forms. For example, the pre-determined value may be single cut-off value, such as a median or mean. The pre-determined value may be established based upon comparative groups, such as, for example, where the risk in one defined group is double the risk in another defined group. The pre-determined value may be a range, for example, where the tested population is divided equally (or unequally) into groups, such as a low-risk group, a medium-risk group and a high-risk group, or into quartiles, the lowest quartile being subjects with the highest risk and the highest quartile being patients with the lowest risk, or into tertiles the lowest tertile being patients with the highest risk and the highest tertile being patients with the lowest risk. The pre-determined value may depend upon the particular population of patients selected. For example, an apparently healthy population will have a different `normal` range of gelsolin than will a population the subjects of which have had a prior infection or other condition. Accordingly, the pre-determined values selected may take into account the category in which a subject falls.
[0038]Although the following figures pertain to the collection of a blood sample from a patient for determining a plasma gelsolin concentration, it should be noted that another body fluid can be collected and the concentration of gelsolin in that body fluid can be compared to a pre-determined baseline value of gelsolin. FIG. 3 is a flowchart showing an embodiment of a method for stratifying patients into subgroups. The method pre-screens patients for entry into a clinical trial. The method is based on collecting a blood sample from each of the patients and comparing a concentration of plasma gelsolin in the blood sample to a pre-determined baseline value of plasma gelsolin. Similarly, the same blood sample may be screened for the presence/absence of F-actin or actin-gelsolin complexes. If the patient has a pGSN level that falls below the established baseline value and/or the presence of F-actin or actin-gelsolin complexes is detected, the patient may be selected to be part of the clinical trial. The expected event rate in these patients, which is measured by the event rate in the placebo group, is more likely to experience poor outcomes than patients either that were not pre-screened or did not have the biomarkers (low pGSN or circulating actin). By increasing the expected event rate, as measured by the event rate in the placebo group, the clinical trial can be completed with fewer patients, as will be described and shown in detail below.
[0039]Step 110 of the method begins with the collection of a blood sample from an interested patient. An "interested patient" is a patient that would be willing to participate in the clinical trial. Blood may be collected by methods known in the art, and include, for example, using ethylenediaminetetraacetic (ETDA)-containing tubes for collection. Blood samples may be, for example, centrifuged, the plasma component removed and diluted with gel sample buffer, and frozen until needed. In step 120, the concentration of pGSN and/or the presence/absence of F-actin or actin-gelsolin complexes is evaluated. Gelsolin activity may be detected in plasma by techniques including, but not limited to, measuring actin filament nucleating activity (a measure of the total gelsolin concentration), measuring filament-severing activity (free gelsolin activity), and quantitative western blotting. Sepharose beads linked to monoclonal anti-gelsolin antibodies may be used to extract actin-gelsolin complexes from the plasma component of the blood sample. Methods for determining the concentration of pGSN and the presence or absence of F-actin or actin-gelsolin complexes in blood samples are known in the literature (see for example, Janmey P A, Stossel T P: Kinetics of actin monomer exchange at the slow growing ends of actin filaments and their relation to the elongation of filaments shortened by gelsolin. J Muscle Res Cell Motil 1986; 7: 446-454, Mounzer K C, Moncure M, Smith Y R, DiNubile M J: Relationship of admission plasma gelsolin levels to clinical outcomes in patients after major trauma. Am R Respir Crit Care Med 1999; 160: 1673-1681, and Lee P S, Drager L, Stossel T P, Moore F D Jr., Rogers S O: Relationship of plasma gelsolin levels to outcomes in critically ill surgical patients. Ann Surg 2006; 243: 399-403). In step 130, the concentration of pGSN is compared to an established baseline value that has been chosen for the particular clinical trial of interest. Similarly, the presence or absence of F-actin or actin-gelsolin complexes in the sample is determined. If the patient tested has a pGSN concentration that is below the baseline value, and/or the presence of F-actin or actin-gelsolin complexes is determined, the method continues to step 140, and the patient is selected to participate in the clinical trial. If, on the other hand, the pGSN concentration is above the baseline value, and/or there is no detectable F-actin or actin-gelsolin complexes in the sample, the method continues to step 145 and the patient is not selected for the clinical trial. The patients that are selected to be part of the clinical trial are more likely to experience a poor outcome.
[0040]The determination of the baseline value for pGSN is arbitrary, and selection requires epidemiologic data with pGSN levels and outcome data of interest. Such data is optimally generated in the patient population of interest but data from other patient populations can be used although the precision of the calculations will be lower than if target-population specific were used. In general, as the baseline pGSN threshold is lowered, the higher the proportion (or numeric measure of outcome) of patients experiencing a poor outcome. However, not all patients presenting with low pGSN will suffer a poor outcome and a few patients with relatively high baseline pGSN will suffer a poor outcome. The number of these "outliers" will depend on the patient population characteristics and the outcome being examined. For each baseline threshold value, the sample size for the potential study can be calculated. Although in general, the smaller the sample size the more "attractive" the study, other considerations such as the inability to analyze the contribution of site effect with few patient per site being enrolled, increased patient heterogeneity with more sites and site for different geographic areas need to be considered. Depending on the constraints of trial being planned (money, number of sites available, number of patients see at each site, amount of drug, and research support including study coordinators, study monitors and project managers), a selection of a baseline threshold value for pGSN level or pGSN level and the presence/absence of F-actin or actin-gelsolin complexes for the specific study is made that provides the most practical patient entry criterion and minimizes the sample size without jeopardizing the statistical analysis.
[0041]It has been shown that typical pGSN levels in healthy patients are between about 150 and about 300 mg/L. To date, for patients at risk of developing sepsis syndrome, the critical baseline threshold value for pGSN levels ranges from about 50 to about 100 mg/L, or from about 1400 to about 3000 mU/mL. Similarly, the critical pGSN level appears to range from about 75 to about 140 mg/L or from about 2000 to about 4000 mU/mL for patients recently started on chronic renal hemodialysis. The established baseline value is chosen based on the outcome studied for the clinical trial, for example, if the clinical trial is carried out to determine whether a new drug for treating sepsis syndrome is effective, the baseline value of pGSN may be chosen to be about 100 mg/L. In another embodiment, if the clinical trial is being carried out to determine whether a new procedure is helpful in chronic renal failure, the baseline threshold value of pGSN may be chosen to be about 140 mg/L.
[0042]FIG. 4A shows a schematic illustration of a prior art method for determining the sample size needed for a clinical trial. Typically, the power, the level of significance and the effect of treatment are selected a-priori based on conventional parameters. As seen in FIG. 4A, a typical value chosen for the level of significance is 5%, for the power is 90%, and for the effect of treatment is a 36% reduction. As described above, the expected event rate for the population measured by the rate in the placebo group is typically established by other means, usually from previous studies, including observational cohorts. For most clinical trials, the event rate in the placebo group is typically selected to be a low value which leads to a high calculation for the sample size needed.
[0043]FIG. 4B shows a schematic illustration of a method for determining the sample size needed for a clinical trial based on the methods of the present disclosure. As described above for FIG. 3, a patient is pre-screened prior to entry into the clinical trial. The patient is pre-screened based on determining the concentration of pGSN in the blood sample and/or the presence/absence of F-actin or actin-gelsolin complexes in the sample. The pGSN level is compared to a baseline value, and the patient is either selected to be in the clinical trial, or alternatively not selected.
[0044]A method for stratifying patients into subgroups includes collecting a blood sample from each of the patients; separating each of the blood samples into a blood component and a plasma component; analyzing the plasma component of each of the blood samples to determine a concentration of gelsolin in the plasma; comparing the concentration of gelsolin in the plasma to a pre-determined baseline value of plasma gelsolin; and stratifying each of the patients into a subgroup based on the comparison, wherein at least one of the subgroups includes patients that have a lower plasma gelsolin concentration than the pre-determined baseline value, resulting in the patients of the subgroup being more likely to experience a poor outcome.
[0045]A method for calculating a sample size for a clinical trial includes choosing values for power, level of significance and size of treatment effect sought for a particular event; selecting a subgroup of people for a placebo arm of the clinical trial from a selection of subgroups, the subgroup having a higher value for an event rate than the remaining subgroups; and calculating the sample size for the clinical trial using the values for power, level of significance, size of treatment effect sought and the subgroup event rate, wherein the people of the subgroup have a lower plasma gelsolin concentration than a pre-determined baseline value of plasma gelsolin.
[0046]A method for calculating a sample size for a clinical trial includes choosing values for power, level of significance and size of treatment effect sought for a particular event; selecting a subgroup of people for the clinical trial from a selection of subgroups, the subgroup having a higher value for an event rate than the remaining subgroups; and calculating the sample size for the clinical trial using the values for power, level of significance, size of treatment effect sought and the subgroup event rate, wherein a blood sample collected from each of the people of the subgroup have detectable levels of actin-gelsolin complexes.
[0047]The following examples are illustrative of the benefits of practicing the methods of the presently disclosed embodiments when determining sample size needed for a clinical trial. The methods are based on pre-screening patients and selecting only those patients that have a higher chance of a poor outcome (i.e., a higher event rate). Patients are pre-screened by collecting blood samples and determining the concentration of pGSN levels and/or the presence/absence of F-actin or actin-gelsolin complexes. If the concentration of pGSN is below a baseline value, the patient is selected to be part of the clinical trial.
Comparison of Sample Size Needed for an Experimental Drug to Reduce Mortality Rate in Patients at Risk of Developing Sepsis Syndrome Admitted to the Hospital
[0048]An epidemiology experimental study collected plasma gelsolin (pGSN) levels on patients admitted to the hospital with pneumonia, peritonitis, multiple traumas, Intensive Care Unit admission with a diagnosis of urosepsis, or immune-compromised patients with severe infection. The patients were followed until discharge or death. The overall mortality rate of the patients was about 14%. Among patients with pGSN levels less than about 100 mg/L, the mortality rate was found to be much higher, about 30%.
[0049]If a clinical trial was being designed for a theoretical experimental drug therapy to reduce the mortality rate in patients at risk of developing sepsis syndrome admitted to the hospital, the above epidemiology experimental data can be used to calculate the sample size needed. The following assumptions could be made for the desired clinical trial: [0050]The statistical threshold for the trial (α; the probability of avoiding a false positive trial) is set at P=0.05 (5%) [0051]The desired power (1-β; the probability of avoiding a false negative trial) is set at 90% [0052]The effect of treatment sought is set at a 36% reduction [0053]A two-side statistical test is desired
[0054]By using a placebo event rate of 14% (the mortality rate seen in all of the patients), the intervention group event rate would be 9% (a relative 36% reduction from the placebo group). Using the above equations,
p1=0.14 (the placebo group event rate of mortality)
q1=(1-p1)=0.86
p2=0.09 (the intervention group event rate of mortality)
[0055]The normal deviates can be calculated or obtained from published tables:
Z.sub.α=Z0.05=1.958, two tailed
Z.sub.β=Z0.10=1.282
[0056]As discussed above, the standard normal approximation calculation for sample size is:
n=(Z.sub.α+Z.sub.β)Z(p1q1+p2q2)/(p.sub- .2-p1)2
[0057]Substituting the values above into the equation, gives:
n=[(1.958×1.282)2]×[(0.140×0.860)+(0.090×0.91- 0)]/[(0.090-0.140)2]
n=[(3.240)2]×[(0.120)+(0.082)]/[(-0.050)2]
n=10.498×0.202/0.0025
[0058]Thus, a total of 849 patients per group would be needed. In a two group study (drug and placebo, for example), the total sample size would be 1698 patients.
[0059]If the same patient entry criteria were used but only patients with pGSN levels less than about 100 mg/L were enrolled, the epidemiology study shows that mortality event rate would be approximately 30%. Using this value as the expected placebo event rate and using the intervention group event rate as 19% (the same relative 36% reduction with treatment), the same sample size calculation is as follows:
p2=0.19 (the drug treated group event rate of mortality)
[0060]Using the same statistical assumption as before, the normal deviates are:
[0061]As before, the standard normal approximation calculation for sample size is:
n=10.498×0.364/0.012
[0063]Thus, the number of patients is calculated to be 316 patients per group, or 632 patients in a two-armed study. The above example shows that by pre-screening patients and selecting only those patients that have a higher chance of a poor outcome (by collecting blood samples and determining the concentration of pGSN levels and/or the presence/absence of F-actin or actin-gelsolin complexes), the sample size needed for the clinical trial can be reduced by 60%.
Comparison of Sample Size Needed for an Experimental Drug to Reduce Mortality Rate in Patients Initiating Hemodialysis with Catheters
[0064]A case-control study collected plasma gelsolin (pGSN) levels on patients with end-stage renal disease within two weeks of starting hemodialysis using a catheter rather than a vascular stent or arteriovenous fistula. The patients were followed for one year and the mortality rate was assessed. The estimated one year mortality rate was 40%. In the case-control study, the odds ratio for patients dying in the first 365 days after starting hemodialysis who had a pGSN level less than the median and with measurable plasma actin was 25.9 compared to that of the group that had pGSN levels above the median and no detectable plasma actin. In this study, approximately 65% of the patients had detectable circulating actin. This suggested that in the high risk group (low pGSN levels and the presence of actin) the expected mortality rate over the first year was 80%.
[0065]If a clinical trial was being designed for a theoretical experimental drug therapy to reduce the mortality rate in patients initiating hemodialysis with catheters, the above experimental data can be used to calculate the sample size needed. The following assumptions could be made for the desired clinical trial: [0066]The statistical threshold for the trial (α; the probability of avoiding a false positive trial) is set at P=0.05 (5%) [0067]The desired power (1=β; the probability of avoiding a false negative trial) is set at 90% [0068]The effect of treatment sought is set at a 25% reduction [0069]A two-side statistical test is desired
[0070]By using a placebo event rate of 40% (the mortality rate assumed for all patients), the intervention group event rate would be 30% (a relative 25% reduction from the placebo group). Using the above equations,
[0071]As above, the normal deviates are:
[0072]Thus, a total of 472 patients per group or a total of 944 patients in a two group study would be needed.
[0073]If the same patient entry criteria were used but only patients with pGSN levels less than the group median value (about 141 mg/L) and the presence of circulating F-actin or actin-gelsolin complexes were enrolled, the expected control rate would be 80%. Using this value as the expected placebo event rate and using the intervention group event rate as 60% (the same relative 25% reduction with treatment), the same sample size calculation is as follows:
n=(Z.sub.αZ.sub.β)2(p1q1+p2q2)/(p2-p1)2
[0075]Thus, the number of patients is calculated to be 105 patients per group, or 210 patients in a two-armed study. The above example shows that by pre-screening patients and selecting only those patients that have a higher chance of a poor outcome (by collecting blood samples and determining the concentration of pGSN levels and the presence/absence of F-actin or actin-gelsolin complexes), the sample size needed for the clinical trial can be reduced by 75%.
[0076]In an embodiment, a method for stratifying patients into subgroups and storing information relating to the subgroups on a database includes collecting a body fluid sample from each of the patients; analyzing each of the body fluid samples to determine a concentration of gelsolin; comparing the concentration of gelsolin in each of the body fluid samples to a pre-determined baseline value of gelsolin; and stratifying each of the patients into a subgroup based on the comparison, wherein at least one of the subgroups includes patients that have a lower gelsolin concentration than the pre-determined baseline value, resulting in the patients of the subgroup being more likely to experience a poor outcome. The method may further include analyzing each of the body fluid samples to detect if one of F-actin or actin-gelsolin complexes are present; and stratifying each of the patients into a subgroup based on the detection, wherein at least one of the subgroups includes patients having F-actin or actin-gelsolin complexes, resulting in the patients of the subgroup being more likely to experience a poor outcome. The concentration of gelsolin can be determined by analyzing a body fluid sample using, for example, ELISA, or immunoassays or other conventional techniques for determining the presence of gelsolin. In an embodiment, the database is accessible over a network such as the internet.
[0077]In an embodiment, a method for calculating on a computer a sample size for a clinical trial includes choosing values from a database for power, level of significance and size of treatment effect sought for a particular event; selecting, on a computer, a subgroup of people for the clinical trial from a selection of subgroups, the subgroup having a higher value for an event rate than the remaining subgroups; and calculating, on a computer, the sample size for the clinical trial using the values for power, level of significance, size of treatment effect sought and the subgroup event rate, wherein the people of the subgroup have a lower gelsolin concentration than a pre-determined baseline value of gelsolin. In an embodiment, the calculated sample size is stored on a database which is accessible over a network such as the internet.
[0078]In an embodiment, a method for calculating on a computer a sample size for a clinical trial includes choosing values from a database for power, level of significance and size of treatment effect sought for a particular event; selecting, on a computer, a subgroup of people for the clinical trial from a selection of subgroups, the subgroup having a higher value for an event rate than the remaining subgroups; and calculating, on a computer, the sample size for the clinical trial using the values for power, level of significance, size of treatment effect sought and the subgroup event rate, wherein a body fluid sample collected from each of the people of the subgroup have detectable levels of actin-gelsolin complexes. In an embodiment, the calculated sample size is stored on a database which is accessible over a network such as the internet.
[0079]The embodiments described herein may be implemented using any appropriate computer system hardware and/or computer system software. In this regard, those of ordinary skill in the art are well versed in the type of computer hardware that may be used (e.g., a mainframe, a mini-computer, a personal computer ("PC"), a network (e.g., an intranet and/or the internet)), the type of computer programming techniques that may be used (e.g., object oriented programming), and the type of computer programming languages that may be used (e.g., C++, Basic, AJAX, Javascript). The aforementioned examples are illustrative and not restrictive.
[0080]For the purposes of this disclosure, a computer readable medium is a medium that stores computer data in machine readable form. By way of example, and not limitation, a computer readable medium can comprise computer storage media as well as communication media, methods or signals. Computer storage media includes volatile and non-volatile, removable and non-removable media implemented in any method or technology for storage of information such as computer-readable instructions, data structures, program modules or other data. Computer storage media includes, but is not limited to, RAM, ROM, EPROM, EEPROM, flash memory or other solid state memory technology; CD-ROM, DVD, or other optical storage; cassettes, tape, disk, or other magnetic storage devices; or any other medium which can be used to tangibly store the desired information and which can be accessed by the computer.
[0081]While a number of embodiments of the present invention have been described, it is understood that these embodiments are illustrative only, and not restrictive, and that many modifications may become apparent to those of ordinary skill in the art. For example, certain methods may have been described herein as being "computer implementable" or "computer implemented". In this regard, it is noted that while such methods can be implemented using a computer, the methods do not necessarily have to be implemented using a computer. Also, to the extent that such methods are implemented using a computer, not every step must necessarily be implemented using a computer. Further still, the various steps may be carried out in any desired order (and any desired steps may be added and/or any desired steps may be eliminated).
[0082]All patents, patent applications, and published references cited herein are hereby incorporated by reference in their entirety. It will be appreciated that several of the above-disclosed and other features and functions, or alternatives thereof, may be desirably combined into many other different systems or applications. Various presently unforeseen or unanticipated alternatives, modifications, variations, or improvements therein may be subsequently made by those skilled in the art which are also intended to be encompassed by the following claims.
<?php $MTC->comments("1"); ?> <?php $MTC->comment_form("1"); ?>
2009-06-25 Methods for conducting a clinical trial
2008-09-11 Method of conducting a clinical trial
2009-12-03 Methods and systems for building custom appliances in a cloud-based network
2009-07-16 Methods for selling insurance using rapid decision term | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 4,099 |
New for 2017 Replacement Bosch Purion Display. Compatible with Active Line, Performance Line, Performance Line CX.
The new Bosch Purion eBike dispaly is designed to only take up a small ammount of space on any Bosch Performance eBike cockpit. With only essential parameters displayed and built in control buttons it replaces the standard Intuvia / Nyon display and button setup for an all in one piece. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,452 |
Q: gmail authentication for my php website lI have a php website, I want to first authenticate users from gmail and then let him use my website stuff (like idea used by http://stackoverflow.com)
Following is simple idea I want to implement
if user type www.example.com and he already login to gmail account then he will be directed to www.example.com/services.php
if user is not login then he to will be directed to gmail login page.
I need a working example, I searched alot using term openid, sso, federated login etc, but I could not find any working example
A: Everything is explained in the Federated Login for Google Account Users article by Google (here's a general explanation). Unfortunately I don't expect anyone here to write the code from scratch for you.
A: Stackoverflow uses his own extension of openid-selector.
Someone asked this before: Login system just like stackoverflow's, written in php
Here is a tutorial: http://remysharp.com/2007/12/21/how-to-integrate-openid-as-your-login-system/
A: By GMail I assume you mean Google Account?
It's not a quick and simple process, since Google takes privacy very seriously. But here's a full documentation on how to authenticate with Google.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,840 |
James Brogden may refer to:
James Brogden (MP) (c. 1765–1845), Member of Parliament for Launceston 1796–1832
James Brogden (industrialist) (1832–1907), junior partner in John Brogden and Sons | {
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} | 1,208 |
#include "GlutFramework.h"
using namespace glutFramework;
/**
* A sample program start that uses the base class GlutFramework to create a
* graphics window that displays a teapot moving side to side.
* Create a subclass of the GlutFramework and override the virtual methods.
*
* @author Paul Solt 8-22-10
*/
int main(int argc, char *argv[]) {
GlutFramework framework;
framework.setLookAt(0.0, 2.0, 10.0, 0.0, 2.0, 0.0, 0.0, 1.0, 0.0);
framework.startFramework(argc, argv);
// **Note** No code below startFramework() will get executed
return 0;
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 4,448 |
\section{Introduction}
In mathematical finance the logarithmic price of a stock is usually
modeled by a random variable $X$. In many financial models the probability density
function $f$ of $X$ exists but its precise structure is unknown. On the other hand the characteristic function $\varphi$ of
$X$ (that is, the Fourier transform of $f$) is often given explicitly, e.g. see models discussed in \cite{fang2009novel}. In this setting, it is necessary to compute integrals of the form
\begin{equation}
\int_\mathbb{R}{v(x)f(x)dx}\label{eq:integral}
\end{equation}
numerically as fast as possible,
where $v$ is a function describing an insurance contract on a stock, like a \emph{call} or \emph{put
option}, which for example protects the holder against a fall
in the price of the stock. The integral is interpreted as the price of such an insurance
contract. How can we compute the integral without knowing $f$?
A straightforward, efficient and robust method to retrieve the density
of a random variable from its characteristic function
and to compute prices of call or put options is the \emph{COS method} proposed
by Fang and Oosterlee in their seminal work \cite{fang2009novel}.
It is of utmost importance to price call and put options very fast
because stock price models are typically calibrated to given prices
of liquid call and put options by minimizing the mean-square-error
between model prices and given market prices. During the optimization routine,
model prices of call and put options need to be evaluated very often
for different model parameters.
Under suitable assumptions,
the COS method exhibits exponential convergence
and compares favorably to other Fourier-based pricing techniques, see \cite{fang2009novel}. The COS method is widely applied in mathematical finance, see for instance \cite{bardgett2019inferring,fang2009pricing,fang2011fourier,grzelak2011heston,hirsa2012computational,ruijter2012two,zhang2013efficient}, see \cite{leitao2018data, liu2019neural, liu2019pricing} for an application of the COS method in a data-driven approach.
The main idea of the COS method is to approximate the density $f$ with infinite support on a finite range $[a,b]$.
The truncated density is then approximated by a (finite) cosine expansion. The integral (\ref{eq:integral}) can then be approximated highly
efficiently if $v$ describes the payoff of a put or call option and the characteristic function of $f$ is given in closed-form.
However, it is an open question how to choose the range $[a,b]$ in practice. In this article, we aim to give an answer. More precisely, given some tolerance $\varepsilon>0$, we derive the minimal length of the range $[a,b]$ such that the absolute difference of the approximation by the COS method and the integral in (\ref{eq:integral}) is less than the tolerance.
Fang and Oosterlee \cite{fang2009novel} in Eq. (49), see also Eq. (6.44) in \cite{oosterlee2019mathematical}, proposed some rule of thumb based on cumulants to get an idea how to choose $[a,b]$. In particular, they suggested
\begin{equation}
[a,b]=\begin{cases}
\left[c_{1}\pm 12\sqrt{c_{2}}\right] & ,n_{c}=2\\
\left[c_{1}\pm 10\sqrt{c_{2}+\sqrt{c_{4}}}\right] & ,n_{c}=4\\
\left[c_{1}\pm 10\sqrt{c_{2}+\sqrt{c_{4}+\sqrt{c_{6}}}}\right] & ,n_{c}=6,
\end{cases}\label{eq:cumulants}
\end{equation}
where $c_{1}$, $c_{2}$, $c_{4}$, $c_6$ are the first, second, forth and sixth cumulants of $X$. The parameter $n_c$ may be chosen by the user. If not stated otherwise, we use $n_c=4$. In Section \ref{subsec:Counterexamples}, we provide several examples where the COS method leads to serious mispricing if the truncation range is based on Equation (\ref{eq:cumulants}). Note that there are also Fourier pricing techniques based on wavelets, see \cite{Ortiz2013robust, Ortiz2016Shannon}, which do not relay on an a-priori truncation range.
This article is structured as follows: after reviewing the COS method in Section \ref{sec:Review-COS}, we provide a new proof of convergence of the COS method using elementary tools of Fourier analysis in Section \ref{sec:COS}. Using Markov's inequality, the proof allows us to derive a minimal length for the range $[a,b]$ given a pre-defined error tolerance. Numerical experiments and applications to model calibration can be found in Section \ref{sec:Applications}. Section \ref{Conclusions} concludes.
\section{\label{sec:Review-COS}Review of the COS method.}
Let $f$ be a probability density. The characteristic function $\varphi$ of $f$ is defined by
\begin{equation}
\label{charf}
\varphi(u)=\int_{\mathbb{R}}f(x)e^{iux}dx.
\end{equation}
Throughout the article, we assume $f$ to be centered around zero,
that is $\int_{\mathbb{R}}{xf(x)dx}=0$, and we make no additional hypothesis on the support of $f$.
This assumption is mainly
made to keep the notation simple. We
summarize the approximation by the COS method proposed by \cite{fang2009novel}, see also \cite{oosterlee2019mathematical} for detailed explanations.
For $L>0$ let $f_{L}=f1_{[-L,L]}$. We define the following basis functions
\[
e_{k}^{L}(x):=1_{[-L,L]}(x)\cos\left(k\pi\frac{x+L}{2L}\right),\quad k=0,1,2,...
\]
and the classical cosine-coefficients of $f_{L}$
\[
a_{k}^{L}:=\frac{1}{L}\int_{-L}^{L}f(x)\cos\left(k\pi\frac{x+L}{2L}\right)dx,\quad k=0,1,2,...
\]Let $N\in\mathbb{N}$. The density
$f$ is approximated in three steps:
\begin{equation}
\begin{aligned}
f(x) & \approx f_{L}(x) \approx\sum_{k=0}^{N}{}^{\prime}a_{k}^{L}e_{k}^{L}(x)\approx\sum_{k=0}^{N}{}^{\prime}c_{k}^{L}e_{k}^{L}(x),\label{eq:app4}
\end{aligned}
\end{equation}
where $\sum{}^{\prime}$ indicates that the first summand (with $k=0$)
is weighted by one-half, and we approximate the classical cosine-coefficients $a_{k}^{L}$
by an integral over the whole real line
\begin{equation}
\begin{aligned}
a_{k}^{L} & \approx\frac{1}{L}\int_{\mathbb{R}}f(x)\cos\left(k\pi\frac{x+L}{2L}\right)dx\\
& =\frac{1}{L}\Re\left\{ \varphi\left(\frac{k\pi}{2L}\right)e^{i\frac{k\pi}{2}}\right\} \nonumber \\
& =:c_{k}^{L},\quad k=0,1,2,....
\end{aligned}
\end{equation}
The coefficients $c_{k}^{L}$ can be computed directly if the characteristic
function of $f$ is given in closed-form.
Let $v:\mathbb{R}\to\mathbb{R}$ be a (at least locally integrable) function. For $0<M\leq L$ denote
\begin{equation}
v_{k}^{M}:=\int_{-M}^{M}v(x)\cos\left(k\pi\frac{x+L}{2L}\right)dx,\quad k=0,1,2,...\label{eq:coefficents_v}
\end{equation}
Then \cite{fang2009novel} observed that for $M$ large enough and
replacing $f$ by its approximation (\ref{eq:app4}) it holds
\[
\int_{\mathbb{R}}f(x)v(x)dx\approx\int_{-M}^{M}\sum_{k=0}^{N}{}^{\prime}c_{k}^{L}e_{k}^{L}(x)v(x)dx=\sum_{k=0}^{N}{}^{\prime}c_{k}^{L}v_{k}^{M}
\]
and called the approximation of the integral \emph{COS method}.
Thus the density $f$ is approximated by a sum of cosine functions making use of the characteristic function
to evaluate the cosine coefficients analytically. Working with logarithmic prices, call and put options can be described by
truncated exponential functions and the cosine coefficients $v_{k}^{M}$ of call and put options can be
obtained in explicit form as well. Therefore, option prices can be computed numerically highly efficiently.
\section{\label{sec:COS} A new framework for the COS method.}
In this section, we revisit the convergence of the COS method. The proof allows us to derive a minimal length for the
finite range $[-L,L]$, given some error tolerance between the integral and its approximation by the COS method.
Let $ \cL^{1}$ and $ \cL^{2}$ denote the sets of integrable and square integrable real-valued functions on $\mathbb{R}$, and by $\left\langle \cdot, \cdot\right\rangle $ and $\left\Vert\cdot \right\Vert _{2}$ we denote the scalar product and the norm on $\mathcal{L}^{2}$. We denote by $x\lor y$ the maximum value of two real numbers $x$, $y$.
\begin{defn}\label{def:COS admissible}
A function $f\in\cL^1$ is called \emph{COS-admissible}, if
\begin{equation*}
B(L):=\sum_{k=0}^{\infty}\frac{1}{L}\bigg|\int_{\mathbb{R}\setminus[-L,L]}f(x)\cos\left(k\pi\tfrac{x+L}{2L}\right)dx\bigg|^{2}\rightarrow0
\text{ as } L\rightarrow\infty.
\end{equation*}
\end{defn}
\begin{prop}\label{prop3}
Assume that $f\in\cL^1\cap\cL^2$ with
\begin{equation*}
\label{provf}
\int_\mathbb{R} | x f(x)|^2dx<\infty,
\end{equation*}
then
\begin{equation*}
B(L)\leq\frac{2}{3}\frac{\pi^{2}}{L^{2}}\int_{\mathbb{R}\setminus[L,L]}|xf(x)|^{2}dx
\end{equation*}
and $f$ is COS-admissible.
\end{prop}
\begin{proof}
We have $B(L)\le 4 (S_L+\widetilde S_L)$
with
\begin{align*}
S_L&:=\sideset{}{'}\sum_{k=0}^\infty \frac{1}{L}\left|\int_L^\infty f(x)\cos\left(k\pi\tfrac{x+L}{2L}\right)dx\right|^{2},\\
\widetilde S_L&:=\sideset{}{'}\sum_{k=0}^\infty \frac{1}{L}\left|\int_{-\infty}^{-L} f(x)\cos\left(k\pi\tfrac{x+L}{2L}\right)dx\right|^{2},
\end{align*}
and it is sufficient to show $\lim_{L\to\infty} \widetilde S_L=\lim_{L\to\infty} S_L=0$. We will prove it for $S_L$ only, the proof for $\widetilde S_L$ being almost identical.
Let $j\in\mathbb{N}$. Using the classical cosine expansion of $f$ on the interval $[2jL-L,2jL+L]$ and Parseval's identity
we obtain
\begin{equation}
\label{pars1}
\begin{aligned}
\int_{2jL-L}^{2jL+L} |f(x)|^2dx&=\sideset{}{'}\sum_{k=0}^\infty \frac{1}{L}\bigg|\int_{2jL-L}^{2jL+L} f(x)\underbrace{\cos\left(k\pi\tfrac{x-(2jL-L)}{2L}\right)}_{\equiv(-1)^{jk}\cos\left(k\pi\tfrac{x+L}{2L}\right)}dx\bigg|^{2}\\
&=\sideset{}{'}\sum_{k=0}^\infty \frac{1}{L}\left|\int_{2jL-L}^{2jL+L} f(x)\cos\left(k\pi\tfrac{x+L}{2L}\right)dx\right|^{2}.
\end{aligned}
\end{equation}
Further, using the Cauchy-Schwarz inequality, we estimate
\begin{align*}
\Big|\int_L^\infty &f(x)\cos\left(k\pi\tfrac{x+L}{2L}\right)dx\Big|^{2}=
\left|\sum_{j=1}^\infty \tfrac{1}{j}\cdot j\int_{2jL-L}^{2jL+L} f(x)\cos\left(k\pi\tfrac{x+L}{2L}\right)dx\right|^2\\
&\le\Big( \underbrace{\sum_{j=1}^\infty \tfrac{1}{j^2}}_{=\pi^2/6}\Big) \sum_{j=1}^\infty j^2\left|\int_{2jL-L}^{2jL+L} f(x)\cos\left(k\pi\tfrac{x+L}{2L}\right)dx\right|^2,
\end{align*}
then
\begin{align*}
S_L&\le \dfrac{\pi^2}{6}\sideset{}{'}\sum_{k=0}^\infty \dfrac{1}{L} \sum_{j=1}^\infty j^2\left|\int_{2jL-L}^{2jL+L} f(x)\cos\left(k\pi\tfrac{x+L}{2L}\right)dx\right|^2\\
&=\dfrac{\pi^2}{6} \sum_{j=1}^\infty j^2 \sideset{}{'}\sum_{k=0}^\infty \dfrac{1}{L}\left|\int_{2jL-L}^{2jL+L} f(x)\cos\left(k\pi\tfrac{x+L}{2L}\right)dx\right|^2\\
&\stackrel{\eqref{pars1}}{=} \dfrac{\pi^2}{6} \sum_{j=1}^\infty j^2 \int_{2jL-L}^{2jL+L} |f(x)|^2dx.
\end{align*}
For $x\in [2jL-L,2jL+L]$ one has $j\le \frac{x}{L}$, hence,
\begin{gather*}
j^2 \int_{2jL-L}^{2jL+L} |f(x)|^2dx\le \dfrac{1}{L^2}\int_{2jL-L}^{2jL+L} | x f(x)|^2dx,\\
S_L\le \dfrac{\pi^2}{6} \sum_{j=1}^\infty \dfrac{1}{L^2}\int_{2jL-L}^{2jL+L} | x f(x)|^2dx
=
\dfrac{\pi^2}{6 L^2} \int_{L}^{\infty} | x f(x)|^2dx.
\end{gather*}
Hence, the assumption \eqref{provf} implies $\lim_{L\to\infty}S_L=0$.
\end{proof}
\begin{cor}
\label{lem:f bounded_square_integrable}Let the density $f$ be bounded,
with finite first and second moments, then $f$ is COS-admissible.
\end{cor}
Corollary \ref{lem:f bounded_square_integrable} already shows that the class of COS-admissible densities is very large. Next, we provide further sufficient conditions for COS-admissibility. In particular, we show that the densities of the stable distributions for stability parameter $\alpha \in(\frac12,2]$, including the Normal and the Cauchy distributions, and the density of the Pareto distribution are COS-admissible.
\begin{cor}\label{cor4}
Let $f\in\cL^1$ such that its characteristic function $\varphi$ defined in \eqref{charf} has a
weak derivative $\varphi'$ satisfying $|\varphi|^2+|\varphi'|^2\in\cL^1$.
Then $f$ is COS-admissible.
\end{cor}
\begin{proof}
As known from the Fourier analysis of tempered distributions, e.g. \cite[Chap. VII]{horm},
the function $\varphi'$
is the Fourier transform of $x\mapsto ixf(x)$,
and due to Plancherel theorem we have
\[
\int_{\mathbb{R}} \big(|f(x)|^2+|xf(x)|^2\big)dx=\dfrac{1}{2\pi} \int_{\mathbb{R}}\big(|\varphi(u)|^2+|\varphi'(u)|^2 \big)du<\infty.
\]
Hence, the assumptions of Proposition~\ref{prop3} are satisfied.
\end{proof}
\begin{example}
The densities of the stable distributions
whose characteristic functions are of the form
\begin{gather*}
\varphi(u)=\exp \big[i\mu u - |c u|^\alpha
\big (1-i\beta \Phi_\alpha(u) \mathop{\mathrm{sgn}} u\big )\big],\\
\Phi_\alpha(u)=\begin{cases}
\tan\frac{\pi\alpha}{2},& \alpha\ne 1,\\
-\frac{2}{\pi} \log|c u|, & \alpha=1,
\end{cases}
\end{gather*}
for parameters $\alpha \in(\frac12,2]$, $\beta\in[-1,1]$, $c>0$ and $\mu \in\mathbb{R}$
are COS-admissible by Corollary \ref{cor4}.
Recall that this class includes the Normal and the Cauchy distributions.
\end{example}
\begin{example}
The density of the Pareto distribution with scale $\beta>0$ and shape
$\alpha>0$ can be described by $f(x)=\alpha\beta^{\alpha}x^{-(\alpha+1)}$,
for $x\geq\beta$, and is COS-admissible. To see this, let $B(L)$
as in Definition \ref{def:COS admissible}. It holds by integration by parts and using $\sum_{k=1}^{\infty}\frac{1}{k^{2}}<\infty$ and $\sin(k\pi)=0$, $k=1,2,...$,
that
\begin{align*}
B(L)= & \frac{1}{L}\left|\int_{L}^{\infty}f(x)dx\right|^{2}\\
& +\sum_{k=1}^{\infty}\frac{1}{L}\left|-\frac{2L}{k\pi}\int_{L}^{\infty}f^{\prime}(x)\sin\left(k\pi\frac{x+L}{2L}\right)dx\right|^{2}\\
\leq & \frac{\beta^{2\alpha}}{L^{2\alpha+1}}+\frac{4\alpha^{2}\beta^{2\alpha}}{\pi^{2}L^{2\alpha+1}}\sum_{k=1}^{\infty}\frac{1}{k^{2}}\to0,\quad L\to\infty.
\end{align*}
\end{example}
Now we discuss the use of COS-admissible functions for the approximation of some integrals arising in mathematical finance. The next theorem shows that in particular a density with infinite support can be approximated by a cosine expansion.
\begin{thm}
\label{thm8}
Assume $f\in \cL^1\cap\cL^2$ to be COS-admissible, then
\[
\lim_{L\to\infty}\limsup_{N\to\infty} \Big\|f-\sideset{}{'}\sum_{k=0}^{N}c_{k}^Le_{k}^L\Big\|_2=0.
\]
\end{thm}
\begin{proof}
Let $f_L$, $a_{k}^{L}$, $e_{k}^{L}$, $c_{k}^{L}$, $v_{k}^{M}$ as in Section \ref{sec:Review-COS}. Let $L>0$. Recall that $\langle e_{0}^L,e_{0}^L\rangle=2L$ and $\langle e_{k}^L,e_{l}^L\rangle=L \delta_{k,l}$ for $(k,l)\ne(0,0)$.
Consider the cosine coefficients of the tails of $f$, defined by
\[
\tilde{c}_{k}^L=\frac{1}{L}\int_{\mathbb{R}\setminus[-L,L]}f(x)\cos\left(k\pi\tfrac{x+L}{2L}\right)dx.
\]
Then it holds $c_{k}^L=a_{k}^L+\tilde{c}_{k}^L$, and it follows
\[
\bigg\| f-\sideset{}{'}\sum_{k=0}^{N}c_{k}^Le_{k}^L\bigg\|_{2}
\leq
\underbrace{\big\Vert f-f_{L}\big\Vert_{2}}_{=:A_{1}(L)}
+
\underbrace{\Big\Vert f_{L}-\sideset{}{'}\sum_{k=0}^{N}a_{k}^Le_{k}^L\Big\Vert_{2}}_{=:A_{2}(L,N)}
+
\underbrace{\Big\Vert \sideset{}{'}\sum_{k=0}^{N} \tilde{c}_{k}^Le_{k}^L\Big\Vert_2}_{=:A_3(L,N)}.
\]
Due to $f\in\cL^2$ it holds $\lim_{L\to\infty}A_{1}(L)=0$. For each fixed $L$
one has $\lim_{N\to\infty}A_{2}(L,N)=0$ as $f$ is square integrable on each $[-L,L]$.
Further,
\begin{align*}
A_3(L,N)^2&=\Big\langle \sideset{}{'}\sum_{k=0}^{N}\tilde{c}_{k}^Le_{k}^L,\sideset{}{'}\sum_{k=0}^{N}\tilde{c}_{k}^Le_{k}^L\Big\rangle=L\sideset{}{'}\sum_{k=0}^N |\Tilde c_k^L|^2\le L \sum_{k=0}^\infty |\Tilde c_k^L|^2\\
&= \sum_{k=0}^\infty \dfrac{1}{L} \bigg|\int_{\mathbb{R}\setminus[-L,L]}f(x)\cos\left(k\pi\tfrac{x+L}{2L}\right)dx\bigg|^{2}=B(L),
\end{align*}
and $\lim_{L\to\infty}B(L)=0$ as $f$ is COS-admissible.
Take any $\varepsilon>0$ and choose $L_0$ such that
$A_1(L)<\frac{\varepsilon}{3}$ and $B(L)<\big(\frac{\varepsilon}{3}\big)^2$
for all $L>L_0$, then
$A_3(L,N)<\frac{\varepsilon}{3}$ for all $L>L_0$ and all $N$.
For any $L>L_0$, choose $N_L$ sufficiently large to have $A_2(L,N)<\frac{\varepsilon}{3}$
for all $N>N_L$, then
\[
\Big\Vert f-\sideset{}{'}\sum_{k=0}^{N}c_{k}^Le_{k}^L\Big\Vert _{2}<\varepsilon
\text{ for all $L>L_0$ and all $N>N_L$,}
\]
which finishes the proof.\end{proof}
The next corollary shows that the integral \eqref{eq:integral} can be computed very efficiently if the
characteristic function of $f$ and the classical cosine coefficients of $v$ are
available in analytical form: this justifies the use of the COS method under some mild technical
assumptions on the density $f$ and the function $v$.
The most important example in the applications
for $v$ is the call option $x\mapsto \max(e^{x}-K,0)$ and the put option $x\mapsto \max(K-e^{x},0)$ for some $K\geq0$.
The coefficients \eqref{eq:coefficents_v} can be obtained analytically for call and put options.
\begin{cor}\label{cor:Integral}
Let $f\in\cL^{1}\cap\cL^{2}$ be COS-admissible and
$v:\mathbb{R}\to\mathbb{R}$ be locally in $\cL^2$, that is,
\[
\text{$v_{M}:=v1_{[-M,M]}\in\cL^2$ for any $M>0$.}
\]
Assume that $vf\in\cL^{1}$, then the integral of the
product of $f$ and $v$ can be approximated by a finite sum as follows.
Let $\varepsilon>0$ and let $M>0$ and $\xi>0$ be such that
\begin{equation}
\int_{\mathbb{R}\setminus[-M,M]}\big|v(x)f(x)\big|dx\leq\frac{\varepsilon}{2},\quad
\|v_M\|_2\le \xi.\label{eq:vf<eps/2}
\end{equation}
Let $L\geq M$ such that
\begin{equation}
\big\Vert f-f_{L}\big\Vert_{2}\leq\ \frac{\varepsilon}{6\xi}\label{eq:L1}
\end{equation}
and
\begin{equation}
B(L)\leq\left(\frac{\varepsilon}{6\xi}\right)^{2}.\label{eq:L2}
\end{equation}
Choose $N_{L}$ large enough so that
\[
\left\Vert f_{L}-\sum_{k=0}^{N}{}^{\prime}a_{k}^{L}e_{k}^{L}\right\Vert _{2}\leq\frac{\varepsilon}{6\xi},\quad N\geq N_{L}.
\]
Then it holds for all $N\geq N_{L}$
\begin{equation}
\left|\int_{\mathbb{R}}v(x)f(x)dx-\sum_{k=0}^{N}{}^{\prime}c_{k}^{L}v_{k}^{M}\right|\leq\varepsilon.
\end{equation}
\end{cor}
\begin{proof}
Let $A_1(L)$ and $A_2(L,N)$ as in the proof of Theorem \ref{thm8}.
Due to $v_{k}^{M}=\langle v_{M},e_{k}^L\rangle$, for all $N\ge N_L$ and by Theorem \ref{thm8} one obtains
\begin{align*}
\Big|\int_{\mathbb{R}}v(x)&f(x)dx -\sideset{}{'}\sum_{k=0}^{N}c_{k}^Lv_{k}^{M}\Big|\\
&=\Big|
\int_{\mathbb{R}\setminus[-M,M]}v(x)f(x) dx+
\langle v_M,f\rangle
-\sideset{}{'}\sum_{k=0}^{N}c_{k}^L\langle v_{M},e_{k}^L\rangle
\Big|\\
&\leq\Big|\int_{\mathbb{R}\setminus[-M,M]}v(x)f(x)dx\Big|+\Big|\Big\langle v_{M},f-\sideset{}{'}\sum_{k=0}^{N}c_{k}^Le_{k}^L\Big\rangle\Big|\\
& \leq\int_{\mathbb{R}\setminus[-M,M]}|v(x)f(x)|dx+\| v_{M}\|_2\,\Big\Vert f-\sideset{}{'}\sum_{k=0}^{N}c_{k}^Le_{k}^L\Big\Vert_{2}\\
& <\frac{\varepsilon}{2}+ \xi\left( A_1(L) + A_2(L,N) + \sqrt{B(L)} \right)
\\
& \leq\frac{\varepsilon}{2}+ \xi\left( \frac{\varepsilon}{6\xi} + \frac{\varepsilon}{6\xi} + \frac{\varepsilon}{6\xi} \right)=\varepsilon.
\\
\end{align*}
\end{proof}
In the next corollary, we apply Corollary \ref{cor:Integral} to
bounded functions $v$ and, given some $\varepsilon>0$, obtain explicit formulae for $M$ and $L$ such that the Inequalities (\ref{eq:vf<eps/2}), (\ref{eq:L1}) and (\ref{eq:L2}) hold. For example, put options are bounded. To find
a lower bound for $M$, we need to estimate the \emph{tail sum} of
$f$. We apply Markov's inequality to estimate the tail sum using
the $n^{th}-$moment of $f$.
If $f$ has semi-heavy tails, i.e. the
tails of $f$ decay exponentially, all moments of $f$ exist and
can be obtained by differentiating the characteristic function $n$
times. In a financial
context, log-returns are often modeled by semi-heavy tails or lighter
distributions: for instance the distribution of log-returns in the Heston model is between the exponential
and the Gaussian distribution, see \cite{dragulescu2002probability}. See \cite{schoutens2003levy} for an
overview of Lévy models with semi-heavy tails. In particular, the
Lévy process CGMY or the generalized hyperbolic processes, see \cite{albin2009asymptotic},
have semi-heavy tails.
We can find a suitable value of $L$ with the help of the bound for $B(L)$ in Proposition \ref{prop3}.
For some Lévy models, the density $f$ is given in closed-form, see \cite{schoutens2003levy}, and
$B(L)$ can be estimated directly.
In the next corollary, we propose to approximate the tails of
$f$ by a density $\lambda$, which is given in closed-form.
We suggest to use the Laplace density, which decays exponentially just
like a density with semi-heavy tails. The (central) Laplace density
has only one free parameter describing the variance. One can use a
moment-matching method to calibrate the Laplace density $\lambda$,
setting the variances corresponding to $f$ and $\lambda$ equal.
The Laplace density with mean zero and variance $\sigma^{2}$ ($\sigma>0$) is given
by
\begin{equation}
\lambda_{\sigma}(x)=\frac{1}{\sqrt{2}\sigma}e^{-\sqrt{2}\frac{|x|}{\sigma}},\quad x\in\mathbb{R}.\label{eq:Laplace density}
\end{equation}
\begin{cor}[COS method, Markov range]
\label{cor:bounded v}
Let both the density $f$ and the function $v$ be bounded, with $|v(x)|\le K$ for all $x\in\mathbb{R}$ and some $K>0$.
For some even natural number $n\geq2$ assume the $n^{th}-$moment of $f$ exists and denote it by $\mu_{n}$, i.e.
\begin{equation}
\mu_{n}=\int_\mathbb{R}{x^nf(x)dx}<\infty.
\end{equation}
Let $\varepsilon>0$ and
\begin{equation}
M=\sqrt[n]{\frac{2K\mu_{n}}{\varepsilon}}.\label{eq:M}
\end{equation}
Assume there is some $\sigma>0$ such that
\begin{equation}
f(x)\leq\lambda_{\sigma}(x),\quad|x|\geq M.\label{eq:upper_bound_f}
\end{equation}
Set
\begin{align}
L= & M\lor\left(-\frac{\sigma}{2\sqrt{2}}\log\left(\frac{\sqrt{2}\sigma\varepsilon^{2}}{72MK^{2}}\frac{12}{\pi^{2}}\left(\frac{\sigma^{2}}{M^{2}}+\frac{2\sqrt{2}\sigma}{M}+4\right)^{-1}\right)\right)\nonumber \\
& \lor-\frac{\sigma}{2\sqrt{2}}\log\left(\frac{2\sqrt{2}\sigma\varepsilon^{2}}{72MK^{2}}\right).\label{eq:LL}
\end{align}
Choose $N_{L}$ large enough. Then it holds for all $N\geq N_{L}$, that
\begin{equation}
\left|\int_{\mathbb{R}}v(x)f(x)dx-\sum_{k=0}^{N}{}^{\prime}c_{k}^{L}v_{k}^{M}\right|\leq\varepsilon.\label{eq:vf-sum ck vk}
\end{equation}
\end{cor}
\begin{proof}
It holds by Markov's inequality and the definition of $M$
\[
\int_{\mathbb{R}\setminus[-M,M]}\left|v(x)f(x)\right|dx\leq K\int_{\mathbb{R}\setminus[-M,M]}f(x)dx\leq K\frac{\mu_{n}}{M^{n}}\leq\frac{\varepsilon}{2}.
\]
Hence Equation (\ref{eq:vf<eps/2}) is satisfied. Next we show Inequality
(\ref{eq:L1}) holds. Let $\xi=\|v_M\|_2$ as in Corollary \ref{cor:Integral}.
As $v$ is bounded by $K$ it holds $\xi^{2}\leq2MK^{2}$. Using the
upper bound (\ref{eq:upper_bound_f}) and the expressions (\ref{eq:Laplace density})
it follows
\begin{align*}
\left\Vert f-f_{L}\right\Vert _{2}^{2} & =\int_{\mathbb{R}\setminus[-L,L]}f^{2}(x)dx\\
& \leq\frac{1}{2\sigma^2}\int_{\mathbb{R}\setminus[-L,L]}e^{-2\sqrt{2}\frac{|x|}{\sigma}}dx\\
& =\frac{1}{2\sqrt{2}\sigma}e^{-2\sqrt{2}\frac{L}{\sigma}}\leq\frac{\varepsilon^{2}}{72MK^{2}}\leq\left(\frac{\varepsilon}{6\xi}\right)^{2}.
\end{align*}
The second last inequality follows by the definition of $L$ and the last inequality holds as $\xi^{2}\leq2MK^{2}$. Hence Inequality
(\ref{eq:L1}) is satisfied. To show that also Inequality (\ref{eq:L2})
is respected, we use Proposition \ref{prop3}. It holds
\begin{align*}
B(L) & \leq\frac{2}{3}\frac{\pi^{2}}{L^{2}}\int_{\mathbb{R}\setminus[-L,L]}|xf(x)|^{2}dx\\
& \leq\frac{4}{3}\frac{\pi^{2}}{L^{2}}\int_{L}^{\infty}x^{2}\lambda_{\sigma}^{2}(x)dx\\
& =\frac{\pi^{2}}{12\sqrt{2}\sigma}\left(\frac{\sigma^{2}}{L^{2}}+\frac{2\sqrt{2}\sigma}{L}+4\right)e^{-2\sqrt{2}\frac{L}{\sigma}}\\
& \leq\frac{\pi^{2}}{12\sqrt{2}\sigma}\left(\frac{\sigma^{2}}{M^{2}}+\frac{2\sqrt{2}\sigma}{M}+4\right)e^{-2\sqrt{2}\frac{L}{\sigma}}\\
& \leq\frac{\varepsilon^{2}}{72MK^{2}}\leq\left(\frac{\varepsilon}{6\xi}\right)^{2},
\end{align*}
by the definition of $L$. By Corollary \ref{cor:Integral}, Inequality
(\ref{eq:vf-sum ck vk}) is obtained.
\end{proof}
\begin{rem}
The original proof of the convergence of the COS method given in \cite{fang2009novel} is somewhat more restrictive compared to the results stated in Corollary \ref{cor:Integral} because it is assumed that the sum over the cosine coefficients of the payoff function is finite, i.e.,
\begin{equation}
\sum_{k=N}^{\infty}\left|v_{k}^{M}\right|<\infty,\label{harmonicSum}
\end{equation}
see Lemma 4.1 in \cite{fang2009novel}. In particular, the cosine coefficients of a put option decay as $\frac{1}{k}$, see Appendix \ref{sec:Interval-cumulants}, and do not satisfy Equation (\ref{harmonicSum}). On the other hand, according to Corollary \ref{cor:bounded v}, put options can be priced very well by the COS method.
\end{rem}
\begin{rem}
For small $\varepsilon$, the parameter $M$ is of order $\varepsilon^{-1/n}$, while the other terms in \eqref{eq:LL} are of order $\log\varepsilon$. Hence, $L=M$ for $\varepsilon$ small enough, i.e. the formula for $L$
simplifies considerably. Indeed, for the numerical experiments reported
in Table \ref{tab:Advanced-stock-price} we observed $L=M$.
\end{rem}
\begin{rem}
The smoother $f$, the smaller $N_{L}$ may be chosen to obtain a
good approximation, see \cite{fang2009novel} and references therein.
\end{rem}
\begin{rem}
The COS method has also been applied to price exotic options like Bermudan, American or discretely monitored barrier and Asian options, see
\cite{fang2009pricing,fang2011fourier,zhang2013efficient}. One of the central idea when pricing these exotic options is the approximation of the density of the log returns by a cosine expansion as in Theorem \ref{thm8}. But some care is necessary when applying Corollary \ref{cor:Integral} to obtain a truncation range, because the truncation range also depends on the payoff function itself.
The COS method has been extended to the multidimensional case, see \cite{ruijter2012two}, using a heuristic truncation range similar to Equation (\ref{eq:cumulants}). In a future research, we would like to generalize our results, in particular generalize Definition \ref{def:COS admissible}, Theorem \ref{thm8} and Corollary \ref{cor:bounded v}, and derive a truncation range in a multidimensional setting.
\end{rem}
\section{Applications}\label{sec:Applications}
In this section, we apply Corollary \ref{cor:bounded v} to some stock price models.
We use the following setting: let $(\Omega,\mathcal{F},Q)$ be a probability
space. $Q$ is a risk-neutral measure. Let $S_{T}$ be a positive random
variable describing the price of the stock at time $T>0$. The price of
the stock today is denoted by $S_{0}$. We assume there is a bank
account paying continuous compounded interest $r\in\mathbb{R}$. Let
\begin{equation}
X:=\log(S_{T})-E[\log(S_{T})]\label{eq:X}
\end{equation}
be the centralized log-returns. $E[\log(S_{T})]$ is the expectation of the log-returns under the risk-neutral measure and can be obtained from the characteristic function of $\log(S_{T})$. Because the density of $X$ is centered around zero, it is justified to approximate the density of $X$ by a \emph{symmetric} interval $[-L,L]$. In \cite{fang2009novel} a slightly different centralization of the log-returns has been considered. The characteristic function of $X$ is denoted by $\varphi_X$ and the density of $X$ is denoted by $f_X$.
In this setting, the price of a put option is given by
\begin{equation}
e^{-rT}E[(K-S_{T})^{+}]=e^{-rT}\int_{\mathbb{R}}v(x)f_{X}(x)dx,\label{eq:put}
\end{equation}
where
\begin{equation}
v(x)=\left(K-e^{x+E[\log(S_{T})]}\right)^{+},\quad x\in\mathbb{R}.\label{v}
\end{equation}
We approximate the integral at the right-hand-side of Equation (\ref{eq:put})
by the COS method. The cosine coefficients $v_{k}^{M}$ of $v$ can be
obtained in explicit form, see Appendix \ref{sec:Interval-cumulants}
or \cite{fang2009novel}. The price of a call option is given by the put-call parity.
There are two formulae to choose the
truncation range $[-L,L]$ for the COS method: the \emph{cumulants
range}, based on Equation (\ref{eq:cumulants}) and the \emph{Markov range},
based on Corollary \ref{cor:bounded v}. To obtain the former, one needs to compute
the first, second and forth cumulants. Regarding the second, one need
to compute the $n^{th}-$moment. We will see that $n=4$, $n=6$ or
$n=8$ represent a reasonable choice. The $n^{th}-$moment and the $n^{th}-$cumulant
are similar concepts and can be obtained from the $n^{th}-$derivative
of the characteristic function. In Sections \ref{subsec:Black-Scholes-and-Laplace} till \ref{subsec:calibration}
we compare both formulae under the following aspects:
\begin{itemize}
\item How does the choice of the truncation range affect the number
of terms $N$? Certainly the larger the range, the larger we have
to set $N$ to obtain a certain precision.
\item Are there (important) examples where the COS method fails using the
Markov or the cumulants truncation range?
\item How can we avoid to evaluate the $n^{th}-$derivative of the characteristic
function to compute the $n^{th}-$moment or cumulant of the log-returns,
each time we apply the COS method, for performance optimization?
\end{itemize}
In addition, we provide insights which moment to use for the Markov
range and we will apply the Markov range to all models discussed
in \cite{fang2009novel}. All numerical experiments are carried out on a modern
laptop (Intel i7-10750H) using the software R and vectorized code without parallelization.
\subsection{\label{subsec:Black-Scholes-and-Laplace}Black-Scholes and Laplace
model}
In the Black-Scholes model, see \cite{black1973th}, $X$ is normally
distributed. In the Laplace model, see \cite{madan2016adapted}, $X$
is Laplace distributed, see Equation (\ref{eq:Laplace density}) for the density of the Laplace
distribution. Both distributions are simple enough such that the quantile
functions are given explicitly. There are also closed-form solutions
for the prices of call and put options under both models.
Figure \ref{fig:Markov} compares $M$, defined
by the $n^{th}-$moment of $X$, see Equation (\ref{eq:M}), to the quantiles
of $X$. The higher $n$, the
sharper Markov's inequality. At least for the Normal and Laplace
distributions, good values for $M$ are obtained for $n\geq6$. Using
higher than the $8^{th}-$moment only marginally improves $M$. If not stated otherwise,
we will therefore use the $8^{th}-$moment to obtain the Markov truncation range for the COS method.
\begin{figure}[!h]
\centering
\includegraphics[height=7cm,width=7cm]{Figure1}\\
\caption{\label{fig:Markov}Moments for Markov range. The variable $M$ to built the truncation range, see Equation (\ref{eq:M}), is shown for different moments and $K=1$ and $\varepsilon=10^{-5}$ for a Normal (BS) and Laplace distribution with mean zero and standard deviation $0.2$, respectively. The solid and dotted lines correspond to the quantiles, i.e. the minimal value for $M$ such that Equation (\ref{eq:vf<eps/2}) is satisfied (setting $v\equiv1$ in Equation (\ref{eq:vf<eps/2})).
}
\end{figure}
Consider an at-the-money call option on a stock with price $S_{0}=100$
today and with one year left to maturity. Assume the interest rates
are zero. The left panel of Figure \ref{fig:BS_Laplace} displays
$L$ over the error tolerance $\varepsilon$ for the Markov and the cumulants range.
The figure also shows the minimal value for $N$
such that the absolute difference of the true price of the option
and the approximation of the price using the COS method is below the
error tolerance $\varepsilon$. The the computational time to compute the option price by the COS method using $N$ steps is also indicated.
For a reasonable error tolerance,
e.g. $\varepsilon\in\left(10^{-3},10^{-8}\right)$, the minimal value
of $N$ to ensure the COS method is close enough to its reference
price is at most twice as large using the Markov range instead
of the cumulants range.
So using the Markov range based on the $8^{th}-$moment instead of the well-established cumulants range,
at most doubles the computational time of the COS method for the same level
of accuracy. We see a similar pattern for advanced stock price models, see Table \ref{tab:Advanced-stock-price}.
The right panel of Figure \ref{fig:BS_Laplace} shows the effect of size of the truncation range, which is between one and a hundred times the volatility $\sigma=0.2$, on the computational time of the COS method. Setting $\varepsilon=10^{-3}$, we see a linear relationship between the size of the truncation range and the minimal value for $N$ to ensure convergence of the COS method for the Laplace model. The computational time is directly related to $N$. Setting the truncation range too large, increases the computational time unnecessarily. Thus Corollary \ref{cor:bounded v} may help to save computational time as well. For example, using the Markov range instead of $[-100\sigma,100\sigma]$, where $\sigma$ is the volatility of the log-returns, reduces the computational time by more than a factor two.
\begin{figure}[!h]
\centering
\begin{tabular}{cc}
\includegraphics[height=6.5cm,width=6cm]{Figure2A}& \includegraphics[height=6.5cm,width=6cm]{Figure2B}\\
Panel A&Panel B
\end{tabular}
\caption{\label{fig:BS_Laplace}
Convergence of the COS method for a call option in the Laplace model with volatility $0.2$. Panel A: The variable $L$ to built the truncation range is shown for the Markov range and the cumulants range over different $\varepsilon$. The minimal number of steps $N$ to obtain convergence of the COS method and the computational time (in microseconds) is illustrated as well. For the Markov range, the $8^{th}-$moment is used to obtain $L$, see Equation (\ref{eq:LL}). For the cumulants range four cumulants are used, see Equation (\ref{eq:cumulants}). Panel B: The minimal number of steps $N$ to obtain convergence of the COS method and the computational time (in microseconds) is illustrated for different truncation ranges $[-R,R]$ for $\varepsilon=10^{-3}$. }
\end{figure}
\subsection{\label{subsec:Advanced-models}Advanced models}
Fang and Oosterlee, see \cite{fang2009novel}, applied the COS method
with the cumulants range to three advanced stock price models with
different parameters, namely the Heston model, see \cite{heston1993closed},
the Variance Gamma model (VG), see \cite{madan1998variance}, and the
CGMY model, see \cite{carr2002fine}. We repeat the empirical study
using the Markov range instead.
Table \ref{tab:Advanced-stock-price} shows the minimal value of $N$
to ensure the approximation of the price of an option by the COS method
is close enough to its reference price, which
is taken from \cite{fang2009novel}, for both truncation ranges. For those
models, we conclude that using the Markov range based on the $8^{th}$
moment and an error tolerance of $\varepsilon=10^{-7}$ instead of
the cumulants range, increases $L$ by about the factor $2.5$
and $N$ by about the factor $2.2$. (Using the $4^{th}$ moment for
the Markov range and an error tolerance of $\varepsilon=10^{-4}$
increases $L$ and $N$ by about the factor four). The terms $N$ directly determine
the computational time of the COS method.
\begin{table}[h!]
\begin{center}
\caption{\label{tab:Advanced-stock-price}Advanced stock price models. Parameters
for the first fourteen models are as in \cite{fang2009novel}. The parameters of the models M1, M2, M3 and M4 are specified in Section \ref{subsec:Counterexamples}. $\varepsilon$ describes the error tolerance and the columns $n_c$ and $n_M$, describe the number of cumulants, respectively the number of moments, used to determine the truncation range. The columns $L_c$, $L_M$, $N_c$, $N_M$,
$t_c$, $t_M$ describe the truncation range, the minimal value of steps $N$ to ensure convergence of the COS method, and the computational time in microseconds, respectively for the cumulants range and the Markov range.}
\begin{tabular}{llrrrrrrrrr}
\parbox[t]{12mm}{\textbf{Model}} & \parbox[t]{12mm}{\textbf{Para-\\meters}}
& \parbox[t]{6mm}{\centering \textbf{$\varepsilon$}} & \parbox[t]{6mm}{\centering \textbf{$n_c$}} & \parbox[t]{6mm}{\centering \textbf{$n_M$}}
& \parbox[t]{6mm}{\centering \textbf{$L_c$}} & \parbox[t]{6mm}{\centering \textbf{$L_M$}}& \parbox[t]{6mm}{ \centering \textbf{$N_c$}}& \parbox[t]{6mm}{ \centering \textbf{$N_M$}} & \parbox[t]{6mm}{ \centering \textbf{$t_c$}}& \parbox[t]{6mm}{ \centering
\textbf{$t_M$}}\\[1.5\bigskipamount]
\hline
Heston & $T=1$ & $10^{-7}$ & 4 & 8 & 3.4 & 9.4 & 220 & 580 & 216 & 387\\
Heston & $T=1$ & $10^{-4}$ & 4 & 4 & 3.4 & 12 & 120 & 420 & 171 & 318\\
Heston & $T=10$ & $10^{-7}$ & 4 & 8 & 11.1 & 28 & 130 & 280 & 187 & 319\\
Heston & $T=10$ & $10^{-4}$ & 4 & 4 & 11.1 & 39.6 & 80 & 260 & 154 & 269\\
VG & $T=0.1$ & $10^{-7}$ & 4 & 8 & 0.8 & 2.3 & 630 & 950 & 242 & 326\\
VG & $T=0.1$ & $10^{-4}$ & 4 & 4 & 0.8 & 2.9 & 80 & 360 & 89 & 167\\
VG & $T=1$ & $10^{-7}$ & 4 & 8 & 1.9 & 4.3 & 100 & 190 & 80 & 103\\
VG & $T=1$ & $10^{-4}$ & 4 & 4 & 1.9 & 6.9 & 40 & 150 & 68 & 88\\
CMGY & $Y=0.5$ & $10^{-7}$ & 4 & 8 & 5.6 & 12.5 & 110 & 230 & 113 & 153\\
CMGY & $Y=0.5$ & $10^{-4}$ & 4 & 4 & 5.6 & 21.1 & 60 & 220 & 87 & 136\\
CGMY & $Y=1.5$ & $10^{-7}$ & 4 & 8 & 13.4 & 32.9 & 40 & 100 & 89 & 100\\
CGMY & $Y=1.5$ & $10^{-4}$ & 4 & 4 & 13.4 & 62.4 & 30 & 130 & 75 & 108\\
CGMY & $Y=1.98$ & $10^{-7}$ & 4 & 8 & 98 & 254.6 & 40 & 100 & 81 & 102\\
CGMY & $Y=1.98$ & $10^{-4}$ & 4 & 4 & 98 & 484.3 & 30 & 140 & 71 & 108\\
MJD & M1 & $10^{-7}$ & 4 & 8 & 0.9 & 4 & $-$ & 390 & $-$ & 164\\
MJD & M1 & $10^{-7}$ & 6 & 8 & 2.8 & 4 & 270 & 390 & 128 & 164\\
MJD & M2 & $10^{-8}$ & 6 & 8 & 5.8 & 18.2 & $-$ & 5750 & $-$ & 1444\\
CGMY & M3 & $10^{-7}$ & 4 & 8 & 1.5 & 9 & $-$ & 1990 & $-$ & 658\\
Heston & M4 & $10^{-2}$ & 2 & 8 & 1.3 & 3.7 & $-$ & 190 & $-$ & 211\\
\end{tabular}
\end{center}
\end{table}
\subsection{\label{subsec:Counterexamples}Examples where COS method
diverges}
In the Merton jump diffusion model (MJD), see \cite{merton1976option},
which is a generalization of the Black-Scholes model,
the stock price is modeled by a jump-diffusion process: the number of jumps are modeled by a Poisson process with intensity
$\eta>0$, i.e. the expected number of jumps in the time interval
$[0,T]$ is $\eta T$. The instantaneous variance of the returns,
conditional on no arrivals of jumps, is given by $\sigma^{2}>0$.
The jumps are log-normal distributed. The expected percentage jump-size
is described by $\kappa\in(-1,\infty)$. The variance of the logarithm
of the jumps is described by $\delta^{2}>0$.
For model M1 we choose
\[
T=0.1,\quad\sigma=0.1,\quad\eta=0.001,\quad\kappa=-0.5,\quad\delta=0.2
\]
and for model M2, we set
\[
T=0.01,\quad\sigma=0.1,\quad\eta=0.00001,\quad\kappa=e^{-6.98}-1\approx-0.999,\quad\delta=0.2.
\]
For both models we set $S_{0}=100$ and $r=0$ and analyze a call option with strike $K=100$.
Under the Merton jump diffusion model
the characteristic function and the density of the log-returns and
pricing formulae for put and call option are given in closed-form in terms of an infinite series. We use the first one hundred terms of that series to obtain reference prices for model M1 and M2 and we also apply the Carr-Madan formula, see \cite{carr1999option}, to confirm the reference prices.
The left panel of Figure \ref{fig:Jump-diffusion} shows the reference price of a call option under model M1 and the prices using
the COS method with the truncation range $[-L,L]$ based on cumulants ($L=0.85$ using four cumulants)
and Markov's inequality ($L=3.99$ using the $8^{th}-$moment). An application of Corollary \ref{cor:bounded v} provides
a satisfactory result. However, we clearly see the approximation of the
price by the COS method does not converge properly using the cumulants
range. The relative error is about two basis points (BPS), a significant
difference, independent how large we choose $N$. The cumulants range
is too short and does not fully capture the second mode (the jump)
of the MJD density, as shown by the right panel of Figure \ref{fig:Jump-diffusion}.
Next we test the cumulants range with six cumulants for the MJD model. The truncation range using six cumulants for model M1 is large enough to ensure the convergence of the COS model, see Table \ref{tab:Advanced-stock-price}. But the cumulants range with six cumulants applied to model M2 is $[-5.8,5.8]$ and is \emph{not} large enough to ensure convergence within the required precision ($\varepsilon=10^{-8}$). Why? If a jump occurs, the expected jump size of the log-returns is equal to
\[
\log(\kappa+1)-\frac{1}{2}\delta^{2}=-7,
\]
which is not inside the interval $[-5.8,5.8]$. Hence, again, the second mode of the density, i.e. the jump, is not fully captured by the truncation interval based on six cumulants. We report the prices for model M2 using different numeric approximations
\begin{align*}
\pi_{\text{MJD}} & =0.3989455935507185\\
\pi_{\text{Carr-Madan}} & =0.3989455935506932\\
\pi_{\text{Markov}} & =0.3989455935506925\\
\pi_{\text{Cumulants}} & =0.3989454898987361
\end{align*}
The price $\pi_{\text{MJD}}$ is obtained by the closed-form solution of the call price in terms of
an infinite series using the first one hundred terms. $\pi_{\text{Carr-Madan}}$
is obtained by the Carr-Madan formula where the damping-factor is
set to $\alpha=0.1$, we use $N=2^{17}$ points and the truncated
Fourier domain is set to $[0,1200]$. $\pi_{\text{Markov}}$ is obtained
by the COS method with $N=10^{6}$ terms and using the $8^{th}-$moment
and $\varepsilon=10^{-8}$ to obtain the truncation range. $\pi_{\text{Cumulants}}$ is obtained
by the COS method also with $N=10^{6}$ terms and using six cumulants
to obtain the truncation range.
Model M3 is a CGMY model with parameters $C=0.005$ and $G=M=Y=1.5$. Consider
an at-the-money call option on a stock with price $S_{0}=100$ today
and with $0.1$ years left to maturity. Assume the interest rates
are zero. Using the cumulative range with four cumulants, the relative error of the
approximation by the COS method and the reference price is about one
basis point, a significant difference, see Table \ref{tab:.-CGMY},
and does not improve when increasing $N$.
Model M4 is the Heston model with the following parameters: speed of mean-reversion
$\kappa=1$, level of mean-reversion $\eta=0.05$, vol of vol $\theta=2$,
initial vol $v_{0}=0.01$ and correlation $\rho=-0.75$. Consider
an at-the-money call option on a stock with price $S_{0}=100$ today
and with $0.5$ years left to maturity. Assume the interest rates
are zero. Set $\varepsilon=10^{-2}$.
Using the cumulative range based only on the second cumulant, the truncation range is $[-1.33,1.33]$
and the price of the option by the COS method is $1.709$.
On the other hand, $1.738$ is the price of the option based on the Markov
range, which is $[-3.71,3.71]$. We used $N=1000$. Using a larger
$N$ does not change the first three digits of the prices anymore.
We also applied the Carr-Madan formula to confirm the price $1.738$.
\begin{figure}[!h]
\begin{centering}
\begin{tabular}{cc}
\includegraphics[height=6.5cm,width=6cm]{Figure3A}& \includegraphics[height=6.5cm,width=6cm]{Figure3B}\\
Panel A&Panel B
\end{tabular}
\end{centering}
\caption{\label{fig:Jump-diffusion}COS method for MJD model. Panel A: convergence of COS call prices for model M1. The approximation of the call option price by the COS method, where the truncation range is based on four cumulants, is $1.263666$ but the reference price, i.e. the exact price, is $1.263921$. Panel B: MJD density and COS approximations. The density approximation by the COS method using four cumulants for the truncation range does not change no matter how large we choose $N$. We set $\varepsilon=10^{-7}$.}
\end{figure}
\begin{table}[h!]
\begin{center}
\caption{\label{tab:.-CGMY}CGMY model. Parameters $C=0.005$ and $G=M=Y=1.5$.
Choose $\varepsilon=10^{-7}$. The reference price is 1.02168477497...,
which we obtained using an approximating range ten times
larger than the Markov range, i.e. $[-89,89]$, and $N=10^{7}$.
Increasing the truncation range or $N$ does not change the first
$10$ decimal digits of the reference price anymore. We set the
difference between the reference price and the COS approximation to
zero if the first $10$ decimal digits coincide.}
\begin{tabular}{p{1cm}p{2cm}p{2cm}p{2.5cm}p{2.5cm}}
\textbf{Terms \textit{N}} & \textbf{Markov abs. error} & \textbf{Cumul. abs. error} & \textbf{Markov rel. error in BPS} & \textbf{Cumul. rel. error in BPS} \\
\hline
$1000$ & $4.8\times10^{-4}$ & $1.07\times10^{-4}$ & $4.74$ & $1.04$\\
$2000$ & $8.4\times10^{-8}$ & $1.07\times10^{-4}$ & $8.2\times10^{-4}$ & $1.04$\\
$4000$ & $0$ & $1.07\times10^{-4}$ & $0$ & $1.04$\\
$8000$ & $0$ & $1.07\times10^{-4}$ & $0$ & $1.04$\\
\end{tabular}
\end{center}
\end{table}
\subsection{\label{subsec:calibration}Application to model calibration}
Usually, one proceeds as follows to calibrate a stock price model,
like the Heston model, to real market data: given a set of market
prices of put and call options, minimize the mean square error between
market prices of the options and the corresponding prices predicted
by the model. During the optimization phase, model prices need to
be computed very often, e.g. by the COS method.
We assume the model is described by $m$ parameters. Let $0<T_{0}\leq T_{1}$
be the smallest and largest maturity of the put and call options, respectively
let $\Theta\subset\mathbb{R}^{m}$ be the space of feasible parameters
of the model.
Let $X_{T}^{\theta}$ be the centralized log returns for the parameter
$\theta\in\Theta$ and maturity $T\in[T_{0},T_{1}]$. To compute the
price of a put or call option with strike $K$ by the COS method via
Corollary \ref{cor:bounded v}, we need to estimate the $n^{th}-$moment
\[
\mu_{n}^{\theta,T}:=E\left[\left(X_{T}^{\theta}\right)^{n}\right],
\]
to obtain an estimate for the truncation range $[-L,L]$ of the
density of $X_{T}^{\theta}$. The $n^{th}-$moment could directly
be determined by differentiating the characteristic function $X_{T}^{\theta}$
exactly $n-$times using a computer-algebra-system.
In general, evaluating the $n^{th}-$derivative of a characteristic
function each time the COS method is called, might slow down the total
calibration time significantly because the $n^{th}-$derivative of
the characteristic function of some models can be very involved. For
fixed $n\in\mathbb{N}$, we therefore let
\begin{align*}
h:\Theta\times[T_{0},T_{1}] & \to[0,\infty)\\
(\theta,T) & \mapsto\mu_{n}^{\theta,T}.
\end{align*}
In the case of the Heston model, the function $h$ is continuous, see Lemma 1 in \cite{ruckdeschel2013pricing}. We propose to identify a function $\hat{h}$ as an approximation of $h$ upfront \emph{before} the calibration procedure. The evaluation of
$\hat{h}$ is expected to be fast. One might for example obtain $\mu_{n}^{\theta,T}$
for a (large) sample in $\Theta\times[T_{0},T_{1}]$ and defined $\hat{h}$
by a non-linear regression.
Training $\hat{h}$ to the sample takes some
time but need to be done only once. The idea is to use $\hat{h}(\theta,T)$ as an approximation of $\mu_{n}^{\theta,T}$
to obtain the truncation range via Corollary \ref{cor:bounded v}. Even if $\hat{h}$
is only a rough estimate of the $n^{th}-$moment, we expect this approach
to work well because Markov's inequality usually overestimates the
tail sum, which provides us with a certain ``safety margin''.
We illustrate this idea for the Heston model. First, we define $\Theta$
for the Heston model, which has five parameters: the speed of mean
reversion $\kappa$, the mean level of variance $\eta$, the volatility
of volatility $\theta$, the initial volatility $v_{0}$ and the correlation
$\rho$. We assume
\[
\Theta\times[T_{0},T_{1}]=\left(10^{-3},10\right)\times\left(10^{-3},2\right)^{3}\times\left(-1,1\right)\times\left(\frac{1}{12},2\right).
\]
We randomly choose $5\times10^{5}$ values in $\Theta\times[T_{0},T_{1}]$
and compute $\mu_{8}^{\theta,T}$ for each of those values by a Monte Carlo simulation. (The $8^{th}-$derivative of the characteristic function of the Heston model is extremely involved).
Then we train a small random forest, see \cite{breiman2001random},
consisting of $50$ decisions trees. We choose a random forest for
interpolation because the calibration is straightforward. We
are confident that other interpolation methods, e.g. based on neural
networks, produce similar results.
Next, we define a test set and choose $1000$ values in $\Theta\times[T_{0},T_{1}]$
randomly. \cite{junikePerformance} calibrated the Heston model to a time series
of $100$ time points in Summer 2017 of real market data of put and
call options, including calm and more volatile trading days. We add
those $100$ parameters to our test set with a maturity of half a
year.
For each parameter set, we compute a reference price\footnote{We computed the reference price by the COS method
using an approximating range two times larger than the Markov range and $N=10^{6}$ terms. We verified the reference price by the Carr-Madan formula, where the damping-factor is set to $\alpha=0.1$, we use $N=2^{17}$ points and the truncated Fourier domain is set to $[0, 1200]$.} of three
call options with strikes $K\in\{75,100,125\}$. We choose $S_{0}=100$, $r=0$ and $\varepsilon=10^{-4}$.
We obtain very satisfactory results approximating prices of call options by the COS method if the truncation range is obtained via Corollary \ref{cor:bounded v}, where the $8^{th}-$moment
is estimated by the random forest $\hat{h}$. The maximal absolute error for the market data test set over all options is less than $\varepsilon$ for $N\geq500$. The maximal absolute error for the random test set is less than $\varepsilon$ for $N\geq2000$.
Last we comment on the CPU time: computing one option price by the
COS method for $N=1000$ takes about $700$ microseconds using the software R and vectorized code
without parallelization. To evaluate $\hat{h}$ on $10^{4}$ parameter sets
using R's package \emph{randomForest,} also without parallelization,
takes on average $60$ microseconds per set.
\section{\label{Conclusions}Conclusions}
The COS method is used to compute certain integrals appearing in mathematical finance by efficiently retrieving the probability
density function of a random variable describing some log-returns from its characteristic function.
The main idea is to approximate a density with infinite support on a finite range by a cosine series.
We provided a new framework to prove the convergence
of the COS method, which enables us to obtain an explicit formula for
the minimal length of the truncation range, given some maximal
error tolerance between the integral and its approximation by the COS
method.
The formula for the truncation range is based on Markov's inequality and it is assumed that the density of the log-returns has semi-heavy tails. To obtain the truncation range, we need the $n^{th}-$moment of the (centralized) log-returns.
The larger $n$, the sharper Markov's inequality. From numerical experiments, we concluded that $n=8$, $n=6$ or even $n=4$ is a reasonable choice.
The $n^{th}-$moment could directly be determined from the $n^{th}-$derivative of the characteristic function. In the case of the Heston model, the characteristic function is too involved and instead we successfully employed a machine learning approach to estimate the $8^{th}-$moment.
\section*{Acknowledgments}
We thank two anonymous referees for many valuable comments and suggestions enabling us to improve the quality of the paper.
| {
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{"url":"https:\/\/math.stackexchange.com\/questions\/610638\/x-1-in-k-mathbbq-implies-x-is-a-root-of-unity","text":"# $||x||=1$ in $K\/\\mathbb{Q}$ implies $x$ is a root of unity.\n\nLet $K\/\\mathbb{Q}$ a finite (i.e. algebraic and finitely generated) extension. Let $x \\in K$, such that $||x||=1$ for all normalized absolute values of $K$ but at most one. Then $x$ is a root of unity.\n\nBy the product formula I get $||x||=1$ for all normalized absolute values of $K$. Hence $x \\in S^1$, seen as the embedding $K \\subset \\mathbb{C}$.\n\nBut now I am a bit confused. Let $x = \\frac{3}{5} + \\frac{4}{5}i$ shows that the conditions $x$ algebraic and $x \\in S^1$ are not sufficient for $x$ to be a root of unity, hence I will have to use that the $p$-adic absolutes are $1$ as well. However, I feel a bit clueless on how to go on. My intuition would tell me that in the example above something will go wrong for $p_i | (5)$, $p_i$ a prime ideal in $\\mathbb{Q}(x)$, then taking the $p_i$-valuation. I don't really know if this is true though, as I don't know any statements about absolute values in fields like the given $K$, but the existence, definition and product formula. Also, understanding the example still doesn't give me a solution.\n\n\u2022 For intuition, it may help to think of the converse, which is much easier. Note that for any root of unity, and any absolute value (e.g. a $p$-adic one) certainly $|x| = 1$ (because $|x|^n = 1$ and $|x|$ is a positive real number.) So what goes wrong in your example is that the $5$-adic valuation (or better, as you say, the $p_i$-adic valuation) is negative, and this property can't change if we take powers. \u2013\u00a0hunter Dec 17 '13 at 17:35\n\u2022 As you say, we can get rid of the superfluous \"all but one\" using the product formula. To proceed from there is a classic result. The easiest way is in terms of adeles; I am leaving this as a comment since you may not know adeles yet. $K$ sits discretely in $\\mathbb{A}_K$. On the other hand, the product $S^1 \\times \\prod_p \\mathcal{O}_{K, p}$ is compact. Therefore their intersection is finite. Taking units on both sides gives us that the set of elements with this property is a finite subgroup of $K^\\times$ and hence is a group of roots of unity. \u2013\u00a0hunter Dec 17 '13 at 17:48\n\u2022 @YACP: Yes, sorry. Q are the rational numbers. I might better change to the mathbb script. \u2013\u00a0Louis Dec 17 '13 at 22:01\n\nThe archimedean places of $K$ correspond to the usual absolute value on $\\Bbb C$, after first applying the embeddings $K\\hookrightarrow\\Bbb C$. Thus if $|x|=1$ for all archimedean places, we know that all of $x$'s conjugates have (the usual) absolute value $1$. Consider the minimal polynomial of $x$. In my Stewart & Tall, the following lemma is used in proving Dirichlet's unit theorem:\n\nLemma. If $p(t)\\in\\Bbb Z[t]$ is a monic polynomial all of whose roots have absolute value $1$, then all of its roots are roots of unity. The proof proceeds as follows:\n\n\u2022 Say $p(t)=(t-a_1)\\cdots(t-a_k)$ and define $p_\\ell(t)=(t-a_1^\\ell)\\cdots(t-a_k^\\ell)$\n\u2022 As symmetric polynomials in $a_1,\\cdots,a_k$, the coefficients of $p_\\ell(t)$ are integers\n\u2022 $t^j$ coeff. of $p_\\ell$ is bounded: $|e_{k-j}(a_1^\\ell,\\cdots,a_k^\\ell)|=|\\sum\\square|\\le\\sum|\\square|=\\sum1=\\binom{k}{j}$ (Vieta's)\n\u2022 Only finitely many $f(t)\\in\\Bbb Z[t]$ satisfying such bounds, so $p_1,p_2,p_3,\\cdots$ has a repeat\n\u2022 Say $p_\\ell(t)=p_m(t)$. So $\\exists\\pi\\in S_k$ such that $a_j^\\ell=a_{\\pi(j)}^m$ (for each $j=1,\\cdots,k$)\n\u2022 Thus $a_j^{\\ell^2}=(a_{\\pi(j)}^m)^\\ell=(a_{\\pi(j)}^\\ell)^m=a_{\\pi^2(j)}^{m^2}\\Rightarrow\\cdots\\Rightarrow a_j^{\\ell^{\\large k!}}=a_j^{m^{\\large k!}}\\Rightarrow a_j^{\\ell^{\\large k!}-m^{\\large k!}}=1$ for each $j$\n\nI am not sure if this is desirable for a homework exercise but it's a fun proof nonetheless.\n\nHere's what I finally did (no guarantee for correctness):\n\nFirst note that as the localizations $R_\\mathfrak{p}$ are given by $\\{y \\in K\\mid ||y||_\\mathfrak{p} \\leq 1\\}$, $x$ is contained in all the $R_\\mathfrak{p}$ hence also in $R$ (for $R$ being the integral closure of $\\mathbb{Z}$ in $K$).\n\nLet $r_1$ be the number of real embeddings of $K$, let $2r_2$ be the number of its complex embeddings into $\\mathbb{C}$.\n\nLet $\\mathfrak{L}: K^\\ast \\rightarrow \\mathbb{R}^{r_1+r_2}, y \\mapsto (\\log(|\\sigma_1(y)|),...,\\log(|\\sigma_{r_1+r_2}(y)|)$\n\nThen we had seen as a Lemma for Dirichlet's unit theorem:\n\nLet $B \\subset \\mathbb{R}^{r_1+r_2}$ compact. Then $\\mathfrak{L}^{-1}(B) \\cap R$ is finite.\n\nNow in our case: take $B = 0 \\in \\mathbb{R}^{r_1+r_2}$.\n\nAs for all $n \\in \\mathbb{N}$: $x^n \\in \\mathfrak{L}^{-1}(B) \\cap R$, we are done.","date":"2019-06-17 03:45:31","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\": 0.9536889791488647, \"perplexity\": 116.12146057655909}, \"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-2019-26\/segments\/1560627998369.29\/warc\/CC-MAIN-20190617022938-20190617044938-00275.warc.gz\"}"} | null | null |
Who Wants to Live in Detroit?
Four decades after 'white flight' began, tens of thousands of African-Americans have now left as well. What does that mean for the city's future?
By Alexia Fernández Campbell and National Journal
Even as Detroit emerges from bankruptcy, the lights are going out in neighborhoods hit by population loss. (National Journal)
DETROIT—In most cities, the opening of a new Starbucks or chain grocery store is no big deal. But this is Detroit.
"No [chain businesses] wanted to move here. It's all about perception. They thought it was too dangerous," says Jordan Twardy, executive director of the Eight Mile Boulevard Association.
Part of Twardy's job involves attracting businesses to the Eight Mile corridor, the notorious dividing line between black and white Detroit made famous in the rap lyrics of Eminem. South of the street, 82 percent of the city's residents are African-American and 41 percent live in poverty. Supermarkets and strip malls are rare.
As Twardy drives along along Eight Mile in his Chevy Cruze, he points to the Gateway Marketplace on the south side of the street, where a Starbucks opened last year and where the city's second national chain grocery store, Meijer, opened a year earlier. Developers are starting to take a chance on neighborhoods they once avoided, he says, and that's a good sign for Detroit.
"We're not hanging banners just yet," says Twardy, who is white and lives in a suburb north of Eight Mile. "Prejudice is still there, and there's still skepticism about how to solve these problems that have been brushed under the rug for the decades."
Detroit represents the starkest example of racial and socioeconomic disparity in the United States, topping the list of most segregated metro areas in the country. Not far from the new Starbucks on Eight Mile is a concrete wall that once separated black and white neighborhoods. But these divisions are starting to blur as more immigrants and predominantly white hipsters move into the city's blighted neighborhoods.
Billions of dollars in corporate investment and federal aid have flowed into Detroit as it recovers from the largest municipal bankruptcy in U.S. history. And even though the population keeps shrinking, Detroiters are eager to point out signs of the coming renaissance. In his State of the City address earlier this month, Mayor Mike Duggan touted the sale of 168 abandoned houses at a public auction and the creation of 200 neighborhood block clubs in the past year.
"We're on our way back." Duggan said to the invitation-only crowd. "Detroit is now on the road to recovery."
It definitely seems that way at first glance. Developers have rescued and restored art deco skyscrapers and other relics from Detroit's golden age. Abandoned warehouses have been converted into lofts. Even the crumbling Packard Plant, a symbol of Detroit's urban ruin, is slated for development.
Now Detroit just needs more people to move in. All the development doesn't change the fact that Detroit's population continues to shrink each year. "For lease" signs hang from most new developments. And the city can't seem to demolish collapsing houses fast enough.
It's unclear what the new Detroit will look like, and how much of it will include the city's low-income, African-American families. Talking about the role of gentrification and race makes Detroiters uncomfortable. For the first time in decades, the percentage of white and Hispanic residents in Detroit has been increasing, from 17 percent in 2010 to 21 percent in 2013, according to census data.
That shift comes after decades of "white flight" to the suburbs, the kind of urban unraveling that crippled Detroit and other Rust Belt cities. Detroit's auto boom and factory work brought thousands of Southern blacks to the city starting in the 1920s as part of the Great Migration. Residents of the predominantly white city resisted integration, and tensions sparked race riots in the 1960s that triggered an exodus of white residents, jobs, and ultimately most of the retail and basic amenities that make up a city.
Detroit became known as a hub of African-American culture in the 1960s and 70s, launching the highly successful Motown Records and the careers of Soul and R&B legends. By 1980, most of the city was African-American. But factory work in the region dried up as companies moved manufacturing overseas. The city's crumbling school system led to a second wave of flight as black families sought better education for their children in the suburbs. The Great Recession pushed Detroit over the edge and thousands of homes into foreclosure.
Now the Tudor houses are burned and boarded up in neighborhoods like the North End, which was once home to Motown legends Diana Ross and Smokey Robinson. About one-quarter of Detroit's buildings are empty or abandoned, and thousands are slated for demolition in the coming months, according to Data Driven Detroit.
Detroit, like other Rust Belt cities, has started to focus on immigration as a solution to repopulating these neighborhoods. Although Mexican immigrants started settling in Southwest Detroit decades ago, a few other neighborhoods have recently seen an influx of foreigners, says Steve Tobocman, director of Global Detroit, a public-private initiative launched in 2012 to help immigrant communities grow and prosper in the region.
"We want to help them build the American Dream as rapidly as possible, picking up property clusters and opening businesses," says Tobocman.
Working-class families from Bangladesh, Iraq, and Yemen have moved into Detroit's blighted neighborhoods as others flee. In the past two years, Global Detroit has launched programs to help immigrants start their own businesses and integrate with other cultures. This month, Global Detroit will host its first homeownership workshop in Spanish to walk immigrants through the process of buying a house through Detroit's Land Bank, which auctions properties seized by the city and county for neglect or unpaid taxes.
In the past, it's been hard to get home loans for low-income immigrants who don't use banks and have no credit history, says Raquel Garcia Andersen, Global Detroit's director of partnerships and community outreach. They are part of Detroit's "shadow economy," she says.
"We have a lot of banks that have been telling us they want to work with newer populations, so they might look at credit in a different way," says Anderson.
Southwest Detroit's Mexicantown is seen as an example of how immigrants can boost declining neighborhoods. Taquerias, coffee shops, and small grocery stores have made Michigan Avenue in Southwest one of the city's busiest commercial corridors.
Residents here recently elected Detroit's first Hispanic city council member. Detroit native Raquel Castañeda-López worked with Welcoming Michigan, a program focused on making immigrants feel at home, to establish Detroit as a "Welcoming City" in 2014.
Since it launched in 2012, Welcoming Michigan has hosted neighborhood dialogues between African-Americans, Latino, and Arab youth. It has even hosted a hijab fashion show and Soul Food night. Getting longtime residents, who are mostly African-American, to meet their new neighbors makes all the difference, says Christine Sauvé, a community coordinator for Welcoming Michigan in Detroit.
"People get anxious about how their community is changing and what does that mean for me?" says Christine Sauvé, of coordinator for Welcoming Michigan in. "We're trying to help people build that human connection."
This article originally misstated Councilwoman Castañeda-López's involvement in the launch of Welcoming Michigan.
Libby Isenstein and Janie Boschma contributed to this article | {
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{"url":"https:\/\/api-project-1022638073839.appspot.com\/questions\/what-are-the-mean-and-standard-deviation-of-2-3-3-5-1-5-4-4-2-6","text":"# What are the mean and standard deviation of {2,3,3,5,1,5,4,4,2,6}?\n\nFeb 17, 2016\n\nMean is $3.5$ and Standard Deviation is $1.83$\n\n#### Explanation:\n\nSum of the terms is $35$, hence mean of $\\left\\{2 , 3 , 3 , 5 , 1 , 5 , 4 , 4 , 2 , 6\\right\\}$ is $\\frac{35}{10} = 3.5$ as it simple average of the terms.\n\nFor Standard Deviation, one has to find average of squares the deviations of the terms from mean and then taking their square root.\n\nThe deviations are $\\left\\{- 3.5 , - 0.5 , - 0.5 , 1.5 , - 2.5 , 1.5 , 0.5 , 0.5 , - 1.5 , 2.5\\right\\}$\nand sum of their squares is\n\n$\\frac{12.25 + 0.25 + 0.25 + 2.25 + 6.25 + 2.25 + 0.25 + 0.25 + 2.25 + 6.25}{10}$ or $\\frac{33.50}{10}$ i.e. $3.35$.\n\nHence Standard Deviation is $\\sqrt{3.35}$ i.e. $1.83$","date":"2021-12-07 09:14:05","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 11, \"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\": 0.972407341003418, \"perplexity\": 417.80293264925547}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-49\/segments\/1637964363337.27\/warc\/CC-MAIN-20211207075308-20211207105308-00322.warc.gz\"}"} | null | null |
Q: AbstractAccountAuthenticator allow only 1 account I have implemented AbstractAccountAuthenticator as a requirement to use SyncAdapter hower my application is supporting only 1 account at a time.
When the user is trying to add another account via settings - Settings crash with an error that it stopped working.
I have seen some application e.g. LinkedIn, Facebook they somehow handle it differently a toast message is shown to the user with a statement that only 1 account is supported. How can I achieve this functionality?
This is my authenticator
class ApplicationAuthenticator(private val context: Context) : AbstractAccountAuthenticator(context) {
// Editing properties is not supported
@Throws(UnsupportedOperationException::class)
override fun editProperties(response: AccountAuthenticatorResponse,
accountType: String): Bundle? {
throw UnsupportedOperationException()
}
// Don't add additional accounts
override fun addAccount(response: AccountAuthenticatorResponse, accountType: String,
authTokenType: String, features: Array<String>,
options: Bundle): Bundle? {
return bundleOf(AccountManager.KEY_INTENT to null)
}
// Ignore attempts to confirm credentials
@Throws(NetworkErrorException::class)
override fun confirmCredentials(response: AccountAuthenticatorResponse, account: Account,
options: Bundle): Bundle? {
return null
}
// Getting an authentication token is not supported
@Throws(NetworkErrorException::class, UnsupportedOperationException::class)
override fun getAuthToken(response: AccountAuthenticatorResponse, account: Account,
authTokenType: String, loginOptions: Bundle): Bundle? {
throw UnsupportedOperationException()
}
// Getting a label for the auth token is not supported
override fun getAuthTokenLabel(authTokenType: String): String {
return context.resources.getString(R.string.application_name)
}
// Updating user credentials is not supported
override fun updateCredentials(response: AccountAuthenticatorResponse, account: Account,
authTokenType: String, loginOptions: Bundle): Bundle? {
return null
}
// Checking features for the account is not supported
@Throws(NetworkErrorException::class)
override fun hasFeatures(response: AccountAuthenticatorResponse, account: Account,
features: Array<String>): Bundle {
return bundleOf(KEY_BOOLEAN_RESULT to false)
}
}
A: When the user clicks the "Add Account" button, Android just calls the addAccount method of your ApplicationAuthenticator. In return it expects either the account to be created, an Intent which launches the account setup or an error.
If you don't allow multiple accounts you have multiple options here:
*
*Return an error with code ERROR_CODE_UNSUPPORTED_OPERATION. Although I've not tried that yet.
*Return your existing account as the result. At this point you can also show a Toast.
To Return the existing account just let addAccount return a Bundle with the following keys and their respective values:
AccountManager.KEY_ACCOUNT_NAME and AccountManager.KEY_ACCOUNT_TYPE of the account that was added, or
*Return an Activity Intent which doesn't create an account but explains to the user that this is an unsupported/unnecessary operation. It's not a requirement that the Activity actually adds an account.
This gives the best user experience IMO.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 7,748 |
What were Caitlan Coleman and Joshua Boyle really doing in Afghanistan?
15 Oct, 2017 04:00 PM 4 minutes to read
Joshua Boyle and son Jonah play in the garden at his parents house. Photo / AP
By Amanda Erickson and Antonio Olivo
In the weeks, months and years after American Caitlan Coleman and Canadian Joshua Boyle went missing in Afghanistan, their families repeated the same story: They were young adventurers, drawn off the beaten track.
"They were interested in cultures that are under-developed," Caitlan's mother Lyn said in 2014. They didn't do things like stay in hotels or visit tourist traps. They were idealists, and also a little naive.
Soon after the pair married in 2011, they spent four months in Guatemala. And in 2012, they jetted off for Russia, Kazakhstan, Tajikistan and Kyrgyzstan. Family members called it a backpacking trek. Afghanistan was not a part of the plan, at least not as far as anyone knew.
But Coleman and Boyle did make their way to a remote area of Afghanistan outside Kabul, where they were kidnapped by the Taleban and later held by the Haqqani network before being rescued last week.
Their captors killed Coleman's infant daughter and allowed Coleman to be raped by a guard, her husband said. The couple and three of their children were rescued in Pakistan, where their captors had taken them from Afghanistan. The operation by the Pakistani military was tipped off by US intelligence. The family arrived in Toronto at the weekend after the five-year ordeal.
Why did Boyle and Coleman, seven months pregnant, decide to go to Kabul? What were they trying to accomplish?
Boyle said he and Coleman went to Afghanistan to try to help "the most neglected minority group in the world, those ordinary villagers who live deep inside Taleban-controlled Afghanistan... where no NGO, no aid worker and no government has ever successfully been able to bring the necessary help." Boyle described himself as a "pilgrim". It's not clear how he and Coleman intended to help, or what they were up to when they were kidnapped.
Coleman's friend suggested to USA Today that she and others had at least a vague notion that the couple intended to do some volunteer work. Sarah Flood said she related to Coleman's travel plans because she had just come back from a service trip to Ukraine. "The idea of going to a country and being helpful is something we absolutely shared," Flood told USA Today. She also said that the trip had been Boyle's idea, but Coleman quickly got excited about it.
The family in a Taleban video. Photo / AP
And then there's the insight of Richard Cronin, who met Coleman and Boyle while they were in Central Asia. The pair befriended Cronin at a hostel in Bishkek. In a blog post from 2012, Cronin wrote that Boyle's excitement about Afghanistan convinced him to go. "I hadn't thought seriously about travelling to Afghanistan until I started talking to Josh," he wrote. "We started talking about Lawrence of Arabia and the explorer Richard Burton. He asked me if I admired these explorers. Of course I did. 'Wouldn't you like to be like one of them?'.He had also said it was safe provided you didn't go to a region where there were foreign troops and the Taleban, namely the south."
After the 9/11 attacks, Boyle became consumed by questions of terrorism and Islam, studying up on the issue and even learning Arabic. A few years later, he got involved in an effort to get Omar Khadr, once the youngest detainee at Guantanamo Bay, released. Khadr pleaded guilty to killing a US Special Forces medic. Boyle briefly married Khadr's sister.
Family held captive by Taliban released after 5 years
Mysterious story of couple rescued from Afghanistan
Man freed after five years says child killed, wife raped
Freed Taliban hostage says children are 'improving'
Boyle's associations with the family led some US intelligence officials to speculate that the visit to Afghanistan may have been part of a larger effort to link up with Taleban-affiliated militants. "I can't say that [he was ever al-Qaeda]," said one former US intelligence official. "He was never a fighter on the battlefield. But my belief is that he clearly was interested in getting into it." Authorities denied that Boyle had any ties to terror.
His "first concern in life has always been helping others," Alex Edwards, a friend of Boyle's since 2002, told Philadelphia. "If things were different, and I was the one being held hostage, Josh wouldn't rest until I was free. He'd stage sit-ins. He'd put up posters. He'd dedicate his life to it. That's just who he is."
Latest from World
Tongan port being checked for arrival of supplies
Ten new cases of Covid-19 recorded in Samoa | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 872 |
using System.Net;
using CassiniDev;
using NUnit.Framework;
namespace Salient.ReliableHttpClient.Tests
{
/// <summary>
/// </summary>
[TestFixture]
public class IntegrationTestFixture : CassiniDevServer
{
[TestFixtureSetUp]
public void Setup()
{
string location = new ContentLocator(@"Salient.ReliableHttpClient.TestWeb").LocateContent();
StartServer(location);
}
[Test]
public void TestServer()
{
var client = new WebClient();
var response = client.DownloadString(NormalizeUrl("Handler1.ashx"));
Assert.AreEqual("Hello World",response);
}
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 7,429 |
Suns still on the hunt for a long-term answer at point guard
Justin Benton
The Phoenix Suns weren't entirely inactive around the trade deadline, but despite adding guard depth with the acquisition of Tyler Johnson, the team still isn't where they want to be. While Johnson adds a little beef to their rotation, Phoenix GM James Jones has made it clear that they still feel in need of a long-term solution at the point guard spot to pair with Devin Booker and DeAndre Ayton. Duane Rankin, who covers the Suns for AZCentral.com, delivered this insight from the Phoenix front office:
"We're going to look for a point guard," James Jones when asked if Tyler Johnson "long-term answer"
Jones' heard cry for #Suns to get a point guard, but also said there are traditional and non-traditional point guards.
'LeBron James, Luka Doncic, they're point guards."
— Duane Rankin (@DuaneRankin) February 8, 2019
The last part of that tweet is very telling of the Suns' decision-making process over the next few months. Headed for the top of the draft lottery, Phoenix could find themselves taking a player like Duke's R.J. Barrett, who has shown ability to be the initiator of an offense despite his size, over someone like Ja Morant, who is a more traditional point guard.
There is the possibility that they choose to opt for a more experienced player in favor of a rookie, though. They will definitely be taking swings at big names in the free agent market come July. Players like Kemba Walker or D'Angelo Russell can probably expect heavy interest from the Suns, as well as less flashy options, like Patrick Beverly, Emmanuel Mudiay, and T.J. McConnell. A player like Tyreke Evans would fit the bill of a nontraditional point somewhat similar to the types mentioned by Jones if the team seeks to go that route.
In the meantime, there will be plenty of free agents on the buyout market that Phoenix could explore and experiment with.
JUST IN: LeBron James has mastered the art of tampering after latest NBA All-Star Draft
Related TopicsJames JonesPhoenix SunsTyler Johnson | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,360 |
A great home in a hot location that is minutes to downtown! This home has an updated kitchen and bathroom with a half bath added, a large front porch, large rear deck. Come check it out today! Dogs considered, no cats.
Lovely rental minutes from Vandy. Two spacious master suites, granite countertops, and stainless steel appliances. 1-car garage. Dogs considered. No cats. | {
"redpajama_set_name": "RedPajamaC4"
} | 9,944 |
=====
Usage
=====
To use Hexes in a project:
.. include:: ../scripts/hexes_example
:code: python
---------------------
Musings on the future
---------------------
What sorts of widgets are important in a terminal app?
* Text areas
* Scrollable text areas
* Auto-scrolling text areas (as for chat or Twitter feed)
* Text input areas
These widgets should be relatively smart, knowing their own dimensions, when to
resize, how to listen to things (some sort of data-binding model here?), how to
style themselves, etc.
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,623 |
When traveling in Europe, tourists are advised to hide their money and credit cards, wear a strap-on-under-shirt wallet pouch, or otherwise do uncomfortable things with their cash. This doesn't ruin the tourist experience, but it does make you question every person who walks by you or stands next to you on the train.
So I suggest that instead, male tourists carry a wallet with a mouse trap-like device inside. When closed (and in your back pocket), it looks like a regular billfold. During the unfolding process, a metal bar is pulled across the inside and latched into place. When fully open, this is just like a mousetrap. Touching the inner part of the wallet will release the spring-loaded bar, clamping it down upon the pickpocket with painful consequences.
Hopefully, enough people will carry these that pickpockets begin to think twice about "liberating" your personal cash, because it might just break their fingers (again).
For added realism, the wallet would come with two dollars, a picture of someone pretty, and two fake credit cards already installed. Attempting to remove them would also trigger the snap-action.
It would come in several styles and colors.
My girlfriend is in Prague right now, and her father was just picked on a crowded train...this serves as the inspiration for the idea.
I'm hesitant to bun this, because I know five seconds after placing it in my pocket I'd absent-mindedly confuse it for the genuine article. Maybe anxiety isn't enough to ruin a vacation, but a series of welts across your hand sure is.
Yeah, it would probably take some practice. But then, many of us learned to drive stickshifts after learning on automatic transmissions, so I'm sure this wouldn't be TOO hard.
//When traveling in Europe// Do you not have pickpockets in the land of 'stickshifters*', then? Us non USians could resemble that remark!.
Not a bad idea though, [shaps], it's a shame that you wouldn't be there for the final denoument though!
Once pickpockets start realizing that people are carrying these, they won't stop taking wallets, they'll just be more carefull opening them.
Actually, that might be kinda funny too...seeing a few people, all seated around a fountain, each of them slowly opening and peering into something soft and leatherlike, with worried expressions on their faces. I'd pay just for that.
[gnomethang]: Yeah, I'm sure we have them, but I never watch travel shows about cities here in the US. I'd much rather learn more about Swiss wines than the glories of Independence Hall.
Swiss wines?? Now i *AM* confused.
The swiss have much better wines than the French - they just export far less. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,337 |
Jaunt is Ready For Friday 13th Spooks With Release of Escape the Living Dead
By Zeena Al-Obaidi Last updated May 13, 2016
Friday 13th is a date that stirs many reactions amongst people: fear, dread, excitement, and even apathy. One thing that cannot be denied is the guilty pleasure of settling down at night to watch a horror film on this early Halloween-esque day. Jaunt, creators of cinematic virtual reality (VR) videos, have hijacked today by giving a makeover to its Jaunt app home as well as releasing its own zombie flick.
The latest piece of content to come out today is Escape the Living Dead, something that the studio described to have a "70s zombie vibe". The retro zombie title has a moving camera throughout the film, simulating the view point of a bitten survivor as a group of heavily-armed neighbours try to escape the undead hoards.
There was a mix of long hours of hard work to create this along with trial and error which ended up as a success as the video was reportedly shot over two days out in the desert near Palm Springs in over 100 degrees farenheit – or the high-end of 37 degrees in celsius – with a stabilised air bladder camera rig used to mount the Jaunt GP16 to the bed of the truck and zombies in hair and makeup for hours before the shoot began. After those two days it took "many months" to add all the visual effects to the piece to in turn transform the location it was shot in.
As well as the film there is also a scary section of the Oculus Store and in the Samsung Gear VR store as well as the Jaunt app having its own horror-makeover.
The experience is currently available if you fancy having a VR horror night in with it available on all devices including iOS, Android, GEar V, HTC Vive, Oculus Rift, and if you don't have a VR head-mounted display (HMD) then it is also available as a 360-degree film on desktop.
Stay with VRFocus for the latest in VR.
Escape the Living DeadJaunt
Zeena Al-Obaidi 923 posts 0 comments
Zeena is a former VRFocus presenter and staff writer. She has always played videogames from the word go, and can't wait to see how the world of VR unfolds. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 1,606 |
Lifestylenewsonline.com
Home » Beauty » The Before-and-Afters for This Acne Scar Removal Cream Are Unbelievably Good
The Before-and-Afters for This Acne Scar Removal Cream Are Unbelievably Good
The whole process of acne is exhausting. There's the emotional toll of constantly worrying about your skin acting up, the frustrating run-around of finding the best acne products for your skin type, and then the damage that a nasty pimple leaves in its wake — red or purple blotches and pockmarks that might only be a few centimeters wide (if you were a saint and didn't pick at them), but that take months to finally vacate your face.
Only discovering miracle pimple products, working with an excellent dermatologist, or being born with genes that love you can get ahead of zits before they happen, and even then nothing's guaranteed. But there are a few things you can use to smooth out the aftermath, and thanks to the incredibly-named skincare brand Cutie Academy, you can get one super-effective scar remedy for less than $15.
The brand's Scar Removal Cream currently has 4,588 five-star ratings, with 2,987 reviews praising its stunning effectiveness. Of those, 535 reviews discuss how well the scar removal cream works for acne marks. Shoppers who say they've suffered from acne for six years claim that in a little over a month, the cream lightened their substantial scarring and hyperpigmentation, with accompanying before-and-after photos that show the transformation from vivid red marks to calm, clear skin.
If you feel like you've tried everything under the sun for your scars, you're not alone. One shopper in a similar situation wrote that she'd tried everything to help with her adult onset acne, from over-the-counter products and prescriptions like spironalatone and retinol to "old wives tale remedies — you name it and I have tried it." While the prescriptions helped, she said nothing fully cleared up her acne until she discovered the Cutie Academy cream.
"After using this product for three weeks, my skin is the clearest and healthiest it has ever been," she wrote. "I have always been one of those women who feels the need to put on concealer and foundation to even leave the house. While I'm not quite ready to let go of my old habits, I could."
Shop now: $14 (Originally 28); amazon.com
More shoppers back her up, saying that on top of smoothing out "pitted problem areas" and scars — even severe "crater face" situations — they find the cream also prevents new breakouts from surfacing in the first place. After a month of twice-daily use, reviewers say that their scars, once "small hills" unaffected by "everything from microdermabrasion to skin peeling," are almost completely gone.
Judging from the cornucopia of reviews that say as much, a month is the sweet spot for seeing your acne scars pack up and leave. Though some people say they've seen results in as little as a week: One exceptionally impressed shopper expanded on his experience, saying that despite doubting that an inexpensive cream could treat 10-year-old acne scars "better than the hundreds I have spent on treatments recommended by dermatologists," after a week of using it twice a day, he's seen "extraordinary" results. "I haven't had this amount of confidence in years," he writes. "I keep getting compliments from people that haven't seen me prior to using this gel, so I definitely know it's not just me thinking it's working."
The Scar Removal Cream is currently on sale for 51 percent off, bringing it from $28 to $14. And if the thought of acne's memory disappearing didn't convince you, the 1,575 five-star reviews discussing more general scars will. If you don't mind a little gore, the before-and-afters people post of their once-sliced skin are mind-blowing. I broke my ankle last summer, and have made my peace with the Frankenstein-esque lines on either side of the break. But after seeing the power of this $14 gel, I might just get rid of it for good.
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Copyright © 2021 Lifestylenewsonline.com. All rights reserved. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,510 |
package me.vilsol.nmswrapper.wraps.unparsed;
import me.vilsol.nmswrapper.NMSWrapper;
import me.vilsol.nmswrapper.reflections.ReflectiveClass;
import me.vilsol.nmswrapper.reflections.ReflectiveMethod;
import me.vilsol.nmswrapper.wraps.NMSItemStack;
import java.util.Random;
@ReflectiveClass(name = "EnchantmentDurability")
public class NMSEnchantmentDurability extends NMSEnchantment {
public NMSEnchantmentDurability(Object nmsObject){
super(nmsObject);
}
/**
* TODO Find correct name
* @see net.minecraft.server.v1_9_R1.EnchantmentDurability#a(net.minecraft.server.v1_9_R1.ItemStack, int, java.util.Random)
*/
@ReflectiveMethod(name = "a", types = {NMSItemStack.class, int.class, Random.class})
public boolean a(NMSItemStack itemStack, int i, Random random){
return (boolean) NMSWrapper.getInstance().exec(nmsObject, itemStack, i, random);
}
/**
* TODO Find correct name
* @see net.minecraft.server.v1_9_R1.EnchantmentDurability#b(int)
*/
@ReflectiveMethod(name = "b", types = {int.class})
public int b(int i){
return (int) NMSWrapper.getInstance().exec(nmsObject, i);
}
/**
* @see net.minecraft.server.v1_9_R1.EnchantmentDurability#canEnchant(net.minecraft.server.v1_9_R1.ItemStack)
*/
@ReflectiveMethod(name = "canEnchant", types = {NMSItemStack.class})
public boolean canEnchant(NMSItemStack itemStack){
return (boolean) NMSWrapper.getInstance().exec(nmsObject, itemStack);
}
/**
* @see net.minecraft.server.v1_9_R1.EnchantmentDurability#getMaxLevel()
*/
@ReflectiveMethod(name = "getMaxLevel", types = {})
public int getMaxLevel(){
return (int) NMSWrapper.getInstance().exec(nmsObject);
}
} | {
"redpajama_set_name": "RedPajamaGithub"
} | 6,475 |
Jamie Travis (nascut el 13 d'agost de 1979) és un cineasta amb seu a Toronto que ha escrit i dirigit curtmetratges, vídeos musicals i anuncis de televisió premiats. Va rebre el reconeixement internacional per les seves dues trilogies de curtmetratges, The Patterns i The Saddest Children in the World.
Els seus sis curtmetratges es van estrenar al Toronto International Film Festival, i la seva obra ha tingut nombroses projeccions retrospectives en festivals i galeries d'art.
Vida i carrera
El curtmetratge de graduació de Travis, Why the Anderson Children Didn't Come to Dinner (2003), és un retrat surrealista de tres germans joves obligats a endurir els estranys abusos culinaris de la seva mare. La pel·lícula va guanyar nombrosos premis, com ara el millor disseny de producció als premis Leo, el millor guió als premis Golden Sheaf i la millor pel·lícula canadenca al Prends ça court de Montreal! Cicle de cinema.
Amb Patterns (2005) un llançament lúdic d'avantguarda del gènere de suspens en què una dona espera ansiosament una trucada telefònica, Travis va ser guardonat amb el premi al Festival Internacional de Cinema de Vancouver al millor director d'un curtmetratge canadenc occidental. A Patterns li van seguir dues seqüeles, Patterns 2 (2006) i Patterns 3 (2006), que van transformar el formalisme auster de la primera entrega en un joc complet de nois i noies. amb cant, dansa i interludis documentals. Altres moments destacats del festival inclouen el Festival de Cinema Gai i Lèsbic de Londres del BFI i el Festival Internacional de Cinema de Hamptons, on Patterns 3 es va emportar el premi al millor curtmetratge.
Una comèdia fosca sobre un nen de nou anys que planeja tancar la seva festa d'aniversari amb un suïcidi, The Saddest Boy in the World (2006) va plantar fermament Travis al mapa internacional. La pel·lícula va rebre una premsa favorable, amb més de 150 projeccions en festivals i múltiples premis, com el millor curt canadenc al Festival Internacional de Cinema de Calgary, el millor curtmetratge al Festival de Cinema de Victoria a la Colúmbia Britànica i Favorit del públic al Festival Internacional de Curtmetratges NextT de Bucarest.
Travis va concloure la seva trilogia Saddest Children in the World amb The Armoire (2009), en què un joc d'amagar surt malament. En la seva estrena, la pel·lícula va rebre una menció honorífica pel Millor curtmetratge canadenc al Festival Internacional de Cinema de Toronto de 2009, i cobejat a la llista del Canada's Top Ten de final d'any del TIFF. També ha guanyat el millor curtmetratge d'acció en directe al Festival de Cinema de Nashville de 2010 i el millor curtmetratge a la 53a Festival Internacional de Cinema de San Francisco.
Travis també ha dirigit vídeos musicals per a reconeguts artistes indie canadencs Tegan i Sara i ha creat anuncis de televisió per a marques i organitzacions destacades. Actualment està desenvolupant el seu debut al llargmetratge.
Travis va viure a Vancouver durant més de 25 anys. És jueu per part del seu pare. S'ha declarat gai.
Filmografia
Why the Anderson Children Didn't Come to Dinner (2003)
Patterns (2005)
Patterns 2 (2006)
Patterns 3 (2006)
The Saddest Boy in the World (2006)
The Armoire (2009)
Kouchibouguac (2011)
For a Good Time, Call... (2012)
Finding Carter (2014)
Faking It (2014-2016)
Scream: The TV Series (2015-2016)
Star (2018)
Angry Angel (2017)
The Bold Type (2017-2019)
Claws (2017-2019)
Charmed (2018)
Dare Me (2020)
Panic (2021)
Yellowjackets (2021)
Vídeos musicals
"Back in Your Head" de Tegan and Sara (2007)
"Hell" de Tegan and Sara (2010)
Referències
Enllaços externs
Directors de cinema canadencs
Artistes de Vancouver | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,859 |
\section{Introduction}
The trace of Sobolev spaces on ${\mathbb R}^n$ to linear subspaces have been studied
in various directions as generalizations of the Sobolev imbedding theorem.
There has also been extensive study how to extend Sobolev, Besov and Lipschitz spaces
from subdomains of ${\mathbb R}^n$ to the whole spaces (see for example, \cite{adh,ste} and
the references therein).
Since 80's, there are generalizations of these problems
for Besov-type spaces on more complicated spaces, namely on the so-called Alfors
$d$-regular sets (\cite{jw,tr}).
On the other hand, recent developments of analysis on fractals give new lights
to these problems. On many fractals such as Sierpinski gaskets and Sierpinski carpets,
diffusion processes and the ``Laplace'' operators are constructed.
It turns out that the domains of
the corresponding Dirichlet forms are Besov-Lipschitz spaces.
\begin{figure}[ht]
\centerline{\epsfig{file=scpent.eps, height=2in}}
\caption{The Sierpinski carpet and the Pentakun}
\end{figure}
In this paper, we consider the following natural question: given a Besov-type space
on a self-similar fractal $K$, what is the trace of the space to a self-similar
subspace $L$? We would indicate two examples in Figure 1. The left figure is
when $K$ is the so-called 2-dimensional Sierpinski carpet (see Section~5~3) for
the definition) and $L$ is the line on the bottom (drawn by the thick line). The right
figure is when $K$ is the Pentakun (a self-similar fractal determined by five contraction
maps; see Section~5~2) for the definition) and $L$ is a Koch-like curve (drawn by the thick
curve). In each case, the domain of the Dirichlet form on $K$ is the Besov-Lipschitz
space, but one cannot obtain the trace using the general theory given by Jonsson-Wallin
(\cite{jw}) and Triebel (\cite{tr}).
This problem was quite recently solved by Jonsson (\cite{jo}) for one typical
case, i.e. when $K$ is the 2-dimensional Sierpinski gasket and $L$ is the bottom line. But his methods
rely strongly on the structure of the Sierpinski gasket and its Dirichlet form, and
they cannot be applied to the so-called infinitely ramified fractals such as Sierpinski
carpets. Instead, we use the self-similarity of the form and some kind of uniform property
of harmonic functions which can be guaranteed by the Harnack inequalities.
Our methods can be applied to the Sierpinski carpets (even to the high dimensional ones) and
we can state the trace theorem under some abstract framework. In fact, we would need
various assumptions for $K$ and for the Dirichlet form on $K$, which are stated in Section~2.
Unless these conditions are satisfied, there may be various possibility of the trace, because of the
\lq\lq complexity\rq\rq\
of the space (see Section~5~4) for an example).
In order to prove our trace theorem, we give a discrete
approximation of our Besov-Lipschitz space in Section~3.1. This approximation result is also new
and is regarded as a generalization of
the main result in \cite{kam}.
The restriction theorem is given in Section~3.2; the key estimate
(Proposition \ref{theo:keyhin}) is based on the idea used by one of the author in \cite{hin}.
The extension theorem is given in Section~3.3, where the classical
construction
of the Whitney decomposition and the extension map is
modified and
generalized to this framework.
\begin{figure}[ht]
\centerline{\epsfig{file=34sc.eps, height=2in}}
\caption{An example of fractal fields}
\end{figure}
Such a trace theorem has an important application to the penetrating process, which is discussed in
Section~6. Let us indicate one concrete example. Given two types of Sierpinski carpets as in Figure 2
(the left carpet is determined by contraction maps with the contraction rate $1/3$ and there is one hole
in the middle, while the right carpet is determined by contraction maps with the contraction rate $1/4$
and there is one bigger hole in the middle). On each carpet, one can construct a self-similar
diffusion; the question is whether one can construct a diffusion which
behaves as the appropriate fractal diffusions within each carpet and which penetrates each fractal.
In order to construct such a diffusion by the superposition of Dirichlet forms on each carpet,
the key problem is whether there is enough functions whose restriction to each carpet is in the domain
of each Dirichlet form. To answer this question, it is crucial to get
the information of the trace of the Dirichlet form on
each carpet to the line, which is the intersection of the two carpets. Indeed,
when one of the author studied this problem on fractals in \cite{kum,hk}, he needed
a very strong assumption on each fractal because of the lack of the information of the trace.
Our trace theorem can be applied here and we can construct penetrating processes on much wider class
of fractals.
Throughout this article, if $f$ and $g$ depend on a variable $x$ ranging
in a set $A$, $f\asymp g$ means that there exists
$C>0$ such that
$C^{-1}f(x)\le g(x)\le C\,f(x)$ for all $x\in A$.
We will use $c$, with or without
subscripts, to denote strictly
positive constants whose values are insignificant.
\section{Framework and the main theorem}
Let $(X,\mathsf d)$ be a complete separable
metric space.
For $\alpha>1$ and a finite index set $W$,
let $\{F_i\}_{i\in W}$ be a family of {\it
$\alpha$-similitudes} on $X$, i.e.\ $\mathsf d(F_i(x),F_i(y))=\alpha^{-1}\mathsf d(x,y)$
for all $x,y\in X$.
Let $S$ be a subset of $W$ and let $N$ denote the cardinality of $S$.
Since
$\{F_i\}_{i\in S}$ is a family of contraction maps,
there exists a unique non-void compact set $K$ such that ${K}
=\bigcup_{i\in S}F_i ({K})$.
We assume that $K$ is connected.
Note that $W$ will be needed in general when we define a self-similar subset $L$ below.
In various important examples such as 1), 3) in Section~5, we can take $W=S$.
We will make the relation to the shift space.
The one-sided shift space $\Sigma$ is defined by $\Sigma = W^{{\mathbb N}}$. For $w \in \Sigma$,
we denote the $i$-th element in the sequence by $w_i$ and write $w =w_1 w_2 w_3 \cdots$.
When $w\in W^n$, $|w|$ denotes $n$.
For $v\in W^m$ and $w\in W^n$, we define $v\cdot w\in W^{m+n}$ by $v\cdot
w=v_1 v_2\cdots v_m w_1 w_2\cdots w_n$. For $A\subset W^m$ and $B\subset W^n$, $A\cdot B$ denotes $\{v\cdot w: v\in A,\ w\in B\}$.
The set $w\cdot A$ is defined as $\{w\}\cdot A$.
By definition, $W^0=\{\emptyset\}$ and $\emptyset\cdot A=A$.
Let ${\mathfrak G}$ be a group consisting of isometries on $K$.
We assume the following.
\begin{itemize}
\item For each $i\in W$, there exist $j=j(i)\in S$ and $\Psi_i\in{\mathfrak G}$ such that $F_i=F_j\circ \Psi_i$.
\item For each $(\Psi,\alpha)\in{\mathfrak G}\times S$, there exists $(\hat\Psi,\hat\alpha)\in{\mathfrak G}\times S$ such that
$
\Psi\circ F_\alpha = F_{\hat\alpha}\circ \hat\Psi
$.
\end{itemize}
Note that, when $W=S$, we can always take as ${\mathfrak G}$ the trivial group consisting of one element.
We write $F_{w_1\cdots w_n}=F_{w_1}\circ F_{w_2}\circ \cdots \circ F_{w_n}$
for $w=w_1w_2\cdots w_n$.
We regard $F_\emptyset$ as an identity map.
For $w\in W^n$ and $A\subset W^n$ for some $n\in{\mathbb Z}_+$, define $K_w=F_w(K)$ and $K_A=\bigcup_{v\in A}K_v$.
\begin{lem}\label{lem:Phi}
There exist maps $\Phi:\bigcup_{n\in{\mathbb Z}_+}W^n \to \bigcup_{n\in{\mathbb Z}_+}S^n$ and $\Psi:\bigcup_{n\in{\mathbb Z}_+}W^n \to {\mathfrak G}$ such that $F_w=F_{\Phi(w)}\circ\Psi(w)$ for each $w\in \bigcup_{n\in{\mathbb Z}_+}W_n$.
In particular, $K_w=K_{\Phi(w)}$.
\end{lem}
\begin{demo}{{\it Proof.\ }}
Set $\Phi(\emptyset)=\emptyset$ and $\Psi(\emptyset)={}$the unit element of ${\mathfrak G}$.
When $i\in W^1$, it suffices to set $\Phi(i)=j(i)$ and $\Psi(i)=\Psi_i$.
Suppose that $\Phi(w)$ is defined for $w\in W^{n}$.
Then, for $w'=w\cdot i$ with $i\in W$,
$
F_{w'}=F_w\circ F_i=F_{\Phi(w)}\circ\Psi(w)\circ F_{j(i)}\circ\Psi_{i}
$.
This is equal to $F_{\Phi(w)}\circ F_{\hat i}\circ\hat\Psi\circ \Psi_i$ for some $(\hat\Psi,\hat i)\in{\mathfrak G}\times
S$.
Therefore, it is enough to define $\Phi(w')=\Phi(w)\cdot\hat i$ and $\Psi(w')=\hat\Psi\circ \Psi_i$.
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
Define $\pi:\Sigma\to K$ by the relation $\{\pi (w)\}=\bigcap_m K_{w_1\cdots w_m}$ for $w=w_1w_2\cdots\in\Sigma$.
Define
\begin{equation}\label{eq:critpc}
C_K := \pi^{-1} \left( \bigcup_{i,j \in S, i \ne j} (K_i \cap K_j)\right),\qquad
P_K := \bigcup_{n \ge 1} \sigma^n (C_K),\end{equation}
where $\sigma : \Sigma \to \Sigma$ is the left shift map, i.e.
$\sigma w = w_2 w_3 \cdots$ if $w = w_1 w_2 w_3\cdots$.
For $v,w\in W^n$, we write $v\stackrel{n,K}{\sim} w$ if $K_v\cap K_w\ne \emptyset$.
For $w\in W^n$ and $A\subset W^n$, $w\stackrel{n,K}{\sim} A$ means that
$w\stackrel{n,K}{\sim} v$ for some $v\in A$.
For $A\subset W^n$, define ${\mathcal N}_0(A)=A$ and ${\mathcal N}_k(A)=\{v\in W^n\mid
v\stackrel{n,K}{\sim} {\mathcal N}_{k-1}(A)\}$ for $k\in{\mathbb N}$ inductively.
We set ${\mathcal N}_k(w)={\mathcal N}_k(\{w\})$ for $w\in W^n$.
Let $I$ be a subset of $W$.
We assume that the cardinality $N_I$ of $I$ is less than $N$.
Let $L$ be a unique non-void compact set such that $L
=\bigcup_{i\in I}F_i (L)$.
Clearly, $L$ is a subset of $K$.
Denote $F_w(L)$ by $L_w$ for $w\in\bigcup_{n\in{\mathbb Z}_+}I^n$.
Let $M\in{\mathbb N}$.
For $v$, $w\in I^n$, we write $v\underset{M}{\stackrel{n,L}{\longleftrightarrow}} w$ if $v \in {\mathcal N}_M(w)$.
We fix $M$ so that for each $i$, $j\in I$, there exist $i_1,i_2,\ldots\in I$ satisfying
$i\underset{M}{\stackrel{1,L}{\longleftrightarrow}}i_1\underset{M}{\stackrel{1,L}
{\longleftrightarrow}}i_2\underset{M}{\stackrel{1,L}{\longleftrightarrow}}\cdots\underset{M}{\stackrel{1,L}{\longleftrightarrow}}j$.
In what follows, we omit $M$ from the notation $\underset{M}{\stackrel{n,L}{\longleftrightarrow}}$.
We assume the following.
\begin{enumerate}
\item[(A1)] $\sup_{n\in{\mathbb Z}_+}\max_{w\in S^n}\#({\mathcal N}_1(w)\cap S^n)<\infty$
and $C_0:=\sup_{n\in{\mathbb Z}_+}\max_{w\in I^n}\#({\mathcal N}_M(w)\cap I^n)<\infty$.
\item[(A2)] There exist $k_1, k_2>0$ such that, for
$x,y\in L$, $n\in{\mathbb Z}_+$ and $v,w\in I^n$ with $x\in K_v$ and $y\in K_w$, $\mathsf d(x,y)< k_1 \alpha^{-n}$ implies
$v\stackrel{n,L}{\leftrightarrow}w$ and $v\stackrel{n,L}{\leftrightarrow}w$ implies $\mathsf d(x,y)< k_2 \alpha^{-n}$.
\item[(A3)] There exist $k_1, k_2>0$ such that, for
$x,y\in K$, $n\in{\mathbb Z}_+$ and $v,w\in S^n$ with $x\in K_v$ and $y\in K_w$, $\mathsf d(x,y)< k_1 \alpha^{-n}$ implies
$v\stackrel{n,K}{\sim}w$ and $v\stackrel{n,K}{\sim}w$ implies $\mathsf d(x,y)< k_2 \alpha^{-n}$.
\end{enumerate}
Let $\hat\mu$ and $\hat\nu$ be the canonical Bernoulli measures on $S^{\mathbb N}$ and $I^{\mathbb N}$, respectively.
That is, they are infinite product measures of $S$ (resp.\ $I$) with uniformly distributed measure.
Denote by $\mu$ the image measures of $\hat\mu$ by the map $\pi|_{S^{\mathbb N}}:S^{\mathbb N}\to K$.
In the same way, the probability measure $\nu$ on $L$ is defined.
By conditions (A1), (A2), and (A3) and \cite[Theorem~1.5.7]{kig}, the Hausdorff dimensions of $K$ and $L$ are equal to $d_f:=\log N/\log\alpha$ and $d:=\log N_I/\log\alpha$, respectively, and $\mu$ and $\nu$ are equivalent to the Hausdorff measures on $K$ and $L$, respectively.
We will further assume the following.
\begin{enumerate}
\item[(A4)] $\mu(\{x\in K: \#
(\pi^{-1}(x)\cap S^{\mathbb N})
=\infty\})=0$ and $\nu(\{x\in L: \#
(\pi^{-1}(x)\cap I^{\mathbb N})
=\infty\})=0$.
\end{enumerate}
Then, by Theorem 1.4.5 in \cite{kig},
$\mu(K_w)=N^{-|w|}$ for every $w\in \bigcup_{n\in{\mathbb Z}_+}S^n$ and $\nu(L_w)=N_I^{-|w|}$ for every $w\in \bigcup_{n\in{\mathbb Z}_+}I^n$.
It also holds that $\mu(L)=0$.
Suppose that we are given a strong local regular
Dirichlet form $({\mathcal E},{\mathcal F})$ on $L^2(K,\mu)$.
${\mathcal F}$ is equipped with a norm $\|f\|_{{\mathcal F}}
=({\mathcal E}(f)+\|f\|_{L^2(\mu)}^2)^{1/2}$.
Here and throughout the paper, for each quadratic form $E(\cdot,\cdot)$,
we abbreviate $E(f,f)$ as $E(f)$.
We assume the following.
\begin{enumerate}
\item[(A5)] (Self-similarity) For each $f\in {\mathcal F}$ and $i\in S$, $F_i^*f\in {\mathcal F}$
where $F_i^*f=f\circ F_i$. Further, there exists $\rho>0$ such that
$${\mathcal E} (f)=\rho\sum_{i\in S}{\mathcal E} (F_i^*f),\quad
f\in{\mathcal F}.$$
\item[(A6)] For every $\Psi\in{\mathfrak G}$, $\Psi^*{\mathcal F}={\mathcal F}$, that is, $\{f\circ \Psi: f\in{\mathcal F}\}={\mathcal F}$.
Further, ${\mathcal E}(\Psi^* f)={\mathcal E}(f)$ for all $f\in{\mathcal F}$.
\item[(A7)] Let $d_w=(\log \rho N)/(\log \alpha)$. Then $d_w>d_f-d$.
\item[(B1)]
The space ${\mathcal F}$ is compactly imbedded in $L^2(K,\mu)$, and ${\mathcal E}(f)=0$ if and only if $f$ is a constant function.
\end{enumerate}
For each subset $A$ of $W^m$ for some $m\in{\mathbb Z}_+$,
let ${\mathcal F}_A$ be a function space on $K_A$ such that $\{f|_{K_A}:f\in{\mathcal F}\}\subset {\mathcal F}_A\subset \{f\in L^2(K_A):
F_w^* f\in{\mathcal F} \mbox{ for all }w\in A\}$.
The space ${\mathcal F}_A$ will be specified later
for some class of Dirichlet forms in Section 4.
Define, for $f,g\in{\mathcal F}_A$,
\begin{equation}\label{eq:energy}
{\mathcal E}_A (f,g)=\rho^m\sum_{w\in A}{\mathcal E} (F_w^*f,F_w^*g).
\end{equation}
We assume that ${\mathcal F}_A={\mathcal F}_{A\cdot S^n}$ for all $n\in{\mathbb N}$ and $({\mathcal E}_A,{\mathcal F}_A)$ is a closed form on
$L^2(K_A,\mu|_{K_A})$.
In what follows, we always consider ${\mathcal F}_A$ as a normed space with norm $\|f\|_{{\mathcal F}_A}=({\mathcal E}_A(f)+\|f\|_{L^2(K_A)}^2)^{1/2}$.
Due to (A5), ${\mathcal E}_A(f)={\mathcal E}_{A\cdot S^n}(f)$ holds for any $f\in{\mathcal F}_A$, and ${\mathcal E}_{\Phi(A)}(f)={\mathcal E}_A(f)$ if $\#\Phi(A)=\# A$ by (A6).
When $A=\{w\}$, we use the notation ${\mathcal E}_w$ in place of ${\mathcal E}_{\{w\}}$.
Functions in ${\mathcal F}$ can be naturally considered as elements in ${\mathcal F}_A$ by the restriction of the domain.
We often write simply $f$ in place of $f|_{K_A}$ when we regard $f\in{\mathcal F}$ as an element of ${\mathcal F}_A$, for notational conveniences.
\begin{defn}\label{defn:good}
Let $A$ be a nonempty subset of $W^m$ for some $m\in{\mathbb Z}_+$.
We say that $A$ is ${\mathcal E}_A$-connected if, for $f\in{\mathcal F}_A$, ${\mathcal E}_A(f)=0$ implies that $f$ is constant on $K_A$.
\end{defn}
\begin{defn}\label{def:same}
Let $A\subset W^m$ and $B\subset W^n$ for some $m$ and $n$.
We say that $A$ and $B$ are of the same type if there exist a homeomorphism $F:K_A\to K_B$ and a bijection $\chi:A\to B$ such that
$F\circ F_u=F_{\chi(u)}$ for all $u\in A$ and $F^*({\mathcal F}_B)={\mathcal F}_A$.
\end{defn}
We assume the following.
\begin{enumerate}
\item[(B2)] There exists $\hat I\subset W$ such that the following hold.
\begin{enumerate}
\item[(1)] $\hat I\supset I$ and $\# \hat I<N$.
\item[(2)] For each $w\in I^n$, ${\mathcal N}_M(w)\cap \hat I^n$ is an ${\mathcal E}_{{\mathcal N}_M(w)\cap \hat I^n}$-connected set.
\item[(3)]
There exist finite elements $u_1,\ldots,u_k\in\bigcup_{n\in{\mathbb Z}_+}I^n$ such that, for any $w\in\bigcup_{n\in{\mathbb Z}_+}I^n$, there exists $j\in\{1,\ldots,k\}$ such that ${\mathcal N}_M(w)\cap\hat I^{|w|}$ and ${\mathcal N}_M(u_j)\cap\hat I^{|u_j|}$ are of the same type, and moreover, $F(L_{{\mathcal N}_M(w)\cap\hat I^{|w|}})=L_{{\mathcal N}_M(u_j)\cap\hat I^{|u_j|}}$ where $F$ is provided in Definition~\ref{def:same}.
\item[(4)] $C_1:=\sup_{n\in{\mathbb Z}_+}\max_{w\in \hat I^n}\#({\mathcal N}_M(w)\cap I^n)<\infty$ and $C_2:=\sup_{n\in{\mathbb Z}_+}\max_{w\in S^n}\#\{v\in\hat I^n: \Phi(v)=w\}<\infty$.
\end{enumerate}
\end{enumerate}
For an open set $U\subset K$, define the capacity of $U$ by
\[\mathop{\rm Cap}(U)=\inf \{ \|u\|_{\mathcal F}^2: u\in {\mathcal F}, u\ge 1
\mbox { $\mu$-a.e.\ on } U\}.\]
The capacity of any set $D\subset K$ is defined as the infimum of the capacity of
open sets that contain $D$.
We denote a quasi-continuous modification of $f\in{\mathcal F}$ by $\tilde f$.
We assume the following.
\begin{enumerate}
\item[(A8)] There exists some $c>0$ such that $\nu(D)\le c\mathop{\rm Cap}(D)$ for every compact set $D\subset K$.
\end{enumerate}
By Theorem~3.1 of \cite{ben}, (A8) is equivalent to the following.
\begin{enumerate}
\item[(A8)'] The measure $\nu$
charges no set of zero capacity
and $f\mapsto \tilde f|_L$ is a continuous map from ${\mathcal F}$ to $L^2(L,\nu)$.
\end{enumerate}
We will provide sufficient conditions for (A8) in Section~4.
For each $n\in{\mathbb Z}_+$, define $Q_n: L^1(L,\nu)\to {\mathbb R}^{I^n}$ as
\[Q_nf(w)={}-\!\!\!\!\!\!\!\int_{L_w}f(y)d\nu (y),~~w\in I^n,\]
where in general $-\!\!\!\!\!\int_A \cdots d\lambda(y):=\lambda(A)^{-1}\int_A \cdots d\lambda(y)$ denotes
the normalized integral on $A$.
Then, one can easily check
\begin{equation}\label{eq:qmqm1}
N_I^{-1}\sum_{j\in I} Q_{m+1}f(w\cdot j)=Q_mf(w),~~w\in I^m.\end{equation}
Let $m\in{\mathbb N}$, $A\subset S^m$, and $J\subset I^m$. Define
${\mathcal F}(J,A)=\{f\in {\mathcal F}: f=0 \mbox{ on }K_{S^m\setminus A},\ Q_m (\tilde f|_{L})=0
\mbox{ on }J\}$,
and define a closed subspace ${\mathcal H}(J,A)$ of ${\mathcal F}$ by
\begin{eqnarray*}
{\mathcal H}(J,A)=\{h\in {\mathcal F}: {\mathcal E}(h,f)=0~~\mbox {for all }
f\in {\mathcal F}(J,A)\}.
\end{eqnarray*}
When $J$ is an empty set, we omit it from the notation.
We assume the following.
\begin{enumerate}
\item[(B3)] There exist some $l_0,m_0\in{\mathbb Z}_+$, $C>0$, a proper subset $D'(w)$ of $S^{|w|}$ with $w\in D'(w)$
for each $w\in\bigcup_{n\in{\mathbb Z}_+}\Phi(\hat I^{n+m_0})$,
a finite subset $\Xi\subset \bigcup_{n\in{\mathbb Z}_+}\Phi(\hat I^{n+m_0})$
and subsets $D^\sharp(v)$ of $D'(v)$ with $v\in D^\sharp(v)$ for each $v\in \Xi$
such that the following hold.
\begin{enumerate}
\item[(1)] For each $w\in\bigcup_{n\in{\mathbb Z}_+}\Phi(\hat I^{n+m_0})$,
\begin{enumerate}
\item[(a)]
$w\in D'(w)$ and $D'(w)\subset{{\mathcal N}_{l_0}(w)}\cap (\Phi(\hat I^n)\cdot S^{m_0})$,
\item[(b)]
there exists $v\in\Xi$ such that
\begin{eqnarray*}
\lefteqn{F_w^*(\{h\in{\mathcal H}(I^{|w|},D'(w)):\int_{K_{D'(w)}}h\,d\mu=0,\ \rho^{-|w|}{\mathcal E}_{D'(w)}(h)\le 1\})}&&\\
&&\subset F_{v}^*(\{h\in{\mathcal H}(I^{|v|},D^\sharp(v)):\|h\|_{{\mathcal F}_{D^\sharp(v)}}\le C\}).
\phantom{\hspace{10em}}
\end{eqnarray*}
\end{enumerate}
\item[(2)] For each $v\in\Xi$,
the operator $F_v^*:{\mathcal H}(D^\sharp(v))|_{K_{D^\sharp(v)}}\to {\mathcal F}$ is a compact operator, where ${\mathcal H}(D^\sharp(v))|_{K_{D^\sharp(v)}}$ is regarded as a subspace of ${\mathcal F}_{D^\sharp(v)}$.
\end{enumerate}
\end{enumerate}
We set $D(w)=D'(\Phi(w))$ for $w\in\bigcup_{n\in{\mathbb Z}_+}\hat I^{n+m_0}$.
We have a sufficient condition concerning (B3); see Section~4.
The following assumption (B4) will be used in the restriction theorem.
\begin{enumerate}
\item[(B4)]
For $f\in{\mathcal F}$, if ${\mathcal E}_{S^m\setminus \Phi(\hat I^m)}(f)=0$ for every $m\in{\mathbb Z}_+$, then $f$ is a constant function.
\end{enumerate}
We next introduce Besov spaces.
\begin{defn}\label{defn:Besov}
For $1\le p<\infty,~1\le q\le \infty,~\beta\ge 0$ and
$m\in {\mathbb Z}_+$, set
\[
a_m (\beta, f):=\gamma^{m\beta}\left(\gamma^{md_f}
\int\!\!\int_{\{(x,y)\in K\times K:\mathsf d(x,y)<c\gamma^{-m}\}}|f(x)-f(y)|^p\,d\mu (x)
d \mu(y)\right)^{1/p}
\]
for $f\in L^p(K,\mu)$,
where $1<\gamma<\infty,~0<c<\infty$. Define a {\it Besov space}
$\Lambda^\beta_{p,q}(K)$ as a set of all $f\in L^p(K,\mu)$ such that
${\bar a}(\beta, f):=\{a_m(\beta, f)\}_{m=0}^{\infty}\in l^q$.
$\Lambda^\beta_{p,q}(K)$ is a Banach space with the
norm $\| f\|_{\Lambda^\beta_{p,q}(K)}
:=\|f\|_{L^p(K)}+\|{\bar a}(\beta, f)\|_{l^q}$.
Let $\hat\Lambda^\beta_{p,q}(K)$ denote the closure of $\Lambda^\beta_{p,q}(K)\cap C(K)$ in $\Lambda^\beta_{p,q}(K)$.
$\Lambda^\beta_{p,q}(L)$ and $\hat\Lambda^\beta_{p,q}(L)$ are defined in the same way by replacing $(K,\mu)$ by $(L,\nu)$.
\end{defn}
We remark that this definition is valid for general Alfors regular compact sets $K$ with normalized Hausdorff measure $\mu$.
We use the notation $\Lambda^\beta_{p,q}(K)$ following \cite{gri}.
$\Lambda^\beta_{p,q}(K)$ was denoted by Lip $(\beta,p,q)(K)$ in \cite{j2,kum}
and by $\Lambda_\beta^{p,q}(K)$ in \cite{stri}.
Note that different choices of $c>0$ and $\gamma>1$ provide the same space $\Lambda^\beta_{p,q}(K)$ with equivalent norms. In what follows, we will take $\gamma=\alpha$.
We are now ready to state our main theorems.
Let $\beta=d_w/2-(d_f-d)/2$.
\begin{theo}\label{theo:mainthm1}
Suppose that \rom{(A1)}--\rom{(A8)} and \rom{(B1)}--\rom{(B4)} hold.
Then, for every $f\in{\mathcal F}$, $\tilde f|_L$ belongs to $\hat\Lambda^\beta_{2,2}(L)$.
Moreover, there exists $c>0$ such that $\|\tilde f|_L\|_{\Lambda^\beta_{2,2}(L)}\le c\|f\|_{{\mathcal F}}$ for every $f\in{\mathcal F}$.
\end{theo}
\begin{theo}\label{theo:mainthm2}
Suppose that \rom{(A1)}--\rom{(A8)} and \rom{(C1)}--\rom{(C2)} hold. (The conditions \rom{(C1)} and \rom{(C2)} will be defined in Section~3.3).
Then, there exists a bounded linear map $\xi$ from $\hat\Lambda^\beta_{2,2}(L)$ to ${\mathcal F}$ such that $\xi(\Lambda^\beta_{2,2}(L)\cap C(L))\subset{\mathcal F}\cap C(K)$ and $\widetilde{\xi f}|_L=f$ $\nu$-a.e.\ for all $f\in\hat\Lambda^\beta_{2,2}(L)$.
\end{theo}
In what follows, we often write ${\mathcal F}|_L=\hat\Lambda^\beta_{2,2}(L)$ to denote the assertions of two theorems above.
\begin{rem}\label{rem:2.5}
In the following two cases, we can prove
$\hat\Lambda^\beta_{2,2}(L)=\Lambda^\beta_{2,2}(L)$.\\
1) $L\subset {\mathbb R}^n$ for some $n\in {\mathbb N}$ and $\beta<1$.~~In this case, the following trace theorem
holds due to \cite{jw}; $B^{2,2}_{\beta+(n-d)/2}({\mathbb R}^n)|_L=\Lambda^\beta_{2,2}(L)$
where $B^{2,2}_{\gamma}({\mathbb R}^n)$ is the classical Besov space with smoothness order $\gamma$. Since
$C_0^\infty({\mathbb R}^n)$ is dense
in $B^{2,2}_{\gamma}({\mathbb R}^n)$ for $\gamma>0$, it follows that functions from
$C_0^\infty({\mathbb R}^n)$ restricted to $L$ are dense in $\Lambda^\beta_{2,2}(L)$.\\
2) $\beta>d/2$.~~
In this case, the following holds due to \cite{gri} Theorem 8.1;
$\Lambda_{2,\infty}^\beta(L)\subset {\cal C}^{\beta-d/2}(L)$, where
${\cal C}^{\lambda}(L)$ is a H\"older space defined as follows. $u\in {\cal C}^{\lambda}(L)$ if
\begin{equation}
\|u\|_{{\cal C}^\lambda(L)}:=\|u\|_{L^\infty(L)}+
\mathop{\nu\rm -esssup}_{x,y\in L,\ x\ne y}
\frac{|u(x)-u(y)|}{\mathsf d(x,y)^\lambda}<\infty.
\end{equation}
Since $\Lambda_{2,2}^\beta(L)\subset \Lambda_{2,\infty}^\beta(L)$, we see that any element in
$\Lambda_{2,2}^\beta(L)$ is continuous in this case.
\end{rem}
\begin{rem}\label{rem:timechange}
Since $\nu$ is smooth with respect to $({\mathcal E},{\mathcal F})$, we can consider the time changed Markov process with respect to the positive continuous additive functional associated with $\nu$ via the Revuz correspondence.
By the general theory of Dirichlet forms, this has an associated regular Dirichlet form $(\check{\mathcal E},\check{\mathcal F})$ on $L^2(L,\nu)$ with
$\check{\mathcal F}=\{f\in L^2(L,\nu): f=\tilde u \ \nu\mbox{-a.e.\ on $L$ for some }u\in{\mathcal F}_e\}$,
where ${\mathcal F}_e$ is the family of $\mu$-measurable functions $u$ on $K$ such that $|u|<\infty$ $\mu$-a.e.\ and there exists an ${\mathcal E}$-Cauchy sequence $\{u_n\}_{n\in{\mathbb N}}$ of functions in ${\mathcal F}$ such that $\lim_{n\to\infty}u_n=u$ $\mu$-a.e.
As is seen in the proposition below, ${\mathcal F}_e={\mathcal F}$ in our framework.
So, our main theorems determine the function space $\check{\mathcal F}$.
\end{rem}
\begin{pro}
Under the condition (B1), ${\mathcal F}_e={\mathcal F}$.
\end{pro}
\begin{demo}{{\it Proof.\ }}
By (B1), there exists some $c>0$ such that
\begin{equation}\label{eq:poincare}
\left\|f-\int_K f\,d\mu\right\|_{L^2(K)}^2\le c{\mathcal E}(f),
\quad f\in{\mathcal F}.
\end{equation}
Let $u\in{\mathcal F}_e$. Take $\{u_n\}_{n\in{\mathbb N}}$ from ${\mathcal F}$ as in the definition of ${\mathcal F}_e$ in Remark~\ref{rem:timechange}.
Define $g_n=u_n-\int_K u_n\,d\mu$ for each $n$.
Then, $\{g_n\}_{n\in{\mathbb N}}$ is ${\mathcal E}$-Cauchy.
Since $\int_K g_n\,d\mu=0$, (\ref{eq:poincare}) implies that $\{g_n\}$ is also $L^2(K)$-Cauchy.
Therefore, $g_n$ converges to some $g$ in ${\mathcal F}$.
By taking a subsequence, we may assume that $g_n\to g$ $\mu$-a.e.
Thus, $\int_K u_n\,d\mu\,({}=u_n-g_n)$ converges to some $C\in{\mathbb R}$.
In particular, $\int_K u_n\,d\mu$ converges to $C$ in ${\mathcal F}$ as a sequence of constant functions.
Therefore, $u_n$ converges to $g+C$ in ${\mathcal F}$.
This implies that $u=g+C$ belongs to ${\mathcal F}$.
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
\section{Proof of main theorems}
\subsection{Discrete approximation}
In this section, we assume (A1)--(A8).
For $n\in{\mathbb Z}_+$, define a bilinear form on $I^n$ as
\[E_{(n)}(g,g)=\sum_{v,w\in I^n,~ v\stackrel{n,L}{\leftrightarrow} w}
(g(v)-g(w))^2~~~\mbox{for }~ g\in {\mathbb R}^{I^n}.\]
We then have the following discrete characterization of $ \Lambda^\beta_{2,q}(L)$ (for related results,
see \cite{kam}).
\begin{lem}\label{theo:lem2}
Let $\beta>0$ and $q\in[1,\infty]$.
Then, there exists $c_1>0$ such that
for each $f\in L^2(L,\nu)$,
\begin{eqnarray}
\lefteqn{c_1\left\|\left\{
\alpha^{n\beta}\left(\alpha^{nd}\int\!\!\!\int_{\{(x,y)\in L\times L:
\mathsf d(x,y)< k_1 \alpha^{-n}\}}|f(x)-f(y)|^2\,d\nu(x)\,d\nu(y)
\right)^{1/2}\right\}_{n=0}^\infty\right\|_{l^q}}\nonumber\\
&\le&\left\|\left\{\alpha^{n\beta}\left(\alpha^{-nd}E_{(n)}(Q_n f)
\right)^{1/2}\right\}_{n=0}^{\infty}\right\|_{l^q}\nonumber\\
&\le& \left\|\left\{
\alpha^{n\beta}\left(\alpha^{nd}\int\!\!\!\int_{\{(x,y)\in L\times L:
\mathsf d(x,y)< k_2 \alpha^{-n}\}}|f(x)-f(y)|^2\,d\nu(x)\,d\nu(y)\right)^{1/2}
\right\}_{n=0}^\infty\right\|_{l^q}.\label{eq:equivalence}
\end{eqnarray}
Here, $k_1$ and $k_2$ are provided in \rom{(A2)}.
\end{lem}
\begin{demo}{{\it Proof.\ }}
Due to the choice of $M$, the exists some $c_2>0$ such that
\[
\sum_{i\in I}\left(g(i)-N_I^{-1}
\sum_{j\in I}g(j)\right)^2\le c_2 E_{(1)}(g),\quad g\in {\mathbb R}^I.
\]
For $f\in L^2(L,\nu)$ and $n\in{\mathbb Z}_+$, we have
\begin{eqnarray*}
\lefteqn{\int\!\!\!\int_{\{(x,y)\in L\times L: \mathsf d(x,y)< k_1
\alpha^{-n}\}}|f(x)-f(y)|^2\,d\nu(x)\,d\nu(y)}\\
&\le& \sum_{(v,w)\in I^n\times I^n,\,v\stackrel{n,L}{\leftrightarrow} w}\int\!\!\!
\int_{L_v\times L_w}|f(x)-f(y)|^2\,d\nu(x)\,d\nu(y)
\qquad\mbox{(by (A2))}\\
&\le& \sum_{(v,w)\in I^n\times I^n,\,v\stackrel{n,L}{\leftrightarrow} w}\int\!\!\!
\int_{L_v\times L_w}3\{|f(x)-Q_nf(v)|^2+|Q_nf(v)-Q_nf(w)|^2\\
&&{}+|Q_nf(w)-f(y)|^2\}\,d\nu(x)\,d\nu(y)\\
&\le& 6C_0 N_I^{-n}\sum_{v\in I^n} \int_{L_v} (f(x)-Q_nf(v))^2\,d\nu(x)
+3 N_I^{-2n}E_{(n)}(Q_n f),
\end{eqnarray*}
where $C_0$ is what appeared in (A1).
Concerning the first term, we have
\begin{eqnarray*}
\lefteqn{\sum_{v\in I^n} \int_{L_v} (f(x)-Q_nf(v))^2\,d\nu(x)}\\
&=& \int_L f(x)^2\,d\nu(x)-N_I^{-n}\sum_{v\in I^n} Q_nf(v)^2\\
&=&\sum_{m=n}^\infty \left(N_I^{-(m+1)}\sum_{v\in I^{m+1}}
Q_{m+1}f(v)^2-N_I^{-m}\sum_{w\in I^m} Q_mf(w)^2\right)\\
&=&\sum_{m=n}^\infty N_I^{-(m+1)}\sum_{w\in I^m}\sum_{i\in I}
\left(Q_{m+1}f(w\cdot i)-N_I^{-1}\sum_{j\in I}Q_{m+1}f(w\cdot j)\right)^2\\
&\le&c_2\sum_{m=n}^\infty N_I^{-(m+1)}\sum_{w\in I^m}E_{(1)}(Q_{m+1}f(w\cdot*))\\
&\le&c_2\sum_{m=n}^\infty N_I^{-(m+1)}E_{(m+1)}(Q_{m+1}f),
\end{eqnarray*}
where the martingale convergence theorem was used in the second equality and (\ref{eq:qmqm1}) was used in the third equality.
Note that $\alpha^d=N_I$.
Suppose $q\in[1,\infty)$. Then,
\begin{eqnarray*}
\lefteqn{\sum_{n=0}^\infty
\alpha^{n(\beta+d/2)q}\left(\int\!\!\!\int_{\{(x,y)\in L\times L:
\mathsf d(x,y)< k_1 \alpha^{-n}\}}|f(x)-f(y)|^2\,d\nu(x)\,d\nu(y)\right)^{q/2}}\\
&\le&\sum_{n=0}^\infty\alpha^{n(\beta+d/2)q}\left(6c_2C_0
N_I^{-n}\sum_{m=n}^\infty N_I^{-(m+1)}E_{(m+1)}(Q_{m+1}f)
+3 N_I^{-2n}E_{(n)}(Q_n f)\right)^{q/2}\\
&\le&c_3\sum_{n=0}^\infty\alpha^{n\beta q}\left(\sum_{m=n}^\infty
\alpha^{-md}E_{(m)}(Q_{m}f)\right)^{q/2}\\
&\le&c_4\sum_{m=0}^\infty\alpha^{m(\beta-d/2)q}E_{(m)}(Q_{m}f)^{q/2}\\
&=& c_4\left\|\left\{\alpha^{n\beta}\left(\alpha^{-nd}
E_{(n)}(Q_n f)\right)^{1/2}\right\}_{n=0}^\infty\right\|_{l^q}^q,
\end{eqnarray*}
where in the third inequality, we used (A7) and the following inequality for $a>0$:
\begin{equation}\label{eq:harheld}
\sum_{i=0}^\infty 2^{ai}\left(\sum_{j\in \Lambda_i} a_j\right)^p
\le c \sum_{j=0}^\infty 2^{aj}a_j^p
\quad\mbox{for }a\ne 0,\ p>0,\ a_j\ge0,
\end{equation}
where $\Lambda_i=\{i,i+1,\ldots\}$ when $a>0$ and $\Lambda_i=\{0,1,\ldots, i\}$ when $a<0$.
When $0<p\le1$, this is obvious since $(x+y)^p\le x^p+y^p$ for $x$, $y\ge0$.
When $p>1$, this is proved by applications of H\"older's inequality; see e.g.\ \cite{lei}.
When $q=\infty$, letting $\gamma=\left\|\left\{\alpha^{n\beta}\left(\alpha^{-nd}
E_{(n)}(Q_n f)\right)^{1/2}\right\}_{n=0}^\infty\right\|_{l^\infty}$, we have for every
$n\in{\mathbb Z}_+$,
\begin{eqnarray*}
\lefteqn{\left|
\alpha^{n(\beta+d/2)}\left(\int\!\!\!\int_{\{(x,y)\in L\times L: \mathsf d(x,y)< k_1
\alpha^{-n}\}}|f(x)-f(y)|^2\,d\nu(x)\,d\nu(y)\right)^{1/2}\right|^2}\\
&\le&c_5\alpha^{n(2\beta+d)}\left(N_I^{-n}\sum_{m=n}^\infty N_I^{-(m+1)}
E_{(m+1)}(Q_{m+1}f)+N_I^{-2n}E_{(n)}(Q_n f)\right)\\
&\le&c_5\alpha^{2n\beta}\sum_{m=n}^\infty \alpha^{-2m\beta}\gamma^2\\
&=&\frac{c_5}{1-\alpha^{-2\beta}}\gamma^2.
\end{eqnarray*}
Thus, the first inequality in (\ref{eq:equivalence}) is proved.
Next, we have
\begin{eqnarray*}
E_{(n)}(Q_n f)
&=&\sum_{(v,w)\in I^n\times I^n,\,v\stackrel{n,L}{\leftrightarrow} w} \left|N_I^{2n}\int\!\!\!
\int_{L_v\times L_w}\{f(x)-f(y)\}\,d\nu(x)\,d\nu(y)\right|^2\\
&\le&\sum_{(v,w)\in I^n\times I^n,\,v\stackrel{n,L}{\leftrightarrow} w} N_I^{2n}\int\!\!\!
\int_{L_v\times L_w}|f(x)-f(y)|^2\,d\nu(x)\,d\nu(y)\\
&\le&\alpha^{2nd}\int\!\!\!\int_{\{(x,y)\in L\times L: \mathsf d(x,y)<
k_2 \alpha^{-n}\}}|f(x)-f(y)|^2\,d\nu(x)\,d\nu(y),
\end{eqnarray*}
which deduces the second inequality of (\ref{eq:equivalence}).
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
\begin{rem} Quite recently, M. Bodin (\cite{bod}) gives a discrete characterization of
$\Lambda_{p,q}^\beta(K)$ for the Alfors $d$-regular set $K$ if it has a
regular triangular system with some property (property~(B) in the thesis).
\end{rem}
\subsection{Proof of the restriction theorem}
In this section, we assume (A1)--(A8) and (B1)--(B4), and prove Theorem~\ref{theo:mainthm1}.
The following lemma is immediately proved by equation~(\ref{eq:energy}).
\begin{lem}\label{lem:scale}
Let $A\subset W^m$, $B\subset W^n$, $f\in{\mathcal F}_A$, and $g\in{\mathcal F}_B$.
Suppose that there exists a bijection $\iota$ from $A$ to $B$ and $F_v^* f= F_{\iota(v)}^* g$ for every $v\in A$. Then, $\rho^{-m}{\mathcal E}_A(f)=\rho^{-n}{\mathcal E}_B(g)$.
\end{lem}
Let $n\in{\mathbb Z}_+$ and $w\in I^n$.
Let $A={\mathcal N}_M(w)\cap \hat I^n$.
Define ${\mathcal G}_w=\{f\in{\mathcal F}_A: Q_n (\tilde f|_{L_A})=0 \mbox{ on }{\mathcal N}_M(w)\cap I^n\}$ and ${\mathcal K}_w=
\{h\in{\mathcal F}_A: {\mathcal E}_A(h,f)=0 \mbox{ for all }f\in{\mathcal G}_w\}$.
Here, we used (and will use) notations $Q_n (\tilde f|_{L_A})$ (on $A$) and ${\mathcal E}_A(f)$
for $f\in {\mathcal F}_A$ in the obvious sense.
\begin{lem}\label{lem:pre}
\begin{enumerate}
\item[$(1)$] There exists some $c>0$ such that $\|f\|_{L^2(K_A)}^2\le c{\mathcal E}_A(f)$ for all $f\in{\mathcal G}_w$.
\item[$(2)$] For each $g\in{\mathcal F}_A$, there exists $h_g\in {\mathcal K}_w$ such that $Q_n (\tilde h_g|_{L_A})=Q_n (\tilde g|_{L_A})$ on ${\mathcal N}_M(w)\cap I^n$ and ${\mathcal E}_A(h_g)\le {\mathcal E}_A(g)$.
\end{enumerate}
\end{lem}
\begin{demo}{{\it Proof.\ }}
(1) Suppose that the claim does not hold.
Then, there exists a sequence $\{f_k\}_{k\in{\mathbb N}}\subset{\mathcal G}_w$ such that $\|f_k\|_{L^2(K_A)}=1$ and $\lim_{k\to\infty}{\mathcal E}_A(f_k)=0$.
We may assume that $f_k$ converges weakly to some $f$ in ${\mathcal F}_A$ and $F_w^* f_k$ converges to $F_w^* f$ weakly in ${\mathcal F}$ for every $w\in A$.
By (B1), $F_w^* f_k$ converges to $F_w^* f$ in $L^2(K)$ for each $w\in A$. Thus, $f_k$ converges to $f$ in $L^2(K_A)$.
We also have
${\mathcal E}_A(f)\le \liminf_{k\to\infty}{\mathcal E}_A(f_k)=0$.
Therefore, ${\mathcal E}_A(f)=0$.
In view of (B2)(2), $f$ is constant on $K_A$.
Since $f$ belongs to ${\mathcal G}_w$ by (A8)', we conclude $f=0$ on $K_A$, which is a contradiction to the fact that $\|f\|_{L^2(K_A)}=\lim_{k\to\infty}\|f_k\|_{L^2(K_A)}=1$.
(2) Let ${\mathcal F}_g=\{f\in{\mathcal F}_A:Q_n (\tilde f|_{L_A})=Q_n(\tilde g|_{L_A})\mbox{ on }{\mathcal N}_M(w)\cap I^n\}$.
Take a sequence $\{h_k\}_{k\in{\mathbb N}}\subset {\mathcal F}_g$ such that ${\mathcal E}_A(h_k)$ converges to the infimum of $\{{\mathcal E}_A(f):f\in{\mathcal F}_g\}$.
Since
\begin{equation}\label{eq:domination}
\|h_k\|_{L^2(K_A)}
\le \|h_k-g\|_{L^2(K_A)}
+\|g\|_{L^2(K_A)}
\le c^{1/2}{\mathcal E}_A(h_k-g)^{1/2}
+\|g\|_{L^2(K_A)},
\end{equation}
we have $\sup_k \|h_k\|_{L^2(K_A)}<\infty$.
There exists a weak limit $h\in{\mathcal F}_A$ of a subsequence of $\{h_k\}_{k\in{\mathbb N}}$ in ${\mathcal F}_A$.
Then $h\in{\mathcal F}_g$ and $h$ attains the infimum of $\{{\mathcal E}_A(f): f\in{\mathcal F}_g\}$.
Dividing by $\epsilon$ both sides of the inequality ${\mathcal E}_A(h+\epsilon f)-{\mathcal E}_A(h)\ge 0$ for $f\in{\mathcal G}_w$ and letting $\epsilon\to0$, we obtain $h\in{\mathcal K}_w$.
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
\begin{lem}\label{lem:qnapp}
There exists some $c_1>0$ such that
\begin{equation}\label{eq:qnapp}
c_1\rho^{-n}{\mathcal E}_{\Phi(\hat I^n)} (f)\ge E_{(n)}(Q_n(\tilde f|_L))~~~\mbox{for all $f\in {\mathcal F}$ and $n\in{\mathbb Z}_+$}.
\end{equation}
\end{lem}
\begin{demo}{{\it Proof.\ }}
First, we prove that ${\mathcal K}_w$ is a finite dimensional vector space.
For each $i\in{\mathcal N}_M(w)\cap I^n$, take a function $g_i\in{\mathcal F}$ such that
$Q_n(\tilde g_i|_L)(j)=\left\{\begin{array}{ll}1& j=i\\0& j\ne i\end{array}\right.$ for all $j\in{\mathcal N}_M(w)\cap I^n$.
Existence of such functions is assured by the regularity of the Dirichlet form $({\mathcal E},{\mathcal F})$.
Define a linear map $\Theta\colon {\mathcal K}_w\to{\mathbb R}^{{\mathcal N}_M(w)\cap I^n}$ by $\Theta(f)=\{{\mathcal E}_A(f,g_i)\}_{i\in{\mathcal N}_M(w)\cap I^n}$.
Suppose $f$ belongs to the kernel of $\Theta$.
Then ${\mathcal E}_A(f,g)=0$ for every $g\in{\mathcal F}_A$, which implies that $f$ is constant on $K_A$ by (B2)~(2).
Therefore, ${\mathcal K}_w$ is finite dimensional.
Since ${\mathcal E}_{A}(h)=0$ implies
$\sum_{v\in A}(Q_n (\tilde h|_{L_A})(v)-Q_n (\tilde h|_{L_A})(w))^2=0$ for $h\in{\mathcal K}_w$, there exists $c_2>0$ such
that $ \sum_{v\in A}(Q_n (\tilde h|_{L_A})(v)-Q_n (\tilde h|_{L_A})(w))^2\le c_2\rho^{-n}{\mathcal E}_A (h)$ for every $h\in{\mathcal K}_w$.
By (B2)~(3) and Lemma~\ref{lem:scale}, we can take $c_2$ independently with respect to $w\in \bigcup_{n\in{\mathbb Z}_+}I^n$.
Therefore, for any $f\in {\mathcal F}$ and $n\in{\mathbb Z}_+$,
by taking $h_f\in{\mathcal K}_w$ as in Lemma~\ref{lem:pre} (2),
\begin{eqnarray*}
\sum_{v\in {\mathcal N}_M(w)\cap I^n}(Q_n (\tilde f|_L)(v)-Q_n(\tilde f|_L)(w))^2
&=&\sum_{v\in {\mathcal N}_M(w)\cap I^n}(Q_n(\tilde h_f|_{L_A})(v)-Q_n(\tilde h_f|_{L_A})(w))^2\\
&\le&c_2\rho^{-n}{\mathcal E}_{A} (h_f)\\
&\le&c_2\rho^{-n}{\mathcal E}_{A} (f).
\end{eqnarray*}
This implies that
\begin{eqnarray*}
E_{(n)}(Q_n(\tilde f|_L))
&=& \sum_{w\in I^n}\sum_{v\in {\mathcal N}_M(w)\cap I^n}(Q_n (\tilde f|_L)(v)-Q_n(\tilde f|_L)(w))^2\\
&\le& c_2\rho^{-n}\sum_{w\in I^n}{\mathcal E}_{{\mathcal N}_M(w)\cap \hat I^n} (f)\\
&\le& c_2C_1\rho^{-n}{\mathcal E}_{\hat I^n} (f)
\le c_2C_1C_2\rho^{-n}{\mathcal E}_{\Phi(\hat I^n)} (f),
\end{eqnarray*}
where $C_1$ and $C_2$ are provided in (B2) (4).
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
Recall finite sets $\Xi$ and $D^\sharp(v)$ for $v\in\Xi$ introduced in (B3).
\begin{lem}\label{lem:compact}
For each $v\in\Xi$,
the operator
$F_v^*\colon {\mathcal H}(I^{|v|},D^\sharp(v))|_{K_{D^\sharp(v)}}\to {\mathcal F}$ is a compact operator.
Here, ${\mathcal H}(I^{|v|},D^\sharp(v))|_{K_{D^\sharp(v)}}$ is regarded as a subspace of ${\mathcal F}_{K_{D^\sharp(v)}}$.
\end{lem}
\begin{demo}{{\it Proof.\ }}
Define $I(v)=\{w\in I^{|v|}: L_w\not\subset K_{S^{|v|}\setminus D^\sharp(v)}\}$.
Note that ${\mathcal H}(I^{|v|},D^\sharp(v))={\mathcal H}(I(v),D^\sharp(v))$.
For each $i\in I(v)$, take a function $g_i$ in ${\mathcal F}(D^\sharp(v))$ such that $Q_{|v|}(\tilde g_i|_L)(j)=\left\{\begin{array}{cl}1&\mbox{if }j=i\\0&\mbox{if }j\ne i\end{array}\right.$ for all $j\in I(v)$.
Define a linear map $\Theta: {\mathcal H}(I^{|v|},D^\sharp(v))|_{K_{D^\sharp(v)}}\to {\mathbb R}^{I(v)}$ by
$\Theta(f)=\{{\mathcal E}(f,g_i)\}_{i\in I(v)}$.
Then, the kernel of $\Theta$ is equal to ${\mathcal H}(D^\sharp(v))|_{K_{D^\sharp(v)}}$.
The homomorphism theorem implies that $\left.{\mathcal H}(I^{|v|},D^\sharp(v))|_{K_{D^\sharp(v)}}\right/{\mathcal H}(D^\sharp(v))|_{K_{D^\sharp(v)}}\simeq \Theta({\mathcal H}(I^{|v|},D^\sharp(v))|_{K_{D^\sharp(v)}})$ as a vector space.
Therefore, there exists a finite dimensional vector space $Z$ of ${\mathcal H}(I^{|v|},D^\sharp(v))|_{K_{D^\sharp(v)}}$ such that ${\mathcal H}(I^{|v|},D^\sharp(v))|_{K_{D^\sharp(v)}}$ is a direct sum of ${\mathcal H}(D^\sharp(v))|_{K_{D^\sharp(v)}}$ and $Z$.
Condition (B3)~(2) concludes the assertion.
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
\begin{lem}\label{lem:harmext}
Let $m\in{\mathbb N}$, $A$ a proper subset of $S^m$, and $J$ a subset of $I^m$.
For $g\in{\mathcal F}$, there exists a unique function $g'$ in ${\mathcal H}(J,A)$ such that $g'=g$ on $K_{S^m\setminus A}$ and $Q_m(\widetilde{g'}|_L)=Q_m(\tilde g|_L)$ on $J$.
Moreover,
there exists $c>0$ such that
\begin{equation}\label{eq:dominationA}
\|g'\|_{{\mathcal F}_A}\le c\|g\|_{{\mathcal F}_A}, \quad
{\mathcal E}(g')\le {\mathcal E}(g)
\end{equation}
for all $g\in{\mathcal F}$.
Further, if $g\ge0$ $\mu$-a.e., then $g'\ge0$ $\mu$-a.e.
\end{lem}
\begin{demo}{{\it Proof.\ }}
First, we prove that there exists some $c'>0$ such that $\|f\|_{L^2(K_A)}^2\le c'{\mathcal E}_A(f)$ for every $f\in {\mathcal F}(A)$.
Suppose this does not hold. Then, there exists a sequence $\{f_n\}_{n\in{\mathbb N}}\subset {\mathcal F}(A)$ such that $\|f_n\|_{L^2(K_A)}=1$ for every $n$ and ${\mathcal E}_A(f_n)$ converges to $0$ as $n\to\infty$.
We may assume that
$f_n$ converges weakly to some $f$ in ${\mathcal F}$.
Then, $f_n$ converges to $f\in{\mathcal F}$ in $L^2(K)$ by (B1), and ${\mathcal E}(f)\le \liminf_{n\to\infty}{\mathcal E}(f_n)=0$.
Therefore, ${\mathcal E}(f)=0$ and $f$ is constant on $K$.
Since $f\in{\mathcal F}(A)$ and $A\ne S^m$, $f$ is identically $0$, which is
contradictory to the fact $\|f\|_{L^2(K)}=1$.
Now, given $g\in{\mathcal F}$, let ${\mathcal F}_g=\{f\in{\mathcal F}: f=g\mbox{ on }K_{S^m\setminus A} \mbox{ and }Q_m(\tilde f|_L)=Q_m(\tilde g|_L) \mbox{ on }J\}$.
Then, in exactly the same way as the proof of Lemma~\ref{lem:pre}~(2), there exists $h\in{\mathcal F}_g$ attaining the
infimum of $\{{\mathcal E}(f):f\in {\mathcal F}_g\}$ and $h\in {\mathcal H}(J,A)$.
Such functions exist uniquely; indeed, if both $h$ and $h'$
attain the infimum above, we have
\[
{\mathcal E}\left(\frac{h-h'}2\right)=\frac12\left({\mathcal E}(h)+{\mathcal E}(h')\right)-{\mathcal E}\left(\frac{h+h'}2\right)\le0,
\]
which implies that $h-h'$ is a
constant. Since $h-h'=0$ on $K_{S^m\setminus A}$, we conclude that $h=h'$.
On the other hand, it is easy to see that $g'$ should attain the infimum above. Therefore, $g'$ is uniquely determined.
By the inequality similar to (\ref{eq:domination}),
we conclude (\ref{eq:dominationA}).
The last assertion follows from the characterization of $g'$ above and the Markov property of the Dirichlet form.
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
The following is the key proposition. Condition (B4) will be used (only) here.
\begin{pro}\label{theo:keyhin}
There exist $0<c_0<1$ and $b_0\in{\mathbb N}$ such that the following holds for all
$n\in{\mathbb Z}_+$ and $h\in {\mathcal H}(I^{n},\Phi(\hat I^n))$:
\[
{\mathcal E}_{\Phi(\hat I^{n+b_0})}(h)\le c_0{\mathcal E}_{\Phi(\hat I^{n})}(h).
\]
Moreover,
for all $i\ge j\ge 1$, $b=0,1,\dots,b_0-1$, and $h\in {\mathcal H}(I^{b_0j},\Phi(\hat I^{b_0j}))$,
\[
{\mathcal E}_{\Phi(\hat I^{b_0i+b})}(h)\le c_0^{i-j}{\mathcal E}_{\Phi(\hat I^{b_0j+b})}(h).
\]
\end{pro}
\begin{demo}{{\it Proof.\ }}
It is enough to prove the first claim.
Recall $l_0$ and $m_0$ in condition (B3).
By (B3), $C:=\sup_{n\in{\mathbb Z}_+}\max_{w\in \hat I^{n+m_0}}\# D(w)$ is finite.
Let $n\in{\mathbb Z}_+$ and $w\in \hat I^{n+m_0}$.
Define
\begin{eqnarray*}
{\mathcal C}_w&=&\{F_w^*f: f\in{\mathcal H}(I^{n+m_0},D(w)),\ \int_{K_{D(w)}}f\,d\mu=0,\ \rho^{-(n+m_0)}{\mathcal E}_{D(w)}(f)\le1\},\\
{\mathcal C}&=&\mbox{the closure of }\bigcup_{w\in \bigcup_{n\in{\mathbb Z}_+}\hat I^{n+m_0}}{\mathcal C}_w \mbox{ in }{\mathcal F}.
\end{eqnarray*}
Then, ${\mathcal C}$ is a compact subset in ${\mathcal F}$ by Lemma~\ref{lem:compact} and (B3).
Let $\delta=1/(4C^2)$ and define
${\mathcal C}(\delta)=\{f\in{\mathcal C}: {\mathcal E}(f)\ge\delta\}$.
Since (B4) holds, for each $f\in{\mathcal C}(\delta)$, there exist $m(f)\in{\mathbb N}$
and $a(f)\in(0,1)$ such that ${\mathcal E}_{\Phi(\hat I^m)}(f)<a(f){\mathcal E}(f)$ for all $m\ge m(f)$.
By continuity, ${\mathcal E}_{\Phi(\hat I^m)}(g)<a(f){\mathcal E}(g)$ for all $m\ge m(f)$ for any $g$ in some neighborhood of $f$ in
${\mathcal F}$. Since ${\mathcal C}(\delta)$ is compact in ${\mathcal F}$, there exist $m_1\in{\mathbb N}$ and $a_1\in(0,1)$
such that ${\mathcal E}_{\Phi(\hat I^{m_1})}(f)<a_1{\mathcal E}(f)$ for every $f\in{\mathcal C}(\delta)$.
In particular,
\begin{equation}\label{eq:dom1}
{\mathcal E}(f)\le a_2 {\mathcal E}_{S^{m_1}\setminus \Phi(\hat I^{m_1})}(f),
\quad f\in{\mathcal C}(\delta)
\end{equation}
with $a_2=(1-a_1)^{-1}>1$.
Now, take $h$ as in the claim of the proposition.
We construct an oriented graph such that the set of vertices is $\Phi(\hat I^{n+m_0})$ and a set of oriented edges is
$E=\{(v,w)\in \Phi(\hat I^{n+m_0})\times \Phi(\hat I^{n+m_0}):
v\in D'(w),\, {\mathcal E}_{w}(h)>0 \mbox{ and }
{\mathcal E}_{w}(h)\ge 2C {\mathcal E}_{v}(h)\}$.
This graph does not allow any loops.
Let $Y$ be the set of all elements $w$ in $\Phi(\hat I^{n+m_0})$ such that ${\mathcal E}_w(h)>0$ and $w$ is not a source of any edges.
For $w\in Y$, define $N_0(w)=\{w\}$, $N_k(w)=\{v\in \Phi(\hat I^{n+m_0})\setminus \bigcup_{l=0}^{k-1}
N_l(w): (v,u)\in E \mbox{ for some }u\in N_{k-1}(w)\}$ for $k\in{\mathbb N}$
inductively, and $N(w)=\bigcup_{k\ge0}N_k(w)$.
It is clear that $\# N_k(w)\le C^k$ and ${\mathcal E}_{v}(h)\le
(2C)^{-k}{\mathcal E}_{w}(h)$ for all $k\ge0$ and $v\in N_k(w)$.
Then, for each $w\in Y$,
\begin{equation}\label{eq:dom2}
{\mathcal E}_{N(w)}(h)
= \sum_{k=0}^\infty\sum_{v\in N_k(w)}{\mathcal E}_{v}(h)
\le \sum_{k=0}^\infty C^k (2C)^{-k}{\mathcal E}_{w}(h)
= 2{\mathcal E}_{w}(h).
\end{equation}
Suppose $w\in Y$ and ${\mathcal E}_w(h)\ge \delta{\mathcal E}_{D'(w)}(h)$. Then, since
\[
F_w^*\left(\left(h-{}-\!\!\!\!\!\!\!\int_{K_{D'(w)}}h\,d\mu\right)\times \rho^{(n+m_0)/2}{\mathcal E}_{D'(w)}(h)^{-1/2}\right)\in{\mathcal C}(\delta),
\]
(\ref{eq:dom1}) implies that ${\mathcal E}(F_w^* h)\le a_2 {\mathcal E}_{S^{m_1}\setminus \Phi(\hat I^{m_1})}(F_w^* h)$, namely,
\[
{\mathcal E}_{w}(h)\le a_2{\mathcal E}_{w\cdot (S^{m_1}\setminus \Phi(\hat I^{m_1}))}(h).
\]
Next, suppose $w\in Y$ and ${\mathcal E}_w(h)< \delta{\mathcal E}_{D'(w)}(h)$.
Since $w$ is not a source of any edges, ${\mathcal E}_v(h)<2C{\mathcal E}_w(h)$ for every $v\in D'(w)\cap \Phi(\hat I^{n+m_0})$.
Then,
\[
{\mathcal E}_{D'(w)\cap \Phi(\hat I^{n+m_0})}(h)
< C\cdot 2C{\mathcal E}_w(h)
< 2C^2\delta {\mathcal E}_{D'(w)}(h)
= \frac12 {\mathcal E}_{D'(w)}(h),
\]
which implies
${\mathcal E}_{D'(w)\cap \Phi(\hat I^{n+m_0})}(h)
< {\mathcal E}_{D'(w)\cap ((\Phi(\hat I^n)\cdot S^{m_0})\setminus \Phi(\hat I^{n+m_0}))}(h)$
by (B3)~(1)~(a).
In particular,
\[
{\mathcal E}_w(h)
<{\mathcal E}_{D'(w)\cap ((\Phi(\hat I^n)\cdot S^{m_0})\setminus \Phi(\hat I^{n+m_0}))}(h).
\]
Therefore, in any cases, we have for $w\in Y$,
\begin{eqnarray}\label{eq:dom3}
{\mathcal E}_w(h)&\le&a_2{\mathcal E}_{w\cdot(S^{m_1}\setminus \Phi(\hat I^{m_1}))\cup((D'(w)\cap ((\Phi(\hat I^n)\cdot S^{m_0})\setminus \Phi(\hat I^{n+m_0})))\cdot S^{m_1})}(h) \nonumber\\
&\le& a_2{\mathcal E}_{(D'(w)\cdot S^{m_1})\cap((\Phi(\hat I^n)\cdot S^{b_0})\setminus \Phi(\hat I^{n+b_0}))}(h),
\end{eqnarray}
where $b_0=m_0+m_1$ and note that
$\Phi(\hat I^{n+b_0})\subset \Phi(\hat I^{n+m_0})\cdot S^{m_1}$.
Then we have
\begin{eqnarray*}
{\mathcal E}_{\Phi(\hat I^{n+b_0})}(h)
&\le& {\mathcal E}_{\Phi(\hat I^{n+m_0})}(h)\\
&\le& \sum_{w\in Y} {\mathcal E}_{N(w)}(h)\\
&\le& 2a_2\sum_{w\in Y}{\mathcal E}_{(D'(w)\cdot S^{m_1})\cap((\Phi(\hat I^n)\cdot S^{b_0})\setminus \Phi(\hat I^{n+b_0}))}(h)\\
&\le& 2a_2 C_3{\mathcal E}_{(\Phi(\hat I^n)\cdot S^{b_0})\setminus \Phi(\hat I^{n+b_0})}(h).
\end{eqnarray*}
Here, we used (\ref{eq:dom2}) and (\ref{eq:dom3}) in the third inequality
and $C_3:=\sup_{n\in{\mathbb Z}_+}\max_{v\in S^{n+m_0}}\#({\mathcal N}_{l_0}(v)\cap S^{n+m_0})$ is finite by (A1).
Hence, the claim of the proposition holds with $c_0=2a_2 C_3/(1+2a_2 C_3)$.
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
\Proofwithname{Proof of Theorem~\ref{theo:mainthm1}.}
Fix $b\in\{0,1,\ldots,b_0-1\}$.
For each $f\in {\mathcal F}$, we can take $g_m\in {\mathcal H}(I^{b_0m+b},\Phi(\hat I^{b_0m+b}))$ such that $g_m=f$ on $K_{S^{b_0m+b}
\setminus \Phi(\hat I^{b_0m+b})}$ and $Q_{b_0m+b}(\tilde g_m|_L)=Q_{b_0m+b}(\tilde f|_L)$ by Lemma~\ref{lem:harmext}.
By using the relations $\|g_m\|_{\mathcal F}\le c\|f\|_{\mathcal F}$, ${\mathcal E}(g_m)\le {\mathcal E}(f)$ (by Lemma~\ref{lem:harmext}), and $g_m\to f$ $\mu$-a.e., we will prove
$g_m\to f$ in ${\mathcal F}$ as $m\to \infty$.
Here, note that the constant $c$ is taken independently of $m$, which derives from the fact that $c$ depends only on $c'$ in the proof of Lemma~\ref{lem:harmext}.
We first obtain that $g_m$ converges weakly to $f$ in ${\mathcal F}$ and $\limsup_{m\to\infty}{\mathcal E}(g_m-f)=\limsup_{m\to\infty}{\mathcal E}(g_m)-{\mathcal E}(f)\le0$.
Therefore, ${\mathcal E}(g_m-f)\to0$ as $m\to\infty$.
By (B1), $g_m-f-\int_K(g_m-f)\,d\mu$ converges to $0$ in $L^2(K)$.
Since $\|g_m-f\|_{L^2(K)}\le c\|f\|_{\mathcal F}+\|f\|_{L^2(K)}$, we have $\int_K(g_m-f)\,d\mu\to0$ as $m\to\infty$, which implies that $\|g_m-f\|_{L^2(K)}\to0$ as $m\to\infty$.
Thus, $g_m\to f$ in ${\mathcal F}$ as $m\to\infty$.
Let $f_m=g_m-g_{m-1}$ where we set $g_{-1}\equiv 0$. Then, $f=\sum_{m=0}^\infty f_m$.
Since $f_i\in{\mathcal F}(I^{b_0 j+b},\Phi(\hat I^{b_0 j+b}))$ for $i>j$
and $f_j\in{\mathcal H}(I^{b_0 j+b},\Phi(\hat I^{B_0 j+b}))$, we have
${\mathcal E} (f_i,f_j)=0$ for $i\ne j$, so that
\begin{equation}\label{eq:orthd}
{\mathcal E} (f)=\sum_{m=0}^\infty {\mathcal E} (f_m).
\end{equation}
Now, for each $f\in {\mathcal F}$,
\begin{eqnarray}
(E_{(b_0i+b)}(Q_{b_0i+b}(\tilde f|_L)))^{1/2}&=& (E_{(b_0i+b)}(Q_{b_0i+b}(\tilde g_i|_L)))^{1/2}=
\left(E_{(b_0i+b)}\left(\sum_{j=0}^i Q_{b_0i+b}(\tilde f_j|_L)\right)\right)^{1/2}\nonumber\\
&\le & \sum_{j=0}^i(E_{(b_0i+b)}(Q_{b_0i+b}(\tilde f_j|_L)))^{1/2}
\le \sum_{j=0}^i(c_1\rho^{-b_0i-b}{\mathcal E}_{\Phi(\hat I^{b_0i+b})}(f_j))^{1/2}\nonumber\\
&\le & \sum_{j=0}^i(c_1\rho^{-b_0i-b}c_0^{i-j}{\mathcal E}_{\Phi(\hat I^{b_0j+b})}(f_j))^{1/2}\nonumber\\
&\le& \sum_{j=0}^i(c_1\rho^{-b_0i-b}c_0^{i-j}{\mathcal E} (f_j))^{1/2},\label{eq:2.2}
\end{eqnarray}
where we apply Minkowski's inequality in the first inequality,
(\ref{eq:qnapp}) in the second inequality, and
Proposition \ref{theo:keyhin} in the
third inequality.
Applying (\ref{eq:2.2}) and noting that $\alpha^{d_w-d_f}=\rho$, we have
\begin{eqnarray*}
\sum_{i=0}^\infty \alpha^{(d_w-d_f)(b_0i+b)}E_{(b_0i+b)}(Q_{b_0i+b}(\tilde f|_L))
&\le &\sum_{i=0}^\infty\rho^{b_0i+b}
\left(\sum_{j=0}^i (c_1\rho^{-b_0i-b}c_0^{i-j}{\mathcal E} (f_j))^{1/2}\right)^2\\
&=&c_1\sum_{i=0}^\infty c_0^i\left(\sum_{j=0}^i (c_0^{-j}{\mathcal E} (f_j))^{1/2}\right)^2\\
&\le &c_2\sum_{j=0}^\infty c_0^jc_0^{-j}{\mathcal E} (f_j)=
c_2\sum_{j=0}^\infty{\mathcal E} (f_j)=c_2{\mathcal E} (f).
\end{eqnarray*}
Here we used (\ref{eq:harheld})
in the second inequality and (\ref{eq:orthd})
in the last equality. Thus, we have
\[\sum_{n=0}^\infty \alpha^{(d_w-d_f)n}E_{(n)}(Q_{n}(\tilde f|_L))\le b_0c_2{\mathcal E}(f).
\]
Combining this with Lemma \ref{theo:lem2} and (A8)', we have
$\| \tilde f|_L\|_{\Lambda^\beta_{2,2}(L)}\le c_3\|f\|_{\mathcal F}$, so that ${\mathcal F}|_L\subset \Lambda_{2,2}^\beta(L)$
and $({\mathcal F}\cap C(K))|_L\subset \Lambda_{2,2}^\beta(L)\cap C(L)$.
Noting that ${\mathcal F}\cap C(K)$ is dense in ${\mathcal F}$ due to the regularity of $({\mathcal E},{\mathcal F})$,
the claim follows by a simple limiting procedure.
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
\begin{rem} Even if (B4) does not hold,
${\mathcal F}|_L\subset \hat\Lambda^\beta_{2,\infty}(L)$ hold.
Indeed, for each $f\in{\mathcal F}$ and $n\in{\mathbb Z}_+$, we have
by Lemma~\ref{lem:qnapp},
\begin{eqnarray*}
c_1{\mathcal E}(f)\ge c_1{\mathcal E}_{\Phi(\hat I_n)}(f)
\ge \rho^{n} E_{(n)}(Q_n(\tilde f|_L))
= \alpha^{(2\beta-d)n} E_{(n)}(Q_n(\tilde f|_L)).
\end{eqnarray*}
Therefore, we have
\[
\left\|\left\{\alpha^{n\beta}\left(\alpha^{-nd}E_{(n)}(Q_n (\tilde f|_L))
\right)^{1/2}\right\}_{n=0}^\infty\right\|_{l^\infty}\le c_1^{1/2}{\mathcal E}(f)^{1/2},
\]
so the same argument as above gives the result.
\end{rem}
\subsection{Proof of the extension theorem}
In this section, we assume (A1)--(A8) and (C1)--(C2), and
prove Theorem~\ref{theo:mainthm2}.
The conditions (C1) and (C2) will be defined below.
In order to construct an extension map $\xi$, we first define a Whitney-type decomposition and
an associated partition of unity.
Let $\Omega^{(n)}=\bigcup_{m=0}^nI^m$ for $n\in{\mathbb Z}_+$.
For $w\in I^0=\{\emptyset\}$, set $A_w=W\setminus{\mathcal N}_2(I)$ and $B_w=W\setminus {\mathcal N}_1(I)$.
For $w\in I^n$ with $n\in{\mathbb N}$,
set $A_w=({\mathcal N}_2(w)\cdot W)\setminus {\mathcal N}_2(I^{n+1})$, $\hat A_w={\mathcal N}_2(w)\cdot W$, $B_w=({\mathcal N}_3(w)\cdot W)\setminus{\mathcal N}_1(I^{n+1})$, and $\hat B_w={\mathcal N}_3(w)\cdot W$.
Clearly
$K_{A_w}\subset K_{B_w}$, $K_{\hat A_w}\subset K_{\hat B_w}$, $K_{A_w}\cap K_{W^{|w|+1}\setminus B_w}=\emptyset$, and $K_{\hat A_w}\cap K_{W^{|w|+1}\setminus \hat B_w}=\emptyset$.
By (A3), we see the following for $w,w'\in\bigcup_{n\in{\mathbb Z}_+}I^n$:
\begin{eqnarray}
& c_1\alpha^{-|w|}\le \mathsf d(L,K_{B_w})\le c_2\alpha^{-|w|}\mbox{ if }B_w\ne\emptyset,\label{eq:whi1}\\
&\mbox{there exists $l>0$ such that
if $|w'|\ge|w|+l$ then $K_{B_w}\cap K_{\hat B_{w'}}= \emptyset$.}\label{eq:whi2}
\end{eqnarray}
For $n\in{\mathbb N}$ and $w\in\Omega^{(n)}$, we set
\[
A^{(n)}_w=\left\{\begin{array}{ll}A_w&\mbox{if }|w|<n\\
\hat A_w&\mbox{if }|w|=n\end{array}\right.,\quad
B^{(n)}_w=\left\{\begin{array}{ll}B_w&\mbox{if }|w|<n\\
\hat B_w&\mbox{if }|w|=n\end{array}\right.,
\]
and $R^{(n)}_w=\{w'\in\Omega^{(n)}: K_{B^{(n)}_w}\cap K_{B^{(n)}_{w'}}\ne \emptyset\}$.
We assume the following.
\begin{enumerate}
\item[(C1)] There exists a finite subset $\Gamma$ of $\bigcup_{n\in{\mathbb N}}(\{n\}\times\Omega^{(n)})$ such that, for any $n\in{\mathbb N}$ and $w\in\Omega^{(n)}$, there exist $(m,v)\in\Gamma$, a bijection $\iota\colon R^{(n)}_w\to R^{(m)}_v$,
and a homeomorphism $F\colon K_{\bigcup_{u\in R^{(n)}_w}B^{(n)}_u}\to K_{\bigcup_{u\in R^{(m)}_v}B^{(m)}_u}$ satisfying that
for every $u\in R^{(n)}_w$, $A^{(n)}_u$ and $A^{(m)}_{\iota(u)}$ are of the same type and so are $B^{(n)}_u$ and $B^{(m)}_{\iota(u)}$, for the homeomorphism $F$.
\end{enumerate}
For each $(m,v)\in \Gamma$, take a function $\bar\varphi^{(m)}_v\in {\mathcal F}\cap C(K)$ such that $0\le \bar\varphi^{(m)}_v\le1$, $\bar\varphi^{(m)}_v(x)=1$ on $K_{A^{(m)}_v}$, and $\bar\varphi^{(m)}_v(x)=0$ on $K_{W^{|v|+1}\setminus B^{(m)}_v}$.
Such a function exists since $({\mathcal E},{\mathcal F})$ is regular.
For $n\in{\mathbb N}$ and $w\in\Omega^{(n)}$, define
$\varphi^{(n)}_w(x)=\left\{\begin{array}{cl} \bar\varphi^{(m)}_v( F(x))& \mbox {if } x\in B^{(n)}_w\\
0 &\mbox{otherwise}\end{array}\right.$,
where $m$, $v$ and $F$ are given in (C1).
We assume
\begin{enumerate}
\item[(C2)] $\varphi^{(n)}_w\in{\mathcal F}\cap C(K)$ for every $n\in{\mathbb N}$ and $w\in\Omega^{(n)}$.
\end{enumerate}
For $n\in{\mathbb N}$ and $w\in\Omega^{(n)}$, define
\[
\psi^{(n)}_w(x)=\frac{\varphi^{(n)}_w(x)}{\sum_{w'\in\Omega^{(n)}}\varphi^{(n)}_{w'}(x)},
\quad x\in K.
\]
This is well-defined since the sum in the denominator
is not less than $1$.
$\psi^{(n)}_w$ is continuous and takes values between 0 and 1.
Since $\varphi^{(n)}_w\in{\mathcal F}$ and vanishes outside of $K_{B^{(n)}_w}$, so does $\psi^{(n)}_w$.
For each $f\in \Lambda^\beta_{2,2}(L)\cap C(L)$, define
\[
\xi^{(n)} f(x)=
\sum_{w\in\Omega^{(n)}}\psi^{(n)}_{w}(x)Q_{|w|}f(w)=\sum_{w\in\Omega^{(n)}}\psi^{(n)}_{w}(x){}-\!\!\!\!\!\!\!\int_{L_w}f(s)d\nu(s).
\]
$\xi^{(n)}$ is a linear map from $\Lambda^\beta_{2,2}(L)\cap C(L)$ to ${\mathcal F}\cap C(K)$.
For $x\in K\setminus L$, $\xi^{(n)}f(x)$ is independent of $n$ if $n$ is sufficiently large because of (\ref{eq:whi2}).
Therefore, for $f\in \Lambda^\beta_{2,2}(L)\cap C(L)$,
\begin{equation}\label{eq:xidef}
\xi f(x):=\left\{
\begin{array}{ll}\displaystyle\lim_{n\to\infty}\xi^{(n)}f(x),& x\in K\setminus L\\
f(x),& x\in L\end{array}
\right.
\end{equation}
is well-defined and $\xi^{(n)}f$ converges to $\xi f$ $\mu$-a.e.
\Proofwithname{Proof of Theorem~\ref{theo:mainthm2}.}
We first prove that $\xi f$ is continuous on $K$.
Since $\xi f$ is continuous on $K\setminus L$ by the construction, it is enough to show for each $x_0\in L$ that
\begin{equation}
\lim_{\begin{subarray}{c}{x\to x_0}\\{x\in K\setminus L}\end{subarray}}\xi f(x)=f(x_0).
\label{eq:need1}\end{equation}
Since $f$ is uniformly continuous on $L$, if we set
$\omega_a(f)=\sup\{|f(s)-f(t)|:s,t\in L,\ \mathsf d(s,t)\le a\}$
for $a>0$, then $\lim_{a\to0}\omega_a(f)=0$.
Let $x_0\in L$, $x\in K\setminus L$, and $\delta=\mathsf d(x,x_0)$.
Suppose that $w\in\bigcup_{n\in{\mathbb N}}I^n$ satisfies $x\in K_{B_w}$.
Then,
$c_1\alpha^{-|w|}\le \mathsf d(L,K_{B_w})\le \mathsf d(x_0,x)=\delta$ by $(\ref{eq:whi1})$.
Next, take $y\in L_w$ and choose $z\in
K_{B_w}$ that satisfies $\mathsf d(y,z)=\mathsf d(y,K_{B_w})\le c_2\alpha^{-|w|}$. Then, since
$\mathop{\rm diam\,}(K_{B_w})\asymp \alpha^{-|w|}$, we have
\[
\mathsf d(y,x_0)\le \mathsf d(y,z)+\mathsf d(z,x)+\mathsf d(x,x_0)
\le c_2\alpha^{-|w|}+c_3\alpha^{-|w|}+\delta\le c_4\delta.
\]
Therefore, ${}-\!\!\!\!\!\!\!\int_{L_w}|f(y)-f(x_0)|\,d\nu(y)\le \omega_{c_4 \delta}(f)$.
Now, take $n$ sufficiently large so that $x\notin\bigcup_{w\in I^n}K_{\hat B_w}$.
Then, $\xi^{(n)}f(x)=\xi f(x)$ and
\begin{eqnarray*}
|\xi f(x)-f(x_0)|
&=& \left|\sum_{w\in\Omega^{(n)}} \psi^{(n)}_{w}(x)
{}-\!\!\!\!\!\!\!\int_{L_w}(f(y)-f(x_0))\,d\nu (y)\right|\\
&\le&\sum_{w\in\Omega^{(n-1)},\ x\in K_{B_w}}\psi^{(n)}_{w}(x)
{}-\!\!\!\!\!\!\!\int_{L_w}|f(y)-f(x_0)|\,d\nu (y)\\
&\le& \omega_{c_4\delta}(f).
\end{eqnarray*}
Thus $(\ref{eq:need1})$ is proved.
Next, we will prove $\{\xi^{(n)}f\}_{n\in{\mathbb N}}$ is bounded in ${\mathcal F}$.
Noting that $\int_K \psi^{(n)}_w(x)\,d\mu(x)\le c_5 \alpha^{-d_f|w|}$ for all $n\in{\mathbb N}$ and $w\in\Omega^{(n)}$ for some $c_5>0$, we have
\begin{eqnarray}
\|\xi^{(n)} f\|_{L^2(K,\mu)}^2
&=&\int_K\left(\sum_{w\in \Omega^{(n)}}\psi^{(n)}_{w}(x){}-\!\!\!\!\!\!\!\int_{L_w}
f(s)\,d\nu (s)\right)^2d\mu (x)\nonumber\\
&\le &\int_K\left(\sum_{w\in \Omega^{(n)}}\psi^{(n)}_{w}(x){}-\!\!\!\!\!\!\!\int_{L_w}
f(s)^2\,d\nu (s)\right)d\mu (x)\nonumber\\
&\le &\sum_{w\in \Omega^{(n)}}c_5 \alpha^{-d_f |w|}\alpha^{d |w|}\int_{L_w}f(s)^2d\nu (s)\nonumber\\
&=& c_5 \sum_{k=0}^n \alpha^{(d-d_f)k}\|f\|^2_{L^2(L,\nu)}\nonumber\\
&\le& \frac{c_5}{1-\alpha^{d-d_f}}\|f\|^2_{L^2(L,\nu)}.
\label{eq:l2hk}\end{eqnarray}
For $n\in{\mathbb N}$, $w\in\Omega^{(n)}$ with $m=|w|$,
let $\bar R^{(n)}_w=\bigcup_{v\in R^{(n)}_w} v\cdot I^{m+l-|v|}\subset I^{m+l}$, where $l$ is provided in (\ref{eq:whi2}).
For $g\in L^2(L,\nu)$, we define
\[
E^{(n)}_w(g)=\sum_{\begin{subarray}{c}u,v\in \bar R^{(n)}_w\\u\stackrel{m+l,L}{\longleftrightarrow}v\end{subarray}}(Q_{m+l} g(u)-Q_{m+l}g(v))^2,\quad
\bar E^{(n)}_w(g)= {\mathcal E}_{\Phi(B^{(n)}_w)}\left(\sum_{v\in R^{(n)}_w}Q_{|v|}g(v)\psi^{(n)}_v\right).
\]
Both $E^{(n)}_w(g)$ and $\bar E^{(n)}_w(g)$ are determined only by the values $\{Q_{m+l}g(u)\}_{u\in \bar R^{(n)}_w}$.
If $E^{(n)}_w(g)=0$, then $Q_{m+l}g$ is constant on $\bar R^{(n)}_w$, which implies that $\bar E^{(n)}_w(g)=0$.
Therefore, there exists $c^{(n)}_w>0$
such that $\bar E^{(n)}_w(g)\le c^{(n)}_w E^{(n)}_w(g)$ for every $g\in{\mathcal F}$.
Due to (C1) and Lemma~\ref{lem:scale}, there exists some $c_6>0$ such that
$\bar E^{(n)}_w(g)\le c_6 \rho^{|w|} E^{(n)}_w(g)$ for all $n\in{\mathbb N}$, $w\in\Omega^{(n)}$ and $g\in{\mathcal F}$.
It also holds that there exists $c_7>0$
independent of $m$ such that
$\sum_{w\in I^m}E^{(n)}_w(g)\le c_7 E_{(m+l)}(Q_{m+l}g)$ for
all $n$ and $g\in L^2(L,\nu)$.
Then, we have
\begin{eqnarray*}
{\mathcal E}(\xi^{(n)}f)
&\le& \sum_{w\in\Omega^{(n)}}{\mathcal E}_{\Phi(B^{(n)}_w)}(\xi^{(n)}f)
= \sum_{w\in\Omega^{(n)}}\bar E^{(n)}_w(f)\\
&\le& c_6\sum_{m=0}^n\sum_{w\in I^m}\rho^m E^{(n)}_w(f)
\le c_6 c_7\sum_{m=0}^n\rho^m E_{(m+l)}(Q_{m+l}f)\\
&\le& c_8\sum_{m=0}^{\infty}\rho^m E_{(m)}(Q_mf).
\end{eqnarray*}
Since $\alpha^{2\beta-d}=\alpha^{d_w-d_f}=\rho$, we obtain ${\mathcal E}(\xi^{(n)}f)\le c_8 \|f\|_{\Lambda^\beta_{2,2}(L)}^2$ by Lemma \ref{theo:lem2}.
By combining this with (\ref{eq:l2hk}), $\{\xi^{(n)}f\}_{n\in{\mathbb N}}$ is bounded in ${\mathcal F}$ and we conclude that $\xi f\in{\mathcal F}$ and
$\|\xi f\|_{\mathcal F}\le c_9\|f\|_{\Lambda^\beta_{2,2}(L)}$ for some $c_9>0$.
Next,
take any $\Lambda^\beta_{2,2}(L)$-Cauchy sequence
$\{f_n\}_{n\in{\mathbb N}}\subset \Lambda^\beta_{2,2}(L)\cap C(L)$ and let $f\in \Lambda^\beta_{2,2}(L)$
be the limit point. By the above result, $\{\xi f_n\}_{n\in{\mathbb N}}\subset {\mathcal F}\cap C(K)$
is a ${\mathcal E}_1$-Cauchy sequence. Let $g\in {\mathcal F}$ be the limit point.
Since $\xi f_n|_L=f_n$
and a subsequence $\xi f_{n_k}$ converges to $\tilde g$ q.e.,
$\tilde g|_L=f$ $\nu$-a.e. Thus, $\xi$ can extend to a continuous map from $\hat\Lambda^\beta_{2,2}(L)$ to ${\mathcal F}$ such that $\widetilde{\xi f}|_L=f$ $\nu$-a.e.\ for
$f\in \hat\Lambda^\beta_{2,2}(L)$.
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
\begin{rem}\label{thm:finunil}
Let $\{L_i\}_{i=1}^m$ be a finite number of self-similar subsets of $K$ where each $L_i$
is constructed by the same number of contraction maps and satisfies (A2),
the second identity of (A4),
(A7), and (A8) in Section~2. Let $L=\bigcup_{i=1}^mL_i$.
With suitable changes for $A_w$, $B_w$ etc., we
can consider conditions
(C1)$^*$--(C2)$^*$ as the corresponding
(C1)--(C2).
Define $\Lambda^\beta_{2,2}(L)$ as in Definition~\ref{defn:Besov}.
Then, under such conditions,
Theorem~\ref{theo:mainthm2}
is still valid, i.e.\ there is a linear map $\xi$ from $\Lambda^\beta_{2,2}(L)$
to ${\mathcal F}$ such that $\xi\bigl(\Lambda^\beta_{2,2}(L)
\cap C(L)\bigr)\subset {\mathcal F}\cap C(K)$, $\widetilde{\xi f}|_L=f$ and
\[
\|\xi f\|_{\mathcal F}\le c_1\sum_{i=1}^m\| f|_{L_i}\|_{\Lambda^\beta_{2,2}(L_i)},
\quad f\in\Lambda^\beta_{2,2}(L).
\]
\end{rem}
\section{Complementary results}
In this section, we give sufficient conditions concerning (A8) and (B3),
and discuss a suitable choice of ${\mathcal F}_A$ for $A\subset W^m$.
We first define fractional diffusions in the sense of \cite{bar} Definition 3.5.
\begin{defn}\label{fradif}
Let $(X,\mathsf d)$ be a complete metric space where $\mathsf d$ has the midpoint property; for each
$x,y\in X$, there exists $z\in X$ such that $\mathsf d(x,y)=\mathsf d(x,z)/2=\mathsf d(z,y)/2$.
For simplicity, we assume $\mbox{diam}\,X=1$.
Let $\mu$ be a Borel measure on $X$ such that there exists $d_f>0$ with $\mu (B(x,r))\asymp r^{d_f}$
for all $0<r\le 1$. A Markov process $\{Y_t\}_{t\ge 0}$ is a fractional diffusion on $X$ if\\
1) $Y$ is a $\mu$-symmetric conservative Feller diffusion,\\
2) $Y$ has a symmetric jointly continuous transition density
$p_t(x,y)~(t>0, x,y\in X)$ which satisfies the Chapman-Kolmogorov equations and
has the following estimate,
\begin{eqnarray*}
&&c_{1}t^{-d_f/d_w}\exp (-c_{2}(\mathsf d(x,y)^{d_w}t^{-1})^{1/(d_w-1)})
\le p_t(x,y)\\
&\le & c_{3}t^{-d_f/d_w}\exp (-c_{4}(\mathsf d(x,y)^{d_w}t^{-1})^{1/(d_w-1)})
~~\qquad\qquad\mbox{ for all}~~0<t<1,~x,y\in X,
\end{eqnarray*}
with some constant $d_w\ge 2$.
\end{defn}
\begin{pro}\label{thm:pr4.2}
\rom{(A8)} holds for the following
three cases.\\
\rom{1)} There exists $c>0$ such that $\|f\|_{L^\infty(K)}\le c\|f\|_{\mathcal F}$ for all $f\in{\mathcal F}$.\\
\rom{2)} The diffusion process corresponding to $({\mathcal E},{\mathcal F})$
is the fractional diffusion and \rom{(A7)} holds.\\
\rom{3)} $K\subset {\mathbb R}^n$,
\rom{(A7)} holds, and ${\mathcal F}=\Lambda^{d_w/2}_{2,\infty}(K)$.
\end{pro}
\begin{demo}{{\it Proof.\ }}
Suppose that 1) holds.
Then, for any nonempty set $D$ of $K$, $\mathop{\rm Cap}(D)\ge c^{-2}$.
Therefore, $\nu(D)\le 1\le c^2\mathop{\rm Cap}(D)$.
The proof when 2) holds is
similar to Lemma 2.5 of \cite{bk},
but we will give it for completeness.
Let $g_1(\cdot,\cdot)$ be the $1$-order
Green density given by
\[E^x\left[\int_0^{\infty}e^{- t}f(X_t)dt\right]
=\int_K g_1(x,y)f(y)d\mu,\]
for any Borel measurable function $f$,
where $\{X_t\}$ is the diffusion corresponding to $({\mathcal E}, {\mathcal F})$. Then,
since $\{X_t\}$ is a fractional diffusion, we have
\begin{eqnarray}
g_1(x,y)&\asymp & \left\{ \begin{array}{ll} c_{1}\mathsf d(x,y)^{d_w-d_f}
&~~\mbox{if}~d_f> d_w,\label{eq:green}\\
-c_{2}\log \mathsf d(x,y)+c_{3} &~~\mbox{if}~d_f=d_w,
\\ c_{4} &~~\mbox{if}~d_f<d_w.
\end{array}
\right.\end{eqnarray}
See \cite{bar} Proposition 3.28 for the proof.
If $d_f<d_w$ then points have strictly positive capacity, and the result is
immediate. We prove the result for $d_f>d_w$: the proof for $d_f=d_w$
is similar.
It is well-known that for each compact set $M\subset K$,
\begin{equation}
\mathop{\rm Cap}(M)=\sup \left\{m (M):\begin{array}{l} m\mbox{ is a positive Radon measure, }
\mathop{\rm supp} m\subset M,\\
G_1 m(x)\equiv \int_M g_1(x,y) m(dy)\le 1\mbox{ for every } x\in K
\end{array}
\right\}.
\label{eq:capca}
\end{equation}
Using the above estimates of $g_1(\cdot,\cdot)$,
\begin{eqnarray*}
\int_M g_1(x,y) \nu (dy) &\le &
\int_K g_1(x,y) \nu (dy)
\le \sum_{n=0}^{\infty}\int_{\alpha^{-n-1}
\le \mathsf d(x,y) <\alpha^{-n}} g_1(x,y)\nu (dy)\\
&\le &c_5\sum_n \alpha^{n(d_f-d_w)}\nu ( \alpha^{-n-1}\le \mathsf d(x,y) <\alpha^{-n} )
\le c_6\sum_{n} \alpha^{n(d_f-d_w-d)} \equiv c_7<\infty,
\end{eqnarray*}
because of the assumption $d_f-d<d_w$.
Thus, setting $\nu_M (\cdot)\equiv \nu (\cdot \cap M)$,
we have $G_1 \nu_M\le c_7$. Using $(\ref{eq:capca})$, $\mbox {Cap}(M)\ge \nu
(M)/c_7$ for each compact set $M$.
For
3), we will use the results by Jonsson-Wallin in \cite{jw} and by Triebel in
\cite{tr}. Denote the Lipschitz and the
Besov spaces in the sense of Jonsson-Wallin by
$\mbox{Lip}_{JW} (\alpha, p,q, K)$ and $B_{\alpha, JW}^{p,q}(K)$
(see page 122--123 in \cite{jw} for definition). Note that
$\mbox{Lip}_{JW} (\alpha, p,q, K)\subset B_{\alpha, JW}^{p,q}(K)$
and they are equal when $\alpha\notin {\mathbb N}$ (page 125 in \cite{jw}).
For each $f\in \Lambda^{d_w/2}_{2,\infty}(K)$,
$(f,0,\ldots,0)\in \mbox{Lip}_{JW} (d_w/2, 2,\infty, K)$. Thus, using
the extension theorem in page 155 of \cite{jw}, we have
\[ \Lambda^{d_w/2}_{2,\infty}(K)\subset \mbox{Lip}_{JW} (d_w/2, 2,\infty, K)
\subset B_{d_w/2, JW}^{2,2}(K)\subset \Lambda_\gamma^{2,\infty}({\mathbb R}^n)|_K,
\]
where $\gamma=(d_w+n-d_f)/2$ and $\Lambda_\gamma^{p,q}({\mathbb R}^n)$ is a classical Besov space
on ${\mathbb R}^n$. Now, since $d_w-d_f>-d$ (due to (A7)), $\Lambda_\gamma^{2,\infty}({\mathbb R}^n)
\subset \Lambda_{(n-d)/2}^{2,1}({\mathbb R}^n)$. Finally, by Corollary 18.12 (i) in \cite{tr},
we have $\mbox{tr}_L\;\Lambda_{(n-d)/2}^{2,1}({\mathbb R}^n)=L^2(L,\nu)$. (Note that this trace
in the sense of Triebel is simply restriction and there is no corresponding extension.)
Combining these facts, we have $ {\mathcal F}|_L\subset L^2(L,\nu)$, which means
$\|\tilde f|_L\|_{L^2(L,\nu)}\le c_9\|f\|_{\mathcal F}$ for all $f\in {\mathcal F}$.
Therefore, (A8)' holds.
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
We now make one concrete choice of ${\mathcal F}_A$ for $A\subset W^m$
and show that such a choice is suitable for Dirichlet forms whose corresponding
processes are the fractional diffusions.
By (A5), (A6), and the self-similarity of $\mu$, for any $w\in\bigcup_{n\in{\mathbb Z}_+}W^n$, there exists $c>0$ such that $\mathop{\rm Cap}(D)\le c\mathop{\rm Cap}(F_w(D))$ for any $D\subset K$.
We assume the converse as follows.
\begin{enumerate}
\item[(A*)]
For any $w\in\bigcup_{n\in{\mathbb Z}_+}W^n$, there exists $c>0$ such that $\mathop{\rm Cap}(F_w(D))\le c\mathop{\rm Cap}(D)$ for any $D\subset K$.
\end{enumerate}
For a subset $A$ of $W^m$ for some $m\in{\mathbb N}$,
we say that
a collection $\{f_w\}_{w\in A}$ of functions in ${\mathcal F}$
is compatible if $\tilde f_v(F_v^{-1}(x))=\tilde f_w(F_w^{-1}(x))$ q.e.\ on $K_v\cap K_w$ for every $v,w\in A$.
Note that this is well-defined by (A*). Define
\begin{equation}\label{eq:cfadef}
{\mathcal F}_A=\{f\in L^2(K_A,\mu|_{K_A}): F_w^* f\in{\mathcal F} \mbox{ for all }w\in A \mbox{ and $\{F_w^* f\}_{w\in A}$ is compatible}\}.
\end{equation}
If we equip $A$ with a graph structure so that $v\in A$ and $w\in A$ are connected if $\mathop{\rm Cap}(K_v\cap K_w)>0$, then $A$ is ${\mathcal E}$-connected when $A$ is a connected graph.
This is verified by using (B1).
\begin{lem}\label{lem:good}
For $A\subset W^m$ with $m\in{\mathbb Z}_+$,
$({\mathcal E}_A,{\mathcal F}_A)$ is a strong local Dirichlet form on
$L^2(K_A,\mu|_{K_A})$.
\end{lem}
\begin{demo}{{\it Proof.\ }}
Let $\{f_n\}_{n\in{\mathbb N}}\subset {\mathcal F}_A$ be a Cauchy sequence in ${\mathcal F}_A$.
Let $g$ be the limit in $L^2(K_A)$.
Let $w\in A$. Since $\{F_w^* f_n\}_{n\in{\mathbb N}}$ is a Cauchy sequence in ${\mathcal F}$, $F_w^* f_n\to\ F_w^* g$ in ${\mathcal F}$.
It is also easily deduced that
$\{F_w^* g\}_{w\in A}$
is compatible. Therefore, $g\in {\mathcal F}_A$ and $f_n\to g$ in ${\mathcal F}_A$.
This implies that $({\mathcal E}_A,{\mathcal F}_A)$ is a closed form on $L^2(K_A)$.
The Markov property and the strong locality are inherited from those of $({\mathcal E},{\mathcal F})$ via relation (\ref{eq:energy}).
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
\begin{cor}\label{cor:a*co}
Assume case \rom{1)} or \rom{2)} in Proposition~\ref{thm:pr4.2}.
Then, \rom{(A*)} holds.
\end{cor}
\begin{demo}{{\it Proof.\ }}
In case~1), non-empty sets have uniform positive capacities, which implies (A*).
In case~2), (A*) is an easy consequence of (\ref{eq:green}) and (\ref{eq:capca}).
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
In the rest of this section, we will discuss
sufficient conditions for \rom{(B3)}.
\begin{lem}\label{lem:Dw}
Let $A\subset W_m$, $m\in{\mathbb Z}_+$.
Then, ${\mathcal F}_A$ is compactly imbedded in
$L^2(K_A,\mu|_{K_A})$.
Suppose that $A$ is ${\mathcal E}_A$-connected.
Then, when we set ${\mathcal A}=\{f\in{\mathcal F}_A: \int_{K_A}f\,d\mu=0,\ {\mathcal E}_A(f)\le C\}$ for a constant $C>0$, ${\mathcal A}$ is bounded in ${\mathcal F}_A$.
\end{lem}
\begin{demo}{{\it Proof.\ }}
Let ${\mathcal B}$ be a bounded subset of ${\mathcal F}_A$.
For each $v\in A$, $\{F^*_v f: f\in{\mathcal B}\}$ is bounded in ${\mathcal F}$.
By (B1), we can take a sequence $\{f_n\}_{n\in{\mathbb N}}$ from ${\mathcal B}$ such that $F^*_v f_n$ converges in
$L^2(K)$.
Therefore, we can take a sequence from ${\mathcal B}$ converging in $L^2(K_A)$.
This implies the first assertion.
By combining this with the ${\mathcal E}_A$-connectedness of $K_A$, there exists $c>0$ such that $\|f-{}-\nobreak\!\!\!\!\!
\int_{K_A}f\,d\mu\|_{L^2(K_A)}^2 \le c{\mathcal E}_A(f)$ for every $f\in{\mathcal F}_A$.
The latter assertion follows from this immediately.
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
We now give a sufficient condition for \rom{(B3)~(2)}.
\begin{pro}\label{prop:ehi}
The following condition \rom{(EHI1)} implies \rom{(B3)~(2)}.
\begin{enumerate}
\item[\rom{(EHI1)}] For any $v\in \Xi$, there exist some $c_1>0$,
subsets $D''(v)$ and $D'''(v)$ of $D^\sharp(v)$ such that
$D'''(v)\subset D''(v)\subset D^\sharp(v)$,
$K_{D''(v)}\cap K_{S^{|v|}\setminus D^\sharp(v)}=\emptyset$,
$K_v\cap K_{S^{|v|}\setminus D'''(v)}=\emptyset$ and
$\mathop{\rm esssup}_{x\in K_{D'''(v)}} h(x) \le c_1\mathop{\rm essinf}_{x\in K_{D'''(v)}} h(x)$ for every $h\in {\mathcal H}(D^\sharp(v))$ with $h\ge0$ $\mu$-a.e.
\end{enumerate}
\end{pro}
\begin{demo}{{\it Proof.\ }}
First, we apply Lemma~\ref{lem:harmext} to $g\in{\mathcal F}$ with $A=D^\sharp(v)$ and
$J=\emptyset$, and denote $g'$ there by $Hg$.
We will follow the proof of Theorem~2.2 in \cite{hin}.
For $h\in {\mathcal H}(D^\sharp(v))$ with $h\ge0$ $\mu$-a.e., we have, by (EHI1),
\[
\mathop{\rm esssup}_{x\in K_{D'''(v)}} h(x) \le c_1\mathop{\rm essinf}_{x\in K_{D'''(v)}} h(x)
\le c_2\| h\|_{L^2(K_{D'''(v)})}.
\]
For $h\in {\mathcal H}(D^\sharp(v))$, let $h_+(x)=\max\{h(x),0\}$ and $h_-(x)=\max\{-h(x),0\}$.
Since $h=Hh=Hh_+-Hh_-$ and $Hh_\pm\ge0$ $\mu$-a.e., we have
\begin{eqnarray}
\mathop{\rm esssup}_{x\in K_{D'''(v)}} |h(x)|
&\le&\mathop{\rm esssup}_{x\in K_{D'''(v)}} Hh_+(x)+\mathop{\rm esssup}_{x\in K_{D'''(v)}} Hh_-(x)\nonumber\\
&\le& c_2(\|Hh_+\|_{L^2(K_{D'''(v)})}+\|Hh_-\|_{L^2(K_{D'''(v)})})\nonumber\\
&\le& c_3(\|h_+\|_{{\mathcal F}_{D''(v)}}+\|h_-\|_{{\mathcal F}_{D''(v)}})\nonumber\\
&\le&2c_3\|h\|_{{\mathcal F}_{D''(v)}}.
\label{eq:dominationw}
\end{eqnarray}
In order to prove (B3) (2), it suffices to prove the following.
\begin{itemize}
\item[$(*)$] If a sequence $\{h_l\}$ in ${\mathcal H}(D^\sharp(v))$ converges weakly to 0 in ${\mathcal F}_{D^\sharp(v)}$, then there exists a subsequence $\{h_{l(k)}\}$ such that $F_v^* h_{l(k)}$ converges strongly to $0$ in ${\mathcal F}$.
\end{itemize}
Indeed, suppose $(*)$ holds.
Let $\{f_m\}$ be a sequence in ${\mathcal H}(D^\sharp(v))$ that is bounded in ${\mathcal F}_{D^\sharp(v)}$.
We can take a subsequence $\{f_{m(l)}\}$ and $f\in{\mathcal F}_{D^\sharp(v)}$ such that $f_{m(l)}$ converges weakly to $f$ in ${\mathcal F}_{D^\sharp(v)}$.
Take $g_l\in{\mathcal H}(D^\sharp(v))$ such that $g_l\to f$ in ${\mathcal F}_{D^\sharp(v)}$.
Applying $(*)$ to $h_l:=f_{m(l)}-g_l$, we can take a sequence $\{l(k)\}$ diverging to $\infty$ such that
$F^*_v f_{m(l(k))}\to F^*_v f$ in ${\mathcal F}$.
This implies (B3)~(2).
In order to prove $(*)$, recall the notion of the energy measure. For $f\in {\mathcal F}\cap L^\infty(K)$, the energy measure $\mu_{\langle f\rangle}$ is a unique positive Radon measure on $K$ such that the following identity holds for every $g\in{\mathcal F}\cap C(K)$:
\[
\int_{K} g\,d\mu_{\langle f\rangle}=2{\mathcal E}(f,fg)-{\mathcal E}(f^2,g).
\]
Now, by (\ref{eq:dominationw}), $C:=\mathop{\rm esssup}_{x\in K_{D'''(v)}} |h_l(x)|$ is bounded in $l$.
Define $\hat h_{l}=(-C)\vee h_l \wedge C$.
Since ${\mathcal F}_{D^\sharp(v)}$
is compactly imbedded in $L^2(K_{D^\sharp(v)})$ by Lemma~\ref{lem:Dw}, $\{h_l\}$ converges to $0$ in $L^2(K_{D^\sharp(v)})$.
Take a subsequence $\{h_{l'}\}$ converging to $0$ $\mu$-a.e.\ on $K_{D^\sharp(v)}$.
Since $({\mathcal E},{\mathcal F})$ is regular, we can take $\varphi\in{\mathcal F}\cap C(K)$ such that $0\le\varphi\le 1$ on $K$,
$\varphi=1$ on $K_w$, and $\varphi=0$ outside $K_{D'''(v)}$.
We have
\[
0=2{\mathcal E}(h_{l'},\hat h_{l'}\varphi)
=2{\mathcal E}(\hat h_{l'},\hat h_{l'}\varphi)
={\mathcal E}(\hat h_{l'}^2,\varphi)+\int_K\varphi\,d\mu_{\langle \hat h_{l'}\rangle},
\]
because $\hat h_{l'}\varphi$ vanishes outside $K_{D'''(v)}$.
Note that ${\mathcal E}(\hat h_{l'}^2)\le 4C^2 {\mathcal E}(h_{l'})$, which is
bounded in $l'$. A suitable subsequence $\hat h_{l''}$ can be taken so that $\{\hat h_{l''}^2\}$ converges weakly to some $g$ in
${\mathcal F}$.
Since $g=0$ on $K_{D^\sharp(v)}$, ${\mathcal E}(\hat h_{l''}^2,\varphi)\to{\mathcal E}(g,\varphi)=0$ as $l''\to\infty$.
On the other hand,
\begin{eqnarray*}
\int_{K}\varphi\,d\mu_{\langle \hat h_{l''}\rangle}
&=&\sum_{z\in S^{|v|}} \rho^{|v|} \int_K F^*_z \varphi\,d\mu_{\langle F^*_z \hat h_{l''}\rangle}\\
&\ge& \rho^{|v|} \int_K F^*_v \varphi\,d\mu_{\langle F^*_v \hat h_{l''}\rangle}\\
&=& \rho^{|v|} \mu_{\langle F^*_v \hat h_{l''}\rangle}(K)
= 2\rho^{|v|} {\mathcal E}(F^*_v \hat h_{l''})
= 2\rho^{|v|} {\mathcal E}(F^*_v h_{l''}).
\end{eqnarray*}
Combining these estimates, we obtain
$\varlimsup_{l''\to\infty}{\mathcal E}(F^*_v h_{l''})\le 0$.
Therefore, $F^*_v h_{l''}$ converges to $0$ in ${\mathcal F}$.
This proves $(*)$.
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
We next give sufficient conditions for \rom{(B3) (1) (b)}.
\begin{pro}\label{prop:wv}
The following conditions imply \rom{(B3) (1) (b)}.
\begin{enumerate}
\item[$(1)$] ${\mathcal F}=\Lambda^\beta_{2,\infty}(K)$ for some $\beta>0$.
\item[$(2)$] For each $v\in \Xi$, $D'(v)$ is ${\mathcal E}_{D'(v)}$-connected.
\item[$(3)$] For each $w\in\bigcup_{n\in{\mathbb Z}_+}\hat I^{n+m_0}$, there exist
subsets $D^\sharp(w)$, $D^{(1)}(w)$, $D^{(2)}(w)$ of $D'(w)$ such that $D^\sharp(w)\subset D^{(1)}(w)\subset D^{(2)}(w)\subset D'(w)$ and the following hold.
\begin{enumerate}
\item[\rom{(a)}] There exists $v\in \Xi$ such that both $D'(w)$ and $D'(v)$, and $D^\sharp(w)$ and $D^\sharp(v)$, are of the same type by the same map $F$.
\item[\rom{(b)}] $K_{D^\sharp(w)}\cap K_{S^{|w|}\setminus D^{(1)}(w)}=K_{D^{(2)}(w)}\cap K_{S^{|w|}\setminus D'(w)}=\emptyset$.
\item[\rom{(EHI2)}] There exists $c>0$ such that $\mathop{\rm esssup}_{x\in K_{D^{(1)}(w)}} h(x) \le c\mathop{\rm essinf}_{x\in K_{D^{(1)}(w)}} h(x)$ for $h\in {\mathcal H}(D^{(2)}(w))$ with $h\ge0$ $\mu$-a.e.
\end{enumerate}
\end{enumerate}
\end{pro}
\begin{demo}{{\it Proof.\ }}
Let $g$ be a function in ${\mathcal F}$ such that $\int_{K_{D'(w)}}g\,d\mu=0$ and $\rho^{|w|}{\mathcal E}_{D'(w)}(g)\le 1$. Let $f(x)=g(F^{-1}(x))$, $x\in K_{D'(v)}$.
Then, $f\in {\mathcal F}_{D'(v)}$, $\int_{K_{D'(v)}}f\,d\mu=0$ and $\rho^{|v|}{\mathcal E}_{D'(v)}(f)\le 1$.
By Lemma~\ref{lem:Dw}, $\|f\|_{{\mathcal F}(D'(v))}\le C$, where $C$ is a constant independent of $w$.
Suppose moreover that $g\in{\mathcal H}(I^{|w|},D'(w))$.
Apply Lemma~\ref{lem:harmext} to $g$ with $A=D^{(2)}(w)$ and $J=\emptyset$ and denote $g'$ there by $g_1$.
Let $g_2=g-g_1$.
By (EHI2) and the same argument in the first part of the proof of Proposition~\ref{prop:ehi}, $g_1$ is bounded on $D^{(1)}(w)$.
Take a function $\psi\in{\mathcal F}$ such that $0\le\psi\le 1$, $\psi=0$ on $K_{S^{|w|}\setminus D^{(1)}(w)}$ and $\psi=1$ on $K_{D^\sharp(w)}$.
Then, $g_1\psi\in{\mathcal F}$.
Since both $g_1\psi$ and $g_2$ vanish on $K_{S^{|w|}\setminus D^{(2)}(w)}$, when we set
$f'(x)=\left\{\begin{array}{ll}(g_1\psi+g_2)(F^{-1}(x)),&x\in K_{D'(v)}\\0,&x\in K\setminus K_{D'(v)}\end{array}\right.$, $f'$ belongs to ${\mathcal F}$ by using the fact ${\mathcal F}=\Lambda^\beta_{2,\infty}(K)$.
Since $f'=f$ on $K_{D^\sharp(v)}$, we have $f'\in{\mathcal H}(I^{|v|},D^\sharp(v))$ and $\|f'\|_{{\mathcal F}_v}=\|f\|_{{\mathcal F}_v}\le C$.
These conclude the assertion.
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
\section{Examples}
In this section, we choose ${\mathcal F}_A$ as in (\ref{eq:cfadef}) for $A\subset W^m$.\\
\noindent
1) Sierpinski gaskets: Let $\{a_0,a_1,\ldots,a_n\}\subset {\mathbb R}^n$ be the vertices
of $n$-dimensional simplex.
Let $W=S=\{0,1,\ldots,n\}$ and let $F_i(x)=(x-a_i)/2+a_i$ for $x\in {\mathbb R}^n$ and
$i=0,1,\ldots ,n$.
Then the unique non-void compact set $K$ which satisfies $K=\bigcup_{i=0}^n F_i(K)$
is the $n$-dimensional Sierpinski gasket.
The map $\Phi$ in Lemma~\ref{lem:Phi} is the identity map.
It is well-known (see \cite{bar,BP,kig} etc.)
that there is a self-similar Dirichlet form $({\mathcal E},{\mathcal F})$ on $L^2(K,\mu)$
where the corresponding diffusion is the fractional diffusion.
In particular, ${\mathcal F}=\Lambda_{2,\infty}^{d_w/2} (K)\subset C(K)$ where $d_w=(\log (n+3))/(\log 2)$.
Note that $d_w-d_f>0$ in this case.
Let $L$ be the $(n-1)$-dimensional gasket determined by $\{F_i\}_{i=0}^{n-1}$.
That is, $I=\{0,1,\ldots,n-1\}$. Let $\hat I=I$, $M=1$.
It is easy to see that
${\mathcal F}_A={\mathcal F}|_{K_A}$ for each $A\subset W^m$.
Then (A1)--(A7), (B2), and (C1)--(C2)
are easy to check with $\rho=(n+3)/(n+1)$. (A8) holds by Proposition \ref{thm:pr4.2} and
(B1) holds by \cite{kig} Lemma 3.4.5.
For (B3), define $l_0=0$, $m_0=0$, $D'(w)=\{w\}$ for $w\in\bigcup_{n\in{\mathbb Z}_+}I^n$, $\Xi=\{\emptyset\}$, and $D^\sharp(\emptyset)=\{\emptyset\}$.
It is easy to check (B3)~(1) by using Lemma~\ref{lem:scale} and Lemma~\ref{lem:Dw}.
Since ${\mathcal H}_{D'(w)}(D'(w))$ is a finite dimensional space,
(B3)~(2) is clearly true.
We will prove (B4).
Let $f\in{\mathcal F}$ and ${\mathcal E}_{S^m\setminus I^m}(f)=0$ for some $m\in{\mathbb Z}_+$.
Then, for each $w\in S^m\setminus I^m$,
${\mathcal E}(F^*_w f)=\rho^{-m}{\mathcal E}_w(f)=0$.
Therefore, $f$ is constant on $K_w$ for each $w\in S^m\setminus I^m$.
We consider an unoriented graph with a vertex set $V=S^m\setminus I^m$ and an edge set $\{(v,w)\in V\times V: \mathop{\rm Cap}(K_v\cap K_w)>0\}$.
Then, $V$ is a connected set. Note that $(v,w)$ is an edge if and only if $K_v\cap K_w\ne\emptyset$.
Therefore, $f$ should be constant on $K_{S^m\setminus I^m}$, thus constant on $K\setminus L$.
This concludes that (B4) holds.
Therefore, we have by Theorem~\ref{theo:mainthm1}, Theorem~\ref{theo:mainthm2} and Remark~\ref{rem:2.5},
\[{\mathcal F}|_L=\Lambda_{2,2}^\beta (L)\qquad\mbox{where }~~\beta=\frac{d_w}2-
\frac{\log (1+1/n)}{2\log 2}.\]
When $n=2$, this relation was obtained in \cite{jo}.
\medskip
\noindent 2) Pentakun:
Let $a_k=e^{2k\sqrt{-1}\pi/5+\sqrt{-1}\pi/2}\in{\mathbb C}$, $k=0,1,2,3,4$.
Let $W=\{0,1,2,3,4,5,6\}$, $S=\{0,1,2,3,4\}$, $I=\hat I=\{2,3,5,6\}$ and $M=1$.
Let ${\mathfrak G}=\{G_k\}_{k=0}^4$ with $G_k:{\mathbb C}\to{\mathbb C}$ defined by $G_k(z)=e^{2k\sqrt{-1}\pi/5} z$.
For $i=0,1,2,3,4$, define a contraction map $F_i:{\mathbb C}\to{\mathbb C}$ by
$F_i(z)=\alpha^{-1}(z-a_i)+a_i$, where $\alpha=\frac{3+\sqrt5}{2}$.
We also define $F_5=F_2\circ G_1$ and $F_6=F_3\circ G_4$.
Then, the resulted nested fractal $K$ is called Pentakun and a subset $L$ is a Koch curve (see Figure~1).
The Hausdorff dimensions of $K$ and $L$ are $(\log5)/(\log \alpha)$ and $(\log4)/(\log \alpha)$,
respectively. There exists a canonical Dirichlet form $({\mathcal E},{\mathcal F})$ on $L^2(K,\mu)$
where the corresponding diffusion is the fractional diffusion (see \cite{bar,kum0,lind} etc.),
so ${\mathcal F}=\Lambda_{2,\infty}^{d_w/2}
(K)$. It is known (see \cite{kum0}) that $d_w=(\log \frac{\sqrt {161}+9}{2})/(\log \alpha)$
and we can check all the assumptions similarly to the case of the Sierpinski gasket.
Note that $C_2$ given in (B2)~(4) is equal to 4.
Thus, by Theorem~\ref{theo:mainthm1}, Theorem~\ref{theo:mainthm2} and Remark~\ref{rem:2.5},
\[
{\mathcal F}|_L=\Lambda_{2,2}^\beta (L)\qquad\mbox{where }~~\beta=\frac{d_w}{2}-\frac{\log 5-\log 4}{2\log \alpha}.
\]
For the Pentakun $K$, let $I'=\{2,3\}$. Then the corresponding self-similar subset $L'$ is a Cantor set
with Hausdorff dimension $(\log 2)/(\log \alpha)$. In this case we should set
$\hat I=\{2,3,5,6\}$, so $I\ne \hat I$.
Again we can check all the assumptions similarly, so
by Theorem~\ref{theo:mainthm1}, Theorem~\ref{theo:mainthm2} and Remark~\ref{rem:2.5},
\[
{\mathcal F}|_{L'}=\Lambda_{2,2}^{\beta'} (L')\qquad\mbox{where }~~\beta'=\frac{d_w}{2}-\frac{\log 5-\log 2}{2\log
\alpha}.
\]
In general, if $K$ is a nested fractal satisfying (A3), then
there is a canonical Dirichlet form on $L^2(K,\mu)$ where the corresponding diffusion is
the fractional diffusion (see \cite{bar,kum0,lind} etc.).
Let $L$ be a self-similar subset of $K$ given in the manner in the first part of Section~2,
and satisfying (A2).
In most cases, all the assumptions
except (B4) can be checked similarly to the case of
the Sierpinski gasket, so that we can use Theorem~\ref{theo:mainthm1} and Theorem~\ref{theo:mainthm2} to
characterize the trace space
if (B4)
holds. However, there are cases where (B4) does not hold -- see 4).
\bigskip
\noindent 3) Sierpinski carpets:
Let $H_0=[0,1]^n$,
$n\ge 2$,
and let $l \in {\mathbb N}$, $l\geq 2$ be fixed.
Set ${\cal Q}=\{\Pi_{i=1}^n [(k_i-1)/l,k_i/l]: 1\le k_i\le l,~k_i\in {\mathbb N}~
(1\le i\le n)\}$, let $N \le l^n$ and $W=S=\{1,\ldots,N\}$.
Let $F_i$, $i\in S$
be orientation preserving affine maps of
$H_0$ onto some element of ${\cal Q}$.
We assume that
the sets $F_i(H_0)$ are distinct.
Set $H_1=\bigcup_{i\in I}F_i (H_0)$.
Then the unique non-void compact set $K$ which satisfies $K=\bigcup_{i=1}^N F_i(K)$
is called the generalized Sierpinski carpet if the following holds:
\begin{enumerate}
\item[(SC1)] (Symmetry) $H_1$ is preserved by all the isometries of the unit cube $H_0$.
\item[(SC2)] (Connected) $H_1$ is connected.
\item[(SC3)] (Non-diagonality) Let $B$ be a cube in $H_0$ which is the union of
$2^n$ distinct elements of ${\cal Q}$. (So $B$ has side length $2l^{-1}$.) Then
if $\mbox{Int} (H_1 \cap B)$ is non-empty, it is connected.
\item[(SC4)] (Borders included) $H_1$ contains the line segment $\{x: 0\leq x_1
\leq 1, x_2=\cdots =x_n=0 \}$.
\end{enumerate}
Here (see \cite{bb}) (SC1) and (SC2) are essential, while
(SC3) and (SC4) are included for technical convenience.
The Sierpinski carpets are infinitely ramified: the critical
set $C_K$ in (\ref{eq:critpc}) is
an infinite set,
and $K$ cannot be disconnected by removing a finite number of points.
It is known (see \cite{bb,bbk,kz} etc.)
that there is a self-similar Dirichlet form $({\mathcal E},{\mathcal F})$ on $L^2(K,\mu)$
where the corresponding diffusion is the fractional diffusion.
In particular, ${\mathcal F}=\Lambda_{2,\infty}^{d_w/2} (K)$ where $d_w=(\log \rho N)/(\log l)$,
$\rho$ given in (A5).
Let ${\mathfrak G}=\{\mbox{the identity map}\}$ and $L=([0,1]^{n-1}\times\{0\})\cap K$ (cf. Figure 1).
Let $I=\{i\in S: F_i(K)\cap L\ne\emptyset\}$, $N_I=\# I$, and assume
\begin{equation}\label{karho}
\rho N_I>1.
\end{equation}
For simplicity, we assume that the $(n-1)$-dimensional Sierpinski carpet $L$ also satisfies the conditions corresponding to (SC1)--(SC4).
Then, (A1)--(A6) and (C1)--(C2) are easy to check with $M=1$.
(A7) holds by (\ref{karho}), because
$\beta=d_w/2-(d_f-d)/2=(\log \rho N_I)/(2\log l)$.
(A8) holds by Proposition \ref{thm:pr4.2}. It is known that the corresponding self-adjoint
operator has compact resolvents (see \cite{bb,bbk,kz} etc.), so (B1) holds.
Letting $\hat I=I$, we can check (B2).
For $w\in I^{m}$, $m\in{\mathbb Z}_+$,
let $x_0(w)\in[0,1]^n$ be the center of $K_w$
and $\Lambda_k(w)$ the intersection of $K$ and a cube in ${\mathbb R}^{n}$ with center $x_0(w)$ and length $(2k+1)l^{-m}$ for $k\in{\mathbb N}$.
In order to assure (B3),
assume for the moment
that there exists some $k\ge6$ such that $\Lambda_k(w)$ is connected for all $w\in \bigcup_{m\in{\mathbb Z}_+}I^{m}$.
Let $l_0=(2k+1)n$ and take $m_0\in{\mathbb N}$ such that $l^{m_0}\ge 2k+1$.
For each $w\in I^{m+m_0}$, $m\in{\mathbb Z}_+$, take $D'''(w)\subset D''(w)\subset D^\sharp(w)\subset D^{(1)}(w)\subset D^{(2)}(w)\subset D'(w)$ so that $K_{D'''(w)}=\Lambda_1(w)$, $K_{D''(w)}=\Lambda_2(w)$, $K_{D^\sharp(w)}=\Lambda_3(w)$, $K_{D^{(1)}(w)}=\Lambda_4(w)$, $K_{D^{(2)}(w)}=\Lambda_5(w)$, and $K_{D'(w)}=\Lambda_k(w)$.
With the use of Proposition~\ref{prop:ehi} and Proposition~\ref{prop:wv}, (B3) can be checked.
Here, the Harnack inequalities (EHI1) and (EHI2) are assured by \cite{bb,bbk,kz}.
To be more precise, let $\hat K=\bigcup_{x\in\{-1,0\}^n}(K+x)$, which is a subset of $[-1,1]^n$.
Then, one can construct a Dirichlet form on $\hat K$ whose corresponding diffusion is the fractal diffusion
in the same way as in \cite{bb,bbk,kz}. Indeed, $\hat K$ has enough symmetry for the coupling arguments in \cite{bb}
to work. In this way, the Harnack inequalities (EHI1) and (EHI2) are assured.
If for each $k$, there exists $w\in \bigcup_{m\in{\mathbb Z}_+}I^{m}$ such that $\Lambda_k(w)$ is not connected, then
take the connected component of $\Lambda_k(w)$
including $K_w$ in place of $\Lambda_k(w)$
and discuss similarly as above. By the covering argument, we
can check (B3).
(B4) is confirmed by an argument similar to the case of Sierpinski gaskets.
Thus, we have by Theorem~\ref{theo:mainthm1} and Theorem~\ref{theo:mainthm2},
\[{\mathcal F}|_L={\hat \Lambda}_{2,2}^\beta (L)\qquad\mbox{where }~~\beta=\frac{d_w}2-\frac 12
\left(\frac{\log N}{\log l}-\mbox{dim}_H L\right).\]
Note that when $\partial [0,1]^n\subset K$, then $0<\beta<1$, so (\ref{karho}) holds
and ${\mathcal F}|_L=\Lambda_{2,2}^\beta (L)$ by Remark \ref{rem:2.5}.
Indeed, let $K_2=[0,1]^{n}$ and $K_1$ be a generalized Sierpinski carpet in ${\mathbb R}^{n}$
with $\partial [0,1]^n\subset K_1$,
which is
determined by $\{F_i\}_{i}$
where $F_i([0,1]^{n})\cap \partial [0,1]^{n}\ne
\emptyset$ for all $i$. Clearly, $K_1\subset K\subset K_2$. For each $K_i$, one can construct the
self-similar Dirichlet form. Let $\rho_i$ be the scaling factor given in (A5).
By the shorting and cutting laws for electrical networks (see \cite{ds}), $\rho_2\le \rho\le
\rho_1$. Then, $\rho_2=l^{2-n}$ and
\begin{equation}\label{bbreses}
\frac{2}{l^{n-1}}+\frac {l-2}{l^{n-1}-(l-2)^{n-1}}\le \rho_1\le
\frac l{l^{n-1}-(l-2)^{n-1}},
\end{equation}
due to (5.9) in \cite{bb}. Since $L=[0,1]^{n-1}\times\{0\}$ and $N_I=l^{n-1}$ in this case,
we have $\rho N_I\ge \rho_2 N_I=l\ge 2$, so (\ref{karho}) holds and $\beta>0$.
Using (\ref{bbreses}),
\[\rho N_I\le \rho_1 N_I\le \frac {l^n}{l^{n-1}-(l-2)^{n-1}}<l^2,\]
where the last inequality is a simple computation. Thus $\beta<1$.
\medskip
\noindent 4) Vicsek sets: Let $a_1=(0,0), a_2=(1,0), a_3=(1,1), a_4=(0,1), a_5=(1/2,1/2)$
be points in ${\mathbb R}^2$ and define $F_i(x)=(x-a_i)/3+a_i$ for $x\in {\mathbb R}^2$ and $i=1,\ldots, 5$.
The unique non-void compact set $K$ which satisfies $K=\bigcup_{i=1}^5 F_i(K)$
is the Vicsek set. As in the case of 1),
there is a self-similar Dirichlet form $({\mathcal E},{\mathcal F})$ on $L^2(K,\mu)$ with $\rho=3$,
where the corresponding diffusion is the fractional diffusion. In particular,
${\mathcal F}=\Lambda_{2,\infty}^{d_w/2} (K)$ where $d_w=(\log 15)/(\log 3)$.
Let $L$ be the line segment from $(0,0)$ to $(1,1)$. Then, (B4) does not
hold. In this case, the trace of the Brownian motion on the Vicsek set is the Brownian motion
on the line segment.
Indeed, one can easily check condition $(H_1) - (H_3)$ in Section~8 of \cite{BP}
on the $1$-dimensional Sierpinski gasket which is the line. So, by \cite{BP} Theorem 8.1,
one sees that the trace of the Brownian motion on the Vicsek set
is a constant time change of the Brownian motion on the line.
Thus,
\[{\mathcal F}|_L=\Lambda_{2,\infty}^1 (L),\]
which is larger than $\Lambda_{2,2}^1 (L)$. This shows that (B4) is necessary for
Theorem \ref{theo:mainthm1}.
\section{Application: Brownian motion penetrating fractals}
In \cite{hk}, one of the authors constructed Brownian motions on {\it fractal fields}, a collection
of fractals with (in general) different Hausdorff dimensions (see also \cite{kum}).
They are diffusion processes which
behave as the appropriate fractal diffusions within each fractal component of the field and
they penetrate each fractal. In \cite{hk},
a restricted assumption (Assumption 2.2 in \cite{hk}) was needed to construct such processes
because we did not know the corresponding function spaces.
Our result in this paper can
be applied here and we can construct such penetrating diffusions without the restricted assumption.
Let $A_0$ be a countable set and
let $\{K_i\}_{i\in A_0}\subset {\mathbb R}^n$ be a family of self-similar sets
together with strong local, regular, and self-similar Dirichlet forms
$({\mathcal E}_{K_i},{\mathcal F}_{K_i})$ on $L^2(K_i,\mu_i)$, where $K_i$ and $\mu_i$ lie in the framework of Section~2.
We also regard $\mu_i$ as a measure on ${\mathbb R}^n$ by letting $\mu_i({\mathbb R}^n\setminus K_i)=0$.
We set $G=\bigcup_{i\in A_0} K_i$.
Let $A_1$ be another countable set and
let $\{D_j\}_{j\in A_1}\subset {\mathbb R}^n$ be a family of disjoint domains
in ${\mathbb R}^n\setminus G$.
Denote the closure of $D_j$ in ${\mathbb R}^n$ by $K_j$
and the Lebesgue measure restricted on $K_j$ by $\mu_j$.
Define $\tilde{G}=G\cup \bigl(\bigcup_{j\in A_1}K_j\bigr)$.
$\tilde{G}$ is called a {\it fractal field} generated by
$\{K_i\}_{i\in A_0}$ and $\{D_j\}_{j\in A_1}$.
(When $G$ is connected as in the introduction, we also call
$G$ a {\it fractal field} or a {\it fractal tiling}.)
Denote by $A$ the disjoint union of $A_0$ and $A_1$.
For $i,j\in A$ with $i\ne j$, let
$\Gamma_{ij}=K_i\cap K_j$.
Define $\Gamma=\bigcup_{i,j\in A,\,i\ne j}\Gamma_{ij}$.
For $x\in \Gamma$, let $J_x:=\{i\in A: x\in K_i\}$
and define $N_x:=\bigcup_{i,j\in J_x,\,i\ne j}\Gamma_{ij}$.
Throughout this section, we impose
the following assumption.\medskip
\noindent{\bf Assumption A}
(1) For each compact set $C\subset {\mathbb R}^n$,
$\#\{i\in A: C\cap K_i\ne \emptyset\}<\infty$.\\
(2) For each $i\in A_1$, $K_i\setminus D_i$ is a null set with respect to the Lebesgue measure on ${\mathbb R}^n$.
\medskip
For each $i\in A_1$, define
${\cal D}({\mathcal E}_{K_i})=\{u\in C_0(K_i): u|_{D_i}\in W^{1,2}(D_i)\}$ and
\[
{\mathcal E}_{K_i} (u,v)=\frac 12\int_{D_i}(\nabla u(x),
\nabla v(x))_{{\mathbb R}^n}\,dx,~~~\mbox{for}~~u,v\in {\cal D}({\mathcal E}_{K_i}).
\]
Then, $({\mathcal E}_{K_i},{\cal D}({\mathcal E}_{K_i}))$ is closable on $L^2(K_i,\mu_i)$.
Its closure will be denoted by $({\mathcal E}_{K_i},{\mathcal F}_{K_i})$.
It is easy to see that $({\mathcal E}_{K_i},{\mathcal F}_{K_i})$ is a strong local regular Dirichlet form.
For $x\in \Gamma$ and $i\in J_x$,
define $\beta_{x,i}=d_w(K_i)/2-(d_f(K_i)-d_f(N_x\cap K_i))/2$.
Here, $d_w(K_i)$ is defined in (A7) for $({\mathcal E}_{K_i},{\mathcal F}_{K_i})$ if $i\in A_0$ and $d_w(K_i)$ is defined as $2$ if $i\in A_1$,
and $d_f(K_i)$ and $d_f(N_x\cap K_i)$ are the Hausdorff dimensions of $K_i$ and $N_x\cap K_i$, respectively.
We will also assume the following throughout this section.
\medskip
\noindent {\bf Assumption B }
(1) For $i\in A_0$,
$({\mathcal E}_{K_i},{\mathcal F}_{K_i})$ is a
strong local regular Dirichlet form on
$L^2(K_i,\mu_i)$ which satisfies (A1),
(A3), the first identity of (A4),
(A5), and (A6) in Section~2.\\
(2) For each $x\in\Gamma$ and $i\in J_x\cap A_0$, $N_x\cap K_i$ is a finite number of union of compact self-similar sets $\{L_j\}$
that are constructed by the same number of contraction maps and each of which satisfies (A2), the second identity of (A4),
(A7), and (A8) in Section~2.
Further, (C1)$^*$--(C2)$^*$ in Remark~\ref{thm:finunil}
holds with $K=K_i$ and $L=N_x\cap K_i$.\\
(3) For each $x\in \Gamma$ and $i\in J_x\cap A_1$, $N_x\cap K_i$ is a closed Alfors $d_{x,i}$-regular set with some $d_{x,i}$.\\
(4)
For every $x\in \Gamma$, $\beta_{x,i}>0$ for all $i\in J_x$, and the set $\Lambda_x:=\{f\in C_0(N_x): f|_{N_x\cap K_i}\in \Lambda_{2,2}^{\beta_{x,i}}(N_x\cap K_i)\mbox{ for all }i\in J_x\}$ is dense in $C_0(N_x)$.
\medskip
We will give several remarks.
When $i\in A_1$, we have $d_f(K_i)=n$ and $d_f(K_i\cap N_x)=d_{x,i}$.
The set $\Lambda_x$ is closed under the operation of the normal contraction; $0\vee f\wedge1\in \Lambda_x$ for $f\in \Lambda_x$.
If $N_x$ itself is an Alfors regular set and $\beta_{x,i}\in (0,1)$ for all $i\in J_x$,
then $\Lambda^{\max_{i\in J_x}\beta_{x,i}}_{2,2}(N_x)\cap C_0(N_x)$ (which is a subset of $\Lambda_x$) is dense
in $C_0(N_x)$ by Chapter~V, Proposition~1 in \cite{jw} and Theorem~3 in \cite{stos}.
The condition $\beta_{x,i}\in (0,1)$ holds, for example, if $i\in N_x\cap A_1$ and $d_{x,i}\in(n-2,n)$, because then $\beta_{x,i}=1-(n-d_{x,i})/2\in (0,1)$.
Define a measure $\tilde{\mu}$ on $\tilde{G}$ by $\tilde{\mu}=\sum_{i\in A}\mu_i$.
We now define a bilinear form $(\tilde {\cal E},{\cal D}
(\tilde {\cal E}))$ on $L^2 (\tilde{G},\tilde{\mu})$ as follows:
\begin{eqnarray*}
\tilde {\cal E} (u,v)&=& \sum_{i\in A} {\mathcal E}_{K_i} (u|_{K_i},v|_{K_i})
~~\mbox{for}~u,v \in {\cal D} (\tilde {\cal E}),\\
{\cal D} (\tilde {\cal E}) &=& \{u\in C_0(\tilde{G}): u|_{K_i}\in{\mathcal F}_{K_i}
\mbox{ for all }i\in A \mbox{ and } \tilde {\cal E} (u,u)<\infty\}.
\end{eqnarray*}
Then, the following is easy to check.
\begin{lem}
\begin{enumerate}
\item[$(1)$] $(\tilde {\cal E},{\cal D} (\tilde {\cal E}))$ is closable in $L^2 (\tilde{G},\tilde{\mu})$.
\item[$(2)$] ${\cal D} (\tilde {\cal E})$ is an algebra.
\item[$(3)$] For $i\in A$, $x\in K_i$, and for $U(x)$ which is a
neighborhood of $x$ in $K_i$, there exists $f\in {\mathcal F}_{K_i}\cap C_0(K_i)$
such that $f(x)>0$ and
$\mathop{\rm supp} f\subset U(x)\cap K_i$, where $\mathop{\rm supp} f$ denotes the
support of $f$.
\end{enumerate}
\label{theo:easy}\end{lem}
Now, let $(\tilde {\cal E}, \tilde {\cal F})$ be the closure of $(\tilde {\cal E}, {\cal D} (\tilde {\cal E}))$.
We then have the following.
\begin{theo}\label{theo:maincon}
$(\tilde {\cal E},\tilde {\cal F})$ is a strong local
regular Dirichlet form on $L^2 (\tilde{G},\tilde{\mu})$.
\end{theo}
Note that the strong local property of $(\tilde {\cal E},\tilde {\cal F})$ can be
easily deduced from those of the original forms on
$\{K_i\}_{i\in A}$. Therefore, it is enough to prove the
regularity of $(\tilde {\cal E},\tilde {\cal F})$.
For the proof of it,
the key part is to prove the following.
\begin{pro}\label{theo:keypro}
\begin{enumerate}
\item[$(1)$] For each $x\ne y\in \tilde{G}$, there exists $g\in {\cal D}(\tilde {\cal E})$
such that $g(x)\ne g(y)$.
\item[$(2)$] For any compact set $L$ in $\tilde{G}$, there exists $f\in {\cal D}(\tilde {\cal E})$
such that $f=1$ on $L$.
\end{enumerate}
\end{pro}
Once this proposition is established, it is easy to prove
the regularity of $(\tilde {\cal E},\tilde {\cal F})$
(see \cite{hk}), so we will only prove the proposition.
\Proofwithname{Proof of Proposition \ref{theo:keypro}.}
Let $B(x,r)$ denote the open ball in ${\mathbb R}^n$ with center $x\in{\mathbb R}^n$ and radius $r$.
When either $x$ or $y$ is in the compliment of $\Gamma$,
then (1) is clear by Lemma \ref{theo:easy}~(3), so we will consider the case
$x,y\in \Gamma$. By Assumption~A~(1), $\# J_x<\infty$.
Since each $K_j$ is closed, by Assumption~A~(1), there exists
$r_x>0$ such that $B(x,r_x)\cap K_j\ne \emptyset$
if and only if $j\in J_x$, and $y\notin B(x,r_x)$.
Since $\Lambda_x$ is dense in $C_0(N_x)$ by Assumption~B~(4),
there exists $u\in \Lambda_x$ such that
$u|_{B(x,r_x/2)}=1$ and $u|_{B(x,3r_x/4)^c}=0$.
Now, by Assumption~B~(1), (2) and the extension theorem (Remark~\ref{thm:finunil}),
for each $i\in J_x\cap A_0$,
there exists ${\hat u}_i\in {\mathcal F}_{K_i}\cap C(K_i)$ such that
${\hat u}_i|_{N_x\cap K_i}=u$.
For each $i\in J_x\cap A_1$, since $N_x\cap K_i$
is a closed Alfors $d_{x,i}$-regular set, we have
\begin{equation}\label{dsetra}
W^{1,2}({\mathbb R}^n)|_{N_x\cap K_i}=\Lambda_{2,2}^{1-(n-d_{x,i})/2}(N_x\cap K_i)
\end{equation}
(see \cite{jw}). By carefully tracing the proof of the extension
theorem in (\ref{dsetra}), we see that there exists
${\hat u_i}\in W^{1,2}({\mathbb R}^n)\cap C_0({\mathbb R}^n)$ such that
${\hat u_i}|_{N_x\cap K_i}=u$
(see, for instance, pages 77--78 in \cite{kum}).
For both cases, since $({\mathcal E}_{K_i},{\mathcal F}_{K_i})$ is regular,
by multiplying a function in ${\mathcal F}_{K_i}\cap C_0(K_i)$ which is
$1$ in $B(x,3r_x/4)$ and $0$ outside $B(x,r_x)$,
we may assume $\mathop{\rm supp} {\hat u_i}\subset B(x,r_x)$.
Define $g\in C_0(\tilde{G})$ as $g|_{K_i}={\hat u}_i$ for $i\in J_x$ and
$g|_{K_i}\equiv 0$ otherwise. Then, $g\in {\cal D} (\tilde {\cal E})$,
$g(x)=1$ and $g(y)=0$. We thus obtain the desired function.
The proof of (2) is quite similar, so we omit it
(see Proposition 2.6~(2) in \cite{hk}).
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
Denote the 1-capacity associated with $({\mathcal E}_{K_i},{\mathcal F}_{K_i})$ and $(\tilde {\cal E},\tilde {\cal F})$ by $\mathop{\rm Cap}_{K_i}$ and $\mathop{\rm Cap}_{\tilde G}$, respectively.
By definition, it is easy to see that $u|_{K_i}\in {\mathcal F}_{K_i}$ for any $i\in A$ and $u\in\tilde {\cal F}$.
Further, $\mathop{\rm Cap}_{K_i}(H)\le \mathop{\rm Cap}_{\tilde G}(H)$ for any $i\in A$ and $H\subset K_i$.
For $i\in A$, let ${\mathcal F}_{K'_i}=\bigl\{f\in {\mathcal F}_{K_i}:\tilde f=0 \mbox{ q.e.\ on } \Gamma\bigr\}$
and $\tilde {\cal F}_{i}=\bigl\{f\in \tilde {\cal F}:\tilde f=0 \mbox{ q.e.\ on }\bigcup_{j\in A\setminus\{i\}}K_j\bigr\}$, where $\tilde f$ is a (corresponding) quasi-continuous modification of $f$.
We will denote by $(\{\tilde{X}_t\}_{t\ge 0},\{\tilde P_x\}_{x\in\tilde G})$ the diffusion process corresponding to $(\tilde {\cal E},\tilde {\cal F})$.
The following proposition shows that $\{\tilde X_t\}$ behaves on $K_i$
in the same way as the diffusion process associated with
$({\mathcal E}_{K_i},{\mathcal F}_{K_i})$ until the process hits $\Gamma$.
\begin{pro}\label{theo:part}
$({\mathcal E}_{K_i},{\mathcal F}_{K'_i})$ and $(\tilde {\cal E},\tilde {\cal F}_i)$ give the same Dirichlet forms
on $L^2(K_i,\mu_i|_{K_i\setminus\Gamma})$,
by identifying the measure space $(\tilde G,\mu_i|_{K_i\setminus\Gamma})$ with
$(K_i,\mu_i|_{K_i\setminus\Gamma})$.
In particular, the corresponding parts of the processes on $K_i\setminus\Gamma$ are the same.
\end{pro}
\begin{demo}{{\it Proof.\ }}
It is easy to see that $f\in\tilde {\cal F}_i$ satisfies that $f|_{K_i}\in {\mathcal F}_{K'_i}$, so we will prove the converse.
Let $f\in {\mathcal F}_{K'_i}$.
By Theorem~4.4.3 of \cite{fot}, we can take an approximation sequence of $f$ from ${\mathcal F}_{K'_i}\cap C_0(K_i\setminus\Gamma)$.
Therefore, the $0$-extension of $f$ outside $K_i$ is an element of $\tilde {\cal F}_i$.
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
For each distinct $i,j\in A$, we denote $K_i\sim K_j$
if $\mathop{\rm Cap}_{K_l}(\Gamma_{ij})>0$ for $l=i$ and $j$.
We now assume the following in addition to
Assumptions~A and B.
\medskip
\noindent{\bf Assumption C}
(1) For each $i\in A_0$,
$({\mathcal E}_{K_i},{\mathcal F}_{K_i})$ is irreducible. \\
(2) For each distinct $i,j\in A$, there exist $k\in{\mathbb N}$ and
a sequence $i_0,i_1,\ldots, i_k\in A$ such that
$K_{i_0}=K_i$, $K_{i_k}=K_j$ and $K_{i_l}\sim K_{i_{l+1}}$
for $l=0,1,\ldots, k-1$.
\\
(3) For each distinct $i,j\in A$ with $K_i\sim K_j$, there exists a positive Radon measure $\nu_{ij}$ on $\Gamma_{ij}$ such that $\nu_{ij}(\Gamma_{ij})>0$ and $\nu_{ij}$ is smooth with respect to both $({\mathcal E}_{K_i},{\mathcal F}_{K_i})$ and $({\mathcal E}_{K_j},{\mathcal F}_{K_j})$.
\medskip
Note that, when $i\in A_1$, $({\mathcal E}_{K_i},{\mathcal F}_{K_i})$ is irreducible since $D_i$ is connected.
(See, e.g.\ Theorem~4.5 in \cite{ou}, for the proof.)
For each nearly Borel set $B\subset {\mathbb R}^n$, define
$\sigma_B=\inf \{t> 0: \tilde {X}_t\in B\}$.
The next proposition shows that ${\tilde X}_t$ penetrates into each $K_i$.
\begin{pro} The following holds
for any nearly Borel set $B$ with
$\mathop{\rm Cap}_{\tilde{G}}(B)>0$.
\begin{equation}
\tilde {P}^x (\sigma_B<\infty)>0
\mbox { for $(\tilde {\cal E},\tilde {\cal F})$-quasi every }x\in \tilde G.
\label{eq:hitinqq}\end{equation}
Especially, if $B$ is
a subset of a certain $K_i$
with $\mathop{\rm Cap}_{K_i}(B)>0$, then
\rom{(\ref{eq:hitinqq})} holds.
\end{pro}
\begin{demo}{{\it Proof.\ }}
By virtue of Theorem 4.6.6 in \cite{fot}, it is enough to prove that $(\tilde {\cal E},\tilde {\cal F})$ is irreducible.
We first recall the following fact.
Let $({\mathcal E},{\mathcal F})$ be a local Dirichlet form. (Here, the locality means ${\mathcal E}(f,g)=0$ if $fg=0$ a.e. All Dirichlet forms appearing in this article are local in this sense; see \cite{sch}.)
Let $Y$ be a measurable subset of the state space and ${\mathcal C}$ a dense set in ${\mathcal F}$.
Then, $Y$ is an invariant set if and only if $1_Y\cdot u\in{\mathcal F}$ for any $u\in{\mathcal C}$.
This is verified by Theorem~1.6.1 in \cite{fot} and a usual approximation argument.
Now, let $M$ be an invariant set for $(\tilde {\cal E},\tilde {\cal F})$.
Fix $i\in A$ and take $u\in {\mathcal F}_{K_i}\cap C_0(K_i)$.
We can take $v\in{\cal D}(\tilde {\cal E})$ such that $v=1$ on $\mathop{\rm supp} u$ by Proposition~\ref{theo:keypro}~(2).
Then, $1_M\cdot v\in\tilde {\cal F}$, which implies that $(1_{M}\cdot v)|_{K_i}\in {\mathcal F}_{K_i}$.
Therefore, $u\cdot(1_{M}\cdot v)|_{K_i}=u\cdot1_{M\cap K_i}$ also belongs to ${\mathcal F}_{K_i}$.
Since ${\mathcal F}_{K_i}\cap C_0(K_i)$ is dense in ${\mathcal F}_{K_i}$, we obtain that $M\cap K_i$ is an invariant set for $({\mathcal E}_{K_i},{\mathcal F}_{K_i})$.
By the irreducibility of $({\mathcal E}_{K_i},{\mathcal F}_{K_i})$, either $\mu_i(M\cap K_i)=0$ or $\mu_i(K_i\setminus M)=0$ holds.
By this argument, there exists a subset $A'$ of $A$ such that $M=\bigcup_{i\in A'}K_i$ $\tilde{\mu}$-a.e.
Assume that $M$ is a nontrivial invariance set.
Then, $A'\ne\emptyset$, $A'\ne A$, and there exist $i\in A'$ and $j\in A\setminus A'$ such that $K_i\sim K_j$ by Assumption~C~(2).
Take a compact set $H\subset \Gamma_{ij}$ such that $\nu_{ij}(H)>0$, and a relatively compact open set $H'$ including $H$.
Take $v\in{\cal D}(\tilde {\cal E})$ such that $v=1$ on $H'$
and let $u=1_M\cdot v\in\tilde {\cal F}$.
Denote by $\tilde u$ the quasi-continuous modification of $u$ w.r.t.\ $(\tilde {\cal E},\tilde {\cal F})$.
Then, $\tilde u|_{K_l}$ is also quasi-continuous w.r.t.\ $({\mathcal E}_{K_l},{\mathcal F}_{K_l})$ for $l=i,j$.
Since $\tilde u=1$ $\mu$-a.e.\ on $H'\cap K_i$, we have $\tilde u=1$ ${\mathcal E}_{K_i}$-q.e.\ on $H\subset H'\cap K_i$.
By Assumption~C~(3), $\tilde u=1$ $\nu_{ij}$-a.e.\ on $H$.
On the other hand, since $\tilde u=0$ $\mu$-a.e.\ on $H'\cap K_j$, we have $\tilde u=0$ ${\mathcal E}_{K_j}$-q.e.\ on $H$.
Therefore, $\tilde u=0$ $\nu_{ij}$-a.e.\ on $H$.
This is a contradiction, which deduces that $(\tilde {\cal E},\tilde {\cal F})$ is irreducible.
\hbox {\rlap{$\sqcup$}{$\sqcap$}}\end{demo}
The fractal field in Figure 2 satisfies
Assumptions~A, B and C, so there is a penetrating
diffusion on the field.
In \cite{hk}, detailed properties of $\tilde {X}_t$ such as heat kernel bounds
and large deviation estimates are established under strong assumptions
such as Assumption $2.2$ in \cite{hk}. Using the results given in this section,
one can relax the assumption and obtain the same results by the same proof given
in \cite{hk},
when each Dirichlet form is the resistance form in the sense of \cite{kig}.
\bigskip
\noindent
{\sc Acknowledgements.} The authors thank Professor R.~S. Strichartz
for fruitful discussions and valuable comments.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,226 |
Q: How to remove empty space in second alinea? I'm tring to remove the extra space and "rebtel.bootstrappedData" in the second alinea but for some reason it won't work.
This is my output
"welcome_offer_cuba.block_1_title":"SaveonrechargetoCuba","welcome_offer_cuba.block_1_cta":"Sendrecharge!","welcome_offer_cuba.block_1_cta_prebook":"Pre-bookRecarga","welcome_offer_cuba.block_1_footprint":"Offervalidfornewusersonly.","welcome_offer_cuba.block_2_key":"","welcome_offer_cuba.block_2_title":"Howtosendarecharge?","welcome_offer_cuba.block_2_content":"<ol><li>Simplyenterthenumberyou'dliketosendrechargeinthefieldabove.</li><li>Clickthe"{{buttonText}}"button.</li><li>CreateaRebtelaccountifyouhaven'talready.</li><li>Done!Yourfriendshouldreceivetherechargeshortly.</li></ol>","welcome_offer_cuba.block_3_title":"DownloadtheRebtelapp!","welcome_offer_cuba.block_3_content":"Sendno-feerechargeandenjoythebestcallingratestoCubainoneplace."},"canonical":{"string":"<linkrel=\"canonical\"href=\"https://www.rebtel.com/en/rates/\"/>"}};
rebtel.bootstrappedData={"links":{"summary":{"collection":"country_links","ids":[null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null,null],"params":{"locale":"en"},"meta":{}},"data":[{"title":"A","links":[{"iso2":"AF","route":"afghanistan","name":"Afghanistan","url":"/en/rates/afghanistan/","callingCardsUrl":"/en/calling-cards/afghanistan/","popular":false},{"iso2":"AL","route":"albania","name":"Albania","url":"/en/rates/albania/
And this is the code I used:
import json
import requests
from bs4 import BeautifulSoup
url = "https://www.rebtel.com/en/rates/"
r = requests.get(url)
soup = BeautifulSoup(r.content, "html.parser")
x = range(132621, 132624)
script = soup.find_all("script")[4].text.strip()[38:]
print(script)
What should I add to "script" so it will remove the empty spaces?
A: Original answer
You can change the definition of your script variable by :
script = soup.find_all("script")[4].text.replace("\t", "")[38:]
It will remove all tabulations on your text and so the alineas.
Edit after conversation in the comments
You can use the following code to extract the data in json :
import json
import requests
from bs4 import BeautifulSoup
url = "https://www.rebtel.com/en/rates/"
r = requests.get(url)
soup = BeautifulSoup(r.content, "html.parser")
script = list(filter(None, soup.find_all("script")[4].text.replace("\t", "").split("\r\n")))
app_data = json.loads(script[1].replace("rebtel.appData = ", "")[:-1])
bootstrapped_data = json.loads(script[2].replace("rebtel.bootstrappedData = ", ""))
I extracted the lines of the script with split("\r\n") and get the wanted data from there.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 2,913 |
\section{Introduction}
In recent years, there has been considerable interest in utilizing graphene as photo-detectors\cite{Xu2009,Gabor2011,Kalugin2011,Sun2012,Yan2012,Vora2012,Yan2012a,Cai2013}. Most of these detectors are based on a hot electron effect, \textit{i.e.} the electronic temperature being substantially higher than the lattice temperature. Two properties of graphene strongly enhance the effect. First, low carrier density gives rise to a very small electron specific heat. Second, weak electron-phonon (e-p) interaction reduces the heat transfer from the electron gas to the lattice. Thus, it is of practical interest to understand the e-p interaction in graphene. Both theoretical and experimental efforts have been devoted to this topic. Earlier work was mainly focused on clean graphene and considered the Dirac spectrum of electrons\cite{Kubakaddi2009,Bistritzer2009,Tse2009,Viljas2010,Betz2012,Baker2012,Baker2013}. As the important role of impurities in electronic transport has been revealed, its effects on the e-p interaction began to draw attention\cite{Vasko2011,Chen2012,Song2012}. For instance, due to the chiral nature of electrons, long range and short range potentials scatter electrons differently in graphene\cite{McCann2006,Nomura2006a,Wu2007}. Recently, a strong enhancement of electronic cooling via e-p interaction in presence of short range disorder has been predicted\cite{Song2012}. This is achieved via a so-called supercollision process. When the carrier density is low, the Bloch-Gr\"uneisen temperature $T_\text{BG}$ can be quite small. Since $T_\text{BG}$ sets the maximum wave vector of phonons that can exchange energy with electrons, when $T_\text{BG} < T $, only a portion of phonons can contribute to the energy relaxation. Interestingly, in presence of short range potentials, the theory has found that a disorder-assisted scattering process can occur, in which all available phonons are able to participate. As a result, the energy relaxation is strongly enhanced. Shortly, two experiments confirmed the supercollision\cite{Betz2013,Graham2013}, although long range potential scattering usually dominates in such samples\cite{Adam2007,Chen2008}. In the case of long range potentials, Chen and Clerk have also predicted an increase of electronic cooling at low temperature for weak screening\cite{Chen2012}. Note that besides the different potential profiles, \textit{e.g.} long range or short range, disorder can be static or dynamic. Despite these studies, in which only static disorder was considered, the dynamics of disorder has not been addressed.
Here, we present an experimental investigation of the effect of vacancy on electronic cooling in both monolayer and bilayer defected graphene. In contrast to typical scattering potentials previously treated in theories or encountered in experiments, which are static, vacancies in our defected graphene are dragged by phonons, hence highly dynamic. By studying the nonlinear electric transport of defected graphene, a strong \textit{suppression} of e-p energy relaxation, instead of an \textit{enhancement} in the case of static potentials, has been observed. The more disordered the graphene film, the stronger the suppression is. Our work provides new experimental insight on the effect of scattering potential on e-p interaction. Moreover, the suppression suggests that the performance of graphene hot electron photo-detectors can be further improved by introducing vacancies.
\section{Experiment}
In this work, we have investigated four exfoliated graphene samples on Si/SiO$_2$ substrates. Thickness of all the monolayer (SM1 and SM2) and bilayer (SB1, SB2) samples were estimated by optical contrast and confirmed by Raman spectroscopy\cite{Ferrari2006}. Graphene flakes were patterned into ribbons, using e-beam lithography. 5 nm Ti/80 nm Au were e-beam deposited, followed by lift-off to form electrodes. Typical sample geometry can be seen in the inset of \rfig{rt}a. In order to introduce vacancies, samples were then loaded into a Femto plasma system and subject to Argon plasma treatment for various periods (from 1 to 5 s)\cite{Chen2013}. Four-probe electrical measurements were carried out in a cryostat using a standard lock-in technique. Room temperature $\pi$-filters were used to avoid heating of electrons by radio frequency noise. Information for four samples are summarized in \rtab{tab:info}.
\begin {table*}[htb]
\caption {\label {tab:info} Sample information of four investigated devices. Different Ar gas flow rates and plasma treatment times have been applied to produce different amount of vacancies. ${V_\text{CNP}}$ is the charge neutrality point (CNP) of samples and $\xi$ is the localization length near the CNP.}
\begin {ruledtabular}
\begin {tabular}{ccccccc}
Devices & Length($\mu$m) & Width($\mu$m) & Ar flow rate(sccm) & Plasma treatment period(s) & ${V_\text{CNP}}\rm(V)$ & $\xi\rm(nm) $ \\
\hline
SM1 & 2 & 3 & 3 & 1 & 14.5 & 156\\
SM2 & 6.7 & 2.7 & 4 & 3 & 30 & 21\\
SB1 & 3 & 2.7 & 4 & 3.5 & 70 & 50\\
SB2 & 6 & 2.7 & 4 & 5 & 57 & 54\\
\end {tabular}
\end {ruledtabular}
\end {table*}
\section{Results and discussion}
Previously, we have already demonstrated a hot electron bolometer based on disordered graphene\cite{Han2013}. It has been shown that the divergence of the resistance at low temperature can be utilized as a sensitive thermometer for electrons. By applying Joule heating, the energy transfer rate between the electron gas and the phonon gas can be obtained. The same method has been employed in this work. As showing in \rfig{rt}a, the resistance of defected graphene exhibits a sharp increase as the temperature decreases. The divergence becomes stronger as one approaches the CNP. The $R-T$ behavior can be well fitted to variable range hopping transport, described as $R \propto \exp[(T_0/T)^{1/3}]$\cite{Mott1968}. Here, the characteristic temperature $T_0=12/[\pi k_\text{B} \nu(E_\text{F}) \xi^2]$, with $k_\text{B}$ the Boltzmann constant, $\nu(E_\text{F})$ the density of states at the Fermi level $E_\text{F}$, and $\xi$ the localization length. By fitting to this formula, the localization length $\xi$ is determined. It is employed as a measure of the degree of disorder. $\xi$ near the CNP for all samples are listed in \rtab{tab:info}.
In the steady state of Joule heating, the electron cooling power equals to the heating power. The corresponding thermal model is sketched in \rfig{rt}c. Two thermal energy transfer pathways are indicated, \textit{i.e.} via electron diffusion into electrodes or e-p interaction into the lattice. In our strongly disordered graphene, the former is significantly suppressed due to a very low carrier diffusivity. It has been found that e-p interaction dominates the energy dissipation in such devices\cite{Han2013}. Then, the electronic temperature can be directly inferred from the resistance. Furthermore, it is estimated that the thermal conductance between the graphene lattice and the substrate is much higher than that due to e-p interaction. Thus, the phonon temperature $T_\text{ph}$ is approximately equal to the substrate temperature $T$\cite{Yan2012,Betz2013,Borzenets2013}. Under these conditions, the energy balance at the steady state of Joule heating can be written as
\begin{equation}
\label{eq1}
P=A(T_\text{e}^\delta-T_\text{ph}^\delta)
\end{equation}
where $P$ is the Joule Heating power, $A$ is the coupling constant and $T_\text{e}$ is the electronic temperature. $\delta$ ranges from 2 to 6, depending on the detail of the e-p scattering process\cite{Viljas2010}.
Upon Joule heating, the electronic temperature is raised, leading to decrease of the resistance, depicted in \rfig{rt}b. Based on the resistance as a function of temperature, we obtain the $P-T_\text{e}$ relation at different carrier densities, plotted in the insets of \rfig{PT}. $P$ is also plotted against $T_\text{e}^3-T_\text{ph}^3$. The linear behavior agrees well with \req{eq1} with $\delta=3$ for both monolayer and bilayer graphene at all carrier densities. It has been theoretically shown that both clean monolayer and bilayer graphene can be described by \req{eq1} with $\delta=4$ at low temperature \cite{Viljas2010,Kubakaddi2009}. In presence of disorder, e-p interaction is enhanced and $\delta$ is reduced to 3\cite{Song2012,Chen2012}. $\delta$ obtained in our result is consistent with these theories, indicating the effect of defects. $T^3$ dependence has also been reported in some other experiments. In the following, we will compare our results in detail with previous theoretical and experimental results.
The e-p interaction is usually considered in two distinct regimes, high temperature and low temperature. In normal metals, Debye temperature $\theta_\text{D}$ demarcates two regimes. Below $\theta_\text{D}$, the phase space of available phonons increases with temperature, while it becomes constant above it(all modes are excited). In graphene, because of its low carrier density, the Bloch-Gr\"uneisen temperature $T_\text{BG}$ becomes the relevant characteristic temperature. It is defined as $2k_\text{B}T_\text{BG}=2hck_\text{F}$. Here $k_\text{B}$ is the Boltzmann constant, $h$ the Plank constant, $c$ the sound velocity of graphene and $k_\text{F}$ the Fermi wave vector. $T_\text{BG}$ stems from the momentum conservation in e-p scattering. Because of it, when $T_\text{ph}>T_\text{BG}$, only a portion of phonons can participate in the process\cite{Fuhrer2010}. Considering the band structure of graphene, we have $ T_\text{BG}=2(c/v_F)E_F/k_B$ in monolayer graphene and $ T_\text{BG}=2(c/v_F)\sqrt{\gamma_1E_F}/k_B$ in bilayer graphene\cite{Viljas2010}. Here $v_F\approx10^6$ m/s is the Fermi velocity, $c\approx2\times10^4$ m/s and $\gamma_1 \approx 0.4$ eV is the interlayer coupling coefficient. Taking into account a residual carrier density $n_0\approx 4\times 10^{11}$ cm$^2$ due to charge puddles\cite{Li2011,Zhang2009}, it can be readily estimated that even at the CNP, $T_\text{BG} > 34$ K. It is much higher than $T_\text{e}=1.5$ K in our experiment. Consequently, we are well in the low temperature regime.
In the low temperature regime, the whole population of phonons can interact with electrons. Thus, the disorder-assisted supercollision is negligible\cite{Song2012}, which rules out it as the origin of the observed $T^3$ dependence. It has been theoretically shown that in the case of weak screening, static charge impurities leads to enhanced e-p cooling power over clean graphene and $\delta=3$\cite{Chen2012}. For comparison, we plot our data, the theoretical cooling power of clean graphene in \rfig{3d}. The theoretical prediction of the cooling power per unit area in clean monolayer graphene is \cite{Viljas2010}
\begin{equation}
P_{\text{clean}}=\frac{\pi^2D^2 E_\text{F} k_B^4}{15\rho \hbar^5 v_\text{F}^3 c^3}(T_\text{e}^4-T_\text{ph}^4)
\end{equation}
where $\rho \approx 0.76\times10^{-6}$ kg/m$^2$ is the mass density of graphene and $D$ is the deformation potential chosen as a common value 18 eV \cite{Chen2008a,Graham2013,Fong2013} (this choice will be discussed later). The theoretical cooling power $P_{\text{clean}}$ as a function of the carrier density and electron temperature is depicted as a transparent surface (with $T_\text{ph}$ =1.5 K) in \rfig{3d}a. It can be clearly seen that the cooling power of our disordered samples SM1 and SM2 (green and blue lines) are well below the surface at all carrier densities. For comparison, we also plot the data from two other experiments in which $T^3$-dependence were observed at low temperatures\cite{Fong2013,Somphonsane2013}. These results (with similar $T_{\text{ph}}$) are either on or above the surface. The suppression of the cooling in \rfig{3d}a is considerable. For instance, at $n=4\times10^{11}$ cm$^2$ and $T_{\text{e}}=20$ K, the theory predicts $P_{\text{clean}}$=4.7 nW/$\mu$m$^2$. In \rref{Somphonsane2013} the cooling power was found to be 27 nW/$\mu$m$^2$. In sharp contrast, our experiment gives a cooling power of 0.33 nW/$\mu$m$^2$ for SM1, over an order of magnitude lower than that in clean graphene. For the more disordered sample, SM2, it is even smaller.
Similar suppression occurs in bilayer graphene samples, too. The cooling power per unit area in clean bilayer graphene is given by \cite{Viljas2010}
\begin{equation}
P_{\text{clean}}=\frac{\pi^2D^2 \gamma_1 k_B^4}{60\rho \hbar^5 v_\text{F}^3 c^3}\sqrt{\frac{\gamma_1}{E_\text{F}}}(T_{\text{e}}^4-T_{\text{ph}}^4)
\label{bilayer}
\end{equation}
\rfig{3d}a shows the plot of \req{bilayer}, the cooling power of the bilayer samples SB1, SB2 and the data from \rref{Somphonsane2013}. Although not as pronounced as monolayer graphene, our data still below the theoretical surface. The weaker suppression may result from the fact that the bottom layer of bilayer graphene has experienced less damage by our low energy plasma than the top one\cite{Chen2013}. Therefore, this less disordered layer provides a channel of substantial cooling.
The e-p coupling strength depends on the deformation potential $D$, which characterizes the band shift upon lattice deformation\cite{Bardeen1950,Herring1956,Suzuura2002}.
For the theoretical cooling power surface in \rfig{3d}, we use $D=18$ eV. Note that $D$ for graphene ranges from 10 to 70 eV in various experiments, but 18 eV is the most common value for graphene\cite{Fong2013}. If the suppression is due to an over-estimated $D$, to account for the small cooling power, one would require $D$ to be only about 5 eV, one-half of the lowest value reported. Therefore, we believe that the suppression cannot be explained by a small $D$.
By linear fits of $P$ versus $T_\text{e}^3-T_\text{ph}^3$, the coupling constant $A$ can be obtained. In \rfig{A}, $A$ is plotted as a function of carrier density $n$. $A$ for all samples decreases when approaching the CNP. This is because fewer carriers at Fermi level could contribute to total cooling power of the sample.
We now take a look at the dependence of the coupling constant on the degree of disorder. As listed in \rtab{tab:info}, the samples have been subject to various periods of plasma treatment. Consequently, the degree of disorder is different, indicated by the localization length $\xi$. For instance, $\xi$ for SM1 and SM2 is 156 nm and 21 nm, respectively. As plotted in \rfig{A}a, the coupling constant $A$ of the less disordered SM1 is only about one-third of the value for the more disordered SM2. The dependence of $A$ on $\xi$ is consistent with the suppression of the e-p scattering by disorder. For the two bilayer samples, SB1 and SB2, the localization lengths are close. The $n$ dependence of $A$ for both samples aligns reasonably well and is consistent with the monolayer samples, see \rfig{A}b.
The Joule heating experiment has also been carried out at different phonon temperatures $T_\text{ph}$. In \rfig{A}c, the coupling constant $A$ is plotted as a function of $T_\text{ph}$. Usually, $A$ is independent of $T_\text{ph}$, which is actually seen at low temperature for SB1. However, as the temperature goes above 7 K, $A$ is enhanced. Later, we will show that the unexpected $T$-dependence is likely related to the dynamic nature of vacancies.
At first glance, the suppression of electronic cooling by vacancies seems surprising, in that previous theories have predicted that disorder would enhance the cooling\cite{Chen2012,Song2012}. Most of earlier experimental results have confirmed the enhancement\cite{Somphonsane2013,Fong2013,Betz2013,Graham2013}. However, there is a key difference between those earlier studies and ours. In the former, disorder is theoretically considered to be static. This is indeed true in other experimental work, in which the dominant disorder is due to charge impurities\cite{Adam2007,Chen2008}. However, in our samples, the dominant disorder is vacancies, which are completely dragged by phonons. The effect of disorder on the e-p interaction has been studied in disordered metals and found to depend on the character of disorder\cite{Schmid1973,Sergeev2000,Lin2002,Zhong2002}. In the case of static disorder, diffusive motion of electrons increases the effective interacting time between an electron and a phonon, leading to an enhancement of interaction. However, dynamic disorder modifies the quantum interference of scattering processes\cite{Sergeev2000}. As a result, the interaction is suppressed, in accordance with the famous Pippard's inefficient condition\cite{Pippard1955}. It is reasonable to believe that the observed suppression results from dynamic disorder, vacancies. Furthermore, since the dynamics of disorder apparently depends on $T_\text{ph}$, the dependence of the coupling constant $A$ on the phonon temperature $T_\text{ph}$ is then conceivable. As described in Schmid's theory\cite{Schmid1973,Sergeev2000}, the e-p scattering is suppressed due to strong disorder. The resultant energy relaxation rate $\tau_\text{e-p}^{-1}$ is of the order of $(q_T l) \tau_0^{-1}$ where $\tau_0^{-1}\propto T^3$ is the relaxation rate in pure material, $q_T$ is the wave vector of a thermal phonon and $l$ is the mean free path. As $q_T \propto T_\text{ph}$, the relaxation rate increases with $T_\text{ph}$, in agreement with our result.
It is also worthy to note that charge impurities are long range potentials that preserve the sublattice symmetry. This is in contrast to vacancies, which are short range potentials and break the sublattice symmetry. The theory for supercollision models disorder as short range potential\cite{Song2012}, while in \rref{Chen2012}, disorder potential is long-ranged. This character of disorder strongly affects scattering of chiral electrons in graphene. Our samples represent a graphene system that is quite different from what was commonly seen, in that dynamic and short-ranged potentials dominate. Therefore, the quantitative understanding of our experimental results, including the power index $\delta$, relies on future theory that takes both the dynamics and the symmetry of disorder into account.
\section{Conclusion}
In conclusion, we have observed significant suppression of electronic cooling in defected graphene. The cooling power of both monolayer and bilayer graphene samples show $T_\text{e}^3$ dependence, consistent with disorder-modified electron-phonon coupling in graphene \cite{Chen2012,Song2012}. However, the magnitude of the cooling power is over an order of magnitude smaller than that of clean graphene predicted by theory\cite{Kubakaddi2009,Viljas2010} and also less than other experiments \cite{Fong2013,Somphonsane2013}. The more disordered a graphene film is, the lower cooling power is observed, confirming the effect of disorder. The suppression of electronic cooling is attributed to the dynamic nature of vacancies, which has not been studied in graphene. This effect can be utilized to further improve the performance of graphene-based bolometer and photo-detector devices.
\begin{acknowledgments}
This work was supported by National Key Basic Research Program of China (No. 2012CB933404, 2013CBA01603) and NSFC (project No. 11074007, 11222436, 11234001).
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 1,516 |
Was the Hamburg knife-attacker a known Islamist?
A migrant that stormed a supermarket in Hamburg, Germany, stabbed one shopper to death and injured several others, according to police.
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The migrant ran into the supermarket and began attacking customers with a "long kitchen knife", Reuters reported. Police identified the attacker as a man born in the United Arab Emirates, but did not give details about his citizenship.
The 26-year-old, who was supposed to leave the country, killed a 50-year-old shopper and left six others injured, police and city officials confirmed. Surprisingly, the homicide squad has taken over the investigation and not the counter-terrorism unit, police authorities announced.
But a video on a website of a German tabloid, showed a helicopter landing outside the supermarket with heavily armed police being deployed.
German tabloid Bild reported late on Friday that armed police forces had also searched the migrant housing center where the attacker had lived. According the newspaper Tagesspiegel, the attacker was known to authorities as an Islamist.
Streets were closed off after the attack and law enforcement asked people to steer clear of the area. The supermarket where the attack took place was located at the intersection of Fuhlsbüttler street and Hermann-Kauffmann street in a northeast neighbourhood in the northern port city.
Man Shouting 'Allah Akbar' Goes on Stabbing Spree in #Hamburg Supermarket. We will never know the motivation! https://t.co/p4e9m02zwk
— Nick Short 🇺🇸 (@PoliticalShort) July 29, 2017
Hamburg Mayor Olaf Scholz said in a statement that the assailant was "apparently a foreigner who was supposed to leave the country," but could not be deported because he did not have any identification papers.
"It's also infuriating that this perpetrator is someone who asked for protection in Germany and then turned his hate against us," Scholz said.
Mutmaßlicher Messerstecher Ahmad A. konnte laut @OlafScholz nicht abgeschoben werden, weil er keine Papiere hatte. https://t.co/RipmK92GhL
— Jörg Diehl (@SponDiehl) July 28, 2017
The blood-soaked attacker was arrested soon after his killing spree on Friday afternoon in the Barmbek neighbourhood of Hamburg. Police captured the suspected killer as he attempted to flee.
According to German weekly Spiegel, the assailant had fled in the direction of the Barmbek metro station but was slowed down by witnesses, allowing police to make their arrest.
"Passersby threw chairs and other objects at the attacker as he fled the scene, enabling plain clothes police officers to take him into custody near the store, according to police and videos posted on Twitter," Reuters reported.
A witness to the attack said that the attacker screamed "Allahu Akbar!" immediately after the attack.
Eye witness to attack on supermarket in Barmbek, Hamburg. Says man stabbed 3 then shouted Allah Akbar. 1/2 pic.twitter.com/1UdpjUKZQO
— Larry Hagler (@IanBennett23) July 28, 2017
The attack at the Edeka Supermarket which killed the 50-year-old woman saw four men aged 64, 57, 56 and 19 suffering from stabbing injuries. A 35-year-old man was injured while tackling the attacker, police said.
Police spokesperson Timo Zill confirmed that a kitchen knife was used in the stabbing, which he described as "indiscriminate" and "totally unexpected."
Further press release in English concerning the knife attack in #Hamburg #Barmbek: https://t.co/YlaAtgznNG
— Polizei Hamburg (@PolizeiHamburg) July 28, 2017
Photographs published by Bild showed the detained suspect, drenched in blood and wearing a spit mask. A spit mask is a restraint device intended to prevent someone from spitting or biting.
The masks protect police from exposure to risk of serious infections. In London, 59 percent of injecting drug users test positive for Hepatitis C.
The spit masks have been criticised in Britain for "breaching human rights guidelines" and critics call the masks "primitive, cruel and degrading".
A decision by the Metropolitan Police Service in London to start using spit hoods was condemned by the human rights group Amnesty International, the civil rights group Liberty and the campaign group Inquest. Many major British police forces have chosen to outlaw spit hoods.
Spit masks can be life threatening, according to Swedish investigators.
Germany is heading for parliamentary elections on September 24. While Chancellor Angela Merkel is set to win a fourth term despite her open-door policy to welcome close to two million migrants, security has been the main focus in the campaign. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 5,945 |
\section{Introduction }
The optimal dividend strategy problem has gained extensive attention. In the diffusion setting, many works concerning dividend optimization use the Brownian motion model for the underlying cashflow process. \cite{Bauerle2004} extends the basic model by assuming that the drift coefficient is a linear function of the level of cashflow and \cite{CadenillasSarkarZapatero2007} uses the mean-reverting model and solves the optimization problem. \cite{HojgaardTaksar2001} considers the optimization problem under the model where the drift coefficient is proportional to the level of cashflow and the diffusion coefficient is proportional to the square root of the cashflow level. \cite{ShreveLehoczkyGaver1984},
\cite{Paulsen2008}, \cite{Zhu2014b} and some references therein address the optimization problems for the general diffusion model where the drift and diffusion coefficients are general functions of the cashflow level.
An interesting and different direction of extension is to include the impact of the changing external environments/conditions (for example, macroeconomic conditions and weather conditions) into modeling of the cashflows. A continuous time Markov chain can be used to model the state of the external environment condition, of which the use is supported by observation in financial markets.
The optimal dividend problem with regular control
for Markov-modulated risk processes has been investigated under a verity of assumptions. \citet{SotomayorCadenillas2011} solves the dividend optimization problem for a Markov-modulated Brownian motion model with both the drift and diffusion coefficients modulated
by a two-state Markov Chain. \cite{Zhu2014c} solves the problem for the Brownian motion model modulated by a multiple state Markov chain.
The optimality results in all the above works imply that distributing dividends according to the optimal strategy leads almost surely to ruin. \cite{DicksonWaters2004} proposes to include capital injections (financing) to prevent the surplus becomes negative and therefore prevent ruin. Under the Brownian motion, \cite{LokkaZervos2008} investigates the optimal dividend and financing problem, and \cite{HeLiang2008} studies the problem with risk exposure control through control of reinsurance rate.
The optimality problem with control in both capital injections and dividend distribution in a Cram\'er-Lundberg model is addressed in \cite{ScheerSchmidli2011}. \cite{YaoYangWang2011} solves the problem for dual model with transaction costs.
The purpose of this paper is to investigate optimal financing and dividend distribution problem with restricted dividend rates in a general diffusion model with regime switching. Under the model, the drift and volatility coefficients are general functions of the level of surplus and the
external environment regime, which is modeled by a Markov process. Similar to the ``reflection problem", the company can control the financing /capital injections process (a deposit process) and the dividend distribution process (a ``withdrawal" process). Both capital injections and dividend payments will incur transaction costs.
Sufficient capital injections must be made to keep the controlled surplus
process nonnegative and the dividend payment rate is capped. This paper can be considered as an extension of the existing works on the dividend optimization problem with restricted dividend rates for the diffusion models with or without regime switching. The model considered is more general as it assumes that 1. the drift and volatility are general functions of the cashflows; and 2. the model risk parameters (including drift, volatility and discount rates) are dependent on the external environment regime.
The rest of the paper is organized as follows. We formulate the optimization problem in Section 2. An auxiliary problem is introduced and solved in Section 3. Section 4 presents the optimality results. A conclusion is provided in Section 5. Proofs are relegated to Appendix.
\section{Problem Formulation}
Consider a probability space
$(\Omega,\mathcal{F},\pr)$. Let $\{W_t;t\ge 0\}$ and $\{\xi_t;t\ge 0\}$ be respectively a standard Brownian motion and a Markov chain with the finite state space $\mathcal{S}$ and the transition intensity matrix $Q=(q_{ij})_{i,j\in\mathcal{S}}$. The two stochastic processes $\{W_t;t\ge 0\}$ and $\{\xi_t;t\ge 0\}$ are independent. We use $\{\mathcal{F}_t;t\ge 0\}$ to denote the minimal complete $\sigma$-field generated by the stochastic process $\{(W_t,\xi_t);t\ge 0\}$. Let $X_t$ denote the surplus at time $t$ of a firm in absence of financing and dividend distribution. Assume that $X_0$ is $\mathcal{F}_0$ measurable and that $X_t$ follows the dynamics, $\dif X_t = \mu(X_{t-},\xi_{t-})\dif t+\sigma(X_{t-},\xi_{s-})\dif W_t$ for $t\ge 0$,
where the functions $\mu(\cdot,j)$ and $\sigma(\cdot,j)$ are Lipschitz continuous, differentiable and grow at most linearly on $[0,\infty)$ with $\mu(0,u)\ge 0$. Furthermore, the function $\mu(\cdot,j)$ is concave and the function $\sigma(\cdot,j)$ is positive and non-vanishing.
The firm must have nonnegative assets in order to continue its business. If necessary, the firm needs to raise money from the market. For each dollar of money raised, it includes $c$ dollars of transaction cost and hence leads to an increase of $1-c$ dollars in the surplus through capital injection. Let $C_t$ denote the cumulative amount of capital injections up to time $t$. Then the total cost for capital injections up to time $t$ is $\frac{C_t}{1-c}$. The company can distribute part of its assets to the shareholders as dividends. For each dollar of dividends received by the shareholders, there will be $d$ dollars of cost incurred to them. Let $D_t$ denote the cumulative amount of dividends paid out by the company up to time $t$. Then the total amount of dividends received by the shareholders up to time $t$ is $\frac{D_t}{1+d}$. We consider the case where the dividend distribution rate is restricted. Let the random variable $l_s$ denote the dividend payment rate at time $s$ with the restriction $0\le l_s\le \bar{l}$ where $\bar{l}(>0)$ is constant.
Then
$D_t=\int_0^tl_s\dif s.$ Both $C_t$ and $D_t$ are controlled by the company's decision makers. Define $\pi=\{(C_t,D_t);t\ge 0\}$. We call $\pi$ a control strategy.
Taking financing and dividend distribution into consideration,
the dynamics of the (controlled) surplus process with the strategy $\pi$ becomes
\begin{eqnarray}
\label{eq:2}
\dif X_t^\pi
&=&(\mu(X_{t-}^\pi,\xi_{t-})-l_t)\dif t+
\sigma(X_{t-}^\pi,\xi_{t-})\dif W_t+ \dif C_t,\ t\ge 0.
\end{eqnarray}
Define
$\pr_{(x,i)}\left(\ \cdot\
\right)=\pr\left(\ \cdot\ |X_0=x,\xi_0=i
\right),$ $\E_{(x,i)}\left[\ \cdot\
\right]=\E\left[\ \cdot\ |X_0=x, \xi_0=i
\right],$
$\pr_{i}\left(\ \cdot\
\right)=\pr\left(\ \cdot\ |\xi_0=i
\right),$ and $\E_{i}\left[\ \cdot\
\right]=\E\left[\ \cdot\ | \xi_0=i
\right].$
The performance of a control strategy $\pi$ is measured by its return function defined as follows:
\begin{align}
R_\pi(x,i)=\E_{(x,i)}\left[\int_0^{\infty} e^{-\Lambda_t}\frac{l_t}{1+d}\dif t-\int_0^{\infty} e^{-\Lambda_t}\frac1{1-c}\dif C_t
\right],\ x\ge 0,i\in \mathcal{S},\label{jf1}
\end{align}
where $\Lambda_t=\int_0^t\delta_{\xi_s}\dif s$ with $\delta_{\xi_s}$ representing the force of discount at time $s$. Assume $\delta_i>0$, $i\in\mathcal{S}$.
A strategy $\pi=\{(C_t,D_t);t\ge 0\}$ is said to be \textit{admissible} if (i) both $\{C_t;t\ge 0\}$ and $\{D_t;t\ge 0\}$ are nonnegative, increasing, c\`adl\`ag, and $\{\mathcal{F}_t;t\ge 0\}$-adapted processes, (ii) there exists an $\{\mathcal{F}_t;t\ge 0\}$-adapted process $\{l_t;t\ge 0\}$ with $l_t\in[0,\bar{l}]$ such that $D_t=\int_0^tl_s\dif s$ and (iii) $X^\pi_t\ge 0$ for all $t>0$. We use $\Pi$ to denote the class of admissible strategies.
Since $\{C_t;t\ge 0\}$ is right continuous and increasing, we have the following decomposition: $C_t=\tilde{C}_t+C_{t}-C_{t-}$, where $\{\tilde{C}_t;t\ge 0\}$ represents the continuous part of $\{C_t;t\ge 0\}$.
For convenience, we use $X$, $X^\pi$, $\xi$ and $(X^\pi,\xi)$ to denote the stochastic processes
$\{X_t;t\ge 0\}$, $\{X_t^\pi;t\ge 0\}$, $\{\xi_t;t\ge 0\}$ and $\{(X_t^\pi,\xi_t);t\ge 0\}$,
respectively.
Note that for any admissible strategy $\pi$, the stochastic process $X^\pi$ is right-continuous and adapted to the filtration $\{\mathcal{F}_t;t\ge 0\}$.
The objective of this paper is to study the maximal return function (value function):
\begin{align}
V(x,i)=\sup_{\pi\in \Pi}R_\pi (x,i),\label{13613-1}
\end{align} and to identify the associated optimal admissible strategy, if any.
Following the standard argument in stochastic control theory
\citep[e.g.][]{FlemingSoner1993}, we can show that the value function
fulfils the following dynamic programming principle: for any stopping time $\tau$,
\begin{align}
V(x,i)=\sup_{\pi\in\Pi}\E_{(x,i)}\Big[\int_0^{ \tau} \frac{l_te^{-\Lambda_t}}{1+d}\dif t-\int_0^{ \tau} \frac{ e^{-\Lambda_t}}{1-c}\dif C_t
+ e^{-\Lambda_{ \tau}} V(X^{\pi}_{ \tau},\xi^\pi_{
\tau})\Big ].\label{60214-3}
\end{align}
\section{An Auxiliary Optimization Problem}\label{sec3}
Motivated by \cite{JiangPistorius2012}, which introduces an auxiliary problem where the objective functional is modified in a way such that only the ``returns" over the time period from the beginning up to the first regime switching are included plus a terminal value at the moment of the first regime switching, we start with a similar auxiliary problem first. The optimality results of this problem will play an essential role in solving the original optimization problem.
Throughout the paper, we define $\underline{\delta}=\min_{j\in \mathcal{S}}\delta_{j}$, $q_i=-q_{ii}$, and $\sigma_1=\inf\{t>0: \xi_t\neq \xi_0\}$. Here, $\sigma_1$ is the first transition time of the Markov process $\xi$. For any function $g:\mathbb{R}^+\times \mathcal{S}\rightarrow \mathbb{R}^+$, we use $g^\prime(\cdot)$ and $g^{\prime\prime}(\cdot)$ to denote the first order and second order derivatives, respectively, with respect to
the first argument. We start with introducing two special classes of functions.
\begin{Definition}
(i) Let
$\mathcal{C}$ denote the class of functions $g: \mathbb{R}^+\times \mathcal{S}\rightarrow \mathbb{R}$ such that for each $j\in\mathcal{S}$, $g(\cdot,j)$ is nondecreasing and $g(\cdot,j)\le \frac{\bar{l}}{\underline{\delta}(1+d)}$.
(ii) Let
$\mathcal{D}$ denote the class of functions $g\in\mathcal{C}$ such that for each $j\in\mathcal{S}$, $g(\cdot,j)$ is concave and $\frac{g(x,j)-g(y,j)}{x-y}\le \frac1{1-c}$ for $0\le x<y$. (iii) Define the distance $||\cdot||$ by
$
||f-g||=\max_{x\ge 0,i\in \mathcal{S}}|f(x,i)-g(x,i)|\ \mbox{ for $f,g\in\mathcal{D}$}.$
\end{Definition}
\begin{Lemma} \label{complete}The metric space $(\mathcal{D},||\cdot||)$ is complete.
\end{Lemma}
Define a modified return function and the associated optimal return function by
\begin{align}
R_{f,\pi}(x,i)
=&\E_{(x,i)}\bigg[\int_0^{ \sigma_1} \frac{l_te^{-\Lambda_t}}{1+d}\dif t-\int_0^{ \sigma_1} \frac{e^{-\Lambda_t}}{1-c}\dif C_t
+e^{-\Lambda_{\sigma_1}}f(X^\pi_{\sigma_1}, \xi_{\sigma_1})
\bigg],\ x\ge 0,i\in\mathcal{S},\label{4214-2}\\
V_f(x,i)=&\sup_{\pi\in \Pi}R_{f,\pi}(x,i),\ x\ge 0,i\in\mathcal{S}.\label{17713-4}
\end{align}
\begin{Lemma}\label{remff9}
For any $f\in\mathcal{C}$,
$V,V_f\in\mathcal{C}$ .
\end{Lemma}
Notice that the un-controlled process $(X,\xi)$ is a Markov process.
For any $f\in\mathcal{C}$ and any $i\in\mathcal{S}$, the following Hamilton-Jacobi-Bellman (HJB) equation for the modified value function $V_f(\cdot,i)$ can be obtained by using standard arguments in stochastic control: for $x\ge 0$\\
{\small$
\max\big\{\max_{l\in[0,\bar{l}]}\left(\frac{\sigma^2(x,i)}{2}V_f^{\prime\prime}(x,i)+
\mu(x,i)V_f^\prime(x,i)-\delta_i V_f(x,i)+l\left(\frac1{1+d}-V^\prime_f(x,i)\right)\right),V^\prime_f(x,i)-\frac1{1-c}\big\}=0$.}
Now we define a special class of admissible strategies, which has been shown in the literature to contain the optimal strategy for the original optimization problem if there is 1 regime only.
Since the return function of the modified optimization includes the dividends and capital injections in the first regime only, this problem can be considered as a problem to maximize the returns up to an independent exponential time for a risk model with 1 regime. It is worth studying the special class of strategies mentioned above to see whether the optimal strategy of the modified problem falls into this class as well.
\begin{Definition}
For any $b\ge 0$, define the strategy $\pi^{0,b}=\{(C^{0,b}_t, D^{0,b}_t);t\ge 0\}$ in the way such that the company pays dividends at the maximal rate $\bar{l}$ when the surplus equals or exceeds $b$, pays no dividends when the surplus is below $b$ and the company injects capital to maintain the surplus at level $0$ whenever the surplus tends to go below $0$ without capital injections.
\end{Definition}
We now investigate whether a strategy $\pi^{0,b}$ with an appropriate value for $b$ is optimal or not for the modified optimization problem. We start with studying the associated return functions.
For convenience, we write $X^{0,b}=X^{\pi^{0,b}}$ throughout the rest of the paper.
\begin{Remark}\label{rem1}
(i) It is not hard to see that $\pi^{0,b}$ is admissible and that both $\pi^{0,b}$ and $X^{0,b}$ are Markov processes. (ii) For any function $f\in\mathcal{C}$ and any $i\in\mathcal{S}$, by applying the comparison theorem used to prove the non-decreasing property of $V(\cdot,i)$ and $V_{f}(\cdot,i)$ in Lemma \ref{remff9} we can show that the function $R_{f,\pi^{0,b}}(\cdot,i)$ is non-decreasing on $[0,\infty)$ as well.
\end{Remark}
For any $f\in\mathcal{C}$, $i\in\mathcal{S}$ and $b\ge 0$, define the operator $\mathcal{A}_{f,i,b}$ by
\begin{eqnarray}
\mathcal{A}_{f,i,b}\ g(x)=\frac{\sigma^2(x,i)}{2}g^{\prime\prime}(x)+
(\mu(x,i)-\bar{l})g^\prime(x)-(\delta_i+q_i) g(x)+\frac{\bar{l}}{1+d}+\sum_{j\neq i}q_{ij}f(x,j)=0.\label{agenerator}
\end{eqnarray}
The following conditions will be required for the main theorems.
\noindent \textbf{Condition 1}: The functions $\mu(\cdot,i)$ and $\sigma(\cdot,i)$ are the ones such that for any given function $f\in\mathcal{D}$ and any given $i\in\mathcal{S}$, the ordinary differential equation $\mathcal{A}_{f,i,b}\ g(x)=0$
with any finite initial value at $x=0$ has a bounded solution over $(0,\infty)$.
A sufficient condition for Condition 1 to hold is that both the functions
$\mu(\cdot,i)$ and $\sigma(\cdot,i)$ are bounded on $[0,\infty)$ (see Theorem 5.4.2 in \cite{Krylov1996}). However, this is far away from necessary. For example, when $\mu(\cdot,i)$ is a linear function with positive slope and $\sigma(\cdot,i)$ is a constant Condition 1 also holds (see section 4.4 of \cite{Zhu2014b}).
\noindent \textbf{Condition 2}:
$\mu^\prime(x,i)\le \delta_i$ for all $x\ge 0$ and $i\in\mathcal{S}$.
Define for any function $f \in\mathcal{C}$ and $i\in\mathcal{S}$, \begin{eqnarray}A_{f,i}
=\frac{\bar{l}/(1+d)+\sum_{j\neq i}q_{ij}f(\infty,j)}{q_i+\delta_i}.\label{12713-7}
\end{eqnarray}
\begin{Lemma}\label{theorem1} Suppose Condition 1 holds. For any function $f\in\mathcal{D}$ , any $i\in \mathcal{S}$, (i)
the function $R_{f,\pi^{0,b}}(\cdot,i)$ for any $b\ge 0$, is a continuously differentiable solution on $[0,\infty)$ to the equations \begin{align}
&\frac{\sigma^2(x,i)}{2}g^{\prime\prime}(x)
+\mu(x,i)g^\prime(x)-(\delta_i+q_i) g(x)+\sum_{j\neq i}q_{ij}f(x,j)=0, \mbox{ $0< x< b,$}\label{1}\\
&\frac{\sigma^2(x,i)}{2}g^{\prime\prime}(x)+
(\mu(x,i)-\bar{l})g^\prime(x)-(\delta_i+q_i) g(x)+\sum_{j\neq i}q_{ij}f(x,j)=-\frac{\bar{l}}{1+d}, \mbox{ $x> b,$}\label{2}\\
&
g^\prime(0+)=\frac1{1-c},\ \ \ \lim_{x\rightarrow\infty}g(x)<\infty, \label{3}
\end{align} and is twice continuously differentiable on $(0,b)\cup(b,\infty)$;
(ii) the function $h_{f,i}(b):=R_{f,\pi^{0,b}}^\prime(b,i)$ is continuous with respect to $b$ for $0<b<\infty$.
\end{Lemma}
Throughout the paper, we use $\frac{\dif^-}{\dif x}g(x,i)$ and $\frac{\dif^+}{\dif x}g(x,i)$ to represent the derivatives of $g$ from the left- and right-hand side, respectively, with respect to $x$.
\begin{Corollary}\label{the2} Suppose Condition 1 holds.
For any $f\in\mathcal{D}$, $i\in\mathcal{S}$ and $b\ge 0$, (i) $R_{f,\pi^{0,b}}(\cdot,i)$ is increasing, bounded, continuously differentiable on $(0,\infty)$, and twice continuously differentiable on $(0,b)\cup(b,\infty)$ with $R_{f,\pi^{0,b}}^\prime(0+,i)=\frac1{1-c}$,
$\left[\frac{\dif^-}{\dif x}R_{f,\pi^{0,b}}^\prime(x,i)\right]_{x=b}=\lim_{x\uparrow b}R_{f,\pi^{0,b}}^{\prime\prime}(x,i)$ and $ \left[\frac{\dif^+}{\dif x}R_{f,\pi^{0,b}}^\prime(x,i)\right]_{x=b}=\lim_{x\downarrow b}R_{f,\pi^{0,b}}^{\prime\prime}(x,i);$ and (ii) if $R_{f,\pi^{0,b}}^\prime(b,i)=\frac1{1+d}$, then $R_{f,\pi^{0,b}}(x,i)$ is twice continuously differentiable with respect to $x$ at $x=b$.
\end{Corollary}
We use $R_{f,\pi^{0,b}}^\prime(0,i)$ and $R_{f,\pi^{0,b}}^{\prime\prime}(0,i)$ to denote $R_{f,\pi^{0,b}}^\prime(0+,i)$ and $R_{f,\pi^{0,b}}^{\prime\prime}(0+,i)$, respectively.
\begin{Lemma}\label{12713-11} Suppose Conditions 1 and 2 hold. For any fixed $f\in\mathcal{D}$, $i\in\mathcal{S}$ and $b\ge 0$, we have $R_{f,\pi^{0,0}}^{\prime\prime}(0+,i)\le 0$, and in the case $b>0$, $R_{f,\pi^{0,b}}^{\prime\prime}(0+,i)\le 0$ if $R_{f,\pi^{0,b}}^\prime(b,i)\le \frac1{1-c}$.
\end{Lemma}
\begin{Lemma}\label{13-2-20} Suppose Conditions 1 and 2 hold.
For any $f\in\mathcal{D}$ and $i\in\mathcal{S}$, (i) $R_{f,\pi^{0,0}}^{\prime\prime}(x,i)\le 0$ for $x\ge 0$, and in the case $b> 0$, $R_{f,\pi^{0,b}}^{\prime\prime}(x,i)\le 0$ for $x\ge 0$ if $R_{f,\pi^{0,b}}^\prime(b,i)=\frac1{1+d}$; and
(ii) for $b>0$, if $R_{f,\pi^{0,b}}^\prime(b,i)>\frac1{1+d}$, $R_{f,\pi^{0,b}}^{\prime\prime}(x,i)\le 0$ for $x\in [ 0,b)$ and $R_{f,\pi^{0,b}}^{\prime\prime }(b-,0)\le 0$.
\end{Lemma}
Let $I\{\cdot\}$ be the indicator function. Define for any fixed $b\ge 0$ and any fixed $\pi\in\Pi$,
\begin{align}
\tau_b^\pi&=\inf\{t\ge 0: X_t^\pi\ge b\},\label{21714-1}\\
{W}_{f,b}(x,i)&=\sup_{\pi\in\Pi}\E_{(x,i)}\Bigg[\int_0^{ \tau_b^\pi\wedge \sigma_1}e^{-\Lambda_s}\frac{l_s}{1+d}\dif s-\int_0^{ \tau_b^\pi\wedge \sigma_1}e^{-\Lambda_s}\frac1{1-c}\dif C_s\nonumber\\
&+e^{-\Lambda_{ \tau_b^\pi}}R_{f,\pi^{0,b}}(X_{\tau_b^\pi}^{\pi},\xi_0)I\{\tau_b^\pi< \sigma_1\}+e^{-\Lambda_{ \sigma_1}}f(X^\pi_{ \sigma_1},\xi_{ \sigma_1})I\{ \sigma_1\le \tau_b^\pi\} \Bigg].\label{450}
\end{align}
\begin{Theorem}\label{the5} Suppose Conditions 1 and 2 hold.
For any $f\in\mathcal{D}$, any $i\in \mathcal{S}$ and any $b>0$, if $R_{f,\pi^{0,b}}^\prime(b,i)>\frac1{1+d}$, then $
R_{f,\pi^{0,b}}^\prime(x,i)> \frac1{1+d}$ for $0< x\le b$ and $R_{f,\pi^{0,b}}(x,i)={W}_{f,b}(x,i)$ for $x\ge 0$.
\end{Theorem}
We show in the following theorems that if $b$ is chosen appropriately, the return function for the strategy $\pi^{0,b}$ coincides with the optimal return function of the modified problem.
\begin{Theorem}\label{23513-9}
Suppose that Conditions 1 and 2 hold. For any $f\in\mathcal{D}$ and any $i\in \mathcal{S}$, (i) if $R_{f,\pi^{0,0}}^\prime(0+,i)\le \frac1{1+d}$, then $V_f(x,i) =R_{f,\pi^{0,0}}(x,i)$ for $x\ge 0$; and (ii) if for a fixed $b>0$, $R_{f,\pi^{0,b}}^\prime(b,i)=\frac1{1+d}$, then $V_f(x,i)=R_{f,\pi^{0,b}}(x,i)$ for $x\ge 0$.
\end{Theorem}
\begin{Lemma}\label{lemff1} Suppose Conditions 1 and 2 hold, $f\in\mathcal{D}$ and $i\in \mathcal{S}$. Let $R_{f,\pi^{0,0}}^\prime(0,i)$ denote $R_{f,\pi^{0,0}}^\prime(0+,i)$.
If $R_{f,\pi^{0,b}}^\prime(b,i)>\frac{1}{1+d}$ for all $b\ge 0$, then $V_f(x,i)=\lim_{b\rightarrow \infty}R_{f,\pi^{0,b}}(x,i)$ for $x\ge 0$.
\end{Lemma}
Again we use $R_{f,\pi^{0,0}}^\prime(0,i)$ to denote $R_{f,\pi^{0,0}}^\prime(0+,i)$.
Define for any $f\in\mathcal{D}$ and $i\in \mathcal{S}$,
\begin{align}
b^f_i=\infty\mbox{ if $R_{f,\pi^{0,b}}^\prime(b,i)>\frac1{1+d}$ for all $b\ge 0$, and $b^f_i=
\inf\{b\ge 0: R_{f,\pi^{0,b}}^\prime(b,i)\le \frac1{1+d}\}$ otherwise.}
\label{23513-10}
\end{align}
We show in the following that the strategy $\pi^{0,b^f_i}$ is optimal for the modified problem.
.
\begin{Theorem} \label{23513-11} Suppose Conditions 1 and 2 hold. For any $f\in\mathcal{D}$ and any $i\in\mathcal{S}$,
(i) $0\le b_i^f<\infty$; and
(ii) $V_f(x,i)=R_{f,\pi^{0,b^f_i}}(x,i)$ for $x\ge 0$.
\end{Theorem}
\section{The Optimality Results}\label{sec4}
We use the obtained optimality results for the modified optimization problem to address the original optimization problem. The starting point is to notice that the optimal return function of the original optimization $V_f$, when the fixed function $f$ is chosen to be the value function of the original optimization, coincides with the value function $V$.
\begin{Theorem} If Conditions 1 and 2 hold,
\label{6214-4}(i) $V\in\mathcal{D}$; (ii) $b_i^V<\infty$ and $V(x,i)=R_{V,\pi^{0,b^V_i}}(x,i)$.
\end{Theorem}
\begin{Theorem} \label{thmj1} Define $\pi^*$ to be the strategy under which, the dividend pay-out rate at any time $t$ is $\bar{l}I\{X_t^{\pi^*}\}$, and the company injects capital to maintain the surplus at level $0$ whenever the surplus tends to go below $0$ without capital injections. If Conditions 1 and 2 hold, then
$V(x,i)=V^{\pi^*}(x,i)$ $i\in E$ and the strategy $\pi^*$ is an optimal strategy.
\end{Theorem}
\section{Conclusion}
We have addressed the optimal dividend and financing problem for a regime-switching general diffusion model with restricted dividend rates. Our conclusion is that it is optimal to inject capitals only when necessary and at a minimal amount sufficient for the business to continue, and to pay out dividends at the maximal rate, $\bar{l}$, when the surplus exceeds the threshold dependent on the environmental state. This result is consistent with the findings for similar problems under simpler model configuration in the literature. For example, the optimal strategy with restricted dividend rates is of threshold type for the Brownian motion (see \cite{Taksar2000}), the general diffusion (see \cite{Zhu2014b}), and the regime-switching Brownian motion (see \cite{Zhu2014c}).
\numberwithin{equation}{section}
\renewcommand{\theequation}{A-\arabic{equation}}
\setcounter{equation}{0}
\section*{APPENDIX}
\subsection*{A.1 \ Proofs for Sections \ref{sec3} and \ref{sec4}}
For any $i\in\mathcal{S}$ and $b\ge0$, define the operator $\mathcal{B}$ by
\begin{eqnarray}
\mathcal{B}\ g(x,i)=\frac{\sigma^2(x,i)}{2}g^{\prime\prime}(x,i)+
\mu(x,i)g^\prime(x,i)-\delta_i g(x,i).
\end{eqnarray}
\noindent {\bf Proof of Lemma \ref{complete}}
Consider any convergent sequence $\{g_n;n=1,2,\cdots\}$ in $\mathcal{D}$ with limit $g$. It is sufficient to show $g\in\mathcal{D}$. As for any fixed $i$ and $n$, $g_n(\cdot,i)$ is nondecreasing and concave, so is the function $g(\cdot,i)$. The inequality $g(\cdot,i)\le \frac{\bar{l}}{\underline{\delta}(1+d)}$ follows immediately by noticing $g_n(\cdot,i)\le \frac{\bar{l}}{\underline{\delta}(1+d)}$. It remains to show that $\frac{g(x,i)-g(y,i)}{x-y}\le \frac1{1-c}$ for $0\le x<y$. We use proof by contradiction. Suppose that there exist $x_0$, $y_0$ with $0\le x_0<y_0$ and $j$ such that
$\frac{g(x_0,j)-g(y_0,j)}{x_0-y_0}> \frac1{1-c}$. Define $\epsilon_0:= \frac12\left( \frac{g(x_0,j)-g(y_0,j)}{x_0-y_0}-\frac1{1-c}\right)$. Clearly,
$\epsilon_0>0$. As $g_n$ converges to $g$, we can find an $N>0$ such that for all $n\ge N$,
$
||g_n-g||\le \epsilon_0(y_0-x_0).
$
Therefore, $
|g_n(y_0,j)-g(y_0,j)|\le \epsilon_0(y_0-x_0)
$ and $
|g_n(x_0,j)-g(x_0,j)|\le \epsilon_0(y_0-x_0).
$
As a result, $g_n(y_0,j)-g_n(x_0,j)\ge g(y_0,j)- \epsilon_0(y_0-x_0)-(g(x_0,j)+\epsilon_0(y_0-x_0))= g(y_0,j)-g(x_0,j)-2\epsilon_0(y_0-x_0)= \frac{y_0-x_0}{1-c}$.
On the other hand, we have $\frac{g_n(y_0,j)-g_n(x_0,j)}{y_0-x_0}<\frac{1}{1-c}$ (due to $g_n\in \mathcal{D}$), which is a contradiction.\hfill $\square$
\noindent {\bf Proof of Lemma \ref{remff9}}
Noting that $l_s\le \bar{l}$ and that $\sigma_1$ is exponentially distributed with mean $\frac1{q_i}$ and $\Lambda_s=\delta_i s$ for $s\le \sigma_1$, the upper-bounds follow easily from \eqref{jf1}, \eqref{13613-1} and \eqref{17713-4}.
Fix $x$ and $y$ with $y>x\ge 0$. Let $\{X_t^x;t\ge 0\}$ and $\{X_t^y;t\ge 0\}$ denote the surplus processes in absence of control with initial surplus $x$ and $y$, respectively. We use $\pi^x=\{(C^x_t,D^x_t):t\ge 0\}$ with $D^x_t=\int_0^t l_s^x\dif s$ to denote any admissible control strategy for the process $\{X_t^x;t\ge 0\}$.
Noting that $\{C_t^x;t\ge 0\}$ is right-continuous and increasing, we have the following decomposition:
$C_t^x=\int_0^t e_s^x \dif s+\sum_{0<s\le t}(C_s^x-C_{s-}^x)$.
Define $\zeta_0=0$, $\zeta_1=\inf\{s>0:C_{s}^x-C_{s-}^x>0\ \mbox{\ or\ } \xi_s\neq \xi_{s-}\}$ and $\zeta_{n+1}=\{s> \zeta_n:C_{s}^x-C_{s-}^x>0\ \mbox{\ or\ } \xi_s\neq \xi_{s-}\}$ for $n=1,2,\cdots$. Note that $\xi_t=\xi_{\zeta_n}$ for $t\in[\zeta_n,\zeta_{n+1})$ and hence,
$\dif X_t^{x,
\pi^x}=(\mu(X_{t-}^{x,\pi^x},\xi_{\zeta_n})-l_t^x+e_t^x)\dif t+\sigma(X_{t-}^{x,\pi^x},\xi_{\zeta_n})\dif W_t$ and $\dif X_t^{y,
{\pi}^x}=(\mu(X_{t-}^{y,{\pi}^x},\xi_{\zeta_n})
-{l}_t^x+{e}_t^x)\dif t+\sigma(X_{t-}^{y,{\pi}^x},\xi_{\zeta_n})\dif W_t$ for $t\in(\zeta_n,\zeta_{n+1}), n=0,1,\cdots$.
By noting $X_0^{x,\pi^x}=X_0^x=x< y=X_0^y=X_0^{y,{\pi}^x}$ and applying the comparison theorem for solutions of stochastic differential equations (see \cite{IkedaWatanabe1977}), we can show that with probability one, $X_t^{x,\pi^x}\le X_t^{y,
{\pi}^x}$ for $t\in[0,\zeta_1)$. Further notice that any discontinuity of a surplus process is caused by a jump in the associated process $C^x$ at the same time and hence, $X_{\zeta_1}^{x,\pi^x}
=X_{\zeta_1-}^{x,\pi^x}+(C_{\zeta_1}^x-C_{\zeta_1-}^x)
\le X_{\zeta_1-}^{y,\pi^x}+({C}_{\zeta_1}^x-{C}_{\zeta_1-}^x)
=X_{\zeta_1}^{y,\pi^x}$ with probability one. As a result, by applying the comparison theorem on $(\zeta_1,\zeta_{2})$ we can see $X_t^{x,\pi^x}\le X_t^{y,
\pi^x}$ for $t\in(\zeta_1,\zeta_{2})$ with probability one. Repeating the same procedure, we can show that $X_t^{x,\pi^x}\le X_t^{y,
\pi^x}$ for $t\in (\zeta_n,\zeta_{n+1}]$ with probability one. In conclusion, $X_t^{x,\pi^x}\le X_t^{y,
\pi^x}$ for all $t\ge 0$ with probability one. Therefore, $\pi^x$ satisfies all the requirements for being an admissible strategy for the risk process $X^y$ and hence,
$R_{f,\pi^x} (y,i)\le V_f(y,i)$ and $R_{\pi^x} (y,i)\le V(y,i)$. Using this and \eqref{4214-2} we can show
$
R_{f,\pi^x} (x,i)
\le R_{f,\pi^x} (y,i)\le V_f(y,i).
$
Similarly we can obtain
$
R_{\pi^x} (x,i)
\le V(y,i).
$
By the arbitrariness of $\pi^x$, we conclude that $V_f(x,i)\le
V_f(y,i)$ and $V(x,i)\le
V(y,i)$ for $0\le x<y$.
\hfill $\square$
For any $f\in\mathcal{C}$ and $i\in\mathcal{S}$, define the function
$
w_{f,i}: \mathbb{R}\times\mathcal{S}\rightarrow\mathbb{R}$ by \begin{align}
w_{f,i}(\cdot,i)=R_{f,\pi^{0,b}}(\cdot,i) \mbox{ and } w_{f,i}(\cdot,j)=f(\cdot,j) \mbox{ if $j\neq i$.}
\end{align}
\begin{Lemma}\label{itolemma} For any $f\in\mathcal{C}$ and $i\in\mathcal{S}$, suppose the function $
w_{f,i}: \mathbb{R}\times\mathcal{S}\rightarrow\mathbb{R}$ with $w_{f,i}(\cdot,j)=f(\cdot,j)$ if $j\neq i$, is bounded, continuously differentiable and piecewise twice continuously differentiable with respect to the first argument on $[0,\infty)$, and the function $w_{f,i}(\cdot,i)$ satisfies the ordinary differential equations \eqref{1} and \eqref{2}. Then, for any $\pi\in\Pi$,
there exists a positive sequence of stopping times $\{\tau_n;n=1,2,\cdots\}$ with $\lim_{n\rightarrow \infty}\tau_n=\infty$ such that
\begin{align}
&w_{f,i}(x,i)=\E_{(x,i)}\bigg[e^{-\Lambda_{ \tau_n \wedge \sigma_1\wedge t}}w_{f,i}(X^\pi_{ \tau_n\wedge \sigma_1\wedge t}, \xi_{ \tau_n\wedge \sigma_1\wedge t})+\int^{
\tau_n \wedge \sigma_1\wedge t}_0l_se^{-\Lambda_s}w_{f,i}^\prime(X^\pi_{ \tau_n\wedge \sigma_1\wedge t}, \xi_{ \tau_n\wedge \sigma_1\wedge t})\dif s\bigg]
\nonumber\\
&-\E_{(x,i)}\bigg[\sum_{0<s\le \tau_n\wedge \sigma_1 \wedge t}e^{-\Lambda_s}\left(w_{f,i}(X^\pi_{s},\xi_{s-})-w_{f,i}(X^\pi_{s-},\xi_{s-})\right)
+\int^{ \tau_n \wedge \sigma_1\wedge t}_0 e^{-\Lambda_s}
w_{f,i}^{\prime}(X^\pi_{s-},\xi_{s-})\dif
\tilde{C}_s\bigg].\nonumber\\
&-\E_{(x,i)}\bigg[\int^{
\tau_n \wedge \sigma_1\wedge t}_0e^{-\Lambda_s}\bar{l}(w_{f,i}^\prime(X^\pi_{s-},\xi_{s-})-\frac{1}{1+d})I\{X^\pi_{s-}\ge b\}\dif s\bigg].
\label{new5}
\end{align}
\end{Lemma}
\proof Note that Applying It\^o's formula yields that
\begin{eqnarray}
&&\E_{(x,i)}\bigg[e^{-\Lambda_{ {\tau_n} \wedge \sigma_1\wedge t}}w_{f,i}(X^\pi_{ {\tau_n}\wedge \sigma_1\wedge t}, \xi_{ {\tau_n}\wedge \sigma_1\wedge t}) -
w_{f,i}(X_{0}^{\pi},\xi_0)\bigg]\nonumber\\
&=&I_1+I_2+I_3+\E_{(x,i)}\bigg[\sum_{0<s\le {\tau_n} \wedge \sigma_1\wedge t}e^{-\Lambda_s}\left(w_{f,i}(X^\pi_{s-},\xi_{s})-w_{f,i}(X^\pi_{s-},\xi_{s-})\right)\bigg],\label{ito}
\end{eqnarray}
where $I_1=\E_{(x,i)}\bigg[\int^{
{\tau_n}\wedge \sigma_1\wedge t}_0e^{-\Lambda_s}\left(\mathcal{B}w_{f,i}(X^\pi_{s-},\xi_{s-})-l_{s}
w_{f,i}^{\prime}(X^\pi_{s-},\xi_{s-}) \right)\dif s \bigg]
$,\\ $I_2=\E_{(x,i)}\bigg[\int^{ {\tau_n}\wedge \sigma_1 \wedge t}_0 e^{-\Lambda_s}\sigma(X^\pi_{s-},\xi_{s-})
w_{f,i}^{\prime}(X^\pi_{s-},\xi_{s-})\dif
W_s\bigg]$ and\\ $I_3=\E_{(x,i)}\bigg[\sum_{0<s\le {\tau_n}\wedge \sigma_1 \wedge t}e^{-\Lambda_s}\left(w_{f,i}(X^\pi_{s},\xi_{s-})-w_{f,i}(X^\pi_{s-},\xi_{s-})\right)
+\int^{ {\tau_n} \wedge \sigma_1\wedge t}_0 e^{-\Lambda_s}
w_{f,i}^{\prime}(X^\pi_{s-},\xi_{s-})\dif
\tilde{C}_s\bigg].
$
Notice that the stochastic processes\\ $\int^{t}_0 e^{-\Lambda_s}\sigma(X^{{\pi}}_{s-},\xi_{s-})
w_{f,i}^{\prime}(X^{{\pi}}_{s-},\xi_{s-})\dif
W_s$ and $
\int^{t}_0e^{-\Lambda_s}\left(q_i w_{f,i}(X^{{\pi}}_{s-},\xi_{s-})-\sum_{j\neq i} q_{ij} w_{f,i}(X^{{\pi}}_{s-},j)\right)\dif s+$\\
$\sum_{0<s\le t}e^{-\Lambda_s}\left(
w_{f,i}(X^{{\pi}}_{s-},{{\pi}}_{s})
-w_{f,i}(X^{{\pi}}_{s-},\xi_{s-})\right)$ are $\pr_{(x,i)}$-local martingales. Hence, we can always find a positive sequence of stopping times $\{\tau_n;n=1,2,\cdots\}$ with $\lim_{n\rightarrow \infty}\tau_n=\infty$ such that both $\int^{t\wedge \tau_n}_0 e^{-\Lambda_s}\sigma(X^{{\pi}}_{s-},\xi_{s-})
w_{f,i}^{\prime}(X^{{\pi}}_{s-},\xi_{s-})\dif
W_s$ and \\
$\int^{ t\wedge \tau_n }_0e^{-\Lambda_s}\left(q_i w_{f,i}(X^{{\pi}}_{s-},\xi_{s-})-\sum_{j\neq i} q_{ij} w_{f,i}(X^{{\pi}}_{s-},j)\right)\dif s$\\
$+\sum_{0<s\le t\wedge \tau_n}
e^{-\Lambda_s}\left(w_{f,i}(X^{{\pi}}_{s-},\xi_{s})
-w_{f,i}(X^{{\pi}}_{s-},\xi_{s-})\right)$ are $\pr_{(x,i)}$-martingales.
Then it follows by the optional stopping theorem that
\begin{align}
&I_2=\E_{(x,i)}\bigg[\int^{t\wedge \tau_n\wedge \sigma_1}_0 e^{-\Lambda_s}\sigma(X^{{\pi}}_{s-},\xi_{s-})w_{f,i}^{\prime}
(X^{{\pi}}_{s-},\xi_{s-})\dif
W_s\bigg]=0,\label{new1}\\
&\E_{(x,i)}\bigg[\int^{ t\wedge \tau_n\wedge\sigma_1 }_0e^{-\Lambda_s}\left(q_i w_{f,i}(X^{{\pi}}_{s-},\xi_{s-})-\sum_{j\neq i} q_{ij} w_{f,i}(X^{{\pi}}_{s-},j)\right)\dif s\nonumber\\
&+\sum_{0<s\le t\wedge \tau_n\sigma_1}
e^{-\Lambda_s}\left(w_{f,i}(X^{{\pi}}_{s-},\xi_{s})
-w_{f,i}(X^{{\pi}}_{s-},\xi_{s-})\right)\bigg]=0.\label{con-1}
\end{align}
Noting that $X_s^{\pi}-X_{s-}^{\pi}=C_s-C_{s-}\ge 0$, $\xi_{s-}=i$, and $w_{f,i}(X^\pi_{s-}, \xi_{s-})=w_{f,i}(X^\pi_{s-},i)$ for $s\le \sigma_1$ given $\xi_0=i$, that the function $w_{f,i}(\cdot,i) $ satisfies both \eqref{1} and \eqref{2} , and that $w_{f,i}(\cdot,j)=f_i(\cdot,j)$ if $j\neq i$, we obtain that for $s\le {\tau_n}\wedge\sigma_1$,
$
\mathcal{B}w_{f,i}(X^\pi_{s-},\xi_{s-})=q_iw_{f,i}(X^\pi_{s-},\xi_{s-})+
\bar{l}(w_{f,i}^\prime(X^\pi_{s-},\xi_{s-})-\frac{1}{1+d})I\{X^\pi_{s-}\ge b\}-\sum_{j\neq i} q_{ij} w_{f,i}(X^\pi_{s-},j)$, which combined with \eqref{ito}, \eqref{new1}, \eqref{con-1} and $\E_{(x,i)}\bigg[w_{f,i}(X_{0}^{\pi},\xi_0)\bigg]=w_{f,i}(x,i)$ implies the final result.\hfill $\square$
\noindent {\bf Proof of Lemma \ref{theorem1}} (i) Let $v_1(\cdot;i)$ and $v_2(\cdot;i)$ denote a set of linearly independent solutions to the equation $
\frac{\sigma^2(x,i)}{2}g^{\prime\prime}(x)+
\mu(x,i)g^\prime(x)-(\delta_i+q_i) g(x)=0,
$ and $v_3(\cdot;i)$ and $v_4(\cdot;i)$ denote a set of linearly independent solutions to the equation $
\frac{\sigma^2(x,i)}{2}g^{\prime\prime}(x)+
(\mu(x,i)-\bar{l})g^\prime(x)-(\delta_i+q_i) g(x)=0. $\\%\label{10513-1}
Define
$W_1(x;i)=v_1(x;i)v_2^\prime(x;i)-v_2(x;i)v_1^\prime(x;i)$, $W_2(x;i)=v_3(x;i)v_4^\prime(x;i)-v_4(x;i)v_3^\prime(x;i)$,
$B_1(x;i)=v_1(x;i)\int_0^x\frac{v_2(y;i)}{W_1(y;i)}\frac{2\sum_{j\neq i}q_{ij}f(y,j)}{\sigma^2(y,i)}\dif y-v_2(x;i)\int_0^x\frac{v_1(y;i)}{W_1(y;i)}\frac{2\sum_{j\neq i}q_{ij}f(y,j)}{\sigma^2(y,i)}\dif y$, and
\begin{align*}
&B_2(x;i)=v_3(x;i)\int_0^x\frac{v_4(y;i)}{W_2(y;i)}
\frac{2\left(\bar{l}/(1+d)+\sum_{j\neq i}q_{ij}f(y,j)\right)}{\sigma^2(y,i)}\dif y\nonumber\\
&\ \ \ \ \ \ \ \ \ \ \ \ \ -v_4(x;i)\int_0^x\frac{v_3(y;i)}{W_2(y;i)}\frac{2\left(\bar{l}/(1+d)+\sum_{j\neq i}q_{ij}f(y,j)\right)}{\sigma^2(y,i)}\dif y.
\end{align*}
Then for any constants $K_1,K_2,K_3$ and $K_4$, the functions, $K_1v_1(\cdot;i)+K_2v_2(\cdot;i)+B_1(
\cdot;i),$ and $
K_3v_3(\cdot;i)+K_4v_4(\cdot;i)+B_2(
\cdot;i),
$
are solutions to the equations \eqref{1} and \eqref{2}, respectively. Define the function $g_{b,i}$ by
$
g_{b,i}(x)=
K_1v_1(x;i)+K_2v_2(x;i)+B_1(x;i)$ for $0\le x< b$ and $g_{b,i}(x)=
K_3v_3(x;i)+K_4v_4(x;i)+B_2(x;i)$ for $x\ge b$,
where $K_1$, $K_2$, $K_3$ and $K_4$ are constants satisfying
\begin{align}
&K_1v_1(b;i)+K_2v_2(b;i)+B_1(b;i)=K_3v_3(b;i)+K_4v_4(b;i)+B_2(b;i),\label{300413-2}\\
&K_1v_1^\prime(b;i)+K_2v_2^\prime(b;i)+B_1^\prime(b;i)=K_3v_3^\prime(b;i)+K_4v_4^\prime(b;i)+B_2^\prime(b;i),\label{300413-03}
\end{align}
\begin{align}
&K_1v_1^\prime(0;i)+K_2v_2^\prime(0;i)=\frac1{1-c},\ \ \lim_{x\rightarrow\infty}(K_3v_3(x;i)+K_4v_4(x;i)+B_2(x;i))<\infty.\label{300413-3}
\end{align}
For $b\ge 0$, we can easily verify that $g_{b,i}^\prime(0+)=\frac1{1-c}$, and that $g_{b,i}(\cdot)$ is continuously differentiable on $[0,\infty)$ and twice continuously differentiable on $[0,b)\cup(b,\infty)$. Hence, the existence of a solution with desired property
has been proven.
It suffices to
show $R_{f,\pi^{0,b}}(x,i)=g_{b,i}(x)$ for $x\ge 0$. Define $w_{f,i}$ by \begin{eqnarray}
w_{f,i}(x,j)=g_{b,i}(x)\ \mbox{ if $j=i$ and, } w_{f,i}(x,j)=f(x,j) \mbox{ if $j\neq i$}.
\label{25614-100}
\end{eqnarray}
Note that the process, $X^{0,b}$, will always stay at or above $0$ and the company injects capital only when the process reaches down to $0$ with a minimal amount to ensure that the surplus never falls below $0$. Further note that $\xi_{s-}=\xi_0$ for $s\le \sigma_1$. Hence, we conclude that the process $C^{0,b}$ is continuous and that given $\xi_0=i$, the following equations hold for $s\le \sigma_1$,
\begin{eqnarray}
&&X^{0,b}_{s}=X^{0,b}_{s-}+(C^{0,b}_{s}-C^{0,b}_{s-})=X^{0,b}_{s-},\ \ \ w_i(X^{0,b}_{s},\xi_{s-})-w_i(X^{0,b}_{s-},\xi_{s-})=0\label{17615-4}\\
&&w_i^{\prime}(X^{0,b}_{s-},\xi_{s-}) \dif \tilde{C}_s^{0,b}=g_{b,i}^{\prime}(0) \dif {C}_s^{0,b}=\frac{\dif {C}_s^{0,b}}{1-c}.\label{23913-1}
\end{eqnarray}
By applying Lemma \ref{itolemma}, we know that for some positive sequence of stopping times $\{\tau_n;n=1,2,\cdots\}$ with $\lim_{n\rightarrow \infty}\tau_n=\infty$, the equation \eqref{new5} holds. Then by setting $\pi=\pi^{0,b}$ in \eqref{new5}, noticing that the dividend payment rate at time $s$ is $\bar{l}I\{X^{0,b}_{s-}\ge b\}$ under the strategy $\pi^{0,b}$ and that $g_{b,i}(x)=w_{f,i}(x,i)$, and using \eqref{17615-4} and \eqref{23913-1},
we arrive at
\begin{eqnarray}
g_{b,i}(x)&=&\E_{(x,i)}[e^{-\Lambda_{\sigma_1\wedge
\tau_n\wedge t}}w_{f,i}(X^{0,b}_{\sigma_1\wedge\tau_n\wedge t},\xi_{\sigma_1\wedge\tau_n\wedge t})]+\E_{(x,i)}\bigg[\int^{\sigma_1\wedge \tau_n\wedge t}_0 \frac{\bar{l}e^{-\Lambda_s}}{1+d}I\{X^{0,b}_{s}\ge b\}\dif s\bigg]\nonumber\\
&&-\E_{(x,i)}\bigg[\int^{\sigma_1\wedge \tau_n\wedge t}_0\frac{e^{-\Lambda_s}}{1-c} \dif C_s^{0,b}\bigg].\label{12713-6}
\end{eqnarray}
Note that the function $w_{f,i}(
\cdot,\cdot)$ is bounded.
By letting $t\rightarrow\infty$ and $n\rightarrow\infty$ on both sides of \eqref{12713-6}, and then using the dominated convergence for the first expectation on the right-hand side and the monotone convergence theorem for the other expectations, we can interchange the limits and the expectation and therefore can conclude that
$
g_{b,i}(x)
=R_{f,\pi^{0,b}}(x,i)$ for $x\ge 0.$
\noindent (ii) Note by \eqref{4214-2} that $\lim_{x\rightarrow \infty} g_{b,i}(x)=\lim_{x\rightarrow \infty} R_{f,\pi^{0,b}}(x,i)=
=A_{f,i},
$
where the second last equality follows by noticing that given $X_0=x$, $X_s^{0,b}\rightarrow \infty$ as $x\rightarrow \infty$ and hence $C_s^{0, b}\rightarrow 0$ as $x\rightarrow \infty$, and the last equality follows by noting that, given $(X_0,\xi_0)=(x,i)$, $\sigma_1$ is exponentially distributed with mean $\frac1{q_i}$, and using the definition of $A_{f,i}$ in \eqref{12713-7}.
So the constants $K_1,K_2,K_3$ and $K_4$ are solutions to the system of linear equations \eqref{300413-2}-\eqref{300413-3} and $K_3v_3(\infty)+K_4v_4(\infty)+B_2(\infty)=A_{f,i}$. Note that the coefficients of the above system of equations are either constants or continuous functions of $b$. Hence, $K_1,K_2,K_3$ and $K_4$ are continuous functions of $b$, denoted by $K_1(b),K_2(b),K_3(b)$ and $K_4(b)$ here. As a result, the function $h_{f,i}(b)=g_{b,i}^\prime(b)=K_1(b)v_1^\prime(b)+K_2(b)v_2^\prime(b)+B_1^\prime(b;i)$ is continuous for $0<b<\infty$. \hfill$\square$
For any $f\in\mathcal{C}$, $i\in\mathcal{S}$ and $b\ge 0$, define the functions $h$ and $\bar{h}$ by
\begin{align}
h_{f,i,b}(x)&=(\delta_i+q_i) R_{f,\pi^{0,b}}(x,i)-\mu(x,i)R_{f,\pi^{0,b}}^\prime(x,i)-\sum_{j\neq i}q_{ij}f(x,j)\nonumber\\
&-\bar{l}\left(\frac1{1+d}-R_{f,\pi^{0,b}}^\prime(x,i)\right)I\{x\ge b\},\label{function}
\end{align}
\begin{align}
\bar{h}_{f,i,b}(x)&=
(\delta_i+q_i) R_{f,\pi^{0,b}}(x,i)-\mu(x,i)R_{f,\pi^{0,b}}^\prime(x,i)-\sum_{j\neq i}q_{ij}f(x,j)\nonumber\\
&-\bar{l}\left(\frac1{1+d}-R_{f,\pi^{0,b}}^\prime(x,i)\right)I\{x> b\}.\label{function1}
\end{align}
\noindent{\bf Proof of Corollary \ref{the2}} (i) is an immediate result of Remark \ref{rem1} and Lemma \ref{theorem1} (i). (ii) By (i) and Lemma \ref{theorem1}(i) we have $\left[\frac{\dif^-}{\dif x}R_{f,\pi^{0,b}}^\prime(x,i)\right]_{x=b}
=\lim_{x\downarrow b}\frac{2h_{f,i,b}(b,i)}{\sigma^2(b,i)}
$ and
$\left[\frac{\dif^+}{\dif x}R_{f,\pi^{0,b}}^\prime(x,i)\right]_{x=b}
=\lim_{x\downarrow b}\frac{2h_{f,i,b}(b,i)}{\sigma^2(b,i)}.$ By noting $R_{f,\pi^{0,b}}^\prime(b,i)=\frac1{1+d},$ we conclude $\left[\frac{\dif^-}{\dif x}R_{f,\pi^{0,b}}^\prime(x,i)\right]_{x=b}=\left[\frac{\dif^+}{\dif x}R_{f,\pi^{0,b}}^\prime(x,i)\right]_{x=b}$. $\square$\\
For any sequence $\{y_n\}$, define
\begin{align}
k_{f,b}(x,i;\{y_{n}\})&=
(\delta_i+q_i-\mu^\prime(x,i))R_{f,\pi^{0,b}}^\prime(x,i)-\sum_{j\neq i}q_{ij}\lim_{n\rightarrow \infty} \frac{f(y_{n},j)-f(x,j)}{y_{n}-x}.\label{newdef-1}
\end{align}
\noindent{\bf Proof of Lemma \ref{12713-11}}
Throughout the proof, we assume $f\in\mathcal{D}$, $i\in\mathcal{S}$ and $b\ge 0$, unless stated otherwise. We use proof by contradiction. Suppose $R_{f,\pi^{0,b}}^{\prime\prime}(0+,i)> 0$.
Since $R_{f,\pi^{0,0}}(\cdot,i)$ is bounded, we can find a large enough $x$ such that $R_{f,\pi^{0,0}}^{\prime}(x,i)<\frac1{1-c}=R_{f,\pi^{0,0}}^{\prime}(0+,i)$, where the last equality is by Lemma \ref{theorem1} (i). Hence there exists an $x>0$ such that $R_{f,\pi^{0,0}}^{\prime\prime}(x,i)< 0$.
In the case $b>0$, notice that $ R_{f,\pi^{0,b}}^{\prime}(0+,i)=\frac1{1-c}\ge R_{f,\pi^{0,b}}^\prime(b,i).$ So for $b>0$ there exists an $x\in (0,b)$ such that $R_{f,\pi^{0,b}}^{\prime\prime}(x,i)\le 0$.
Define $x_1=\inf\{x>0: R_{f,\pi^{0,b}}^{\prime\prime}(x,i)\le 0\}$. Then $x_1>0$ in the case $b=0$ and $x_1\in(0,b)$ in the case $b>0$, and for $b\ge 0$, \begin{eqnarray}
R_{f,\pi^{0,b}}^{\prime\prime}(x_1,i)=0,\ \ \mbox{ $R_{f,\pi^{0,b}}^{\prime\prime}(x,i)>0$ for $x\in[0,x_1)$.}\label{12713-9}
\end{eqnarray}
As a result, for $b\ge 0$, \begin{eqnarray}
R_{f,\pi^{0,b}}^{\prime}(x,i)>R_{f,\pi^{0,b}}^{\prime}(0+,i)=\frac1{1-c}\ \mbox{ for $x\in(0,x_1]$.}\label{12713-10}
\end{eqnarray} Write $R_{f,\pi^{0,b},i}(x) =R_{f,\pi^{0,b}}(x,i)$. It follows by Lemma \ref{theorem1} that for $b\ge 0$, $\mathcal{A}_{f,i,b}R_{f,\pi^{0,b},i}(x)=0$ for $x>0$.
Therefore, it follows by \eqref{12713-9} and \eqref{function} that for $b\ge 0$,
$h_{f,i,b}(x)=
\frac{\sigma^2(x,i)}{2}R_{f,\pi^{0,b}}^{\prime\prime}(x,i)>0$ for $0<x<x_1$ and $h_{f,i,b}(x_1)
=\frac{\sigma^2(x_1,i)}{2}R_{f,\pi^{0,b}}^{\prime\prime}(x_1,i)=0$.
Hence, we obtain that for $b\ge 0$, \begin{eqnarray}
\frac{h_{f,i,b}(x,i)-h_{f,i,b}(x_1,i)}{x-x_1}<0,\ \ 0<x<x_1.\label{12713-12}
\end{eqnarray}
Note that $x_1>b$ in the case $b=0$, and that $x_1<b$ in the case $b>0$. Therefore, we can find a non-negative sequence $\{x_{1n}\}$ with $b<x_{1n}\le x_1$ in the case $b=0$, $x_{1n}\le x_1<b$ in the case $b>0$, and $\lim_{n\rightarrow \infty}x_{1n}=x_1$ such that $\lim_{n\rightarrow \infty} \frac{f(x_{1n},j)-f(x_1,j)}{x_{1n}-x_1}$ exists.
By replacing $x$ in \eqref{12713-12} by $x_{1n}$ and then letting $n\rightarrow \infty$ on both sides of \eqref{12713-12} gives $ k_{f,b}(x_1,i;\{x_{1n}\})-(\mu(x_1,i)-\bar{l}I\{b=0\})R_{f,\pi^{0,b}}^{\prime\prime}(x_1,i)
\ge 0,$
which combined with \eqref{12713-9} implies
$ \left(\sum_{j\neq i}q_{ij}\lim_{n\rightarrow \infty} \frac{f(x_{1n},j)-f(x_1,j)}{x_{1n}-x_1}
-q_iR_{f,\pi^{0,b}}^\prime(x_1,i)\right)$\\$+\left(\mu^\prime(x_1,i)-\delta_i\right)R_{f,\pi^{0,b}}^\prime(x_1,i)
\le 0.
$
It follows by this inequality, $R_{f,\pi^{0,b}}^\prime(x_1,i)>\frac1{1-c}$ (see \eqref{12713-10}) and $\lim_{n\rightarrow \infty} \frac{f(x_{1n},j)-f(x_1,j)}{x_{1n}-x_1}\le \frac1{1-c}$ (due to $f\in\mathcal{D}$) that
$\left(\mu^\prime(x_1,i)-\delta_i\right)R_{f,\pi^{0,b}}^\prime(x_1,i)
>0$, which combined with \eqref{12713-10} implies $\mu^\prime(x_1,i)-\delta_i> 0$. This contradicts the assumption that $\mu^\prime(x_1,i)\le \delta_i$ (Condition 2).
\hfill $\square$
\noindent{\bf Lemma \ref{13-2-20}} We consider any fixed $f\in\mathcal{D}$ and $i\in\mathcal{S}$ throughout the proof.
We first show that there exists a positive sequence $\{x_n\}$ with $\lim_{n\rightarrow \infty}x_n=\infty$ such that for $b\ge 0$, \begin{eqnarray}
R_{f,\pi^{0,b}}^{\prime\prime}(x_n,i)\le 0.\label{23513-1}
\end{eqnarray}
Suppose the contrary: for some $M>0$, $R_{f,\pi^{0,b}}^{\prime\prime}(x,i)>0$ for all $x\ge M$. This implies $R_{f,\pi^{0,b}}^\prime(x,i)>R_{f,\pi^{0,b}}^\prime(M+1,i)>R_{f,\pi^{0,b}}^\prime(M,i)\ge 0$ for $x>M+1$, where the last inequality follows by the increasing property of $R_{f,\pi^{0,b}}(\cdot,i)$ (see Corollary \ref{the2}(i)). As a result,
$
R_{f,\pi^{0,b}}(x,i)>R_{f,\pi^{0,b}}(M+1,i)+R_{f,\pi^{0,b}}^\prime(M+1,i)(x-M-1)$ for $x>M+1$,
which
implies $\lim_{x\rightarrow \infty} R_{f,\pi^{0,b}}(x,i)=\infty$. This contradicts the boundedness of $R_{f,\pi^{0,b}}(\cdot,i)$ (Corollary \ref{the2}(i)).
Write $R_{f,\pi^{0,b},i}(x)=R_{f,\pi^{0,b}}(x,i).$ By Lemma \ref{theorem1} it follows that
\begin{eqnarray}
\mathcal{A}_{f,i,b}R_{f,\pi^{0,b},i}(x)=0
\mbox{ for $x> 0$}.\label{13-2-4}
\end{eqnarray}
\noindent (i) By Lemma \ref{theorem1} and Corollary \ref{the2} we can see that
$R_{f,\pi^{0,b},i}(\cdot)$ is twice continuously differentiable on $[0,\infty)$ with the differentiability at $0$ referring to the differentiability from the right-hand side.
It follows by noting $R_{f,\pi^{0,b},i}^{\prime}(b)=R_{f,\pi^{0,b}}^{\prime}(b,i)=\frac1{1+d}\le \frac1{1-c}$ for $b>0$, and Lemma \ref{12713-11} that
\begin{eqnarray}
R_{f,\pi^{0,b},i}^{\prime\prime}(0+)\le 0\ \mbox{ for $b\ge 0$.} \label{13-2-1}
\end{eqnarray}
We use proof by contradiction to prove the statement in (i).
Suppose that the statement in (i) is not true. Then there exists a $b\ge 0$ and a $y_0> 0$ such that $R_{f,\pi^{0,b},i}^{\prime\prime}(y_0)=R_{f,\pi^{0,b}}^{\prime\prime}(y_0,i)>0$. Let $\{x_n\}$ be the sequence defined as before. We can find a positive integer $N$ such that $x_N>y_0$. By noting $R_{f,\pi^{0,b},i}^{\prime\prime}(x_N)=R_{f,\pi^{0,b}}^{\prime\prime}(x_N,i)\le 0$ (due to \eqref{23513-1}), \eqref{13-2-1} and the continuity of $R_{f,\pi^{0,b},i}^{\prime\prime}(\cdot)$, we can find $y_1, y_2$ with $0\le y_1<y_0<y_2\le x_N$ such that \begin{eqnarray}
R_{f,\pi^{0,b}}^{\prime\prime}(y_1,i)=0,\ \ R_{f,\pi^{0,b}}^{\prime\prime}(y_2,i)=0,\ \ \mbox{and }\ R_{f,\pi^{0,b}}^{\prime\prime}(x,i)>0\ \mbox{ for $x\in (y_1,y_2)$.} \label{13-2-5}
\end{eqnarray}
Hence,
\begin{eqnarray}
R_{f,\pi^{0,b},i}^{\prime}(y_2)> R_{f,\pi^{0,b},i}^{\prime}(y_1).\label{13-2-2}
\end{eqnarray}
It follows by \eqref{13-2-4} and \eqref{function} that $-\frac{\sigma^2(x,i)}2R_{f,\pi^{0,b},i}^{\prime\prime}(x)=h_{f,b,i}(x)$ for $x> 0$.
Note that for $x>0$, $I\{x\ge b\}=I\{x>b\}$ in the case $b=0$, and that in the case $b>0$, $\frac1{1+d}-R_{f,\pi^{0,b}}^\prime(b,i)=0$ and hence, $\bar{l}\left(\frac1{1+d}-R_{f,\pi^{0,b}}^\prime(x,i)\right)I\{x\ge b\}=\bar{l}\left(\frac1{1+d}-R_{f,\pi^{0,b}}^\prime(x,i)\right)I\{x> b\}$ for $x>0$.
Therefore, for $x> 0$,
$
\frac{\sigma^2(x,i)}2R_{f,\pi^{0,b}}^{\prime\prime}(x,i)=\bar{h}_{f,i,b}(x)$, which combined with \eqref{13-2-5} implies that for $x\in(y_1,y_2)$,
\begin{eqnarray}
&&\bar{h}_{f,i,b}(y_1)
=\frac{\sigma^2(y_1,i)}2R_{f,\pi^{0,b}}^{\prime\prime}(y_1,i)
=0
<\frac{\sigma^2(x,i)}2R_{f,\pi^{0,b}}^{\prime\prime}(x,i)=
\bar{h}_{f,i,b}(x),\label{13-2-8}\\
&&\bar{h}_{f,i,b}(y_2)
=\frac{\sigma^2(y_2,i)}2R_{f,\pi^{0,b}}^{\prime\prime}(y_2,i)
=0
<\frac{\sigma^2(x,i)}2R_{f,\pi^{0,b}}^{\prime\prime}(x,i)=
\bar{h}_{f,i,b}(x). \label{13-2-9}
\end{eqnarray}
Let $\{y_{1n}\}$ and $\{y_{2n}\}$ be two sequences with $y_{1n}\downarrow y_1$ and $y_{2n}\uparrow y_2$ as $n\rightarrow \infty$ such that $\lim_{n\rightarrow\infty}
\frac{f(y_{1n},j)-f(y_1,j)}{y_{1n}-y_1}$ and $\lim_{n\rightarrow\infty}\frac{f(y_{2n},j)-f(y_2,j)}
{y_{2n}-y_2}$ exist for all $j\in \mathcal{S}$. It follows by \eqref{13-2-8} and \eqref{13-2-9} that $\frac{\bar{h}_{f,i,b}(y_{1n})
-\bar{h}_{f,i,b}(y_{1})}{y_{1n}-y_1}>0>
\frac{\bar{h}_{f,i,b}(y_{2n})-\bar{h}_{f,i,b}(y_{2})}{y_{2n}-y_2}
$.
By letting $n\rightarrow \infty$, we obtain\\
$k_{f,b}(y_1,i;\{y_{1n}\})-\mu(y_1,i)R_{f,\pi^{0,b}}^{\prime\prime}(y_1,i)
+\bar{l}R_{f,\pi^{0,b}}^{\prime\prime}(y_1,i)I\{y_1> b\}\ge 0$\\ and
$k_{f,b}(y_2,i;\{y_{2n}\})
-\mu(y_2,i)R_{f,\pi^{0,b}}^{\prime\prime}(y_2,i)
+\bar{l}R_{f,\pi^{0,b}}^{\prime\prime}(y_2,i)I\{y_2> b\} \le 0. $
Therefore, by noting $R_{f,\pi^{0,b}}^{\prime\prime}(y_1,i)=0=R_{f,\pi^{0,b}}^{\prime\prime}(y_2,i)$ (see \eqref{13-2-5}) we have
\begin{eqnarray}
&&k_{f,b}(y_1,i;\{y_{1n}\})\ge 0\ge k_{f,b}(y_2,i;\{y_{2n}\}).
\label{13-2-14}
\end{eqnarray}
On the other hand, note that $0<\delta_i+q_i-\mu^\prime(y_1,i)\le\delta_i+q_i-\mu^\prime(y_2,i)$ (due to the concavity of $\mu(\cdot,i)$), $R_{f,\pi^{0,b}}^{\prime}(y_1,i)<R_{f,\pi^{0,b}}^{\prime}(y_2,i)$ (see \eqref{13-2-2}), $\lim_{n\rightarrow\infty}\frac{f(y_{1n},j)-f(y_1,j)}{y_{1n}-y_1}\ge\lim_{n\rightarrow\infty}\frac{f(y_{2n},j)-f(y_2,j)}{y_{2n}-y_2}$ (due to the concavity of $f(\cdot,j)$). As a result,
$
k_{f,b}(y_1,i;\{y_{1n}\})< k_{f,b}(y_2,i;\{y_{2n}\})$, which is a contradiction to \eqref{13-2-14}.
\noindent (ii) We distinguish two cases: (a) $R_{f,\pi^{0,b}}^{\prime\prime}(b+,i)>0$ and (b) $R_{f,\pi^{0,b}}^{\prime\prime}(b+,i)\le 0$.\\
(a) Suppose $R_{f,\pi^{0,b}}^{\prime\prime}(b+,i)>0$. By \eqref{23513-1} we can find $N>0$ such that $x_N>b$ and $R_{f,\pi^{0,b}}^{\prime\prime}(x_N,i)\le 0$. Then by the continuity of the function $R_{f,\pi^{0,b}}^{\prime\prime}(\cdot,i)$ on $(b,\infty)$ (see Corollary \ref{the2}(i)) we know that there exists a $y_2\in (b,x_N]$ such that
$
R_{f,\pi^{0,b}}^{\prime\prime}(y_2,i)=0\ \mbox{ and }\ R_{f,\pi^{0,b}}^{\prime\prime}(x,i)>0\ \mbox{ for $x\in(b,y_2)$}.$
We now proceed to show that $R_{f,\pi^{0,b}}^{\prime\prime}(b-,i)\le 0$. Suppose the contrary, i.e., $R_{f,\pi^{0,b}}^{\prime\prime}(b-,i)> 0$. By noting $R_{f,\pi^{0,b}}^{\prime\prime}(0+,i)\le 0$ (see \eqref{13-2-1}), it follows that there exists a $y_1\in (0, b)$ such that
$
R_{f,\pi^{0,b}}^{\prime\prime}(y_1,i)=0$ and $R_{f,\pi^{0,b}}^{\prime\prime}(x,i)>0$ for $x\in(y_1,b)$. In summary, \eqref{13-2-5} holds for $x\in(y_1,y_2)-\{b\}$.
Rrepeating the argument right below \eqref{13-2-5} in (i), we obtain a contradiction.
\noindent (b) Suppose $R_{f,\pi^{0,b}}^{\prime\prime}(b+,i)\le 0$. It follows by \eqref{13-2-4} and the assumption $R_{f,\pi^{0,b}}^\prime(b,i)>\frac1{1+d}$ that
\begin{eqnarray}
&&R_{f,\pi^{0,b}}^{\prime\prime}(b-,i)=\lim_{x\uparrow b}\frac{2h_{f,i,b}(x,i)}{\sigma^2(x,i)}< \lim_{x\downarrow b}\frac{2h_{f,i,b}(x,i)}{\sigma^2(x,i)}
=R_{f,\pi^{0,b}}^{\prime\prime}(b+,i)\le 0.\label{23513-7}
\end{eqnarray}
We now show that $R_{f,\pi^{0,b}}^{\prime\prime}(x,i)\le 0$ for all $x\in[0,b)$. Suppose the contrary. That is, there exists some $x\in[0,b)$ such that $R_{f,\pi^{0,b}}^{\prime\prime}(x,i)> 0$. Then by noting $R_{f,\pi^{0,b}}^{\prime\prime}(0+,i)\le 0$ (see \eqref{13-2-1}) and $R_{f,\pi^{0,b}}^{\prime\prime}(b-,i)< 0$ (see \eqref{23513-7}), we can find $y_1$ and $y_2$ with $0\le y_1<y_2< b$ such that
$
R_{f,\pi^{0,b}}^{\prime\prime}(y_1,i)=0$, $R_{f,\pi^{0,b}}^{\prime\prime}(y_2,i)=0$ and $R_{f,\pi^{0,b}}^{\prime\prime}(x,i)>0$ for $x\in(y_1,y_2)$.
Repeating again the argument right after \eqref{13-2-5} in (i), we can obtain a contradiction.
\hfill $\square$
\noindent{\bf Theorem \ref{the5}} Note that $\tau_b^\pi=0$ given $X_0^\pi\ge b$. Hence, it follows from the definition \eqref{450} that \begin{eqnarray}
{W}_{f,b}(x,i)=\sup_{\pi\in\Pi}\E_{(x,i)}\left[R_{f,\pi^{0,b}}(X^\pi_{0},\xi_0) \right]=R_{f,\pi^{0,b}}(x,i)\ \mbox{ for $x\ge b$ and $b=0$}.\label{13-2-40}
\end{eqnarray}
We consider the case $b>0$. By Lemma \ref{13-2-20} (ii) we know that $R_{f,\pi^{0,b}}^{\prime\prime}(x,i)\le 0$ for $x\in [0,b)$, and $R_{f,\pi^{0,b}}^{\prime\prime}(b-,i)\le 0$.
Therefore, it follows by Corollary \ref{the2}(i) that
\begin{eqnarray}
\frac1{1-c}= R_{f,\pi^{0,b}}^\prime(0+,i)\ge R_{f,\pi^{0,b}}^\prime(x,i)\ge R_{f,\pi^{0,b}}^\prime(b,i)> \frac1{1+d}\ \mbox{ for $0<x\le b$}.\label{42}
\end{eqnarray}
Define $
w_{f,i}(y,j)=
R_{f,\pi^{0,b}}(y,i) \mbox{ if $j=i$, and $w_{f,i}(y,j)=
f(y,j)$ if $j\neq i$}.
$ Then by Corollary \ref{the2}(i) and Lemma \ref{theorem1} we know that $w_i(\cdot,j)$ satisfies the conditions in Lemma \ref{itolemma}. Then by applying Lemma \ref{itolemma} we know that for some positive sequence of stopping times $\{\tau_n;n=1,2,\cdots\}$ with $\lim_{n\rightarrow \infty}\tau_n=\infty$, the equation \eqref{new5} holds. By letting $t$ in \eqref{new5} be $\tau_b^\pi\wedge t$, noting that $X_s^{\pi}-X_{s-}^{\pi}=C_s-C_{s-}\ge 0$, and that given $(X_0,\xi_0)=(x,i)$, $X^\pi_{s-}\in[0,b)$ and $w_i(X^\pi_{s-}, \xi_{s-})=R_{f,\pi^{0,b}}(X^\pi_{s-},i)$ for $s\le \sigma_1\wedge \tau_b^\pi$, that $\sum_{0<s\le \tau_n\wedge \sigma_1 \wedge \tau_b^\pi \wedge t}e^{-\Lambda_s}\frac{X^\pi_{s}-X^\pi_{s-}}{1-c}
+\int^{\tau_n\wedge \sigma_1 \tau_b^\pi \wedge\wedge t}_0
\frac{e^{-\Lambda_s}}{1-c}\dif
\tilde{C}_s=
\int_{0}^{\tau_n\wedge \sigma_1 \tau_b^\pi \wedge\wedge t}\frac{e^{-\Lambda_s}}{1-c}\dif
C_s$, and using\eqref{42}, we derive that for any $\pi\in \Pi$, $t>0$ and $0\le x\le b$,
\begin{eqnarray}
&&\E_{(x,i)}\bigg[\int_0^{\tau_n\wedge \sigma_1\wedge \tau_b^\pi \wedge t}\frac{l_se^{-\Lambda_s}}{1+d}\dif s-\int_0^{\tau_n\wedge \sigma_1\wedge \tau_b^\pi \wedge t}\frac{e^{-\Lambda_s}}{1-c}\dif C_s\nonumber\\&+&e^{-\Lambda_{\tau_n\wedge \sigma_1\wedge \tau_b^\pi \wedge t}}w_i(X^\pi_{\tau_n\wedge \sigma_1\wedge \tau_b^\pi \wedge t},\xi_{\tau_n\wedge \sigma_1\wedge \tau_b^\pi \wedge t}) \Bigg]
\le
R_{f,\pi^{0,b}}(x,i).\label{45001}
\end{eqnarray}
Note that the functions $R_{f,\pi^{0,b}}(\cdot,j)$ and $f(\cdot,j)$ $j\in\mathcal{S}$ are all bounded. Hence, the functions $w_i(\cdot,j)$ $j\in\mathcal{S}$ are also bounded.
By letting $\tau_n\rightarrow\infty$ and $t\rightarrow \infty$ on both sides of \eqref{45001}, using the monotone convergence theorem and the dominated convergence theorem and noticing that due to $\xi_{s}=\xi_0$ for $0\le s<\sigma_1$ we have
$\E_{(x,i)}\bigg[e^{-\Lambda_{ \tau_b^\pi\wedge\sigma_1}}w_{f,i}(X^\pi_{ \tau_b^\pi\wedge \sigma_1},\xi_{ \tau_b^\pi\wedge \sigma_1})\bigg]=\E_{(x,i)}\bigg[e^{-\Lambda_{ \tau_b^\pi}}R_{f,\pi^{0,b}}(b,\xi_0)I\{\tau_b^\pi< \sigma_1\}+e^{-\Lambda_{ \sigma_1}}f(X_{\sigma_1},\xi_{\sigma_1})I\{\sigma_1\le \tau_b^\pi\}\bigg]$ and that $\pi$ is an arbitrary admissible strategy and \eqref{450},
we can conclude
\begin{eqnarray}
W_{f,b}(x,i)\le
R_{f,\pi^{0,b}}(x,i)\ \mbox{ for } 0\le x\le b.\label{4220}
\end{eqnarray}
Note that $\{(X_t^{0,b},\xi_t);t\ge 0\}$ is a strong Markov process and that by the Markov property it follows that
\begin{align}
R_{f,\pi^{0,b}}(x,i)
&=\E_{(x,i)}\bigg[\int_0^{ \tau_b^{\pi^{0,b}}\wedge \sigma_1}\frac{\bar{l}e^{-\Lambda_s}}{1+d}I\{X^{0,b}_s\ge b\}\dif s-\int_0^{ \tau_b^{\pi^{0,b}}\wedge \sigma_1}\frac{e^{-\Lambda_s}}{1-c}\dif C_s\nonumber\\
&+e^{-\delta( \tau_b^{\pi^{0,b}}\wedge \sigma_1)}R_{f,\pi^{0,b}}(X_{ \tau_b^{\pi^{0,b}}\wedge \sigma_1}^{0,b},\xi_{ \tau_b^{\pi^{0,b}}\wedge \sigma_1}) \bigg]\le W_{f,b}(x,i) \mbox{ for $ x\ge 0$},\label{510}
\end{align}
where the last inequality follows by noting $\pi^{0,b}\in \Pi$ and the definition \eqref{450}.
Combining \eqref{13-2-40}, \eqref{4220} and \eqref{510} completes the proof.
\hfill$\square$
\noindent {\bf Proof of Theorem \ref{23513-9}}
We first show that
\begin{eqnarray}
&& R_{f,\pi^{0,b}}^\prime(x,i)\le R_{f,\pi^{0,b}}^\prime(b,i)= \frac1{1+d}\ \mbox{ for $x>b$, $b\ge 0$}.\label{13-2-31}
\end{eqnarray}By Lemma \ref{13-2-20}(i) it follows that $R_{f,\pi^{0,0}}^{\prime\prime}(x,i)\le 0$ for $x\ge 0$. As a result, \eqref{13-2-31} holds for $b=0$. Now suppose $b>0$. By Lemma \ref{theorem1} (i) we know that $R_{f,\pi^{0,b}}^\prime(0+,i)=\frac1{1-c}$. Since $R_{f,\pi^{0,b}}^\prime(b,i)=\frac1{1+d}$, it follows by Corollary \ref{the2} (ii) that $R_{f,\pi^{0,b}}(\cdot,i)$ is twice continuously differentiable on $[0,\infty)$ and by Lemma \ref{13-2-20} (i) that $R^{\prime\prime}_{f,0,b}(x,i)\le 0$ for $x\ge 0$. Hence, \eqref{13-2-31} holds for $b>0$ as well, and
\begin{eqnarray}
&&\frac1{1-c}=R_{f,\pi^{0,b}}^\prime(0+,i)\ge R_{f,\pi^{0,b}}^\prime(x,i)\ge R_{f,\pi^{0,b}}^\prime(b,i)= \frac1{1+d}\ \mbox{ for $x\in [ 0,b]$}.\label{17713-5}
\end{eqnarray}
It follows by using \eqref{13-2-31} and \eqref{17713-5}, and noting $\bar{l}\ge l_s$ for $s\ge 0$ we obtain that for $b\ge 0$,
\begin{eqnarray}
&&\bar{l}I\{X^\pi_{s}\ge b\}\left(R_{f,\pi^{0,b}}^{\prime}(X^\pi_{s-},i)-\frac1{1+d}\right)
-l_sR_{f,\pi^{0,b}}^{\prime}(X^\pi_{s-},i)\nonumber\\
&=&(\bar{l}-l_s)I\{X^\pi_{s}\ge b\}R_{f,\pi^{0,b}}^{\prime}(X^\pi_{s-},i)-\frac{\bar{l}}{1+d}I\{X^\pi_{s}\ge b\}-l_sI\{X^\pi_{s}< b\}R_{f,\pi^{0,b}}^{\prime}(X^\pi_{s-},i)\nonumber\\
&\le&\frac{\bar{l}-l_s}{1+d}I\{X^\pi_{s}\ge b\}-\frac{\bar{l}}{1+d}I\{X^\pi_{s}\ge b\}-\frac{l_s}{1+d}I\{X^\pi_{s}< b\}=-\frac{l_s}{1+d},\label{0fs9}
\end{eqnarray}
By \eqref{13-2-31} again we can obtain
\begin{eqnarray}
&& R_{f,\pi^{0,b}}^\prime(x,i)\le \frac1{1-c}\ \mbox{ for $b\ge 0$ and $x>b$}.\label{13-2-313}
\end{eqnarray}
Further, note that for $b\ge 0$ and any $t\ge 0$,
\begin{align}
&\E_{(x,i)}\bigg[\int_{0<s\le \sigma_1 \wedge
t}e^{-\Lambda_s}R_{f,\pi^{0,b}}^\prime(X^\pi_{s},\xi_{s-})\dif \tilde{C}_s
+ \sum_{0<s\le \sigma_1 \wedge
t}e^{-\Lambda_s}\left(R_{f,\pi^{0,b}}(X^\pi_{s},\xi_{s-})-R_{f,\pi^{0,b}}(X^\pi_{s-},\xi_{s-})\right)\bigg]\nonumber\\
&\le \E_{(x,i)}\bigg[\int_{0}^{ \sigma_1 \wedge
t}\frac{e^{-\Lambda_s}}{1-c}\dif \tilde{C}_s+ \sum_{0<s\le \sigma_1 \wedge
t}\frac{e^{-\Lambda_s}}{1-c}(X^\pi_{s}-X^\pi_{s-})\bigg]
=\E_{(x,i)}\bigg[ \sum_{0<s\le \sigma_1 \wedge
t}\frac{e^{-\Lambda_s}}{1-c}\dif C_s\bigg],\label{17713-7}
\end{align}
where the last inequality follows by \eqref{17713-5}, \eqref{13-2-313}, $\dif\tilde{C}_s\ge 0$, $X_s^{\pi}-X_{s-}^{\pi}=C_s-C_{s-}\ge 0$ and $\dif C_s=\dif \tilde{C}_s+C_s-C_{s-}$.
Define $
w_{f,i}(y,j)=R_{f,\pi^{0,b}}(y,i)$ if $j=i$, and $w_{f,i}(y,j)=f(y,j)$ if $j\neq i$.
Then by Corollary \ref{the2}(i) and Lemma \ref{theorem1} we know that the conditions in Lemma \ref{theorem1} are satisfied. By applying Lemma \ref{itolemma} we know that for some positive sequence of stopping times $\{\tau_n;n=1,2,\cdots\}$ with $\lim_{n\rightarrow \infty}\tau_n=\infty$, the equation \eqref{new5} holds for any $\pi\in\Pi$, any $b,t>0$ and any $n\in\mathbb{N}$. By using
\eqref{new5}, \eqref{0fs9} and \eqref{17713-7} (setting $t=t\wedge \tau_n$) we arrive at
$R_{f,\pi^{0,b}}(x,i)\ge \E_{(x,i)}\bigg[\int^{ \sigma_1\wedge
t\wedge \tau_n}_0\frac{l_se^{-\Lambda_s}}{1+d}\dif s-\sum_{0}^{ \sigma_1 \wedge
t\wedge \tau_n}\frac{e^{-\Lambda_s}}{1-c}\dif C_s
+e^{-\Lambda_{ \sigma_1\wedge
t\wedge \tau_n}}w_{f,i}(X^\pi_{ \sigma_1\wedge t\wedge \tau_n},\xi_{ \sigma_1\wedge t\wedge \tau_n})\bigg]$ for $b\ge 0$.
By noting that the functions $R_{f,\pi^{0,b}}(\cdot,i)$ and $f(\cdot,j)$, $j\in \mathcal{S}$ are bounded and letting $t\rightarrow \infty$ and then $n\rightarrow \infty$ and then using the monotone convergence theorem for the first two terms inside the expectation and the dominated convergence theorem for the last term, we obtain that for $b\ge 0$,
$
R_{f,\pi^{0,b}}(x,i)
\ge\E_{(x,i)}\bigg[\int^{ \sigma_1}_0\frac{l_se^{-\Lambda_s}}{1+d}\dif s-\int_0^{ \sigma_1 }\frac{e^{-\Lambda_s}}{1-c}\dif C_s
+e^{-\Lambda_{ \sigma_1}}w_{f,i}(X^\pi_{ \sigma_1},\xi_{ \sigma_1})\bigg]
.$
By noting $w_{f,i}(X^\pi_{ \sigma_1},\xi_{ \sigma_1})=f(X^\pi_{ \sigma_1},\xi_{ \sigma_1})$ given $\xi_0=i$, the arbitrariness of $\pi$ and the definition of $V_f$ in \eqref{17713-4} we conclude
$
R_{f,\pi^{0,b}}(x,i)\ge
V_f(x,i)\ \mbox{ for $x\ge 0$.}$
On the other hand, $R_{f,\pi^{0,b}}(x,i)\le V_f(x,i)$ for $x\ge 0$ according to the definition \eqref{17713-4}. Consequently, $R_{f,\pi^{0,b}}(x,i)=V_f(x,i)$ for $x\ge 0$.
\hfill$\square$
\noindent {\bf Proof of Lemma \ref{lemff1}} Recall that $\tau_b^\pi$ is defined in \eqref{21714-1}. By Theorem \ref{the5} it follows that for any large enough $b$ and any $x\ge 0$,
\begin{align*}
R_{f,\pi^{0,b}}(x,i)&=W_{f,b}(x,i)=\sup_{\pi\in\Pi}\E_{(x,i)}\bigg[\int^{ \sigma_1 \wedge
\tau_b^\pi}_0\frac{l_se^{-\Lambda_s}}{1+d}\dif s-\int^{ \sigma_1 \wedge
\tau_b^\pi}_0\frac{e^{-\Lambda_s}}{1-c}\dif C_s\nonumber\\
&+e^{-\Lambda_{\tau_b^\pi}}R_{f,\pi^{0,b}}(b, \xi_0)I\{\tau_b^\pi< \sigma_1\}+e^{-\Lambda_{\sigma_1}}f(X^\pi_{\sigma_1}, \xi_{\sigma_1})I\{\sigma_1\le \tau_b^\pi\} \bigg] \nonumber\\
\ge& \sup_{\pi\in\Pi}\E_x\Bigg[\int^{ \sigma_1\wedge
\tau_b^\pi}_0\frac{l_se^{-\Lambda_s}}{1+d}\dif s-\int^{ \sigma_1 \wedge
\tau_b^\pi}_0\frac{e^{-\Lambda_s}}{1-c}\dif C_s+e^{-\Lambda_{\sigma_1}}f(X^\pi_{\sigma_1}, \xi_{\sigma_1})I\{\sigma_1\le \tau_b^\pi \}\Bigg].
\end{align*}
Note $\lim_{b\rightarrow\infty}\tau_b^\pi=\infty$ and $f$ is bounded. Then it follows by letting $b\rightarrow \infty$ on both sides, and then using the monotone convergence theorem twice and the dominated convergence that
$\liminf_{b\rightarrow\infty}R_{f,\pi^{0,b}}(x,i)\ge\sup_{\pi\in\Pi}\E_{(x,i)}\left[\int^{ \sigma_1 }_0\frac{l_se^{-\Lambda_s}}{1+d}\dif s-\int^{ \sigma_1 }_0\frac{e^{-\Lambda_s}}{1-c}\dif C_s+e^{-\Lambda_{\sigma_1}}f(X^\pi_{\sigma_1}, \xi_{\sigma_1})\right]=V_f(x,i)$ for $x\ge 0.$
This combined with the fact $R_{f,\pi^{0,b}}(x,i)\le V_f(x,i)$ for $x\ge 0$ completes the proof.\hfill $\square$
\noindent{\bf Proof of Theorem \ref{23513-11}}
(i) $b_i^f\ge 0$ is obvious by the definition. We just need to prove $b_i^f<\infty$. Suppose the contrary. Then by \eqref{23513-10} we have $R_{f,\pi^{0,b}}^{\prime}(b,i)> \frac1{1+d}$ for all $b\ge 0$. Hence, it follows by Lemma \ref{lemff1} that $V_f(x,i)=\lim_{b\rightarrow \infty}R_{f,\pi^{0,b}}(x,i)$ for $x\ge 0$.
For any $b\ge 0$, by Theorem \ref{the5} we know $R_{f,\pi^{0,b}}^\prime(x,i)>\frac1{1+d}$ for $x\in(0,b]$, which implies
$R_{f,\pi^{0,b}}(x,i)>R_{f,\pi^{0,b}}(0,i)+\frac{x}{1+d}$ for $x\in(0,b]$. Hence, for any $x\ge 0$, we can find a $b>x$ such that $V_f(x,i)\ge R_{f,\pi^{0,b}}(x,i)>R_{f,\pi^{0,b}}(0,i)+\frac{x}{1+d}$. Hence, $\lim_{x\rightarrow\infty}V_f(x,i)=+\infty$, which contradicts $V_f(x,i)\le \frac{\bar{l}}{\underline{\delta}(1+d)}$ for $x\ge 0$ (see Lemma \ref{remff9}).
\noindent (ii) is a result of (i) and Theorem \ref{23513-9}.
\hfill $\square$
\noindent {\bf Proof of Theorem \ref{6214-4}}
(i) Define an operator $\mathcal{P}$ by
\begin{eqnarray}
\mathcal{P}(f)(x,i):=V_f(x,i),\ x\ge 0,i\in\mathcal{S}\ \mbox{ and $f\in\mathcal{C}$}.\label{6214-2}
\end{eqnarray}
Then by Theorem \ref{23513-11} we have, \begin{eqnarray}
\mathcal{P}(f)(x,i)=V_f(x,i)=R_{f,\pi^{0,b_i^f}}(x,i),\ x\ge 0,i\in\mathcal{S}\ \mbox{ and $f\in\mathcal{C}$}.\label{6214-1}
\end{eqnarray}
Recall that $\mathcal{D}\subset\mathcal{C}$ and $(\mathcal{D},||\cdot||)$ is a complete space.
We will first show that $\mathcal{P}$ is a contraction on $(\mathcal{D},||\cdot||)$. Consider any $f\in\mathcal{D}$. It follows by Lemma \ref{remff9} and \eqref{6214-1} that $\mathcal{P}(f)=V_f\in\mathcal{C}$. Note that for any $f\in\mathcal{D}$ and $i\in \mathcal{S}$, $b_i^f<\infty$ according to Theorem \ref{23513-11}. Further notice that by Lemma \ref{theorem1} (ii), we know $R_{f,\pi^{0,b}}^\prime(b,i)$ is continuous in $b$ and $R_{f,\pi^{0,0}}^\prime(0+,i)=\frac1{1-c}>\frac1{1+d}$ by Corollary \ref{the2} (i). Hence, according to the definition of $b_i^f$ in \eqref{23513-10}, we have $R_{f,\pi^{0,b^f_i}}^\prime(b_i^f,i)=\frac1{1+d}$. Therefore, it follows by Corollary \ref{the2} that
for any $i\in\mathcal{ S}$, the function $R_{f,\pi^{0,b^f_i}}(\cdot,i)$ is twice continuously differentiable on $(0,\infty)$ and by Lemma \ref{13-2-20} (i) that $R_{f,\pi^{0,b^f_i}}(\cdot,i)$ is concave. Notice that by Corollary \ref{the2} (i) again $R_{f,\pi^{0,b^f_i}}^\prime(0+,i)=\frac1{1-c}$. Hence, $\frac{\dif}{\dif x}\mathcal{P}(f)(x,i)=R_{f,\pi^{0,b^f_i}}^\prime(x,i)\le R_{f,\pi^{0,b^f_i}}^\prime(0+,i)=\frac1{1-c}$ for $x>0$, which results in $\frac{\mathcal{P}(f)(x,i)-\mathcal{P}(f)(y,i)}{x-y}\le \frac1{1-c}$ for $0\le x<y$. Therefore, we can conclude $\mathcal{P}(f)\in\mathcal{D}$. For any $f_1,f_2\in\mathcal{D}$, it follows by \eqref{6214-2} that
\begin{eqnarray}
&& ||\mathcal{P}(f_1)-\mathcal{P}(f_2)||\nonumber\\
&=&\sup_{(x,i)\in\mathbb{R}^+\times \mathcal{S}}|V_{f_1}(x,i)-V_{f_2}(x,i)|
=\sup_{(x,i)\in\mathbb{R}^+\times \mathcal{S}}\left|\sup_{\pi\in\Pi} R_{f_1,\pi}(x,i)-\sup_{\pi\in\Pi}R_{f_2,\pi}(x,i)\right|\nonumber\\
&\le&\sup_{(x,i)\in\mathbb{R}^+\times \mathcal{S}}\sup_{\pi\in\Pi} |R_{f_1,\pi}(x,i)-R_{f_2,\pi}(x,i)|
\sup_{(x,i)\in\mathbb{R}^+\times E}E_{(x,i)}\left[e^{-\Lambda_{\sigma_1}}||
f_1-f_2||\right]\nonumber\\
&=&||
f_1-f_2||\int_0^\infty q_ie^{-q_i t}e^{-\delta_i t}\dif t=\max_{i\in E}\frac{q_i}{q_i+\delta_i}||f_1-f_2||,\label{jth2}
\end{eqnarray}
where the last inequality follows by \eqref{4214-2} and the last equality follows by noting that $\sigma_1$ is exponentially distributed with mean $\frac1{q_i}$. Therefore, $\mathcal{P}$ is a contraction on the space
$(\mathcal{D},||\cdot||)$.
Note that for any $f\in\mathcal{C}$ and $i\in\mathcal{S}$, $f(\cdot,i)$ is non-decreasing. Hence, it follows by \eqref{4214-2} and \eqref{6214-1} that the operator $\mathcal{P}$ is non-decreasing. Consider two functions $g_1,g_2$ defined by $g_1(x,i)=0$ and $g_2(x,i)=\frac{\overline{l}}{\underline{\delta}(1+d)}$. It is not hard to verify that $g_1,g_2\in\mathcal{D}$ and $g_1\le V\le g_2$. Hence, $\mathcal{P}(g_1)\le\mathcal{P}(V)\le \mathcal{P}(g_2)$. Note that by \eqref{60214-3} $\mathcal{P}(V)=V$. Hence, $\mathcal{P}(g_1)\le V\le \mathcal{P}(g_2)$. Apply the operator $\mathcal{P}$ once again, we have $\mathcal{P}^2(g_1)\le V\le \mathcal{P}^2(g_2)$. By repeating this $n-2$ more times, we obtain $\mathcal{P}^n(g_1)\le V\le \mathcal{P}^n(g_2)$. As a result, $\lim_{n\rightarrow}\mathcal{P}^n(g_1)\le V\le \lim_{n\rightarrow \infty}\mathcal{P}^n(g_2)$. Since $\mathcal{P}$ is a contraction on the complete space $(\mathcal{D},||\cdot||)$, there is a unique fixed point in $\mathcal{D}$ and is identical to both $\lim_{n\rightarrow \infty}\mathcal{P}^n(g_1)$ and $\lim_{n\rightarrow \infty}\mathcal{P}^n(g_2)$. Consequently, $\lim_{n\rightarrow \infty}\mathcal{P}^n(g_2)=V=\lim_{n\rightarrow \infty}\mathcal{P}^n(g_2)$. As a result, $V\in\mathcal{D}$.
\noindent (ii) The results follow immediately by (i) and Theorem \ref{23513-11}.
\hfill $\square$
\noindent {\bf Proof of Theorem \ref{thmj1}}
Since, $ b^V_i< \infty$ for all $i\in \mathcal{S}$,
we can define an operator $\mathcal{Q}$ by
\begin{eqnarray}
\mathcal{Q}(f)(x,i)=R_{f,\pi^{0,b^V_i}}(x,i)\ \mbox{
for $f\in\mathcal{C}$, $x\ge 0$, and $i\in \mathcal{S}$}.\label{jth6}
\end{eqnarray}
The function $R_{f,\pi^{0,b^V_i}}$ is obviously nonnegative according to its definition. It follows by Lemma \ref{remff9} that $R_{f,\pi^{0,b^V_i}}\le V_f\le \frac{\bar{l}}{{\underline{\delta}(1+d)}}$ and by Corollary \ref{the2} that the function $R_{f,\pi^{0,b^V_i}}(\cdot,i)$ is increasing. Therefore, $R_{f,\pi^{0,b^V_i}}\in\mathcal{C}$. Then by \eqref{jth6} we know $\mathcal{Q}(f)\in\mathcal{C}$.
It follows by \eqref{4214-2} that
\begin{eqnarray*}
||\mathcal{Q}(f_1)-\mathcal{Q}(f_2)||
&=&\sup_{(x,i)\in\mathbb{R}^+\times \mathcal{S}}|R_{f_1,\pi^{0,b^V_i}}(x,i)-R_{f_2,\pi^{0,b^V_i}}(x,i)|\nonumber\\
&\le&\sup_{(x,i)\in\mathbb{R}^+\times E}E_{(x,i)}\left[e^{-\Lambda_{\sigma_1}}||
f_1-f_2||\right]\nonumber\\
&=&||
f_1-f_2||\int_0^\infty q_ie^{-q_i t}e^{-\delta_i t}\dif t=\max_{i\in E}\frac{q_i}{q_i+\delta_i}||f_1-f_2||.
\end{eqnarray*}
Consequently, $\mathcal{Q}$ is a contraction on $(\mathcal{C},||\cdot||)$. Hence, there is a unique fixed point of $\mathcal{Q}$ on $(\mathcal{C},||\cdot||)$.
Note by \eqref{jth6} we have
$ \mathcal{Q}(V)(x,i)=R_{V,\pi^{0,b^V_i}}(x,i)=V(x,i)$, where the last equality follows by Theorem \ref{6214-4} (ii). Therefore, $V$ is a fixed point.
By \eqref{jth6} and noticing that $\pi^{0,b_i^V}$ and $\pi^*$ are identical before $\sigma_1$, we have
\begin{align}\mathcal{Q}(R_{\pi^*})(x,i)&=R_{R_{\pi^*},\pi^{0,b_i^V}}(x,i)
\\
&=\E_{(x,i)}\bigg[\int_0^{ \sigma_1} e^{-\Lambda_t}\frac{l_t^*}{1+d}\dif t-\int_0^{ \sigma_1} e^{-\Lambda_t}\frac1{1-c}\dif C_t^*\nonumber\\
&+e^{-\Lambda_{\sigma_1}}R_{\pi^*}(X^{\pi^*}_{\sigma_1}, \xi_{\sigma_1})
\bigg],\ x\ge 0,i\in\mathcal{S},\label{4214-21}
\end{align}
where the last equality follows by \eqref{4214-2}.
It is not hard to see that the process $(X^{\pi^*},J)$ is a Markov process. Hence, it follows by the Markov property that
\begin{align}R_{\pi^*}(x,i)
&=\E_{(x,i)}\bigg[\int_0^{ \sigma_1} e^{-\Lambda_t}\frac{l_t^*}{1+d}\dif t-\int_0^{ \sigma_1} e^{-\Lambda_t}\frac1{1-c}\dif C_t^*\nonumber\\
&+e^{-\Lambda_{\sigma_1}}R_{\pi^*}(X^{\pi^*}_{\sigma_1}, \xi_{\sigma_1})
\bigg],\ x\ge 0,i\in\mathcal{S}.\label{4214-20}
\end{align}
Combining \eqref{4214-21} and \eqref{4214-20} we obtain
$
\mathcal{Q}(R_{\pi^*})(x,i)=R_{\pi^*}(x,i), \ x\ge 0,i\in\mathcal{S}.$
Therefore, $R_{\pi^*}$ is also a fixed point. As there is a unique fixed point, we conclude $V=R_{\pi^*}$.\hfill $\square$\\
\noindent {\bf Acknowledgements}
This work was supported by the University of New South Wales Australian Business School Special Research Grants.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 8,147 |
Published by The History Press
Charleston, SC 29403
www.historypress.net
Copyright © 2011 by Frank Jump
All rights reserved
First published 2011
e-book edition 2013
Cover design by Julie Scofield
Manufactured in the United States
ISBN 978.1.62584.144.5
Library of Congress Cataloging-in-Publication Data
Jump, Frank.
Fading ads of New York City / Frank Jump.
p. cm.
ISBN 978-1-60949-438-4
1. Advertising--New York (State)--New York--History. 2. Brick wall signs--New York (State)--New York--History. 3. Street art--New York (State)--New York--History. I. Title.
HF5841.J86 2011
659.13'42--dc23
2011038134
_Notice_ : The information in this book is true and complete to the best of our knowledge. It is offered without guarantee on the part of the author or The History Press. The author and The History Press disclaim all liability in connection with the use of this book.
All rights reserved. No part of this book may be reproduced or transmitted in any form whatsoever without prior written permission from the publisher except in the case of brief quotations embodied in critical articles and reviews.
_This book is dedicated to Tucker and Eric Ashworth_.
_Both would have been instrumental in the making of this book and ecstatic about its completion_.
Contents
Introduction, by Wm. Stage, author of _Ghost Signs: Brick Wall Signs in America_ , 1989
On Advertising Legend Douglas Leigh, by Tod Swormstedt, founder of the American Sign Museum
Foreword. Fading Ads: An Urban Archaeology of Life and Photography, by Dr. Andrew Irving, visual anthropologist
Acknowledgements
New York City's Fading Ad Campaign: The Chrome Age, by Frank H. Jump
SNAKE OILS, ELIXIRS, TONICS, CURE-ALLS AND LAXATIVES
Days with Art: Lingering in Frank Jump's Images, by Amy Sadao, Visual AIDS
FOOD, SNACKS AND CANDY
A COKE AND A SMOKE
BREWERIANA
City of Immigrants and the Mack Sign Company, by Walter Grutchfield— _14 th to 42nd Street_
SHOES, HATS AND ASSORTED GARMENTS—WHOLESALE AND RETAIL
Might as Well jUmP! Reflections on the Color Blue, by Kevin Walsh, _Forgotten New York_
LAUNDRY PRODUCTS, WASHING MACHINES AND REAL ESTATE
JEWELRY AND ACCESSORIES
SAVINGS, LOANS AND FUR VAULTS
PAINT AND HARDWARE
HORSE AND CARRIAGE
FUEL, OIL, GAS AND COAL
Fading Light, by Julian Seery Gude, great-great-grandson of O.J. Gude
MUSIC AND ENTERTAINMENT
HOTELS, SPEAKEASIES AND SAUNAS
Rosario Dawson and Her Uncle Frank
PROSTHESES AND UNDERTAKERS
Fading Ads and a Transatlantic Relationship, by Sam Roberts, _UK Brickads_
Epilogue. The Collaged City, by Kathleen Hulser, public historian, New-York Historical Society
Notes
About the Author
Introduction
Frank Jump is a commercial archaeologist, plain and simple. There is no school, no curriculum for this discipline; one earns his degree by taking an abiding interest in the subject, be it Route 66 filling stations, vintage brick pavers or neon signs, and one delves into the matter. One gets involved. Frank's particular passion is one that I share as well: documenting fading advertisements on brick walls, visual whispers from the past found most commonly in the warehouse and factory districts and tucked away in the older neighborhoods of our cities.
Getting involved means canvassing these areas, locating the best examples of fades—Frank's sobriquet—deciphering their texts, often obscured or jumbled; perhaps researching some extinct brand of coffee, flour or stove polish; capturing these ads on camera; and, finally, presenting them in an accessible and hopefully enjoyable manner. I am happy to report that Frank Jump has done all this, exceeding expectations in every way. No one paid him to do this; he embarked on this ambitious undertaking without thought of compensation other than the joy that comes from a job well done.
In 1997, Frank Jump's Fading Ad Campaign began as a 35mm chrome photographic project documenting vintage commercial advertisements on brick faces of buildings in the New York City area. Some of these ads, he was delighted to learn, were more than a century old, touting all-purpose patent medicines, horseshoeing services, long-forgotten cigars and whiskeys. He began roaming the city, photographing them in earnest.
Obtaining good photographs is not as easy as one might think. One soon learns that walls are not to be photographed at just any time of day. Depending on which direction they face, sunrise or sunset may work best for bringing out that warm cast on the brick. For other ads that are faded to the point of near-illegibility—ghost signs—an overcast day may serve to best bring out the outlines of the letters. North-facing walls can be a bother, the sunlight never quite bending enough to fully illuminate the old ad painted there. Moreover, the conscientious documentarian is not content with shooting large ads on the sides of tall buildings from street level; he wants elevation to get the angle right. Frank has more than a few stories about talking his way into lofty apartments and businesses in order to get those shots that otherwise would be impossible. And, if pressed, he would also admit to scaling rickety fire escapes; stopping in the No Stopping zone of the interstate; and taking an unauthorized stroll on the elevated tracks of the J Train, between stations, snapping pictures, ignoring the conductor's commands to get the hell off. Had the MTA police arrested him for trespassing it would have been poetic indeed; all for the love of the sign.
Frank's pictures and experiences have been funneled into a website and blog, fadingad.com, an interactive and readily navigable format showcasing, as mentioned, the vintage wall signs of New York City and surrounding locales. And there are some pretty good ones, ones that will make you smile, ones that may have escaped our attention had it not been for Frank's intrepid odyssey that continues to this very day. Interestingly, certain signs found in Frank's pictorial survey have transcended media, attracting artists who render them as watercolor paintings. In these instances, the sign has come full circle: from being painted on a wall, originally, to being photographed to being painted and placed back on a wall. A gallery wall.
And so it was that Frank's Fading Ad Campaign would be recognized as something unique and valuable, the popular press, both here and abroad, running features about these hoary old signs and the man who captures them. Now that Frank has this book coming out, I must say that it is about time, for there is a fraternity of commercial archaeologists out there who anticipate turning its pages and drinking its contents like a parched pilgrim at a desert well. Frank once mentioned that of the thousands of ads he's photographed, many have already been covered up, vandalized or destroyed. Yet he took heart that many others "still silently cling to the walls of buildings, barely noticed by the rushing passersby." Well, this book will get people to notice.
Wm. Stage
St. Louis, July 2011
Author of _Ghost Signs: Brick Wall Signs in America_ , 1989
On Advertising Legend Douglas Leigh
Although I had learned about and come to respect Frank Jump's work in documenting ghost signs, it wasn't until the summer of 1999 that I had the opportunity to meet him in person. Jump happened to be on a road trip, heading back to New York City, and took time to stop and see what I was doing as founder of the American Sign Museum here in Cincinnati. The museum was very much in its infancy then, and I had just begun to assemble a collection of vintage signs and sign-related items.
Several months later, I had the occasion to visit with Frank and his partner, Vincenzo, at their Brooklyn home. That opportunity was all about our mutual interests, and we've remained friends as both of our projects progressed. I will never forget that first visit to see Frank...
My trip to New York was a last-minute mission of mercy. The urgency had been created by a phone call I received from Ilaria Borghese, the great-granddaughter of Douglas Leigh, the creative genius behind Times Square's Great White Way. As she explained, Leigh's widow (and second wife), Elsie, was planning to clean out their former Upper East Side apartment in the next two days, and all was going in a dumpster. She said, "If you want anything, you'd better get up here and grab it."
I couldn't believe it—Douglas Leigh's incredible legacy being tossed in a dumpster. I booked the next flight I could get to LaGuardia. As I was scrambling to get details together, I remembered Frank's invitation from the summer before to stop by. I called him and, in a rather frantic voice, tried to explain my dilemma, asking if he could pick me up at the airport and let me stay overnight. "Sure," he said without hesitation. "You can tell me all about it when you get here."
He picked me up that evening, and over dinner, I rehashed my conversation with Ilaria and told him my plan was to get over to the apartment and save as much as I could. Frank said he would drive me over to the former Leigh penthouse first thing in the morning. He told me he had to be at work at noon that day but he'd do whatever he could to help up until he had to leave.
What Frank and I found when we exited the seventh-floor elevator was an expansive apartment with boxes piled everywhere. It was just like Ilaria said it would be: a bunch of workmen gathering up the boxes indiscriminately and loading them onto the same elevator for disposal in the dumpster waiting street-side. Frank and I were able to put the workmen off for a time while we scurried around taking stock of the various piles and trying to segment the archival items from the clothes, furniture and other personal items. Toward the end, we were actually grabbing boxes from the workmen's arms and stacking them to the side.
At one point, Elsie asked if we wanted "the Snowflake," and we both looked at each other and said in unison, "The Snowflake?" Unfortunately, I was not equipped to ship the several-ton illuminated snowflake that had hung over the intersection of 57th and Fifth Avenue every holiday season. By the end of the day, we were able to save a little more than seven hundred items, dominated by historic photographs, slides and sketches of Leigh's work and nearly three hundred cans of 16mm promotional films. We were even fortunate to save such things as Leigh's Rolodex and several personally annotated scrapbooks of newspaper and magazine clippings documenting Leigh's career.
When I founded the American Sign Museum, the mission was to inform and educate the general public, as well as businesses and special interest groups, about the history of the sign industry and its significant contribution to commerce and the American landscape. Frank Jump has played a part as preservationist in this mission, having spent the last two decades urgently documenting the history of mural advertisements throughout the five boroughs of New York City with his Fading Ad Campaign.
Tod Swormstedt
Founder of the American Sign Museum, Cincinnati, Ohio
Former editor and publisher of _Signs of the Times Magazine_
FOREWORD
Fading Ads
_An Urban Archaeology of Life and Photography_
This book tells two stories: that of New York City and its obsession with money, advertising and renewal over the last 150 years; and the story of the life of a teacher and photographer who has dedicated much of his time to documenting and archiving the hundreds of gigantic advertisements that were painted, often by hand, on the sides of walls and buildings. These advertisements mostly date from the late 1800s and early 1900s, but their traces can still occasionally be glimpsed today throughout different parts of the city. Many have disappeared; they either perished when the buildings they were painted on were knocked down or they were simply covered up by bigger, newer buildings going up as part of New York's ever-present desire to reinvent and rebuild itself. The destiny of other advertisements, however, was much less dramatic and more gradual. For regardless of the thickness of their paint or intensity of their original colours, their fate has been to s-l-o-w-l-y fade out of existence while exposed to the city's scorching summers and freezing winters, remaining open to the relentless cycles of sun, rain, snow and ice in a dense urban climate of pollution and humidity that inevitably takes its toll.
It is remarkable how many of these advertisements refused to succumb to the forces of time, fashion and urban planning and still remain visible from different vantage points around the city. From Carriages, Coupes and Hansoms in Chelsea (page 161) to Omega Oil in Harlem (pages 36–37), a good number of these enormous painted advertisements have endured through the years and continue to look down on New York's neighborhoods. Some are more than a century old and have graced the sides of the same bricks, walls and buildings since they were first painted. Others may have recently perished, but thanks to Frank H. Jump's intense interest and commitment to photographing and archiving these unique signs over the last fifteen years of his life, we are left with a rich historical record of this essential component of New York City's commercial and visual past.
Reckitt's Blue (page 12), for example, dates from about 1890, and Jump photographed it in 1997. The advert still survives today but can no longer be seen, as it was covered over by a new building in 2004. As such, Reckitt's Blue is an advertisement that connects radically different eras in the time it has stood there. When it was first painted, standing tall as an ultramarine blue attempt to entice the commuters walking by to purchase laundry whitener and keep their clothes bright and clean, the world was a completely different kind of place. For the most part, people did not travel at more than the speed of horse-drawn cart; electricity and running water were not yet established in people's homes; and the average life expectancy at birth was about forty-three. Medicine, as we know it, had not been developed; neither women nor black citizens were seen as being responsible enough to vote; and colonialism was still in the process of subjugating vast swathes of the world's population.
Reckitt's Blue. Taken July 1998.
It is no exaggeration, therefore, to claim that the course of the advertisement's lifespan was not just as extraordinary in New York's history but also in the world's. For in the time that Reckitt's Blue has stood proud selling its wares to successive generations of New Yorkers, the world has undergone unprecedented and radical changes in its social, cultural and technological constitution. Indeed, the advert has witnessed the invention of the film camera, the automobile, the first airplanes, two world wars and the Great Depression, television, the jazz age, the jet engine, the rise and fall of Nazism and the Soviet Union, McCarthyism, JFK, the discovery of DNA, the Beatles, nuclear fusion, the civil rights movement, space travel, Picasso, the first men on the moon, punk and hip-hop, postmodern architecture, portable computers, the Internet, 9/11, the gentrification of New York and much else besides. Who would have thought a simple advertisement would endure the rise and fall of nations and ideologies, while New York and the world changed beyond recognition? Certainly not the men who painted it, whose livelihoods depended on their ability to make citizens look up and desire the goods on show to the extent that they became convinced that _their_ lives would be better with _that_ particular soap powder, _those_ particular shoes, _these_ particular tailor's shears.
Often painted up to fifty feet tall and twenty feet wide, in bold, attention-seeking colors designed to stir New York's citizens from their reverie and make them lift their eyes from the ground, some of the advertisements documented in this book were considered eyesores in their day. Mostly, they advertise products that can no longer be bought, made by companies that no longer exist, painted on buildings whose original occupants are forgotten, by men long since departed. What remains are the faded traces of these men's manual labor, which was creatively marshaled into a form of advertising and rhetoric that was popular in its day, and which above all else used _color_ to convey its message to the people walking below. And what _we_ are left with, by way of the photographs contained in this book, is a rich historical testament to the city's past. It is a story told through images of a city that has seen many changes in character over the last two hundred years, not least the rapid shifts in fortune from commercial booms to near bankruptcy and a population increase from fewer than 50,000 citizens distributed across the five boroughs in 1790 to over 8 million today. However, the book is also the story of a man who has been present to much that life has to offer and has gone through a similar series of ups and downs over the course of his own lifetime, including having to negotiate the looming shadow of life and death from a young age, and who now teaches technology in a Brooklyn elementary school. A person who transcended this personal battle with life and death in order to document an important and vibrant part of the city's social, cultural and visual heritage.
I first got to know Frank H. Jump through the familiar lettering of Arial typeface during the late 1990s, by way of the e-mails that bounced back and forth between London, where I was based, and New York, where JuMp! lived. By 1999, I had gotten to know Frank Jump as a voice whose words were not just located on the other end of a telephone line but on the other side of the Atlantic, which had the curious effect of never being fully coordinated in time. Accordingly, the distinct ambience of a late-night conversation in London was met by a Frank Jump living in the full light of day in New York, his mind set to five hours earlier due to the time difference. Likewise, Frank Jump's late-night musings about the nature of life, art and photography at 3:00 a.m. were always placed in the incongruous context of watching the sun and defeated morning commuters pass by my apartment window. However, a few months later, on June 22, 1999, I was due to meet Frank in person for the first time at a conference and art exhibition at the HERE Gallery in SoHo, where he had some photographs on show. For once, we were coordinated in both time _and_ space, which are, of course, the two most important themes of his photography.
In an e-mail I sent to introduce my thesis to JuMp!, I asked if he would be interested in discussing his work with me as part of a research project I was conducting in Europe, North America and Africa. The aim of the project was to document AIDS as a global phenomenon in terms of how people in different cultures experience, live with and understand the disease, particularly in relation to how experiences of HIV/AIDS affect the perception of time and existence; how different cultural beliefs affect people's experiences of the disease; and the limitations of language and the extent to which people's experiences, perceptions and emotions can be communicated or understood by others. I was particularly interested in the way art speaks to and is used to explore and understand these issues, not just in terms of how experiences of AIDS demand that the world is seen differently and how this manifests in art but also in tracing the trajectory of perception through art as the experience of living AIDS becomes normalized.
Underlying my research is a simple fact that is well known but often forgotten—namely, how the persistence of consciousness is associated with the body, the present and the passage of time. Obviously, time cannot be perceived directly, but nevertheless, there is an implicit recognition and heightened awareness in the way nature inscribes time—say in a flower, the changing seasons, someone's face, a building or the surfaces of the human body—that seem prevalent throughout your photographs and their subject matter. It seemed to me that JuMp! was working on many of these themes by way of a different medium, namely photography, and so I was hoping he would be interested in meeting up and discussing the issue of time and existence as it appears both in his work and life. I was particularly interested in understanding how perception of time, existence and one's environment is not just altered through living with AIDS, but how time and nature might resonate in different ways, perhaps, as part of a general reorientation away from death and back toward life. JuMp!'s photographs suggest that death is not the ultimate otherness after all but life (contrary to what some philosophers have implied), and in some photographs, this dialogue is made quite explicit, whether this is through their associations with memory and the body.
Time and space were also the subject matter of the PhD I was researching at the time. More specifically, I was undertaking a study about the perception of time and space as experienced through the framework of people close to death. The research attempted to understand how the world appeared to people confronting mortality, including the sometimes radical transformations in perceptions of time, space and existence that people undergo when living under such circumstances. These often extend into changes in self-identity, self-understanding and body image; in people's religious beliefs and commitments and concepts of the afterlife, nonexistence and eternity; changes in the type of imaginary worlds people inhabit in relation to their surroundings; changes in the character and meaning of nature and humanity; and of everyday social roles and interactions while entering a new existential territory or attempting to reclaim life.
I first e-mailed Frank after coming across his photographs of the enormous fading advertisements that adorned New York's skyline. His work spoke to me of many things, of the cityscape, of urban advertising and the scale, height and changing vitality of the metropolitan experience over the last hundred years. But most of all, his photographs spoke to me about time.
Attempts to understand the close and complicated relationship between time and human experience stretch far back in history, beyond St. Augustine (354–430), Epicurus (341–270 BC) and the pre-Socratic philosophers' early meditations on time, to ancient Greek mythology in the form of the titan Chronos. By legend, Chronos was said to live on the horizon at the farthest edges of the world and, as such, was a constant but elusive presence on the periphery of human awareness who by some accounts ended up eating his own children. The implication here seems straightforward: time is both a creative and destructive phenomenon that is necessary for birth, life and death but largely exists beyond the sphere of human influence and understanding. The world and its contents are brought into existence by time—including human life and consciousness, which could not exist if they remained static—but ultimately, things born into the world are devoured by it. Simply put, without time there would be no sentience, no body, no life, no death and no social existence, at least not in a form that we could recognize, insofar as human beings are formed as persons and raised in time; they carry out their domestic and working lives in time; they think, move and act in time; experience moods, feelings and emotions in time.
Since its earliest origins as a word and concept, time has been understood in relation to an _other_. Thus, human time is contrasted with the time of gods, trees, other-persons, rocks, the planet, science, infinity, stasis, eternity and the universe. Often the time of one society or culture—be that local, national or historical—is defined and imagined in relation to other cultures and other places. New York, for example, is popularly seen as a fast city whose citizens are fleet of foot and mind and symbolizes progress and momentum, in contrast to the slow pace of life in the village, countryside or the past. Nevertheless, the most resonant of _others_ remains the phenomenon of death, for human beings always exist in relation to death and people's understandings and experiences of time stand against the human organism's mortality and finitude. A long line of philosophers—including Heidegger, Levinas and Derrida—have each ventured that death is the ultimate otherness. But while disciplines such as philosophy, psychology and psychoanalysis gravitate toward a search for universal themes and arguments that are shared across humanity, sociologists and anthropologists have suggested that death is largely understood as existing in social and cultural forms that mediate its otherness. Consequently, anthropological conceptions of death are relational to the life of the social group and focus on the relationship between an individual and the wider society and how time and death are understood through social, cultural, religious and material forms, including art.
It goes without saying that time is present in all artworks, from being able to look at a painting and see the movement of the wrist, the sweep of the arm and physiology of the elbow in a painter's brushstrokes as he moves his brush over the canvas to the rhythm and tempo of jazz or the dramatic pauses and silences in one of Shakespeare's plays. Likewise, a photograph cannot exist outside of time any more than painting, music and film can. The long-standing, but erroneous, cliché of a photograph as freezing a moment in time is called into question not just by variations in shutter speed but, as is beautifully demonstrated in Frank Jump's photographs, by the temporal passage and fading colors of the once pristine advertisements he documents. The cognitive and physiological constraints of the human eye mean that we can only see an image through time. Our bifocal eyes placed on the front of the head take in a panorama of nearly 180 degrees that encompasses almost everything in front of the shoulders; however, less than 1/1000 of the visual field is actually in focus, while the vast majority remains vague and blurry. This is why our eyes are in constant motion, by way of saccadic movements that occur at an average rate of three times a second and ensure that dramatic variations in focus and acuity are not noticed. Consequently, when we look at a photograph, as when we look at a landscape or during our normal everyday encounters with people and things, the eye constantly and involuntarily scans the world in front of it so as to present a picture to the retina.
In Frank Jump's illuminating and detailed photographs of New York's fading advertisements, there are at least three different registers of time at play. There is the time that is contained within the fading advertisements themselves: a kind of social and historical time that bears witness to the passing of fashions, commercial possibilities and successive years of weathering, which in some cases took place over the whole of the twentieth century. Then there is the time of the artist, that precise moment when Jump took the original photograph, the particular assemblage of light, weather and aesthetic judgment out of which the photograph emerged. As Roland Barthes writes in _Camera Lucida_ , "Of all the objects in the world, why choose [why photograph] this object, this moment, rather than some other?" Lastly, there is the time of the viewer who is looking at it, as if through a corridor of time, in which he or she is not just witnessing the world in the moment JuMp! released the shutter but the history of New York that is contained in the advertisement's fading colors as seen from the present. Nowhere is this more present than in Jump's documentation of the seasons in his series of images of Eaglo Paints—an advertisement that is no longer there—in which the different modes of linear and cyclical time are simultaneously at work. At once inexorably moving forward, time also returns in the familiar forms and colors of spring, summer, autumn and winter, highlighting the persistent relationship between time, metaphysics and art. Here, metaphysics is not meant in the modern, sometimes spiritual, sense but in terms of its original meaning of that which exists beyond the physical, including the phenomenon of time. We cannot "see" time. We cannot touch, taste or hear time. We cannot carry it or weigh it. In that sense, time has no physical basis. But we can detect the traces of time in JuMp!'s art, in the way that his photographs attempt to represent time at work and provide us with a visual and material record of its effects and passage.
On the day I eventually got to meet JuMp!, the location was a church on West 4th Street where a one-day AIDS conference and art exhibition was being held. The theme concerned the air of optimism now that the future appeared to have opened up, thanks to the new antiretroviral drugs. The stated purpose was to ask _what next?_ However, I was struck by the fact that no matter how hard people tried to think about the future, about what was next and about where life would take them, they kept returning to the past. _What on earth was the past all about?_ everyone wondered in unison. The crowd swayed together, occasionally moved forward again in time, but then would fall back into a sense of shock when its members looked back to the past: to the pre-antiretroviral era of the 1980s and '90s when thousands of young men and women were gravely ill, dying or dead.
New York's nighttime descended on the city for Frank's generation in the early 1980s, when HIV/AIDS began to impinge upon the city. It not only exposed many thousands of people—gay and straight—to death but cut across the dominant cultural narrative of life as something that extends into old age. The strange discrepancy of young, healthy people succumbing in large numbers to disease and death in relation to an overall population getting steadily older reversed the seemingly established order of city life. By the early 1990s, AIDS patients took up 8.5 percent of all hospital beds. And by March 31, 2000, there were 72,207 known deaths from AIDS in New York out of 116,316 known people diagnosed, including almost 10,000 infants. Moreover, for every diagnosed person, numerous others are affected, including friends, partners, parents, children, neighbors, relatives, acquaintances, counselors, medical staff, volunteers and work colleagues. These persons are also "living with HIV/AIDS" insofar as their lives become intertwined with a complex and life-threatening disease, forming a polythetic population of infected and affected persons across New York that crosses sexualities, genders, generations and neighborhoods and composes a substantial proportion of the city's entire population.
Given such a history, it is unsurprising that the conference and exhibition contained a sense of trepidation about the present and betrayed a complex mixture of anxiety, uncertainty and hope about the future: _what next?_ As I walked around the exhibition, I remember overhearing one guy who said that because he was from the South and his mama had brought him up to be polite, he was going to treat the disease as a guest. He said he was going to be the best host possible, make it comfortable in his body and look after it right until it was time for him or his guest to leave. The man he was talking with said he hated AIDS from the depth of his guts to the top of his head. He said he'd always hated it and always would, he hated having to acknowledge it, hated having to talk about it, hated taking medication and hated telling people he had the disease. As such, simply walking around the exhibition and capturing snippets of conversation was a highly revealing process—almost as revealing as anything in the artworks themselves—and provided a tangible sense of the diversity of responses, attitudes and ideas about HIV/AIDS. However, what made the atmosphere dense with emotion were the various stories, testimonies and life histories that were being spoken over the main microphone to the room. Some of these voiced descriptions of life with and affected by HIV/AIDS contained a casual abandon and some good jokes, while others were raw, angry and wrung out.
I had arranged to meet JuMp! after the conference on the church steps outside. Frank H. Jump, however, was early and had ventured inside; thus, the element of surprise was on his side. I heard someone call my name from somewhere out of the milling crowd. I heard the voice again, but I still couldn't locate the face. Then, out of nowhere came a face that declared itself "Frank Jump." Tall, handsome, wearing a set of pirate's earrings and an orange Hawaiian shirt came Frank Jump in person and extended his hand. As we shook hands, I caught sight of his Dick Tracey Special Edition Millennium watch. His early arrival caught me slightly off guard, and it took me a few seconds to regain my balance, by which time Frank Jump (assured and cool) was conspiratorially close and sharing his thoughts about art and life, a consequence of which meant an instant level of intimacy was demanded from myself. No cold formalism or stiff and stilted introductions here. We sat down to catch the last talk of the conference and then went out into the bright sunlight of Greenwich Village. Immediately to our left, a heavily bearded man stared. The man was surrounded by posters and declarations, wearing a sandwich board declaring that AIDS was God's cure for faggots, and as we walked past, he shouted out that he was glad so many gays were dying of AIDS and would be going to hell. Do I remember being shocked?
Not by the sentiments, as they were a form of tabloid currency that was common enough through the late 1980s and early 1990s. But I remember I was surprised that somebody could be bothered to get out of bed and drag himself and all his sandwich boards down to such an event. After all, by 1999, AIDS had been part of popular consciousness for well over a decade and formed part of the everyday landscape of many major cities, including New York and, of course, the West Village. I remember looking into the heavily bearded man's eyes and finding them strangely docile, with a smattering of self-righteousness. I was curious more than anything, although no doubt he probably interpreted this as confrontational rather than curious. Frank H. Jump, meanwhile, carried on his conversation without a hint of acknowledgement. I kept meaning to find out the level of self-conscious performance involved or whether JuMP! had even noticed the man, his comments or his posters but never did; the conversation had moved elsewhere.
New York City is in large parts a municipality that was founded on trade and immigration, and as such, the city and its buildings and architecture testify to the complex, historical relationship among people, money, material goods, architecture and advertising. The city's vast surfaces provide a seemingly irresistible temptation to businesses and advertisers and provide a visible index of the products, fashions and moral standards of different eras. These somehow combine to form part of the content and character of the city we know today: diverse, contested and always in process, in the thrall of capital, often overly sentimental and, above all, full of nervous commercial and artistic energy. Indeed, like the human body, the nervous system of New York is never in the same state twice. It also ages and renews itself over time, betraying the effects not just of its past but its hopes and desires for the future. New York's surfaces, therefore, provide an ideal medium for inscribing advertising and other promises of life that the photographs in this book describe and document. Because many of the fading advertisements are more than a century old, they are often an attempt to sell goods, lifestyles and moral values that no longer exist and can no longer be bought. They remain weathered and betray the passing of the years on the sides of vast buildings, while the products that gave birth to them have been reclaimed by Chronos.
Taken together, the seventy-two or so fading ads presented in this book demonstrate how the surfaces of New York City combine to provide an ever-changing complexion that is made up of different hues and tones that change and offer a visible measure of time. Indeed, the close link between complexion and complex, which both come from the Latin _con_ that denotes "with" or "together" and _plectare_ (to plait or twine), suggests how the fading advertisements that make up such an integral part of the city's complexion can be read as the meeting point of life stories, urban dreams and hard-nosed and competitive attempts to sell aspirations and goods for profit. Although in modern usage "complexion" refers to the surfaces of the body, if we trace the term's meaning beyond a few hundred years, we see how complexion was also used to describe the interior constitution and attributes of the person and their body or, in this case, the soul, content and character of New York City, a city founded on trading, traditions and artistic and entrepreneurial creativity, as well as corruption, decay, false promises and much else besides. If New York is a city in which prestige is dreamt about, worked for and occasionally wrestled from other people, it is telling that in its original incarnation the word "prestige" meant to achieve by illusion, delusion, trickery, deception, enchantment and glamour. Prestige and complexion: two of the dominant faces of New York that combined to create the social, cultural and economic circumstances for the production of advertisements on the size and scale of those that can barely be contained by JuMp!'s camera lenses. Here is the idea of the city as combining people's transient activities and ambitions with highly durable materials such as brick, stone, concrete and glass, in which working and consuming bodies live out their lives. It is not a new idea but one that stretches at least as far back as Aristotle's proclamation that "a city is composed of different men; similar people cannot bring a city into existence." However, when Aristotle made this statement, he was envisaging a city of nearer to ten thousand people rather than ten million, and so we are talking about a design for living that is qualitatively and not just quantitatively different. The modern-day city is frequently understood by sociologists and anthropologists as being modeled on the corporeal body—with the various constituent parts corresponding to different body parts and performing corresponding functions. If so, then the walls of buildings offer a layer of protective skin, practical in design and function but open to countless aesthetic and commercial possibilities and forms of advertising and decoration that have been enthusiastically taken up by businesses and good sellers.
The advertising possibilities presented by New York City's surfaces, which for more than a century have been painted by men in order to inscribe, ingrain and testify to the varieties and qualities of different products, were perhaps unequaled in any other part of the world. These advertisements are not just symbols in their own terms for a particular social group or generation of people but simultaneously condense cultural values and meanings that cannot always be articulated or put into language. Particular products may have helped define a sense of collective integrity, cultural identity and understanding and influenced people's economic choices and decisions within a certain neighborhood or generation. As such, they are also an index of class difference and inequality, whereby certain products are advertised in particular places to particular kinds of persons. Of course, given the demographic shifts and changes within the city, many of the fading advertisements documented by JuMp! don't tell us about New York City's neighborhoods as they are now but as they _were_. They contain within their paints the historical traces and demographic patterns of the city's population as it was rather than as it came to be. The paints accumulate dust, dirt and grime alongside rain, sunlight and the moral and aesthetic values of different generations over time, thereby creating a resonant and colorful epidermis that covers the city and can be studied to offer an understanding of the past. Even in New York City, bricks and buildings generally last longer than the generations that occupy them, and seemingly old advertisements are encountered anew by each generation of new citizens, migrants and relocaters. As a generation dies or simply moves out to the suburbs, the dead skin peels off and is eventually sloughed away, only to reveal the advertisements anew to succeeding groups and generations. It is evocative of the difference between _cutis_ and _pellis_ , between the living skin that breathes and renews itself and the dead, discarded skin that has become scoured, loosened or simply fallen away from the body.
New York City, which until 1898 meant Manhattan and parts of the Bronx, was founded in 1626 and built on an island about 12.5 miles long and 2.5 miles across, covering only 22.2 square miles of contemporary New York City's 469 square miles. The city was built from south to north, with construction beginning at Manhattan's southern tip and expanding northward. By 1653, the city had only reached modern-day Wall Street, which marked the city's northernmost extension. By the first census in 1790, the population only consisted of 33,111 sometimes feverish souls; building had only just reached the modern-day city hall; Chelsea and Greenwich were outlying villages; Midtown was farmland; and large parts of the island were covered in thick woods, with many roads meandering in accordance with the population's agricultural needs. Ten years later, the population had doubled to 60,489. By the 1810 census, it had reached 96,373, and with a further massive, tenfold population increase anticipated over the coming century, the city councilors decided to "unite regularity and order with the Public convenience and benefit, and in particular to promote the health of the city," in a way that would contribute to the "free and abundant circulation of air" and help regulate odors and people. A major consequence of the potentially massive population increase was the city's current grid system, insofar as the newly formed city commission's first important act was to commission a plan for the future layout of the city. They engaged a surveyor, John Randel, to survey the entire island, with the purpose of transforming its woods, swamps and grasslands into a place "composed principally of the habitations of men, and that straight-sided and right-angled houses." Randel and his assistants spent the next three years painstakingly measuring and mapping Manhattan's entire topography, with a resulting 7.8- by 2.0-foot map, which offered unprecedented levels of detail about the island. Randel's map is even more interesting because it does not simply map Manhattan's topography, streets and buildings of the time but also imposes a design for the island's future, in that a grid system is laid over the land, determining where future streets would be built.
As often is the case, even the best made predictions, including the massive population increase of the city's commissioners, turned out to be wrong. The actual turn of events, scale and height of building far exceeded even their most extreme expectations, and the population grew from 96,373 persons on Manhattan Island at the time of Randel's map to 2,284,103 in the 1920 census. Thus, by the 1920s, a new sense of scale and structure had emerged, against which an individual born when farmland covered most of Manhattan could compare his agricultural practices and muddy, organic desires, especially after the invention of pressured water pipes and the elevator allowed for taller and taller buildings, simultaneously laid out on the logical principles and Enlightenment rationality of the grid system of 1811. But while the city's commissioners may have imposed their will on all subsequent generations through their urban blueprint and its subsequent implementation in practice and architecture of their extended, straight avenues, they could not have anticipated the scale or verticality of the buildings. Instead, the rapidly shifting and feverish cityscape of New York provides both the impetus and background to the advertisements that became an integral part of the city and its history. A history that for the last fifteen years Frank Jump has attempted to document though his photographs and own unique brand of urban archaeology.
Today, the iron, brick, concrete, steel and glass materials of Manhattan's contemporary buildings combine with contemporary advertising and the painted adverts of the past to provide the dominant visible and aesthetic material of city life. The perspectival horizons formed by the city's buildings continually offer a series of extended surfaces and straight edges that enable New Yorkers to look far into the distance, much like a Euclidean drawing or one of Canaletto's paintings. A frame of receding perspectives and diminishing angles continually guides the eye toward a vanishing point on the horizon: a point distant in time and space, mathematically known in advance, subject to scientific reason, embedded in perspectival painting but lined by commercial advertising. It was during New York's most frenetic periods of change that the fading advertisements documented in this book were born and came into being, selling goods from the sides of buildings and the ever-increasing brick space of a city built not just along but upward. How curious it is that when citizens look up toward the sky it is not just buildings that look down upon them but huge, glaring advertisements selling products as diverse as hair cream and music. In modern New York at midday, 1.2 million people, a population greater than some nations, cram into the square mile of midtown Manhattan around the 34th Street area where many of these advertisements were, or are still, located. They provide something more permanent than the temporary patterns and ever-changing configurations of the crowded streets below, where the frenetic pace set by walkers at times can average up to 300.0 feet per minute along 22.5-foot-wide sidewalks.
Frank Jump still spends many of his evenings and weekends roaming around the city, looking for evidence of the passing of time. He still occasionally wears Hawaiian shirts and takes pictures of a whole array of persons and places. However, his main obsession remains documenting the fading advertisements painted on the sides of buildings. He captures them before they fade away and puts them on his website, and in this book, he has selected seventy-two among the many hundreds of slides to tell a story of New York as it is and as it was.
Frank was twenty-six when his illness was diagnosed, and he was told that he had "a couple of good years left." So he took himself out of the workforce and filled in all the offers for new credit cards and bank accounts that came to his door in the junk mail, thinking, "I've never got to pay any of this back." But Frank did not die. When we first met, he had been HIV-positive for thirteen years, had gone bankrupt and had reenrolled in college. Frank was acutely aware of himself as a body that might disappear; a body in an urban landscape that, like the advertisements, was fading; a body that, like them, was not supposed to last long but somehow remained part of the city. Accordingly, Frank sees his reflection not in the mirror but in the fading colors of the advertisements that provide him with evidence of his existence.
JuMp! in front of Eaglo Paints on Nostrand Avenue (pages 154–155) in 2001.
Frank has mostly remained healthy, and HIV has been more a psychological burden and a brooding potentiality that overshadows his existence and calls his future into question. His body has responded to the disease in an extraordinary way, so much so that his body became an object of medical curiosity and doctors keep devising challenges for it.
In late 1999, some doctors wanted to know what would happen if they took him off his medications. Ordinarily, when someone stops taking antiretroviral medications, the disease returns stronger than before. This is one of the pitfalls of triple-combination medication. Once on it, you are on it for life—or you risk a more potent infection. You live with the side effects: distended belly, nausea, diarrhea, low testosterone, nightmares and the possibility of Crixivan (a protease inhibitor) crystals forming in your kidneys.
Frank agreed to stop his medications so his doctors could observe what would happen. They told him that he would become sick but that they wanted to know if the conventional wisdom was right—namely, that in chronic infection the body's ability to mount an immune response is lost forever—or if instead Frank' s body would develop an immune response. After eight weeks Frank would resume his medications, and then the process would be repeated. Afterward, the amount of virus in Frank's blood would be measured to see if his body had fought back. Frank likes to throw himself into the fire, and this time he faced three bouts of miserable and painful illness. Because we kept in contact by e-mail, I already knew that Frank had volunteered to come off the combination therapy so the doctors could see how his body responded. Up went the viral load, down went the T-cell count and d-d-down went Frank Jump.
Frank e-mailed me:
_My viral load has shot up from 0 to two million in less than 6 weeks. My T-cells have dropped to my lowest ever and I panicked. When I discovered these results I went back on antivirals (but only after I visited my physician who didn't check my viral load until my normal appt). The study apologized to me for not keeping me abreast of the T-cell numbers and let me know that my viral load was tremendously high. I said I'd try this twice more. If after the second time, the viral load increases as high again, I won't try it a third time. If the numbers peak at a considerably lower number, indicating some kind of "intelligence" on behalf of my immune system, I will try a third time. If this doesn't work—tant pis—I'm ready for another theory_.
Now he's back and looks better than ever. In 2011, at the age of fifty-one, and exactly two hundred years after John Randel's map outlining his grid design for the city, Frank has been living with the disease for half of his life. He still hasn't documented every fading advertisement in New York.
Dr. Andrew Irving
Visual Anthropologist
University of Manchester, UK
Acknowledgements
With special thanks to my mother, Willy Jump, and my husband, Vincenzo Aiosa, for their unconditional love and support. Many thanks to Nelson Santos and Thomas Devaney for their insights; Amy Sadao, Jenny Ricketts, Lemmy Caution, Sandra Walker R.I., Kevin Walsh, Walter Grutchfield, Julian Seery Gude, Sam Roberts, Kathleen Hulser, Dr. Andrew Irving, Tod Swormstedt and Wm. Stage for their collaboration; and Lucy Winner, Mel Rosenthal, Joe Goldberg and Jim Greenberg for their guidance. For their generous contributions to the Fading Ad Campaign, I would like to thank the following people: Donn and Bob Middleton (Mack signs), Professor Mary Brabeck and Dr. Michael Brabeck, Jay Lesiger and Tom Klebba, Frank Ligtvoet and Nanne Dekking, Andy Hughes, Isabel and Rosario Dawson, Sean Strub, Tama Starr (ArtKraft Strauss), Caroline Kim and Luis Santiago, Robert French, Raphael Carty, David E. Oppliger, Ronald J. Nigro, Neil Boxer, Patricia DeLeeuw, Kenneth J. Figueroa, Lindsay M. Wright, Brady R. Galan, Judith Ann Coffman, Phillip Brian Harper and Thomas Freedman, Jane K. Rushton, Samuel H. Havens, Kathleen Barnes, R.B. Caldwell and James Jaxxa, Elizabeth J. Misek, Lawrence Martin and Angela Martin-Fehr, Richard C. Richardson Jr. and Patricia B. Richardson, Margaret More Hunt, Apurva N. Shah and Chetna R. Shah, Malina Sahu, Saumil S. Doshi, Marion L. Vetter, Budd M. Heyman and Mary Heyman, Patricia Hedieh Eshaghian, Sawsan K. Abdel-Razig, Celia Faye Stewart, Nathan H. Thompson and Robin Freeman. For their continued support, thank you to Betsy Gotbaum, Leslie Nolan, Barbara Snow, Jacqueline Girardi, Kevin Langley, Jon Nalley, Malaga Baldi, Bobby Rivers, David W. Dunlap, Nick Hirshon, Pete Anderson, James Lileks, Rory Albanese, James Duncan, Gaia Son and Bob Kovel, Eric Sawyer, Andy Humm and Terry Washington. With the warmest gratitude for sharing their family histories, I'd like to thank Beth Beeman, Aileen Schaefer, DeDe Burke and Michael Hughes.
New York City's Fading Ad Campaign
_The Chrome Age_
_If I leave behind an act which I have accomplished, it becomes a thing by falling into the past. It is no longer anything but a stupid and opaque fact. In order to prevent this metamorphosis, I must ceaselessly return to it and justify it in the unity of the project in which I am engaged. Setting up the movement of my transcendence requires that I never let it uselessly fall back upon itself, that I prolong it indefinitely. Thus I can not genuinely desire an end today without desiring it through my whole existence, insofar as it is the future of this present moment and insofar as it is the surpassed past of days to come. To will is to engage myself to persevere in my will_.
_—Simone de Beauvoir_ , The Ethics of Ambiguity
Fading ads are a beacon in the navigation of an urban life. In the late twentieth century, the Fading Ad Campaign began as a 35mm chrome photographic project documenting vintage mural ads on buildings in New York City. Quickly, it became a metaphor for survival since, like myself, many of these ads had long outlived their expected life span. Although this campaign doesn't deal directly with HIV/AIDS, it is no accident that I chose to document such a transitory and evanescent subject. Of the thousands of ads I've photographed, many have faded out of existence, been covered over or destroyed. But still many silently cling to the walls of buildings, barely noticed by the rushing passersby.
In 1986, I found out I was HIV-positive. At age twenty-six, with experts giving me possibly four more years to live, I was immediately thrust into a midlife crisis and the curvature of space-time around me got more curved. Like a figure skater, I started to spin faster as I pulled my arms in toward my center, my speed increased until—I finally approached the speed of life.
Time passed so quickly that others around me seemed to grow old and die before my eyes. My urgency to leave my mark as a person and an artist who had walked this planet became intensified. I spent lots of money, none of it mine. I accessed all my available credit lines and bought a home recording studio. I wrote lyrics and music for underground theatre and film on the subjects of AIDS and homelessness. Much of the material was based on my experiences as an activist with the newly formed New York City–based group ACT UP (AIDS Coalition to Unleash Power) and my personal dealings living with this new, deadly virus.
By 1990, long-term plans started to creep back into my consciousness after being banished by a dubious prognosis. By now I had met my lifetime companion, husband and collaborator, Vincenzo Aiosa, who played an active role in signs scouting for this campaign and later became a photo contributor for the Fading Ad Blog. In 1995, after almost a twenty-year hiatus, I returned to college to finish my bachelor of arts in music, theatre and film at Empire State College (SUNY), much of it based on my life experiences as a recording artist and activist. In February 1997, when I first discovered the vintage sign Omega Oil in Harlem, I had an artistic epiphany, realizing this was to be my next documentary photo project. What was not evident was to what extent this undertaking would alter my life's trajectory.
In November 1997, I agreed to exhibit this work in a public art space at the Gershwin Hotel for the annual Visual AIDS Day Without Art event as an "HIV+ artist." While politicizing my HIV status for the public good wasn't a problem for me, the project still seemed one of historical documentation devoid of personal experience and expression: the focus was the signs and not my life condition framing it. And even if I personified each sign and ensured to photograph its best side—like an aging diva that was once greatly admired—these images were primarily windows to our commercial and industrial past. They represent a time in American history that was filled with enterprising ideas and burgeoning global marketing, and I couldn't see how they would fit in the context of an "AIDS art show" alongside talented artists whose work clearly and provocatively portrayed their struggles with the virus. It wasn't until my degree program mentor, Lucy Winner at Empire State, said, "Frank, there's a connection between your survival, the survival of these signs and your fervent passion to photograph them," that it all clicked.
Dr. Andrew Irving, professor of visual anthropology at Manchester University, describes how HIV/AIDS often transforms people's perceptions of time, existence and the body. Visual artists with HIV/AIDS often experience an accelerated life and reflect this acceleration of time in their work. Many of the artists I've known who are categorized as HIV-positive artists or AIDS artists, either by self-proclamation or by their audience, often express their experience with accelerated time in profound and daring ways.
Subsequently, I found a home in the Visual AIDS archive of past and present artists with HIV, whose works are displayed on the Estate Project website in conjunction with the Museum of Modern Art. Thanks to this site, I became aware of a vast body of work to which I would otherwise never have been exposed. Today, I'm grateful to be included and associated with such a collection of talented visual artists and their brave expressions of personal struggle, including the challenge of making art in the face of mortality and finite time. Together they document a major historical event of the last century—the age of HIV/AIDS.
The sense of urgency I felt every day drove me to document New York's fading advertisements and capture the marks left by artists and artisans, most long since dead, who spent their lives painting huge commercial murals over the last 150 years. Some signs are over 100 years old and were never expected to survive this length of time. Like my own body, they weren't meant to last this long, and as such, there is a direct correlation between the signs and my own life experience and subsequent attempt to document their presence through photography. After accumulating an abundance of vintage ad images across America, I realized I was now committed to this campaign through thick and thin, health and sickness, for the rest of my life and was determined never to let my archive "uselessly fall back upon itself."
Within the last century, we have witnessed the constant rise and fall of countless businesses. Some manufactured popular products that are still being produced today, while others ceased production and only remain as a name on the side of a building, slowly fading in the sun. Many of the vintage ads I have documented were preserved because buildings were built next to them that covered them over and protected them from the elements. They remained hidden from the eye for decades at a time, only to be revealed again many years later when an adjacent building was torn down. Other images survived through the sheer luck of having a northern exposure—away from the intensity of the sun.
The images in this book provide a visual archaeology that reminds us of a bygone era in advertising and illustrates the past lifestyles, commercial tastes and social trends of New York City (and by extension much of America and other cities around the world) that use the sides of brick buildings to advertise goods and services. The images also speak to current advertising strategies and provide a priceless historical context for understanding the rise of commerce, socioeconomic trends and the demographic shifts and displacements caused by these economic and social overreactions in New York, such as the white flight of the late 1950s and early 1960s to the suburbs. As such, the importance of these huge graphic landmarks must be recognized in the ongoing effort to foster interest toward their preservation.
Within a couple of years of conceiving and publicizing this campaign, I was asked to give public testimony in support of newer advertising before the New York City Landmarks Preservation Committee, where I compared historic ads with today's ads. I spoke on how preservation projects have the potential to establish a productive dialogue between ad agencies, local artists, local historical societies and preservation groups. Such collaborative efforts have since produced new site-specific, culturally sensitive advertisements in historic districts, for example, the work of Colossal Media in SoHo and Williamsburg, Brooklyn.
Embarking on the campaign has taken me to many places and opened up many wonderful dialogues with people around the world. On July 9, 1998, the _New York Times_ followed up my exhibit at the New-York Historical Society with a front-page spread in the "Metro" section that brought widespread attention to the historical significance and contemporary importance of preserving fading advertisements. After the show came down, I decided to create a permanent home for this collection of images that would allow the advertisements to reach a much wider, even global, audience. So I launched the Fading Ad Campaign website in February 1999. The site has subsequently been featured in many online and print media, including: Archaeology Online, POZ Magazine Online, _USA Today_ , Yahoo, Forbes, the _Chicago Sun Times_ , the _London Guardian/Observer Magazine_ 8 and several radio and TV spots with the BBC, Canadian TV, Brooklyn Independent Television, _Leonard Lopate Show_ (NPR) and _City Arts_ (PBS).
Original Fortunoff's store sign on Pennsylvania Avenue near Livonia Avenue. Painted by Albert (Israel) Adelson for Concord Co. (a family business) in the 1950s
The campaign has afforded me a much wider audience and has facilitated an incredible network of correspondence with other fading ad enthusiasts and historians like Kevin Walsh ( _Forgotten New York_ ), Walter Grutchfield ( _14 th to 42nd Street_) and Sam Roberts (UK _Brickads_ ). These correspondences and links to other related sites have opened up a whole new perspective of "lost America" and made me aware of the growing interest in historical ad images in urban and roadside environments. It has also led me to some incredibly supportive people around the globe who have become lifelong friends and colleagues. Sharing the campaign with our global community has been exciting and humbling. It is like putting a message into a bottle and tossing it out on a sea of electrons. It seems so vast, yet it feels like home. I'm always astounded when I try to fathom humanity reaching out across the vast electronic ocean of the Internet in search of one another, finding like minds, finding one's self and sometimes finding beauty. Never before have we been so isolated from nature and desensitized to violence and callousness as in our urban environment, yet we still are searching for connections to a greater population, a global voice.
Today, we can look at a vintage Reckitt's Blue or Omega Oil mural advertisement and feel transported. Back when they were freshly painted, some people regarded them as eyesores. A century ago, there was a public outcry in New York City against the proliferation of billboard advertising, not unlike today's acrimonious battle against the new "Times Square–style" ad campaigns along Sixth Avenue, Houston Street bordering SoHo and in lower Manhattan. I do not think it is a question of whether or not we should continue to surround ourselves with larger-than-life images whose sole purpose is to move us to consume. It is a question of what images we may find enduring or possibly want to endure. When Joseph Berger of the _New York Times_ interviewed Kathleen Hulser, public historian at the New-York Historical Society, she observed how "they [ads] evoke the exuberant period of American capitalism. Consumer cultures were really getting going and there weren't many rules yet, no landmarks preservation commission or organized community saying: 'Isn't this awful? There's a picture of a man chewing tobacco on the corner of my street.'" Berger adds, "While a century ago, preservation groups viewed the signs as vulgar interlopers, some now want to sustain them."
Nonetheless, today's gargantuan temporary vinyl backdrops are disposable and very portable in comparison to the painted advertisements of other eras. Images printed in digitally pointillist perfection onto sheer canvases stretch across building faces the size of football fields. Some cover and protect older painted signs. A walk up Sixth Avenue from West 23rd Street to 34th Street is like a stroll through the land of the giants; overwhelming at times, it can also be comforting to feel like an insignificant grain of sand.
Advertising is and has always been the realm of the modern artist. When your canvas is the entire side of a building in a landmark neighborhood, there is a giant opportunity to be creative. If your target neighborhood is Chelsea, an area now known for its art galleries, trendy retail shops and restaurants, why not draw attention to its placement and make a bold artistic statement? There are some painted sign companies that meet this challenge to produce images that are not only enduring but also provocative and at the same time classy, as in some liquor ads that walk the thin line between poking fun at the art world and homage. Some new ads even copy the styles of vintage ads and classical art ( _Colossal Media_ ).
**A H...NOSTALGIA!**
In 1966, Pier Paolo Pasolini said, "If you know that I am an unbeliever, then you know me better than I do myself. I may be an unbeliever but I am an unbeliever who has a nostalgia for a belief." The allure of past expressions in ads runs deeper than nostalgia. It is a belief that clutching onto the familiar as we plunge into the new millennium may inform our future decisions. These ads are something we can physically touch. They are markers of the passing of time. The rate at which technology is growing elicits new forms of social development. These signs are part of us and remain closer to us in terms of time and body than our grasp of what may come along. The Internet can either augment or attenuate the impact of advertising and human society depending on how much more or less prominent electronic advertising becomes on the Internet.
What other forms will advertising take in the next century? Will ads be even more temporary? Will they impact the environment in monumental ways, or will they become increasingly tenuous and miniscule? Will they be projected, holographic, canyon-sized images that disappear with the flick of a switch? Will they continue to beckon us toward the things we need; the things we don't have; or the things we can't do without? And will we ever again believe words like "the purest and the best"?
Gloria Steinem said on HBO's _Real Time with Bill Maher_ , "Nostalgia is a form of obstructionism." It is a moot point where we can move forward without looking back. Some individuals, communities and nations seem to progress with relative ease without looking where they've been. Others maintain the past as an ongoing reference to the present. Nonetheless, as the old forms surely fade, the new and upcoming ways of being call for new forms of adaptation. The challenge is keeping oneself ahead of the learning curve.
Recently, Professor Gerald Torres of Texas University in Austin said, "Nostalgia is corrosive and you need to banish nostalgia in my view," leading us to ask why Torres might say such a thing about the "good old days." Nostalgia is a response to the loss of discernible landmarks, within either one's external or internal landscapes. The loss of a familiar building or even a city block can be as devastating to an individual as a subdural hematoma. Our ability to effectively navigate our internal schemas underpins our sense of well-being and usefulness and to a large part is coupled with our ability to effectively navigate through an urban space-time. What makes nostalgia corrosive is if one soaks in it too long, it will dissolve our determination to recognize and give significance to those new landmarks that foster growth and equilibrium toward future proprio-coherent senses of well-being. Often we are anchored to our past through recollections of a younger, more vital self, which creates a dissonance when we confront our current self. Perhaps it is wise to banish this cause of stagnation—the corrosive eating away at our tenacity to move forward in life—which is an obstruction, often manifested in anxieties over becoming outmoded. Art forms and the media on which they are recorded are also becoming outmoded. Is the art of the "wall dog" quickly drifting toward obsolescence as well?
Gloria Steinem on HBO's _Real Time with Bill Maher_ (Episode 82, September 15, 2006). Jump photographed the screen with closed captioning on. _Courtesy of HBO, Inc_.
Jerry "Orange" Johnson, originally from Cleveland, Ohio, worked as a wall dog—an ad sign painter—for Seaboard Outdoor before establishing Orange Outdoor Advertising, Brooklyn's premier hand-painted sign company, in 1977. Johnson is perhaps better known for his "spoof" advertising murals, often depicting nonexistent products with classic nostalgic images accompanied by sublime ironical text. The following sign caught the eye of filmmakers for a documentary segment on PBS called _City Arts_ , on which the campaign was also featured. Agility Nut writes about Johnson's _Penmanship Stupid_ ad on a personal website about roadside ephemera. The ad depicts Zsa Zsa Gabor as a pen manufacturing company representative:
_The Penmanship sign was located in Boerum Hill and was painted in 1997. It was a faux billboard created by Jerry Johnson of Orange Outdoor Advertising. Johnson painted satiric, retro-style paintings on this wall annually for about 15 years. Previous signs poked fun at subjects including: oleo margarine, electric companies, the Williamsburgh Savings Bank, plates, the power of cash and Ebbets Field. Sadly, this "ad" was painted over in 2003 (with red paint) and probably marks the end of the series_.
Over the years, several journalism students from Columbia University have contacted me about the wall dogs and the photo-documentation of the art of this dying breed of sign painters. One such student, Leila Abboud, in an article written in 2005 entitled "The Wall Dogs' Last Stand: Technology Puts Sign Painters Out of Work," focused on the quandary of older sign painters in a changing industry. Wall dog Alberto Gonzalez practiced his craft of outdoor sign painting on brick walls and billboards all over New York City for decades, balanced on two-foot-wide scaffolds that were hundreds of feet in the air and "braving the blazing sun, wind and cold to paint advertisements," only to face becoming superseded by a newer and cheaper process. Abboud further stated:
_The advent of digitally printed vinyl ads over the past decade has rendered painted ads nearly obsolete. Vinyl ads are cheaper and faster to produce, and neon and electric ads have spread. The union to which Gonzalez belongs once had hundreds of members. Now there are a dozen. Similar shrinkage has occurred across the nation_.
_With the sign painters will disappear the last traces of an era of American advertising when itinerant sign painters ruled. Nicknamed "wall dogs," these men traveled the country from the 1920s to the 1950s spreading the first national advertising campaigns. They emblazoned the sides of barns with logos for products like Mail Pouch Tobacco and Coca-Cola. The men earned a reputation for being wild, said St. Louis–based photographer William Stage, who published a book about the "wall dogs" and their work. "They would drink beer as they hung from rope scaffolds high above the street, and spill paint on cars and people below," said Stage_.
_In many cities, including New York, traces of the wall dogs' handiwork can still be seen. The lead-based paint of the old ads survived time and weather. Although Gonzalez may think his craft is nearly extinct, his work and that of other "wall dogs" may not be forgotten. A small but devoted band of photographers and urban archeologists across the country has tried to preserve and document the remaining ghost signs_.
_And in Los Angeles, outdoor advertising companies have seen an increased demand for painted signs after the city outlawed large vinyl draped signs two months ago. In Fort Dodge, Iowa, a building was torn down revealing a red, white and blue Coca-Cola ad on the adjacent building. The town is debating whether to restore the sign_.
_"People are drawn to them because it reminds them of another time," said Frank Jump, a New York–based documentary photographer who has photographed thousands of old ads over the past five years. Restoring old ads for historic or nostalgic value has become something of a trend in the Midwest, said Jump_.
When traveling throughout the United States, you will often see these fading relics "restored" by local preservationists and arts organizations. Personally, I feel these ads should be left alone to fade into imperceptibility, which is part of their natural life cycle. Although I understand the motivation behind re-creating these images to remind us of their former glory, for me, the fading quality of these images is a beautiful process to behold. Although I do not want to stand in the way of progress when newer signs cover these relics, I prefer to see them untouched. They've survived this long; they deserve to be left undisturbed.
Accordingly, the aim of this book is to present a photographic archive of some of the vintage mural advertisements I photographed during the first four years of the campaign in the five boroughs of New York City, with each image serving as a window to our past. Ironically, _Fading Ads of New York City_ is in itself a documentation of obsolescence, for not only have many of the products these adverts touted as "newest" or "the best" become obsolete, but the medium through which I documented them has become obsolete as well. Kodachrome was my first love for the way that chrome slides produce a luxurious color representation of the world. The 35mm negative of the image is actually a "positive." The "positive to positive" reproduction was called the Cibachrome or Ilfochrome print—a high-contrast and rich color photograph that uses azo dyes on a polyester medium that does not fade. It was the preferred medium of artists, collectors and museums. Unfortunately, with the advent of digital photography and output, this nonpareil process has become discontinued.
Inspired by the philosophy of Simone de Beauvoir, I cannot let this archive "become just a thing." It must move forward. I am committed to expressing the challenge and experience of living with HIV/AIDS in relation to the images of the campaign. These enduring images represent a surviving culture, shellshocked by famine and depression; world wars, struggle and emancipation; rhetorical scourges and over-the-top ad campaigns. They are living proof that we are even surviving capitalism, if only by the skin of our teeth. It is almost as if the human race was told we only have a few good years left to live. We are all living like it is the last days—strangely anticipatory and post-apocalyptic. Fortunately, I've gained a lasting view of my life and the life of this campaign, from more than just the sheer luck of a northern exposure.
_Amateur archeologists can still unearth them, faded and weathered as they are, by walking the streets of the five boroughs_.
_—Joe Berger_ , New York Times
A series of twenty-four images hung in the corridors of the New-York Historical Society (NYHS) from August through November 1998 and can be seen on the Fading Ad Campaign website. How they got there is quite fortuitous. Just a year earlier, these images were captured on 35mm chrome. My photography professor at SUNY–Empire State College, Mel Rosenthal, always told us that as photographers, you should shoot what's familiar. Start at home in your neighborhood; start with what you know. So off I went to document "The Rise & Fall or The Fall & Rise of New York City" for our final project. Having grown up in New York, the city was a neighborhood that I thought I knew well. But I never dreamed I would discover a hidden world that existed in silent decay. Rosenthal also said, "Pound the pavement to try to get a show." So I did.
At the Museum of the City of New York on Fifth Avenue on the northeastern corner of Central Park's Museum Mile, a lovely woman named Leslie Nolan told me that these images were worthy of a show but they didn't have time in their schedule for one. Nolan recommended that I go to the New-York Historical Society on the opposite side of Central Park, just a block south of the American Museum of Natural History. Nolan said the executive director, Betsy Gotbaum, was very receptive to exhibiting new documentations of "the city." Dale Neighbors, who was in charge of exhibitions at the New-York Historical Society, almost immediately responded, "Sure, we can exhibit these images this summer into the fall. I know exactly where we can hang them. How about twenty-four images?" My jaw dropped.
When I returned to class and told Rosenthal that I would have a show at NYHS, he demanded, "Get it in writing!" So I did. I faxed the press release to the _New York Times_ , and the rest is history. When I saw Rosenthal again, the front page of the _Times_ hung proudly on his office's front door, and he greeted me with a "you son-of-a-bitch" and a giant bear hug and a kiss. I am forever grateful for Rosenthal's guidance as a young emerging artist.
Much of what I learned about these images, the companies that were being advertised, their products and the sign companies that painted them was from standing behind people who came to the museum to see the show. I often went and listened intently. I went to the NYHS on many occasions that summer as a "fly on the wall," just to see my work discussed by the crowd. What I thought was a minor exhibit ended up getting major press attention that led to me making connections with fellow fading ad enthusiasts around the world.
For the sake of organization and narrative, the original twenty-four images have been integrated into this book's collection by categories that are arranged to tell the story of the human body and how it moves through the city of New York's size and scale: forever propelled by desires and needs, we only need to look up to see how the story of the body is made explicit through the vast amounts of visible advertising on the city's surfaces. Indeed, it has become apparent that one of the major themes of this book is the relationship between people's bodies, advertising and the New York urban landscape over the past century.
The questions and challenges presented by the human body are a narrative that is writ large and continually played out on the surfaces of New York's buildings and walls. By focusing on such signs, we are offered a glimpse into the cultural, moral and medical values of past eras. Our bodies break down, a human failure that warrants cures and preventative medicine. Our bodies have needs that sometimes get us in trouble, either by the results of excesses or the consequences of social and religious constricts. Our bodies need comfort, shelter and security. Our bodies need pleasure and sensual stimulation. Our actions to fulfill these needs also pit our bodies against conventional, social and religious mores. Ultimately, our tired bodies decay and die. All the trials, tribulations, joys and pain of the body can be mirrored in these ubiquitous signs that envelop us.
Frank H. Jump
Snake Oils, Elixirs, Tonics, Cure-alls and Laxatives
Omega Oil—West 147th Street, and Frederick Douglass, Harlem, New York City. Taken March 1997. Ad circa 1910. Hung in NYHS Exhibition.
**O MEGA OIL**
_For Sun Burn, For Weak Backs, For Stiff Joints, For Sore Muscles, For Athletes. Trial Bottle 10¢_.
Marketed as an all-purpose miracle oil, the company was incorporated from February 3, 1870, through April 2, 1924. Snake oil in American vernacular has a negative connotation that suggests medical quackery and hoax cures. In China, snake oil is used to treat rheumatoid arthritis and is proven to contain high concentrations of eicosapentaenoic acid or EPA, one of the omega-3 fatty acids. Although the actual compounds in snake oil may have an analgesic effect, it has generally come to mean a panacea. An elixir—derived from the Arabic word _al-iksir_ , meaning "effective recipe"—is usually made with alcohol, not unlike a tincture that contains one particular active ingredient dissolved in alcohol, and is usually pleasant tasting. Again, the use of elixirs suggests miraculous cures and panacea. Many liniments were highly aromatic and contained salicylic acid (aspirin) and capsaicin (active ingredient in hot chili pepper). All of these purported medicinal products were marketed and sold during the turn of the twentieth century all over the country, and perhaps first in New York City. One just has to begin a journey on an urban yellow brick road in a spiral search of one of these ancient hucksters, and he will soon hear their tireless and faded siren's song beckoning to him.
The giant wall mural of Omega Oil is one Emerald City example of the guaranteed claim to remove your pain and cure your ills. Today's equivalent would be television commercials promising to remedy erectile dysfunction, baldness or depression. "Ask your doctor about..." I'm sure physicians just love their patients coming in with self-diagnosed and self-prescribed ailments they are sure they possess. At least one hundred years ago, you could have gone to the chemist or druggist and said, "Give me a bottle of Sloan's Liniment. Got a bad flare-up of the rheumatiz." Perhaps the trusted balm would work. We have more control over our illnesses than we fully comprehend. Today, people walk into their physician's office with a litany of self-diagnosed maladies that were suggested to them on the tube. "Are you tired, run-down, listless? Do you poop out at parties? Are you _unpoopular_?" Lucille Ball and Desi Arnaz nailed it back in the 1950s. The problem "back in the day" (I promised myself I would use this meme only once) was that there was very little regulatory measures for these "medications."
The origins of the Food and Drug Administration, the U.S. federal government's "oldest comprehensive consumer protection agency," can be traced back to about 1848 with the appointment of Lewis Caleb Beck to the Patent Office. Beck carried out chemical analyses of agricultural products, a function that was inherited in 1862 by the newly created Department of Agriculture. Not known by its present name until 1930, "the FDA's modern regulatory functions began with the passage of the 1906 Pure Food and Drugs Act, a law a quarter-century in the making that prohibited interstate commerce in adulterated and misbranded food and drugs."
Don't you feel more at ease now? Here we have the root of the word disease. One can be diagnosed with a life-threatening illness and still feel "at ease" with one's self. We are _not_ our disease. We _are_ our ease. For some time now, especially in gay publications like the _Advocate_ or on subway billboards in target neighborhoods, there would be advertisements for AIDS medications about which "you should ask your doctor." Sustiva was one of them. Having personally sat in focus groups to name antiretroviral drugs in the early 2000s, I know the power of a brand name's influence on your desire to take that drug. Sustiva. Sustain Life. One of the side effects of Sustiva is "wild, hallucinatory dreams." After falling through the fabric of my bed into an unknown dimension every night, I knew this was not the medication that would allow me to be adherent. What was astounding was I started receiving instant messages from younger people of Gay AOL chat rooms who knew I was HIV-positive asking if I wanted to sell them my Sustiva because it had fast become the latest party drug. OMG! NO! NO!
Since flushing these toxic medications down the toilet hurts the fish in the ocean, I'd recycled them through donor programs when they made me irrational—at best, suicidal, and, at worst, starting to present with lipodystrophic-acquired disorders characterized by either complete or partial lack of adipose tissue (lipoatrophy). Over the last twenty-one years, I've become a virtual clinical study in both the deleterious and beneficial effects of AIDS medications. Living as long as I have with this virus, I've tried every imaginable AIDS medication, from antiretrovirals to protease-inhibitors.
Fortunately now, the most benign and minimal amount of chemical agents keeps "my virus" at non-detectable levels. In 1999, a doctor who was conducting a drug trial suggested no medications at all. After a year's round of self-injections with interleukin-2, my T-cells, the immune cells that fight disease, started rebounding at levels only Superman could possess. The stuff was like kryptonite. So the next step in this experiment was "a drug holiday," which in the '70s meant a whole different thing. Of course, this process was closely monitored by medical professionals at the esteemed New York Hospital–Cornell Medical Center on York Avenue within a block or so of the Mecca Cigarettes ads (page 90), which I visited on my daily walks to the clinic. A week into my "drug holiday"—a total cessation of my AIDS meds—the expected outcome was that my viral load would increase dramatically and then decrease on its own. As expected, my levels shot up to about ten thousand viral particles per unit. Within six weeks, my viral counts were way up over two million, and my preciously nurtured and cultivated T-cells began to disappear to the lowest levels yet measured.
Within a few weeks, upon receiving a rectal cancer diagnosis in August 2000, I quit the study to have the lentil-sized "mass" surgically removed. In the hospital waiting rooms, there were medical journals with advertisements of different chemotherapies for breast and cervical cancers. After undergoing vicious and barbaric rounds of chemotherapy and radiation treatments, I was left with third-degree burns to my groin that left permanent scarring and suffered permanent hearing loss and tinnitus from _Cisplatin_ ototoxicity. Just last week, I heard there is now a shortage of this cancer treatment.
With all this said and done, perhaps I would have fared better with a dose of Dr. Tucker's 59 for All Pain (page 55). At least I would have gotten a good buzz instead of this incessant ringing in my ears. What a royal pain in the ass that all was, but I'm glad that bit of history is behind me now. Ba-ddum-bum. So much for the "wonders of modern medicine." Concurrently, I am still here in spite of and as a result of modern science.
Fluoruracil Chemotherapy Bag and Hospital Bracelet Collage/ID Card. January 2001.
Caroline Rance, the writer of the blog _The Quack Doctor_ , discusses Omega Oil's secret ingredients:
_The miraculous little green herb was possibly henbane, but fortunately it was in very small quantities—the main ingredients were chloroform, oil of wintergreen and mineral oil. Oil of wintergreen (methyl salicylate) is still a component of deep-heat liniments today and I imagine that sufferers did feel the benefit of this sweet-smelling emerald liquid. It got on the wrong side of the FDA in 1942, when it was judged misbranded because of the exaggerated claims, but this didn't kill off the product—it is still available in some countries today_.
Omega Oil was the first fading ad that I ever photographed. I don't remember noticing ghost signs before this. A friend from my photography class called me up and asked if I would walk with him through Harlem to photograph friezes and other architectural details since I had lived there before and he wasn't comfortable walking with his camera alone. When I noticed the sign, I asked Arthur if he had seen other signs like this before. Terms like ghost signs were not in my lexicon. I was a novice. I quickly climbed the construction scaffolding that still surrounded the newly built police department precinct building that was adjacent to this emerald beauty. Before the cops came out to yell at me to climb down, I got this shot.
Watercolor of Omega Oil. _Courtesy of Sandra Walker, RI_.
What was unimaginable at the time was the kaleidoscopic journey on which this sign would take me, from scouring the five boroughs with Vincenzo in 1997, to a 1999 cross-country road trip that would have me sleeping on the floors and couches in the homes of journalist and ghost signs author Wm. Stage and Tod Swormstedt (American Sign Museum). This sign afforded me the invaluable and instructive collaborations with other urban archaeologists like Kevin Walsh ( _Forgotten New York_ ) and Walter Grutchfield ( _14 th to 42nd Street_). Who would have thought that I'd become acquainted with Julian Seery Gude, the great-great-grandson of outdoor media giant and pioneer O.J. Gude, through the digital medium of the Internet if I hadn't found Omega Oil on the day I began my search for a photographic subject.
Another fortunate outcome of discovering this sign was receiving generous attention from journalists around the globe. In May 1999, the _London Observer_ did a lovely two-page spread that got the attention of an expatriate American photorealist watercolorist living in London, Sandra Walker, RI, whose works are "represented in numerous public and private collections and embassies throughout the world." In 2003, I received a wonderful and unexpected card in the mail from the "internationally renowned artist," who paints watercolors of city scenes, including fading ads. Walker wrote to me about how in 1999, a friend had sent her a newspaper clipping from the _Observer_ , and that my photography mirrored the work she does in watercolor. Walker had gone to Harlem in search of the Omega Oil building, found it and painted it. Walker said, "I just wanted to thank you for the inspiration."
Perhaps what has made this image so enduring for me is the actual endurance of the ad itself. Omega Oil is still standing tall, in all its emerald splendor, a tireless beacon that continues to illuminate our remedial and commercial past. This one may just outlive us all.
**S YRUP OF FIGS**
"So. Are you regular?" A question only an alterkaker would understand. Quite literally, this means "old shitter" in Yiddish. This often overheard question made by our grandparents and aunts and uncles while sitting at the dais of a wedding or bar mitzvah can be considered small talk to some, unmentionable to others. While thumbing carefully through a vintage turn-of-the-century magazine, you will notice an abundance of ads for products to help you go a little easier. One would think that constipation was a much bigger problem back then, or perhaps people seemed much more at ease about talking about their bowel movement or "unmovement."
On a website called _The Museum of Menstruation_ , there's a tongue-in-cheek anecdote taken from a 1904 women's medical journal called "The 20th Century Song Book": "Menstrual, kidney & liver problems, constipation & bowels, tiredness, indigestion, colic, colds, chills, fever, childbirth, rheumatism, arthritis, menarche, leuchorrhea, dizziness, pain, headache, Cardui, Wine of Cardui, tonic, 'female weakness,' etc."
Syrup of Figs—Greenpoint Avenue (near corner of Franklin Street), Brooklyn. Taken March 1998. Ad circa 1900s. Hung in NYHS Exhibition.
Perhaps this should be sung to a popular "list song" tune like the "Twelve Days of Christmas" or Billy Joel's "We Didn't Start the Fire." At first I thought Syrup of Figs was a dessert topping. I stood corrected when one of my readers e-mailed me that this was a remedy for constipation and was the main ingredient in Fletcher's Castoria as well. This was a big "a-ha moment" for me. Google Images has quite a collection of marvelous ads for the sweet viscous extract, mostly all claiming to be made from California figs. The etymological root for the word "syrup" actually comes from the Arabic _sharab_ (beverage) via the Latin _siropus_. According to Wiktionary, syrup of figs is a laxative and the Cockney rhyming slang is "a wig," which brings up my childhood hysteria over "Merkin Hall."
As a kid, hemorrhoid commercials were equally hilarious to me. Not so funny anymore. An Israeli MD claims on his website that "hemorrhoids are the consequence of the sitting defecation posture." After 150 years of sitting, the "sitting toilet bowl" has caused physical damage. Apparently, physiologically we were meant to squat. Our diets over 150 years ago were much less processed, and we did get our daily dose of dietary fiber, depending on where you lived and, of course, what you ate. I recall Archie Bunker yelling at Edith on _All in the Family_ , "I don't need roughage! I need smoothage!" As American consumers, we are obsessed with what we eat and what we don't eat. The big push (no pun intended) for adding dietary fiber to our diets is over a century old. On a government website sponsored by the National Institute of Health, the benefits for individuals who eat dietary fiber are provided:
_Significantly lower risk for developing coronary heart disease, stroke, hypertension, diabetes, obesity, and certain gastrointestinal diseases_ [and] _lowers blood pressure and serum cholesterol levels. Increased intake of soluble fiber improves glycemia and insulin sensitivity in non-diabetic and diabetic individuals. Fiber supplementation in obese individuals significantly enhances weight loss_ [and] _benefits a number of gastrointestinal disorders including the following: gastroesophageal reflux disease, duodenal ulcer, diverticulitis, constipation, and hemorrhoids...More effective communication and consumer education is required to enhance fiber consumption from foods or supplements_.
Perhaps this abstract is a bit more than you can chew, but the trend in American diets is inching bit by bit (or bite by bite) toward eating better than we have in the past one hundred years. Organic, whole foods are not just brand names but have become a desired option for more people who can afford to eat better. We all can afford to eat better, but not everyone has the means or proper dietary education. First Lady Michelle Obama has recently piloted the Let's Move campaign, aimed at helping end the problem of childhood obesity by "Creating a healthy start for children; Empowering parents and caregivers; Providing healthy food in schools; Improving access to healthy, affordable foods; Increasing physical activity."
The school where I teach has a cooking workshop for second graders that teaches them how to prepare simple yet healthy snacks like black bean salsa, guacamole and fruit salads—all with adult supervision, of course. Ideally, when they go home with the recipes and the steps for making the snacks, they can share them with their parents. Every time you turn on the TV, open your laptop or walk out your door, you're being asked to make a food choice, so they may as well be good choices. Aren't you hungry for... _something_ now?
**F LETCHER'S CASTORIA**
_Children cry for Fletcher's_.
With the original intent to be seen by people crossing the Manhattan Bridge over Chinatown, this sign was clearly visible from the D train until its demise over a decade ago. In 1999, when I first launched the campaign website, I was informed by fading ads enthusiast Barbara Roberts that Castoria was not a castor oil at all but a prune-flavored laxative for children. Roberts informed me the ingredients were cascara sagrada, syrup of figs and extract of prunes.
Remnants of Castoria ads still pepper the five boroughs and are by far one of the more commonly seen vintage ads in New York City, from the Bowery to Jamaica, Queens. Chris Riley wrote to me in May 2000 that Fletcher's Castoria was manufactured by Charles Henry Fletcher beginning in 1871 and had been patented in 1868. The initial patent for Castoria included "senna, sodium bicarbonate, essence of wintergreen, taraxicum, sugar, and water." Riley said it was modified later to include other things like pumpkin, anise and wormseed to improve the taste to be more like root beer.
Fletcher's Castoria—Market and Henry Streets, Chinatown, New York City. Taken March 1997. Ad circa 1900s. Hung in NYHS Exhibition.
Within months of the website launching, I was contacted by the Centaur Company, which still owns the patent for Castoria, and asked if we could trade links. For more information about Castoria products, visit the Centaur Company website
**R ADWAY'S READY RELIEF**
Perhaps Radway's Ready Relief is a precursor to today's pain-management addiction problems, as can be seen in this sales promotion for the cure-all that started out with a patriotic pill-popping pitch in the _New York Times_ on October 23, 1861:
_A F REE GIFT; TO THE AMERICAN PEOPLE. WHAT THE FEDERAL TROOPS ARE FIGHTING TO SUSTAIN. HOW OUR SOLDIERS SHOULD BE PREPARED TO FIGHT. DRS. RADWAY & CO. RADWAY & CO., No. 23 John-st., New-York_.
_H OW OUR SOLDIERS SHOULD BE PREPARED TO FIGHT. Health and discipline are the most important elements. RADWAY'S READY RELIEF. IN DR. RADWAY'S REMEDIES RADWAY'S READY RELIEF IN HIS KNAPSACK, WHEN SICK, DUTY OF CIVILIANS. ARMY INDORSEMENT. NOT ONE IN THE HOSPITAL_.
_I MPORTANT TO FAMILIES. DR. RADWAY'S PILLS. RADWAY'S REGULATING PILLS RADWAY'S RENOVATING RESOLVENT. IT CURES, WITH ASTONISHING RAPIDITY. HUMORS AND SORES OF ALL KINDS, LADIES NOTICE_.
Two years later, this early "info-mercial" appeared in the _New York Time_ s touting new medical benefits and cures with an apparent topical application of this miraculous medication:
_A New Method of Curing Certain Diseases, by the Use of Radway's Ready Relief Has been recently discovered, and is calculated to supersede the use of all electromagnetic machines, external applications, liniments, plasters, emenagogues, medicines for fever and ague, rheumatism, paralysis, numbness, neuralgia, &c. The application is simple, the effects pleasant, and the cure positive and infallible. It proves what we have for nearly a quarter of a century maintained, that Radway's Ready Relief is the best, quickest, and most important remedial agent ever discovered. It is, however, only recently that we have discovered its immense curative power in certain diseases by this new process_.
_T HE SECRET OF ITS APPLICATION was suggested to us by Professor Reid, Professor and Lecturer of Chemistry for many years in the New-York College of Physicians, the New-York Hospitals, Edinburgh University, &c., &c. We have tried it in a number of cases that had previously resisted all other treatments. Every case yielded with astonishing rapidity. The success of the Ready Relief, in its ordinary application, has been greater than all other remedies in use. This can be vouched for by thousands who have used it_.
Radway's Ready Relief—Delancey Street, New York City. Taken August 1997. Ad circa 1890s.
Radway's close-up.
_Let those afflicted with the following complaints use it under the new method, and we guarantee an immediate cure: R HEUMATISM, NEURALGIA, Gout, Sciatica, Nervousness, Fever and Ague, INDIGESTION, SMALL POX, MEASLES. Cramps, Spasms, Lumbago, Headache, Heart Disease, FEMALE COMPLAINTS, Retention, Suppressions, or Misplaced Menstruation, URINAL ORGANS, Diseases of the Kidneys, Bladder, Urethra, Weakness, Spermatorrhea, Pains, Aches, Spasms in the Back, Thighs, Hips, &c. THE APPLICATION in these cases must be made TO THE SPINE_.
_Rub the entire length of the spine and across the small of the back with R ADWAY'S READY RELIEF for ten or fifteen minutes, night and morning, and a cure will follow. We do not recommend its application in this form as an experiment, but with a full knowledge that it will cure either of the above-named complaints_.
_Newspaper clipping courtesy of Melissa Ulloa_.
_We are recommending no new or untried medicine—its merit and efficacy as a specific for many of the pains, aches and infirmities that afflict mankind, is acknowledged by thousands—but suggest to our readers and patients a new method of its application for certain complaints, which, if followed, will insure a more positive and rapid cure than any known treatment in vogue. In all cases ask for R ADWAY'S READY RELIEF. Patients are invited to call at Dr. RADWAY'S Medical Office, No. 87 Maiden-lane, New-York. RADWAY'S RELIEF is sold by Druggists_.
There was a spine-tingling jingle that went with this pain medication and cure-all with lyrics and sheet music published in 1883 (at right).
No wonder this cure-all was so popular, with all those claimed medicinal benefits! Currently, I'm working on having this jingle re-created by singers with HIV by the fall of 2011. I can hear those bass _ba-ba-ba-bas_ sung with ecstatic exuberance already.
As sung by the Apollo Club, of Boston
Radway's Ready Relief (from Paine) Male Quartet with Bass solo
_Twenty years of sleepless nights_ ,
_Twenty years of sleepless nights_ ,
_William Sydney Myers, Myers_
_Esah of Havana Cuba, Of Havana_
_Cuba, Cuba, ba, ba, ba, ba_ ,
_Cuba, Cuba_ ,
_The correspondent of the_ Times,
_The_ London Times
_The_ London, London Times.
_Suffer'd with acute and chronic rheumatiz_ ,
_Acute, acute_
_Acute and chronic rheumatiz_
_For five and twenty years he had not enjoy'd one whole nights calm rest_
_One night's calm rest_.
_Radway's Read Relief_ ,
_Radway's Read Relief_ ,
_He applied_ ,
_It immediately gave him rest_
_And secur'd him the first calm_
_And undisturbed sleep during the twenty years_
_The continued use of Radway's Relief cured him_
_Radways Ready Relief_ ,
_Always ask for_
_Radways Ready Relief_ ,
_Take no other!_
_Price per bottle twenty-five cents_ ,
_Radway, Radway, Radway, Radway_ ,
_Only, oh! Only_ ,
_Utterly Too cheap! Cheap!_
_And for sale, for sale, ev'rywhere_
_Radways Ready Relief_ ,
_O always ask for Radways Ready Relief_ ,
_Radways Ready Relief_ ,
_Ev'rywhere for sale by all apothecaries_
_All apothecaries in the land_ ,
_Including Chelsea Beach_ ,
_O always ask for Radway's Ready Relief_
_Ev'rywhere for sale by all apothecaries_
_All apothecaries in the land_ ,
_Including Chelsea Beach_ ,
_Ask for Radway's Ready Relief_ ,
_His Ready Relief_
_Radway's ready relief: Male quartet with bass solo By John Knowles Paine_
From Harvard University, January 31, 1960
Eda Kohn Loeb Music Library
**A MAROSA SCALP RUB**
_For Your Hair_
Another fading ad under the EL along Broadway in Brooklyn—a veritable treasure-trove of signs of life from Brooklyn's commercial past. On a walking tour that I had co-sponsored with the Brooklyn Historical Society during my show at the Williamsburg Art & Historical Center, along with Kevin Walsh's _Forgotten New York_ crew, we took a large number of people down Broadway.
Amarosa Scalp Rub—Broadway, Williamsburg, Brooklyn. Taken August 1997. Ad circa 1900s.
**(?) O IL—CURES SORE THROAT**
Another snake oil that claimed to "cure sore throat as soon as applied. For sale by druggist." The red-and-white holding tank in the distance was one of two that were known as the Elmhurst gas tanks, a popular landmark, often mentioned in traffic reports on the radio. KeySpan (now National Grid Energy) dismantled the twin tanks in 2002 for the construction of a new community park.
(?) Oil—Cures Sore Throat—Williamsburg, Brooklyn. Taken August 1997. Ad circa 1900s.
**D AVID BLUSTEIN & BROTHER FOR RAW FURS & GINSENG**
Here is an example of the business address marker as ad. This form of storefront sign was common in the garment district and other areas of New York City, where the simple "sign above your store" was impossible given the immensity of these buildings and the large population and diversity of businesses within. Along the length of some buildings, you will see the traces of these address markers on every floor. Walter Grutchfield's website encompasses the Garment District, and you can see a multitude of these markers and his excellent documentation of them. At the top you see Photo-Stemmerman. Below that is a name I can't make out, but directly under it, it says MFG (manufacturing) FURRIERS. Right below that it says "Karp Bros. Coats & Suits." But clear as day it says "David Blustein & Brother for Raw Furs & Ginseng" below that.
David Blustein & Brother for Raw Furs and Ginseng—Seventh Avenue, New York City. Taken March 1998. Ad circa 1910.
This Blustein remnant of the later years of the Far East trade was documented on the authority of Grutchfield with the following details:
_David Blustein & Bro. was founded around 1905 by the brothers David Blustein and Isadore Blustein. Isadore Blustein (1868–192?) was the older brother who lived in West Virginia (David Blustein & Bro. had an office in Charleston) from as early as 1907 until at least the mid-1920s...David Blustein (1874–1932) was born in Triski, Russia, immigrated in 1891, was naturalized (in Charleston, West Virginia) in 1896, lived in Manhattan (West End Ave. in 1920) and later on Long Island (Hempstead in 1930). The business was located here at 158–164 W. 27th St. from 1913 until 1948. In 1940 the name was changed to Blustein Furs...This refers to the branch office in Charleston, West Virginia, and to exporting ginseng and other botanical roots_. _A photograph sent to me by Dennis A. Stone, a grandson of David Blustein, shows David (the man at the far right) and two other men gazing at a large pile of ginseng...In 1945 Dennis Stone's father was serving in the U.S. Army in Frankfurt, Germany, when he annotated this price list of Blustein offerings for raw furs. The annotation reads, "I was in contact with my office by cable and air mail. I wrote this price list from Frankfort." Dennis notes, "He was apparently running the business in 1945 while serving in the Army." The verso expands on various offers, including prices for ginseng_.
_...Photo Stemmerman at the top refers to the Stemmerman Photo Craft Laboratories located here from 1913 to 1918. William H. Stemmerman (1867–1951) was a druggist and physician in Passaic, New Jersey, and the original Stemmerman Photo-craft Laboratories was located in that city from around 1911. Stemmerman retired from his businesses before 1920. (He appears in the US Census of 1920 living in Pasadena, Calif. with employment listed as "none.") The New York branch of the photo lab re-located to the east 20s in 1918, and Joseph G. Taylor (1891/92–?) became the owner. In the early 1930s the name changed to Taylor Photo Co_.
It was not uncommon to see companies advertising constituent products for a final larger product as you see in signs for "coat fronts" and here for raw furs. It is surprising to see ginseng sold next to animal furs. I've often wondered about the history of this trade. Not surprisingly, many of the things we traded came from China. Today, goods produced and bought from the Chinese are driving our whole consumer economy, which to some is a blessing and others a curse. American consumers enjoy lower prices for goods while American producers are literally digging for ginseng. What's fascinating is it all began with furs and ginseng. On the _JSTOR Plant_ website, a bit of history and etymology of ginseng is offered:
_Ginseng enters the English and French vocabularies in the ages of exploration. In the French as early as 1750, in English in 1654, and for Portuguese, the term ginsao appeared even earlier (Holmes, 283). In Chinese_ , _(phonetically rénsh_ n) is the name for ginseng and that roughly translates as "man root." Even the Chinese character _is meant to look like a man with two legs. So there you have it. A little linguistic lesson for the day_.
_So, the ginseng trade has been well established for centuries and has turned a tidy profit for many, including that beacon of 19 th century New York City life, John Jacob Astor. Astor was one of the more ambitious people ever to set foot on American soil. Astor, German by birth, landed in the United States in 1784 just at the end of the Revolutionary War, immediately established himself in the fur trade with Quebec, and became incredibly wealthy. Not being content relying on middlemen, Astor bought his own ship for sending his furs to London, learned of the East India Company trade with China, and immediately capitalized in the Chinese demand for furs_.
Hard to imagine that this woody root—once used as an aphrodisiac and currently used as a mild organic stimulant in alternative energy drinks, with claims to countless other benefits—was also the root of New York's trade with China. Apparently, George Washington had heard about the Chinese need for ginseng and learned there was a huge supply growing indigenously in North America. Since the only crop we could grow that would be valuable to the Chinese at the time was ginseng, which wasn't being used widely in the United States, it became a valued U.S. export. On a Chinese cultural website, the following historical facts were provided:
_Ginseng helped promote the formation of the notion of international trade in the US. The entire country was connected to trade with China. Not only merchants in New York, Boston and Philadelphia, but isolated farmers in deep mountains had learned that they could be paid by something grown in the northern slope of the mountains. About the same time when the Empress of China unloaded the Ginseng at China, George Washington (1732–1799) met some people who were doing Ginseng business in Virginia. He recorded it in his diary, "I met numbers of Persons & Pack horses going in with Ginseng: & for salt & other articles at the Market Below."_
In today's market, there is still a great demand for ginseng. Although our more lucrative exports lay in automobiles and other technology, China produces its own. In the American producer's dig for a _new_ ginseng, let's hope the only commodity left to trade with China isn't just our national debt.
**D R. TUCKER'S 59 FOR ALL PAIN**
This sign on Broadway in the Bushwick section of Brooklyn advertises a turn-of-the-century pain medication. On the corner is a business marker for S. Huffman—Jeweler & Optician. Recently in Park Slope, Brooklyn, another Dr. Tucker's sign was revealed while a restaurant called Sidecar was renovating its space. Brothers John and Bart DeCoursy had this to say about their summer of 2007 discovery on their website:
_When the brothers first signed the lease on 560 5 th Ave, they had no idea that a gem lay waiting to be discovered. As they transformed the building from a children's clothing store into a restaurant, they pulled back layer upon layer of wallpaper and sheetrock in search of exposed brick. Instead, they found what looked like the letter A. They furiously scraped the wall down until the whole image was on display: "DR. TUCKER'S 59 CURES COUGHS, COLDS. FOR ALL PAIN." Since they wanted a classic decor that would feel warm and comfortable, they knew immediately that it would be incorporated into the design. With a little research, they found that Dr. Tucker's 59 was a popular elixir during the late 1800's and was eventually taken off the shelves in the early 1900's due to the fact that its main ingredient, cocaine, had been outlawed. Seen as a sign from above that the boys had found the right location, they incorporated this beautiful work of art into their logo; Sidecar—Food & Drink Cures All Pain. Eventually, Bart also created a cocktail in its honor. All of the present cocktails' ingredients, however, are perfectly safe and legal_.
I'll drink to that!
Dr. Tucker's 59 for All Pain—Brooklyn's Broadway Elevated J Line. Taken June 1998. Ad circa 1900s.
Days with Art
_Lingering in Frank Jump's Images_
The poignancy in Frank Jump's chosen subject matter, his disappearing ads, transforms the language of advertising into a poetics of signs. And there is nothing forgettable about either the images or Frank's inspired pursuit of art, and of living. One thinks of Atget's photographs of the façades and storefronts in a disappearing Paris, but Jump's compositions are decidedly less formal. Or perhaps it can be said that there is still room to breathe in Jump's images, as Atget's are sealed off (if brilliantly so).
Much has been made of the connections between Jump's photographs and his own biography. Both elucidate the culture of a specific moment. Both survive, beautifully so, even surrounded by loss. Both are deeply inspiring. One work that continues to hold me is the triptych Confrontation (Dr. Tucker's 59 for All Pain). It documents an advertisement painted in white text on brown, forming a banner running the length of a windowless brick building. Read left to right, the three photographs shift from sunlight-heightened contrast to an overcast, slow fadedness and ultimately include the elevated subway track, underscoring the proximity of the building (and photographer) to the approaching train. In the last, Jump has shifted the color of the clouds and the sign to an impossible luminosity. In each, the presence of 59 and PAIN and the sometimes legible FOR ALL form an unshakeable chant, not unlike the raps and beat poetry Jump composed for the early days of ACT UP.
With the generosity and leadership of artists, Visual AIDS utilizes visual art to promote dialogue about HIV. We document the work of HIV-positive artists and pay tribute to the creative contributions of AIDS activism—and we are proud to honor the extraordinary photography of Frank Jump.
After almost a decade of attempting to fathom and alter the human devastation, in 1989, Visual AIDS inaugurated a Day Without Art. Early exhibitions of Jump's Fading Ad Campaign were part of a shift in this landmark art action that coincided with the World Health Organization's AIDS Awareness Day on December 1. Originally, the day was to be a cultural intervention: shrouding works of art and darkening the galleries in the face of the AIDS crisis. The gestures were resonant. Tom Sokolowski, a founding member of Visual AIDS, described the event's importance to the _New York Times_ : "The language of art speaks in different ways from normal discourse. Perhaps those of us who are engaged in the making and displaying of works of art can in some way use the medium to dispel ignorance and bigotry that have surrounded what began simply as a medical problem." After the advent of drug therapies that extended the lives of those who had access, it became more urgent to share the creative contributions of HIV-positive artists. Jump's show at the Gershwin Gallery in 1997 and his 2000 exhibition at the Williamsburg Art and Historical Center were leading examples of a new Day With[out] Art.
Dr. Tucker's 59 for All Pain—up on elevated train track, confronted by conductor. Taken March 1999.
Like the Visual AIDS Archive Project, in which he is a long-standing artist member, Frank Jump's art practice creates a record of ephemeral histories. Even without the awareness of his roots in formative New York gay and AIDS activism, it's impossible not to characterize Frank as innately collaborative. At the height of publicity and interest in Fading Ad, Frank took the opportunity to speak about HIV. He has always been open about his longtime survivor status and was instrumental in linking his photographs with the message that "AIDS is not over."
It's worth mentioning that Jump's early adoption of the web to share his photographs challenged an individualized idea of art and the singular, marketable, finite work of the artist. When Frank opened the Fading Ad Gallery in Brooklyn in the mid-2000s, he programmed exhibitions of various Visual AIDS Archive Project members, and not just on Day With[out] Art but year round. And through it all, there are the photographs. Like a private moment in a public space, the image of 59 for All Pain sometimes lingers on with me for days. Holding this image is an active experience. It is one of the things art can do, and particularly in Frank Jump's hands, as he does it so lovingly and so well.
Amy Sadao
Executive Director, Visual AIDS
August 1, 2010, New York
Food, Snacks and Candy
Hams & Capocolli/Wallabout Provisions Co.—Brooklyn Navy Yard. Taken September 1997. Incorporated from July 28, 1925, through October 31, 1933. Hung in NYHS Exhibition.
**H AMS AND CAPOCOLLI**
One hundred and twenty five years before this sign was painted in its block sans serif lettering, the navy yard was a bustling harbor where merchant vessels were built. Earlier still in the 1630s, Wallabout Bay was the site where the first French-speaking Walloon settlers from the Netherlands relocated. It was from this very early Dutch settlement that the city of Brooklyn sprang forth. Embedded in shores of this waterfront is a fascinating and gruesome history of colonial skirmishes with the Canarsee Indians, the cultivation of the first tobacco plantations in the New World and the British mooring their prison ships during the American Revolution, throwing over ten thousand American soldiers who died of neglect overboard to decay on the Wallabout Basin beaches.
This sign stared west over the Brooklyn Navy Yard for almost a century to witness the navy yard's height of activity during World War II, when over seventy thousand people were employed. Hams and Capocolli also witnessed the goings-on at the notorious JJ's Navy Yard Cocktail Lounge just a couple of blocks south on Flushing and Washington Avenues. The lounge seems to have been there forever and is the subject of some Brooklynites' bravado stories, according to Nicole Brydson: "Dubbed everything from secret stripper bar to the scariest bar in Brooklyn, neighbors often wear a visit to this dive, on the corner of Washington and Flushing Avenues, like a badge of honor; the bar as portal to a forgotten yesteryear."
Ham production is documented to have originated between 300 and 400 BC during the Roman times. Prosciutto ham and capocolli are cured meats popular in Italian sandwiches called panini. A single sandwich is called a panino. In the United States, they are known as cold cuts, and in earlier days they were considered meat "provisions," since they were ready-made and had a long shelf life due to the curing process that preserves the meat. In Brooklyn, the slang term for a capocola (singular) is gabagool, which derives from the Neapolitan accent and other southern Italian and Sicilian dialects. Vincenzo insists on pronouncing capocola with his Torinese accent. Not a grinder or sub goes past his lips without at least a few slices of the sweet and the hot dry capocolli in combination, and to this day it is one of his favorite sandwiches (and this one of his favorite ads). This ad also caught the eye of my friend Sandra Walker, who created this watercolor based on the photograph, which for me as an artist is the highest form of flattery.
Watercolor of Hams & Capocolli. _Courtesy of Sandra Walker, RI_.
From the height of this ad, Hams and Capocolli could have also spied the goings-on at the Cumberland Packing Company, which is just a few blocks south.
**S WEET'N LOW**
_Everyone in my family tells this story, but everyone starts it in a different way. My mother starts it in the diner across from the Brooklyn Navy Yard, where my grandfather Benjamin Eisenstadt, a short-order cook, invented the sugar packet and Sweet'N Low, and with them built the fortune that would be the cause of all the trouble_.
_—Rich Cohen_ 51
Everyone I know has a story about the pink packet, especially if you are from Flatbush. I didn't start out in Brooklyn though. Usually after a few beers, I'll admit I'm from Queens, famous for its...diners! As early as I can remember, on every table in every diner, there was a rectangular glass container filled with an assortment of sweeteners. I never knew people actually bought Sweet'N Low. As a kid, when going out to the Crossbay Diner with the families of friends, invariably I would witness with horror their grandmother surreptitiously dumping loads of the pink packets into her unsnapped clamshell pocketbook, her eyes darting back and forth, looking to see if anyone was spying on her grand pink theft as she snapped her purse closed. So when I noticed the product on grocery store shelves, it seemed odd that anyone would actually buy it. When I first drove by the Cumberland Packing Company along the Brooklyn Navy Yard on Flushing Avenue, I imagined thousands of old ladies with their pocketbooks grabbing what they could while quickly ambling down the aisles of warehouse shelves. Could make for a great marketing strategy. Instead of winning a year's supply on _The Price Is Right_ , you could win "A five-minute trip through the Cumberland Packing Company, Home of Sweet'N Low!" Five minutes alone in the factory and all you can stuff in your purse—a grandmother's dream. Now that we're over fifty, I'm continually mortified as Vincenzo stuffs a few pink packets into his shirt pocket before leaving a restaurant, although sometimes—he steals the salt shaker.
Not very long after I took this shot, the Cumberland Packing Company painted a new Sweet'N Low ad, covering the lovely cursive script and block serif lettering of "The Perfect Sugar Substitute." The newer ad is a bit disappointing in its style, but subconsciously, it seems aesthetically justifiable to me, since there really is "no perfect sugar substitute." Back in the day, the options were limited to saccharine or cyclamates. The cyclamates scare and subsequent ban of the early 1960s is etched into my consumer DNA. I can remember packages of Funny Face drink mix touting "no cyclamates" as I went shopping for groceries with my mom at Bohack's on Merrick Boulevard in Laurelton, Queens.
I've often wondered what ever happened to cyclamates, since they seem to have faded from the American consumer consciousness. On the Elmhurst College website, the history of this sweetener was provided by Professor Charles E. Ophardt:
_Michael Sveda, while a graduate student at the University of Illinois, discovered cyclamate by smoking a cigarette. While working on the synthesis of anti-pyretic (anti-fever) drugs in the laboratory in 1937, he put his cigarette down on the lab bench. When he put it back in his mouth, he discovered the sweet taste of cyclamate (unsanitary lab technique)_.
Sweet'n Low—the Perfect Sugar Substitute. In back of the Cumberland Packing Company. Brooklyn Navy Yard. Taken May 1997 Ad circa early 1950s. Hung in NYHS Exhibition.
_Cyclamate was initially marketed as tablets that were recommended for use as a tabletop sweetener for diabetics. In 1958, cyclamates were classified as GRAS (Generally Recognized as Safe). A mixture of cyclamate and saccharin, which had been found to have synergistic sweetening properties and improved taste, was subsequently marketed for use in special dietary foods. In the 1950's diet drinks were introduced using a cyclamate/saccharin blend. The market grew rapidly and soon accounted for about 30% of the soft drink sales_.
_In 1969, the result of a chronic toxicity study with a mixture of cyclamate and saccharin was interpreted as implicating cyclamate as a bladder carcinogen in rats. Cyclamate was removed from GRAS status and eventually in 1970 banned in the United States from use in foods, beverages and drugs, and is still currently banned. However, many other countries did not act on this incomplete data, and cyclamate continued to be used as a sweetener in those countries. Today over 55 countries have approved the use of cyclamate_.
So cyclamates are alive and well and living in food products around the globe—just not here in the United States. Apparently, there's been a lot of controversy over saccharine in the last few years, and according to Rich Cohen, the Sweet'N Low family has had their hand in it. Here is another excerpt from Rich Cohen's "bittersweet" family story:
_My father starts the story in downtown Brooklyn, in the courtroom where my Uncle Marvin, the first son of the patriarch, a handsome, curly-haired man who insists on being called Uncle Marvelous, is facing off against federal prosecutors. After assuming control of the Cumberland Packing Company, which makes Sweet'N Low, Sugar in the Raw, Nu-Salt, and Butter Buds, Marvin, among other things that caused a scandal, put a criminal on the payroll, a reputed associate of the Bonanno crime family. That criminal made illegal campaign contributions to Senator Alfonse D'Amato, who sponsored legislation that kept saccharin on the market. Saccharin, a key ingredient of Sweet'N Low, had been found to cause cancer. In the end, Marvin cut a deal with prosecutors, testifying for the government and keeping himself out of prison_.
Never could I imagine so much controversy and drama over a food additive. Oops. I forgot about red dye #2. In a 2008 _Newsweek_ article about the five top controversial food additives, another sweetener that goes by the brand name Splenda, otherwise known as sucralose (acesulfame potassium), makes that list. But for the sake of brevity, let's stick to one sweet scandal at a time, and this one's a saccharine story to say the least. On the NPR website that is currently featuring Rich Cohen's book _Sweet and Low_ , it goes into detail about the sweet deal that was made with the FDA and its attempt to ban the pink stuff:
_Saccharin was discovered in 1879, and was used during both world wars to sweeten foods, helping to compensate for sugar shortages and rationing. It is 300 times sweeter than sugar. In 1977, a Canadian study that looked specifically at the role of impurities—and of other suspected tumor causes, such as parasites in test animals—showed convincingly that saccharin itself was causing bladder cancer in rats. That same year, FDA proposed to ban saccharin for all uses except as an over-the-counter drug in the form of a tabletop sweetener. Congress responded by passing the Saccharin Study and Labeling Act, which placed a moratorium on any ban of the sweetener while additional safety studies were conducted. The ban was officially lifted in 1991 and saccharin continues to have a fairly large appeal as a tabletop sweetener, particularly in restaurants, where it is available in single-serving packets under trade names such as Sweet 'n Low. Because it has a good shelf life, saccharin is used widely in fountain sodas, and its stability at high temperatures makes it an option for sweetening baked goods, unlike aspartame, which degrades when heated. Saccharin also is favored economically because it can be made inexpensively_.
After a long and protracted attempt to ban saccharine as an artificial sweetener, in 2010, the EPA finally released this cloying statement about this substance:
_In the late 1990s, the National Toxicology Program and the International Agency for Research on Cancer re-evaluated the available scientific information on saccharin and its salts and concluded that it is not a potential human carcinogen. Because the scientific basis for remaining on EPA's lists no longer applies, the agency has removed saccharin and its salts from its lists. EPA proposed the removal of saccharin and its salts from the lists on April 22, 2010 and did not receive any comments opposing the proposal_.
Phew! I sure feel safer now. There are now so many choices for artificial sweeteners—the pink stuff, the yellow stuff, the blue stuff. Surely you remember the Equal commercials, with Cher so eloquently and glamorously touting that she uses the "Blue Stuff." I always imagined a counter-ad featuring Roseanne for Sugar in the Raw (also packaged at Cumberland factory) with her claim in that Midwest twang, "Heck, I use the Brown Stuff." I've stopped using sugar since my nutritionist told me our bodies (especially mine) convert it almost immediately into triglycerides. Stevia seems to be the latest sweetener craze. Less sugar still is the best sugar substitute, as far as I'm concerned. Childhood obesity and a rise in people becoming diabetic in this country are problems that need to be tackled. Just don't knock me over trying to get to the Sweet'N Low, please.
**F ULTON STREET FISH MARKET**
_If I invite some boy some night To dine on my fine finnan haddie—Cole Porter_
Historically located along the East River in Lower Manhattan just south of the Brooklyn Bridge and only six blocks north of Wall Street, the Fulton Fish Market was one of the oldest continuously running fish markets in the country, second only to the Maine Avenue Market in Washington, D.C. After 180 years of operation in Manhattan, the famous fish market relocated its operations to the Hunts Point in the Bronx on November 14, 2005. According to the New Fulton Fish Market website:
_The market first opened on that site in 1807 on land donated to New York City, and at first was a general market for both fish and goods other than fish. In 1822 the fish merchants occupied a new Fulton Fish Market building, located on South Street between Fulton and Beekman Streets_.
Fulton Street Fish Market—New York City. Taken June 1997. Storefront ads circa 1940s.
_Prior to 1850, housekeepers from Brooklyn and nearby areas would purchase fish directly from the market. However, since that time, wholesale customers were the primary buyers. The market gradually gained in importance, and in 1924 the market sold 384 million pounds of fish, 25 percent of all seafood sold in the United States_
_The Fulton Fish market was primarily located in two open air structures, the "Tin Building" and the "New Building," in which various dealers rented stalls from the Port Authority of New York with closed offices at the back of the stalls. The New Building was opened in 1939 by Mayor La Guardia, after pilings of the old market building gave way in 1936 and the entire building slid into the river. Not only was the marketplace old and established, but many of the wholesalers at the Fulton market were well-established firms_.
An ultramodern, 400,000-square-foot, fully refrigerated indoor fish market and facility with all the latest amenities, including fish wholesalers, retailers and restaurants, was completed at a cost of $86 million without any loss of employees. The project generated new construction and retail jobs in an economically depressed area in the Bronx while opening up prime real estate in Manhattan for luxury residential housing. One of the unspoken reasons why the fish market moved, other than the smell of fish that permeated the area for over a century, was the stranglehold that organized crime had on the Manhattan market. The new facility is run under the supervision of the City of New York Business Integrity Commission in an attempt to "eliminate organized crime and other forms of corruption and criminality from the industries it regulates." Sounds like a new and more effective organization should be regulating Wall Street, which gives new meaning to the term organized crime.
**G RADE A MILK**
This "got milk" ad heralds from a time when this part of Brooklyn had the lion's share of dairies. It may seem hard to imagine a pastoral Brooklyn with cows grazing as far as the eye could see, but in its heyday, much of the area's milk was produced along what is now Atlantic Avenue in East New York and along Flushing Avenue alongside the Brooklyn Navy Yard, straight through Bushwick, the most eastern division of Brooklyn. The early Dutch and English settlers of the late seventeenth century in Flatlands raised dairy cows. The following history was provided by a historical resource study prepared by the Department of the Interior on this precursor to western Long Island's nascent dairy industry:
_Other Dutch towns, such as Bushwick, and English communities, such as Jamaica, Flushing and those at the eastern end of Long Island, had much larger herds of sheep than the Dutch settlements of the bay. In Flatlands, swine were almost as equally scarce, there being twenty-one in 1676; Flatbush at that time had fifty-two. Eleven oxen were to be found at Flatlands in 1676 and twenty-five at Flatbush. The main draft animals were horses, both Flatlands and Flatbush having each approximately one hundred horses in 1676_.
Grade A Milk—Waverly Place, Clinton Hill, Brooklyn. Taken August 1998. Ad circa 1900s.
_The most important livestock were cattle. Flatlands reported 209 cows in 1676, and Flatbush. It would appear that Jamaica Bay farmers were as much dairy farmers as growers of crops. This gives special importance to the meadows bordering the bay, since it provided forage for numerous cattle as well as horses. Dankers informs us that the Dutch mowed the salt meadows and moved the hay presumably to their barns. In the same year as the Danker's visit, 1679, horses were grazing on Barren Island. That salt hay was mowed and sold in Brooklyn in 1826 is reported in Baxter's journal. Thus the bay's meadows for two hundred years provided grazing for livestock and cut hay for use at farms in the area and also for sale to others_.
Apparently, by the mid-nineteenth century, Brooklyn dairy farmers were faced with a lung plague that afflicted their cattle. According to a report of the commissioner of agriculture for the year 1879:
_In 1849, William Meakim of Bushwick, Long Island (New York) kept a large dairy and employed a man with a yoke of oxen in drawing grain from the New York and Brooklyn distilleries. A milkman on the way, who had lung fever in his herd persuaded this man to use his oxen in drawing a dead cow out of his stable. Soon after the oxen sickened and died; and the disease extending to his dairy cows, Mr. Meakim lost forty head in the short space of three months. The stables have thus become infected, Mr. Meakim continued to lose from six to ten cows yearly for the succeeding twenty years, or as long as he kept in the milk business. This, which is but one instance out of a hundred, covers fifteen years of the plague in the Skillman stables, and brings the record down to 1869_.
Despite this twenty-year outbreak of bovine pneumonia, by the late 1890s, the Brooklyn dairy businesses had experienced a boom. Names like Axelrod, Breakstone, Renken, Borden and Sheffield Farms all shared their roots in Brooklyn dairy farms during this time. Many of these dairies were in production until the early 1960s. I remember drinking milk from a bottle from the Renken Dairy as a child. According to Kevin Walsh on his _Forgotten New York_ site, the following defunct dairy-related buildings still stand:
_R.H. Renken Dairy at SW corner of Classon & Myrtle Avenues, Clinton Hill Brooklyn_. [The Art Deco–designed] _Building although presently unused, was constructed in the 1930s by Koch & Wagner. The M.H. Renken Dairy of Brooklyn, NY was established in 1927 in the eastern part of the borough, continuing operation until 1962_.
_Sheffield Farms, a major milk distributor in the Northeast in the late 1800s and early 1900s, was acquired by National Dairy Products (now Kraft, Inc.) in 1925. It has left a number of impressive buildings around town, not least what is now the CBS Broadcast Center at 524 West 57 th Street at the northern end of Hell's Kitchen_.
_...Sheffield Farms in Brooklyn. This facility was on south side of Fulton Street between Brooklyn & New York Avenues. The building was transformed to The Restoration Center in the early 1970s. The Restoration center contains retail stores that includes a Duane-Reade drugstore, Baskin-Robbins, a branch of Citibank, a supermarket that was formerly Pathmark, a skating rink and the Billie Holiday Theater where off-off Broadway shows are presented_.
Even the lesser-known products like Axelrod's Dairy, established in the late nineteenth century and catering to New York's kosher community, had their headquarters in Brooklyn. In 1930, Axelrod established another branch in Freeport, New York. According to Rabbi Jeffrey A. Marx in a historical outline of the Breakstone (Breghshtein) Brother's Dairy:
_It should be noted that Joseph and Isaac's enterprise was, by no means, a novel one. In fact, the history of Axelrod's Dairy is closely parallel to Breakstone Bros.: In 1896, Wolf Axelord had established both a dairy retail store and wholesale distribution business on Madison St. A few years later, he moved his business to Brooklyn, where Axelrod's Dairy sold sour cream, and Pot Cheese products which were delivered by horse and wagon. At some point, Axelrod's began manufacturing their own products, which were produced in up-state NY dairies and then, packed in ice, shipped by train to NY. In 1932, just a few years after Breakstone's Dairy, Axelrod's was also bought out by National Dairy Products_ Co.
According to Marx, in 1889, Joseph Breakstone was selling milk from 203 Division Street in Williamsburg, Brooklyn. His brother Isaac Breakstone, "still listed as Brekstone," was operating a butter business at 602 Myrtle Avenue in Brooklyn, with other outlets at 3 Wallabout Market, as well as operating other locations in Greenwich Village and Harlem. In 1907, the Breakstone Brothers moved their operation to 3 Washington Avenue (now within the confines of the Brooklyn Navy Yard) and also established locations in Queens.
The W.M. Evans Dairy Co. at 3480 Fulton Street and Eldert Lane in Cypress Hills, Brooklyn (on the Queens border), had, according to Kevin Walsh, "the highest numbered address on Brooklyn's Fulton Street." Who knew Brooklyn was once renowned for its dairy production? Brooklyn is also famous for a local beverage documented as having originated in 1890: the egg cream, which used Grade A milk or _echt_ cream (Yiddish and Dutch for "real" cream), chocolate syrup and seltzer. Queens is also famous for its dairies, including Queensboro, Dairylea and the Elmhurst Dairy, Inc., which was founded in 1919 and is still based in Jamaica, Queens.
**B ABY RUTH AND BUTTERFINGER CANDIES**
Another sign that is hiding behind a tall, modern, multistory high-rise building that was recently built on Delancey Street is this delectable morsel of an ad, which is the second ad I captured after Omega Oil. This beauty of a sign gave me so much momentum and was also recommended by Vincenzo, my partner, who found many of these images while riding his construction van through New York City. Needing an eye-level shot for this, I talked the manager of the now disappeared five-and-dime department store to let me get on the roof. Sadly, four years later, this campaign would have been much more difficult to photograph due to world events that changed the lives of New Yorkers and people around the world forever. But part of the wonder of exploring my urban landscape was the adventures of getting access to an interior to get to an exterior of a building. For weeks before this ad was finally re-obscured, the building in front had been dug to its foundation, and the Butterfinger ad was finally visible by street passersby once again for a brief hungry bite.
Baby Ruth Candy & Butterfinger—Delancey Street, New York City. Taken March 1997. Ad circa 1930. Hung in NYHS Exhibition.
In a phone call with Nestlé, which then had the patent for this early twentieth-century candy bar, its historian shared some tidbits about the bar. I learned that Baby Ruth was developed in 1921 and named after President Grover Cleveland's daughter, who died at the age of twelve. Curtis's Candy of Chicago (1921–64) sold Baby Ruth to Standard Brands when it dissolved and was then bought by Nabisco. The Nestlé Food Corporation finally acquired the candy bar in 1990. The Nestlé historian deduced that the ad must have been painted in 1930, since after the stock market crash, Curtis's stopped its advertising for a number of years. Butterfinger Candy was another invention of the Curtis's Candy Company of Chicago, Illinois, in 1923, with the name being chosen by a public contest. Company founder Otto Schnering had both Butterfinger and Baby Ruth bars dropped out of planes in the days before the stock market crash of 1929, which helped them gain their wide appeal.
Still a Halloween favorite in the tiny bite-size wrapped version, Baby Ruth candy has staying power. The sign still dwells in the shadows of a modern structure, no longer able to watch the frantic Williamsburg Bridge traffic or candy being dropped from the sky.
**P LANTERS PEANUTS**
This is another dubious vintage mural of which I've questioned its veracity. The sign faces the rising sun and would surely have been much more faded if it were indeed authentic.
Kevin Walsh wrote me this e-mail in 1999 in response to my posting this image on the campaign website: "Hey, we've both been taken in by that Planters Peanuts ad in Ridgewood. A few days ago I got the news that that ad was painted in 1985 when they were filming _Brighton Beach Memoirs_...I may have to start a page with amazing re-creations, since I have three of them now." Walsh writes this on his site about this sign:
_Planters Peanuts Always Fresh! Your friendly Planters Peanuts salesman, along with the ever-present Mr. Peanut, has been preaching the gospel of nuts from the Seneca Avenue elevated platform in Ridgewood since at least the 1930s, and possibly before that. At least I thought it did. I've been informed that this ad has only been there since 1985, when the Seneca Avenue station appeared in the movie_ Brighton Beach Memoirs _! It's so realistic, it has me completely fooled. Note the grinding/repairing and typewriter and cutlery ads above the Planters ad—those are apparently real. Other Forgotten Fans have told me that a Planters' ad was indeed on this site before the movie was shot. So for me, this ad is shrouded in mystery_.
Planters Peanuts—Ridgewood, Queens. Taken August 1997. Ad's authenticity in question.
Honestly, I don't think the signs above the ad are authentic either. Compared to other Planters Peanuts signs I've photographed, like the ones in Wilkes-Barre, Pennsylvania, where the brand originates, this Mr. Peanut figure isn't really a good likeness (see Missing Mr. Peanut in Wilkes-Barre, Pennsylvania). The original Mr. Peanut was tall, with lanky legs.
On one of our fading ad tours through Scranton and Wilkes-Barre, we came across the original building where Italian immigrant Amedeo Obici built his peanut empire. Obici and his brother-in-law Mario Peruzzi founded the unincorporated Planters Peanuts & Chocolate Company in Wilkes-Barre, Pennsylvania, in 1906. According to the Planters Peanuts website, a grade-school artist, Antonio Gentile, submitted his sketch of Mr. Peanut and won a brand icon contest. A commercial graphic artist later added a top hat, monocle and cane. A three-story image of Mr. Peanut was painted on the side of the original factory on the Main Street entrance, with another large Mr. Peanut painted on the back facing the Lackawanna Railroad line. Mr. Peanut stood tall, watching the trains pass for decades until, in 2006, on the 100th anniversary of the company's founding, Mr. Peanut had a catastrophic encounter with progress and a wrecker's ball.
In August 2006, several organizations in the Wilkes-Barre area that knew of the plans of local developer Marvin Slomowitz to demolish this historical landmark and build a forty-thousand-square-foot strip mall tried to save the building from destruction. Thom Greco, a local businessman who, according to Denise Allabaugh of the _Citizen's Voice_ , "owns a plethora of Planters' memorabilia," led a campaign with the help of a local radio DJ named Jenn to sell "Save Mr. Peanut" wristbands for $5 each. Allabaugh also reported that Greco hoped the $1,200 raised by his campaign might be used to ensure the proper preservation of the building's grand façade, which was agreed upon to remain.
An appeal by the Young Professionals Association of Northeast Pennsylvania, in which my photos of Mr. Peanut were used to try to include the warehouse behind the building where the Mr. Peanut sign was painted and the adjacent wall to the façade where another Mr. Peanut sign was painted, also fell on deaf ears. According to the developer, the historic façade "encompasses the front of the building and did not apply to Mr. Peanut's picture on the side, according to the agreement." Slomowitz categorized the warehouses with the signs as "anything but historical." According to Allabaugh, Wilkes-Barre city councilman Jim McCarthy, along with a crowd of supporters of preserving the warehouses on which these beautiful antique signs were painted, watched as the building where Mr. Peanut stood for half a century was demolished "with lumps in their throats."
In Joni Mitchell's 1970 hit "Big Yellow Taxi," she croons how they "paved paradise and put up a parking lot." Sometimes instead of a parking lot, they put up a strip mall. I have yet to visit this strip mall. I miss my visits to Mr. Peanut. "Don't it always seem to go that you don't know what you've got till it's gone?"
**U NEEDA BISCUIT**
In 1997, I spoke with a Nabisco historian who asserted that Uneeda Biscuit was introduced in 1898 and was the first product to bear the name of the National Biscuit Company, later known as Nabisco Foods. That same year, according to _Advertising Age dot com_ , the Philadelphia advertising agency N.W. Ayer & Son launched the first prepackaged biscuit, Uneeda, with the slogan "Lest you forget, we say it yet, Uneeda Biscuit." Nabisco went on to launch the first million-dollar advertising campaign for Uneeda. Due to the proliferation of remnant Uneeda Biscuit ads nationwide (the French Quarter in New Orleans has a famous one), it is evident that this painted sign was part of this million-dollar advertising budget that was allocated to outdoor advertising (also see page 212).
Uneeda Biscuit—Malcolm X Boulevard (formerly Reid Avenue), Brooklyn. Taken July 1998. Ad circa 1905. Hung in NYHS Exhibition.
**G OLD MEDAL FLOUR**
Gold Medal Flour ads are up there with Fletcher's Castoria ads for staying power, as the grades of lead paint used allowed them to weather so well. You can make out above the Gold Medal logo and to the right of a steel support rivet the names Washburn-Crosby—an 1877 partnership between the flour's founder, Cadwallader C. Washburn, and John Crosby. After a flour dust explosion that destroyed the old Minneapolis mill, a new one was built and was the first in the world to use automatic steel rollers. By 1907, Washburn & Crosby launched its long-running advertising slogan, "Eventually...Why not now?" According to the Gold Medal Flour website:
_B.S. Bull, the company's advertising manager, is credited with the creation of the slogan. As the story goes, he was editing a wordy text about the superior quality of Gold Medal Flour and found, that when he was finished he had edited out all the words except 4: "Eventually."He then added, "Why Not Now?" Having had this brilliant idea, he was struck with self-doubt and tossed the paper into the wastebasket. It was said to have been found by a young member of the firm, James Ford Bell, who later became the first president of General Mills, Inc. (The slogan was used on billboards, company trucks, train cars, flour bags and in the company's printed advertisements, appearing as late as the early 1950's. Other companies adopted the slogan as their own; it was seen in political cartoons; the slogan was the title for a Sunday sermon; and it even appeared as the front-page headline of the_ Cincinnati Times-Star _with a small-print notation, "With apologies to Gold Medal Flour.")_ 71
Gold Medal Flour—Ferry Terminal, Staten Island. Taken August 1998. Ad circa 1910.
According to the Chicago Advertising Federation website, in 1921,
_a Gold Medal Flour ad created by the agency, Blackett-Sample-McFarland, developed a picture puzzle with answers so obvious the company was flooded with mail from consumers. To answer the letters, company officials thought it would be best to respond with a letter signed by a woman. The character, Betty Crocker, was created with a signature only. By 1940 Betty Crocker was the second best known woman in America, running only behind Eleanor Roosevelt_.
Unfortunately, Staten Island has never been a good source of fading ads. Compared to Queens, which still sports more than a handful of beauties, this is the only ad to represent this borough. But it is one heck of a sign. I will need to scour the borough of Richmond carefully with Kevin Walsh and his crew again someday soon I hope. Eventually...Why not now?
**C ADET DOG FOOD**
Pet owners take pet food for granted. Most people think prepared pet foods have been around since canned baked beans, but the invention of the kibble and dried chunks (or extruded food pellets) and canned moist morsels we feed to our dogs isn't quite half a century old. An organic pet food retail website provided the following historical context:
_Before the advent of pet foods, most dogs and cats lived off of grains, meats, table scraps and homemade food from their owners. It wasn't until the mid-1800's that the world saw its first food made specifically for dogs. An American electrician, James Spratt concocted the first dog treat. Living in London at the time, he witnessed dogs around a ship yard eating scraps of discarded biscuits. A light bulb went off in his head and shortly thereafter he introduced his dog food, made up of wheat meals, vegetables and meat. His company flourished and by 1890 he was taken over by a large corporation and production had begun in the United States as well_.
Cadet Dog Food—Third Avenue, Brooklyn. Taken November 1998. Ad circa 1950.
_But it wasn't until the early 1900's that pet food really caught on. Canned horse meat was introduced in the United States under the Ken-L-Ration brand after WWI as a means to dispose of deceased horses. The 1930's saw the introduction of canned cat food and dry meat-meal dog food by the Gaines Food Co. During WWII metal used for cans was set aside for the war effort, which nearly ruined the canned pet food industry. But by the time WWII ended, pet food was off and running again, and sales had reached $200 million...At first, canned pet food was the primary type sold, but by 1956 the first extruded pet foods were hitting store shelves. Extrusion is the process by which pet foods are formed into pellets, and then sprayed with synthetic nutrients to compensate for nutrition lost during processing_.
According to the Pet Food Institute, pet food companies flourished even during the lean years of the early twentieth century. On its website it claims the following:
_The Depression of the '30's may have meant hard times for industry, but the merchandising of dog food germinated and spread countrywide. One report accounted for 221 brands of canned dog food with about 200 of this total produced at a half dozen canning plants. The plants did the canning; the merchandisers supplied labels. The situation was similar with dry pet foods. All that was needed was a brand name and empty bags_.
_Working capital was minimal and the market wide open. Metropolitan New York saw Cadet brand canned ration (Re-Dan Packing Co.) and Snappy canned dog food (Foster Canning Co.) as the two leading sellers_.
Daniel Pearlstein was the owner of Cadet Dog Food and Re-Dan Packing. I learned this from comments from his granddaughter Leslie Sanders Axelrod on the _Fading Ad_ Blog, where she was reunited with a cousin with whom she used to spend holidays at their home in Rego Park, Queens. I posted pictures Richard Boll had sent me of a Cadet Ad that was briefly exposed on the wooden slats of a house in Maspeth, Queens. It was from this ad we learned that one of the ingredients of Cadet Dog Food was "chlorophyllin." I imagine this additive was to promote better dog breath. The Cadet Dog Food image pictured here was from a painted ad on Third Avenue in Boerum Hill, Brooklyn, not far from the terminus of the Gowanus Canal. The building on which it was painted has since been demolished, and a new high-rise will soon be in its place.
A Coke and a Smoke
Bronx Coca-Cola—241st Street and White Plains Road, Wakefield, the Bronx. Taken August 1997.
**C OCA-COLA AND MECCA SMOKES PENTIMENTO**
In explaining a layered fading ad, I've always used the term _pentimento_ , a painterly term that describes evidence of a previous work on a canvas seen through an existing upper layer. Viewing these works under varied wavelengths of light, like ultraviolet, infrared and even X-ray scanning, can aid scientists in deciphering both palimpsests and pentimenti. The use of the word pentimento in "street and photography" has also been cited on the Internet as a term "used in a modern sense to describe the appearance of the sides of buildings with painted advertising." Often when newer ads are painted over older ads, "the paint wears away to reveal the older layers." Examples of this can be seen in the work I did in the Netherlands in 1998 while photographing fading ads in Amsterdam.
In the last fifteen years that I've been documenting vintage advertising on brick, I still get exhilarated when I find a Coca-Cola ad. It doesn't matter how faded, or how skillfully or sometimes not so skillfully painted, this graphic mark transcends time—this is a logo for the ages. Not only is it the most popular soft drink in the world, it is also one of the most often found remnants of the early twentieth-century outdoor advertising campaigns. Here is a description of the product and trademark taken from the _Logo Blog_ :
_Originally_ [it was] _intended as a "patent medicine" when it was invented in the late 19 th century by pharmacist John S. Pemberton as a "coca wine." Coca-Cola has dominated the worldwide soft drink market for decades now. The Coca-Cola logo, like the product itself, is rated among the most recognized logo design and brands in the world. The first Coca-Cola logo was created by John Pemberton's partner and bookkeeper, Frank Mason Robinson, in 1885. Thinking that the two Cs would look well in advertising, it was Robinson who came up with the name and chose the logo's distinctive cursive script_.
Coca-Cola and Mecca Smokes pentimento—Bleecker and Carmine Streets, New York City. Taken April 1997. Ad circa early 1900s. Hung in NYHS Exhibition.
_The typeface used, known as Spencerian script, was developed in the mid 19 th century and was the dominant form of formal handwriting in the United States during that period. The red and white colored scheme in the Coca-Cola logo was kept simple and distinctive to lure young minds. Even the Coca-Cola bottle symbolized the "youthful exuberance of America." Since then, various designs of the Coca-Cola bottle had been released over the decades. But the ever popular version is the famous 1915's curved-vessel bottle called the "contour bottle," better known to many as the "hobble skirt" bottle. Though mistakenly designed as cacao pod, the bottle like Coca-Cola logo has been highly popular and is often regarded as the best design ever. The Coca-Cola logo was first advertised in the_ Atlanta Journal _in 1915 and also appeared on the display of Pemberton's pharmacy. A Coca-Cola dispenser with the popular logo design was later created by Raymond Loewy. The Coca-Cola logo got registered as a trademark in 1887 and has since then become the brand's corporate identity_.
Almost imperceptibly overlaying this Coke ad was perhaps the oldest Mecca Cigarettes ad in New York City. From the price of five cents, and from a similar sign on the East Side (page 90), I can date this sign between 1910 and 1912. When I first glanced at this sign on the corner of Carmine and Bleecker Streets in Greenwich Village where Sixth Avenue also intersects, I could vaguely make out the distinctive Coca-Cola logo, but the Mecca Cigarettes ad was still a mystery. The building wall faced south, explaining the intensely faded aspect of this pentimento. I had to get on the roof to take this. The pizza joint downstairs did not have access to the roof. So I went next door and rang the bell. "Who is there?" "Photographer!" I got buzzed in. The woman who lived there had roof access, but we had to move her refrigerator away from the staircase to the roof that was being blocked. I was amazed that she let a total stranger do all this to get on her roof with a camera. Once I was up there, I could clearly see the pack of ten unfiltered cigarettes open for the taking.
In the course of documenting these signs, I have become aware of the profound specificity within certain niches of advertising enthusiasm. There are people who only collect Coca-Cola ads. There are the breweriana fans. Tobaccoiana fans are a breed of their own. Given that there was so much outdoor advertising in the sale of tobacco products (Mail Pouch and Owl Cigars, just to name two), there surely are still thousands of remnants scattered across the globe. An even more specific fan base is the enthusiasts of the colorful card collectibles that were generated from the nicotine ad campaigns. James A. Shaw offers his insight about Mecca Cigarettes on his colorful and comprehensive website:
_The centuries-old city of Mecca is nestled among the ancient hills of Arabia. Mecca was the birthplace of the prophet Muhammad, and has been the destination of millions of Moslem pilgrims. Mecca Cigarettes was named after this holy city when introduced by the Kinney Bros. Tobacco Company in 1878. Mecca sales benefited from a heavy dose of advertising dollars following the dissolution of The American Tobacco Company trust in 1911. Magazine and newspaper ads, a wood framed sign, plus several sets of attractive cigarette insert cards boosted consumer awareness. Ads published in 1915 claimed Mecca "the largest selling brand in America today." America's fascination with the mystic of the Mideast would last until the First World War. When Turkey allied itself with the enemies of the United States, plus the 1913 introduction of the modern "American Blend" Camel cigarette, the popularity of the different Turkish and Egyptian brands plummeted. Even though the Turkish brands would no longer play an important role with smokers, exotic Turkish tobacco would continue to be imported and used as a "seasoning" in the Camel, Lucky Strike, and Chesterfield brands_.
**B RONX COCA-COLA**
( _pages 82–83_ )
If I recall correctly, I distinctly remember scaling the wall of this subway platform with the help of Vincenzo to take this shot.
The following is from Kevin Walsh's website:
_Thanks to the MTA's renovations of several stations along the White Plain's Road line in the Bronx, this sign can now be seen in its full magnificence. Before renovation this sign could only be seen from the catwalk on the west side of the elevated structure just south of the station's platform or from the street. Taken from the #2 terminus at 241 st Street & White Plains Road. Wakefield, The Bronx_.
**P HILIP MORRIS**
The _New York Times_ quoted me in 1998 as calling these signs "tombstones." Although I was taken out of context (I see most of these signs as signs of life, metaphors for survival), this particular sign does have a tombstone quality. In retrospect, the ghostly image of the World Trade Center in the distance is indeed haunting.
As a child, I enjoyed watching cigarette commercials, the music always drawing me in. My love affair with cigarette commercials started with the Marlboro theme, so _Bonanza_ -like in its bravado. The Marlboro man was pretty hot, too. Later, I was hooked on the Benson & Hedges theme that was performed by a Tijuana Brass sound-alike (and album cover look-alike) group called the Brass Ring. The song was entitled "The Disadvantages of You," and I actually went out and bought the single in 1967. I played it constantly and danced around my room with an imaginary cigarette that, of course, was "a silly millimeter longer" than the other brands, so I often got my imaginary cigarette caught up my nose or slammed in my bathroom door by accident, just like in the commercials. I also remember loving the ironic title; within the context of the cigarette commercial, the disadvantage was meant to be the clumsy length of the cigarette, not the fact that smoking the cigarette actually killed you. Fortunately for me, I got hooked on TV commercial music and not the product it was selling. Philip Morris owns both cigarette brands Marlboro and Benson & Hedges.
In the United States, Benson & Hedges was acquired by Philip Morris in 1953, when "the boards of directors of Benson & Hedges and Philip Morris agreed to a merger of the two firms." Today, Philip Morris (strategically rebranded as Altria Group in 2003) is the largest shareholder of familiar American food brands such as Kraft (which merged with General Foods), Nabisco and an exhaustive list of popular cigarette brands and wineries worldwide.
Philip Morris—Fifth Avenue and East 20th Street. Taken June 1998. Ad circa 1940s.
Reporter for _Time_ magazine Randy James stated, "The first tobacco advertisement in the United States ran in 1789 when what is now the Lorillard Tobacco Company promoted their snuff in a local New York newspaper." In this article, James goes on further to explain the dramatic increase of the use of tobacco products in the United States due to advertising:
_By the late 19 th century, two innovations had helped launch cigarette companies to national prominence. The first, a cigarette-making machine introduced in the 1880s, dramatically increased production; instead of producing some 40,000 hand-rolled cigarettes a day, a company with one of these machines could produce 4 million cigarettes daily. The second development came in the late 1870s with the invention of color lithography, which revolutionized advertising and packaging and helped developing brands strengthen their identities. Using this new technology, companies began including small cigarette cards in every box as premiums. These collectible trading cards depicted movie stars, famous athletes and even Native American chiefs_.
In 1964, the U.S. surgeon general warnings on cigarettes were introduced. Cigarette commercials were totally banned from television in the United States in the early 1970s. According to the World Health Organization (which wants a ban on all tobacco advertising), there are an estimated 1.3 billion cigarette smokers worldwide. Of these smokers, 33 to 50 percent will be killed by their habit. By the end of the twenty-first century, if smoking trends continue, 1 billion people will die due to smoking-related illnesses. This is not including deaths caused by secondhand smoke. I can think of safer, more enjoyable ways of controlling the planet's exponential growth in population, can't you?
**G OLD MEDAL FLOUR/MECCA SMOKES EAST SIDE PENTIMENTO**
Another great pentimento with nature doing her tricks!
Gold Medal Flour/Mecca Smokes pentimento—East 70th Street and York Avenue, New York City. Taken June 1997. Ad circa 1910.
**M ECCA SMOKES EAST SIDE**
_Persistent Satisfaction_.
Today that means "insistent addiction."
Mecca Smokes—East 70th Street and York Avenue, New York City. Taken June 1997. Ad circa 1910.
**S OHO COCA-COLA**
The translucent "water tower" on the left is a sculpture by Rachel Whiteread. It is a resin cast of the interior of an actual water tower. In a 2001 piece called _Chronology_ , Mickey Stretton compares my work with Whiteread's while formulating the following theory about the exploration of time as a fourth dimension in visual arts:
_I also have a great fascination for Installation art, such as that of Rachel Whiteread. Whiteread creates sculpture of casts from negative space that the object occupies, and the spatial language she uses is another great example of looking closer at something to see beyond the expected. Had I a tin-opener close by (not to mention a few extra pages) I would probably open a whole can of worms about using space as a method of communication. I haven't though, so I won't. Besides, the whole issue of time is much more interesting and so with time in mind, I'll move swiftly on_.
_Before I can talk about the works of artists whose use of time I find inspirational and thought-provoking, I guess I must first try to describe what I find so intriguing about the concept of time itself. Whilst I don't let myself lose sleep over it, the whole concept of time poses so many unanswerable questions that I find completely fascinating. How do we understand the notion of time, when we can't see or feel it? Why does a minute go faster when you're having fun than when you're waiting for a bus? Is time elastic, able to be stretched or otherwise manipulated? What relationships exist between space and time?_
_Apologies if you think that last bit got a bit heavy, but stick with me and I'll see if I can explain what I mean. Of any roll of film to come out of my camera, at least four or five shots will invariably be of rusting signs, derelict shop frontages or similar urban typography, so when a friend introduced me to the work of Frank Jump I felt an immediate empathy. Jump is a photographer whose work (www.frankjump.com) concentrates on the ageing ads painted onto the sides of factories or shops around his native city of New York, and reveals a shared preoccupation and fascination with found imagery, typography and urban architecture_.
_But more than that, the work also relies inherently on the influence of time, and as such provokes thought and begs the question: what story do they tell about the building/advert/product? Is it art or is it graphic design, but these pieces are often selling products that are no longer on sale and are complete strangers to today's world of commercialism, so surely they must be art? If they are art, how much of their beauty is a result of their age? What did they look like a year ago? A decade or a century ago? Would they have been considered beautiful then?_
_Jump isn't attempting to answer these questions, but instead just trying to get the viewer to ask themselves the questions it provokes. A couple of paragraphs ago I proposed the question "How do we understand the notion of time, when we can't see it or feel it?" In this case we can visualize time by its effects on the ageing sign, but we can't see time itself—only a vehicle or language that visually represents it. Of course, you could argue that a clock is an adequate way to see time, but even then we are using numbers, in standard measurements of hours, minutes and seconds, as a means to visualize it. And besides, how accurate is a clock as a way of seeing time (they run fast or slow, or stop completely—time never stops, does it?)_.
SoHo Coca-Cola—Grand Street facing West Broadway, New York City. Taken September 1998. Ad circa 1910.
Which leads me to postulate that progress, like time, never stops—or does it? In this photograph, Duckett & Adler is still visible, as is Wintergarden. Although partly responsible for its partial demise, the Colossal Media outdoor advertising company has been using a fraction of this wall for new advertising. As in the Liberty Beer sign (page 97), there is the ambivalence of having a perfect fade interrupted, even in the name of progress. Thus is the cycle of the urban landscape continuing in its dynamistic unfolding—a collage of commercial artistic endeavors building layer upon layer for future fading.
Breweriana
Guinness Beer—Long Island City, Queens. Taken September 1997. Ad circa 1960s.
**L IBERTY BEER**
In my search of breweriana sites of trays and other ephemera, I found a tray with the name "Liberty Beer American Brewing Co. Rochester NY." This site also gives a timeline for this Rochester brewery, and the date range of 1889–1920 seems to be when this ad was executed. During Prohibition, the company changed its name to Rochester Food Products Company. Rochester, New York, is on the Genesee River and has the magnificent High Falls adorning its waterfront before it empties into Lake Ontario.
Shortly after the Iroquois tribes were forced out of this area, Colonel Nathaniel Rochester founded the town in 1803. Rochester is nicknamed the "Flour City," but it easily could have been nicknamed the "Brew City." According to Rochester city historian Ruth Rosenberg-Naperstack, "The earliest documented distillery opened in Rochester in 1815." According to Rosenberg-Naperstack:
_Intoxication was a social problem that arrived along with the earliest settlers; an individual lifestyle carried from communities left behind. Distilleries were among the earliest industries in the Genesee country. Farmers grew corn and rye for sale to distilleries because the profits were higher than wheat sales to flour mills. A jug of hard liquor was commonly passed when a roof was raised or a field of wheat was cut by community effort_.
_Taverns were among the earliest businesses built, finding ready use among travelers moving into the new country. It became a friendly gathering place for the men to tell stories of the Revolution, of wild animals and adventures_.
_Excessive drinking was not unnoticed, but it took decades for the attitudes towards drinking and hospitality to evolve. The liquor jug was considered indispensable when dozens of men gathered to raise a roof and running a distillery in the days when liquor was perceived as medicinal was not frowned upon_.
Rosenberg-Naperstack stated that there were many mergers and partnerships of breweries in the nineteenth century. The German population in 1812 was scarce, but by what was called the 1848 German Revolution, thousands of Germans had immigrated to Rochester. This explains the many types of German city brews found in this ad. "Culmbacher" was most likely "Kulmbacher." The American Brewing Company evolved from a partnership between Meyer and Loebs (1861) and then was renamed the Lion Brewing Company (1879) and was bought out by the Loebs Brothers in 1885 before finally being incorporated as the American Brewing Company in 1889. Rosenberg-Naperstack goes on further to say that before stopping production in 1950, "In its heyday in 1890 the American Brewing Company on Hudson near Drayton Street covered a full block with its fireproof six-story brick building."
Liberty Beer—Canal and Lafayette Streets, Chinatown, New York City. Taken August 1998. Ad circa 1910.
Rosenberg-Naperstack also claims that the temperance movement started in 1827 by the Rochester Presbytery. Thus the brewing and distillery industry and the temperance movement in Rochester grew hand in hand. Rochester was home to many forward thinkers. This socially influential list includes temperance and women's rights pioneer Susan B. Anthony, abolitionist Frederick Douglass and anarchist Emma Goldman, just to name a few. This jewel of a boomtown was also the home of Kodak, a company that profoundly altered the course of the use of the photographic image in media and social communications. A bulk of my photographic archive was documented using Kodak products.
The Liberty Beer sign was uncovered in the summer of 1998. Previously, it had remained peering over handbag hawkers and Excellent Dumpling eaters at the intersection of Lafayette and Canal Streets for thirteen years until it was finally covered by a new T-Mobile ad, commissioned to the Colossal Media outdoor advertising company. Colossal Media is known for its slick, hand-painted ads, often done in the style of the early twentieth-century billboards. Here is an excerpt from a letter I wrote to the Landmarks Preservation Commission on behalf of Colossal in December 2007 in support of its efforts to gain the airspace rights to a building in SoHo with a vintage Coca-Cola ad (page 92):
_Colossal Media produces modern ads that are visually exciting—ads that will become tomorrow's classic fading ads. With the hand-painted brickface ad medium, Colossal Media is continuing the tradition of the painted ad that has become an indelible symbol of New York's urban landscape. Tourists come to New York, the mecca for world commerce—and expect to see both the historic and modern—the old classic and the new classic. Sign enthusiasts all over the world, like Sam Roberts UK Brick Ads Blog 95 marvel over Colossal's painted mural ads, which have become a tourist attraction. No one expects fading ads to last forever. Just like fashion trends, they come and go. I'd rather see a vintage ad covered by a classic Colossal modern work of art than with one of those crass outdoor illuminated billboards or a dingy nylon fabric hung ad. The pride Colossal Media takes in their work is evident in their finished product_.
There really is a lot going on in this image, even the MetLife blimp peeking through on the left of the sign. From what I can extract using my Photoshop "Hue/Saturation Centrifuge," the wording across the top seems to spell "Hollenbeck Company," with the body of the text stating, "Importers and Bottlers of Beer—Imported Pilsener, Humbser, Kaiser, Munchener, Culmbacher, and Warsteiner Beers." To the upper left of the script written "Liberty" is the name "Rochester." I'm quite ambivalent about seeing this sign covered, as much as I am all for progress.
**G UINNESS BEER**
I believe these images are on what was the Guinness brewery in Long Island City (pages 94–95). The intent of painting this sign was so commuters on the Long Island Rail Road and drivers on the Long Island Expressway would see it, where it is still clearly visible. The following history was provided on a Guinness beer enthusiast's website:
_In 1934, long time Irish bottler of Guinness Stout (and US importer of Guinness, Bass Ale and Perrier water), E. &J. Burke, Ltd., opened a brewery in Long Island City (Queens), NY. They brewed and marketed their own Burke's Beer, Ale and Stout and, as war in Europe loomed in the late '30's, Guinness and Burke began discussions about brewing Guinness in the Burke facility for the US market_.
_In 1943, Guinness bought the Burke brewery from the failing Burke and in 1948 began brewing Guinness Extra Stout for the US market. It was only the second brewery Guinness would run outside of Ireland. (The first being the long-lived UK Park Royal brewery in London which closed in 2005). The Long Island City brewery closed in 1954_.
Guinness Beer close-up.
City of Immigrants and the Mack Sign Company
_As apparitional as sails that cross_
_Some page of figures to be filed away_
_—Hart Crane, "To Brooklyn Bridge"_
New York's faded vintage signs constitute an important element in the city's business history. They remind us pointedly that the New York of an earlier time was a city of industry and business. It was a city of manufacturing, Whitman's lusty place of work and sweat. The "Manhatta" of Paul Strand and Charles Sheeler. A city that puffed smoke. It was also a city of immigrants. Many of the surviving signs bear the names of people born in Russia, Germany, Italy, Switzerland. Almost incredibly, these men (and some women) left their names on the walls we walk beneath today. They were little men, obscure figures. They lived in tenements downtown on the East Side, in Brooklyn, Harlem or the Bronx. They went to work. They left their names behind.
The stories were as varied as life itself. There is Harriet Hubbard Ayer, who abandoned her life as a wealthy society matron to found a cosmetics company that manufactured facial cream and later became the highest-paid woman journalist in America. Many were successful businessmen who retired into wealth and security. They received obituaries in the _New York Times_ and were cited for their many philanthropies and benefits to mankind. Such were, for instance, Sol Mutterperl (popular-priced ladies' handbags), who endowed scholarships at the Jewish Theological Seminary; Samuel Rubel (coal and ice), who entertained more than ten thousand orphans at his expense at Coney Island; Benjamin Lowenstein (Nassau Smelting & Refining), for whom a students' lounge at Columbia University was named; Louis Friedman (Friedman & Herskovitz, Furriers), "active in the affairs of the Hebrew National Orphans Home"; Israel Cummings, president of the Alumni Association of the Education Alliance, part of the Federation of Jewish Philanthropies; Louis Philippe, "a pioneer in the development of indelible lipstick"; William Olden (Olden Camera), identified with Jewish philanthropic and communal activities; Harry Shulman (Baker Brush Co.), who received a special citation from the United Jewish Appeal in recognition of his philanthropic work; and Max Berley (Berley Real Estate), who was treasurer of the University Settlement House. Other prominent figures include Charles B. Sheridan, who made ninety-one trips to Europe, and Hamilton Disbrow, who was known as the "dean of commuters" of the Delaware, Lackawanna & Western Railroad, having traveled almost daily to New York from his home in New Jersey on this line for more than seventy years.
Others were active in politics, like James J. Riordan (Mohahan Express), who was an officer in the election campaigns of James J. Walker (New York City mayor) and Al Smith (New York State governor); Otto Shulhof (American Neon Light & Sign Corp.), who numbered among his friends "former Governor Smith, former Mayor James J. Walker, Postmaster General James A. Farley and Senator Robert F. Wagner"; and Ernest F. Eilert (Eilert Printing), who ran for borough president in Manhattan in 1921.
Many were family businesses that went through several generations, such as the three generations of Richard A. Bachia who made fine Havana cigars; the three generations of the Trischett family engaged in bias bindings; and the three (or more) generations of the Gross family of S.G. Gross, pawnbrokers.
There are tragic stories also, like William Klatzco, who committed suicide by inhaling illuminating gas in his office at 105 East 29th Street; and Hyman A. Posner, who swallowed poison while on the "Jungle Walk" at the Bronx Zoo. He left a note: "I have lost my nerve and pep because business has been so bad." Moses Gootman was murdered in his home in the middle of the night by an intruder. Louis Reiff committed suicide by throwing himself from a window at the New Yorker Hotel on New Year's Eve.
The importance of New York's vintage signs is particularly pertinent to the period 1900 to 1930, when the painted wall sign was used extensively to advertise products and business locations. These signs were a primary medium of the time, and thousands have survived (past their time) into the twenty-first century. In midtown New York, they cluster heavily in the garment area, where a single wall will often have a dozen or more individual signs. They were furriers, manufacturers of cloaks (coats), suits, underwear, waists (blouses), dresses, skirts, raincoats, neckwear, hats, embroidery, leather goods, knitwear, silk and cotton goods. They manufactured cigars, toys and bedding. There are signs for restaurants, pawnbrokers, hotels, banks, moving companies, commission merchants, import and export companies, butchers, paper and twine dealers...You name it, we got it! In more recent times, these are the yimin, the leftover men of Yuan, China.
I bought my first 35mm camera in March 1986 and took a picture of my first wall sign on April 19, 1986. This was a sign overlooking the Holland Tunnel exit near the corner of Varick and Canal Streets. For many years, I called this sign the "Hot Cow." It showed a cow with the letters HOT and the rest erased. Years later, Frank Florianz sent me a photo of the sign in its original state. It was an advertisement for Hotel Bar Butter. It was painted out totally many years ago. Many others from that early period are no longer with us. Some early favorites were Charles Hellmuth Printing Ink on West 18th Street, Chelsea Warehouses on Broadway near 125th Street, Dinoto's Bread near Yankee Stadium, the Penn View Hotel on West 34th Street, Admiration Cigars/Bowery Dept. Store on the Bowery near Canal Street and Hartz Mountain (altered to Art Mountain) on Cooper Square.
Fleeting phenomena of the rainbow. We grab our cameras and run to get the photo before it fades...
To Frank Jump, these signs are the symbols of our own human frailty, our very temporary presence in a life that ends in death. I first heard of Frank in 1998, when he exhibited photos at the New-York Historical Society. I was not able to attend the opening but saw the show later, managed to get his telephone number, talked with him about our mutual interest and then sent him a book (by a German author) that showed an earlier artist's interest in the wall sign. (We were not the first.) Several years later, I met Frank when he had an exhibition at the Williamsburg Art & Historical Center. I remember he was very personable, very likable. But out the window was an angle on the Williamsburg bridge, and all I could really think about was how to get a photo of that. (Photographers are a funny lot and seldom very sociable.)
_But often when the laugh was loud_ ,
_And highest gleamed the circling bowl_ ,
_I saw what unseen passed the crowd_ ,—
_The shadow on his soul_.
_—Fitz-James O'Brien_
About 2002, I put up a website devoted to signs in the Midtown Manhattan area from 14th to 42nd Streets. This drew the attention of a man named Bob Middleton, who had been the owner of the Mack Sign Company, commercial sign painters. Bob Middleton followed his father, Harry Middleton, in the business. In 1930, Harry Middleton bought the Mack Sign Company for seventy-five dollars from persons unknown, who were located in the Inwood section of Manhattan. Subsequently, Harry Middleton painted two of the most beautiful surviving signs in New York: Griffon Cutlery on West 19th Street and Necchi Sewing Machines on West 25th Street. His son, Bob Middleton, sent me material relating to the Mack Sign Company, such as layouts used by the painters when painting the signs. After much arm-twisting, I convinced Bob to write a description of how the work was done. This can be read on 14to42's Mack Sign page. Recently, Bob was included in a documentary, _UpThere_ , directed by Malcolm Murray (2010). [Signs identifiably by the Mack Sign Company are noted in Frank Jump's collection of images.]
_J'ai seul la clef de cette parade sauvage_.
_—Rimbaud_
_I alone have the key to this savage_
_sideshow.—Varèse_
Walter Grutchfield, July 2011
_14 th to 42nd Street_
Shoes, Hats and Assorted Garments–Wholesale and Retail
United Thread Mills—East Houston Street, New York City. Taken March 1997. Ad circa 1960s. Hung in NYHS Exhibition.
**U NITED THREAD MILLS**
_Manufacturers of sewing thread, cotton, nylon, synthetics_.
According to Walter Grutchfield, Bob Middleton and Joseph Piscitelli of Mack Sign Company painted this sign in the period between 1965 and 1969. The United Thread Mills manufactured sewing thread in this textile mill at 146 Greene Street from 1962 to 1982. Originally located at 656 Broadway in 1942 for seven years, this company then moved to 54 Bleecker Street, where it remained from 1949 to 1962. In 1982, United Thread Mills moved from its Greene Street location to Rockville Centre, Long Island.
This image appears as a city backdrop in Kubrick's _Eyes Wide Shut_ , which was filmed in London but made to look like New York City. I've often wondered from where they lifted that image. The New York Landmarks Preservation Committee fought to have this sign left untouched but made a deal with an ad agency to only cover 15 percent of the visible ghost sign. You can see examples of this sign being painted over at "United Thread Mills in Transition" on the _Fading Ad Blog_.
**S EELY SHOULDER SHAPES**
Incorporated from November 17, 1953, through December 15, 1959.
According to the _Manhattan Ghost Signs Digital Collection_ , an attractively designed new website created by Queens College graduates Otto Luna and Dana Rubin to catalogue the fading ads of the Garment District (using the extensive photo archive of Walter Grutchfield), Seely Shoulder Shapes was originally called Seely Shoulder Pad Corp., located at 263–5 West 40th Street from 1945 until 1956. Shoulder shapes or pads are still produced today for the fashion industry, but no longer at this address.
Seely Shoulder Shapes—West 40th Street, Midtown, New York City. Taken March 1997. Hung in NYHS Exhibition.
In a _New York Times_ article in which I was featured and provided the front-page image, David W. Dunlap stated the following about the fate of this sign: "Another vintage sign that is not destined to last much longer trumpets Seely Shoulder Shapes, a garment business from the 1950's. Painted byArtkraft Strauss, which is still in operation, the mural is at 265 West 40th Street, on the site where _The New York Times_ is planning its new headquarters." After the New-York Historical Society show came down, I tried to sell the collection. One of my benefactors is the lovely Tama Starr, president of the Artkraft-Strauss outdoor advertising company, whom I had the pleasure of spending an hour or more with, talking about the sign-painting industry. Starr was very encouraging of the campaign and gave me a copy of her seminal book about advertising, _Signs & Wonders: The Spectacular Marketing of America_.
In Starr's book, she speaks primarily on the illuminated sign industry, although she does touch on the origins of advertising and the innate ability of humans to create signs and symbols. In a section she calls "From Cave Artists to Wall Dogs," Starr addresses questions about which I've always speculated, like what was the first advertisement and who invented the wall ad? Starr writes:
_The story of outdoor advertising traditionally begins with the first symbolic marketers, the cave dwellers during the Upper Paleolithic period, starting about 40,000 years ago. At Lascaux, France, and elsewhere, elaborate action murals depicting a variety of animals and lively hunting scenes portray the dynamic relationship between hunters and the migratory beasts who represented fundamental economic forces: food and clothing, the entire constellation of blessings that Nature could either bestow or withhold_.
_Anthropologists identify these Stone Age rendering—the first wall-mounted messages—as the earliest examples of both art and writing. They speculate that, like modern media, the messages were intended to influence as well as reflect the viewer's life. Like advertisements, they depict dreams fulfilled—the animals rushing into the hunters' traps—and urge specific, concrete action—Hunt! Be hunted!—on both parties to the economic transaction_.
_All known cultures use signs, in one form or another, to convey straightforward messages with immediacy. Anthropologist Ashley Montagu defines a sign as a "concrete denoter" with an inherent, specific meaning: "This is it; do something about it!" He points out the essentially human character of signs by noting that while many types of animals respond to signals—interruptions in an energy field for the purpose of communicating—only a few intelligent and highly trained animals can understand even the simplest signs_.
_History's first known poster bulletin was a notice of a reward for a runaway slave posted on a wall in the Egyptian city of Thebes more than 3,000 years ago. Egyptian merchants of the same periods chiseled sales messages into tall, square stone obelisks and roadside stone tablets called stelae, and painted them in bright colors to attract the attention of passersby_.
_In Pompeii, billboard-like walls covered with advertisements were preserved in the lava that engulfed them when Mount Vesuvius erupted in A.D. 79. Excavations there have revealed wall messages on the shady side of the marketplace too, offering enticing invitations along the lines of "For a Good Time, See Cora." And even earlier, in ancient Greece, innovative practitioners of what may arguably be the world's second-oldest profession (symbol-making being necessarily precedent) expanded their out-of-home client base by carving the message "_ EΠO MOI, _" "Follow Me," in the bottoms of their leather sandals, leaving an impression in the clay pavement as well as in the imaginations of potential customers. The connection between the two ancient occupations was not limited to amateurs, however. Modern visitors to Ku adasi in Turkey are shown magnificent Byzantine mosaics that once served as on-premises business signs for houses of pleasure_.
_The Romans brought the use of whitewashed walls with painted ads on them on their conquests throughout Europe. They also developed artful on-premise business signs specially designed for the illiterate, such as a friendly-looking bush denoting a tavern. Some ancient trade symbols—such as the three golden balls of the pawnshop, the giant key of the locksmith, the big shoe of the shoemaker, and the red and white stripes of the barber—have remained in use for a thousand years and more_.
Tama Starr loved the image of Seely Shoulder Shapes and bought it. As I signed the photo for her, she signed her book for me:
_Frank Jump! It's always a good sign to meet a new friend. Best wishes always, Tama Starr 7/14/00_
I believe this image still hangs in Tama Starr's office.
**C ROWN COAT FRONTS**
This collection of signs still overlooks the northeast corner of Union Square atop the New York Film Academy. I was fortunate enough to have been allowed on the roof by my friend Michael J. Young, who is now the provost, director of education at NYFA. Grutchfield provided the following background information about the Crown Coat Front Company, as well as additional information gleaned from archival photographs:
_Crown Coat Front was located at 105 E. 16 St. from 1947 to 1957/58. A coat front is defined by George E. Linton...as "Trade term for a built-up stiffening or shape-retaining interlining for the fronts of coats, made of stitched layers of haircloth, felt and canvas." A few details have been lost and the color is considerably faded. Frank Jump has a nice photograph of Crown Coat Front and the Carl Fischer Musical Instruments sign next to it on his website_.
Crown Coat Fronts—Union Square East, New York City. Taken April 1997. Ad circa 1950s.
_...An image dated 1933 on the New York Public Library's Digital Gallery 103 shows that prior to Crown Coat Front a sign painted in the same position read Hyman & Oppenheim/Human Hair Goods. Hyman & Oppenheim were longtime occupants of 105 E. 16th St. (from 1910 to 1931). The original Hyman was Gerson Hyman (1852–1911), a native of Wirballen, Russian Poland (now Lithuania). His two sons, Louis Hyman (1882–?) and Joseph Hyman (1886–1932), succeeded him in the business. The original Oppenheim was Manuel Oppenheim (1851–1917). He was born in Neustadt, Suwalki province, also in Russian Poland (now Lithuania). He was in the hair business with his brother, Heyne (or Heine) Oppenheim, in the 1870s before going into partnership with Gerson Hyman around 1885_.
_Manuel Oppenheim's son, Jesse Oppenheim (1876–1936), succeeded him in the business. When Jesse Oppenheim applied for a passport in 1920, a letter signed by Joseph Hyman was attached stating that Oppenheim had at that time been with the corporation 27 years...Around 1939 Hyman & Oppenheim changed its name to Hyman & Hyman, and they were located at 38 W. 48th St...The business had also changed and seems to have dealt mostly in distribution of beauty products rather than strictly hair goods. Hyman & Hyman were in business until the early 1970s_.
To the far right of the Crown Coat Front Company Inc., MFGR'S (Manufacturers) of Civilian & Military Coat Fronts, it says "Carl Fischer Musical Instruments," which according to the New York Department of State, Division of Corporations, was incorporated on September 20, 1971, and dissolved on April 13, 1987. Carl Fischer Music Publishing was incorporated on July 31, 1923, and is still active. Underneath that, it says "York Band Instruments Company Est. 1892." Below that it reads what I believe says "Mogi, Momomoi & Company, Importers of Japanese Goods" (also visible on the NYPL archival photo), a company mentioned in the January 17, 1923 _The Jewelers' Circular_ 85, no. 2. Underneath on the far right, it reads "Far East Trading Company Inc. N.Y."
**J EWISH SHOE FACTORY**
On this cryptic ad in Williamsburg, the writing on this wall is in Yiddish, which uses Hebrew characters and a German syntax. Evidently, the last line says "produkten." Any other interpretations would be greatly welcomed. According to Moshe Rappoport from Zurich in a recent e-mail to me, "Apparently (I can't make out the first letter very clearly) it says 'Scouring Powder Products.' That would be like Ajax or Comet white powder cleanser products for washing dishes and pots. Since it was used with food it would have had to have been Kosher. Do they still sell those tubular cans?"
Oh yes, Moshe, they do sell those tubular cans. However, the tubular gas silos seen in the distance near East Elmhurst are no longer on the landscape of the Long Island Expressway, but this sign still remains visible from the Hewes Street subway stop on the elevated J train platform on Broadway nearing the Brooklyn waterfront. A few of my friends who are scholars of Judaic studies and are fluent in both Hebrew and Yiddish have been trying to decipher this ad. Of course, not all the letters are clear and in Yiddish, the vowels are omitted and every letter becomes more significant in denoting the meaning. The one letter that always drew my attention, as it was the most clearly visible, is the Hebrew letter aleph , which is thought to be derived from the West Semitic word for "ox," and the shape of the letter derives from a Proto-Sinaitic glyph based on a hieroglyph depicting an ox's head, .
Jewish Shoe Factory—Hewes Street, Williamsburg, Brooklyn. Taken August 1997 Ad circa 1920s. Hung in NYHS Exhibition.
As a very young child, I would get dangerously high fevers and would become delirious. My mother would have to give me Phenobarbital to prevent seizures. I knew when I was getting the fevers when I would have the same recurring dream where this letter would appear to me on top of a large wooden gate to a horse corral that was closed and I could not open. This letter would sit defiantly on top of that gate. Since I did not grow up Jewish, I'm not quite sure how this letter appeared in my subconscious so early, although growing up in New York, I was exposed to Jewish culture by neighbors and friends. I also went to Jewish summer camps in the Catskills, where our counselors would tell us "Cropsey maniac" stories to scare us around the campfire at night. I also remember hearing stories about a golem.
In Jewish folklore, the golem is an anthropomorphic being made from inanimate stuff that could be created to protect you in a time of great need. The creature came to life when the aleph was first carved into the golem's head to spell the word truth, or _emet_. To deactivate or kill the golem, you must remove the aleph, which then spells the word death, or met. In a May 2009 _New York Times_ article, it was said that "the Golem, according to Czech legend, was fashioned from clay and brought to life by a rabbi to protect Prague's 16th-century ghetto from persecution, and is said to be called forth in times of crisis."
_New York Magazine_ used this image to feature my show at the Williamsburg Art & Historical Society in December 2000, which was part of the worldwide Day without Art event on December 1 that was sponsored by the group Visual AIDS. During the preparation for this show, I was very debilitated from chemotherapy and radiation treatment for rectal cancer. In the show, I used very large reproductions (four feet by six feet), and hanging them was a very strenuous activity. Of course, I could not have done it without Vincenzo's help. I grappled with the concepts of truth (with a small "t") and death. Although the timing of this show seemed like it would have made it impossible for me to mount it, the exhibition became my golem. I didn't focus on my physical weakness from the treatment, nor did I focus on the pain from my surgery and the radiation. Instead, I remained focused on getting this show on the road. To this day, I keep an aleph under my bed just in case.
**S UZY PERETTE'S DRESSES**
_Gigi Young originals_.
Incorporated on April 6, 1938, as Lombardy Dresses, the company became known as Perette Dresses on January 18, 1950, and as Suzy Perette from January 31, 1958, to April 14, 1960.
"The Perette Silhouette" is a headless and limbless torso in a dress that hangs silently over the Garment District's West 37th Street like a two-dimensional coquette effigy. Meant to represent a dress on a dressmaker's mannequin, this towering image seems a bit uncanny. One of my favorite overheard comments at the NYHS was a conversation a woman had with her husband about Suzy Perette Dresses. When she walked in front of this image, she gasped and said, "I wore Suzy Perette Dresses. Anybody who was anybody wore Suzy Perette Dresses." She continued to describe some of the dresses she wore and the occasions for which she wore them. This elderly woman was transported back to a time when she was a young ingénue going out to a party with her friends. Many of these dresses were one of a kind and were only worn once.
Suzy Perette's Dresses—West 37th Street, Garment District, New York City. Taken March 1997. Ad circa 1950s. Hung in NYHS Exhibition.
**W EBER & HEILBRONER**
_Stein Bloch clothes in the New York manner_.
Weber & Heilbroner was a men's haberdashery from March 19, 1913, to June 27, 1979.
I overheard several people commenting about this establishment while this image hung at the New-York Historical Society show. Weber & Heilbroner was a rather posh haberdashery. One of the visitors actually worked there as a bookkeeper in the 1950s. In an interview with Manhattan borough historian Cal Jones, he fondly remembers shopping at the Weber & Heilbroner at Maiden Lane for a black woolen sweater with white trim. Cal described how the collar could be tucked in to make it look like a crewneck. According to Grutchfield:
Weber & Heilbroner—West 35th Street, Herald Square, Garment District, New York City. Taken April 1997. Ad circa late 1950s. Hung in NYHS Exhibition.
_Weber & Heilbroner, Haberdashers, were founded in 1902 by Milton Weber (1876–1936) and Louis Heilbroner (1878–1924). Louis Heilbroner, men's furnishing, was located at 920 3rd Ave. (corner of 53rd St.) in 1899. The original (1902) Weber & Heilbroner stores were located at 920 3rd Ave. and 757 Broadway (corner of 8th St.). Within a few years the business expanded to stores downtown at 58 Nassau St. (corner of Maiden Lane), 150 Nassau St., and 369 Broadway. A display ad from the_ New York Times, _29 July 1904, mentions 58 Nassau St. & 757 Broadway. Also this one from the_ New York Times, _17 Sep. 1904. Weber & Heilbroner opened a store at 6th Ave./Broadway at 34th St. around 1923. Space in the Marbridge Building became available in 1922 when Rogers, Peet & Co., who had occupied the building from its inception, moved across 6th Ave. to the former Herald Building (_New York Times, _18 April 1922). Weber & Heilbroner's store at 34th St. and 6th Ave./Broadway was one of four or five that survived until the 1970s when Weber & Heilbroner went out of business_.
Many of the incorporation dates I've included were found through the NYS Corporation's website (in the late '90s, there was an automated phone bank) and include when businesses were incorporated and why they went bankrupt. Many of these companies went bankrupt when they failed to pay their state corporate taxes. Weber & Heilbroner fell to this malady. The sign is still behind a large, stretchy mylar ad opposite Macy's at Herald Square. It is a bit hard to image just how large this sign is, until you look at the tiny yellow taxicabs and white moving trucks on the street below. I counted how many stories up I needed to be in order to shoot this sign from above. I counted lucky thirteen.
Upon ringing the service bell in the building lobby across the street on the north side of West 35th Street, I scanned the directory of the building while waiting for the doorman and noticed a business on the thirteenth floor called "Magnus" and told him I had an appointment with the manager. Naturally, the doorman called up and asked who I spoke with, and I said, "The girl at the front desk." The manager, Sheila, told him to let me up. When the elevator door opened up, Sheila was standing there with a young African American gay man who, before I could say hello, said, "Hi. I am the front desk girl, and WHO are YOU?" We all busted out laughing, and Sheila invited me in as I began to tell her about my desire to hang out one of her windows to get this shot. Although I was expecting resistance, she turned and said, "Thank goodness someone is documenting this. I love this sign. My office is right opposite it and I look at it every day. I think it's beautiful." She told me this as she started clearing coffee cups and inventory manifests from the windowsill. The window hadn't been opened in years, but she insisted on getting it open so I could get an unobscured shot. Sheila ordered pizza and soda. As we waited for lunch to arrive, I hung out of Sheila's window and snapped away. It was a "magnusly" memorable lunch engagement.
**M ISS WEBER MILLINERY**
_No. 48, Take Elevator_.
Miss Weber had a millinery shop with an electric elevator. Miss Weber designed and manufactured hats in suite number forty-eight, and you didn't have to walk the stairs to get to her. _Inventors_ writer Mary Bellis affirms that the electric elevator came into use toward the end of the nineteenth century, with the first one built in 1880 by the German inventor Werner von Siemens.
According to Grutchfield, Ida L. Weber (1879–1932) was born in New York City, the daughter of Swiss immigrants Jacob and Regula Weber, and kept this hat-making shop here from 1911 through 1913 and then moved to 66 West 39th Street in New York's Millinery District. Miss Weber continued making hats there until about 1923, when she went out of business. According to the University of Wisconsin Digital Collection, millinery and dressmaking were popular professions for women around the turn of the century, with the 1900 census reporting 338,144 female dressmakers, 138,724 seamstresses and 82,936 female milliners in the United States. This report stated that dressmaking ranked third, seamstressing ninth and millinery fourteenth "among occupations for women breadwinners."
Miss Weber Millinery—West 22nd Street, Flatiron District, New York City. Taken March 1997. Ad circa 1910. Hung in NYHS Exhibition.
Oddly enough, the term "mad as a hatter" comes from eighteenth-century England, where millinery workers who used felt (which used mercury in processing the material) to make hats were subjected to long-term mercury exposure, resulting in dementia. Countless women in New York City were subjected to perilous safety conditions making hats, shirts, dresses and other garments in factories that were run and owned by men. Miss Ida L. Weber ran her own company in a time when industry was totally dominated by male ownership, and I tip my hat to her. I worked as a dental office manager around the block from Miss Weber when I found this sign in the spring of 1997 tucked quietly in the entryway of 48 West 22nd Street, where she still greets passersby with a quiet downward glance.
**R.H. M ACY'S UPTOWN STABLES**
_Orders for goods taken here_.
Try to imagine being a resident of Harlem at the turn of the century. Up until the 1880s, Harlem was like living in the suburbs, way north of all of the hustle and bustle of "the city." So I can only imagine that shopping for goods may have been a challenge, being so far away from the center of commerce. The elevated train lines were just being extended during a boom of development, but most of the major goods arrived on horse-drawn carriages from Midtown. To have a Macy's Uptown depot where you could place an order for a set of china and silverware must have been some luxury. The concept of a department store at the time was revolutionary. You didn't have to go to several stores to get what you needed; it was all under one roof. It was an invention of a late nineteenth-century entrepreneur who, like most Americans, didn't get it right the first time. Rowland Hussey Macy made a fifth attempt at opening a retail store in Manhattan in 1858 on Sixth Avenue near Fourteenth Street, which was "far north of the traditional retail market." Macy's Fourteenth Street location sold a whopping $85,000 worth of merchandise within one year. However, Macy's previous four attempts at establishing similar stores had "failed resoundingly," including the bankruptcy of his Haverhill store in Massachusetts.
Apparently, Macy's success wasn't won alone, for at age thirty-six, the Quaker from Nantucket, Massachusetts, invited the Bavarian Jewish immigrant Lazarus Straus to run a kitchenware concession in the basement of his rapidly growing store, which was already "selling toys, candy, jewelry, hardware and men's furnishings." Straus, having already demonstrated "a sound reputation as a merchant in Georgia before moving with his family to New York in the aftermath of the Civil War," began selling china, porcelain, glassware and crockery in the Macy's cellar in 1873 at the behest of Lazarus's son Nathan. In 1884, the Straus family became part owners of the Macy's business. By 1896, the Lazarus family was sole owner of R.H. Macy's & Company. Lazarus's son Isidor "continued to run Macy's until his death, along with that of his wife Ida, on the 'Titanic' in 1912."
R.H. Macy's Uptown Stables—West 148th Street, Harlem. Taken March 1997. Ad circa 1900s. Hung in NYHS Exhibition.
After their demise on the _Titanic_ , Isidor's sons took over the business and began the nationwide expansion of Macy's department stores after World War I. Nathan Straus went on to establish Abraham & Straus in 1893, which ultimately acquired Macy's & Co. a century later. Today, at a Walmart, for example, you can buy all the food that you need for the week, some underwear for the kids, a sit-on power lawnmower and some shrubs to plant in your garden, get a pedicure and have your eyes examined for a new pair of glasses all in one giant consumer cavern, all on one floor! No stairs! And we owe our right to the one-stop shopping experience to the brainchild of R.H. Macy and, perhaps even more, to the business acumen of the Straus family.
**H ARRY'S DEPARTMENT STORE/AUFRECHT INSURANCE & REAL ESTATE**
_It's Harry's Department Store for the Greatest Values—Graham & Metropolitan Avenues_
I've spent more time staring at this image in particular than almost any other sign I've documented. It is not for any other reason than that the four- by six-foot reproduction that hung at the WAH Center Exhibition in 2000 hangs in my office in my country home. I've often wondered who the man on the left was and where the large woman with the teal-colored sweat suit was going. At first, I didn't even realize there was a man on the left since the slide positive was scanned initially in the cardboard frame in which it is housed, which cuts almost a millimeter of information from the image, thus cropping it slightly. When the positive was removed from the cardboard sleeve to be scanned for this large-scale reproduction, suddenly the man in the short-sleeved polo shirt and gray slacks appeared.
Department store owner Harry also has been a mystery to me. There doesn't seem to be any mention of Harry's Department Store in any of the online archives I've searched. Kevin Walsh conveniently provided a link to the wedding announcement of Jacob M. Aufrecht that was scanned and uploaded by Tom Tryniski in an extensive online archive he calls Old Fulton NY Post Cards. Normally, this wedding announcement, which states the usual familial and temporal information, would seem quite unremarkable:
_Berger—Aufrecht_
_Miss Elise Berger, daughter of Mr. & Mrs. Emanuel Berger of 660 West 180th st., Manhattan, and Jacob M. Aufrecht, son of Mr. and Mrs. Marcus Aufrecht of 551 East 53d st., Brooklyn, will be married at the Hotel Astor on Sunday evening Oct 27, by Rabbi Alexander Lyons of the Eighth Avenue Temple_.
_The bride will be attended by Miss Helen Welkersheimer. Max Abrams of Brooklyn will be the best man. Following a motor trip to Canada, the couple will reside at 551 East 53 d st., Flatbush_.
_Miss Berger is a graduate of Columbia University. Mr. Aufrecht is engaged in the real estate business_.
_—Brooklyn Daily Eagle, Sunday October 10, 1929_ 123
Totally mundane in its details, this wedding announcement would never have struck a chord unless I noticed the date that the bride and groom were to be wed. The Thursday afternoon before their wedding is known as Black Thursday (Black Friday in Europe due to the time difference). The Monday and Tuesday after their wedding are remembered as "Black" days as well. One could only wonder if the newlyweds ever went on their Canadian road trip after a nuptial weekend that landed smack in the middle of the stock market crash of 1929, precipitating the Great Depression. As I write this on a similar weekend, Standard & Poors downgraded the United States from a Triple A to an AA+ credit rating. Life goes on. Doesn't it?
Harry's Department Store/Aufrecht Insurance & Real Estate—Metropolitan and Grand Street Station, Williamsburg, Brooklyn. Taken August 1997. Ad circa 1940s.
Might as Well jUmP!
_Reflections on the Color Blue_
In early July 1998, I was seated in my office at a well-known direct marketer on Long Island when someone—I forget who—left a _New York Times_ article on my desk. I was enraptured as I read about a man who was just as fascinated by the fading remnants of a forgotten New York as I was and documented his discoveries on the worldwide web, as it was known in those days. His name was Frank Jump, and he ran, and still runs, a website dedicated to the "faded ads" that dot New York City's landscapes.
The year 1998 was still the wild west days for what we now know as the Internet, but the web was beginning to assert itself as the Number One disseminator of information; where previously, amateur chroniclers had to finance and print up periodicals known as "zines" to get across their obsessions and desires, here was a golden opportunity for a cheap means of getting across what you wanted to say. The word "blog" hadn't been invented yet, but thousands of mavens were beginning to poke their heads above the muck and make their thoughts known worldwide. Today, bloggers influence elections, elect players to all-star games and influence the entertainment industry and everything else in every corner of life you can name, but in the late 1990s, it was mainly a hobbyists' forum.
So it was this incredible Frank Jump photograph of Reckitt's Blue that prompted me to sketch out on scrap paper what I wanted for _Forgotten New York_ that memorable day in that direct marketing office. The circa 1890 ad for a laundry product manufactured by Reckitt's known simply as "Blue" was hidden for many years behind a building on Washington Avenue and Dean Street in Brooklyn; when the building was torn down, lo and behold: there it was. Reckitt's Blue happens to be my favorite shade of blue, by the way—and according to a _Forgotten_ fan who wrote in to inform me, the color of the ad is: C-67.45 percent, M-34.9 percent, Y-7.84 percent, K-1.57 percent, taken from percentages from RGB monitor samples. The original color, considering the fade, may have been closer to C-74.51 percent, M-48.63 percent, Y-23.92 percent, K-10.59 percent.
I conceived of _Forgotten New York_ out of the floating images of rusted lampposts, hidden alleys, bricked streets, ancient business signs, New York City neighborhoods that the guidebooks don't acknowledge such as Georgetown, Eastchester, Throgs Neck, Winfield and Eltingville, as well as the ghost ads advertising long-deceased businesses that can still be found on walls all over the city. After seeing Frank's article and sketching out what components would make up _Forgotten New York_ , I grabbed a camera (one I hadn't used for years) and set out snapping everything I remembered in my head that fit the _Forgotten New York_ format. If you know what my early web pages looked like from those years ago, you also know that I was a pretty crappy photographer in those days. I have since become a better one; I merely shoot on the sunny side now.
Frank H. Jump and Kevin Walsh on May 1, 2005, at the Fading Ad Gallery.
Of course, I soon contacted Frank, and he has become a staunch _Forgotten New York_ ally over the years. The day I met him for the first time, what became Forgotten Tour #1 on June 1, 1999, he related the story of how he walked the el tracks on the J line in Bedford-Stuyvesant to better get a picture. I posted an image of ancient ad aficionado and daredevil Frank Jump as he stands on top of a scaffold he had ascended in order to get better views of the Bushwick RKO. Frank's pictures of the theatre also appear on his own website. I have met many other people who have performed such daring feats to acquire just the right shot. Later that fall in Coney Island, at a lot on Surf Avenue and West 37th Street, a couple of ancient autos were for sale. (They needed a lot of rehabilitation.) Forgottoners gathered at a 1940s Olds, where Frank tried to give the old clunker a "jump" start.
In May 2005, author and friend Dawn Eden published an article for the _Daily News_ about fading ads, on which Jump and I collaborated. Frank continues to limn ancient ads on his newer site, the _Fading Ad Blog_. Frank, a New York City teacher, has beaten cancer, is beating the HIV virus and married his lifelong partner, Vincenzo Aiosa, in 2004. He is also the uncle of Rosario Dawson—we'll have to have Dawson on a Forgotten Tour sometime!
Kevin Walsh
_Forgotten New York_
July 10, 2008
Laundry Products, Washing Machines and Real Estate
Reckitt's Blue—622 Washington Avenue, Brooklyn. Taken March 1997. Ad was re-obscured as of April 2004. Ad circa 1890s. Hung in NYHS Exhibition.
**R ECKITT'S BLUE**
_The Purest and Best_.
Ultramarine. A perfect expression for what I felt the magical moment I first glimpsed this magnificent relic of the late nineteenth century—an azure monument to a pioneering advertising and global marketing campaign. Reckitt's Blue, a laundry whitener, was one of the first widely marketed laundry products manufactured by Reckitt & Sons, which was established in 1840 in Hull, England, by a Quaker named Isaac Reckitt. This laundry starch company began producing laundry blue in 1852 by using a combination of a synthetic ultramarine and sodium bicarbonate. Previously, the semiprecious stone lapis lazuli was ground to produce the active ingredient, so the advent of this synthetic made the product affordable to the masses. The naturally occurring blue pigment found in lapis was first discovered to be sodium aluminosulfosilicate in 1824 by the French chemist Jean-Baptiste Guimet. Guimet realized he could synthesize ultramarine by combining china clay, soda ash and trace sulfur.
Originally, the synthetic was imported from France or Germany, but after Isaac's death in 1862, Reckitt's sons began manufacturing it in Hull. After its introduction to the nascent global market, Reckitt's Blue soon outsold their original laundry starch and stove-blacking products. Stove blacking was produced from lead and used as a hearth polish. Reckitt & Sons also produced other liquid metal polishes (Brasso) and boot polishes. Reckitt's Blue is no longer manufactured in the United States.
My sister Jackie happened to be with me when I discovered this sign and suggested that I take the shot from the side. I kissed her goodbye and got in my car to have my film developed in Manhattan. When I got to the Manhattan Bridge, I got caught in a major mid-morning traffic jam. Anxious to get the film to the developers before noon for a two-hour turnaround, I started to get stressed out. I did not want the anxiety attack that I felt coming on, so I took a deep breath in and exhaled slowly in an act of surrender. Suddenly, I shot out of the top of my head and was hovering in the cables of the Manhattan Bridge with a 360-degree view of Brooklyn and Manhattan. No longer in my body, I was free. Staring down at the car, I was immediately sucked back inside looking at the Manhattan skyline with the sun glinting off the windows of the World Trade Twins. Never have I felt such euphoria before without chemical assistance. It was a fleeting moment of bliss that seemed to last an eternity and a nanosecond all at once. I knew then and there that something very important was happening in my life. No longer would I obsess about my own mortality. Even four years later, when I was diagnosed with rectal cancer, I knew I would survive it if I just surrendered my anxiety.
_Courtesy of Reckitt & Colman, Inc_.
Sadly, Reckitt's Blue has been re-obscured, sandwiched between the late nineteenth-century wall upon which it still remains and an eyesore some call urban renewal. This image has become iconic for me since it was the one the _New York Times_ used for its feature on the campaign exhibit.
**P. Z ACCARO REAL ESTATE/BENDIX HOME LAUNDRY PENTIMENTO**
In a stunning piece of prose he wrote for the _New York Times_ called "Olde York," David W. Dunlap describes the multilayered nature of New York as the following: "The city is not so much a tidy pile of building blocks as it is a chambered nautilus with walls of palimpsest on which the etchings of the present never entirely obscure the patterns beneath. Forever visible behind the newest layer, no matter how faint, are remnants of the civic past. New York does not forget."
Usually, it is a challenge to decipher the two or more ads in a palimpsest—a word usually used to describe when a previously written text starts to ghostly appear from under another text written on top—but this P. Zaccaro/Bendix one was very clear and remains clear to this day. This four-story mural dominates the Delancey Street access to the Williamsburgh Bridge while traveling east on Kenmare Street near the corner of Elizabeth. Although it roughly faces south, it has been protected by the sun's destructive power by the shade of the building across the street.
The Bendix Home Appliance Corp was founded in 1936 by Judson S. Sayre and became the largest U.S. washing machine manufacturer by 1950. Bendix introduced the first automatic front-loading washing machine in 1937 and secured the patent in 1939 with inventors John W. Chamberlin and Rex Earl Bassett Jr.
P. Zaccaro's son, John Zaccaro—the husband of the late Geraldine Ferraro, former 1984 Democratic vice presidential nominee and U.S. House of Representatives member from New York—currently owns this real estate firm. Halfway down you can read the Bendix ad for washing machines that were probably sold at J. Eis and Sons. The telephone exchange Orchard-4 is a seven-digit number that was adopted in the 1940s, which is about when I would date this sign.
Here is an excerpt of a piece I wrote called "False Memories or Twisting at Two":
_As a child I remember being fascinated by the washing machine in our kitchen. The rhythmic sound of the agitator going back and forth, back and forth, back and forth would mesmerize me. I would chant, "Wash-a-sheena, wash-a-sheena, wash-a-sheena..." while doing "the twist" all around the apartment. My mother Willy said I would throw all kinds of things in that old Bendix: pillows, forks, toasters, 45s, plants, cats—always transfixed by the washing machine blades going back and forth, back and forth...The spinning clothes became a cylindrical wall of blur that was like its own form of matter like a plasma phase or excited energy state—an Einstein-Bose condensate. The centrifuge could not separate what was real from what was imagined. The forces of attraction were too strong_.
P. Zaccaro Real Estate/Bendix Home Laundry pentimento—Kenmare and Elizabeth Streets, New York City. Taken March 1997. Ad circa 1950s. Hung in NYHS Exhibition.
_Once, I got away from my mother at a State Fair in NY. My mother searched for me frantically. Finally she turned towards the stage at the prompting of a cheering crowd and there I was doing the twist to Chubby Checker's "Let's Twist Again." I also remember doing the twist on top of a bar in Amsterdam for American soldiers always chanting to myself—Wash-a-sheena, wash-a-sheena, wash-a-sheena...I have other false memories_.
Today, Photoshop has become my centrifuge for separating fused memories. Usually when I'm trying to decipher a pentimento, I will select the section of the text and alter the hue and saturation, which sometimes helps to reveal the underlying ad. If you look carefully, just underneath the words "Real Estate, Brokerage, Management," the cubic image of a Bendix front-loading machine is clearly visible.
**B ENDIX HOME LAUNDRY**
This is one of the many Bendix Home Laundry fading ads I've captured. A young family from Guyana was kind enough to let me climb out of their window onto an adjacent roof for this shot. Ironically, and with much delight, the Fading Ad Gallery was located directly across the street from this billboard and painted sign at 880 Bedford and 679 Myrtle for the almost two years that it was open. This also is one of the many MACK signs throughout the city. In February 2005, _Swindle Magazine_ —a now defunct bi-monthly arts and culture publication founded in 2004 by artist Shepard Fairey—did a story on the gallery and this campaign. The photographer, Chris Glancy, thought it would be fun to get me to climb the billboard's catwalk and sit on the edge. This image can be seen on the campaign website.
Bendix Home Laundry—Bedford and Myrtle Avenues, Brooklyn. Taken September 1998. Ad circa 1950s.
**E MIL TALAMINI REAL ESTATE**
_ALgonquin 4-1817_
In the 1954 film _It Should Happen to You_ , starring Judy Holliday as Gladys Glover, an unemployed model uses the $1,000 she has saved up to rent a billboard in Columbus Circle, where she writes her name, motivated solely by the desire "to make a name for herself." The billboard is usually rented by the Adams Soap Company, which was late in reserving it. Through several twists and turns, she agrees to give the space over to the soap company in exchange for ten other billboard spaces around town, upon which she has her name written. New York City passersby start noticing Gladys's name and wonder, "Who is Gladys Glover?"
Who was Emil Talamini? It has been a great mystery to me since before I took this photograph. As a teenager, I would frequent Greenwich Village often. The view uptown on Sixth Avenue from Christopher Street was a familiar landscape that usually implied a night of fun in the city or perhaps a day of protest, since the first gay and lesbian rights marches went up Sixth Avenue from Christopher Street. So when I took this photograph with the moon shooting out of the black stovepipe vent like from the barrel of a cannon, the mystery of _who was Emil Talamini_ persisted to intrigue me. Finally, I have a clue. On October 29, 1970, the _New York Times_ obituaries wrote about the death of Mr. Talamini:
_Emil Talamini, of 70 East 10 th Street, a real estate broker and investor long active in the Greenwich Village area, died yesterday in Memorial Hospital after a long illness. He was 64 years old_.
_Mr. Talamini was a 1928 graduate of Brown University. He was with Charles F. Noyes, Inc., and the William A. White & Sons before establishing his own real estate business. He was a member of the Real Estate Board, the New York Yacht Club and the Dunes Club of Narragansett, R.I_.
_He leaves his wife, the former Jeanette Jackson, and a stepson, Ronald Wills_.
Little else is available about Mr. Talamini besides his obituary and this sign for his business and telephone number showing the old New York City exchange—ALgonquin 4-1817. Mr. Talamini may be gone, but his name and old telephone number live on. No one who answers that telephone number knows anything about Mr. Talamini, and I'm sure they would love for the sign to be covered up by another billboard ad or perhaps even the name Gladys Glover, unless I'm the only curious and compulsive crackpot who calls this number asking about him.
Emil Talamini Real Estate—Sixth Avenue, Greenwich Village, New York City. Taken August 1997. Ad circa 1940s.
Jewelry and Accessories
Shields Fine Jewelry—Fifth Avenue and West 33rd Street, Midtown, New York City. Taken April 1997. Original ad circa 1940s.
**S HIELDS FINE JEWELRY**
_I could have been_
_One of these things first_
_I could have been a sailor_
_Could have been a cook_
_A real live lover_
_Could have been a book_
_I could have been a signpost_
_Could have been a clock_
_As simple as a kettle_
_Steady as a rock_
_—Nick Drake, "One of These_
_Things First"_
_Bryter Layter LP, 1970_
Thanks to Grutchfield's tireless and remarkable research of the seemingly countless signs between 14th Street and 42nd Street, this ad's significance and the history of Shields Fifth Avenue's as a men's jewelry outlet was revealed. From the late 1940s until 1958, Shields was located at 302 Fifth Avenue near 31st Street. Shields then moved uptown to 401 Fifth Avenue at 37th Street. Shields closed its doors in New York City in the late 1960s; however, the name Shields Fifth Avenue lived on as a registered trademark of Watchbands Inc. of North Attleboro, Massachusetts. Wall dog Harry Middleton of Mack Sign Company painted this sign in the late 1940s. Grutchfield wrote that Bob Middleton said he had worked on this sign circa 1955, and according to Middleton, "with no windows, it was a rough job to paint." Under this sign and somewhat obscured by the water tower, it also reads "Victor Porting Company."
In Tama Starr's _Signs and Wonders_ , the first sentence on the first page she simply states, "Our signs tell us who we are." This is a level of signification that I've been tuned into for over fifteen years now. To some, other objects in our environment may also communicate an identity, or perhaps we may see ourselves personified in a building, a park or even a signpost. Looking south down Fifth Avenue with the Flatiron Building in the distance, you can still see the Bob Pins ad (see next ad) on the billboard facing north on the lower left. In 1979, Katherine Hepburn was asked in an interview with Morley Safer if she could be any building, which one would she be? She replied, "Most definitely, the Flatiron." Also documented later in the Andrew Britton biography of _Katharine Hepburn: Star as Feminist_ , Hepburn stated further: "Me, I'm like the Flatiron Building. All I can say is I could never be anyone else. I don't want to be anyone else, and I've never regretted what I've done in my life, even though I've had my nose broken a few times doing it."
In 1987, when I was living with my former partner, activist and real estate developer Eric Sawyer, he told me about the time he ran into Ms. Hepburn in Morningside Park after the city had completed a major landscape renovation project there. Sawyer thought he had seen Hepburn turning a corner to go down the stairs into the park, and then heard a cane being dropped and banging down the stairs. Sawyer rushed around the corner to look and see if Hepburn was OK, and there she was, picking up the cane and exclaiming in her faltering graciousness, "Oh, this was once such a glorious park. We used to come here, Spencer and I, and eat a boxed lunch on a blanket over there." She pointed to a tree in the distance and continued, "This park had fallen into such disrepair for so long, I had to come see what they have done to restore it."
Hepburn asked Sawyer what he was doing in the neighborhood, and he told her he had recently bought a Harlem brownstone in an auction and was restoring it. Hepburn asked if he minded telling her how much he had paid for it. Without flinching, Sawyer told her, and she shared how much she had paid for her Turtle Bay residence back in the 1930s. Hepburn then asked if she could see the brownstone, and Sawyer, rather amazed, said, "Sure! Now?" Hepburn signaled for her driver, and a late 1970s blue Chevy Impala pulled up, and off they went to his brownstone on Manhattan Avenue, which was under reconstruction.
One Saturday morning when I was living with Sawyer, the phone rang and woke me up. I answered groggily, and I heard that unmistakable voice on the other end: "Good morning, this is Kate Hepburn, is Eric Sawyer available?" I said, "Yes, Ms. Hepburn, I'll get him right away." She said, "Please, call me Kate." I jumped out of bed and got him.
Hepburn's identification with a New York City landmark building is quite astounding, since I, too, have identified myself with fading ads, which are landmarks in their own right. Having to choose one sign in particular would be a hard decision since so many of these signs mean so many different things to me. Here's my dilemma—there is my first sign, Omega Oil, and my first true love, Reckitt's Blue. Hepburn's ability to be so sure about the Flatiron speaks to her self-confidence and self-awareness—the recognition of the power she had as a person, as a woman and as a public figure. It also speaks to the power a building, or even a signpost, may have on an individual. These signs as part of our urban landscape have both external and internal significance and, as Starr states, "tell us who we are." Which makes it so much more evident how the disappearance of these landmarks can be devastating. Hepburn was and continues to be a human landmark, and I think of her every time I pass by or even think about the Flatiron Building. She will be sorely missed. In retrospect, since there isn't any one particular sign with which I could identify myself, I believe I need to err on the side of Christopher Isherwood's sentiment—"I am a camera."
**B OB PINS FRESCO**
Bob pins or bobby pins were first used in the 1920s to keep the new bob-style hairdo in place. The trademark for the bobby pin was held for several decades by the Bob Lépine Corp of Buffalo. This cheerful fresco must have been from the early1940s, as it contains deco-esque style elements. High above Fifth Avenue in the shadows of the Empire State Building, this ad sits above a roof of a carpet retailer that was only accessible due to my political proclivities. Upon entering the business, I was greeted by the owner, a short, well-dressed elderly Jewish man. I started my spiel. "Are you aware of the vintage sign outside of your building?" Sign? What sign? So I explained what I perceived as a bobby pins ad from the '40s that I could only get access to if he let me onto his roof for just a minute. I was wearing one of my Dinkins campaign buttons, and he exclaimed, "You look like a nice boy and you're a Democrat." Ironically, the man's yarmulke was being held down by a bobby pin. Last I looked, these two happy Caucasian faces are still smiling faintly northward at the Empire State Building.
**G RIFFON SHEARS AND MACK SIGN COMPANY SPOTLIGHT**
This is one of the most photographed fading ads that the Mack ad company produced. Grutchfield wrote that the Griffon Cutlery Works was founded by Albert L. Silberstein in 1888. Griffon manufactured "razors, nail files and a great many types of scissors, including pinking shears, nippers and manicure sets, as well as 'Ladies' Button Hole' scissors." According to the Griffon trademark registration, embroidery scissors, poultry shears, barber shears, tweezers, pushers, blackhead removers and nose scissors were also manufactured goods. Harry Middleton painted this colossal and magnificent ad about 1939 or later. In the 1940s, Griffon became incorporated, and the sign was repainted to represent the change in the company name from "Works" to "Corp." During the 1960s, Griffon left New York City for Inwood, New York, on Long Island while maintaining a showroom at 385 Fifth Avenue.
In a July 9, 1998 article, "Saving Images of 'Dead Sign' on Old Walls," written about me by Randy Kennedy for the _New York Times_ , he opens his "Public Lives" piece with the following:
_"Tweezers, nippers, manicure sets," says Frank H. Jump, reading slowly, his head craned back in a familiar stance. He is on West 19 th Street near Seventh Avenue staring up at a fading, chiaroscuro advertisement painted on a building by a long-gone company called Griffin_ [sic] _Shears. The ad outlasted the business, but like hundreds of the city's so-called dead signs—murals for extinct goods or companies, painted high on brick walls—it is fast disappearing_.
Bob Pins fresco—West 32nd Street, Midtown, New York City. Taken August 1997. Hung in NYHS Exhibition.
Griffon Shears and Mack Spotlight—West 19th Street and Seventh Avenue, Chelsea, New York City. Taken September 1997. Ad circa 1950s.
Savings, Loans and Fur Vaults
Ash's Certified Cold Storage—Bronx. Taken August 1997. Ad circa late 1950s.
**T HE GREATER NEW YORK SAVINGS BANK**
This is a remnant of a time when smaller local savings banks flourished. The Greater New York Savings Bank was established in 1897. In 1964, it merged with the City Savings Bank of Brooklyn, which was then acquired by the Flatbush Savings Bank in 1970. Finally, in 1997, Flatbush Savings was acquired by Astoria Federal Savings Bank. The fingers logo is reminiscent of the "let your fingers do the walking" Yellow Pages ad campaign, which didn't make the top ten best slogans of the century but did place as honorable mention for slogans of the century at _Advertising Age_ 's website. On September 25, 2004, the _New York Times_ printed this obituary about the creator of this campaign:
_Stephen Baker, who created the "Let your fingers do the walking" ad campaign for the Yellow Pages and advised readers on how to live with a neurotic dog, died in Manhattan on Sept. 13. He was 83. The cause was cancer of the gastrointestinal tract, his son Scott, said. As an art director for the advertising agency Cunningham & Walsh and later as founder of his own firms, Mr. Baker tried to create sassy ads that steered clear of the hard sell_.
_For one commercial in the "Let your fingers do the walking" campaign to promote AT &T's Yellow Pages, a woman's well-manicured hand saunters down a street, its long red fingernails resembling high heels. AT&T used the slogan for at least six years_.
_"Baker was part of what was then known as the creative revolution," said Fred Danzig, a former editor of Advertising Age. "He was one of the people who I suppose you might say was at the forefront of it."_ 143
My original caption for this was "A picturesque cobblestone street in one of Brooklyn's more isolated neighborhoods known for its tough waterfront residents and fabulous views of Lady Liberty." Since then, this neighborhood has become a hotbed of shopping opportunities, with a Fairway market being built on the waterfront, as well as a giant IKEA. The Brooklyn Waterfront Artists Coalition also opened a waterfront gallery since this building with the Fingers Walking logo was torn down. Now in its place is a community garden, another urban green feature that is always at risk of becoming new real estate.
The Greater New York Savings Bank—Red Hook, Brooklyn. Taken June 1998. Building now nonexistent. Ad circa early 1960s.
**J.J. F RIEL LOANS**
Joseph John Friel was born on March 15, 1853. He emigrated from County Donegal, Ireland, in 1875 and came to the United States with five dollars in his pocket. Soon, Friel got a job as a ditch digger in a construction company in Bedford-Stuyvesant, Brooklyn. When digging a ditch one day in the hot sun, he looked up at a beautiful house at 699 Willoughby Avenue and proclaimed that he would one day own it. His supervisor thought Friel must have been suffering from heat prostration and made him sit down and rest. After being a ditch digger, Friel worked faithfully for several years for a pawnbroker on Grand Street. Friel had made arrangements with his boss to buy the business from him "on time." By the time his boss died, Friel had begun to grow this brokerage company into a million-dollar business.
According to the Brooklyn Genealogy Website, J.J. Friel ran a pawnbroker business between the years 1880 and 1890 at 86 Myrtle Avenue off Duffield Place, where the new Metrotech Building high-rise casts its shadow on the Flatbush Avenue extension. An additional office was listed at 989 Myrtle Avenue between Sumner and Throop where there is now a New York City Housing Project. Both addresses no longer exist. A _New York Times_ obituary states that Joseph John Friel started in the pawn brokerage business on Grand Street in 1870. Numerous signs for this business can be found from Park Slope, Brooklyn, to Jamaica Queens. This sign on Coney Island Avenue in the Kensington section of Brooklyn, however, no longer sees the light of day.
J.J. Friel Loans—Coney Island Avenue near Caton Avenue. Taken March 1999. Ad circa 1950s.
Recently, a family member of Friel, Michael Hughes (great-grandson), of Detroit, Michigan, contacted me about the whereabouts of the Park Slope J.J. Friel sign I had posted on my blog. Hughes spoke about the possible restoration of the sign and got me in touch with his aunts, Friel's surviving grandchildren—DeDe Burke of Mount Kisco, New York, and Aileen Schaefer of Islip, New York. Almost all the historical and genealogical information on Friel was gleaned through these telephone interviews with Friel's descendants.
In 1898, Friel married Frances Noonan, and by 1903, at age fifty, he and his wife had a daughter, Mary Margaret Friel. J.J. Friel died in May 1914 at age sixty from pneumonia in his home at 699 Willoughby Avenue. Mary Margaret, who inherited the family fortune upon her father's death, went on to graduate from Manhattanville College in 1924 and to marry Henry Mannix in 1926. Henry Mannix became a partner in the law office of White & Case, which was already a legendary Wall Street firm. Mary Margaret and Henry Mannix had ten children, nine of whom lived to adulthood. According to Friel's granddaughter Aileen, the Friel business continued to be run by the family well into the 1970s. Friel was buried in Holy Cross Cemetery in Brooklyn.
After getting off the phone with a family member and realizing how closely Friel was buried to where I lived, I immediately cut some flowers from my garden and headed over to the cemetery on my Vespa. After a three-minute ride, I picked up a map from the cemetery office, where they had kindly written the names of the family members buried at the plot with the years of their birth and death, and placed the flowers on the Friel-Mannix family burial ground. It seems almost unfathomable that this man, whose name I've known for over fifteen years and about whom I knew next to nothing, was buried 1.2 miles from my home, and I now have contact with his family ninety-seven years after his death.
Solemnly, I stood in front of the Friel tombstone while "Taps" was played at a funeral procession nearby. I cannot begin to describe how deeply profound and moving this experience was for me. The tombstone bore the many names of members of the Friel-Mannix family, beginning with the Friels' first child, a son named James who died at birth in 1899. Mary Margaret Friel was thereby their second child and their only child. Having lost her father at the age of ten, she lived a rich and full life with forty-eight grandchildren to recount their great-grandfather's legacy. I returned home and spoke on the phone with the eldest living daughter of Mary Margaret, Aileen Schaefer. We spoke about faith, trust and surrender. We spoke about the remarkable circle of life that brought us to this telephone conversation and life's mysteries. I feel honored to take part in the telling of their story. Perhaps one day in the next century this sign will be exposed again and the story of J.J. Friel will come to light yet again.
**L OANS**
_Diamonds, Jewelry, Clothing, Furs_
Will building New York ever be finished? As long as there are loans, there will be renovation. What is remarkable about this image is it documents a time when the Brooklyn waterfront was still not very accessible to the public. Since this picture was taken, so many changes have occurred in this part of Brooklyn. Williamsburg has become a youth mecca, mostly students from NYU and other prestigious educational institutions, the neighborhood has exploded with bars, restaurants and galleries. The waterfront is seeing some of the more upscale new residential buildings, and the area has become one of the hottest neighborhoods in America, let alone New York City.
Still, many lower-income families have become displaced by this rapid and meteoric gentrification. The neighborhood suffered initially, like most Brooklyn waterfront neighborhoods such as Red Hook and Sunset Park, from the building of the Brooklyn-Queens Expressway and the Gowanus Expressway. The block behind this was where Gabila Knishes originated. I went back there today, and the entire block has been bulldozed. Gabila's moved to Copaigue, Long Island. Like most other industries on Brooklyn's waterfront, it has been moved to somewhere more affordable. Brooklyn's waterfront is finally allowing public access, with parks that stretch along the East River from Red Hook to Greenpoint. Luxury apartments are also dotting the riverfront, rivaling Manhattan's riverfront assortment of posh digs.
Loans—Flushing Avenue, Brooklyn waterfront in Williamsburg. Taken August 1997.
**A SH'S CERTIFIED COLD STORAGE**
_"Safe as a Bank" Cypress 2—0200_
The National Cold Storage building on Brooklyn's waterfront, one of the largest cold storage warehouses in New York, was recently torn down to build the Brooklyn Bridge Park at Old Fulton Street Landing. An image of this building can be seen on the _Fading Ad Blog_. Not to be confused with refrigeration of edible perishables, cold storage was a service offered after World War II across North America for women's fur preservation. This April 11, 1944 ad from the _Ottawa Evening Citizen_ best explains the need for "cold storage" with its persuasive ad copy:
_Proper Storage Protects Your Fur Coat Investment Furs lose the oils that keep skins supple, in warm air. They require careful cleaning to remove dust and possible moth larvae, then storage in low temperature with the right degree of humidity. Such care lengthens the life of your coat, protects your fur investment, as no other care will do_.
_When you buy a bond you immediately put it in a safety deposit box or some other place equally safe and accessible. What about your furs? They represent a VERY CONSIDERABLE sum of money, yet, perhaps, you leave them in your apartment during the warm summer months, without special cleaning and other care, and wonder why that lovely coat appears dull and lifeless and brittle-feeling when you come to don it again in the autumn_.
_Give us the opportunity to store your coat. Our fur storage vaults give perfect protection. Insurance covers every loss possibility and is world wide. The cost? Very low indeed. Only 2% of your own valuation.—M. Caplan Ltd. 195½ Sparks St_.
Paint and Hardware
Brush Up Business with Paint, Paste, Paper & Push—Tribeca, New York City. Taken September 1998.
**E AGLO PAINT SEASONAL QUADRIPARTITE**
_Signs and vines weather and grow_.
_Brick, pigment, plant and lime—_
_Tenuously intertwined through time_.
_As paint degrades and image fades_ ,
_Soft tones evolve_
_From salmon pinks and jades—_
_Into sand and grime_.
_—Frank H. Jump, July 2000_
This sign on Nostrand Avenue and Glenwood Road, near Brooklyn College, illustrates the effects of nature on a fading ad, making it a dynamic reminder of the passing of time—not to mention one's own mortality. The last image was taken on June 22, 1999, and is an unusual summer state for this entwining symbiosis of signage and vinage. Apparently, the roots were cut, and the healthy, lush leaves have withered in the summer sun—another unexpected death.
George Carey Simos (University of Wales, Aberystwyth) published on the web in 2000 a very comprehensive thesis on photography called "Are Photographs Copies of the World?" that includes the Eaglo Paint Quadripartite. Simos's introduction:
_Are photographs copies of the world, or do they represent the photographer's interpretation of the world, as Susan Sontag mentions in her book, "On Photography"? The answer to this question does not only lie in our daily lessons about visual perception, but also in the written theories about how we read images or interpret them. I will be examining these later_.
_It is clear, that the world around us is extremely complex and visually mind-boggling. What is seen every day forms every individual's interpretation of his or her world. Every day across the globe, people decide to frame a part of this "interpretation" to make it last forever—or at least as long they are around to see it. The role of the photographer is the same, just like the painter who "captures" a moment in time, in reality or imagination, with his paintings. The photographer is also such an artist, who may capture "a moment," "a moment in time" as he or she perceived it—in this case how he or she "interpreted" it. When the camera lens clicks and burns the image onto photographic film, it is as if the photographer is "painting," or "drawing" his or her interpretation of the moment. This takes place because of the variables involved in taking a photograph. These may include the camera, the lighting, the angle, the shutter speed, the film, or the film speed. Choosing all the above, the photographer becomes the artist choosing his or her paints, paper, and brushes_.
_In this essay, I would like to delve deeper into the idea of photographs as "interpretations" of the world as Susan Sontag once famously wrote, and will attempt to describe the considerations that the artist/photographer makes in each instance. At this point I would like to introduce my examples, their uses, and their characteristics_.
_This series of seasonal photographs was taken by Frank Jump, a photographer in New York, who wanted to show his interpretation of the fading memories of the 1930's adverts in New York City. He chose to freeze these moments from the same angle every season, to illustrate how we interpret the same subject in different seasonal states. These photographs perhaps represent the artist's nostalgia for the past. Because of this we can assume that he was making a record of something, a selection from his interpretation of the environment around him_. _The moment he froze on four occasions is unique to itself as is the way the photographer perceived it when he took the photographs_.
_—Making a Record—_
_Making a record is one of the most popular purposes for the photographic medium. The photographer decides that what he or she sees and perceives, and what is worth recording. The photographer follows this purpose (recording), photographing events, or processes. The final product is therefore a record of the photographer's interpretation of the event, or process. A good example of making a record, is the collection of photographs by both Frank Jump, and David Modell found on the picture-page of this study. Frank H. Jump photographs the building at different seasons, to record the changes that the seasons bring about. He is thus recording a process. David Modell on the other hand is recording social conditions in a particular time period in history. He is recording an event_.
When I first noticed and photographed the Eaglo Paint sign in Flatbush in June 1998, it was covered in a green carpet of leaves, with just the bottom left peeking out. This sign became a friendly landmark on my walk home from the train station at Brooklyn College, and I was happy to see it every day, waiting patiently for me to notice it. By the fall, the leaves had turned a bright crimson, and I stood in the same spot where I had shot it in June to document its fiery display. In December, the vines were already defoliated, and the detail of the sign became even more apparent. I took another shot in the dead of winter from the same spot. By the spring of 1999, I thought about taking the shot. In the interim, someone, perhaps the owner of the building, had severed the roots from the stem of the vines, and the leaves turned quickly to a pallid sepia. So I took the shot again with the dying vine, not knowing what was planned for that wall. Within a few weeks, the vines were scraped from the wall and the sign was mortared over and then covered with a water-resistant silver paint, its process of fading into obscurity stopped.
Often, people have asked why so many of the older ads have lasted so long, and I usually tell them it's the lead paint that was used back then. In examining the lists of chemicals that constituted what was in paints marketed for home use, the toxicity levels are staggering. I learned from a buddy of mine with a vast knowledge of chemicals and dyes, Robert Baptista ( _The Despair of Port Arthur, TX_ ), whom I had met through the _Fading Ad Blog_ , that the Eaglo Paint and Varnish Corp had a home right here in New York City. Baptista wrote the following information to me in an e-mail:
Eaglo Paint—Brooklyn. Ad circa 1950s.
Taken summer 1998; fall 1998; winter 1998; spring 1999.
_The Eaglo Paint and Varnish Corporation was located at 49-20 Fifth St. in Long Island City. It was established in the early 1900s. Herbert E. Hillman was Technical Director for many years_.
_Attached [below] is an ad for Eaglo Paint that appeared in the_ Bridgeport Post _on May 11, 1973_.
_The New York–New Jersey metropolitan area was home for many of the largest paint manufacturers such as Sherwin-Williams and Benjamin Moore. The raw materials for paint, namely pigments, solvents, resins and additives, were readily available from local industries_.
As director of Eaglo Paints, Herbert E. Hillman was named in 1953 in a lawsuit involving lead poisoning, in which twenty cases of childhood lead poisoning resulted in five fatalities. In another publication about house paints, Harriet A.L. Standeven wrote that Eaglo Paint and Varnish Corporation's Magic Satin product, which was introduced in October 1952, contained a highly carcinogenic substance called alkyd-modified styrene-butadiene, which, while making the paint slower to dry, increased its durability. Standeven concluded by saying, "Today the addition of an alkyd resin is considered deleterious, although the practice continued until the 1990s."
**B RUSH UP BUSINESS WITH PAINT, PASTE, PAPER & PUSH**
(pages 150–151)
This beauty of a sign remains one of the best untouched ads in New York City. Just a few blocks north of Ground Zero, this TriBeCa treasure is perhaps a remnant of an early home improvement and office supplies campaign. Or is it? The style of this ad, with the block font and shading, is indicative of the 1910 style of ads. But most of the ads I've documented from this period, especially the ones facing south or west toward the blazing sun, have faded beyond recognition. As much as I wanted this to be an authentic ad, I've always been dubious. In a May 2009 blog posting, Jeremiah Moss of _Jeremiah's Vanishing New York_ made the following observations:
_This is a landmark building. The blog Haute Notes wrote extensively on its history, which dates back to 1860: "By the time of the First World War, photos show fine etched-glass entry doors and a sweeping canopy sheltering Vogric's Café. Its Slovenian owner advertised the Knickerbocker beers and ales brewed in Manhattan by Colonel Jacob Ruppert."_
_Most curious are the seeming ghost signs on the facade, which show a giant hand holding a paint brush and the words: "Brush Up Business with Paint, Paste, Paper, & Push." (Here, "push" means to sell, writes Forgotten NY.)_
_Frank Jump, in his excellent ghost sign blog, dates the signage to the 1910s. But Haute Notes writes, "the signs don't appear in any of the historic photos, even those from the 1940s."_
_Was the ghost sign somehow uncovered? Or was it put there later on, maybe for a 1970s movie, and made to look like the 1910s?_
_Whatever the story, it seems one of two things will happen to this building: Bouley will eventually move in or it will be demolished. Either way, we'll probably lose the signage, as we lost the Delphi, a once-strong survivor in a vanishing part of town_.
Now that the fading ad enthusiasts' network has grown, the detective work has become a shared task. I value the collaboration, as loose as it may be at times, with other documentarians. Of course, the Internet has revolutionized the way we disseminate images and information about these images.
**W.W. G RAINGER**
Unless you are a building contractor, or married to one like me, you probably have never heard about W.W. Grainger, an industrial supplies company established in 1924—that still is in business! It is also another company that hired the prolific Mack team to paint its sign.
W.W. Grainger—Canal Street, SoHo, New York City. Taken August 1998. Ad circa late 1960s.
Horse and Carriage
J.A. Keals Carriage Manufactory & Repair—47th and Broadway, New York City. Taken April 1998. Ad circa 1877. Hung in NYHS Exhibition.
**J.A. K EAL'S CARRIAGE MANUFACTORY & REPAIR**
When this sign was exposed, I received a call from David W. Dunlap at the _New York Times_ to get a shot of this before it was covered up again. Joseph Berger of the _Times_ wrote this about the sign in an article called "Fading Memories" in 2005:
_The tearing down of old buildings in Times Square has given some once-obscured old signs a fleeting exposure. One of the last signs in the area—for J.A. KEAL'S CARRIAGE MANUFACTORY REPAIRING on Broadway and 47 th Street—was suddenly revealed around 1998 when the adjacent building was torn down. But it was concealed again the next year once a new building began rising on the same spot_.
Of all the ads I've photographed, J.A. Keal's Carriage Manufactory & Repair best illustrates the collision of two advertising eras and serves as a time capsule. This image also resonates with the motivation and urgency I have that drives this campaign. The banner on the lower left reads: "Broadway Cares/Equity Fights AIDS," which is "one of the nation's leading industry-based, nonprofit AIDS fundraising and grant-making organizations." Since 1988, BC/EFA "has raised over $195 million for essential services for people with AIDS and other critical illnesses across the United States by drawing upon the talents, resources and generosity of the American theatre community."
Above that on a building is an ad that was designed to mimic a ticker-tape news caption, which documents the shows that were being advertised in this theatre-driven neighborhood. The dazzling, optically illusive Levi's ad is just to the right of that. The image is split almost in half, with the left side a portal on the late twentieth century and the right side a window onto the late nineteenth century. Kevin Walsh makes these comments on his _Forgotten New York_ website: "It stood there like a ghost, as if to mock the high tech/neon ads that currently decorate the Great White Way. The contrast between advertising techniques of the 1880s and 1990s is striking. Keal's ad was painted there in the era of gaslights, when the auto was yet to be invented and horses were still the prime means of transportation."
**C ARRIAGES, COUPES AND HANSOMS**
Built in 1870, 109 West 17th Street is another example of how New York was once a horse town. On Walter Grutchfield's site, for which I provided this image and an additional image adjacent to this window, he claims, "The building was, in fact, a livery stable kept by Patrick Logan from approximately 1900 to 1905." Grutchfield, who had been diligently documenting and researching vintage ads from 14th to 42nd a decade before me, said, "Until I saw these on Frank's web site, I had been unaware of their existence. But there they were, and on a building right next door to the old B&H! I must have walked past them a hundred times without seeing them."
The elusiveness of fading ads is what makes this campaign so intriguing. In an August 2010 article by Nick Hirshon of the _New York Daily News_ , Kathleen Hulser of New-York Historical Society said, "Ghost signs draw your attention to things that are often below the level of consciousness." An older gentleman who lives in Manhattan and walks down this street every day made me aware of this sign. Eager to get the best shot of it, I had to wait until a double-parked Fed Ex truck moved out of the way. Then it dawned on me what I needed to do. This moment was documented in Randy Kennedy's "Public Lives" article about me in the _New York Times_ :
_On West 17 th Street, he_ [Jump] _climbed atop a Federal Express truck while the driver was away to get a shot of a tiny mural for a kind of rent-a-car garage of its day, on a red-brick building that now houses a store selling expensive Japanese screens. The sign says, "To Let, Carriages, Coupes, Hansoms," and "Victoria's Light Wagons, Horses Taken in Board by the Month."_
_"The driver came back and he said 'You get the hell off there.' And I said very nicely, 'I just need a minute or two more, thanks.'"_ 161
Often, I have to just go for it, because asking permission doesn't always work. Once, I asked a business owner in Bed-Stuy Brooklyn if I could get on the roof of her antique shop, which she told me she had owned for over thirty-five years, to shoot a vintage cigarette ad. The woman looked at me dubiously and asked, "What ad?" I took her by the hand and led her outside and pointed to the sign, to which she exclaimed, "Oh snap." Then she wanted to know what was so important about photographing these signs. I told her she had owned this shop for thirty-five years and never realized this sign was above her. That's why this is such an important subject to document. Unfortunately, the FedEx driver didn't want to hear it when he saw me on top of his truck. C'est la guerre.
Carriages, Coupes and Hansoms—109 West 17th Street, Chelsea, New York City. Taken August 1997. Ad circa early 1900s. Hung in NYHS Exhibition.
Fuel, Oil, Gas and Coal
Diana Coal & Oil—Atlantic Avenue, Brooklyn. Taken March 1997. Hung in NYHS Exhibition.
**D IANA COAL & OIL**
Incorporated from December 7, 1955, through March 25, 1981.
If you've ever taken Atlantic Avenue going east toward South Conduit on your way to John F. Kennedy Airport or out to the Long Island beaches, you would have seen this silo on the south side as you were turning right onto Conduit. Written in a message thread in a forum on Trains dot com: "Their coal silos were a landmark on Atlantic Avenue for decades—a quick check of Bing shows the silos are no more, the address (3298 Atlantic Ave) being a big empty field roughly where Conduit intersects Atlantic."
I used to teach in a middle school nearby and witnessed the demolition of Diana Coal & Oil. The documentation of the process was included in the school yearbook for I.S. 171 Abraham Lincoln Middle School in 2003. Diana Coal & Oil's demise was also documented on the _Fading Ad Blog_. Six years earlier, I had flipped the bird to angry drivers as I stood in the middle of Atlantic Avenue to get this shot, car horns blaring from all sides. Speaking of birds, Diana Coal & Oil was the home of thousands of pigeons. The silos, aside from being a landmark, were a health risk, since they were easily accessible from the street and children would climb inside, where bird droppings literally coated every surface available. The eight cylindrical cement towers stood at least ten stories high and were seen long before you got to the intersection.
**P ARAGON OIL**
Oil certainly has been in the news these days. Between the wars being fought over the control of oil, oil company mergers, the price of oil going up dramatically and down slightly and, worst of all, the horrific spills in the gulfs and bays around the world, you are bound to be slipping on some oil story somewhere. In doing research for this book, I've found that I'm partially responsible for personally polluting the ocean of information on the Internet, as whenever I do a search for a particular product or business, my photographs come up rather early in the results. I'm not complaining, since many a webmaster has paid good money to get his or her website placed in the top Google searches. So when I went to do my research on Paragon Oil, my photo made the top ten.
Interestingly enough, the Paragon Oil website hit number one. So when I went to its site to find out about the history of this family business, there was another immigrant story looking to live the American dream, right next to my photograph—without a photo credit! When I called the company to tell them I would be quoting their family story, the receptionist said, "Don't you need permission for that?" So I made a trade.
[Paragon] _was founded in the early twentieth century in New York City by five brothers: Henry, Irving, Robert, Benjamin, and Arnold Schwartz. The brothers, and their sister Bess, were first-generation Americans; their parents were Jewish immigrants from the town of Belaya Tserkov, near Kiev, today located in the Ukraine. Their father Sam was a blacksmith, and relatives back in Ukraine were involved in the whale oil business_.
Paragon Oil—Long Island City, Queens. Taken September 1997. Ad circa 1950s.
_The combination of these factors led to the brothers experimenting with building the first oil heaters designed for residential buildings_.
_Paragon Oil was primarily a fuel oil distribution business, operating a fleet of trucks around the Northeastern United States. It continued in operation until the 1950s, when it was modernized. Today Paragon Oil leads the industry with a fleet of 57 fuel oil trucks delivering to homes and businesses_.
My photo still adorns their history page and now says "photocredit" next to my image. Now _that's_ progress.
**O IL HEATS BEST (COSTS LESS)**
Oil heat claims to be more ecologically friendly. The fuel is more efficient and can cost less than gas. According to an energy website, "Because oil requires more sophisticated burner systems than gas does to be fired efficiently, oil-fired heating appliances typically cost more than their gas-fired competitors." This sign, a remnant of the gas v. oil heats best ad campaigns, is a great example of what I call an urban ediglyph—the interplay between a fading ad and graffiti. Usually, an ediglyph is when the graffiti and fading ad are part of the same wall, creating a synthesis of building wall (edi, as in ediface) and glyph (as in petroglyph) with the graphic of an ad and the graphic of a graffiti tag or street art.
This block on White Plains Road in the Bronx served as a giant canvas for local graffiti artists. What is fascinating is how these voracious graffiti taggers left the Oil Heats Best sign alone. With few exceptions, these street artists tend to leave the work of the wall dogs uncovered and much prefer to cover the work of rival taggers. Where the two works of art collide—the ancient painted message of a long deceased commercial artist and the cryptic message or cartoon image of a spray paint artist—a uniquely significant ediglyph is created. The best example of this is a wall in DUMBO (Down Under the Manhattan Bridge Overpass) near the Fulton Ferry Landing on Adams Street. A large brick face with an old warehouse name painted on the side of the building was used as a canvas for a graffiti artist who painted what looked like schematic drawings of surveying instruments, gear-cranked machines and the barrel of a shotgun. These images are on the Exhibition Archives page that documents the shows we had at the Fading Ad Gallery.
Oil Heats Best (Costs Less)— White Plains Road near 241st Street, Wakefield, the Bronx. Taken May 1997. Ad circa 1950s. Hung in NYHS Exhibition.
**B UY TEXACO/SPLITS CAFÉ CHAMPAGNE BAR**
I can't say I ever had the pleasure to enjoy a split of champagne at this long defunct champagne bar, nor did I ever fill up my tank with Texaco gas at this location. This sign was stuccoed over—a common death for a fading ad—shortly after I took this shot. This sign and service station presided over the busiest approach to the Holland Tunnel, the best route out of town when going to Jersey City, for decades. Gas filling stations have slowly begun to disappear in Manhattan. If you are low on gas, fill up before driving over a bridge or through a tunnel to get to the Big Apple. The filling stations here are few and far between, and there aren't any Texaco filling stations left in the New York City tri-state area.
Today on this same corner is built the most acutely sharp, triangular, five-story glass and black brick apartment building with a roof deck I have ever seen. Although a Greenwich Village apartment is a desirable location in the city, this particular corner is one of the noisiest and busiest. I shot this while coming out of class at SUNY Empire State College—Manhattan location—which used to be housed diagonally across from this ghost gas station.
Buy Texaco/Splits Café Champagne Bar—Seventh Avenue South, New York City. Taken June 1998. Ad circa 1960s.
Fading Light
Today, advertising signs in New York grace towering buildings that block out the light and display the names of Duane Reed and McDonald's, when, in former times, outdoor signs were for products like Omega Oil for "weak backs" and GiGi Young Originals from Lombardy Dresses. We've learned of these long obsolete ads because Frank Jump took time to notice them. My connection to Frank resulted from two tectonic shifts in technology: the first in the late nineteenth century, when electricity transformed our cities with light, and the second at the birth of the twenty-first century, when blogging unexpectedly emerged from the blossoming of the Internet. My blog, _Julian's Name_ , and Frank's curiosity of this industry would enable him to find my story about my great-great-grandfather O.J. Gude, whose signs were among the first captured by Jump throughout his early photo documentation of New York City.
Oscar James Gude started his outdoor advertising firm, the O.J. Gude Co., in 1878 in Brooklyn with just $100 in capital. O.J. went on to pioneer the first use of the electric light bulb on a billboard in May 1892—just thirteen years after Thomas Edison invented the light bulb. But it wasn't just the lights that attracted advertisers like H.J. Heinz; it was the signs' movement. The electric sign's animation was made possible by tabulating machines that would go on to make International Business Machines a three-letter acronym in households around the world. O.J. used this nascent computing technology to precisely switch individual bulbs off and on, creating moving lightscapes. These lightscapes are what gave Broadway its name—the Great White Way—the dazzling and dancing white light cast from light bulbs in O.J.'s advertising "spectaculars" on Times Square. These were so novel and foreign in their day that in Theodore Dreiser's _Sister Carrie_ , he called them "fire signs." Throughout its history, The O.J. Gude Sign Company of New York would produce over ten thousand painted wall ads and billboards around the United States.
Like most American success stories, O.J. came from humble immigrant beginnings. The son of a tailor from Hanover, Germany, who died when O.J. was fourteen, he went to work to support his mother and five young siblings. By the early twentieth century he had become a multimillionaire. O.J. made his money not only with his legendary spectaculars but also with ordinary painted and paper billboards. It is the hand-painted signs that have survived to be captured and chronicled by Frank, while the far more dazzling electric spectaculars of today's Times Square are lost forever.
Which brings me to discuss Frank's favorite photo of an O.J. Gude sign, M. Rappoport's Music Store, uncovered during building demolition. The ad features an early phonograph and record, along with the still famous RCA Victor Dog Nipper, who listens, head cocked, to "his master's voice" coming from the horn-shaped speaker. I can picture Frank looking at the sign and wondering about the music store and the people inside browsing through the newly invented gramophone recordings and marveling at their new playback devices.
I'm sure Frank wasn't aware of O.J.'s humble beginnings at the time he took this shot, or that later in O.J.'s life, after becoming successful, he turned away business from a fellow innovator who had recently developed a new way of mass producing cars that would prove far more important to the world than O.J.'s own use of light bulbs in signs. According to family lore, O.J. refused business from Henry Ford because Mr. Ford had "grease on his hands," and therefore, he thought that Ford would never be successful. Like many Victorians, O.J. valued appearances, saying that if down to his last twenty-five cents he would have chosen a shave over a meal because it could increase his chances of success. For all of O.J.'s business acumen, our family have passed down stories and letters that chronicle the many personal mistakes typical of this _nouveau riche_ , including spoiling his children to the point that many of them were unable to hold a job or attain even a piece of their father's business success, much less a life well lived.
Ironically, I ended up in the advertising business, not because O.J.'s business was passed down through generations but by accident. The O.J. Gude Co. of New York was sold off in the '20s; it didn't even live long enough to be passed down to O.J.'s children. As for myself, I had dreams of being involved with computer technology. In the late 1980s, I "went west," bound for Silicon Valley in an '85 Saab loaded full of clothes and dreams. But while searching for tech jobs in the _San Jose Mercury_ classifieds, a position for a Yellow Pages advertising rep caught my eye. Before I could interview for any of my dream tech companies, I received a job offer making more money than I could imagine. Dreams of O.J.'s advertising riches swirled in my head.
Years later, I found a way to join the family trade of advertising with my love of technology by helping Yellow Pages, and later a traditional newspaper company, transition to online advertising. Soon, I was publishing my first personal blog, which eventually led me to form my own digital advertising and design company. When I wrote an article about O.J., it provided the connection to the old signs that Frank was capturing with his 35mm SLR. In a recent telephone conversation, Frank and I discussed how future generations might discover the wall art of people from earlier times since so much of our work has become digitized. We also discussed the fragility of our digitally stored world. Will much of it be forgotten on old computer servers and hard drives or just simply be victim of binary decay or obsolescence? Will these digital archives become the modern equivalent of finding a roll of undeveloped film in an attic trunk?
O.J. left behind work that vanishes from surfaces left from his time in a way that only has significance to us with the benefit of hindsight. These monumental forms of commercial art—O.J.'s advertising legacy—have a finite shelf life, and with the passage of time, they have become transformed. The transformation continues when contemporary visual images are captured by a camera, further extending the lives of these fading relics, giving them resonance in past, present and future time (an invisible abstraction). Long after we are gone, perhaps future generations will unearth these photographic images, not unlike the domestic wall art of Pompeii preserved beneath the volcanic ash of Vesuvius, and view them with the same sense of wonder and curiosity as the photographer who first produced this body of work, the Fading Ad Campaign—giving insight into the life of an artist who documented the work of past artisans. I have learned many things about my great-great-grandfather's work because of Frank's archive. His fading ads provided a bridge for me to link my life with my family history, along with a new appreciation of their place in time. I hope these images put a smile on your face, as they have for me.
Julian Seery Gude
Great-great-grandson of O.J. Gude
Music and Entertainment
M. Rappoport's Music Store—89-10 Jamaica Avenue, Queens. Taken August 1997. Ad circa 1905. Hung in NYHS Exhibition.
**M. R APPOPORT'S MUSIC STORE**
4109 Jamaica Avenue near Woodhaven Avenue.
This image depicts both opulence and obsolescence. The M. Rappoport's Music store ad is one of the most elaborate and beautifully colored painted ads of the Borough of Queens and was created by the O.J. Gude Company of New York. According to Gude's great-great-grandson Julian Seery Gude, "O.J. made his mark in a time of great change, where modern technological advances were having life changing impacts on people and societies the world round." Prominently featured on this mural ad are the "victrola" and the flat disk phonograph, both examples of the most technologically advanced medium for playing sound recordings in its day.
In 1901, the Victor Phonograph Company sold its portable Victor for roughly $3. The Monarch Deluxe pictured in this ad sold for $60, an equivalent to $1,200 in today's economy. Even today, these machines can be bought at flea markets and antique shops all over America for a couple of hundred dollars at most. Millions of brittle gramophone records (made of 25 percent shellac, powdered slate, cotton fibers and wax lubricant) still are piled heavily in attics, bookshelves and warehouses across America and around the world. In 1939, the first flexible and "unbreakable" vinyl record for 78 rpm playback was introduced. Within forty years, the medium advertised on this mural became obsolete, replaced by the vinyl long-playing record that was played at 33.33 rpm.
If you think about the span of time that gramophone recordings were in vogue and compare them to the more recent compact disc, perhaps the former were longer-lived. Most people today download their music onto a computer and transfer the digital file (an AIFF, which is uncompressed, or an MP3, which is compressed), which is essentially an unfathomably long string of zeros and ones, to a digital recording device or MP3 player. Avid music listeners used to need wall space to store their recordings, and now we need disk space.
In the '60s, I loved the way a 45 felt in my hand, thick and thin between my index finger and thumb. Back in the 1970s, I could not wait to buy an LP so I could handle the cover to read the lyrics and ogle over the cover art. The tactile and visual pleasure of playing music today is gone.
The M. Rappoport's Music ad was revealed briefly while building a pharmacy. One day I took Vincenzo's niece Concetta around with me in search of signs. When we busted into this construction site, the neighbors around this construction pit started coming out into their backyards to talk about this sign with me. Some people were confused as to how the sign "got there." Others were happy I was there to document it. Although it was only exposed for a couple of months in the summer of '97, the sign looks from its quality that it had not seen the light of day for a very long time in the first place. The expansion of that section of Woodhaven along Jamaica Avenue went at a staggering pace. By the time the first box of replacement needles was used up for one of these Monarch Deluxe players, the adjacent building had already broken ground for construction. By the time the wall dog who painted this ad completed it, innovations were speeding toward the electronic recording.
M. Rappoport's Music Store may be gone, but the ad still lurks silently behind a boxy, late 1990s Rite Aid Pharmacy construction. Given time, perhaps this corporation will cease to exist and the functionality for this building will also grow obsolete. Who knows, twenty years from now the neighbors of 89th and 90th Streets may come out of their back doors to stand on their backyard porches to ponder the arrival of this messenger from the past once again.
**42 ND STREET/THE WORLD'S GREATEST MOVIE CENTER/BICKFORD'S**
_Little nifties from the fifties, Innocent and sweet._
_Sexy ladies from the eighties, Who are indiscrete oh!_
_—Al Dubin, "42 nd Street"_
The former Selwyn Office Building was scheduled to be demolished in the spring of 1998. It was being used as a visitors' center until it suddenly collapsed on December 30, 1997. The building's historic façade was to be incorporated into the New 42nd Street Studio building. The Selwyn Theatre was also going to be demolished and rebuilt as the American Airlines Theatre where the Roundabout Theatre Company now resides. According to the Buena Vista University (Iowa) website, Edgar and Arch Selwyn opened the Selwyn Theatre in 1918 at the height of 42nd Street's hustle and bustle. Designed by theatre architect George Keister, the Selwyn was built in the Italian Renaissance style, "decorated in gold and blue with large murals adorning the walls," and had seating for over one thousand people with two levels of box seats.
Next door was a shuttered, closed Bickford's. Almost like a domino effect, the entire block has been transformed since this picture was taken. Anyone coming out of a fourteen-year coma would not be able to recognize 42nd Street today. In the 1970s and 1980s, the kinds of films that played in these theatres were not what one would call family-oriented films. What was once a dicey, somewhat seedy and always challenging street to walk down is now a Disney ride. This is the block where I saw my first adult film and where I bought my first fake ID at age fourteen. My father finally gave it back to me last year after confiscating it in 1974. The excitement of 42nd Street is still there, but the sleaze has been completely whitewashed.
42nd Street/The World's Greatest Movie Center/Bickford's—Selwyn Office Building, 229 West 42nd Street, New York City. Taken April 1997. Ad circa 1950s. Hung in NYHS Exhibition.
The sign reads "Cooped up? Feelin' low? Enjoy a movie today." Although many of the original movie theatres have been torn down, rebuilt or expanded, the block can still boast to having close to forty movie screens between the AMC Theatres–Empire 25 and Regal Cinemas E-Walk Stadium. A majority of this block's transformation took place within five years from November 1995, when all adult theatres on the north side of 42nd Street were closed down, to July 2000, when the American Airlines Theatre opened its doors.
Prior to the 1970s, as evidenced from this photograph, this was a place where you could sit at a cafeteria counter for breakfast, lunch or dinner and not spend an arm and a leg. The business just to the right of this ad was a Bickford's Restaurant, which can be seen written on the ceramic tiled marquis. David W. Dunlap writes the following about Bickford's:
_If you lived in New York anytime from the 1930's through the 1960's, chances are you knew Bickford's. They were up and down Broadway, on Fordham Road and the Grand Concourse in the Bronx, Nostrand Avenue and Fulton Street in Brooklyn, Main Street and Jamaica Avenue in Queens_.
_"Breakfast at Bickford's is an old New York custom," a 1964 guidebook said. "In these centrally located, speedy-service, modestly-priced restaurants a torrent of traffic is sustained for a generous span of hours with patrons who live so many different lives on so many different shifts."_
_To say the least. The best minds of Allen Ginsberg's generation "sank all night in submarine light of Bickford's," he wrote in "Howl."The Beat Generation muse, Herbert Huncke, practically inhabited the Bickford's on West 42 nd Street. Walker Evans photographed Bickford's customers, and Andy Warhol rhapsodized about Bickford's waitresses. Bickford's make its way into the work of writers as diverse as Woody Allen and William Styron_.
_"Death (being edged to the doorway): Where's a good hotel? What am I talking about hotel, I got no money. I'll go sit in Bickford's. (He picks up the News)."_
—Getting Even, _Woody Allen_
_"How vividly there still lingers on my palate the suety aftertaste of the Salisbury steak at Bickford's, or Riker's western omelette, in which one night, nearly swooning, I found a greenish, almost incorporeal feather and a tiny embryonic beak."_
—Sophie's Choice, _William Styron_
**WABC M USIC RADIO 77**
Ron Lundy, Harry Harrison, Cousin Brucie—these were the most influential voices of my childhood. These men were the DJs of my era. My fondest memories were those sunny days when Mom said, "Pack up a pail and shovel, we're goin' to the beach. And don't forget the transistor radio!" The sounds of Motown filled the salty air of the Rockaways. And the Beatles, Beach Boys and all those fabulous hits we now call classic oldies flew over the radio waves and floated above the surf. I can still hear the roar of the rollercoaster of Rockaway's Playland. Maybe I'm just getting old, but they don't make 'em like that anymore.
WABC Music Radio 77—Harlem. Taken August 1999. Ad circa early 1970s.
**O DEON THEATRE**
In an interview with Harlemite and Manhattan borough historian Celedonia "Cal" Jones, he said he remembered sneaking into the Odeon in the late 1930s to watch movies as a child. Cal, who lived two or three blocks north of the Odeon, said, "When you go shopping for theatres to sneak into, you perfected your talents on the local theatres. We had it down pretty pat." Cal recounted, "One of us snuck in and he'd come around and crack the door and the rest of us would crawl upstairs." Sometimes he would "back into the theatre," which is the remarkable talent of walking backward while people were exiting. Cal also remembers that the Odeon used to give out sets of jelly glasses to the first hundred paying customers who got in the theatre or some other kind of promotional gift. Usually the Odeon showed a double feature but not first-run films. "After it hit downtown, it would work its way uptown, so for the area, they were first runs." Cal's buddy Sonny Neal remembered the newsreels between the films. Sonny said they would show the black newsreels after the general newsreels, like the clips about the all-black troops during World War II. Cal remembers he had heard the theatre once had live vaudeville performances, but by the time he was "coming up" in Harlem, during its renaissance, the Odeon was a straight movie house.
Odeon Theatre—West 145th Street, Harlem. Taken August 1997. Sign circa 1920s.
In a letter written by the New York City Landmarks Preservation Committee to restore the Apollo Theatre (formerly known as the Hurtig & Seaman's New Burlesque Theatre), the following information about the Odeon Theatre was provided:
_Brecher and Schiffman were white businessmen who played a major role in the history of Harlem's entertainment industry as owner-operators of a number of Harlem's leading theatres featuring black entertainers. Brecher was owner of several Broadway and Harlem clubs and theatres, while Schiffman was a theatre operator and motion picture distributor. They first became business partners in 1920, converting the Odeon Theatre (1910, Van Buren & Lavelle, 256 West 145th Street) to show motion pictures_.
Many of the old theatres and movie houses had elaborate organs that were played in between vaudeville acts or during films, which up to then were still silent films. There is an interesting connection between the owners of the Odeon and a young jazz organist by the name of Thomas Waller. While reading some selected passages from the book _Black and Blue: The Life and Lyrics of Andy Razaf_ by Barry Singer, required reading for a course at the Center for the Humanities at Washington University at St. Louis about the American poet and jazz lyricist Andy Razaf, I was happy to discover that his musical partner, twenty-year-old Thomas "Fats" Waller, was employed by Brecher and Schiffman.
_Wildly unsuited to a wife who'd finally divorced her unreliable musician-husband in late 1923, and still sorely missing his late mother, young Waller reveled in the camaraderie of musicians, loving to jam with them, loving to drink with them. The orchestra rows surrounding his organ post at the Lincoln Theatre often were filled with musician friends drinking in Waller's astonishing keyboard talent, passing the bottle, laughing with him as he often provided outrageous musical accompaniment to the action onscreen. Willie "the Lion" Smith would later suggest that Waller's unsuspected, painful shyness was the cause of his early alcoholism. Everyone, including young Waller, loved the person he became boozed up. The drinking escalated rapidly_.
_In March 1925 Marie Downes, owner of the Lincoln Theatre, informed her organist that she had decided to sell the old film house to Frank Schiffman and his partner Leo Brecher—two white men, operators of the Odeon Theatre on 145 th Street and both the Harlem Opera House and Loew's Seventh Avenue burlesque theatre on 124th Street, who were about to attempt an ambitious consolidation of black vaudeville entertainment in Harlem_.
_The news stung Waller, who viewed the Lincoln in much the same light as he did the revered memory of his late mother. Schiffman and Brecher, however, cordially brought the young keyboardist up to their offices, where they openly informed him of their plans to turn the Lincoln into a straight movie house while transferring the old theatre's successful vaudeville programming to the larger Lafayette, which, after undergoing continual policy reconfigurations over the previous few years, was for the moment a straight movie house presenting only sporadic stage shows. The two men then offered Waller the Lafayette organist's job for $50 a week (double his Lincoln Theatre salary), crowning the news with the information that if he accepted, Waller could expect to play a Robert Marston–model organ at the Lafayette, the first grand organ to be found in Harlem_.
Cal Jones informed me that the Odeon is currently in use as St. Paul's Community Church. Since organ music is also common in church ritual and the seating capacity can afford a larger audience, it explains why so many of these old theatres in New York made the transition to becoming churches. On the New York City Chapter of the American Guild of Organists' website, some additional information was provided about the organ and the dramatic shift of tenor that occurred at the Odeon from the profane to the sacred. Established in 1947, St. Paul's Community Church was founded after a controversy within the African Methodist Episcopal Church. After the removal of the Reverend Hale B. Thompson as pastor of the Bethel AME Church (originally located on Sullivan Street in Greenwich Village where Cal's grandfather and mother were married in 1893), St. Paul's Community Church was formed and called upon Thompson to head their church. The former Odeon 145th Street Theatre, designed by Thomas W. Lamb and built in 1910, was purchased by this church group in 1952. The theatre's organ was originally built by M.P. Möller as Op. 3963 for the Mecca Temple (now the City Center 55th Street Theatre) at 130 West 55th Street in 1924 and had "four manuals and 37 ranks." In 1955, Möller rebuilt this organ, providing "several new ranks and a new three-manual drawknob console."
The movement from vaudeville theatres to movie houses and then to churches all came about due to the economic and demographic shifts in Harlem's population. Cal Jones remembers from his research of his family history that in the 1920s, Harlem was undergoing rapid changes when whites were moving from Harlem to the suburbs. In 1916, when a row of buildings on 135th Street was bought up by St. Philip's Episcopal (formerly the Free African Church of St. Philip), this "opened the floodgates" to African Americans moving to Harlem. Cal recounts his teachers in elementary school (the poet Countee Cullen was his English teacher) telling him that when they would show performers in black face on the films at the RKO Alhambra Theatre on 126th Street, people would pelt the screen with rotten eggs from the balcony, where the blacks were relegated to sit. The Odeon's building sign is yet another reminder of Harlem's rich and vibrant past.
**H ARDMAN DUO PLAYER PIANO**
_Twice the Fun_.
The sounds of ragtime piano being played from a scratchy 78 rpm gramophone record constitute the perfect soundtrack to a documentary film about fading ads. So often when being presented images from the turn of the century, we watch silent films from a time that seems slightly sped up and out of sync as Model Ts dart around jerkily through crowded intersections filled with quickly paced pedestrians, or we gaze with nostalgia at the television as a camera pans slowly over a vintage sepia-tone albumen print of a 1900s Orchard Street scene in a style now called "The Ken Burns Effect" while listening to an old player piano roll churning out a popular piano tune from that day.
Before the advent of the gramophone record player, player pianos witnessed a dramatic rise in popularity in the late nineteenth century. There were several key players who led to the development and mass production of the player piano. Thaddeus Kochanny of the Chicago Chapter of the Automatic Musical Instrument Collectors' Association (AMICA) wrote the following synopsis about these developments:
_In 1863, the Frenchman_ [Henri] _Fourneaux invented the player piano. He called it, "Pianista," the first pneumatic piano mechanism, which was introduced at the Philadelphia Centennial Exhibition in 1876. In 1887, a year after_ [Edwin] _Votey invented the Pianola; Edwin Welte_ [Welte-Mignon system] _introduced the perforated paper roll in Germany. The perforated roll mechanism to make music was based on the Jacquard punch cards used to weave designs into cloth_.
Votey's Pianola is the player piano that became most popular in the United States. The technology was such that mass production of these costly instruments (about $250, today's equivalent of about $6,000) made them readily available to those who could afford real-time acoustic piano music in their home or business. On the Pianola Institute's website, dedicated to the history of the pianola and Votey's achievements, it is explained that "the basic principal upon which Votey's system operated subsequently became the standard for virtually all roll operated piano playing systems."The Aeolian Corporation acquired the U.S. rights from Votey to market his piano player system from 1897 as the pianola and later became "the world's leading manufacturer of roll operated instruments."Votey's main achievement was melding the best of preexisting player piano designs, creating "an instrument that would enjoy mass market appeal for three decades."
The brightest star of the piano roll production industry was Jean Lawrence Cook, an African American musician and arranger (and World War I veteran) who transcribed the first ragtime tunes in the early days of jazz. J. Lawrence Cook was to the music roll industry what Quincy Jones was to the modern recording industry. After freelancing for several of the leading music roll companies, Cook was hired in May 1923 by the QRS Music Roll Company (later QRS Records). John Farrell, a contributor for a website called the Monrovia Sound Studio, documented the following about Cook:
Hardman Duo Player Piano—the Bronx, down by the waterfront. Taken June 1997. Ad circa 1950s.
_Cook was a musical chameleon—he could produce convincing keyboard impressions of Art Tatum, "Fats"Waller, Teddy Wilson, Erroll Garner and several other leading pianists of the day. However, a major part of his QRS career was spent producing arrangements of commercial pop songs, as required for the catalogue of a commercial roll producer, and it is to Cook's credit that his unique skill injected musicality into otherwise unexceptional material_.
At the height of the production of player pianos in the early 1920s, literally hundreds of thousands of piano rolls were distributed across the United States and Europe. In a _Time_ magazine article published on February 15, 1943, the pianola was remembered as such:
_In the heyday of 1923, when 197,252 pianolas (more than 50% of all the pianos sold in the U.S.) were sold in a single year, the pianola industry hired the greatest pianists, such as Paderewski, to record their performances on perforated paper. It also hired such early jazzers as J. Lawrence Cook and Harlem's historic James P. Johnson. But as the pianola gave ground to the phonograph, the pianola industry could no longer afford to pay for personal recordings_.
With the advent of affordable radios and the crash of 1929 came the near annihilation of the player piano industry, as "uncounted thousands of these instruments were chopped up and used for fuel." Throughout the Depression, largely due to the brilliance of J. Lawrence Cook, and with a new financier, Max Kortlander (himself an accomplished pianist), the QRS Company struggled to maintain itself after a brief production stoppage. It is said that Cook's most celebrated arrangements occurred in the period between the 1940s and 1950s. The player piano industry witnessed its resurgence in the 1950s, with newer models of player pianos being produced by a handful of companies, one of which was the Hardman DUO.
_An exciting innovation in piano design and engineering has led to the creation of the Hardman DUO, the amazing new player-piano developed and manufactured exclusively by Hardman, Peck & Co. Unveiled in the Spring of 1957, the DUO is actually two pianos in one. At once an incomparable Hardman Console famed for acoustical richness is changed from manual to a player-piano, ready to play any of the hundreds of melodies on music rolls—everything from classics to rock 'n roll_.
Professor Alan Wallace, also a jazz musician, contributed the following comments to the Monrovia Studios Website that were gleaned from the liner notes of a Mercury Records recording featuring Cook:
_In 1959 Mercury Records released a 12-inch long-playing stereo record titled PIANO-ROLL ROCK'N ROLL. The record features 12 music rolls by J. Lawrence Cook. Guitar, string bass and drums accompany the numbers. Mercury SR 60083_.
_The liner notes on the reverse side of the record cover give no indication that Lawrence was involved in the actual recording. However, his music rolls, played on a Hardman Duo Player Piano, are in good company with accompanying musicians, Milton Hinton, Tony Mottola, George Duvivier and Osie Johnson_.
_In the mid 1950's, the Hardman Piano Company, introduced the "Hardman Duo," an electrified player piano in a modern "console" case. The company slogan was "It's Twice The Fun when Your Piano is 2-in-1—See The Hardman Duo today!"_ 186
J. Lawrence Cook produced over twenty thousand arrangements for the pianola within a mile from where this sign was photographed. Cook's legacy is immortalized in countless sheets of perforated paper, now perhaps deteriorating in dusty attics or destroyed due to obsolescence. Even the vinyl recordings of his work (which were never reissued as CDs) are destined to decay. In an August 20, 2010 _Buffalo News_ article entitled "The Day the Music Died," reporter Mark Sommer wrote, "When the last piano roll was manufactured at QRS Music Technologies in Buffalo, NY, The remark scribbled at the end of the production sheet said simply, 'End of era.'"
**C AMERA MART**
This ad was also painted by the Mack Sign Company. Camera Mart was incorporated on July 16, 1948, and dissolved its corporation on April 15, 1991. One of its most notable rental gigs was for the music documentary _Woodstock_ in 1970. Other notable films include _Glengarry Glen Ross_ (1992), _The Prince of Tides_ (1991), _Green Card_ (1990), _New York Stories_ (1989), _Big_ (1988), _Sophie's Choice_ (1982), _Kramer vs. Kramer_ (1979) and _Pumping Iron_ (1970).
_WE MAKE VIDEO EASIER_
_Whether you're new to video or not, we at Camera Mart can help. With the most comprehensive line of video equipment. For every type of industrial, educational or medical production. Camera, recorders, monitors, receivers, production and editing systems. For sale, rent or long-term lease_.
_But equipment is just the half of it. The other half is our people. Knowledgeable, dedicated professionals who make sure you get the right components or system for any application. Factory-trained people who inspect and test everything and provide the best service in the business_.
_If you're looking for the answer to your video equipment needs, look no further than Camera Mart. It's that easy! Send for our video catalogue today_.
_—ad in the November 1980 issue of_ Videography
Camera Mart—Tenth Avenue and 456 West 55th Street, New York City. Taken August 1999. Ad circa 1960s.
Hotels, Speakeasies and Saunas
Hotel Irvin—West 30th Street and Eighth Avenue, Midtown, New York City. Taken August 1998.
**H OTEL IRVIN**
_From 2.50 up_
Grutchfield wrote that the Hotel Irvin for Women was named for Mary M. Irvin, president of the group that worked for years to establish a women's residence. As early as 1916, the organization planned a hotel "where self-supporting girls and women with small incomes could be accommodated comfortably and well at little cost." By 1924, the organization had "managed to acquire the land and begin construction" at 308 West 30th Street. Asher Mayer (president) and Charles H. Strong (treasurer) opened the hotel in 1925 "for exclusive occupany by business women [with apartments] arranged in small flexible units with facilities for self-housekeeping [and rents] adjusted on a basis to meet the big demand that exists for this type of housing." According to Grutchfield, the Irvin seemed to drop the women-only policy in the 1940s and went out of business by the mid-1950s. This sign has been well documented by other fading ad enthusiasts on the Internet on photo-sharing sites like Flickr. I seemed lucky to find it with a rainbow flag waving atop.
**H ENRY HUDSON HOTEL**
In 1998, Ian Schrager, former Studio 54 partner and hotelier (Royalton Hotel), made it known that he was going to renovate the old Henry Hudson Hotel at 353 West 57th Street and reopen it as an affordable boutique hotel by 2000. Originally built about 1929 and financed by Anne Morgan, the daughter of the financier J.P. Morgan, it was to serve as the American Women's Association clubhouse and also function as a residence for young New York City women. According to Christopher Gray of the _New York Times_ , Morgan "used her huge inheritance to advance causes of social justice among her crowd of socially advantaged women."
Henry Hudson Hotel—West Side Highway, New York City. Taken June 1998. Ad circa 1950s.
According to Gray, the clubhouse did pretty well considering it was built around the year of the stock market crash, but by 1941, it was succumbing to bankruptcy, so the savvy Ms. Morgan decided to convert the clubhouse into a hotel for both men and women and call it the Henry Hudson. Gray stated that during the hotel's grand opening, "an 11-year-old Dutch girl smashed a bottle of Dutch milk against the doorway." Gray reported that the United Nations Security Council met in the hotel in 1946. It was later the headquarters of WNET, New York City's local public television station, from where the _MacNeil/Lehrer NewsHour_ was broadcast, but WNET moved after the recent renovation by Schrager to its new location on West 33rd Street. Gray also commented, "As for the American Woman's Association, it withered and was defunct by 1980, and its cornerstone—with its predictions for the future—sits undisturbed."
**Y OUNG & SCHMUCK**
_Fine Wines, Liquors and Cigars—Pool & Billiard Parlor_
On August 27, 2000, Daniel B. Schneider of the _New York Times_ reported on this five-story tenement building at 772 Eighth Avenue, at Hell's Kitchen's eastern border, which, in its time was "a broad-shouldered factory and tenement district that by the late 19th century was one of the city's grittier slums," populated by a mostly Irish demographic that included Scots, Germans and African Americans. In Schneider's interview with the architectural historian and _New York Times_ writer Christopher Gray, it was revealed that:
_Fred Schmuck was the original owner of the building, which was built in 1873. The wide, handsome tenements now being demolished to the north were built in 1897; the "Young & Schmuck" sign was no doubt painted during this 24-year period, when pool halls were becoming popular around the city. Mr. Gray added, however, that census records and city directories during these years do not show a Young or a Schmuck at the address, or associated with a billiard parlor_.
_Mr. Gray also said newspaper reports beginning in the 1890's record many police raids of liquor, gambling and prostitution dens in the area, but Young & Schmuck's billiard parlor was not mentioned. Further research is called for_.
_As recently as last year, 772 Eighth Avenue retained an astonishingly unaltered, turn-of-the-century wooden storefront, beneath layer upon layer of chipped and peeling paint_.
According to James T. Young of Gray, Tennessee, in a telephone interview on July 22, 2011, his great-grandfather, James T. "Pop"Young, was born in 1869 in the United States and at age four returned to Scotland to be raised by his aunt. In recounting his great-grandfather's saga, Young said he had returned to the United States divorced from his first wife Ann Lowe, with whom he had three children. Young said that "his great-great grandfather was also in an early billiard parlor and gin joint business in 'The Kitchen' in the 1890's or thereabouts in New York City."
Young & Schmuck—Eighth Avenue and 47th Street. Taken September 2000.
In addition to running the family business, Pop Young was also in real estate and the coal-supplying business, as well as being "a barman, carpenter and Union President the 1900s in New York City and Florida." On James T. Young's genealogy page, he shares the following anecdotes about his great-grandfather as told by his grandmother, Emma Bartko, who was born on March 17, 1872, near Miskolc, Hungary, and immigrated to New York City in 1885. Bartko claimed Young
_was known to own a racehorse at old Aqueduct in NY. Him being voted out of office as Union president during the Depression was probably part due to this horse. He originated in New York City's "Kitchen"; only outsiders called it "Hell's Kitchen." His past position as a Labor man offered an opportunity to his grandson Jimmy to choose almost any trade Union job including working as a "Checker" on the "Docks."_
_When he was dying in a Catholic hospital he became a Catholic. His son (my father) at the church services said "He always was a gambling man, we're lucky he wasn't dying in a Jewish hospital, else we might be standing with Yarmulke's on and in a Synagogue." His wife was less gracious, who as a Catholic believed he was entitled an instant trip to heaven, having just been baptized. "I can't believe GOD is letting that SOB get directly into heaven after the life he led, leaving me, drinking, gambling and what ever else!" He sure doesn't deserve it. The Mass was even funnier...his Scot Rite Mason friends stood outside the Church with Masonic salutes wondering how the Catholics had won him over?_ 197
James T. "Pop" Young died on January 16, 1951, in New York City and was buried on January 19, 1951, in Evergreen Cemetery in Brooklyn (Burial #366578). Emma Bartko died in August 1974 in a nursing home in Queens and was buried in Maple Grove Cemetery in Queens.
**M OUNT MORRIS BATHS—STEAM & TURKISH**
_LEhigh 4-9004_
According to Aviva Stampfer, a writer on the Place Matters website, a joint project of City Lore and the New York Municipal Art Society, the Mount Morris Baths
_was founded in 1898 by a group of Jewish doctors, when Turkish (hot air) baths were an important part of the religious and social traditions of Eastern European Jews. The doctors lived on the upper floors, using the basement as a professional spa. In the 1920s, Finnish immigrant Hugo Koivenon bought the baths and incorporated Finnish features such as "needle showers" and vitea treatments. East Harlem residents (especially those living in the neighborhood's many cold water flats) came for the sauna, steambath and therapeutic pool_.
This was the sign to the Mount Morris Baths as you looked down the stairs at its entrance on the basement level, below the street and somewhat out of view of the sidewalk passersby. The plastic illuminated sign that hung high up over its entrance and said "Turkish Baths—Mt. Morris—Men Only" harkened back to a time when there weren't many legal challenges based on gender discrimination for entering a public place as this. As you walked in, there were safe-sex brochures and free condoms available, although the signs prohibiting explicit sex on the premises juxtaposed to the posters about safe sex seemed contradictory. The place had a musty smell, and I imagine that there were still some of the original water molecules circulating in the fetid, steamy mist since its maiden _shvitz_ of 1898.
Mount Morris Baths—Steam & Turkish—Harlem. Taken April 1997. Sign circa 1950s.
In a January 2003 article for the _New York Times_ , journalist Alan Feuer provided the following more recent historical context about this bathhouse:
_Twenty years ago, at the height of the AIDS epidemic, the gay bathhouse scene was nearly run out of town when state officials enacted a raft of laws banning many homosexual gathering places. The New St. Marks Baths in the East Village, for example, was shut in 1985 by the City Department of Health and was replaced nine years later by a video rental store_.
_The Mount Morris bathhouse, the only one in the city that caters to gay blacks, has been operating continuously since 1893 and survived the crackdown essentially for two reasons. First, it is far from the city's gay meccas, on a quiet, unassuming block of Madison Avenue at East 125 th Street, across the street from the offices of the Rev. Al Sharpton. Second, it has matured through the years, remaining a place to meet new people and enjoy a steam, but with the reality of the city health code's prohibition on open sex_.
Apparently, the owner at the time, Walter Fitzer, a retired mechanical engineer and volunteer firefighter from Lynbrook, New York, seemed "an unlikely candidate [to Feuer] to be running a bathhouse known for attracting gay black men." Notwithstanding, it was my experience growing up gay in New York City that most of the bars and bathhouses were owned by straight, white men. Fitzer told Feuer in his interview, "'I always tell the clients, 'If I can't bring my wife down here, it isn't right.'" Having been a patron of this establishment in the late 1980s when I was living in Harlem before it was destroyed by what the city called "urban renewal," I couldn't imagine anyone bringing their wife to Mount Morris. It was by no means a Plato's Retreat, which was a sex club that opened in 1977 in the basement of the Ansonia Hotel that did cater to a more "ecumenical" crowd. One of my favorite understatements from Fitzer in this interview is: "Bathhouses have been gay since the days of the Greeks. It's no big secret."
According to Feuer, Fitzer also claimed, "Harlem royalty like Joe Louis and Sam Cooke used to sweat here years ago, and it is nothing to see French tourists, straight businessmen and Hasidic Jews perspiring in the steam room, side by side."
On the Place Matters website, Stampfer also presented the following:
_Mount Morris attracted a mixed clientele that included area residents and patients of nearby North General Hospital. Mount Morris became known as well for its emphasis on sex education, providing condoms, lubricant, and brochures, and also hiring an education director who held a lecture series five nights a week on topics of interest to gay men, and ran a popular G.E.D. program_.
Despite the discrepancies in the year this _mikva_ or ritual Jewish bath was founded, for at least seventy of the over one hundred years this establishment was operating, it was frequented chiefly by gay African American men. Many people, like myself, wondered why this sauna was overlooked for nearly a decade when gay bathhouses were systematically closed during the '80s by the New York City Department of Health in its hasty response to the AIDS crisis. And why had it survived unscathed? Didn't New York City health commissioner Stephen Joseph and the Koch administration care enough about _black_ male homosexuals? I don't believe it was left open out of any consideration by Koch for the services Mount Morris provided. For the most part, the city was totally unprepared for the AIDS crisis when it hit with a vengeance.
I remember challenging Koch in August 1987 during his obligatory momentary appearance at the New York City chapter of Parents of Gays annual awards dinner when I asked him why there wasn't a public service campaign on safe sex aimed at New York City's LGBT community, as there was in San Francisco. Koch's typical flippant response was, "Oh, the gays here know what to do." So I began chanting, "You're full of shit" and was joined by my friend Andy Humm and others until Koch stormed out of the banquet hall. Urban legend has it that later that evening on the news, it was said that Koch collapsed in Chinatown after overeating at one of his favorite restaurants.
In a recent telephone conversation with my longtime friend and journalist Andy Humm ( _Gay City News_ ), he commented to me that it was fortuitous that Mount Morris had remained open as long as it did after the bathhouse closings since it provided much-needed services to its community. In addition, the pioneering and exemplary work of the Minority AIDS Task Force (1985), Harlem United (1988) and other grass-roots community organizations that targeted black and Latino populations that weren't publicly gay helped an ailing community that was for the most part in denial. Sadly, I was alerted by e-mails through my website of the sauna's closing in 2003 and wondered why there wasn't the same uproar in the gay community as there was over the closing of the Wall Street Sauna in February 2004. Of course, south of 110th Street there were private AIDS organizations like Gay Men's Health Crisis (1981) and the AIDS Resource Center (Bailey House, 1983) that had been mobilized since the onset of the epidemic and provided services initially for self-identified gay men, usually white, with regard to education about AIDS prevention, medical and financial counseling and advocacy.
Humm also reminded me that in the early days of ACT UP, there were two camps with totally divergent ideologies: one, those who wanted to aid the City of New York in creating guidelines for establishments where public sex was a potential in an attempt to keep them open; and two, those who wanted no restrictions at all on public spaces because any limitations would be an infringement of their personal freedoms. Ultimately, both camps lost the battle because many of these sex establishments that provided the only reliable sources of HIV/AIDS prevention materials were closed in spite of their attempts to work with the failures of the Koch administration. Today, I have heard, the sex clubs are opening up again and are filled with young people who did not experience the horror of disease, loss and grief as we did as young people living through the height of the AIDS epidemic in the '80s and '90s. Remember, folks—the AIDS crisis is not over!
Rosario Dawson and Her Uncle Frank
_Born and raised in NYC, Rosario began her film career by accident when she was cast off the street to act in the 1995 indie production "Kids." The film propelled Rosario in the acting world. She went on to attend the Lee Strasberg Theater Institute and soon after started appearing in movies such as "He Got Game," "Josie and the Pussycats," "Men in Black II," "The Adventures of Pluto Nash," "25 th Hour," "Alexander," and most recently "Sin City" and "RENT."_
_At the age of 16, while in high school, Rosario decided to live with Uncle Frank and partner Vincenzo Aiosa. Frank is photographer/composer...and a NYC Public School Teacher. Rosario has great memories of living with Frank and Vincenzo, who—by the way—got married in Toronto in 2004. Frank has been a great influence in her life and she treasures their relationship tremendously_.
_Why This Campaign is So Important: It is estimated that one in 4 families includes someone that is gay, lesbian, bisexual, or transgendered. 69% of GLBT youth report experiencing some form of harassment or violence, with 46% reporting verbal harassment, 36% reporting sexual harassment, 12% reporting physical harassment, and 6% reporting physical assault_.
_Gay and lesbian youths are two to three times more likely to commit suicide than other youths. Of 1.3 million homeless children on America's streets, 500,000 are thought to be GLBT kids thrown out by their parents_.
_How This Campaign Came About: These facts could not be ignored any longer. So in 2002, PFLAG NY took the lead in developing an awareness campaign to get the word out about the PFLAG organization to parents, families, teachers, clergy, politicians, GLBT people, and the general public. The goal was to reach people who would most benefit from PFLAG's services and, ultimately, to increase acceptance, reduce bigotry, and change hearts and minds_.
_PFLAG NY recruited talented individuals from various fields (advertising, PR, law, media) to work pro bono on the awareness effort, which became known as the "Stay Close" campaign. After three years in the making, PFLAG NY launched "Stay Close" featuring straight celebrities with their gay relatives. The message is simple: Stay Close to your loved ones because relationships are too precious to lose_.
_Courtesy of StayClose dot org_.
I was so grateful when Rosa took the time to participate in this incredibly important campaign while she was in production for the film _Rent_ , for which she played an HIV-positive young woman named Mimi. Rosa also knew how important this organization was to my mother, Willy. For over twenty-one years, Willy Jump helped parents cope with learning about their LGBT children at the NYC chapter of PFLAG (Parents, Families and Friends of Lesbians and Gays) alongside her dear friends Amy and Dick Ashworth. After marching with me in the first LGBT March on Washington in 1979 and meeting the Ashworths for the first time, my mom vowed to march with me in New York City, which she did proudly behind the PFLAG banner in the NYC LGBT Pride Parade for over twenty years since 1980.
The morning of the event in New York City in 1980—which traditionally took place on the last Sunday in June to commemorate the Stonewall Inn riots of 1969—I told my mom to meet me on the corner of Bedford and Christopher Streets an hour before the march was to begin its "illegal lurch" uptown toward Central Park, thinking it wouldn't be that crowded yet. (I'm not sure when the first legally obtained permit for the march was, but it was a _march_ long before it became a _parade_.)
So here I am desperately looking for my mom, peering over the throngs of leather queens, drag queens, dykes on bikes and twinks, screaming, "MOM! MOM!" while perched on a nearby lamppost. Almost immediately, this dashing older gentleman with an impish _diastemaed_ smile came up to me and tugged my pant leg while shouting at me over the din in his incredibly coarse voice that seemed incongruous to his appearance. "You really aren't looking for your MOM, but some big queen you call MOM—right?" "No," I said, slowly realizing to whom I was responding. "I really _am_ looking for my mom."
Slowly putting his fingernail up to his mouth to hide his incredulity, he began sputtering like a typewriter on steroids in a rapid-fire breathy dragon voice: "OH MY GOD! If my mother would just even _acknowledge_ my being gay, let alone come _marching_ with me! COME MARCHING WITH ME? I could die right now a happy drag queen. Do you know how lucky you are? I have to meet this WOMAN! MOM! MOM! MOM!"
And almost as soon as he began calling, my mother happily appeared, exclaiming, "Hi, Frankie. Who is your friend?" I told her that this was the inimitable Harvey Fierstein. "Points! Points! You are scoring here," Harvey raspily whispered. "And _this_ is my mother, Willy Jump," I proudly exclaimed. Harvey grabbed my mother around the neck and planted a wet one on her cheek.
Not surprisingly, the two of them would run into each other for the next decade at LGBT events on panel discussions. I ran into Harvey repeatedly over the years, from book signings to random rides on the subway while he was going to the theatre to perform _Torch Song_ to spotting him on parade floats—always with a warm, gravelly greeting, "How's your mother?"
Needless to say, the '80s were a very difficult time for parents of gay men in the organization, particularly the Ashworths, who lost two sons to AIDS. My continued wish is that through the efforts of my mom, my niece, parents and friends of the LGBT community across the country, and brilliantly audacious public figures like Harvey Fierstein, this cause doesn't become just another fading ad campaign.
Prostheses and Undertakers
Assorted signs on 32nd and West 32nd Street, New York City. Taken September 1998. Ads circa 1910–1930s.
**P OMEROY TRUSSES, ELASTIC STOCKINGS, ABDOMINAL BELTS, ARTIFICIAL LEGS**
The history of this company was revealed to me through its advertising in various medical journals and business directories from 1880 to 1941. Arthur C. Pomeroy claimed he would "attend personally to the careful adjustment and adaptation to each case o[f] his specialties in Trusses." These products were offered in a print advertisement in the 1880 _Proceedings of the Medical Society of the County of Kings_ , locating the business at 746 Broadway on the corner of Astor Place:
_POMEROY'S TRUSSES. The "Finger Pad,"The "Water Pad,"The "Frame Trusses,"The "Jointed Spring Trusses,"The "Elastic Rupture Belt," &c_.
_Dr. Richardson's Clavicle and Let Splints, Elastic Stockings, Knee Caps, Belts, &c., Crutches, Shoulder Braces, Suspensories, Abdominal Supporters, Club Foot Shoes, Leg Braces, and other Surgical Appliances_.
_Ladies may avail themselves of the services of a skillful and experienced female attendant, with Mr. Pomeroy's advice and aid, in the selection and preparation of such appliances as may be required...Instruments for prolapsus ani, Trusses for inguinal and umbilical hernia of infants, and other surgical appliances. COMMON TRUSSES FITTED AT LOW PRICES_ 205
_Prolapsus ani_. Ouch. Pomeroy offered quite an impressive list of prosthetic devices and services to aid the troubled body, with special consideration for female patients. Pomeroy placed another ad in the _Pennsylvania Medical Journal_ 9, no. 2, published by the Medical Society of the State of Pennsylvania from 1905 to 1906. By 1906, Pomeroy had moved his office location to 17 Union Square and had opened a second office at 414 Fulton Street in Brooklyn. In this ad, Pomeroy claimed: "Our Specialty is the fitting of trusses by the Frame method. We guarantee, with this truss, to retain securely and comfortably the hernia of any person referred to us. Difficult cases that have been considered unholdable, are especially solicited. We have such coming to us from all parts of the country, and invariably give them complete satisfaction."
Pomeroy Trusses—208 Livingston Street, Brooklyn. Taken March 1999. Ad circa 1910.
By 1910, the Pomeroy "Frame" Truss franchise had opened offices throughout the Tri-State area. Several locations in the New York City area were opened to service the needs of an ailing and rapidly growing population. A medical directory from 1910 revealed the following information about the company's founding:
_ARTHUR C. POMEROY, Pres. & Treas. HENRY M. DEAN. Sec. Established 1867—Incorporated 1875 Pomeroy Company 34 EAST 23d STREET (Bet. B'way & 4th Ave.) Telephone, 6881 Gramercy 330 LENOX Ave., Near 126th Street Telephone, 2388 Harlem 208 LIVINGSTON STREET, BROOKLYN 825 BROAD STREET, NEWARK, N.J. Telephone, 2433 Market_
_The Pomeroy Frame Truss FOR HERNIA SUPPORTING BELTS MADE, TO FIT...Skillful and experienced women will attend ladies at our application rooms, or, if the physician so requests, at their residences_.
Again, it is advertised that female specialists would attend to female patients. Also, the house call is offered as an available option for services. On further investigation, I found a much later ad in the July 1941 _Bulletin of the New York Academy of Medicine_. By then, Mr. Arthur C. Pomeroy had offices in Newark, Boston, Springfield, Detroit and Wilkes-Barre and had had many locations in Brooklyn, the Bronx and Manhattan. By 1941, the Pomeroy "empire" heralded:
_The Pomeroy Frame Truss embodies the knowledge and experience of seventy years. Its time-proven effectiveness in retaining herniae through passive resistance, rather than through active pressure, has won the recognition and approbation of countless physicians through three generations_.
_There is no guarantee of truss satisfaction greater than the combination of POMEROY skill and experience as exemplified in the POMEROY FRAME TRUSS_.
My search to connect the disparate pieces of information on the Internet ended with a pleasant surprise. Beth Beeman, the great-great-granddaughter of Mr. Pomeroy, confirmed that the grave site in Green-Wood Cemetery in Brooklyn, with the birth and death dates of December 12, 1873, and December 30, 1956, belongs to the same man who helped sustain a prosthetic limb and abdominal belt empire in the state with the same moniker. Beeman also confirmed that his father, Daniel Pomeroy, founded the business in 1867 and passed it along to his son, who graduated from Wesleyan University with a bachelor's degree in philosophy in 1895.
In our e-mail exchanges, it was also confirmed that the May 3, 1903 _New York Times_ obituary for three-year-old Ruth Elizabeth Pomeroy, daughter of Arthur Cleveland Pomeroy who died of scarlet fever, was indeed the oldest sister of Beeman's grandmother and for whom Beeman's mother, a published poet, was named. At eighty-three years of age, Mr. Pomeroy experienced the loss of a young child, completed a valued education in a prestigious university, inherited a thriving business and lived way past the expected lifespan of an American male in his time. This sign is a testament to Pomeroy's legacy.
**A SSORTED SIGNS ON 32ND**
(pages 200–201)
_G. Schoepfer Glass Eyes & Taxidermist Supplies_
_Cottage Restaurant—Home Cooked Food_
_Fotoshop—Motion Picture Equipment—Same Day Service_.
According to Grutchfield, Gustav Schoepfer (1891–1988) first opened his business on East 95th Street in Brooklyn about 1917 and "manufactured, exported and imported glass eyes and other items for toy makers, opticians and taxidermists." In his research, Grutchfield was surprised to discover the eyes were used by milliners and for dolls and stuffed toys. Beaks were manufactured for taxidermists. In 1921, Schoepfer opened his first Manhattan location at 106 East 12th Street and relocated to several sites in the West 30s area during the late 1920s before settling at 132 West 32nd Street from 1931 until 1939. This sign was at that location, dating from the 1930s. Grutchfield asserts that as of June 2003, "G. Schoepfer, Inc. still exists (in Cheshire, Conn.) as a major supplier of glass and acrylic eyes; optical goods and optometrists equipment and supplies."
Grutchfield goes on to say that the first Cottage Restaurant was referred to as a tearoom and was located on Lexington Avenue near Bloomingdale's on East 58th Street from about 1924. The restaurant's president, George H. Rice, was born in Mississippi, and his wife, Miriam (secretary and treasurer), hailed from Louisiana. The 1930 U.S. Census listed their ages as twenty-nine and twenty-eight, respectively. The Cottage Restaurant opened at 132 West 32nd Street in 1926 and remained open until the early 1970s.
The sign below and to the right says "Fotoshop, Cameras, Motion Picture Equipment, Photo Finishing, Same Day Service." It's ironic that decades after this sign was painted and the store it advertised closed, I am using a computer program called Photoshop to decipher its contents.
**W M. W. PECAN UNDERTAKER & EMBALMER**
_1666 B'way. 2 nd Door Above Meat Market_
Is it just me, or does this sound like the plot for a Stephen Sondheim musical? According to Brooklyn Genealogy Information dot com, W.W. Pecan started his undertaker establishment at 570 Grand Street in the Eastern Division. Pecan's livery stables were located at 279 Graham Avenue near Grand Street. In addition to his service to the community as undertaker and embalmer, he also served for five years as the assistant foreman of the Volunteer Hook & Ladder Co., No. 2. Pecan's business listing and short bio were transcribed for this website:
Wm. W. Pecan Undertaker & Embalmer—under the Broadway el in Bushwick, Brooklyn. Taken March 1999. Sign circa late 1890s.
_Furnishing Undertaker, Livery and Boarding Stables, Coffin Wareroom...Among those following the undertaking business in Brooklyn no one is better known or more universally esteemed than Mr. Wm. W. Pecan. This gentleman established his business here in 1859, and has been located at his present address since 1862. At this point he occupies a fine office and wareroom, and carries a complete assortment of coffins, caskets, and general funeral furnishing requisites. He does an extensive business as a funeral director and embalmer, and his services are in frequent demand_.
_In addition to his undertaking business Mr. Pecan carries on a large livery and boarding stable business. His stables...are thoroughly equipped throughout. He runs two coaches, five light wagons, and ten fine horses, and also has accommodations for boarding twelve horses. Mr. Pecan is a native of New York City_.
The building where Wm. W. Pecan's original office was located, on the corner of Lorimer and Grand Streets, is still standing. However, Pecan's Graham Avenue stables have since been demolished, and a new building is in their place. This particular building pictured in the sign was located farther south and east on the south side of Broadway, even farther toward the eastern border of Brooklyn and Queens, closer to Trinity Cemetery and Cemetery of the Evergreens. This building has since been remodeled and stuccoed and has had a new building built adjacently to it, obscuring any evidence that this was once a thriving funeral parlor— _and_ meat market.
As you drive along the scenic Jackie Robinson Parkway (formerly the Interborough), with the best skyline Manhattan views, you are surrounded by the hundreds of thousands of graves of the likes of the Dutch modernist painter Pieter Cornelis Mondriaan of Amersfoort; jazz musician and composer Eubie Blake; and the countless known and lesser-known veterans from the Revolutionary and Civil Wars to the major wars and conflicts of the twentieth century. James T. "Pop"Young (Young & Schmuck) is buried at the Evergreens, as well as jazz tenor sax player Lester Young and dancer Bill "Bojangles" Robinson of the song "Mr. Bojangles" and the eight women listed as unknown from the Triangle Shirtwaist Fire of 1911. And who knows of the countless bodies that were interred by Mr. Wm. W. Pecan, Undertaker & Embalmer of the Eastern Division of Brooklyn, during his long and esteemed tenure. The countless stories of the people who struggled, conquered, lost and died still linger on the sides of brick faces throughout New York City's five boroughs in fading ads. One day, just take a brisk or leisurely stroll with your head up and perhaps you will see this magnificent and ever-changing city through these eyes. At the risk of being nostalgic, I'd say do it for old times' sake.
Fading Ads and a Transatlantic Relationship
I have never met Frank Jump, but he has, perhaps unknowingly, been a major influence on my life over the last five years. It all started for me in Stoke Newington, an area of North London in the UK where I grew up. In early 2006, I looked up and saw something I had never noticed before. "Fount Pens Repaired," it said, a sign painted on the first and second floors of the building in front of me. Farther down the street, I spotted another, this time promoting "Alf, the Purse King." At the time I just walked on, unaware of how important this writing on the walls would become to me.
In the summer that year, I mentioned these pieces of hand-painted advertising to my girlfriend (now wife), saying that I thought they should be photographed before they disappeared. She encouraged me to "just get on with it," words she would later come to (partially) regret. I set about taking pictures and sending these to friends to ask if they knew of further locations. I also tried to research their history. Two things happened at this point: first was an overwhelmingly positive reaction from the people I had e-mailed; second was discovering Frank Jump's Fading Ad site.
Frank's site and my communications with him encouraged me to pursue this new interest further. I was surprised how little information existed online about what Frank refers to as "fading ads." His diligent work documenting signs in the United States, and across the world, gave me the idea to do something similar for signs in my own country. Frank is more than just an enthusiast; he is a passionate educator, and I learned a lot from him as I considered what to do next.
Things soon started to move quickly. I was inundated with locations and photos from friends and family and soon had a sizable collection of material. Again inspired by Frank, I set up my first blog called _Brickads_ , a name I gave to what I would later refer to as _Ghostsigns_. The blog allowed me to publish the material I was gathering. Press and public interest followed. This led to me forging a link with the History of Advertising Trust and starting a project to create a permanent online archive of hand-painted advertising in the UK and Ireland. From announcing the project to launching the archive took exactly one year. It now includes over seven hundred images, each recording a piece of advertising and social history for future generations. It takes them back to a time when people had their fountain pens repaired and Alf was the self-proclaimed "Purse King."
Like Frank, I have dedicated a significant part of my life to photographing and researching these fading ads. In doing so, Frank finds meaning at a very personal level, providing a metaphor through which to explore his own survival against the odds with HIV/AIDS. For me, I think it started with a belief that something should be done to document these historical artifacts and then a passion developing in response to people's reactions to the idea. In the absence of anyone else taking on the task, I felt an obligation to do so.
Both Frank and I personify the signs in our writing, suggesting an emotional connection with our subjects. Perhaps this is what stimulates me to care, working to save their memory before they pass away from natural or man-made causes. I am not a supporter of legal protection measures, as the signs have managed for long enough without them. However, through photography, Frank and others like him can extend their sell-by date indefinitely.
Sam Roberts
_UK Brickads_
www.ghostsigns.co.uk
EPILOGUE
The Collaged City
Painted wall signs in cities beckoned nineteenth-century Americans to soothe their nerves with Omega Oil, brighten their clothes with Reckitt's Blueing, snack on Uneeda Biscuits and restore their energy with Coca-Cola. The rise of national brands and consumer culture made commercial images dance in bright colors at a giant scale up and down city walls. Some thought it was part of the reckless energy of the new world and a byproduct of the hectic press of American business. But others drew back in disgust, decrying the visual nuisance that seized the public's eye for constant huckstering. This clash between commerce and culture found a powerful expression in the City Beautiful movement of the early twentieth century that sought to reform cities aesthetically, as well as morally. The City Beautiful improvement program included creating sculpture competitions, fine municipal buildings, parks and wide boulevards but also targeted signs and billboards that would compete with its program of civic pride and uplift. At the very same time, national advertisers were organizing coast-to-coast campaigns and the outdoor advertising industry was consolidating its political power in a trade association to advance its interests and defend itself from City Beautiful–inspired regulations.
Wrigley's Spearmint Gum created giant electric signs in Times Square, painted walls and bought space in the streetcars that whizzed city dwellers from home to work. Not only did the habit of chewing gum alarm the upper crust, but they also feared that the sight of large, luscious lips on gum ads might topple the moral order. The National Biscuit Company, with its pioneering packaged crackers, lavished $7 million on advertising in the first decade of the twentieth century. Its iconic kid in the yellow slicker aimed to remind readers that these wrapped national brand crackers were sealed against moisture. The accompanying jingle, "lest you forget, we say it yet, Uneeda biscuit," played a similar memory strategy with words: blithely coining a name and spelling for the cracker that made teachers cringe. The ubiquitous advertisements charmed enough of the public to ensure a healthy corporate future for Nabisco, but they also provided visual and grammatical fuel to the firestorm of cultural criticism of outdoor advertising. Another famous example of visual vinegar was the giant lighted Heinz pickle on the Hotel Cumberland at Madison Square, which gave off an orange glow in 1899. What especially annoyed critics in this case was how the electric sign outshone the nearby Dewy Arch, a typical City Beautiful piece of civic uplift celebrating the victory in the Spanish-American War in Cuba and the Philippines. The juxtaposition of civic art and advertisement was common: advertisers, of course, sought the most prominent placements in areas with high visibility. For example, Christopher Columbus, presiding over his circle at 59th Street, overlooked in 1913 a giant sign for Kelly tires with a cute gal smiling from within the rim, and on a taller building behind it, an enormous rubric proclaimed "Gainsborough Studios."
Uneeda Biscuit—Fifteenth Street and Fourth Avenue, Park Slope, Brooklyn. Taken August 2003.
These examples illustrate the clash between sign haters and advertisers at a time when the American public was beginning to develop a range of responses to ads. Initially, the public often seemed delighted by the signs: they were novel, a form of free entertainment, one of the amusing sights to eyeball in cities. But as national brands and national advertising campaigns became more and more assertive, a growing opposition movement turned to politics and government to fight the ads. Just before World War I, reformers of New York's Municipal Art Society had proposed a Billboard Commission to regulate signs, hoping to decrease their size, control their placement and tone down the "glaring colors" that so offended the city's aesthetic police. Commission members spoke of their duty to protect "the public's sense of sight" and charged that "disgusting billboard advertisments...rob the people of their rightful heritage of natural beauty." O.J. Gude, who was head of one of the biggest outdoor advertising companies in New York, was clever enough to have joined a committee of the Municipal Art Society, which was pushing for reform. He commented that tourists were coming to Times Square to see the electric signs, which had admirers even in France, and implied that "beauty" was in the eyes of the beholder rather than a legislative concept. Gude's compatriots in the business had been practicing the art of populism for some years already in their trade journal, _The Billposter_. "But it is simply ridiculous to howl about the nuisance of billboards in those locations where the monotonous dullness and deadness of the view is enough to give one the blues...far from debasing the tastes of passersby, they are refreshing as signs of life amid the brick and stone Saharas of the great city," wrote one advertiser in 1903.
Heinz Factory—Bergen Street and Franklin Avenue, Prospect Heights, Brooklyn. Taken July 1999.
O.J. Gude Company of NY—Select Brand—Grade A Milk—Coney Island Avenue, Flatbush, Brooklyn. Taken August 1999.
Maybe reformers who argued against painted wall signs using the cultural terms generated by the conventions of painting were missing the cultural dynamics behind the impact of commercial imagery. As advertising began to use testing and research, the gap between the language City Beautiful reformers used to describe the experience of the urban dwellers and the actual experience of those seeing these large advertisements demonstrated that advertisers were understanding something important about evolving habits of perception. Walter Dill Scott, a professor of psychology at Northwestern University, studied Omega Oil advertisements in the early twentieth century. He wrote in his influential 1902 _Theory of Advertising_ : "Omega Oil is not only a thing which can be applied and felt by the skin, but it is also a thing that can be seen and smelt." Scott emphasized that the Omega headline "It's Green," which appeared meaningless, actually sparked a pleasant association and mood and concluded that Omega Oil ads succeeded by appealing to the senses, which evoked strong mental images in the consumer. Dill's analysis of the emotive power of advertising seemed novel and startling at the time and draws our attention to the seductive power of the imagery, rather than the notion of coercive selling that made reformers so indignant.
Indeed, it is that seductive power that has persisted long after the advertising messages have lost their meaning. And the seduction derives from the urban contexts of billboards, light displays and wall signs—the dynamics of viewers and signs—as much as from the individual compositions. In cities that are constantly being torn down and rebuilt, old things can become the object of considerable nostalgic affection, quite apart from their overt content, such as cigarette advertising or long-ago patent medicine pitches. Where once a patent medicine make suggested "children cry for Castoria," his sweet laxative syrup, nowadays the ad itself is the syrup. Furthermore, I would suggest that the gigantic scale of the wall signs gives them a kind of supernatural power at an unconscious level, as though the line were spoken by something larger than life, a force bigger than an individual such as the state, a society or even subliminally—the gods. Cultural critic Susan Stewart discusses the gigantic as something experienced in relation to the scale of the human body, and thus the gigantic becomes "a metaphor for the abstract authority of the state and the collective, public life." These psychological dimensions of the scale, color and familiarity of advertising in urban contexts move us from the literal meanings of signs to a consideration of how they appear in a welter of tall buildings, crisscross patterns of light and vistas framed by the canyons of avenues and belts of illuminated elevated trains.
City observers perceive signs as part of a collage of sensations rather than as individual and discrete messages and objects. Imagine the city observer as a stroller—what the nineteenth-century Parisians celebrated as a _flaneur_ —who watches the ever-changing spectacle of scurrying people against an unplanned scenography of building angles, sign fragments, windows silhouetted like movie stills and flash frames created by traffic lights. This is the visual and psychological collage that gives the old wall signs their persistent significance in the city. Thus, when we isolate rare and now precious images of wall signs, we may lose sight of the fundamental texture of how people experience them in the city. Fortunately, Frank Jump's astute framing of the remains of wall signs around New York City often has clues to those contexts. For example, "Philip Morris, America's Finest Cigarette" includes a rich deep background with the World Trade Center towers glimpsed in the distance down the avenue. This framing implies a kind of looking that a period passerby might well have done: catching a cigarette pitch from the side of the eye, while keeping an eye for action on the street and scanning the horizon for clouds massing off the harbor behind the towers at the tip of Manhattan. In other cases, Jump wittily reminds us that the signs and the buildings are transient objects, apt to be draped with ivy, or scorched by the sun, or obscured by the cheeky spray of graffiti writers. Or he simply lets us savor the irony of the juxtaposed "Cottage Restaurant's home cooked food" next to the "taxidermist's supplies." And for modern-day gals who never pressed a sewing machine treadle to the ground, "Seely Shoulder shapes" is a tongue twister, not a dressmaker's notion.
The phrases and shapes jumbled into an urban vocabulary of looking have become part of the rhetoric of the city, a part of the urban texture we expect in places built before the car, when brick was red and paint was so full of lead it lasted for generations (while making the billposters tremble with the side effects of lead poisoning over a lifetime). As people have learned to appreciate cities as layered accumulations of civic infrastructure, built environment, vernacular expression, commercial culture, populist marking and yes, even formal architecture, the place of the faded wall sign has assumed a role as an essential part of the fabric. According to urban historian Kevin Lynch, old signs can help make the city legible and may form part of the saturated memories that give a city meaning and function as landmarks, helping people to orient themselves. In a built environment as rapidly changing as New York, the persistence of old signs that peek from behind sleek new skyscrapers can offer a reassuring sense of continuity. Old signs have become part of a new rhetoric of urban preservation in which not just architecture but the visual look of neighborhoods becomes the objects of historic marking. It is fascinating to realize that in New York, the Municipal Art Society, which in the Progressive era fought the sign, became attuned to the role of signs as part of a common visual heritage and helped write rules to _require_ them in the redeveloped Times Square area in the 1980s.
In the course of more than 150 years of walls signs painted tall upon city brick, urban dwellers have come to like them. From targets of ire, derided as visual junk and architectural desecration, wall signs have become objects of appreciation and preservation. And in the process of learning to look and learning to like them, the public has shaped notions of common visual rights. Spotting old signs and interpreting them as interesting pieces of the urban fabric has become a populist history sport—something that would have astounded a City Beautiful advocate and yet might, somehow, please him in the long run, as he saw his goal of city appreciation being ultimately fulfilled.
Kathleen Hulser, public historian
New-York Historical Society
July 2011
Notes
ON ADVERTISING LEGEND DOUGLAS LEIGH
. American Sign Museum, www.signmuseum.net.
NEW YORK CITY'S FADING AD CAMPAIGN
. Read: Frank H. Jump, Interview #33, ACT-UP Oral History Project, Sarah Schulman, www.actuporalhistory.org/interviews/images/jump.pdf; or watch video excerpt, _Are You on the List?_ www.actuporalhistory.org/interviews/interviews_06.html#jump.
. The Estate Project: for artists with AIDS, Alliance for the Arts, www.artistswithaids.org.
. Fading Ads: The Photography of Frank H. Jump, _Archaeology Online_ (October 20, 1998), www.archaeology.org/online/features/ads.
. Stephen Greco, "Signs of Life: Photographer Frank Jump Captures Traces of a Lost World, _POZ Magazine_ (November 1998), www.poz.com/articles/233_1666.shtml.
. Web Guide: Hot Sites, Fading Ads, _USA Today_ (updated April 2003), www.usatoday.com/tech/webguide/hotsites/2002-09-11-hotsites.htm.
. Missy Sullivan, "Artist Power," _Forbes Magazine_ (May 22, 2000), www.forbes.com/forbes/2000/0522/036.html.
. _London Observer Magazine_ , "Signs of the Times," (May 3, 1999), www.frankjump.com/londonobserver.pdf.
. BIT, "Brooklyn Review: The Fading Ad Blog, Best of Brooklyn Independent Television," December 29, 2010, www.fadingad.com/fadingadblog/?p=7639.
. Kevin Walsh, _Forgotten New York_ , www.forgotten-ny.com.
. Walter Grutchfield, _New York City Signs: 14 th to 42nd Street_, www.14to42.net.
. Sam Roberts, _Ghostsigns Blog_ , www.ghostsigns.co.uk/blog.
. Joseph Berger, "Fading Memories," _New York Times_ , November 5, 2005, query.nytimes.com/gst/fullpage.html?res=9B07E6DF143EF936A35752C1A9639C8B63.
. Gregory and Maria Pearce, "Pasolini: Quo Vadis? The Fate of Pier Paolo Pasolini," www.cinemaseekers.com/Pasolinitext.html.
. Nostalgia: Fading Ad Wiki, fadingad.wikispaces.com/Nostalgia.
. PBS, "Channel Thirteen's Life Part Two, with Professor Gerald Torres of The University of Texas (Austin)," www.pbs.org/lifepart2/watch/season-1/adapting-change/adapting-change-transcript.
. _Fading Ad Blog_ , "Penmanship Stupid—Faux Ad—Jerry "Orange" Johnson, www.fadingad.com/fadingadblog/?p=8732.
. _Agility Nut_ , "Super Signage: Penmanship Sign (gone)," www.agilitynut.com/signs/nyc.html.
. Leila Abboud, "The Wall Dogs' Last Stand: Technology Puts Sign Painters Out of Work," Columbia School of Journalism Website, www.jrn.columbia.edu/studentwork/cns/2002-03-20/244.asp (dead link).
SNAKE OILS, ELIXIRS, TONICS, CURE-ALLS AND LAXATIVES
. Wikipedia, "Snake oil," en.wikipedia.org/wiki/Snake_oil.
. Online Etymology Dictionary, "Elixir," www.etymonline.com/index.php?term=elixir.
. U.S. Department of Health and Human Services, U.S. Food and Drug Administration History, www.fda.gov/AboutFDA/WhatWeDo/History/default.htm.
. Up To Date, "lipodystrophic syndromes," www.uptodate.com/contents/lipodystrophic-syndromes?source=search_result&selectedTitle=1~2.
. Kelly Pilson, MEd, _Your Health Online Magazine_ , "Chemotherapy, Ototoxicity & the Inner Ear," September 20, 2007, www.yourhealthmagazine.net/articles/cancer-awareness/1064-Chemotherapy,-Ototoxicity-.html?-the-Inner-Ear=.
. Caroline Rance, The Quack Doctor, "Omega Oil," March 10, 2010, thequackdoctor.com/index.php/the-omega-oil.
. Julian Seery Gude, "O.J. Gude and Me," _Julians.name_ , December 18, 2005, www.blog.julians.name/2005/12/18/oj-gude-and-me.
. Sandra Walker, RI, sandrawalkerri.com/about_sandra.htm.
. Fading Ad Campaign, "Sandra Walker," www.frankjump.com/sandrawalker.html ( _Omega Oil_ , watercolour, 2003).
. _Urban Dictionary_ , "alter kaker," www.urbandictionary.com/define.php?term=altercocker.
. _The 20 th Century Song Book_, Chattanooga Medicine Company, 1904, www.mum.org/songbk1.htm.
. Wiktionary, "Syrup of figs," en.wiktionary.org/wiki/syrup_of_figs.
. D. Sikrov, MD, "Hemorrhoids Are the Consequence of the Sitting Defecation Posture," www.sikiron.com/pages/150_bowl.php.
. PubMed, "Health benefits of dietary fiber," U.S. National Library of Medicine National Institutes of Health, www.ncbi.nlm.nih.gov/pubmed/19335713.
. Let's Move, "America's Move to Raise a Healthier Generation of Kids," Childhood Obesity Task Force Report, www.letsmove.gov/about.
. The Centaur Company, www.centaur.com (established 1871).
. _New York Times_ , "A free gift; To the American people. What the Federal troops are fighting to sustain," October 23, 1861 (archive), www.nytimes.com/1861/10/23/news/free-gift-american-people-what-federal-troops-are-fighting-sustain-our-soldiers.html?pagewanted=2.
. _New York Times_ , "A new method of curing certain diseases by the use of Radway's Ready Relief," December 4, 1863 (archive), www.nytimes.com/1863/12/04/news/a-new-method-of-curing-certain-diseases-by-the-use-of-radway-s-ready-relief.html.
. John Knowles Paine, "Radway's Ready Relief," Music Library of the Harvard College Library, January 31, 1960, books.google.com/books?id=HGERAAAAYAAJ&pg=PP11#v=onepage&q&f=false.
. Walsh, "I'm still standing, RKO Bushwick," _Forgotten NY_ , www.forgotten-ny.com/STREET%20SCENES/RKO%20Bushwick/RKO.html.
. Robert Pozarycki, "Elmhurst Park Grows: Working to Replace Tanks with Greenspace," _Times Newsweekly Online_ , August 19, 2010, www.timesnewsweekly.com/news/2010-08-19/Local_News/ELMHURST________PK_GROWS.html.
. See Blustein ad: www.14to42.net/images/blustein-1920.jpg.
. Grutchfield, _New York City Signs: 14 th to 42nd Street_, www.14to42.net/27street3.5.html.
. Michael Gallagher, "Ginseng, John Jacob Astor, and the Chinese trade," _JSTOR Plant Science_ , February 25, 2011, jstorplants.org/2011/02/25/ginseng-john-jacob-astor-and-the-chinese-trade.
. Dr. David Wang, "Ginseng: The herb that helped U.S. commerce," _World Huaren Federation_ , www.huaren.org/Text/1204731076687-9411/pC/1204726261750-6644.
. Sidecar Brooklyn, www.sidecarbrooklyn.com/wall.html.
DAYS WITH ART
. "Dr. Tucker's 59 for all pain, Confrontation," Fading Ad Campaign, frankjump.com/045.html.
FOOD, SNACKS AND CANDY
. George Taylor, _Martyrs to the Revolution in the British Prison-ships in the Wallabout Bay_ (New York: W.H. Arthur & Company Stationers, 1855) books.google.com/books?id=dR1CAAAAIAAJ&dq=Wallabout&pg=PA24#v=onepage&q&f=false.
. Nicole Brydson, "Navy Yard dive not dead yet, building for sale," _Brooklyn The Borough_ , February 2009, www.brooklyntheborough.com/2009/02/navy-yard-dive-not-dead-yet.
. Better Living New Zealand, "What are other kinds of ham," www.betterliving.co.nz/content/athome/cooking/what-are-other-kinds-of-ham.aspx.
. Urban Dictionary, "gabagool," www.urbandictionary.com/define.php?term=gabagool.
. Richard Cohen, _Sweet and Low: A Family Story_ (New York: Farrar, Straus, Giroux, 2006) www.npr.org/templates/story/story.php?storyId=5418961.
. Wikipedia, "Sodium cyclamate," en.wikipedia.org/wiki/Sodium_cyclamate.
. Charles E. Ophardt, Virtual Chembook, "cyclamate," Elmhurst College, Illinois (2003), www.elmhurst.edu/~chm/vchembook/549cyclamate.html.
. Cohen, _Sweet and Low_.
. _Daily Beast_ ( _Newsweek_ ), "Five controversial food additives," www.newsweek.com/2008/03/12/five-controversial-food-additives.html.
. "An FDA guide to sweeteners on the market," NPR Books, www.npr.org/templates/story/story.php?storyId=5418961.
. United States Environmental Protection Agency, "EPA Removes Saccharin from Hazardous Substances Listing," yosemite.epa.gov/opa/admpress.nsf/d0cf6618525a9efb85257359003fb69d/ea895a11ea50a56d852577f9005e2690!OpenDocument.
. New Fulton Fish Market Cooperative at Hunts Point, Inc., "History," www.newfultonfishmarket.com/history.html.
. The City of New York Business Integrity Commission, www.nyc.gov/html/bic/html/home/home.shtml.
. Frederick R. Black, "Jamaica Bay: A History," Gateway National Recreation Area New York, New Jersey–Washington, D.C., 1981, www.nps.gov/history/history/online_books/gate/jamaica_bay_hrs.pdf.
. Report of the Commissioner of Agriculture for the Year 1879 (Washington, D.C.: Government Printing Office, 1880), books.google.com/books?id=XGgTAAAAYAAJ&pg=PA438&sig=oyaxBA84xc5AYVb00nH02Gq8iQw&hl=en#v=onepage&q&f=false].
. Walsh, "Dairies," _Forgotten New York_ , www.forgotten-ny.com/SIGNS/dairies/dairies.html.
. Rabbi Jeffrey A. Marx, "Historical Outline of Breakstone Brothers Dairy," Breghshtein Family Website, December 2007, www.breakstone.us/Dairyhistory.htm#_edn37.
. _Fading Ad Blog_ , "W.M. Evans Dairy Co. Inc.—3480 Fulton Street—Cypress Hills, Brooklyn," March 26, 2008, www.fadingad.com/fadingadblog/?p=1120.
. Wikipedia, "Baby Ruth," en.wikipedia.org/wiki/Baby_Ruth.
. Fading Ad Campaign, "Planter Peanuts Repro Collage," March 1999, frankjump.com/037.html.
. Walsh, "Forgotten Ads of Queens," _Forgotten New York_ , www.forgotten-ny.com/ADS/Ads%20in%20Queens/queens.html.
. _Fading Ad Blog_ , "Missing Mr. Peanut in Wilkes-Barre, PA," November 30, 2008, www.fadingad.com/fadingadblog/?p=2572.
. Denise Allabaugh, "Former Planters Building on South Main Street Being Razed for Strip Mall," _Citizen's Voice_ , August 15, 2006, www.myspace.com/nepa_ypa/blog/287250044.
. _Advertising Age_ , "Advertising history timeline," 2005, adage.com/century/timeline.
. Gold Medal Flour, "Our Heritage," www.gmflour.com/gmflour/ourheritage.aspx.
. Chicago Advertising Federation, "Greatest Ads," www.chicagoadfed.org/i4a/pages/index.cfm?pageid=3281.
. For more Gold Medal Flour ads: www.frankjump.com/goldmedal.html.
. Sojourner Farms, "The History of Pet Food," www.sojos.com/historyofpetfood.html.
. Pet Food Institute, "Pet Food History," www.petfoodinstitute.org/petfoodhistory.htm.
. _Fading Ad Blog_ , Cadet Dog Food—Maspeth, Queens—Richard Boll," June 29, 2009, www.fadingad.com/fadingadblog/?p=4167.
A COKE AND A SMOKE
. Wikipedia, "pentimento," en.wikipedia.org/wiki/Pentimento.
. _Logo Blog_ , "Coca-Cola logo," www.logoblog.org/coca_cola_logo.php.
. James A. Shaw, "'Perfect Satisfaction' from Arabia 1912–1915," Jim's Burnt Offerings, www.jimsburntofferings.com/packsmecca.html.
. Walsh, _Forgotten New York_ , www.forgotten-ny.com/ADS/newgaryads/ads.html.
. "The Disadvantages of You," the Brass Ring, Dunhill Records (1967), youtu.be/0lr_s4bGnoA.
. Smoker's History, "Philip Morris's Predecessor, Benson & Hedges," last modified on June 13, 2011, www.smokershistory.com/Benson.html.
. Wikipedia, "Altria," en.wikipedia.org/wiki/Altria.
. Randy James, "Cigarette Advertising," _Time_ magazine, June 15, 2009, www.time.com/time/magazine/article/0,9171,1905530,00.html#ixzz1SHmZhuK2.
. Ibid.
. Information Research Lab, "Smoking Deaths Worldwide," www.inforesearchlab.com/smokingdeaths.chtml.
. Mickey Stretton, et al., "Chronometry," New Masters of Flash Annual, www.frankjump.com/mastersflash1.pdf.
BREWERIANA
. Advertising Beer Tray, "Liberty Beer—Tamo' Shanter Ale 'American Brewing Co., Rochester, N.Y.'" trayman.net/TrayDetail/Script/Liberty%20Beer.htm.
. _Fading Ad Blog_ , "Knopf Clothier as Seen from High Falls Mills—Genesee River—Rochester, NY," August 8, 2010, www.fadingad.com/fadingadblog/?p=6935.
. Ruth Rosenberg-Naperstack, "A Brief History of Brewing in Rochester," _Rochester History_ 54, no. 2 (Spring 1992): 9, www.rochester.lib.ny.us/~rochhist/v54_1992/v54i2.pdf.
. Ibid., 3–4.
. Ibid., 16.
. Ibid., 4.
. Wikipedia, "Rochester, New York," en.wikipedia.org/wiki/Rochester,_New_York.
. fadingad.wordpress.com/2007/06/24/sam-roberts-uk-ghost-ads-on-colossal-media.
. Jess Kidden, "Guinness in America," sites.google.com/site/jesskidden/guinnessinamerica.
CITY OF IMMIGRANTS AND THE MACK SIGN COMPANY
. Grutchfield, _New York City Signs: 14 th to 42nd Street_, "Mack Brothers Sign Co.," www.14to42.net/mack.html.
SHOES, HATS AND ASSORTED GARMENTS—WHOLESALE AND RETAIL
. Grutchfield, _New York City Signs: 14 th to 42nd Street_, "United Thread Mills," www.14to42.net/unitedthread.htm.
. _Fading Ad Blog_ , "United Thread Mills in Transition—East Houston Street, NYC 1998," March 6, 2008, www.fadingad.com/fadingadblog/?p=1042.
. Dana Rubin and Otto Luna, "Seely Shoulder Shapes," Manhattan Ghost Signs Digital Collection, ghostsignsnyc.omeka.net/items/show/47.
. Artkraft Strauss, "Senior Management," www.artkraft.com/staff.html.
. Tama Starr and Edward Heyman, _Signs & Wonders_ (New York: Doubleday, 1998), 19–21.
. New York Public Library Digital Collection, "Manhattan: Union Square (East)," digitalgallery.nypl.org/nypldigital/dgkeysearchdetail.cfm?strucID=413425&imageID=723622F.
. Grutchfield, _New York City Signs: 14 th to 42nd Street_, "Crown Coat Front Company," www.14to42.net/16street.html.
. NYPL Digital Library, digitalgallery.nypl.org/nypldigital/dgkeysearchdetail.cfm?strucID=413425&imageID=723622F.
. "New York Notes," _The Jewelers Circular_ 85, no. 2, January 17, 1923, books.google.com/books?id=4EEcAQAAMAAJ&lpg=RA7-PA99&ots=GC0bdtAxOH&dq=Mogi%2C%20Momomoi%20%26%20Company%2C&pg=RA7-PA99#v=onepage&q=Mogi,%20Momomoi%20&%20Company,&f=false.
. Wikipedia, "aleph," en.wikipedia.org/wiki/Aleph.
. Libby Tucker, "Cropsey at Camp," New York Folklore Society, _Voices_ 32 (Fall–Winter 2006), www.nyfolklore.org/pubs/voic32-3-4/gspirits.html.
. Wikipedia, "Golem," en.wikipedia.org/wiki/Golem.
. Dan Bilefsky, "Hard Times Give New Life to Prague's Golem," _New York Times_ , May 10, 2009, www.nytimes.com/2009/05/11/world/europe/11golem.html.
. Grutchfield, _New York City Signs: 14 th to 42nd Street_, "Weber & Heilbroner ad," www.14to42.net/images/weber&heilbroner1904.jpg.
. Grutchfield, _New York City Signs: 14 th to 42nd Street_, "Weber & Heilbroner," www.14to42.net/34streetw047.html.
. Mary Bellis, "History of the Elevator," inventors.about.com/library/inventors/blelevator.htm.
. Grutchfield, _New York City Signs: 14 th to 42nd Street_, "Miss Weber Millinery," www.14to42.net/22street2.html.
. University of Wisconsin Digital Collection, "All Sewn Up: Millinery, Dressmaking, Clothing and Costume," digital.library.wisc.edu/1711.dl/HumanEcol.MillineryBooks.
. Wikipedia, "Mad as a hatter," en.wikipedia.org/wiki/Mad_as_a_hatter.
. Funding Universe, "R.H. Macy & Co., Company History," www.fundinguniverse.com/company-histories/RH-Macy-amp;-Co-Inc-Company-History.html.
. Jason Maoz, "Retail Kings," _Jewish Press_ , January 25, 2006, www.jewishpress.com/printArticle.cfm?contentid=18350.
. Straus Historical Society, "Lazarus and Sara Straus Family," www.straushistoricalsociety.org/l_ss.php.
. Fund Universe, www.fundinguniverse.com/company-histories/RH-Macy-amp;-Co-Inc-Company-History.html.
. Walsh, _Forgotten New York_ , "Walking on Graham Avenue," www.forgotten-ny.com/STREET SCENES/manhattan.graham/graham.html.
. _Brooklyn Daily Eagle_ , "Berger-Aufrecht," October 20,1929, B5 fultonhistory.com/Newspaper%205/Brooklyn%20NY%20Daily%20Eagle/Brooklyn%20NY%20Daily%20Eagle%201929%20Grayscale/Brooklyn%20NY%20Daily%20Eagle%201929%20a%20Grayscale%20-%201219.pdf.
. Ibid.
MIGHT AS WELL JUMP!
. Walsh, _Forgotten New York_ , www.forgotten-ny.com/STREET SCENES/RKO Bushwick/RKO.html.
. Fading Ad Campaign, "RKO Bushwick," www.fadingad.com/bushwick.html.
. Walsh, Forgotten New York, www.forgotten-ny.com/forgottoners/coney3.html.
. Fading Ad Campaign, www.fadingad.com/dailynews.html.
. Fading Ad Campaign, www.fadingad.com/rosariodawson.html.
. Walsh, Forgotten New York, www.forgotten-ny.com/SLICES/jump/jump.html.
LAUNDRY PRODUCTS, WASHING MACHINES AND REAL ESTATE
. David W. Dunlap, "Old York; Look Close, and in This Ever-New Town You Will See Traces of the Past Peeking Through," _New York Times_ , December 12, 2000, section 14, page 1, column 5, www.nytimes.com/2000/12/10/nyregion/old-york-look-close-this-ever-new-town-you-will-see-traces-past-peeking-through.html.
. Bendix, "Bendix Spirit of Invention," www.bendixappliances.com/about.
. "Bendix Home Appliances, Inc.," patent search, Espace, worldwide.espacenet.com/publicationDetails/biblio?CC=US&NR=2165884&KC=&FT=E&locale=en_EP.
. Fading Ad Campaign, "Bendix Home Laundry," www.frankjump.com/031.html.
. _New York Times_ , "Emil Talamini," obituary, October 29, 1970, query.nytimes.com/mem/archive/pdf?res=F70B17FD3C5F1B7493CBAB178BD95F448785F9.
JEWELRY AND ACCESSORIES
. Grutchfield, _New York City Signs: 14 th to 42nd Street_, "Shield's Jewelry," www.14to42.net/31street1.5.html.
. New York First, www.newyorkfirst.com/gifts/7064.html.
. Andrew Britton, _Katherine Hepburn: Star as Feminist_ (New York: Columbia University Press: 1984), 238, books.google.com/books?id=PP4S2-VBXYoC&lpg=PA238&ots=wT9hpXCEdA&dq=Katherine%20Hepburn%20the%20Flatiron&pg=PA238#v=onepage&q&f=false.
. Wikipedia, "bobby pin," en.wikipedia.org/wiki/Bobby_pin.
. Grutchfield, _New York City Signs: 14 th to 42nd Street_, "The Griffon Cutlery Works," www.14to42.net/19street3.html.
. Randy Kennedy, "Saving Images of 'Dead Signs' on Old Walls," _New York Times_ , July 9, 1998, section B, page 2, column 4, www.nytimes.com/1998/07/09/nyregion/public-lives-saving-images-of-dead-signs-on-old-walls.html.
SAVINGS, LOANS AND FUR VAULTS
. Scripophily, "New York Bank History," www.scripophily.com/nybankhistoryg.htm.
. _Advertising Age_ , "Top Ten Slogans of the Century," adage.com/century/slogans.html.
. Jennifer Bayot, "Stephen Baker, Ad Executive, Dies at 83," obituary, _New York Times_ , September 24, 2004, www.nytimes.com/2004/09/24/obituaries/24baker.html.
. Brooklyn Genealogy Information, "J.J. Friel," www.bklyn-genealogy-info.com/Town/Eastern/G.html.
. _New York Times_ , "Joseph John Friel," obituary, May 6, 1914, query.nytimes.com/gst/abstract.html?res=FB0713FC3C5E13738DDDAF0894DD405B848DF1D3.
. _Fading Ad Blog_ , "National Cold Storage—Brooklyn Bridge Park—Lawns Closed," April 4, 2010, <http://www.fadingad.com/fadingadblog/?p=5959>.
. _Evening Citizen_ Ottawa, Ontario], "Proper Storage Protects Your Fur Coat Investment," April 11, 1944, 16, [news.google.com/newspapers?id=LfYuAAAAIBAJ&sjid=9NsFAAAAIBAJ&dq=keeping%20your%20fur%20safe&pg=3012%2C1670749.
PAINT AND HARDWARE
. George Carey-Simos, "Are Photographs Copies of the World?" University of Wales, Aberystwyth, www.aber.ac.uk/media/Students/gss99/gss9901.html.
. George Carey-Simos, "Are Photographs Copies of the World?" University of Wales, Aberystwyth, www.gcsimos.com (updated website).
. Robert Baptista, "Colorants Industry History," Colorant History, colorantshistory.org.
. _Fading Ad Blog_ , "Sunday's Feature Fade: The Despair of Port Arthur, Texas—Robert Baptista," March 9, 2008, www.fadingad.com/fadingadblog/?p=1053.
. _Fading Ad Blog_ , "Eaglo Paint Update from Robert Baptista," March 27, 2008, www.fadingad.com/fadingadblog/?p=1124.
. Christian Warren, _Brush with Death: A Social History of Lead Poisoning_ (Baltimore, MD: Johns Hopkins University Press, 2000), 23, books.google.com/books?id=bowkr3SNfLIC&pg=PA323&lpg=PA323&dq=%22Eaglo+Paint+and+Varnish+Corporation%22&source=bl&ots=ZNdF-aCNsk&sig=dLZ1-S0bez6beYoEvGpiv3S0SWY&hl=en&ei=rPIlTq-AC-aw0AGjiOjgCg&sa=X&oi=book_result&ct=result&resnum=5&ved=0CC4Q6AEwBA#v=onepage&q=%22Eaglo%20Paint%20and%20Varnish%20Corporation%22&f=false.
. Harriet A.L. Standeven, _House Paints, 1900–1960: History and Use_ (Hong Kong: Getty Publications, 2011), 93–94, books.google.com/books?id=1WviUh1u5UYC&pg=PA93&dq=%22Eaglo+Paint%22&hl=en&ei=3vAlTt__Jqio0AHYiMSvCg&sa=X&oi=book_result&ct=result&resnum=7&ved=0CGAQ6AEwBg#v=onepage&q=%22Eaglo%20Paint%22&f=false.
. Jeremiah Moss, _Jeremiah's Vanishing New York_ , "Tomb of Delphi," May 4, 2009, vanishingnewyork.blogspot.com/2009/05/tomb-of-delphi.html.
HORSE AND CARRIAGE
. Joseph Berger, "Fading Memories," _New York Times_ , November 5, 2005, query.nytimes.com/gst/fullpage.html?res=9B07E6DF143EF936A35752C1A9639C8B63&pagewanted=2.
. Broadway Cares, www.broadwaycares.org/about_us.
. Walsh, _Forgotten New York_ , "Times Square Relic," www.forgotten-ny.com/ADS/Carriage/carriage.html.
. Grutchfield, _New York City Signs: 14 th to 42nd Street_, "Livery Stable," www.14to42.net/17street.html.
. Nicholas Hirshon, "Fading Away on Brick Buildings throughout Queens, 'Ghost Signs' Advertise the Businesses and Product," _New York Daily News_ , August 24, 2010, www.nydailynews.com/ny_local/queens/2010/08/24/2010-08-24_fading_away_on_brick_buildings_throughout_queens_ghost_signs_advertise_the_busin.html#ixzz1SheQkL2b.
. Kennedy, "Saving Images of 'Dead Signs,'" _New York Times_.
FUEL, OIL, GAS AND COAL
. "Coal Heating," _Model Railroader Magazine_ , November 6, 2010, message thread, cs.trains.com/TRCCS/forums/t/181933.aspx#Diana%20Coal%20&%20Oil%20Co.
. _Fading Ad Blog_ , "Diana Coal & Oil Loses Her Top—East New York, Brooklyn," February 24, 2010, www.fadingad.com/fadingadblog/?p=5624.
. Paragon Oil, "History," paragonoilco.com/?p=about.
. von Wentzel Family Website, "Fuel Choices," www.vonwentzel.net/HVAC/FuelChoices/index.html.
. See Fading Ad Wiki, fadingad.wikispaces.com/Fading+Ad+Vocabulary.
. See Fading Ad Gallery Exhibitions Archive, www.fadingad.com/exhibit2.html.
MUSIC AND ENTERTAINMENT
. Julian Seery Gude, _Julians.name_ , "O.J. Gude and Me," www.blog.julians.name/2005/12/18/oj-gude-and-me.
. Victor-Victrola, "Timeline of the Victor Phonograph Company," www.victor-victrola.com/Timeline.htm.
. Wikipedia, "gramophone record," en.wikipedia.org/wiki/Gramophone_record.
. The Digital Journalist, "M. Rappaport's Music Store, Jamaica, Queens, August 1997," October 2004, digitaljournalist.org/issue0410/disjpg04.html.
. Dr. Michael Whitlatch, "Selwyn Theatre," Buena Vista University (Iowa), web.bvu.edu/faculty/whitlatch/42nd/selwyn.htm.
. The New 42nd Street, "Our Theatres," www.new42.org/new42/new42_theaters.html.
. Dunlap, "Old York," _New York Times_ , www.nytimes.com/2000/12/10/nyregion/old-york-look-close-this-ever-new-town-you-will-see-traces-past-peeking-through.html.
. Jay Shockley, "Apollo Theatre," Letter from the Landmarks Preservation Commission, June 28, 1983, www.neighborhoodpreservationcenter.org/db/bb_files/AT003.pdf.
. Barry Singer, _Black and Blue: The Life and Lyrics of Andy Razaf_ (New York: Schirmer, 1992), 142–43, cenhum.artsci.wustl.edu/files/cenhum/Monday_July_19_Afternoon_Session.pdf.
. The NYC Chapter of the American Guild of Organists, "St. Paul's Community Church," www.nycago.org/Organs/NYC/html/StPaulCommunity.html.
. Thaddeus Kochanny, "History: Heyday of the Player Piano," Automatic Musical Instrument Collectors' Association (Chicago Chapter), www.amica.org/Live/Instruments/Player_Pianos/Player_Piano_History.htm.
. Wikipedia, "player piano," en.wikipedia.org/wiki/Player_piano.
. Pianola, "Player Piano History and Development: Edwin Votey," www.pianola.com/pphist.htm.
. Monrovia Sound Studio, www.doctorjazz.co.uk/index.html.
. John Farrell, "Jean Lawrence Cook," Monrovia Sound Studio, March 2002, www.doctorjazz.co.uk/page11.html.
. "Roll On, Imperial," _Time_ magazine 41, no. 7, February 15, 1943, www.doctorjazz.co.uk/page11.html.
. Kochanny, "History," www.amica.org/Live/Instruments/Player_Pianos/Player_Piano_History.htm.
. Player Pianos, "Duo Art Player Piano," www.playerpianos.com/2009/04/duo-art-player-piano.html; "Steinway Duo Art Player Piano, Unusual," youtu.be/5t5kX1l3FZw.
. Prof. Alan Wallace, "Piano Roll, Rock 'n Roll," Monrovia Sound Studio, www.doctorjazz.co.uk/page11.html.
. Mark Sommer, "The Day the Music Died," _Buffalo News_ , January 3, 2009, www.buffalonews.com/incoming/article134934.ece.
. New York State Department of State Division of Corporations, entity search, "Camera Mart," appext9.dos.state.ny.us/corp_public/CORPSEARCH.ENTITY_INFORMATION?p_nameid=436597&p_corpid=374326&p_entity_name=Camera%20Mart&p_name_type=%25&p_search_type=CONTAINS&p_srch_results_page=0.
. Internet Movie Database, "Camera Mart," www.imdb.com/company/co0028623.
. _Videography Magazine_ ,"Camera Mart," November 1980, www.experimentaltvcenter.org/history/pdf/AnconaGorewitz_74.pdf.
HOTELS, SPEAKEASIES AND SAUNAS
. Grutchfield quoting _New York Times_ , February 21,1916, 11.
. Grutchfield quoting _New York Times_ , July 4, 1924, 20.
. Grutchfield, _New York City Signs: 14 th to 42nd Street_, "Hotel Irvin," www.14to42.net/quik08.html.
. Christopher Gray, "Streetscapes/The Henry Hudson Hotel, 353 West 57th Street; from Women's Clubhouse to WNET to $75 a night," _New York Times_ , January 4, 1998, www.nytimes.com/1998/01/04/realestate/streetscapes-henry-hudson-hotel-353-west-57th-street-women-s-clubhouse-wnet-75.html.
. Ibid.
. Daniel B. Scheider, "Hell's Kitchen in the Raw," _New York Times_ , August 27, 2000, www.nytimes.com/2000/08/27/nyregion/fyi-829021.html?pagewanted=print&src=pm.
. James T. Young Family Genealogy Page, "Descendants of James T. 'Pop' Young," October 27, 2000) www.genealogy.com/users/y/o/u/James-T-Young/FILE/0006page.html.
. Aviva Stampfer, "Mt. Morris Turkish Baths (Former): Popular Bathhouse and One of the Last in NYC, Founded in 1898," Place Matter, July 2010, www.placematters.net/node/1368.
. Alan Feuer, "Mount Morris Journal; a Gay Bathhouse in Harlem? Hey, It's No Secret," _New York Times_ , January 19, 2003, www.nytimes.com/2003/01/19/nyregion/mount-morris-journal-a-gay-bathhouse-in-harlem-hey-it-s-no-secret.html.
. Ibid.
. Stampfer, "Mt. Morris Turkish Baths."
. See Kristin Loomis, "AIDS: The Numbers Game," _New York Magazine_ , March 6, 1989, books.google.com/books?id=K-gCAAAAMBAJ&lpg=PA44&ots=laDh3zMoKJ&dq=Who%20was%20NYC%20health%20commissioner%20in%201988%20Stephen%20Joseph&pg=PA44#v=onepage&q&f=false
ROSARIO DAWSON AND HER UNCLE FRANK
. Stay Close PFLAG, "Rosario Dawson and Uncle Frank," stayclose.org/campaign/celebrity.asp?id=9.
. Stay Close PFLAG, "The Stay Close campaign," stayclose.org/campaign/default.asp.
PROSTHESES AND UNDERTAKERS
. "Pomeroy's Trusses," _The Proceedings of the Medical Society of the County of Kings_ (1880), dspace.sunyconnect.suny.edu/bitstream/handle/1951/43289/proc1880.index.pdf?sequence=1.
. "Pomeroy Truss Company," _The Pennsylvania Medical Journal_ 9, no. 2 (1906), books.google.com/books?id=J6JEAAAAYAAJ&pg=PA153&lpg=PA153&dq=Pomeroy+Truss+Company+Brooklyn&source=bl&ots=Jl8D83QCKe&sig=VG6F3sw8lkB5a9jXG89vUsuFUs4&hl=en&ei=IkMkTurjKojV0QGZurjPAw&sa=X&oi=book_result&ct=result&resnum=1&ved=0CDgQ6AEwAA#v=onepage&q=Pomeroy%20Truss%20Company%20Brooklyn&f=false; "Pomeroy Truss Co – Livingston Street – March 1999" (July 19, 2011) Fading Ad Blog, <http://www.fadingad.com/fadingadblog/?p=8699>.
. "Pomeroy Truss Co.," _Medical Directory of New York, New Jersey and Connecticut_ (1910), www.archive.org/stream/medicaldirectory12medi/medicaldirectory12medi_djvu.txt.
. Pomeroy Truss Company ad, National Institute of Health, archived, <http://www.ncbi.nlm.nih.gov/pmc/articles/PMC2625107/pdf/jnma00778-0002.pdf>
. "Pomeroy ad," _Bulletin of the New York Academy of Medicine_ (1941), <http://www.ncbi.nlm.nih.gov/pmc/articles/PMC1933685/pdf/bullnyacadmed00568-0002.pdf>.
. "Arthur C. Pomeroy," Green-Wood Cemetery, Find a Grave, www.findagrave.com/cgi-bin/fg.cgi?page=gr&GRid=58706599.
. _New York Times_ , "Pomeroy," obituary, May 1, 1903, query.nytimes.com/gst/abstract.html?res=F50F14FC385D16738DDDA80894DD405B838CF1D3.
. Ruth Bavetta, "Mother's Day, 1964," Good Reads, www.goodreads.com/story/show/24995-mother-s-day-1964.
. Grutchfield, _New York City Signs: 14 th to 42nd Street_, "G. Schoepfer (glass eyes) and Cottage Restaurant," www.14to42.net/32street6.html.
. "Wm. W. Pecan," The Eastern District of Brooklyn, Brooklyn Genealogy, www.bklyn-genealogy-info.com/Town/Eastern/G.html.
. "Wm. W. Pecan," Leading Manufacturers & Merchants: City of Brooklyn, Brooklyn Genealogy, www.bklyn-genealogy-info.com/Business/Progress/P/pecan473.html.
. Evergreens Cemetery, "Stories," www.theevergreenscemetery.com/stories.
. Evergreens Cemetery, "The Triangle Shirtwaist Fire," www.theevergreenscemetery.com/stories/shirtwaist-fire/the-triangle-shirtwaist-fire.
About the Author
Frank H. Jump is a New York City artist and educator. A native of Queens, New York, Jump has lived in Brooklyn with his husband, Vincenzo Aiosa, since 1989. Jump's first major photo exhibition ran at the New-York Historical Society from August to November 1998. After launching the Fading Ad Campaign website in February 1999, the debut of vintage hand-painted advertising on the Internet had a noticeable effect on popular culture, as evidenced by the subsequent proliferation of similar websites and blogs and the use of vintage advertising in television commercials, films and modern hand-painted ads. In the mid-2000s, Jump and Aiosa opened the Fading Ad Gallery in Brooklyn, where Jump's photography was on display for nearly two years, as well as having their curatorial debut with several shows featuring other HIV-positive visual artists and local Brooklyn artists of various media. Jump continues his documentation of these remnants of early advertising with the acclaimed _Fading Ad Blog_ , a daily photo blog featuring images he and Aiosa have taken of ads worldwide, as well as the work of other fellow urban archaeologists. Jump teaches instructional technology, guitar, digital photography and other interdisciplinary studies at an elementary school in Flatbush, where he also resides.
<http://www.fadingad.com/fadingadblog/>
_Visit us at_
www.historypress.net
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composer install --prefer-dist --no-interaction
# Generate application key
php artisan key:generate
# Verify environment config
cat .env
# Install dependencies
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npm run dev
# Execute PHPUnit tests
vendor/bin/phpunit
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,500 |
Luan Rafael Domingos Santana (Campo Grande, Mato Grosso do Sul, 13 de marzo de 1991), conocido como Luan Santana es un cantante y compositor brasileño de la música sertaneja y pop.
Biografía
Vida antes de la fama
Luan Rafael Domingos Santana nació el 13 de marzo de 1991 en Campo Grande, en Mato Grosso do Sul, donde vivió hasta sus ocho años de edad, durante este periodo se trasladó varias veces incluyendo las ciudades de Maringá y Manaus y Ponta Pora a causa de la transferencia de trabajo de su padre, que trabajaba en un banco. Él es el hijo de Marizete Santana y Amarildo Domingos, y tiene una hermana menor Bruna Santana.
Luan siempre disfrutaba tocando la guitarra, y aprendió de niño por el aliento de su padre, que se dio cuenta del talento del niño con el instrumento. Luan comenzó su escuela en la escuela NS Ayuda y también ha sido objeto de varias escuelas, debido las mudanzas de su familia por otra ciudad. Santana solía llevar su guitarra a la escuela y en las clases libres le gustaba tocar guitarra, así como en los recreos. Santana también solía jugar en la iglesia a la que asistió durante meses en la ciudad de Maringá.
Especificar en la historia del cantante Luan Santana comenzó es difícil. Ya a la edad de tres años en su ciudad natal de Campo Grande, en Mato Grosso do Sul, que llamó la atención de toda la familia, con los acordes de la música country melodía no paraba de cantar. Clásicos como "Cambio de vida", "Chico Mineiro" y "Cabocla Tereza" fueron interpretadas por Luan, sin error en la letra. Al darse cuenta de su talento, su padre se lo dio a la guitarra, para fomentar aún más al pequeño cantante. A partir de entonces las presentaciones adquirido un atractivo añadido, Luan cantó y "trató" rasguear algunas notas en el instrumento musical, que se ha vuelto inseparable desde allí.
Ante la insistencia de amigos y familiares, a los 14 años Luan Santana hace una fiesta en la que hizo su primera grabación. La sede fue la ciudad de Jaraguari también en Mato Grosso do Sul, ciudad natal de sus padres, y la vecina Campo Grande. El repertorio de la música principal elegido para ese día fue "Falando Serio", que hasta entonces sin precedentes y el buque insignia en sus presentaciones.
Carrera musical
2008 - 2009: Inicios de su carrera
Con un escritor aficionado, Luan Santana hizo sus shows en el 2008 que se registró como el primer CD. Pero él no estaba de acuerdo con el resultado final, y acabó por romper el CD no le gustaba la calidad de sonido. Pero un amigo, tenía una copia y terminó poniendo en Youtube que rápidamente terminó la difusión y adopción por público comenzó a pedir a la música en la radio de Mato Grosso do Sul, Goiás, Paraná y Rondônia.
El 11 de agosto de 2007 en la pequeña ciudad de Bela Vista MS, Luan Santana tomó la primera etapa. En ese momento, Luan no cantaba profesionalmente, nunca había grabado una canción en el estudio, pero fue contratado para este espectáculo por el éxito que tuvo en la radio en la región con la canción "Falando en Serio". A partir de entonces, el programa de Luan comenzó a dividirse entre los estudios y los espectáculos comenzaron a marcarse. Con una agenda llena de conciertos durante todo el año 2008.
Su primer álbum fue Tô de Cara, que vendió más de 50.000 copias en todo el Brasil. La canción más exitosa del álbum es "Meteoro", que fue el segundo álbum, con la canción de la cantante se ha convertido en uno de los artistas más sonado de la radio de Brasil, de acuerdo con Billboard Brasil. Al año siguiente, en 2009 Luan lanza su primer álbum en vivo, grabado en Campo Grande, en Mato Grosso do Sul, con una gran audiencia de 85 mil personas. Las canciones que hicieron más éxito fue vocè nao sabe o que é amor, "Sinais" y "Chocolate". De las 17 canciones presentes en el álbum, siete son parte de su álbum anterior. Este álbum fue certificado Platina por ABPD por 85.000 copias vendidas en Brasil.
2010 - 2011: En vivo en Rio de Janeiro
Su segundo álbum en vivo fue grabado en diciembre de 2010 en el HSBC Arena en Río de Janeiro y puesto en libertad en abril de 2011. El primero single fue "Adrenalina", una canción que alcanzó la primera posición en la tabla de la Billboard Brasil en 2010. La segunda fue "Química do Amor ", con la participación de Ivete Sangalo. Tercero fue "Um Beijo". En 2011 la canción "Amar Nao é Pecado" se convirtió en el tema de la novela Morde e Assopra, Rede Globo.
Em 10 de abril de 2011, Luan Santana lanzó su cuarto álbum y su segundo en vivo, titulado Luan Santana Ao Vivo no Rio. El espectáculo fue grabado en el HSBC Arena el 11 de septiembre de 2010. Incluso un año después de lanzar el álbum alcanzó la primera posición en las listas de veinte Portugués Inglés Gráfico de DVD y Albums Chart. El álbum cuenta con diecinueve canciones, siendo dieciséis inéditas y cuatro canciones son parte de su DVD. "Adrenalina", "Um Beijo", "As lembranças vao na Mala" y "Amar não É Pecado" las mejores músicas del álbum, Santana se ha asociado con la cantante español naturalizados mexicana, Belinda en la canción "Meu menino, minha menina".
2012: Quando Chega a Noite
En febrero de 2012 Luan estaba en el estudio terminando su nuevo disco, y se había tomado un descanso de su apretada agenda de conciertos por Brasil. En marzo, ' el álbum contó con canciones que fueron éxitos en varias partes de Brasil, como "Te Vivo", "Nega" y "Incondicional". Siendo el más reciente álbum de Luan, sin lugar a dudas es uno de los más hablado de la carrera de la cantante, que tuvo críticas positivas. Su actuación con las canciones eran geniales, que Luan fue nominado al Grammy Latino 2012. Además, el álbum fue el más vendido de 2012, con más de 300.000 copias.
En 2012 Santana hizo más de 200 shows en Brasil conjunto, con el más alto compromiso para el mes de junio, que fue presentado junto con Michel Teló en las fiestas de junio. Santana tuvo el mayor número de nominaciones en la 18 ª edición del Multishow. En 2010 ganó como la revelación del año. En 2011, el cantante fue ganador del "Melhores do Ano" programa llevado a cabo por Domingão Faustão la categoría de Mejor Cantante, y también será recompensado con la mejor música, "Meteoro".
2013: O Nosso Tempo é Hoje
El 24 de abril de 2013, Santana ganó junto a Roberto Carlos el premio al Mejor Cantante, realizado por Troféu Imprensa del SBT.
El 7 de julio de 2013 Luan Santana grabó su nuevo DVD en la Arena Maeda, "O Nosso Tempo é Hoje", en el estado de São Paulo. El primer sencillo es "Tudo o Que Você Quiser". Otras canciones que tuvieran éxitos fueron "Garotas Não Merecem Chorar", "Cê Topa", "Te esperando" y "Donzela", entre otras
Discografía
Estudio
2008: Tô de Cara
2012: Quando Chega a Noite
2016: 1977
En Vivo
2009: Luan Santana - Ao Vivo
2011: Luan Santana - Ao Vivo no Rio
2013: O Nosso Tempo É Hoje
2014/2015: Luan Santana Acústico
Recopilatorio
2013: As Melhores... Até Aqui
2014: Duetos
DVD
2009: Luan Santana - Ao Vivo
2011: Luan Santana - Ao Vivo no Rio
2013: O Nosso Tempo É Hoje
EP
2013: Te esperando
Singles
2009: "Tô de Cara"
2009: "Meteoro"
2010: "Você não Sabe o que É Amor"
2010: "Sinais"
2010: "Adrenalina"
2011: "Química do Amor" (con Ivete Sangalo)
2011: "Um Beijo"
2011: "Amar Não é Pecado"
2011: "Nêga"
2012: "Você de Mim Não Sai"
2012: "Incondicional"
2012: "Te Vivo"
2013: "Sogrão Caprichou"
2013: "Te Esperando"
2013: "Garotas Não Merecem Chorar"
2013: "Tudo Que Você Quiser"
2014: "Cê Topa"
2014: "Tanto Faz"
2014: "Eu não merecia isso"
2015: "Escreve aí"
2015: "Chuva de Arroz"
2016: "Cantada"
2016: "Eu, Você, o Mar e Ela"
2016: "Dia, Lugar e Hora"
2016: "E Essa Boca Aí" (con Bruninho & Davi)
2017: "Acordando o Prédio"
2017: "Não Vou Mais Chorar" (con Nego do Borel)
2018: "Sofazinho" (con Jorge & Mateus)
2018: "Vingança" (con MC Kekel)
2018: "Menina"
2018: "Próximo Amor" (con Alok)
Premios y nominaciones
Giras
Turnê Meteoro 2009-2010
Turnê Adrenalina 2011
Turnê Quando chega a noite 2012
Turnê Te esperando 2013
Turnê Te esperando 2.0 2013
Turnê O nosso tempo é hoje 2014
Referencias
Enlaces externos
Nacidos en Campo Grande (Mato Grosso del Sur)
Cantantes masculinos de Brasil
Cantantes en portugués
Cantantes de pop de Brasil
Presentadores de televisión de Brasil
Católicos de Brasil
Cantantes de Mato Grosso del Sur | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 8,137 |
When you are promoting a sale, campaign or special event, you want to catch peoples' attention, but don't want to spend all the time and money on remaking all your signs, you may wonder if there is a solution for you. Temporary exterior signs from AlphaGraphics Denver can be just what you need to promote your next event or sale. And it won't break the bank either.
Having great signs for your business is critical to getting noticed. With limited time to get the attention of people who walk or drive by your business, every moment counts, and you want to make sure your message hits home quickly. This is challenging with permanent signs and can be even more difficult with temporary signs. But sometimes short-term signs are the solution you need, especially if your event is only for a limited time.
No matter what your event is, we have options that are sure to please.
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To get started, give us a call at (303) 691-0626 or request a quote online to learn more. | {
"redpajama_set_name": "RedPajamaC4"
} | 7,966 |
{"url":"https:\/\/stats.stackexchange.com\/questions\/155889\/how-to-implement-kernel-density-estimation-in-multivariate-3d","text":"# How to implement Kernel density estimation in multivariate\/3D\n\nI have dataset like the following fromat and im trying to find out the Kernel density estimation with optimal bandwidth.\n\ndata = [[[1, 4, 3], [2, .6, 1.2], [2, 1, 1.2]],\n[[2, 0.5, 1.4], [5, .5, 0], [0, 0, 0]],\n[[1, 4, 3], [5, .5, 0], [2, .5, 1.2]],\n.........]\n\n\nbut I couldn't figure out how to approach it. also how to find the \u03a3 matrix. I tried sklearn kde for univariate kde like,\n\n# kde function\ndef kde_sklearn(x, x_grid, bandwidth):\nkde = KernelDensity(kernel='gaussian', bandwidth=bandwidth).fit(x)\nlog_pdf = kde.score_samples(x_grid[:, np.newaxis])\nreturn np.exp(log_pdf)\n\n# optimal bandwidth selection\nfrom sklearn.grid_search import GridSearchCV\ngrid = GridSearchCV(KernelDensity(), {'bandwidth': np.linspace(.1, 1.0, 30)}, cv=20)\ngrid.fit(x)\nbw = grid.best_params_\n\n# pdf using kde\npdf = kde_sklearn(x, x_grid, bw)\nax.plot(x_grid, pdf, label='bw={}'.format(bw))\nax.legend(loc='best')\nplt.show()\n\n\nCan any one help me to extend this to multivariate \/ in this case 3D data?\n\n\u2022 As I understand it, the sklearn approach does allow you to run CV on multivariate data, but you can only specify one bandwidth, so it is applied in all dimensions. You could run this on data normalised by the standard deviation in each dimension - but it will still just be a single scaling parameter for the bandwidth. I also want a more flexible approach, but haven't found a ready-made one yet. \u2013\u00a0Gabriel Jul 21 '15 at 10:26","date":"2019-10-17 22:57:31","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\": 0.47989052534103394, \"perplexity\": 2557.010639545837}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2019-43\/segments\/1570986677230.18\/warc\/CC-MAIN-20191017222820-20191018010320-00022.warc.gz\"}"} | null | null |
package net.finmath.marketdata.model.volatility.caplet.tenorconversion;
import java.time.LocalDate;
import net.finmath.exception.CalculationException;
import net.finmath.marketdata.model.AnalyticModel;
import net.finmath.marketdata.model.volatility.caplet.CapVolMarketData;
import net.finmath.marketdata.model.volatility.caplet.CapletVolBootstrapping;
import net.finmath.marketdata.model.volatility.caplet.smile.LinearSmileInterpolater;
import net.finmath.marketdata.model.volatility.caplet.smile.SmileInterpolationExtrapolationMethod;
import net.finmath.marketdata.products.Swap;
import net.finmath.rootfinder.NewtonsMethod;
import net.finmath.time.Schedule;
import net.finmath.time.ScheduleGenerator;
import net.finmath.time.ScheduleGenerator.DaycountConvention;
import net.finmath.time.ScheduleGenerator.Frequency;
import net.finmath.time.ScheduleGenerator.ShortPeriodConvention;
import net.finmath.time.businessdaycalendar.BusinessdayCalendar.DateRollConvention;
import net.finmath.time.businessdaycalendar.BusinessdayCalendarExcludingTARGETHolidays;
import net.finmath.time.businessdaycalendar.BusinessdayCalendarExcludingWeekends;
/**
* This class implements a correlation provider based on iCap market data.
* The weekly iCap delivery consists of 3M and 6M caplet volatilities.
* Using those and the formulas from the Schlenkrich paper we can calculate the correlation
* between forward rates of the form F(0,0.5*i,0.5*i+0.3) and F(0,0.5*i+0.25,0.5*(i+1)).
* Those aren't enough though to convert tenors from anything else than 3M to 6M.
*
* @TODO Calibrate a parameterization as suggested by Schlenkrich given the calculated 3M correlations and use this parameterization to extract all missing correlations.
*
* @author Daniel Willhalm
*/
public class CorrelationProviderTenorBasis implements CorrelationProvider {
private final CapVolMarketData iCap3MCapVolMarketData;
private final CapVolMarketData iCap6MCapVolMarketData;
private CapletVolBootstrapping iCap3MCapletVolBootrapper;
private CapletVolBootstrapping iCap6MCapletVolBootrapper;
private double[][] iCap3MCapletVolMatrix;
private double[][] iCap6MCapletVolMatrix;
private double[][] correlationMatrix1M;
private double[][] correlationMatrix3M;
private double[][] correlationMatrix6M;
public CorrelationProviderTenorBasis(final CapVolMarketData iCap3MCapVolMarketData, final CapVolMarketData iCap6MCapVolMarketData) {
this.iCap3MCapVolMarketData = iCap3MCapVolMarketData;
this.iCap6MCapVolMarketData = iCap6MCapVolMarketData;
}
public double get6MCorrelation(final double firstForwardFixingTimeVectorInYears, final double secondForwardFixingTimeVectorInYears, final AnalyticModel analyticModel) throws CalculationException {
//todo: Kalibriere auf Basis der 3M Correlationsdaten die zur Verf�gung stehen eine Parametrisierung der Korrelationen wie sie im Schlenkrich zu finden ist.
return 0.0;
}
public double get3MCorrelation(final double firstForwardFixingTimeVectorInYears, final double secondForwardFixingTimeVectorInYears, final AnalyticModel analyticModel) throws CalculationException {
//Auf Basis der 3M und 6M Icap daten wird mit der Formel aus Schlenkrich die Korrelation bestimmt. Achtung f�r Umrechung auf 12M br�uchte man mehr 3M Korrelationen die man sich �ber die Parametrisierung holen m�sste.
final double[] capletFixingTimeVectorInYears = new double[iCap3MCapVolMarketData.getMaxExpiryInMonths()/iCap3MCapVolMarketData.getUnderlyingTenorInMonths()-1];
for (int i = 0; i < capletFixingTimeVectorInYears.length; i++) {
capletFixingTimeVectorInYears[i] = (i+1)*iCap3MCapVolMarketData.getUnderlyingTenorInMonths()/12.0;
}
int rowIndex = 0;
int columnIndex = 0;
double distanceToFixingTimeRow = Double.MAX_VALUE;
double distanceToFixingTimeColumn = Double.MAX_VALUE;
for (int i = 0; i < capletFixingTimeVectorInYears.length; i++) {
if (Math.abs(firstForwardFixingTimeVectorInYears - capletFixingTimeVectorInYears[i]) < distanceToFixingTimeRow) {
rowIndex = i;
distanceToFixingTimeRow = Math.abs(firstForwardFixingTimeVectorInYears - capletFixingTimeVectorInYears[i]);
}
if (Math.abs(secondForwardFixingTimeVectorInYears - capletFixingTimeVectorInYears[i]) < distanceToFixingTimeColumn) {
columnIndex = i;
distanceToFixingTimeColumn = Math.abs(secondForwardFixingTimeVectorInYears - capletFixingTimeVectorInYears[i]);
}
}
//lazy initialization
if (correlationMatrix3M != null) {
return correlationMatrix3M[rowIndex][columnIndex];
}
iCap3MCapletVolBootrapper = new CapletVolBootstrapping(iCap3MCapVolMarketData, analyticModel);
iCap6MCapletVolBootrapper = new CapletVolBootstrapping(iCap6MCapVolMarketData, analyticModel);
if(iCap3MCapletVolMatrix == null || iCap6MCapletVolMatrix == null) {
iCap3MCapletVolMatrix = iCap3MCapletVolBootrapper.getCapletVolMatrix();
iCap6MCapletVolMatrix = iCap6MCapletVolBootrapper.getCapletVolMatrix();
}
correlationMatrix3M = new double[iCap3MCapVolMarketData.getMaxExpiryInMonths()/iCap3MCapVolMarketData.getUnderlyingTenorInMonths()-1][iCap3MCapVolMarketData.getMaxExpiryInMonths()/iCap3MCapVolMarketData.getUnderlyingTenorInMonths()-1];
for (int i = 0; i < correlationMatrix3M.length; i++) {
correlationMatrix3M[i][i] = 1;
}
//todo: give interpolation method as an argument
final SmileInterpolationExtrapolationMethod smileInterExtrapolater3M = new LinearSmileInterpolater(iCap3MCapletVolMatrix, iCap3MCapVolMarketData.getStrikeVector());
final SmileInterpolationExtrapolationMethod smileInterExtrapolater6M = new LinearSmileInterpolater(iCap6MCapletVolMatrix, iCap6MCapVolMarketData.getStrikeVector());
final double[][] K = new double[correlationMatrix3M.length/2-(1-correlationMatrix3M.length%2)][2];
final double[][] nu = new double[correlationMatrix3M.length/2-(1-correlationMatrix3M.length%2)][2];
final double[] sumNu = new double[correlationMatrix3M.length/2-(1-correlationMatrix3M.length%2)];
for (int i = 0; i < correlationMatrix3M.length/2-(1-correlationMatrix3M.length%2); i++) {
final LocalDate localDate = iCap3MCapletVolBootrapper.getDiscountCurve().getReferenceDate();
final Schedule schedule3M = ScheduleGenerator.createScheduleFromConventions(
localDate,
localDate.plusMonths(2*(i+1)*iCap3MCapVolMarketData.getUnderlyingTenorInMonths()),
localDate.plusMonths(2*(i+2)*iCap3MCapVolMarketData.getUnderlyingTenorInMonths()),
Frequency.QUARTERLY, DaycountConvention.ACT_365, ShortPeriodConvention.FIRST, DateRollConvention.MODIFIED_FOLLOWING,
new BusinessdayCalendarExcludingTARGETHolidays(new BusinessdayCalendarExcludingWeekends()), -2, 0);
final Schedule schedule6M = ScheduleGenerator.createScheduleFromConventions(
localDate,
localDate.plusMonths(2*(i+1)*iCap3MCapVolMarketData.getUnderlyingTenorInMonths()),
localDate.plusMonths(2*(i+2)*iCap3MCapVolMarketData.getUnderlyingTenorInMonths()),
Frequency.SEMIANNUAL, DaycountConvention.ACT_365, ShortPeriodConvention.FIRST, DateRollConvention.MODIFIED_FOLLOWING,
new BusinessdayCalendarExcludingTARGETHolidays(new BusinessdayCalendarExcludingWeekends()), -2, 0);
//tests have shown that the correlations that we get close to the strike are the most meaningful
//The way the strike is approximated isn't perfect. That's why the first correlation in the test class is > 1
final double strikeATM = Swap.getForwardSwapRate(schedule3M, schedule3M, iCap3MCapletVolBootrapper.getForwardCurve(), iCap3MCapletVolBootrapper.getParsedModel());
final double[] shortTenorTau = new double[nu[0].length];
for (int k = 0; k < shortTenorTau.length; k++) {
shortTenorTau[k] = schedule3M.getPeriodLength(k);
}
final double longTenorTau = schedule6M.getPeriodLength(0);
for (int k = 0; k < nu[0].length; k++) {
nu[i][k] = shortTenorTau[k]
* (1.0 + longTenorTau * iCap6MCapletVolBootrapper.getForwardCurve().getForward(
iCap6MCapletVolBootrapper.getParsedModel(), capletFixingTimeVectorInYears[i * 2 + 1]))
/ (longTenorTau * (1.0 + shortTenorTau[k] * iCap3MCapletVolBootrapper.getForwardCurve()
.getForward(iCap3MCapletVolBootrapper.getParsedModel(),
capletFixingTimeVectorInYears[i * 2 + 1] + schedule3M.getPeriodStart(k))));
sumNu[i] += nu[i][k];
}
for (int k = 0; k < K[0].length; k++) {
K[i][k] = (strikeATM - (iCap6MCapletVolBootrapper.getForwardCurve().getForward(
iCap6MCapletVolBootrapper.getParsedModel(), capletFixingTimeVectorInYears[i * 2 + 1])
- sumNu[i] * iCap3MCapletVolBootrapper.getForwardCurve()
.getForward(iCap3MCapletVolBootrapper.getParsedModel(),
capletFixingTimeVectorInYears[i * 2 + 1]
+ k * iCap3MCapVolMarketData.getUnderlyingTenorInMonths() / 12.0)))
/ sumNu[i];
}
//newtons method is not really necessary since we search for the root of a linear function but due to convenience we use one iteration of newtons method to find it
final double guess = 1.0;
final NewtonsMethod newtonsMethod = new NewtonsMethod(guess);
double derivativeSumToMinimize = 0.0;
double secondDerivativeSumToMinimize = 0.0;
double rightHandSideSum = 0.0;
double derivativeRightHandSideSum = 0.0;
for (int k1 = 0; k1 < 2; k1++) {
for (int k2 = 0; k2 < 2; k2++) {
if (k1 == k2) {
rightHandSideSum += nu[i][k1]*nu[i][k2]*smileInterExtrapolater3M.calculateInterpolatedExtrapolatedSmileVolatility(K[i][k1], i*2+1+k1)*smileInterExtrapolater3M.calculateInterpolatedExtrapolatedSmileVolatility(K[i][k2], i*2+1+k2);
}
else {
rightHandSideSum += nu[i][k1]*nu[i][k2]*guess*smileInterExtrapolater3M.calculateInterpolatedExtrapolatedSmileVolatility(K[i][k1], i*2+1+k1)*smileInterExtrapolater3M.calculateInterpolatedExtrapolatedSmileVolatility(K[i][k2], i*2+1+k2);
derivativeRightHandSideSum += nu[i][k1]*nu[i][k2]*smileInterExtrapolater3M.calculateInterpolatedExtrapolatedSmileVolatility(K[i][k1], i*2+1+k1)*smileInterExtrapolater3M.calculateInterpolatedExtrapolatedSmileVolatility(K[i][k2], i*2+1+k2);
}
}
}
derivativeSumToMinimize += 2*(smileInterExtrapolater6M.calculateInterpolatedExtrapolatedSmileVolatility(strikeATM, i)*smileInterExtrapolater6M.calculateInterpolatedExtrapolatedSmileVolatility(strikeATM, i)-rightHandSideSum)*(-derivativeRightHandSideSum);
secondDerivativeSumToMinimize += 2*derivativeRightHandSideSum*derivativeRightHandSideSum;
newtonsMethod.setValueAndDerivative(derivativeSumToMinimize, secondDerivativeSumToMinimize);
correlationMatrix3M[i*2+1][i*2+2] = newtonsMethod.getNextPoint();
correlationMatrix3M[i*2+2][i*2+1] = newtonsMethod.getNextPoint();
}
return correlationMatrix3M[rowIndex][columnIndex];
}
double get1MCorrelation(final double firstForwardFixingTimeVectorInYears, final double secondForwardFixingTimeVectorInYears, final AnalyticModel analyticModel) {
return 0;
}
public double[][] getiCap3MCapletVolMatrix() {
return iCap3MCapletVolMatrix;
}
public double[][] getiCap6MCapletVolMatrix() {
return iCap6MCapletVolMatrix;
}
public double[][] getCorrelationMatrix3M() {
return correlationMatrix3M;
}
public double[][] getCorrelationMatrix6M() {
return correlationMatrix6M;
}
public CapletVolBootstrapping getICap3MCapletVolBootrapper() {
return iCap3MCapletVolBootrapper;
}
public CapletVolBootstrapping getICap6MCapletVolBootrapper() {
return iCap6MCapletVolBootrapper;
}
@Override
public double getCorrelation(final int oldTenor, final double firstForwardFixingTimeVectorInYears, final double secondForwardFixingTimeVectorInYears, final AnalyticModel analyticModel, final String indexForDiscount) throws CalculationException {
if(oldTenor == 6) {
return get6MCorrelation(firstForwardFixingTimeVectorInYears, secondForwardFixingTimeVectorInYears, analyticModel);
}
if(oldTenor == 3) {
return get3MCorrelation(firstForwardFixingTimeVectorInYears, secondForwardFixingTimeVectorInYears, analyticModel);
}
if(oldTenor == 1) {
return get1MCorrelation(firstForwardFixingTimeVectorInYears, secondForwardFixingTimeVectorInYears, analyticModel);
}
throw new IllegalArgumentException("Wrong Tenor for the iCap correlation provider");
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 7,108 |
Alexandre Stassievitch (born 20 September 1950) is a French former professional footballer who played as a defender. He played club football for Lens, Poissy, Montluçon, Nœux-les-Mines, and Saint-Omer, and represented France at the 1976 Summer Olympics.
Club career
Born in Libercourt, Stassievitch began playing football as a center forward for local side Ostricourt. In 1972, he signed for RC Lens and began playing as a defender for the second team. He made his Ligue 1 debut for Lens in 1974, making several more appearances during his time with the club. He was on the bench as Lens lost the Coupe de France 1974-75 final.
Stassievitch spent the following seasons in Ligue 2, one with AS Poissy and two with ÉDS Montluçon. In 1980, Gérard Houllier recruited him for US Nœux-les-Mines. In his two seasons with the club, they entered the promotion playoffs and reached the 1/16-finals of the Coupe de France.
He would spend the next five years playing amateur football for US Saint-Omer before retiring in 1987.
International career
Stassievitch captained France at the 1976 Summer Olympics in Montreal, where the team reached the quarterfinals.
References
External links
Biography at Sports-reference.com
Profile at Afterfoot.fr
1950 births
Living people
French footballers
Footballers at the 1976 Summer Olympics
Olympic footballers of France
RC Lens players
AS Poissy players
Montluçon Football players
Ligue 1 players
Ligue 2 players
Association football defenders
Mediterranean Games silver medalists for France
Mediterranean Games medalists in football
Competitors at the 1975 Mediterranean Games
US Saint-Omer players
US Nœux-les-Mines players | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 9,554 |
\section{Introduction}
Relationships between information theory and
statistical physics have been widely recognized in the last few decades,
from a wide spectrum of aspects. These include conceptual aspects, of
parallelisms and analogies
between theoretical principles in
the two disciplines, as well as technical aspects, of
mapping between mathematical formalisms in both fields and borrowing
analysis techniques from one field to the other. One example
of such a mapping, is between the paradigm of random codes for channel coding
and certain models of magnetic materials,
most notably, Ising models and spin glass
models (see, e.g.,
\cite{Baierlein71},\cite{Merhav10},\cite{MM09},\cite{Nishimori01}, and many
references therein).
Today, it is quite widely believed
that research in the intersection between information theory and statistical
physics may have the potential of fertilizing both disciplines.
This paper is more related to the former aspect mentioned above, namely,
the relationships between the two areas in the conceptual level.
However, it has also ingredients from the second aspect.
In particular, let us consider two
questions in the two fields, which at first glance, may seem completely
unrelated, but will nevertheless turn out later to be very related. These are
special cases of more general questions that we study later in this paper.
The first is a simple question in statistical mechanics, and it is about
a certain extension of a model described in \cite[page 134, Problem 13]{Kubo61}:
Consider a one--dimensional chain of $n$ connected elements (e.g., monomers or
whatever basic units that form a
polymer chain), arranged along a straight line (see Fig.\ \ref{chain}),
and residing in thermal equilibrium at
fixed temperature $T_0$. The are two types of elements, which will be referred
to as type `0' and type `1'.
The number of elements of each type $x$ (with $x$ being either `0' or `1') is given by $n(x)=nP(x)$,
where $P(0)+P(1)=1$ (and so, $n(0)+n(1)=n$).
Each element of each type may be in one of two different states, labeled by $\hx$, where
$\hx$ also takes on the values `0' and `1'.
The length and the internal energy of an element of type $x$ at state $\hx$
are given by $d(x,\hx)$ and
$\epsilon(\hx)$ (independently of $x$), respectively. A contracting force
$\lambda < 0$ is applied to
one edge of the chain while the other edge is fixed. What is the minimum
amount of mechanical work $W$ that must be carried out by this force, along an isothermal
process at temperature $T_0$, in order
to shrink the chain from its original
length $nD_0$ (when no force was applied) into a shorter length,
$nD$, where $D <
D_0$ is a given constant?
\begin{figure}[ht]
\hspace*{1cm}\input{chain3.pstex_t}
\caption{A chain with various types of elements and various lengths.}
\label{chain}
\end{figure}
The second question is in information theory. In particular, it is
the classical problem of
lossy source coding, and some of the notation here will deliberately be chosen
to be the same as
before: An information source emits a string of $n$ independent
symbols, $x_1,x_2,\ldots,x_n$, where each $x_i$ may either be `0' or
`1', with probabilities $P(0)$ and $P(1)$, respectively.
A lossy source encoder maps the source string,
$(x_1,\ldots,x_n)$, into a shorter (compressed) representation
of average length $nR$, where $R$ is the
coding rate (compression ratio), and the compatible decoder maps
this compressed representation into a reproduction string,
$\hbx=(\hx_1,\ldots,\hx_n)$, where each $\hx_i$ is again, either `0' or `1'.
The fidelity of the reproduction is measured in terms of a certain distortion
(or distance) function, $d(\bx,\hbx)=\sum_{i=1}^n d(x_i,\hx_i)$, which should
be as small as possible, so that $\hbx$ would be as `close' as possible to
$\bx$.\footnote{For example, in lossless compression, $\hbx$ is required to be
strictly identical to $\hbx$, in which case $d(\bx,\hbx)=0$. However, in some
applications, one might be willing to trade off between compression and fidelity,
i.e., slightly increase the distortion at the benefit of reducing the
compression ratio $R$.}
In the limit of large $n$, what is the minimum coding rate $R=R(D)$
for which there exists an encoder and decoder such that the average distortion,
$\left<d(\bx,\hbx)\right>$, would not exceed $nD$?
It turns out, as we shall see in the sequel,
that the two questions have intimately related answers. In particular,
the minimum amount of work $W$, in the first question, is related
to $R(D)$ (a.k.a.\ the {\it rate--distortion function}), of the second
question, according to
\begin{equation}
W=nkT_0R(D),
\end{equation}
provided that the Hamiltonian, $\epsilon(\hx)$,
in the former problem, is given by
\begin{equation}
\epsilon(\hx)=-kT_0\ln Q(\hx),
\end{equation}
where $k$ is Boltzmann's constant, and $Q(\hx)$ is the
relative frequency (or the empirical probability) of the symbol $\hx\in\{0,1\}$ in the
reproduction sequence $\hbx$, pertaining to an optimum lossy encoder--decoder
with average per--symbol distortion $D$ (for large $n$).
\begin{figure}[ht]
\hspace*{1cm}\input{rd.pstex_t}
\caption{Emulation of the rate--distortion function $R(D)$ by a physical system.}
\label{rd}
\end{figure}
Moreover, the minimum amount of work $W$, which is simply the free energy difference
between the final equilibrium state and the initial state of the chain, is achieved by a
reversible process, where the
compressing force $\lambda$ grows very slowly from zero, at
the beginning of the process, up to a final level of
\begin{equation}
\lambda = kT_0 R'(D),
\end{equation}
where $R'(D)$ is the derivative of $R(D)$ (see Fig.\ \ref{rd}).
Thus, physical compression is strongly related to data compression, and the
fundamental physical limit on the minimum required work is intimately
connected to the fundamental information--theoretic limit of the minimum
required coding rate.
This link between the the physical model and the lossy source coding problem
is obtained from a large deviations perspective.
The exact details will be seen later on, but in a nutshell, the idea is this:
On the one hand, it is
possible to represent $R(D)$ as the large deviations
rate function of a certain rare event, but on the other hand, this large
deviations rate function, involves the use of the Legendre transform, which
is a pivotal concept in thermodynamics and statistical mechanics. Moreover,
since this Legendre transform is applied to the (logarithm of the) moment generating
function (of the distortion variable), which in turn, has the form a partition
function, this paves the way to the above described analogy. The Legendre
transform is associated with the optimization across a certain parameter,
which can be interpreted as either inverse temperature (as was done, for
example, in
\cite{McAllester},\cite{Merhav08},\cite{Rose94},\cite{Shinzato}) or as a
(generalized) force, as proposed here. The interpretation of this parameter
as force is somewhat more solid, for reasons that will become apparent
later.
One application of this analogy, between the two models, is a
parametric representation of the rate--distortion function $R(D)$ as
an integral of the minimum mean square error (MMSE) in a certain Bayesian estimation
problem, which is obtained in analogy to a certain variant of
the fluctuation--dissipation theorem. This representation opens the door
for derivation of upper and lower bounds on the rate--distortion function via bounds on the MMSE,
as was demonstrated in a companion paper \cite{Merhav10b}.
Another possible
application is demonstrated in the present
paper: When the setup is extended to allow information sources with memory
(non i.i.d.\ processes), then the analogous physical model consists of
interactions between the various particles. When these interactions are
sufficiently strong (and with high enough dimension), then the system exhibits
phase transitions. In the information--theoretic domain, these phase
transitions mean irregularities and threshold effects in the behavior of the
relevant information--theoretic function, in this case, the rate--distortion
function. Thus, analysis tools and physical insights are `imported' from
statistical mechanics to information theory.
A particular model example for this is worked out in Section 4.
The outline of the paper is as follows. In Section 2, we provide some
relevant background in information theory, which may safely be skipped
by readers that possess this background. In Section 3, we establish the
analogy between lossy source coding and the above described physical model,
and discuss it in detail. In Section 4, we demonstrate the analysis for a
system with memory, as explained in the previous paragraph.
Finally, in Section 5 we summarize and conclude.
\section{Information--theoretic background}
\label{bkgd}
\subsection{General overview}
One of the most elementary roles of Information Theory is to provide fundamental
performance limits pertaining to certain tasks of information processing,
such as data compression, error--correction coding, encryption, data hiding, prediction,
and detection/estimation
of signals and/or parameters from noisy observations, just to name a few (see
e.g., \cite{CT06}).
In this paper, our focus is on the first item mentioned --
data compression, a.k.a.\ {\it source coding},
where the mission is to convert a piece of information
(say, a long file), henceforth referred to as the {\it source data}, into a shorter
(normally, binary) representation, which enables either perfect recovery of
the original information, as in the case of {\it lossless compression}, or non--perfect
recovery, where the level of reconstruction
errors (or distortion) should remain within pre-specified
limits, which is the case of {\it lossy data compression}.
Lossless compression is
possible whenever the statistical characterization of the
source data inherently exhibits some level of {\it redundancy} that
can be exploited by the compression scheme, for example, a binary file,
where the relative frequency of 1's is much larger than that of the 0's,
or when there is a strong statistical dependence between consecutive bits.
These types of redundancy exist, more
often than not, in real--life situations. If some level of errors and
distortion are allowed, as in the lossy case, then compression can be made
even more aggressive.
The choice between lossless and lossy data compression depends on the
application and the type of data to be compressed. For example, when it comes
to sensitive information, like bank account information, or a piece of important text,
then one may not tolerate any reconstruction errors at all. On the other hand, images and
audio/video files, may suffer some degree of harmless reconstruction errors (which may
be unnoticeable to the human eye or ear, if designed cleverly) and thus allow
stronger compression, which would be very welcome, since images and video
files are typically enormously large. The {\it compression ratio}, or the {\it
coding rate}, denoted $R$, is
defined as the (average) ratio between the length of the compressed file (in
bits) and the length of the original file.
The basic role of Information Theory, in the context of lossless/lossy
source coding, is to characterize the fundamental limits of compression: For a
given statistical characterization of the source data, normally modeled by a certain
random process, what is the minimum achievable compression ratio $R$ as a function
of the allowed average distortion, denoted $D$, which is defined with respect
to some distortion function that measures
the degree of proximity between the source data and the recovered data.
The characterization of this minimum achievable $R$ for a given $D$,
denoted as a function $R(D)$, is called the {\it rate--distortion function} of
the source with respect to the prescribed distortion function.
For the lossless case, of course, $D=0$.
Another important question is how, in principle, one may achieve (or at least
approach) this fundamental limit of optimum performance, $R(D)$? In this context, there is a big
gap between lossy compression and lossless compression. While for the lossless case,
there are many practical algorithms (most notably, adaptive Huffman codes, Lempel--Ziv
codes, arithmetic codes, and more), in the lossy case, there is unfortunately,
no constructive practical scheme whose performance comes close to $R(D)$.
\subsection{The rate--distortion function}
The simplest non--trivial model of an information source is that of an i.i.d.\
process, a.k.a.\ a {\it discrete memoryless source} (DMS), where the source
symbols, $x_1,x_2,\ldots,x_n$, take on values in a common finite set (alphabet)
$\calX$, they are
statistically independent, and they are all drawn
from the same probability mass function, denoted by $P=\{P(x),~x\in\calX\}$.
The source string $\bx=(x_1,\ldots,x_n)$ is compressed into a binary
representation\footnote{
It should be noted that in the case of
variable--length coding, where $\ell=\ell(\bx)$ depends on $\bx$, the code
should be designed such that the running bit-stream (formed by
concatenating compressed strings corresponding
to successive $n$--blocks from the source) could be uniquely parsed in the correct manner
and then decoded.
To this end, the lengths $\{\ell(\bx)\}$ must be collectively large enough so as to satisfy the
Kraft inequality. The details can be found, for example, in \cite{CT06}.}
of length $\ell=\ell(\bx)$ (which may or may not depend on $\bx$), whose
average is $\left<\ell(\bx)\right>$, and the compression ratio is
$R=\left<\ell(\bx)\right>/n$.
In the decoding (or decompression) process, the
compressed representation is mapped into a reproduction string
$\hbx=(\hx_1,\hx_2,\ldots,\hx_n)$, where each $\hx_i$, $i=1,2,\ldots,n$, takes
on values in the {\it reproduction alphabet} $\hat{\calX}$ (which is typically
either equal to $\calX$ or to a subset of $\calX$, but this is not necessary).
The fidelity of the reconstruction string $\hbx$ relative to the original source
string $\hx$ is
measured by a certain distortion function
$d_n(\bx,\hbx)$, where the function $d_n$ is defined
additively as $d_n(\bx,\hbx)=\sum_{i=1}^n d(x_i,\hx_i)$, $d(\cdot, \cdot)$
being a function from $\calX\times\hat{\calX}$ to the non--negative reals.
The average distortion per symbol is $D=\left<d_n(\bx,\hbx)\right>/n$.
As said, $R(D)$ is defined (in general) as the infimum of all rates $R$ for which there exist a
sufficiently large $n$ and an encoder--decoder pair for $n$--blocks,
such that the average distortion per symbol would not exceed $D$.
In the case of a DMS $P$, an elementary coding theorem of Information Theory
asserts that $R(D)$ admits the following formula
\begin{equation}
\label{rdc}
R(D)=\min I(x;\hat{x}),
\end{equation}
where $x$ is a random variable that represents a single source symbol (i.e., it is
governed by $P$),
$I(x;\hx)$ is the mutual information between $x$ and $\hx$, i.e.,
\begin{equation}
I(x;\hx)=\left<\log\frac{W(\hx|x)}{Q(\hx)}\right>\equiv
\sum_{x\in\calX}\sum_{\hx\in\hat{\calX}}P(x)P(\hx|x)\log\frac{W(\hx|x)}{Q(\hx)},
\end{equation}
$Q(\hx)=\sum_{x\in\calX}P(x)W(\hx|x)$ being the marginal distribution
of $\hx$, which is associated with a given conditional distribution
$\{W(\hx|x)\}$,
and the minimum is over all these conditional probability
distributions for which
\begin{equation}
\left<d(x,\hx)\right>\equiv
\sum_{x\in\calX}\sum_{\hx\in\hat{\calX}}P(x)W(\hx|x)d(x,\hx) \le D.
\end{equation}
For $D=0$, $\hx$ must be equal to $x$ with probability one (unless
$d(x,\hx)=0$ also for some $\hx\ne x$), and then
\begin{equation}
R(0)=I(x;x)=-\left<\log P(x)\right>\equiv H,
\end{equation}
the Shannon entropy of $x$, as expected.
As mentioned earlier, there are concrete compression algorithms
that come close to $H$ for large $n$.
For $D > 0$, however, the proof of achievability of $R(D)$ is
non--constructive.
\subsection{Random coding}
The idea for proving the existence of a sequence of codes
(indexed by $n$) whose performance approach $R(D)$ as $n\to\infty$, is based on the notion of
{\it random coding}: If we can define, for each $n$, an ensemble of codes of (fixed) rate
$R$, for which the average per--symbol distortion (across both the randomness of $\hx$ and
the randomness of the code) is asymptotically less than or equal to $D$,
then there must exist at least one
sequence of codes in that ensemble, with this property. The idea of random
coding is useful because if the ensemble of codes is chosen wisely, the average ensemble
performance is surprisingly easy to derive (in contrast to the performance of
a specific code) and proven to meet $R(D)$ in the limit of large $n$.
For a given $n$, consider the following ensemble of codes: Let $W^*$ denote the
conditional probability matrix that achieves $R(D)$ and let $Q^*$ denote the
corresponding marginal distribution of $\hx$. Consider now a random selection
of $M=e^{nR}$ reproduction strings, $\hbx_1,\hbx_2,\ldots,\hbx_M$, each of
length $n$, where each $\hbx_i=(\hx_{i,1},\hx_{i,2},\ldots,\hx_{i,n})$,
$i=1,2,\ldots,M$,
is drawn independently (of all other reproduction
strings), according to
\begin{equation}
Q^*(\hbx_i)=Q^*(\hx_{i,1})Q^*(\hx_{i,2})\cdot\cdot\cdot Q^*(\hx_{i,n}).
\end{equation}
This randomly chosen code is generated only once and then revealed to the
decoder. Upon observing an incoming source string $\bx$, the encoder seeks the
first reproduction string $\hbx_i$ that achieves $d_n(\bx,\hbx_i)\le nD$, and
then transmits its index $i$ using $\log_2 M=nR\log_2e$ bits, or equivalently,
$\ln M=nR$ {\it nats}.\footnote{While $\log_2M$ has the obvious interpretation of the
number of bits needed to specify a number between $1$ and $M$, the natural
base logarithm is often mathematically more convenient to work with. The
quantity $\ln M$ can also
be thought of as the description length, but in different units, called nats, rather
than bits, where the conversion is according to $1$ nat $=\log_2 e$ bits.} If
no such codeword exists, which is referred to as the
event of {\it encoding failure}, the encoder sends an arbitrary sequence of
$nR$ nats, say, the all--zero sequence.
The decoder receives the index $i$ and simply outputs the corresponding reproduction
string $\hbx_i$.
Obviously, the per--symbol distortion would be less than $D$
whenever the encoder does not fail, and so, the main point of the proof is to
show that the probability of failure (across the
randomness of $\bx$ and the ensemble of codes)
is vanishingly small for large $n$,
provided that $R$ is slightly larger than (but
can be arbitrarily close to) $R(D)$, i.e., $R=R(D)+\epsilon$ for
an arbitrarily small $\epsilon > 0$. The idea is that for any source string
that is {\it typical} to $P$ (i.e., the empirical relative frequency of
each symbol in $\bx$ is close to its probability), one can show (see, e.g.,
\cite{CT06}) that the
probability that a single, randomly selected reproduction string $\hbx$
would satisfy $d_n(\bx,\hbx)\le nD$, decays exponentially as $\exp[-nR(D)]$.
Thus, the above described random selection of the entire codebook together
with the encoding operation, are
equivalent to conducting $M$ independent trials in the quest for having at
least one $i$ for which
$d_n(\bx,\hbx_i)\le nD$, $i=1,2,\ldots,M$. If $M = e^{n[R(D)+\epsilon]}$, the
number of trials is much larger (by a factor of $e^{n\epsilon}$)
than the reciprocal of the probability of
a single `success', $\exp[-nR(D)]$, and so, the probability of obtaining at
least one such success (which is case where the encoder succeeds) tends to unity as
$n\to\infty$. We took the liberty of assuming that source string is typical to
$P$ because the probability of seeing a non--typical
string is vanishingly small.
\subsection{The Large deviations perspective}
From the foregoing discussion,
we see that
$R(D)$ has the additional interpretation of the exponential rate
of the probability of the event
$d_n(\bx,\hbx)\le nD$, where $\bx$ is a given string typical to $P$ and $\hbx$
is randomly drawn i.i.d.\ under $Q^*$. Consider the following chain of
equalities and inequalities for bounding the probability of this event from above.
Letting $s$ be a parameter taking an arbitrary non--positive value, we have:
\begin{eqnarray}
\mbox{Pr}\left\{d_n(\bx,\hbx)\le nD\right\}&=&
\mbox{Pr}\left\{\sum_{i=1}^n d(x_i,\hx_i)\le nD\right\}\nonumber\\
&\le&\left<\exp\left\{s\left[\sum_{i=1}^n
d(x_i,\hx_i)-nD\right]\right\}\right>\nonumber\\
&=&e^{-nsD}\left<\prod_{i=1}^n e^{sd(x_i,\hx_i)}\right>\nonumber\\
&=&e^{-nsD}\prod_{i=1}^n\left<e^{sd(x_i,\hx_i)}\right>\nonumber\\
&=&e^{-nsD}\prod_{x\in\calX}\prod_{i:~x_i=x}\left<e^{sd(x,\hx_i)}\right>\nonumber\\
&=&e^{-nsD}\prod_{x\in\calX}\left[\left<e^{sd(x,\hx)}\right>\right]^{nP(x)}\nonumber\\
&=&e^{-nI(D,s)}
\end{eqnarray}
where $I(D,s)$ is defined as
\begin{equation}
I(D,s)=\exp\left\{-n\left[sD-\sum_{x\in\calX}P(x)\ln\left(
\sum_{\hx\in\hat{\calX}}Q^*(\hx)e^{sd(x,\hx)}\right)\right]\right\}.
\end{equation}
The tightest upper bound is obtained by minimizing it over the range $s\le 0$,
which is equivalent to maximizing $I(D,s)$ in that range. I.e., the tightest
upper bound of this form is $e^{-nI(D)}$, where $I(D)=\sup_{s\le 0}I(D,s)$
(the Chernoff bound).
While this is merely an upper bound, the methods of large deviations theory
(see, e.g., \cite{DZ93}) can readily be used to establish the fact that the
bound $e^{-nI(D)}$
is tight in the exponential sense, namely, it is the correct asymptotic exponential
decay rate of $\mbox{Pr}\{d_n(\bx,\hbx)\le nD\}$. Accordingly,
$I(D)$ is called the {\it large deviations rate function} of this event. Combining this with the
foregoing discussion, it follows that $R(D)=I(D)$, which means that an
alternative expression of $R(D)$ is given by
\begin{equation}
\label{ldrdo}
R(D)=\sup_{s\le
0}\left[sD-\sum_{x\in\calX}P(x)\ln\left(
\sum_{\hx\in\hat{\calX}}Q^*(\hx)e^{sd(x,\hx)}\right)\right].
\end{equation}
Interestingly, the same expression was obtained in \cite[Corollary
4.2.3]{Gray90} using completely different considerations (see also
\cite{Rose94}). In this paper,
however, we will also concern ourselves, more generally, with the rate--distortion
function, $R_Q(D)$, pertaining to a given reproduction distribution $Q$, which may
not necessarily be the optimum one, $Q^*$. This function is defined similarly as in
eq.\ (\ref{rdc}), but with the additional constraint that the marginal
distribution that represents the reproduction would agree with the given $Q$,
i.e., $\sum_{x}P(x)W(\hx|x)=Q(\hx)$.
By using the same large deviations
arguments as above, but for an arbitrary random coding distribution $Q$, one
readily observes that $R_Q(D)$ is of the same form as in eq.\ (\ref{ldrdo}),
except that $Q^*$ is replaced by the given $Q$ (see also \cite{Merhav10b}).
This expression will now be used as a bridge to the realm of equilibrium
statistical mechanics.
\section{Statistical mechanics of source coding}
Consider the parametric representation of the rate--distortion function
$R_Q(D)$, with respect to a given reproduction distribution $Q$:
\begin{equation}
\label{ldrd}
R_Q(D)=\sup_{s\le
0}\left[sD-\sum_{x\in\calX}P(x)\ln\left(
\sum_{\hx\in\hat{\calX}}Q(\hx)e^{sd(x,\hx)}\right)\right].
\end{equation}
The expression in the inner brackets,
\begin{equation}
Z_x(s)\equiv\sum_{\hx\in\hat{\calX}}Q(\hx)e^{sd(x,\hx)},
\end{equation}
can be thought of as the partition function
of a single particle of ``type'' $x$, which is defined as follows.
Assuming a certain fixed temperature $T=T_0$, consider the Hamiltonian
\begin{equation}
\epsilon(\hx)=-kT_0\ln Q(\hx).
\end{equation}
Imagine now that this particle may be in various
states, indexed by $\hx\in\hat{\calX}$. When a particle of type $x$ lies in state
$\hx$ its internal energy is $\epsilon(\hx)$, as defined above, and its length
is $d(x,\hx)$. Next assume that instead of working with the
parameter $s$, we rescale and redefine
the free parameter as $\lambda$, where $s=\lambda/(kT_0)$. Then,
$\lambda$ has the physical meaning of a force that is conjugate to
the length. This force is stretching for $\lambda > 0$
and contracting for $\lambda < 0$. With a slight abuse of notation, the
Gibbs partition function \cite[Section 4.8]{Kardar07}
pertaining to a single particle of type $x$ is then given by
\begin{equation}
Z_x(\lambda)=\sum_{\hx\in\hat{\calX}}\exp\left\{-\frac{1}{kT_0}[\epsilon(\hx)-\lambda
d(x,\hx)]\right\},
\end{equation}
and accordingly,
\begin{equation}
G_x(\lambda)=-kT_0\ln Z_x(\lambda)
\end{equation}
is the Gibbs free energy per particle of type $x$. Thus,
\begin{equation}
G(\lambda)=\sum_{x\in\calX}P(x)G_x(\lambda)
\end{equation}
is the average per--particle Gibbs free energy (or the Gibbs free energy density)
pertaining to a system with a total of
$n$ non--interacting particles, from $|\calX|$ different types, where the number
of particles of type $x$ is $nP(x)$, $x\in\calX$.
The Helmholtz free energy per particle is then given by the Legendre transform
\begin{equation}
\label{legendre}
F(D)=\sup_{\lambda}[G(\lambda)+\lambda D].
\end{equation}
However, for $D < D_0\equiv \sum_{x\in\calX}\sum_{\hx\in\hat{\calX}}P(x)Q(\hx)d(x,\hx)$
(which is the interesting range, where $R_Q(D) > 0$), the maximizing $\lambda$
is always non--positive, and so,
\begin{equation}
F(D)=\sup_{\lambda \le 0}[G(\lambda)+\lambda D].
\end{equation}
Invoking now eq.\ (\ref{ldrd}), we readily identify that
\begin{equation}
F(D)=kT_0R_Q(D),
\end{equation}
which supports the analogy between the lossy data compression problem and
the behavior of the statistical--mechanical model of the kind
described in the third paragraph of the Introduction: According
to this model, the physical system under discussion is a long chain
with a total of $n$ elements, which is composed of $|\calX|$ different types of
shorter chains (indexed by $x$), where the number of elements in the short
chain of type $x$
is $nP(x)$, and where each element of each chain can be in various states,
indexed by $\hx$. In each state $\hx$, the internal
energy and the length of each element are $\epsilon (\hx)$ and $d(x,\hx)$,
as described above. The total length
of the chain, when no force is applied, is therefore
$\sum_{i=1}^n\left<d(x_i,\hx_i)\right>|_{\lambda=0}=nD_0$. Upon applying a contracting
force $\lambda < 0$,
states of shorter length become
more probable, and the chain shrinks
to the length of $nD$, where $D$ is related to $\lambda$ according to
the Legendre relation\footnote{
Since $G(\lambda)$ is concave and $F(D)$ is convex,
the inverse Legendre transform holds as well, and so, there is one--to--one
correspondence between $\lambda$ and $D$.}
(\ref{legendre}) between $F(D)$ and $G(\lambda)$, which is given by
\begin{equation}
\label{fl}
\lambda = F'(D)=kT_0R_Q'(D),
\end{equation}
where $F'(D)$ and $R_Q'(D)$ are, respectively, the derivatives of $F(D)$ and
$R_Q(D)$ relative to $D$. The inverse relation is, of course,
\begin{equation}
D=-G'(\lambda),
\end{equation}
where $G'(\lambda)$ is the derivative of $G(\lambda)$. Since $R_Q(D)$ is
proportional to the free energy, where the system is held in equilibrium at
length $nD$, it also means the minimum amount of work required in order
to shrink the system from length $nD_0$ to length $nD$, and this minimum is
obtained by a reversible process of slow increase in $\lambda$, starting from
zero and ending at the final value given by eq.\ (\ref{fl}).\\
\noindent
{\it Discussion}\\
This analogy between the lossy source coding problem and the
statistical--mechanical model of a chain, may suggest that physical insights
may shed light on lossy source coding and vice versa. We learn, for
example, that the contribution of each source symbol $x$ to the
distortion, $\sum_{i:~x_i=x}d(x_i,\hx_i)$, is analogous to the
length contributed by the chain of type $x$ when the contracting force $\lambda$ is
applied. We have also learned that the local slope of $R_Q(D)$ is
proportional to a force, which must increase as the chain is contracted more
and more aggressively, and
near $D=0$, it normally tends to infinity, as $R_Q'(0)=-\infty$ in most cases.
This slope parameter also plays a pivotal role in theory and practice of lossy
source coding: On the theoretical side, it gives rise to a variety of
parametric representations of the rate--distortion
function \cite{Berger71},\cite{Gray90}, some of
which support the derivation of important, non--trivial bounds. On the more
practical side, often data compression schemes are designed by optimizing
an objective function with the structure of
$$\mbox{rate}+\lambda\cdot\mbox{distortion},$$
thus $\lambda$ plays the role of a Lagrange multiplier.
This Lagrange multiplier is now understood to act like a physical force,
which can be `tuned' to the desired trade--off between rate and distortion.
As yet another example, the convexity of
the rate--distortion function can be understood from a
physical point of view, as the Helmholtz free energy is also convex, a fact
which has a physical explanation (related to the fluctuation--dissipation
theorem), in addition to the mathematical one.
At this point, two technical comments are in order:
\begin{enumerate}
\item We emphasized the fact that the reproduction distribution $Q$ is fixed.
For a given target value of $D$, one may, of course, have the freedom to
select the optimum distribution $Q^*$ that minimizes $R_Q(D)$, which would
yield the rate--distortion function, $R(D)$, and so, in principle, all the
foregoing discussion applies to $R(D)$ as well. Some caution, however, must
be exercised here, because in general, the optimum $Q$ may depend on $D$ (or
equivalently, on $s$ or $\lambda$), which means, that in the
analogous physical model, the internal energy $\epsilon(\hx)$ depends on the
force $\lambda$ (in addition to the linear dependence of the term
$\lambda d(x,\hx)$). This kind of dependence does not support the above described
analogy in a natural manner. This is the reason that we have defined the
rate---distortion problem for a fixed $Q$, as it avoids this problem.
Thus, even if we pick the optimum
$Q^*$ for a given target distortion level $D$, then this $Q^*$ must be
kept unaltered throughout the entire process of increasing $\lambda$ from zero
to its final value, given by (\ref{fl}), although $Q^*$ may be sub--optimum for
all intermediate distortion values that are met along the way from $D_0$ to $D$.
\item An alternative interpretation of the parameter $s$, in the
partition function $Z_x(s)$, could be the (negative) inverse
temperature, as was suggested in \cite{Rose94} (see also \cite{Merhav08}).
In this case, $d(x,\hx)$ would be the internal energy of an element of type
$x$ at state $\hx$ and $Q(\hx)$, which does not include a power of $s$,
could be understood as being proportional to the degeneracy (in some
coarse--graining process). In this case, the distortion would have the meaning
of internal energy, and since no mechanical work is involved, this would also
be the heat absorbed in the system,
whereas $R_Q(D)$ would be related to the entropy of the system.
The Legendre transform, in this case, is the one pertaining to the passage
between the microcanonical ensemble and the canonical one.
The advantage of the interpretation of $s$ (or $\lambda$) as force, as
proposed here, is that it lends itself naturally to
a more general case, where there
is more than one fidelity criterion. For example, suppose there are two fidelity criteria,
with distortion functions $d$ and $d'$. Here, there would be two conjugate
forces, $\lambda$ and $\lambda'$, respectively
(for example, a mechanical force and a magnetic force), and the physical analogy
carries over. On the other hand, this would not work naturally with the
temperature interpretation approach since there is only one temperature parameter in
physics.
\end{enumerate}
We end this section by providing a representation of $R_Q(D)$ and $D$ in an integral form,
which follows as a simple consequence of its representation as the Legendre
transform of $\ln Z_x(s)$, as in eq.\ (\ref{ldrd}).
Since the maximization problem in (\ref{ldrd}) is a convex problem ($\ln Z_x(s)$ is convex in
$s$), the minimizing $s$ for a given $D$ is obtained by taking the derivative
of the r.h.s., which leads to
\begin{eqnarray}
D&=&\sum_{x\in\calX}P(x)\cdot\frac{\partial\ln Z_x(s)}{\partial s}\nonumber\\
&=&\sum_{x\in\calX}P(x)\cdot\frac{\sum_{\hx\in\hat{\calX}}Q(\hx)d(x,\hx)e^{sd(x,\hx)}}
{\sum_{\hx\in\hat{\calX}}Q(\hx)e^{sd(x,\hx)}}.
\end{eqnarray}
This equation yields the distortion level $D$ for a given value of the
minimizing $s$ in eq.\ (\ref{ldrd}). Let us then denote
\begin{equation}
\label{ds}
D_s\equiv\sum_{x\in\calX}P(x)\cdot\frac{\sum_{\hx\in\hat{\calX}}Q(\hx)d(x,\hx)e^{sd(x,\hx)}}
{\sum_{\hx\in\hat{\calX}}Q(\hx)e^{sd(x,\hx)}},
\end{equation}
which means that
\begin{equation}
\label{rds}
R_Q(D_s)=sD_s-\sum_{x\in\calX}P(x)\ln Z_x(s).
\end{equation}
Taking the derivative of (\ref{ds}), we readily obtain
\begin{eqnarray}
\label{derds}
\frac{\mbox{d}D_s}{\mbox{d}s}&=&\sum_{x\in\calX}P(x)\frac{\partial}{\partial
s}\left[\frac{\sum_{\hx\in\hat{\calX}}Q(\hx)d(x,\hx)e^{sd(x,\hx)}}
{\sum_{\hx\in\hat{\calX}}Q(\hx)e^{sd(x,\hx)}}\right]\nonumber\\
&=&\sum_{x\in\calX}P(x)
\left[\frac{\sum_{\hx\in\hat{\calX}}Q(\hx)d^2(x,\hx)e^{sd(x,\hx)}}
{\sum_{\hx\in\hat{\calX}}Q(\hx)e^{sd(x,\hx)}}-\right.\nonumber\\
& &\left.\left(\frac{\sum_{\hx\in\hat{\calX}}Q(\hx)d(x,\hx)e^{sd(x,\hx)}}
{\sum_{\hx\in\hat{\calX}}Q(\hx)e^{sd(x,\hx)}}\right)^2\right]\nonumber\\
&=&\sum_{x\in\calX}P(x)\cdot\mbox{Var}_s\{d(x,\hx)|x\}\nonumber\\
&\equiv&\mbox{mmse}_s\{d(x,\hx)|x\},
\end{eqnarray}
where $\mbox{Var}_s\{d(x,\hx)|x\}$ is the variance of $d(x,\hx)$ w.r.t.\
the conditional probability distribution
\begin{equation}
W_s(\hx|x)=\frac{Q(\hx)e^{sd(x,\hx)}}
{\sum_{\tilde{x}\in\hat{\calX}}Q(\tilde{x})e^{sd(x,\tilde{x})}}.
\end{equation}
The last line of eq.\ (\ref{derds}) means that the
expectation of $\mbox{Var}_s\{d(x,\hx)|x\}$ w.r.t.\ $P$ is exactly the
MMSE of estimating $d(x,\hx)$ based on the `observation' $x$ using the conditional mean
of $d(x,\hx)$ given $x$ as an
estimator. Differentiating both sides of eq.\ (\ref{rds}), we get
\begin{eqnarray}
\frac{\mbox{d}R_Q(D_s)}{\mbox{d}s}&=&s\cdot\frac{\mbox{d}D_s}{\mbox{d}s}+D_s-
\sum_{x\in\calX}P(x)\cdot\frac{\partial\ln Z_x(s)}{\partial s}\nonumber\\
&=&s\cdot\mbox{mmse}_s\{d(x,\hx)|x\}+D_s-D_s\nonumber\\
&=&s\cdot\mbox{mmse}_s\{d(x,\hx)|x\},
\end{eqnarray}
or, equivalently,
\begin{equation}
R_Q(D_s)=\int_0^s s'\cdot \mbox{mmse}_{s'}\{d(x,\hx)|x\}\mbox{d}s',
\end{equation}
and
\begin{equation}
D_s=D_0+\int_0^s\mbox{mmse}_{s'}\{d(x,\hx)|x\}\mbox{d}s'.
\end{equation}
In \cite{Merhav10b}, this representation was studied
extensively and was found quite useful.
In particular, simple bounds on the MMSE were shown
to yield non--trivial bounds on the rate--distortion function in some
cases where an exact closed form expression is unavailable. The physical
analogue of this representation is the fluctuation--dissipation theorem,
where the conditional variance, or equivalently the MMSE, plays the
role of the fluctuation, which describes the sensitivity, or the linear
response, of the length of the system to a small perturbation in the
contracting force. If $s$ is interpreted as the negative inverse temperature,
as was mentioned before, then the MMSE is related to the specific heat of the
system.
\section{Sources with memory and interacting particles}
The theoretical framework established in the previous section extends, in
principle, to information sources with memory
(non i.i.d.\ sources), with a natural correspondence to a physical
system of interacting particles. While the rate--distortion function for a
general source with memory is unknown, the maximum rate achievable by
random coding can still be derived in many cases of interest.
Unlike the case of the memoryless source, where the best random coding
distribution is memoryless as well, when the source exhibits
memory, there is no apparent reason to believe that good random coding
distributions should remain memoryless either, but it is not known what
the form of the optimum random coding distribution is. For example, there is
no theorem that asserts that the optimum random coding distribution for
a Markov source is Markov too. One can, however examine various forms
of the random coding distributions and compare them. Intuitively, the stronger
is the memory of the source, the stronger should be the memory of the random
coding distribution.
In this section, we demonstrate one family of random coding distributions, with a
very strong memory, which is inspired by the Curie--Weiss model of spin
arrays, that possesses long range interactions. Consider the random coding distribution
\begin{equation}
Q(\hbx)=\frac{\exp\left\{B\sum_{i=1}^n
\hx_i+\frac{J}{2n}\left(\sum_{i=1}^n\hx_i\right)^2\right\}}{Z_n(B,J)}
\end{equation}
where $\hat{\calX}=\{-1,+1\}$, $B$ and $J$ are parameters, and $Z_n(B,J)$ is
the appropriate normalization constant.
Using the identity,
\begin{equation}
\exp\left\{\frac{J}{2n}\left(\sum_{i=1}^n\hx_i\right)^2\right\}=
\sqrt{\frac{n}{2\pi J}}\int_{-\infty}^{+\infty}\mbox{d}\theta
\exp\left\{-\frac{n\theta^2}{2J}+\theta\sum_{i=1}^n\hx_i\right\},
\end{equation}
we can represent $Q$ as a mixture of i.i.d.\ distributions as follows:
\begin{equation}
Q(\hbx)=\int_{-\infty}^{+\infty}\mbox{d}\theta \pi_n(\theta)Q_{\theta}(\hbx)
\end{equation}
where $Q_\theta$ is the memoryless source
\begin{equation}
Q_{\theta}(\hbx)=\frac{\exp\{(B+\theta)\sum_{i=1}^n\hx_i\}}{[2\cosh(B+\theta)]^n}
\end{equation}
and the weighting function $\pi_n(\theta)$ is given by
\begin{equation}
\pi_n(\theta)=\frac{1}{Z_n(B,J)}\sqrt{\frac{n}{2\pi
J}}\exp\left\{-n\left[\frac{\theta^2}{2J}-\ln[2\cosh(B+\theta)]\right]\right\}.
\end{equation}
Next, we repeat the earlier derivation for each $Q_\theta$ individually:
\begin{eqnarray}
Q\left\{\sum_i d(x_i,\hx_i)\le nD\right\}&=&
\int_{-\infty}^{+\infty}\mbox{d}\theta
\pi_n(\theta)Q_{\theta}\left\{\sum_{i=1}^nd(x_i,\hx_i)\le
nD\right\}\nonumber\\
&\le&
\int_{-\infty}^{+\infty}\mbox{d}\theta
\pi_n(\theta)e^{-nR_{\theta}(D)},
\end{eqnarray}
where $R_\theta(D)$ is a short--hand notation for $R_{Q_\theta}(D)$, which is
well defined from the previous section since $Q_\theta$ is an i.i.d.\ distribution.
At this point, two observations are in order: First,
we observe that a separate large deviations analysis for each
i.i.d.\ component $Q_\theta$ is better than applying a similar analysis
directly to $Q$ itself,
without the decomposition, since it allows a different optimum choice of $s$
for each $\theta$, rather than one optimization of $s$ that compromises all values of
$\theta$. Moreover, since the upper bound is exponentially tight for
each $Q_\theta$, then the corresponding mixture of bounds is also
exponentially tight. The second observation is that
since $Q_\theta$ is i.i.d., $R_{\theta}(D)$ depends on
the source $P$ only via the marginal distribution of a single symbol
$P(x)=\mbox{Pr}\{x_i=x\}$, which is assumed here to be independent of $i$.
A saddle--point analysis gives rise to the following
expression for $R_Q(D)$, the random--coding rate distortion function
pertaining to $Q$, which is the large deviations rate function:
\begin{equation}
R_Q(D)=\min_{\theta}\left\{\frac{\theta^2}{2J}-\ln[2\cosh(B+\theta)]+R_{\theta}(D)\right\}
+\phi(B,J)
\end{equation}
where
\begin{equation}
\phi(B,J)=\lim_{n\to\infty}\frac{\ln Z_n(B,J)}{n}.
\end{equation}
We next have a closer look at $R_{\theta}(D)$, assuming
$\calX=\hat{\calX}=\{-1,+1\}$,
and using the Hamming distortion
function, i.e.,
\begin{equation}
d(x,\hx)=\frac{1-x\cdot\hx}{2}=\left\{\begin{array}{ll}
0 & x=\hx\\
1 & x\ne \hx \end{array}\right.
\end{equation}
Since
\begin{eqnarray}
\sum_{\hx}Q_{\theta}(\hx)e^{sd(x,\hx)}&=&
\sum_{\hx}\frac{e^{(B+\theta)\hx}}{2\cosh(B+\theta)}\cdot
e^{s(1-x\hx)/2}\nonumber\\
&=&\frac{e^{s/2}\cosh(B+\theta-sx/2)}{\cosh(B+\theta)},
\end{eqnarray}
we readily obtain
\begin{eqnarray}
R_\theta(D)&=&\max_{s\le
0}\left[s\left(D-\frac{1}{2}\right)-\sum_xP(x)
\ln\cosh\left(B+\theta-\frac{sx}{2}\right)\right]\nonumber\\
& &+\ln\cosh(B+\theta).
\end{eqnarray}
On substituting this expression back into the expression of $R_Q(D)$, we
obtain the formula
\begin{eqnarray}
R_Q(D)&=&\min_{\theta}\left(\frac{\theta^2}{2J}+\max_{s\le
0}\left\{s\left(D-\frac{1}{2}\right)-\right.\right.\nonumber\\
& &\left.\left.\sum_xP(x)\ln\left[2\cosh\left(B+\theta-\frac{sx}{2}\right)\right]\right\}\right)
+\phi(B,J),
\end{eqnarray}
which requires merely optimization over two parameters. In fact, the
maximization over $s$, for a given $\theta$, can be carried out in closed form,
as it boils down to the solution of a quadratic equation. Specifically,
for a symmetric source ($P(-1)=P(+1)=1/2$),
the optimum value of $s$ is given by
\begin{equation}
s^*=\ln\left[\sqrt{(1-2D)^2c^2+4D(1-D)}-(1-2D)c\right]-\ln[2(1-D)],
\end{equation}
where
\begin{equation}
c=\cosh(2B+2\theta).
\end{equation}
The details of the derivation of this expression are omitted as they
are straightforward.
As the Curie--Weiss model is well known to exhibit phase transitions (see,
e.g., \cite{Honerkamp02},\cite{MM09}), it is expected that $R_Q(D)$,
under this model, would consist of phase transitions as well. At the very
least, the last term $\phi(B,J)$ is definitely subjected to phase transitions
in $B$ (the magnetic field) and $J$ (the coupling parameter).
The first term, that contains the minimization
over $\theta$, is somewhat more tricky
to analyze in closed form. In essence,
considering $s^*\equiv s^*(\theta)$ as a function of $\theta$,
substituting it back into the expression of $R_Q(D)$, and finally,
differentiating w.r.t.\ $\theta$ and equating to zero (in order to minimize),
then it turns out that
the (internal) derivative of $s^*(\theta)$
w.r.t.\ $\theta$ is multiplied
by a vanishing expression (by the very definition of $s^*$ as a
solution to the aforementioned quadratic equation). The final result of
this manipulation is that the minimizing $\theta$ should be a solution
to the equation
\begin{equation}
\theta=J\sum_xP(x)\tanh\left(B+\theta-\frac{s^*(\theta)x}{2}\right).
\end{equation}
This is a certain (rather complicated) variant
of the well--known magnetization equation in the mean field model,
$\theta=J\tanh(B+\theta)$, which is well known
to exhibit a first order phase transition
in $B$ whenever $J > J_c=1$. It is therefore reasonable to expect
that the former equation in $\theta$, which is more general, will
also have phase transitions, at least in some cases.
\section{Summary and Conclusion}
In this paper, we have drawn a conceptually simple analogy between
lossy compression of memoryless sources and statistical mechanics
of a system of non--interacting particles. Beyond the belief that this
analogy may be interesting on its own right, we have demonstrated its
usefulness in several levels. In particular, in the last section,
we have observed that the analogy between the information--theoretic
model and the physical model is not merely on the pure conceptual
level, but moreover, analysis tools from statistical mechanics can be
harnessed for deriving information--theoretic functions. Moreover, physical
insights concerning phase transitions, in systems with strong interactions,
can be `imported' for the understanding possible irregularities in these
functions, in this case, non--smooth dependence on $B$ and $J$.\\
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,658 |
the mess created in the living room on Christmas day.
Don't clean it up too quickly.
Thank you for your love and support for our growing business this holiday season! Wishing you all peace and love this holiday season. | {
"redpajama_set_name": "RedPajamaC4"
} | 4,604 |
Invite Him
Obama Appreciates
Join / Contact
About Sankarshan Das
The Whole World is Waiting to Be Awakened by His Message
Sankarshan Das is a luminary singer-songwriter, sometimes described as another Bob Dylan, rooted in the love and peace, counter-cultural era of the 60's and the early 70's, when he shared the stage with the Jefferson Airplane and the Grateful Dead and was an instant sensation with his regular daily concerts on the University of Texas campus. Seeking to deepen his message he became a disciple of the great spiritual master, His Divine Grace A.C. Bhaktivedanta Swami, who was also the guru of George Harrison and was the inspiration of George's mega hit song, "My Sweet Lord."
For many years now he has been circling the globe twice a year to share his message songs of peace, love, and joy with enraptured audiences all over the world in every continent except for Antarctica. (The penguins aren't ready for it yet.)
His burning desire is to bring genuine peace and happiness to every corner of our ever-increasingly troubled planet and he hopes to meet you personally as well as the leaders of your country on his next around the world tour.
Copyright 2019. Sankarshan Das. All Rights Reserved. | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 8,791 |
Store, Opening hours & Service Area
London - Opening hours
Swapfiets store
58 Commercial Street, Spitalfields, London E1 6LT UK
Mon. 10:00 - 13h30 & 14h00 - 18:30
Tue. 10:00 - 13h30 & 14h00 - 18:30
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Our store has different opening hours on these days:
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You can reach us for swaps and repairs within these times.
Mon. 10:00 - 18:30
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Our customer service has different service hours on these national holidays:
Christmas eve: Friday 24th of December 2021 - 08:00 til 16:30
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New year's eve: Friday 31st of December 2021 - 08:00 til 16:30
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Costs per month Power 7
What kind of model is the Swapfiets Original?
Does Swapfiets have retail stores?
When will the first payment for my Swapfiets be collected?
I have received a surcharge. How can I pay this? | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,903 |
class Renderer;
class LoadSavePanel : public Panel
{
public:
enum class Type { Load, Save };
private:
std::vector<TextBox> saveTextBoxes;
ScopedUiTextureRef backgroundTextureRef, cursorTextureRef;
LoadSavePanel::Type type;
public:
LoadSavePanel(Game &game);
~LoadSavePanel() override = default;
bool init(LoadSavePanel::Type type);
};
#endif
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,686 |
Select a City All Cities Los Angeles New York San Francisco
PRINT Summer 1972
Gianni Pattena
John Weber Gallery
Gianni Pettena is an Italian artist whose work has features in common with both earth works and Conceptual art, thoughhe stands significantly apart from their tendencies to suspend value and meaning. Pettena is a trained architect and has taught in schools of architecture, but what he himself makes is only "architecture" of a cleverly nefarious sort. In the last few years artists have tried innumerable ways of dealing with (or overlooking) a world that grows increasingly piggish. Pettena's response is one of the subtlest I have yet encountered. In his work, political considerations are not at variance or in competition with—or even withheld from—the esthetic operation.
I imagine Pettena as an architect actively on strike. Like those artists who have No Comment on the life that surrounds us, Pettena is reticent and guarded. But there aren't many people who are "on strike" now whose artistic response is as constructive as this. Pettena's works deserve the respect due art objects, though they are not mere properties. His works are neither the residue of desperation, nor pseudo-philosophical games. They are not "gesture" and not theater. While they comment on the rising temperature of life, they are real works of visual art.
On Wednesday, April 13th, the John Weber Gallery showed a film made by Pettena of some of his recent "situations," as he calls them. The film begins with a Smithson-likesequence of aerial views of the great Kennecot open-pit copper mine in Utah, Ordinarily Pettena is more urban than Smithson, which is part of the meaning in both cases: the American finds room for a decent 'and sublime neutrality ("I am interested in the politics of the Triassic period"—Smithson), and usually works with a kind of landscape that is both "unimproved" and not likely to be "spoiled" by artistic modification, while the Italian is more at home in the townscape with all its social and political torque (a Futurist legacy?). The seductive approaches to the copper mine are in themselves Smithsonesque, not only in terms of the choice of motif but even in stylistic relation to Smithson's Spiral Jetty film. The difference will be apparent in the fact that Pettena's, as attractive as it is, is a raped landscape. Yet Pettena has a gift for the most polite, discrete subversion. There is a legend in Utah that it was the owners of this very mine who were responsible for the frame-up of Joe Hill, the labor organizer and songwriter. Pettena may not have known this, but the subversive tone of other pieces by him suggests, that the reference is apt.
After The mine the film shows three of Pettena's own works made in America this year. In Clay House; Situation No. 4 a common frame house in Minneapolis was completely hand-coated with clay mud. The result has an unexpected power and solemnity. As with Robert Smithson, this is not "avant-garde," and ordinary people can get to the heart of the matter without the priestly assistance of theoreticians—ordinary people like the kid who stopped and asked Pettena, "Are you making something or destroying something?" The sheer beauty of the resultant non-house is a real surprise: it is evocative of Frank Lloyd Wright, what with the (literally) earthy naturalism of the "paint-job" that wholly blankets this overstuffed down-home bungalow. And the finished object has something of the leaden numbness of works by Jasper Johns and Robert Morris. Other Smithson-like concerns enter as well, especially where we see a series of beautiful close-ups of the rich brown clay coat on the side of the house, and particularly when the sun cracks the clay—like the surface of an old master painting. Then something delicately subversive happens. When the assertive qualities of this archetypically middle-American house have been effectively silenced by the seeming redirection of a natural effect (the artist has also frozen buildings in ice), insects begin to claim it as a natural part of the earth: they inconspicuously and inoffensively crawl over the walls, suggesting the population of a bombed city in which no property boundaries or title deeds survive.
The next section shows the construction of Tumbleweed Catcher, Situation No. 5. This work has a general affinity with Oldenburg's projected urban monuments. It is conceived as a giant foursquare wooden, towerlike scaffolding that has "caught" clusters of tumbleweed. The overall form of the tower has a sort of freaked resemblance to Mies van der Rohe, with "homemade" and irregular angles. What is of significance here is that in the midst of a committed activism (much more than a "passivism") the artist has found grounds for pleasure and enjoyment. This would not be remarkable if it weren't true that art is largely polarized today between a sometimes irresponsible indulgence and a puritanical, uptight radicalism that gropes for revolutionary direction. Sol LeWitt is a good example of the latter alternative, and I found an appropriate irony and frustration in the fact that his recent works at the Museum of Modern Art seemed to me so suggestive of designs for some newfangled paper currency. Pettena escapes this split because, I think, he has a fair understanding of the politics—rather than the history—of life in the biosphere. But the fact that he knows what is wrong does not disqualify him from guiltless pleasure in the meantime.
All the features of Pettena's art are lucid and explicit. They are not all obvious, but they are all set out aboveboard. Siege (A Red Plot); Situation No. 6, which is covered in the last segment of the film, was sufficiently obvious for the governor of Utah to fully grasp. His Excellency said he liked the idea but not the color. The piece consists of the painting (or "plotting") of a line entirely around the city limits of Salt Lake City. Altogether obvious, you might say, yet the more you think the more you understand. The line starts and finishes at the University of Utah (itself somewhat suspect because the Utah equivalent of An Eton Boy really ought to go to Brigham Young U.). Actually, it begins and ends right between the university and a U.S. Army base—a base established during the campaign to get rid of the Mormons. Well, Ingres did say that line was the probity of art.
The film is not in itself a work of art, as Smithson's Spiral Jetty is; simply a record, it is just something to tack down the facts. But it does serve a real artistic function, insuring and guaranteeing that these works are not sterile "concepts." They are acts, considered and responsible perpetrations which tax the esthetic airwaves with about as much political content as they can be expected to bear without causing a shutdown, or, what might be worse, a premature showdown. I admire Pettena's work very much because he has along with comprehension of political urgencies a lively interest in making things that, without capitulating, can be engaging and worthwhile . . . meanwhile.
—Joseph Masheck | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 3,772 |
\section{Introduction}
The concept of spontaneous symmetry breaking
is an important one in modern theoretical physics.
It is known in statistical physics, condensed matter, and nuclear physics,
as well as in relativistic field theory. An example
of model subject to such a phase transition is given
by an ideal fluid coupled to local external perturbation, \cite{1}.
For example, one may consider a pressure pulse
localized in space of inviscid incompressible unbounded fluid
which rises a fluid current with a net momentum flux ${\cal P}.$
Then ${\cal P}$ plays the role of an order parameter, and the
symmetry is spontaneously broken when ${\cal P}\ne 0.$
Statistical properties of the {\em symmetrical phase}
of water response for the local external perturbation
were discussed in \cite{1} in details.
As an example of such a behavior one imagines an
eddy risen around a point wise distinct
pressure perturbation.
It was shown that the energy of hydrodynamic perturbation is confined
within the eddy formed around the pulse region and
demonstrates a power like decay
$\sim r^{-4} $ outside this region, \cite{1}.
In the present paper we consider the water response in the
case of symmetry broken, {\em i.e.,} when
${\cal P}\ne 0$. Since the symmetry breaking
does not affect the scaling properties of the theory, the universal
quantities for the long-range asymptotic behavior of the correlation
functions are the same as in the symmetrical phase.
We demonstrate that if ${\cal P}\ne 0$ the leading order of
long-range water response for perturbation is not determined by
the scaling degrees of freedom, but by the anomalies risen due to an
explicit symmetry breaking.
The results achieved
would give a key for better understanding of various aspects in
the studies of the ocean surface. For example, one can
consider the problem
of generating of circulation motions (meanders) close to
ocean currents. The grown up meanders is about to separate from the
main current
forming the stable closed rings
(which averaged lifetime is about 2-3 years)
of hundreds of kilometers in diameter drifting slowly along the
main current (with an averaged speed of 2-10 cm/s), \cite{MK}-\cite{K}.
The leading order of correlation functions stationary spectra in the
fully developed turbulence theory of Kolmogorov \cite{Monin} as well as
in the statistical theory of waves based on
the Zakharov's kinetic equations \cite{MK} can be found from pure
phenomenology in principle. However, that is not the case for the
water coupled to perturbation.
The matter is of presence of redundant
degrees of freedom connected to perturbation as well as
to hydrodynamic equations themselves.
To get rid of these degrees of freedom we propose a physically
relevant hypothesis on the
mechanism coupling water to perturbation.
In \cite{1} we used the renormalization group (RG)
technique to justify such an additional assumption for the
case of $O(3)-$symmetrical perturbation.
In the framework of RG method the physical degrees of freedom are to be
replaced by the scaling degrees which are related to the physical degrees
through the RG transformations of fields and parameters of the theory.
Since the properties of scaling degrees of freedom are governed by
a group structure
(the renormalization group) one investigates them much easier
then those of
the origin problem. The results obtained from RG-analysis are
considered as
somewhat statistical steady state limit of the physical system.
The renormalized correlation functions are
distinguished from their physical analogies only by normalization
conditions, so that they can be also used for the analysis of
asymptotic behavior
of the physical system. The investigation of
the relevant scaling degrees of freedom which has been
brought about in \cite{1}
demonstrates that the asymptotic behavior
$r>>l_p$ ($l_p$ is the perturbation scale)
of water response can be determined unambiguously
if one supposes a coupling
between translation and rotation components of the velocity of fluid.
The stationary spectra can have place for those scale intervals which are
transparent for currents of conserved quantities.
The conserved energy current from the large-scale region of pumping
into the small-scale range of viscous dissipation
allows one to adjust the well known
Kolmogorov's spectrum in the fully developed turbulence (the Five
Thirds Law).
However, the {\em inertial} range of Kolmogorov lies apart from the
scale spectrum of our interest.
The transparency interval related to the enstrophy current
(the squared averaged vorticity) ${\cal E}$ is exactly
the scale of the problem considered, though the enstrophy current
gives us the spectrum of vorticial
component of the velocity field only,
\begin{equation}
{\cal E}^{1/2}k \simeq A(k),
\label{a}
\end{equation}
where $\bf A$ is the vector velocity potential:
\begin{equation}
\bf v (\bf x,t)= -\bf{grad} {\ } \phi(\bf x,t)+\bf{rot} {\ } \bf A(\bf x,t),
\label{v}
\end{equation}
and $\phi$ is the scalar velocity potential.
However, the spectrum for $\phi(k)$ is still
unknown from these phenomenological considerations.
In language of the critical phenomena theory the spectrum
(\ref{a}) determines the {\em critical dimension} $\Delta [A]$ of the
field $A$, $\Delta[A]=1.$ The use of critical dimension allows to compute
the spectrum of any correlation function of the field $A$ by simple
dimensional counting.
For example, for the pair correlation function in Fourier representation
$D_A(k)\equiv
\langle\bf A (\bf k)\bf A (-\bf k)\rangle,$ one has the asymptotics
\begin{equation}
D_A(k)\sim k ^{\Delta[A]}, \quad \Delta [D_A]=2\Delta [A]-d,
\label{pair}
\end{equation}
where $d$ is the dimension of space.
Further phenomenological considerations
allow to determine the spectra of quantities which can be measured in
experiments,
for example, for the energy as a function of distance apart
from the perturbation point,
$E(r),$ one can obtain:
\begin{equation}
{\cal E}r^{-4}\sim E(r).
\label{e}
\end{equation}
These results were derived in \cite{1} for the case of $O(3)-$symmetrical
perturbation
and justified within the framework of RG-approach. Formally,
they are still valid for the case of symmetry broken, ${\cal P}\ne 0.$
The assumption on the mechanism coupling water to perturbation
proposed in \cite{1}
allows to fix the spectrum for $\phi$ in the form
\begin{equation}
\phi(k)\sim k^{7/12}.
\label{f}
\end{equation}
The confinement of energy within the region of $O(3)-$symmetrical
perturbation could be interpreted as a kind of {\em short-range} order
(in analogy with infinite ferromagnets), but the {\em long-range} order
is suppressed. In case of symmetry broken spontaneously there is
a nontrivial expectation value for $\langle \phi\rangle=
\alpha(\bf x)$
(in analogy to the arising of spontaneous magnetization in ferromagnets)
engaged in long-range order with large-scale current motion.
In a regular way $\alpha (\bf x)$ is to be determined from the
equation of state
\begin{equation}
(\alpha-\alpha_0)=f(j),
\label{es}
\end{equation}
with some function $f(j)$ calculating usually in the framework of
Feynman graph expansion, \cite{Br}. However, the power-like
asymptotic solution for (\ref{es}) can be derived readily from
phenomenology: considering ${\cal P}$ as a new
dimensional parameter of the theory
with symmetry broken, one obtains the spectrum $\alpha(k)$ in the form
\begin{equation}
\alpha(k)\sim k.
\label{fsb}
\end{equation}
Continuing the analogy with ferromagnets, one can conclude that the
ordered phase (of broken symmetry)
water response can be described by a quantity analogous to the
longitudinal susceptibility
\begin{equation}
\chi_L=\frac{\partial \alpha}{\partial j},
\label{sus}
\end{equation}
determined by
\begin{equation}
\chi_L\equiv \int d \bf x {\ } \langle\left[\phi(\bf x) - \alpha(\bf x)\right]
\left[\phi(0) - \alpha(0)\right]\rangle.
\label{susdef}
\end{equation}
By the way, from the critical phenomena theory point of view
the main problems of the theory in the non-symmetrical phase are
to determine
an explicit form for the function $f(j)$ in (\ref{es}) and to justify
the phenomenological result (\ref{fsb}).
The plan of the paper follows:
First, in the second section,
for the convenience of readers
we briefly reproduce the main result of \cite{1}
on model of ideal unbounded fluid coupled to perturbation
concluding the section by a discussion on the explicit symmetry
breaking. The statistical properties of hydrodynamical system
can be described by a partition function of statistical mechanics with
a classical euclidean action. As a result we derive the classical
action functional designed to describe the long-range
asymptotic behavior of the water coupled to perturbation.
The relevant functional turns out to be analogous to that one
of the {\em abelian Higgs model} well-known in relativistic field theory
and superconductivity.
In the next section we observe the basic properties of the theory
in case of an explicit $U(1)-$symmetry breaking and consider
the physical consequences for the asymptotic behavior of the
model. These properties are dramatically different from those
demonstrated by the model in the symmetrical case discussed in \cite{1}.
In the latter phase scaling degrees of freedom were completely
determined by the vector velocity potential $\bf A,$ though
in presence of comparably strong fluid flow the vector velocity
components are confined in the flux and their contributions
are irrelevant for the asymptotic behavior of the system.
Looking for the stable stationary
solutions, we obtain an infinite countable set of such solutions
distinguished one from the other by energy gaps. In particular,
a "ground state" solution can be interpreted as a pure
laminar flow ($\bf A\equiv 0$).
In the fourth section we consider the asymptotic for $\alpha(r)$
in case of the laminar flow; it is provided by a Goldstone
asymptotics arising by an explicit $U(1)-$symmetry breaking
which has place in the real physical system.
To account the contributions of eddies into flux for
"excited states" we construct an instanton solution
for the theory of water coupled to perturbation and find
the interval of validity for the phenomenological result (\ref{fsb})
in the fifth section.
In the Section 6, we discuss the results obtained from the
point of view of dynamical systems theory. We conclude in the last section.
\section{The Model of Ideal Unbounded Fluid
Coupled to Perturbation}
In \cite{1} it was shown that after elimination of all
redundant degrees of freedom the problem considered
possesses the entire $U(1)-$gauge symmetry \cite{gft}.
In case of $ j\ne 0$ this symmetry turns out to be hidden
by an explicit symmetry breaking term appearing in the effective
action functional. To reveal the hidden symmetry of hydrodynamics of
ideal fluid we first suppose $ j=0$.
Then the equation of hydrodynamics takes the form
\begin{equation}
\bf {div}{\ } \bf v(\bf x,t)=0, \quad
p(\bf x,t)= \int {\ }d\bf y \frac{\partial_iv_j(\bf y,t)
\partial_jv_i(\bf y,t)}
{|\bf x-\bf y|}.
\label{ce}
\end{equation}
Here $\bf v(\bf x,t)$ is the velocity of fluid and
$p(\bf x,t)$ is the pressure distribution. The trivial
boundary conditions for the fields at infinity are implied.
The equations (\ref{ce}) do not lead to a hamiltonian in
the usual way, since it is not possible to define
the conjugated moments. Nevertheless, these equations can be
derived from the classical Lagrangian ${\cal L}(\bf \varphi)$
formulated in favor of the scalar and vector velocity potentials, \cite{1}:
\begin{equation}
{\cal L}(\phi, A_i, p) =\frac 12\int d{\bf x}\left[
(\partial p)^2+(\partial \phi)^2+\frac 12 F^2+p\partial_iv_j\partial_jv_i +J_iA_i
\right],
\label{lagr}
\end{equation}
where we have introduced the eddy component of velocity
field as a {\em gauge} invariant tensor
$F^k=(\bf {rot}{\ }\bf A)_k=\partial_iA_j-\partial_jA_i,$
$J_i=\partial_jF^k-\partial_kF^j$ ($i\ne k\ne j$) is the
{\em vorticity} conserved for ideal fluid, and
the tensor $v_iv_j\equiv (\partial_i\phi)(\partial_j\phi)+
F^iF^j-\partial _i\phi F^j-\partial_j\phi F^i$ (the last two terms
are vanished when integrated over $\bf x$ with trivial boundary conditions).
With no coupling to pressure field ($p=0$)
(\ref{lagr}) is invariant under the following
transformations of fields
\begin{equation}
\left\{
\begin{array}{l}
A_i(\bf x)\to A_i(\bf x)-\partial_i \Lambda(\bf x), \\
\phi(\bf x)\to \phi(\bf x)e^{iu\Lambda(\bf x)},
\end{array}
\right.
\label{gi}
\end{equation}
where $u\Lambda(\bf x)$ is an arbitrary continuous, differentiable phase function.
The relations (\ref{gi}) express the $U(1)-$gauge symmetry of the
functional (\ref{lagr}) ($U(1)$ is the group of multiplication
by complex numbers). In accordance to the Noether's theorem
(\ref{gi}) relates to two conserved currents,
vorticity, defined above,
\begin{equation}
\partial_iJ_i=0,
\label{cc1}
\end{equation}
and a current related to rotations in the complex plain $U(1)$:
\begin{equation}
I_i=\phi^*\partial_i\phi-\phi\partial_i\phi^*, \quad \partial_iI_i=0.
\label{cc2}
\end{equation}
The statistical properties of mechanical system of an infinite number
of degrees of freedom can be derived from the partition function
of statistical mechanics $Z= Tr (e^{-S})$ with somewhat classical
euclidean action functional,
\begin{equation}
S(\phi,A_i,p)=\frac 12 \int\left[ (\partial p)^2+(\partial\phi)^2+
\frac 12 F^2 +p\partial_iv_j\partial_jv_i \right]
\label{s}
\end{equation}
The last term in (\ref{s}) does not meet the entire symmetry (\ref{gi}),
since the pressure field as it is included in the action functional contains
somewhat redundant degrees of freedom, we therefore can integrate it over
in the partition function $Z$. The result of functional integration
does not depend on $p(\bf x,t)$. This procedure is reduced to elimination
of the quadratic term proportional to $(\partial p)^2$ from
(\ref{s}) and the replacement of the $U(1)-$symmetry breaking
term by
\begin{equation}
\frac 12 \int d{\bf x} {\ } \partial_iv_j(\bf x,t)
\partial_jv_i(\bf x,t)\int _{V_p}d \bf y \frac
{\partial_iv_j(\bf y,t)\partial_jv_i(\bf y,t)}
{|\bf x-\bf y|},
\label{t}
\end{equation}
which relates the fluctuations of velocity fields risen by the
perturbative pulse in the perturbed region $V_p$ to those
fluctuations apart from $V_p$.
In \cite{1} we investigated possible contributions of (\ref{t})
into the action functional considering the insertions of various
power like composite operators. It was shown that
the only component which is important from the RG point of view have to be
$U(1)$-gauge symmetrical, {\em i.e.,}
\begin{equation}
m^2\phi^2
\label{m2f2}
\end{equation}
in the first order,
where $m^2$ is somewhat mass parameter (the coefficient of the relevant
RG-invariant composite operator). The use of Ward identities which
express the $U(1)-$gauge invariancy of the theory allows to demonstrate
that all other combinations of quadratic operators are ultra-violet
(UV) finite, {\em i.e.,} the relevant correlation functions do not
have UV-divergencies, and then they do not participate in scaling degrees
of freedom.
Instantly close to the region of perturbation the pressure pulse
rises the wave motions with eigenmodes $k>k_0\simeq V_p^{-1/3}.$ Due
to strong nonlinearity of interaction in hydrodynamical
equations the eigenmodes of oscillations spread very fast from a
band of order $V_p^{-1/3}$ over the whole spectrum, and various multipole
oscillations of any type are arisen with time. Clearly, the long-range
fluid behavior will depend to some extent on the statistical
properties of wave mode coupling. Following \cite{1} we suppose
the simplest model for the coupling mechanism by inclusion of
the $\varphi^4$-type interaction term into (\ref{s}) with a wave
modes coupling constant $g>0$. In accordance with the general
critical phenomena approach we note that the accounting of highest
oscillation harmonics, {\em i.e.,} $\varphi^6$, $\varphi^8,$
and so on cannot alter the large-distance asymptotic behavior of
water response if $g\ne 0.$ Again, since we are interested in
$U(1)-$gauge symmetrical term the only amendments into action
functional have to be of the form, $\sim g\phi^4.$
As a resulting hypothesis we obtain the effective action
functional to be:
\begin{equation}
S(\phi, A_i)=\frac 12 \int d\bf x \left[(\partial \phi)^2+
\frac 12 F^2+m^2\phi^2+\frac 13 g\phi^4\right]
\label{se}
\end{equation}
which is designed to describe the asymptotic properties of water response.
The action (\ref{se}) has some redundant degrees of freedom, the
gauge degrees, with unknown dynamics. As a consequence it
is not renormalizable, and it has no solutions in the massless limit $m^2=0,$
\cite{Br}.
To construct a renormalizable theory we introduce an abelian gauge
geometrical structure:
(i) $\phi({\bf x})$ and $\phi^*({\bf x})$ are vectors
for $U(1)$ transformations,
(ii) The derivative $\partial_i$ is replaced by the
covariant derivative $\nabla_i$:
\begin{equation}
\nabla_i=\partial_i+iu_0A_i,
\label{A7}
\end{equation}
where
$u_0$ is the coupling constant of interaction between
the scalar and rotational components of the velocity potential $\varphi$
(analogous to the electron charge $e$ in electrodynamics).
(iii) It follows that the curvature tensor
is $iu_0F_{ij}:$
$$ iu_0F_{ij}=[\nabla_i,\nabla_j]=iu_0(\partial_iA_j-\partial_jA_i).$$
(iv) Since the $U(1)$-gauge group is abelian
($\bf A(\bf x,t)$ is a translation invariant), one can write
the parallel transporter $U(C)$ along
any continuous contour $C$ which is an element of $U(1).$ In
terms of a line integral:
\begin{equation}
U(C)=\exp\left[-iu_0\oint_C A_i(s) {\ } ds_i\right]
\label{pt}
\end{equation}
as a consequence of vorticity conservation for ideal fluid.
Thus, the rotational component of velocity potential just
carries on the
fluctuations of the scalar potential field $\phi({\bf x})$.
By the way, two solutions for different points $\phi({\bf x},t )$
and $\phi ({\bf y},t)$ are related through the parallel
transporter (\ref{pt}), where $C$ is an integration path
connecting the points $\bf x$ and $\bf y$, \cite{gft}.
Now, the gauge degrees of freedom (correspondent to invariancy
of velocity with respect to the $\partial_i\Lambda$-shifts of
vector potential $A_i$) can be taken into account by the usual
procedure analogous to the Faddeev-Popov quantization \cite{Br}.
In particular, it leads to inclusion of a gauge dependent term into
the action functional,
\begin{equation}
S (A_i,\phi)=\frac 12 \int d\bf x {\ } \left[ |\nabla_i\phi|^2+ \frac 12 F^2_{ij}
+\zeta^{-1} (\partial_i A_i)^2+m^2\phi^2+ \frac 13 g \phi^4\right],
\label{A9}
\end{equation}
where $\zeta$ is an arbitrary valued ($ \zeta\in [0,\infty) $)
auxiliary gauge parameter of the theory.
The model
(\ref{A9}) demonstrates the existence of a statistically
steady state independently of the details of velocity evolution.
In \cite{1} the model (\ref{A9}) has been investigated in the symmetrical
phase, $m^2>0$. The crucial distinction between symmetry implementation in
the cases of positive and negative signs for $m^2$ lies though
in the structure of the "ground state" ({\em i.e.,} the expectation
value of velocity potential).
Suppose, first, that
$\bf A=0$ in (\ref{A9}), then one has the standard model of a scalar unharmonic
oscillator. For $m=0$ the oscillator is subject to a phase transition.
At the classical level in the symmetrical phase ($m^2>0$) the oscillator
model describes the fluctuations having the trivial expectation value
of the field, $\langle\phi \rangle=0$ (see Fig. 1.a).
If $m^2<0$, the system allows
an infinite number of possible expectation values
related to each other by the unitary transformation group $U(1).$
In particular, if one fixes a phase parameter of the group $U(1)$
under certain physical conditions, then for the field $\phi$ there are
two possible mean values (see Fig. 1.b)
\begin{equation}
\alpha(\bf x)\equiv\langle\phi \rangle=\pm
\sqrt{ m^2 /g}= \alpha_0(\bf x).
\label{alpha}
\end{equation}
The latter situation is usually referred to as spontaneously broken symmetry.
Rise of a net fluid current from the region of initial perturbation
into outside (see Fig.2) one can treat as a result of spontaneously
symmetry breaking which can be described by (\ref{A9}) with $m^2<0$.
If we held $\bf A=0,$ the lagrangian is still invariant under the set of
$U(1)$-transformations with no gauge section. When $j=0$,
the Goldstone theorem predicts the appearance of a massless
degree of freedom corresponding to unphysical "angular motion" for which
there is no restoring force. The physical interpretation of such a
degree of freedom would be the following: the quantity (\ref{sus})
diverges as $j\to 0, $ {\em i.e.,} an infinitely small initial fluid flow
$j$ risen by perturbation generates the nontrivial expectation value
for vector potential, $\alpha\ne 0.$
If we assume that $j \ne 0,$ then the action (\ref{A9}) has the
$U(1)$-symmetry breaking explicitly by the new term
\begin{equation}
-\int d\bf x {\ } j(\bf x) Re[\phi(\bf x,t)],
\label{sbt}
\end{equation}
where $Re[\phi(\bf x,t)]$ is the real part of the complex valued field $\phi$.
This symmetry breaking term gives in the first order in $j$
a mass proportional to
$j^{1/2}\partial_iI_i$ (the axial current $I_i$ is no more conserved)
to the unphysical angular degree of freedom correspondent to
rotations in the complex plain ({\em i.e.,} (\ref{sus}) has no
more divergent).
The situation is though to be changed dramatically if one includes the vorticial
velocity component into consideration ($\bf A\ne 0$). Due to
so called {\em Higgs mechanism} the angular degree of freedom does not
produce divergences in (\ref{sus}) even in the zero order in $j$, and
the gauge field $A_i$ acquires a mass without spoiling the gauge invariance
and renormalizability of the theory, \cite{gft}-\cite{7}.
These ideas which are quite familiar in the weak interactions theory and
superconductivity allow an heuristic interpretation also in hydrodynamics:
In the symmetrical phase ($j=0$) $\bf A$ plays
the purely transporting role for scalar velocity potential fluctuations from one point to
another in accordance with (\ref{pt}). When the symmetry is broken spontaneously,
$\bf A$ acquires the longitudinal polarization degree of freedom
giving it a mass; as a direct consequence a vector potential field can penetrate
only exponentially into the fluid flow with a range proportional to
the inverse of the acquired mass. Like a superconductor expels a magnetic field
from its interior, except for a thin layer at the surface over which the field
decreases exponentially, the fluid flow ousts the eddies from its interior
onto the periphery. The microscopic origin of the Higgs phenomenon in
hydrodynamics lies in screening currents of fluid compensating the
external velocity rotational component (see Fig.3).
In \cite{1} it was shown that the scaling degrees of freedom proportional to $g$
are vanished from the asymptotic behavior of fluid involved into eddy motion. By the way,
considering $ j=0$ (say, on the periphery skin of a large-scale current),
one can omit the term $g\phi^4$ from (\ref{A9}) to obtain
the relevant effective action functional. However, if $j\ne 0$ (within the current),
from the heuristic point of view it is obvious that the statistical steady state should be
free of coupling to the vector potential $\bf A$, and the solution for $\alpha(\bf x)$
({\em i.e.,} (\ref{fsb})) is to be determined by a Goldstone asymptotics, \cite{Nal}.
\section{The Gauge-Invariant $U(1)$ Theory of the
Water Coupled to Perturbation}
In the present section we develop the heuristic ideas of the preceding one.
We derive the action functional for the theory in case of symmetry
spontaneously broken and investigate its properties.
Implementing the local transformation
\begin{equation}
\left\{
\begin{array}{c}
\phi(\bf x) \to\left[\alpha(\bf x)+ \phi(\bf x)\right]
e^{ iu\Lambda(\bf x)/\alpha(\bf x) }, \\
A_i(\bf x)\to A_i(\bf x) - \frac u{\alpha(\bf x)}\partial_i\Lambda(\bf x)
\end{array}
\right.
\label{gi2}
\end{equation}
to the model action functional relevant to the
system of equations (\ref{ce}), we
fix the gauge in such a way that
\begin{equation}
u\Lambda(\bf x)=2 \pi n \alpha (\bf x), \quad n\in Z.
\label{ug}
\end{equation}
The parameter $u$ which characterize the coupling strength of
vorticial and translational velocity components in this gauge
is related to a {\em circulation,} $\Gamma$,
\begin{equation}
\Gamma_n \equiv \oint \bf A{\ } d\bf x = \frac {2n \pi}{u}.
\label{circ}
\end{equation}
Physical degrees of freedom are become clear now (since $\Lambda(\bf x)$ is gauged away
from the theory):
\begin{equation}
\begin{array}{c}
S (A_i,\phi)=\frac 12 \int d{\!}x \left[ |\nabla_i\phi|^2+ \frac 12 F^2_{ij}
+\zeta^{-1} (\partial_i A_i+\sqrt{2}u\alpha\cdot Im[\phi])^2+m_{\phi}^2\phi^2+\right.
\\ \left.+m_A^2A^2 +\frac 13 g \phi^4
+u^2_1\phi A^2 +\frac 43 g_1 \phi^3 - j\phi + S(\alpha)
\right],
\end{array}
\label{s2}
\end{equation}
where we have denoted
$u^2_1=u^2\alpha$, $g_1=g\alpha,$ $m^2_A\equiv 2\alpha u$, $m^2_{\phi}=2g\alpha^2-m^2,$
and $Im[\phi]$ is an imaginary part of $\phi$.
Comparing (\ref{s2}) and (\ref{A9}), one can see that the vector field $\bf A$
obtains the longitudinal polarization degree of freedom for
which is expressed in (\ref{s2}) as the new mass term $m^2_A A^2$. The longitudinal
components of the vector fields $A_i$ and $Im[\phi]$ are ghosts, which both cancel against the
Faddeev-Popov ghost \cite{deWitt} all having the same mass $m_A.$
The behavior of (\ref{s2}) is in a way very different from (\ref{A9}):
the unphysical imaginary part of the scalar velocity potential $\phi$
disappears and the vector field $\bf A$ obtains a mass so that the vorticial
velocity component is short-range only, {\em i.e.,} it is repelled completely
from the flux.
The standard way to illustrate the last sentence is to
demonstrate that the response of the flux for an elementary vortex
immersed in is equal to zero \cite{Pol}.
Consider the constant shift transformation (a purely vorticial
constant velocity component)
\begin{equation}
F_{ij}\to F_{ij}+f_{ij}
\label{st}
\end{equation}
for the gauge section $F_{ij}$ in the disordered theory (\ref{A9}).
Then the partition function $Z[f_{ij}]$ is invariant
under gauge transformations:
\begin{equation}
f_{ij}\to f_{ij}+\partial_i\lambda_j-\partial_j\lambda_i,
\label{gt2}
\end{equation}
since it can be compensated by the appropriate change $A_i\to A_i+\lambda_i.$
Therefore, in a gauge-invariant theory (\ref{A9}) one has
\begin{equation}
Z[F_{ij}+f_{ij}]=Z[F_{ij}];
\label{gi3}
\end{equation}
furthermore, one notes that the constant $f_{ij}$
can be removed from (\ref{A9}) by the transformation:
\begin{equation}
A_i\to A_i +x_jf_{ij}.
\label{tr2}
\end{equation}
In the phase with long-range correlations (\ref{A9}),
the change (\ref{tr2}) is equivalent to somewhat change of trivial
boundary conditions for the equations (\ref{ce}) at infinity.
In particular, this yields the new term into the partition function
\begin{equation}
Z[f_{ij}]-Z[0]=\frac{\varepsilon_A}{u^2}\int d\bf x {\ }f^2_{ij},
\label{yr}
\end{equation}
where $\varepsilon_A$ (the Lagrange multiplier) would be some function which has a natural
interpretation as an amplitude of the response of the flux for an elementary vortex.
Obviously, if the circulation $\Gamma< \Gamma_{0},$ where $\Gamma_{0}$ is some critical
value correspondent to the phase transition point, the vector potential
becomes short-range correlated, and the partition function $Z$
should not depend on $f_{ij}.$
Thus $\varepsilon_A=0$ in the theory (\ref{s2}).
In case of the vector velocity potential $\bf A$ is strong enough
then, because of circulation conservation, it can be allowed in the
fluid flow in the form of narrow flux tubes.
The relation (\ref{circ}) in this context means that there are always an integer number $n$
of such vorticial tubes in the flow, {\em i.e.,} that each of them have
a source and a sink (see Fig.4).
Varying (\ref{s2}) with respect to $A_i$ and $\phi^*\phi$ with
the boundary condition (\ref{circ}) ({\em i.e.,} fixing the circulation
$u^{-1}$
to be constant in ideal fluid), one can easily estimate the energy of
a flux with length $l_f$ as
\begin{equation}
E\sim u^2m^2_A l_f,
\label{fe}
\end{equation}
which demonstrates the property inherent to a confinement phenomenon.
Note, that (\ref{fe}) could be derived rigorously by considering of
a Wilson's loop
operator for a point-wise vorticial current,
\begin{equation}
{\cal J}_i(\bf x)=-iu\oint \delta(\bf x-\bf y) {\ }d\bf y,
\label{vc}
\end{equation}
for (\ref{s2}) (see, for example \cite{2}).
This situation is analogous to that of superconductors \cite{hooft}: if
electrically charged bosons (Cooper's bound state of an electron pair)
Bose-condensed then there the electric fields become short-range,
and the magnetic
fields are ousted from the interior. If finally a magnetic field is
admitted inside
a superconductor it can only come in some multiple vortices, never spread
out because of Meissner effect. Following the analogy with
superconductors, one
can say that the source and the sink vortices confined in the
fluid flow are kept together in a potential well, and the potential
is being linearly proportional to their separation.
As it well known, \cite{7}-\cite{col}, the hidden symmetry begets the
hidden renormalizability: the divergence structure of renormalizable theory
(\ref{A9}) is unaffected by spontaneous symmetry breaking, and the counterterms
needed in (\ref{s2}) remain those of the symmetrical theory (\ref{A9}).
Consequently, the critical dimensions calculated in \cite{1} for
the quantities in (\ref{A9}) are still valid formally also for (\ref{s2}).
\section{ Goldstone Asymptotics
of the Water Flux Coupled to Perturbation}
In the present section we construct (\ref{es}) explicitly and
justify the phenomenological result (\ref{fsb}). We shall consider
the theory in the ordered phase assuming that the velocity
field has no vorticial components ({\em i.e.,}
the flux contains no vorticial pairs, $n=0$ in (\ref{ug})). Therefore,
to describe the statistical properties of the system we
can integrate the partition function $Z[\phi,\bf A]$ over $\bf A$
eliminating the vector field $\bf A$ from the theory. The resulting
partition function will depend solely on the scalar velocity
potential, $Z[\phi],$ and the relevant action functional
will be identical to those of scalar $\phi^4-$theory in the ordered phase
(nonlinear $\sigma-$model):
\begin{equation}
S=-\frac 12\int d\bf x \left[(\partial\phi)^2+\tau \phi^2+
\frac{g}{3!}(\phi^2)^2-j{\ } Re[\phi]\right].
\label{sf4}
\end{equation}
The distinguishing feature of (\ref{sf4}) is the presence of
Goldstone singularities which arise due to an explicit
$U(1)-$symmetry breaking. The physical origin of these
singularities is following, \cite{Pol}: the scalar velocity potentials
with different orientation in the complex plane, however,
correspond to the same fluid velocity and though to the
same energy. The relevant conserved current meets the Ward identity
in the momentum representation:
\begin{equation}
k_i\langle I_i(k)\phi(-k)\rangle=\langle\phi(0)\rangle.
\label{wi}
\end{equation}
Taking $k\to 0$, one concludes that $\langle I_i(k)\phi(-k)\rangle$
must be singular in this limit:
\begin{equation}
\langle I_i(k)\phi(-k)\rangle_{k\to 0}=\langle\phi(0)\rangle\frac{k_i}{k^2}+
\ldots
\label{sing}
\end{equation}
The general solution for Goldstone asymptotics in (\ref{sf4})
was given in \cite{Nal} for the unbounded theory and then
generalized in \cite{Nal2} to the theory in a half-space.
In particular, the hypothesis \cite{pp} was proven in
\cite{Nal} for any order of $\epsilon-$expansion
($2\epsilon=4-d$) with $j, k\to 0:$
the equation of state (\ref{es}) has the form
\begin{equation}
(\alpha-\alpha_0)=aj^{1-\epsilon}+bj+\ldots,
\label{ess}
\end{equation}
and the longitudinal susceptibility (\ref{sus}) is to be
\begin{equation}
\chi_L=a_1j^{-\epsilon} + b_1+\ldots,
\label{sus1}
\end{equation}
the numerical coefficients $a, b, a_1, b_1$ are specified in
\cite{Nal}. For the transversal susceptibility, $\chi_T\sim j^{-1}$ as
it follows from the Goldstone theorem. Following the discussion in
\cite{Vas}, formulae (\ref{ess}) and (\ref{sus1}) can be interpreted as a
Goldstone scaling (by analogy to critical scaling) for which
$j$ and $k\sim 1/r$ play the role of significant parameters.
The certain Goldstone dimensions $\Delta^G$ belong to $k$, $j$,
and $\alpha(\bf x)|_{j=0}$:
\begin{equation}
\Delta^G[k]=1,\quad \Delta^G [j]=2,
\quad \Delta^G[\alpha]=d-2.
\label{gr}
\end{equation}
In contrast with critical dimensions (\ref{gr}) are known
precisely as well as the normalized scaling functions of the
simplest correlation functions \cite{Vas}.
The last relation in (\ref{gr}) justifies the result (\ref{fsb}):
performing the inverse Fourier transformation, one obtains
at three dimensions
\begin{equation}
\alpha(r)\sim \frac 1{r^2}.
\label{fsb1}
\end{equation}
\section{The Instanton Solutions for the Theory of Water
Coupled to Perturbation}
In the previous sections we have considered
the stationary (with no time dependence) stable solutions
of (\ref{s2}) which correspond to the
saddle points (solutions of the hydrodynamical equations).
However, in the case
discussed, $j\ne 0,$ the actual hydrodynamical equations,
posses the non-constant solutions also.
In the previous sections we have shown that there is
a countable set of possible stable stationary solutions
(enumerated by an integer number $n$)
for the system of flux coupled to perturbation
distinguished one from the other by the energy gaps (\ref{fe}).
Obviously, the non-constant statistically steady solutions
are related to a specific mechanism of gap generation
(the generation of new pairs of eddies in the flux),
{\em i.e.,} they describe possible transitions
between constant solutions with different $\Gamma_n$ (\ref{circ}).
Another interpretation can be used: since
the source and sink eddies are confined together in the
potential well in the fluid flux, one can consider a tunneling process
of the eddy pair into another potential well.
This tunneling process can be provided by an
{\em instanton } solution \cite{Pol}.
The contribution of instantons into the correspondent partition functional
$Z$ is indeed irrelevant if we are interested in
relatively short periods of time $t<t_0,$
where $t_0$ is a "tunnelling time". However, for $t>t_0,$
it becomes very large.
Consider the action (\ref{sf4}) in case of the symmetry
broken spontaneously. Classical minima of this action defined from
the equation:
\begin{equation}
\Delta \phi_a-m^2\phi_a+\frac g2 \left(\sum_1^2\phi^2_b\right)\phi_a=j,
\label{e2}
\end{equation}
where $\phi_{1,2}$ are the real and imaginary parts of the field $\phi$,
$\phi=(\phi_1+\phi_2)/\sqrt{2}$.
We use the anzatz
\begin{equation}
\phi=\mu (r)e^{i\frac{\Lambda}{\alpha}},
\label{anzatz}
\end{equation}
which gives the equation for $\mu(r)$ in the form,
\begin{equation}
\mu''-m^2\mu-\frac 2{r^2}\mu +g\mu^3=j.
\label{eq1}
\end{equation}
There exists a solution to (\ref{eq1}) with the properties:
\begin{equation}
\mu(r\to 0) \to 0, \quad \mu(r\to \infty)\to \alpha_0.
\label{bc}
\end{equation}
The problem of existence and stability of the solution of (\ref{eq1})
with (\ref{bc}) were discussed in \cite{Pol}. The effective asymptotical
solution is given in the preceding section by the Goldstone asymptotics.
Suppose now that one has introduced a set of vortices, placed at the points
$\bf x_a$ with circulations $\Gamma_a$ into the flux (\ref{sf4}).
The partition function of statistical mechanics $Z$ is then to be
presented in the form (in case of $ \alpha_0\gg u^2$)
\begin{equation}
Z=Z_0Z_{inst},
\label{pf}
\end{equation}
where $Z_0$ is the standard partition function of the theory (\ref{sf4})
and
\begin{equation}
Z_{inst}= Tr\left( \exp \left[\frac{\alpha_0}{2u^2}\int d{\bf x}{\ }
\sum_{a\ne b} \frac{2\pi \Gamma_a\Gamma_b L}{|\bf x_a-\bf x_b |}
+ C\sum_a \Gamma^2_a
\right] \right)
\label{zinst}
\end{equation}
($L$ being the size of the flux pattern considered; the second term
is the vortex self-energy). One can see that in case of large fluid flux
$\alpha_0$ and, consequently, strong confining property
the vortices revolving alternatively are combined into pairs. Such
pairs have very small influence on the correlation functions and are
irrelevant in case of large $\alpha_0.$ The asymptotics provided by the
instanton solutions is just the same as (\ref{fsb1}).
\section{Discussion from the Point of View of
Dynamical Systems Theory}
The Navier-Stockes
equation for an ideal fluid can be replaced by the relation for
the pressure field, and the Galilean invariance
of hydrodynamical equations is manifested
as a $U(1)-$gauge invariance (\ref{gi}).
We shall concern with the phase space (of infinite dimensionality)
relevant to the dynamical system of water coupled to perturbation
and limit ourselves to a qualitative consideration.
Picturing the instant states of the system in the phase space,
we obtain its phase diagram.
The stable stationary solutions discussed in \cite{1}
and in the present paper can be interpreted as the attraction regions
or fixed points of
trajectories of the system in the phase space.
Consider the manifold of initial conditions correspondent to the only
solution of disordered phase (${\cal P}=0$). In \cite{1}
we have shown that it can be realized exceptionally
as an eddy risen around a point-wise distinct
perturbation. One can imagine this manifold
as a torus covered
by the trajectories tending to some stable cycle (see Fig.5).
If we chose a point apart from the torus as
the initial condition (for example, the points $A$ or $B$ on the
diagram Fig. 5), the system will leave the vicinity
of the torus and tends to some region of attraction
which is closed in a sense that there are no trajectories
going out of it. This behavior represents a phase transition in the
language of statistical theory.
Within the attraction region the trajectory passes consequently through an
infinite set of fixed points distinguishing
by the $\Gamma_n$ values (\ref{gi}). Most of these points are
{\em hetreoclinic}, and so that they are unstable in a sense
that the smallest deviation from the certain set of initial
conditions will make the system trajectory
to jump to some other point. These processes, in principle,
are to be described by the instanton solutions
(see Sec. 5).
This technique would provide us with information on the transition
probabilities between the particular {\em heteroclinic } fixed
points. Such a quantity could be naturally interpreted within the context
of the dynamical systems theory. Let us surround each
{\em heteroclinic }
fixed point by a ball of radius $\varepsilon$ and consider a
fixed point $n_0$ which corresponds to
the solution with $\Gamma_{n_0}$ (see Fig. 5). Taking
$\varepsilon$ to be small enough,
we can make the volume of each ball to be finite. Denote the
volume of a ball sector starting from
which the trajectory of the system drops
into the $\varepsilon$-vicinity
of the other point, $n_i$, as $V_ \varepsilon (n_0\to n_i)$.
Then, one can introduce the quantity
\begin{equation}
P(n_0\to n_i) =\frac{V_ \varepsilon (n_0\to n_i)}{\sum_{k, k\ne 0}
V_ \varepsilon (n_0\to n_k)}
\label{pr}
\end{equation}
which is analogous to a transition probability defined in the
statistical theory.
If the point which we have chosen is a {\em homoclinic}
attractive fixed point, the probability (\ref{pr}) then tends
to zero. $0<P<1$ for {\em
heteroclinic} points, and $P=1$ for a repelling point.
We do not know {\em a priori} whether there are some {\em homoclinic}
attractive fixed points in the region of attraction
(see Fig. 5) or there are only the {\em
heteroclinic} points.
We expect though that in case of
$\alpha\gg u$ there is a degeneracy of solutions in a sense that
they are predicted by
the Goldstone asymptotics.
\section{Conclusion}
In a conclusion one can say that the flux of ideal fluid coupled
to local external perturbation in the region
$r> l_p$ demonstrates asymptotically some properties
similar to those of superfluids.
In \cite{1} and in the present paper we have considered
the statistically steady asymptotic solutions of the model
by various nonperturbative techniques of the quantum field theory.
The results on RG-analysis, Goldstone asymptotics, and
instanton-type solutions are, by the way, exact, and they
demonstrate that the long-standing hydrodynamical problem
of water coupled to perturbation, in principle, can be treated as a
critical phenomenon.
The relevant physical system
contains too many redundant degrees of freedom.
To fix the statistically stable behavior
in the system one needs to add
some extra assumptions on the character of perturbation
as well as on the character of wave modes coupling.
The model describing such a behavior is subject to
a phase transition managed basically by the
perturbation symmetry. Roughly speaking, the
symmetry properties of the initial perturbation define whether
the vorticial or translational fluid velocity
components is the most important one for
the long-range asymptotic fluid response.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 3,520 |
Disneyland Paris reopens revamped Pirates of the Caribbean
One of Disneyland Paris' most famous attractions, Pirates of the Caribbean, was officially reopened in late July.
The blockbuster Pirates of the Caribbean films provided the inspiration for the Imagineering team to come up with new twists and turns in the story and a chance to introduce new characters and magic to this classic attraction.
Visitors to the resort can now discover new surprises as well as a band of new buccaneers.
The first surprise is the presence of Jack Sparrow himself in the adventure. Indeed, he is present in two scenes – in one he hides in a barrel behind a pirate who holds the famous treasure map and in the other he sits triumphantly in an elaborately carved chair among the rich mounds of gold and jewels that he has found.
Captain Barbossa, from the popular films, also appears in the attraction for the first time, joining a skeletal crew in a scene unique to Disneyland Paris.
Additionally, ghostly visages of Davy Jones and Blackbeard warn that "dead men DO tell tales!"
Throughout the attraction, the show has been enhanced with new Audio-Animatronics figures, costumes, special effects, lighting and sound, including some of the now-classic musical themes from the films. The well-known A Pirates Life for Me! song continues to underscore the attraction.
Disneyland Paris guests will also discover some changes to the Blue Lagoon Restaurant, re-opened as Captain Jack's. The interior of the eatery has been re-themed and it has been staffed with a fun-loving band of pirates.
disneyland paris france pirates of the caribbean Theme Parks
Hong Kong Disneyland starts multi-year expansion
Ferrari World Abu Dhabi wins 'World's Leading Theme Park' title at WTA 2019...
Ferrari World Abu Dhabi wins 'World's Leading Theme Park' title at WTA 2019 | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 2,344 |
Source: IT News / DIPR
Imphal, June 22 2019: Chief Minister N.Biren Singh officially launched the Chief Minister's Green Manipur Mission by planting a tree sapling at the complex of Socio Economic and Environment Development Society (SEEDS) located near Saheibung village in Kangpokpi District today.
A brainchild of Chief Minister N Biren Singh, the mission has been introduced to develop and protect forests in each and every village of the State.
An open gym each worth Rs.5 to 6 lakhs will be installed as incentive at all the villages, which are willing to set aside at least 5 acres of land for development of forest.
Speaking at the launching function organised at the complex, the Chief Minister said that Manipur once known for its salubrious climate has also started facing the impact of environmental degradation.
As such, it is high time to protect and develop forests in each and every locality for a sustainable living, he observed.
Stating that our forefathers used to have a farsighted scientific tradition of conserving forests at Umang Lai (Sylvan Deity) complexes in every locality of Manipur, the Chief Minister said that all these wooded areas had now been destroyed by the current generation in the name of development.
He said that the State Government is even ready to issue Patta in the name of the village concerned if the land where forest would be developed is a wasteland.
Rampant felling of trees in the hills of Manipur has created soil erosion leading to siltation of rivers and lakes, the Chief Minister said.
He further said that accumulation of top soil can be seen instead of pebbles in the catchment areas of different rivers these days.
N Biren further opined that in many hill areas, the soil has become too soft and weak due to excessive use of chemical fertilizers in poppy plantation.
The Chief Minister also urged the Departments concerned and public to plant maximum number of saplings during this monsoon season.
Upon the demand of Saheibung villagers, the Chief Minister assured to consider opening of a Primary Health Sub-Centre (PHSC) in the village after discussing the matter in the State Cabinet.
He also said that the Government is also ready to construct a market shed at Chiru Lamkhai near Sangaithel Park if necessary land/space is provided by the villagers.
State Planning Board Deputy Chairman S Rajen Singh, Principal Chief Conservator of Forests (PCCF) K Angami, All Manipur Working Journalists Union (AMWJU) President Bijoy Kakchingtabam, SEEDS president A Mobi and Saheibung village chief Athang also attended the function as dignitaries.
The function was organised by Department of Forest with supports from Directorate of Environment and Socio Economic and Environment Development Society (SEEDS) . | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 7,652 |
Q: What's the difference between 'group' and 'grouping'? I'm not a native English speaker and I was wondering the difference between those two terms.
From what I understood so far 'group' is a generic word used to denote a number of persons/things considered related in some way.
'grouping' is used to denote a group of people sharing a common intent or interest.
Is this the case? It seems that 'grouping' has a more specific meaning and that 'group' can be used always in place of 'grouping' (because it's a more general term).
So in which case do you use 'group' and in which one do you use 'grouping' instead?
A: When you use the word group your emphasis should be on the collection of things in the group.
How many people do you have in your group?
When you use the word grouping your emphasis should be on the act of forming the group rather than the group itself.
Which grouping would be better- girls in one group and guys in another, or else adults vs children?
A: Perhaps the thing that is confusing, is that "group" can be both a noun and a verb, and "grouping" likewise.
"Group" as a noun means the collection of things.
"Grouping" as a noun, is a gerund. A Gerund, in English is :"As applied to English, it refers to the usage of a verb (in its -ing form) as a noun."
So, it's a verb being used as a noun! Maybe that's what's confusing you?
There is no difference in meaning whatsoever between these two nouns. The only difference, is their usage. That is, there are situations where one would use "group", but not "grouping" E.g.:
The group of people gradually grew in size.
I don't like the grouping system we have at work....
We cannot say, "I don't like the group system..." It's grammatically incorrect.
"Group" as a verb is to arrange into a group. To form a collection of people or items.
"Grouping" as a verb is the act of arranging the group in present time, E.g.: "I am grouping these letters together." Instead of "I am group..."
I hope that clears up any ambiguity.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,484 |
{"url":"https:\/\/inductive.examinationport.com","text":"Does mart do exams?\n\nFACILYMES OF ENGLISH During those days in click this seventies there were those who came to see me a few months before the advent of the Internet. Many of them were just having an afternoon coffee together, and you might be aware of the fact that the \u201cinternet\u201d was so widely used that it had the effect of creating an embarrassing and inconvenient situation. In some ways, during those days you got used to the idea that nowadays \u201cblogs\u201d get along with you, and that you must be keen on seeing each other\u2014or the \u201cinternet\u201d\u2014spending time in public. No matter what the news is, the Internet is the world\u2019s best source, most likely the \u201cgreatest source of information\u201d and its own best invention. Even if the news program is carried, that still remains the news source. Almost all of the information you get is sent back into the user. And finally, as you\u2019ve always said, because even \u201cinformation\u201d is too many to list here, too fast to follow all text, you need to be careful to appreciate the real world: no, they don\u2019t really care what you see. And they don\u2019t get you what you\u2019re looking for by the hours in a private room. Just look at the Internet users, people who manage the net for free. You\u2019ve seen lots of them. Don\u2019t worry. There\u2019s no evil that\u2019s coming for them, and everything that\u2019s in your life will be a little harder for you if you don\u2019t see the good. There\u2019s a lot on the Internet which isn\u2019t something you\u2019d expect from a conventional blog, but there\u2019s also a number of valuable resources which you must have (actually, there\u2019s one place in the world where you should go to when you have to). And in a typical day I\u2019d look for a place for people to visit people who are in a discussion with each other, and a few of them would go into them to say \u201chi,\u201d or \u201chi,\u201d so that you\u2019d at least take advantage of the opportunity to have them share with you. More and more people are going to be interested in them, and I\u2019ll think of them, and let them know it\u2019s a worthwhile venture, especially as they can buy and sell it fairly freely. While the truth is, the things done by the internet vary widely, and it\u2019s common that people who really like to find or get out of the connected world have to opt for them, particularly during those really hot days of the week (Monday to Friday) when the internet is so often the most valuable thing. So, without being a big fan of you or a single person at the same level, that\u2019s why I\u2019ll write here, for those interested in good news. At the same time, I\u2019m not a big fan of blogs, not a great fan of bloggers, but I do absolutely love to think of them and I\u2019m very much interested in seeing the impact eBooks will have (after having seen one chapter). So, I\u2019m going to write here to show you how awesome public viewing the Internet is and how useful it can be, and this is in addition to the fact that you can get to meet up with current, seemingly everyday people and get insight into what\u2019s really new and interesting, and so on. So I\u2019ll explain I _wanted_ to find a place for web users who are interested inDoes mart do exams? A: Because you can use and code samples and the code sample here isn\u2019t worth it.\n\nExam Help Sri Lanka\n\nYour write-up says: Please ask a reviewer like mine to review the original research file. No comments required. but then when I see this line: \\_document(SUMPRODUCTDATASURE, [${X_\\_=_\\_}B, MATCH${X_\\_=_\\_}=B, MATCH $DCCDSP_ADAPT], [WITH$GLOBAL=${\\_\\_}[\\_\\_]{], MATCH$DCCDSP_ADAPT`]) I can\u2019t find references about how this may change or not yet, but does it change _any_ things in the code? I saw this link, if you remember correctly: https:\/\/help.angular.com\/articles\/write-up\/code-and-the-testing If you write files like this: \\documentclass[hand sides=l]{wingshow} \\usepackage[hand sides=l]{geometry} \\usepackage[troubleshoot=yes]{babel} \\usepackage[deconstruction=no]{font-variant} \\usepackage{csiv} \\usepackage[mat]{elements} and those you can use it easily: \\csivlocale{l} \\csbox{ \\dimexpr{\\textbf{d_g,}\\text{}\\begin{itemize}[t]} \\fontsize\\smallnumber { \\displaystyle #\\textbf{Z_cmcl}\\indent\\size#\\textbf{a} \\fontsize\\smallnumber {\\hbox{\\end{itemize}{a}}} \\textbf{F_cmcl}{\\textbf{A}\\mathrm{[} $\\Big(\\psi^{E(\\lceil{\\alpha_C}}{\\textbf{A}}^{-1\\rceil})^{-1}$-L\\mathrm{R}{[}${\\left\\vert\\mathrm{a\\hspace{-\\frac{+}{\\psi} \\right\\rceil}\\mathrm{(}\\psi)^*{\\mathrm{[}\\psi]_{X,\\alpha_C}}$} \\right\\vert_{{\\psi}=1}$\\paratum{\\textstyle \\textbf{A}{[}${\\left\\vert\\mathrm{a\\hspace{-\\frac{+}{\\psi} \\right\\rceil}\\mathrm{(}\\psi)^*{\\mathrm{[}\\psi]_{X,\\alpha_C}}$} \\right\\vert_{\\psi=1}$ \\paratum{\\textstyle \\frac{+}{\\psi} \\right\\rceil}$} ]}${\\mathrm{X}\\rceil}:{\\textbf{A}}_{\\alpha_C}{\\mathrm{[}\\psi]_{\\mathrm{D}^{\\circ}}}{\\mathrm{}\\mathrm{d}_\\oplus {\\mathrm{0\\textbf{\\\u2018}}}}}}}{{{\\mathbb Z}}}{\\cdot}}\\chi_E\\cdot\\lambda {\\mathrm{d}_\\oplus {\\mathrm{0\\textbf{\\\u2018}}}}\\$ and so there isn\u2019t a need for any writing and getting. \\end{diagram} } \\cshape \\begin{diagram}[sas=3,dspace,hbox=\u201dblack\u201d, sep=\\dimexpr, l1=2pt] \\end{diagram} A: Here\u2019s a link to HowDoes mart do exams? Quiz: What do you do for an MA? Answer: My exams take 5-6 weeks to prepare! A: You only do exams after they have already begun, so you can\u2019t change the plan Me: I do tests after they get done. By this time I find out what results the most people would expect, for example, that I\u2019m on my first C in. A: You only work full-time. Me: You can only do ten consecutive days see this site weeks for one program (not for three or more) according to the exam rules. A: If you have more than three or more programs, you have to work for less than the C in for one program. Like any other program, things like doing a month for students and other things like getting a student into B\/C\/D. So where do you recruit my students? At what point at what? A: I think people ask, what are you expecting a school assignment to look like? Sometimes the school assignment is actually starting things going to a class or the school. Is this good or bad? A: The answer is: don\u2019t give up. This is what you should do. Don\u2019t give up and don\u2019t quit. But is the assignment going to be good (or bad)? A: No. If I got a couple of kids to spend two hours on it after class, something like \u201cHow many hours did your third, Get More Info 2012 semester?\u201d would probably work better. Me: Should I give them some ideas about how to fix this? A: They shouldn\u2019t be teaching in class. They should be working at hand-table meetings. I think the education system is all about the school group and the teacher, This Site and most of us wouldn\u2019t complain if we weren\u2019t worried.\n\nExamination Center List 2021\n\nIf you want to send a class back from the classroom to see how the school group has improved, if you want to send a class back that your best interest would be to have your best interest assigned to it. A: Exactly. Me: I ask that you use a teacher\u2019s group to accommodate the requirements of the school group. The kids don\u2019t have to sit next to each other \u2014 because our group always has to do the two-thirds of a 1-on-1 assignment. A: It should be okay. Me: I told you it was okay. Really. A: I don\u2019t feel obliged to send your best interests from your group to any school group. Then again, some other people who\u2019ve been teaching a class, even to students under 14, and heard of the school group haven\u2019t been teaching a class to all of you for well over a year. So this is not how the school group works: every one of the teachers, their groups, how they approach the group, and then also the leaders and the faculty from the school as a whole, learn have a peek at this website group wants and what they need. I know they have other problems, so I want you to respect and appreciate that. Because I\u2019m the one who did that day and I didn\u2019t know that school group taught more than our teachers do, me and my colleagues, are looking to help the school group know what is the best interest best for you and their students and why you should join us if you are going to hand down a senior thesis at the school. Even if you are not doing that or you have other problems that might affect, say, the academic work you are doing anyway, and would you like to know the best interest of your peers so that you can see what their work is. What they need, what they can get, what they believe in. Look for somebody to explain it to you. You don\u2019t want that. You want it to stay true to itself. Talk to somebody about it. I don\u2019t expect you to believe me. A: Really.\n\nExam Help With Shan Sir\n\nMe: You want to take our school group to the school for us. 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\section{Introduction}
The Bibliometric-enhanced Information Retrieval (BIR) workshop series has started at ECIR in 2014~\cite{Mayr2014} and serves as the annual gathering of IR researchers who address various information-related tasks on scientific corpora and bibliometrics~\cite{Mayr2015}. The workshop features original approaches to search, browse, and discover value-added knowledge from scientific documents and related information networks (e.g., terms, authors, institutions, references).
The current incarnation is a continuation of the evolution of our workshop series. The first BIR workshops set the research agenda by introducing the workshop topics, illustrating state-of-the-art methods, reporting on current research problems, and brainstorming about common interests. For the fourth workshop, co-located with the ACM/IEEE-CS JCDL 2016, we broadened the workshop scope and interlinked the BIR workshop with the natural language processing (NLP) and computational linguistics field~\cite{Cabanac2016}. This joint activity has been continued in 2017 at SIGIR in the second BIRNDL workshop~\cite{mayr-SIGIRforum2017}.
This 7th full-day BIR workshop at ECIR 2018\footnote{\url{http://bit.ly/bir2018}} aimed to foster a common ground for the incorporation of bibliometric-enhanced services (including text mining functionality) into scholarly search engine interfaces. In particular we addressed specific communities, as well as studies on large, cross-domain collections. This workshop strived to feature contributions from core bibliometricians and core IR specialists who already operate at the interface between scientometrics and IR.
\section{Overview of the papers}
This year's workshop hosted two keynotes as well as a set of regular papers and two demos\footnote{Workshop proceedings are available at: \url{http://ceur-ws.org/Vol-2080/}}. The publications are briefly outlined in the following subsections.
\subsection{Keynotes}
This workshop featured two inspirational keynotes to kick-start thinking and discussion on the workshop topic. They were followed by paper presentations and demos (Fig.~\ref{fig:collage}) in a format that we found to be successful at previous BIR workshops.
Cyril Labbé tackled a hot topic in his keynote titled ``Trends in gaming indicators: On failed attempts at deception and their computerised detection''~\cite{Labbe2018}. He outlined various efforts to manipulate indicators by tricking the scientific community (e.g., by submitting automatically generated papers). Other issues undermining the trust we place in peer-reviewed science were examined, such as data--results mismatch impeding the reproduction of results in cancer research. Labbé surveyed his recent work in these areas while reflecting on the potential of B+IR (bibliometrics and information retrieval) to address these critical issues.
Ralf Schenkel presented in his keynote ``Integrating and exploiting metadata sources in a bibliographic information system''~\cite{Schenkel2018} an in-depth summary of recent metadata activities in the computer science bibliography DBLP, which is maintained by Schloss Dagstuhl and University of Trier. He outlined procedures for monitoring, selecting, and prioritizing computer science venues for inclusion in the DBLP bibliography. A special focus was given to author disambiguation and utilization of citation data.
\begin{figure}\centering
\includegraphics[width=\linewidth]{BIR_Collage_20pct}
\caption{A sense of the atmosphere at the BIR workshop.}\label{fig:collage}
\end{figure}
\subsection{Regular papers}
Sarol, Liu, and Schneider proposed a citation and text-based publication retrieval framework~\cite{Sarol2018}. After the user provides some seed articles, the system collects papers connected by citations and applies a combination of citation- and text-based filtering methods. The framework is evaluated in a systematic reviewing task.
Ollagnier, Fournier, and Bellot highlighted the central references of a paper based on the mining of its fulltext, quantifying the occurences of all in-text references~\cite{Ollagnier2018}. They benchmarked this approach compared to a system in production at OpenEdition,\footnote{\url{https://www.openedition.org}} and discuss the results in terms of enhanced relevance.
In their article on query expansion, Rattinger, Le Goff, and Guetl combined word embeddings and co-authorship relations~\cite{Rattinger2018}. The set of documents used for pseudo-relevance feedback was enriched by similar documents from co-authors, applying a locally trained Word2Vec model. Adding similar documents from co-authors significantly improved the baseline.
Bertin and Atanassova reported on the construction of the InTeReC dataset~\cite{Bertin2018}. Utilising different section types from PLOS articles, InTeReC consists of within-text references and their surrounding sentences. Additionally, verb phrases were extracted, providing an idea of the nature of the reference.
Kacem and Mayr investigated the usage and influence of a specific search stratagem -- the Journal Run -- in an academic search engine log file \cite{Kacem2018}. They studied the frequency and stage of use of journal run as well as its impact on sessions. The authors found that the frequency of usage of the analyzed journals is not related to the impact factor within these sessions and that the size of the journal (Bradford Zones) has an insignificant correlation.
\subsection{Demo papers}
Cataldi, Di Caron, and Schifanella designed the $d$-index to evaluate the degree of dependence of a researcher with respect to his/her co-authors over time. They implemented this indicator and demonstrate it online\footnote{\url{http://d-index.di.unito.it}} with DBLP as a bibliographic datasource~\cite{Cataldi2018}.
The demo paper by Bessagnet presented a framework combining thematic, temporal, and spatial features of Twitter tweets in the field of Human and Social Sciences \cite{Bessagnet2018}. The author promoted 5~W dimensions (who, when, what, where, why) for the analysis of tweets.
\section{Outlook}
While the past workshops laid the foundations for further work and also made the benefit of bringing information retrieval and bibliometrics together more explicit, there are still many challenges ahead. One of them is to provide infrastructures and testbeds for the evaluation of retrieval approaches that utilise bibliometrics and scientometrics. To this end, a focus of the proposed workshop and the discussion was on real experimentations~(including demos) and industrial participation. This line was started in a related workshop at JCDL~(BIRNDL~2016) and continued at SIGIR~(BIRNDL~2017), but with a focus on digital libraries and computational linguistics. Given the complex information needs scholars are usually facing, we emphasized on information retrieval and information seeking and searching aspects.
In July 2018 we will run the third iteration of the BIRNDL workshop\footnote{\url{http://wing.comp.nus.edu.sg/birndl-sigir2018/}} at the 41st SIGIR conference in Ann Arbor, MI, USA. In conjunction with the BIRNDL workshop, the 4th CL-SciSumm Shared Task in Scientific Document Summarization\footnote{\url{http://wing.comp.nus.edu.sg/cl-scisumm2018/}} will be hold.
\section{Further Reading}
In 2015 we published a first special issue on ``Combining Bibliometrics and Information Retrieval'' in the \emph{Scientometrics} journal~\cite{Mayr2015}.
A special issue on ``Bibliometrics, Information Retrieval and Natural Language Processing in Digital Libraries'' will appear in 2018 in the \emph{International Journal on Digital Libraries} \cite{BIRNDL-SP-IJDL}.
Another special issue on ``Bibliometric-enhanced Information Retrieval and Scientometrics'' is in preparation for the \emph{Scientometrics} journal.
Since 2016 we maintain the ``Bibliometric-enhanced-IR Bibliography''\footnote{\url{https://github.com/PhilippMayr/Bibliometric-enhanced-IR_Bibliography/}} that collects scientific papers which appear in collaboration with the BIR/BIRNDL organizers. We invite interested researchers to join this project and contribute related publications.
\section{Acknowledgement}
We wish to thank all those who have contributed to the workshop proceedings: all the contributing authors and the many reviewers who generously offered their time and expertise\footnote{The list of PC members can be found at \url{https://www.gesis.org/en/services/events/events-archive/conferences/ecir-workshops/ecir-workshop-2018/}}.
\bibliographystyle{splncs}
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è una stazione della metropolitana di Osaka situata nell'area est della città. La stazione serve la Linea Chūō e si trova non lontano dal castello di Osaka e dalla stazione di Hanaten della JR West.
Struttura
La stazione è dotata di due marciapiedi con due binari sotterranei.
Stazioni adiacenti
Altri progetti
Fukaebashi | {
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{"url":"https:\/\/aimsciences.org\/article\/doi\/10.3934\/dcdss.2011.4.247","text":"American Institute of Mathematical Sciences\n\nApril\u00a0 2011,\u00a04(2):\u00a0247-271. doi:\u00a010.3934\/dcdss.2011.4.247\n\nGlobal and exponential attractors for a Ginzburg-Landau model of superfluidity\n\n 1 Facolt\u00e0 di Ingegneria, Universit\u00e0 e-Campus, 22060 Novedrate (CO), Italy 2 Dipartimento di Matematica, Universit\u00e0 di Bologna, 40126 Bologna, Italy 3 Dipartimento di Matematica e Informatica, Universit\u00e0 di Salerno, 84084 Fisciano (SA), Italy\n\nReceived\u00a0 October 2008 Revised\u00a0 June 2009 Published\u00a0 November 2010\n\nThe long-time behavior of the solutions for a non-isothermal model in superfluidity is investigated. The model describes the transition between the normal and the superfluid phase in liquid 4He by means of a non-linear differential system, where the concentration of the superfluid phase satisfies a non-isothermal Ginzburg-Landau equation. This system, which turns out to be consistent with thermodynamical principles and whose well-posedness has been recently proved, has been shown to admit a Lyapunov functional. This allows to prove existence of the global attractor which consists of the unstable manifold of the stationary solutions. Finally, by exploiting recent techinques of semigroups theory, we prove the existence of an exponential attractor of finite fractal dimension which contains the global attractor.\nCitation: Alessia Berti, Valeria Berti, Ivana Bochicchio. Global and exponential attractors for a Ginzburg-Landau model of superfluidity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (2) : 247-271. doi: 10.3934\/dcdss.2011.4.247\nReferences:\n [1] R. A. Adams, \"Sobolev Spaces,\", Academic Press, (1975). Google Scholar [2] A. V. Babin and M. I. Vishik, \"Attractors of Evolution Equations,\", North-Holland, (1992). Google Scholar [3] V. Berti and S. Gatti, Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations,, Quart. Appl. Math., 64 (2006), 617. Google Scholar [4] V. Berti and M. Fabrizio, Existence and uniqueness for a mathematical model in superfluidity,, Math. Meth. Appl. Sci., 31 (2008), 1441. doi:\u00a010.1002\/mma.981. Google Scholar [5] V. Berti, M. Fabrizio and C. Giorgi, Gauge invariance and asymptotic behavior for the Ginzburg-Landau equations of superconductivity,, J. Math. Anal. Appl., 329 (2007), 357. doi:\u00a010.1016\/j.jmaa.2006.06.031. Google Scholar [6] M. Brokate and J. Sprekels, \"Hysteresis and Phase Transitions,\", Springer, (1996). Google Scholar [7] M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity,, Commun. Pure Appl. Anal., 4 (2005), 705. doi:\u00a010.3934\/cpaa.2005.4.705. Google Scholar [8] Q. Du, Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity,, Appl. Anal., 53 (1994), 1. doi:\u00a010.1080\/00036819408840240. Google Scholar [9] A. Eden, C. Foias, B. Nicoalenko and R. Temam, \"Exponential Attractors for Dissipative Evolution Equations,\", John-Wiley, (1994). Google Scholar [10] L. C. Evans, \"Partial Differential Equations,\" Graduate Studies in Mathematics, 19,, American Mathematical Society, (1998). Google Scholar [11] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\\mathbbR^3$,, C.R. Acad.Sci. Paris Ser. I Math., 330 (2000), 713. doi:\u00a010.1016\/S0764-4442(00)00259-7. Google Scholar [12] M. Fabrizio, Ginzburg-Landau equations and first and second order phase transitions,, Internat. J. Engrg. Sci., 44 (2006), 529. doi:\u00a010.1016\/j.ijengsci.2006.02.006. Google Scholar [13] M. Fabrizio, A Ginzburg-Landau model for the phase transition in Helium II,, Z. Angew. Math. Phys., 61 (2010), 329. doi:\u00a010.1007\/s00033-009-0011-5. Google Scholar [14] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation,, Discrete Contin. Dynam. Systems, 10 (2004), 211. doi:\u00a010.3934\/dcds.2004.10.211. Google Scholar [15] J. Fleckinger-Pell\u00e9, H. Kaper and P. Takac, Dynamics of the Ginzburg-Landau equations of superconductivity,, Nonlinear Anal., 32 (1998), 647. doi:\u00a010.1016\/S0362-546X(97)00508-7. Google Scholar [16] S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors,, Proc. Amer. Math. Soc., 134 (2006), 117. doi:\u00a010.1090\/S0002-9939-05-08340-1. Google Scholar [17] J. K. Hale, \"Asymptotic Behavior of Dissipative Systems,\", Amer. Math. Soc., (1988). Google Scholar [18] H. G. Kaper and P. Takac, An equivalence relation for the Ginzburg-Landau equations of superconductivity,, Z. Angew. Math. Phys., 48 (1997). doi:\u00a010.1007\/s000330050054. Google Scholar [19] K. Mendelssohn, Liquid Helium,, in, XV (1956), 370. Google Scholar [20] R. Nibbi, Some generalized Poincar\u00e9 inequalities and applications to problems arising in electromagnetism,, J. Inequal. Appl., 4 (1999), 283. doi:\u00a010.1155\/S1025583499000405. Google Scholar [21] A. Rodriguez-Bernal, B. Wang and R. Willie, Asymptotic behaviour of the time-dependent Ginzburg-Landau equations of superconductivity,, Math. Meth. Appl. Sci., 22 (1999), 1647. doi:\u00a010.1002\/(SICI)1099-1476(199912)22:18<1647::AID-MMA97>3.0.CO;2-W. Google Scholar [22] Q. Tang Q and S. Wang, Time dependent Ginzburg-Landau superconductivity equations,, Physica D, 88 (1995), 130. Google Scholar [23] R. Temam, \"Infinite-Dimensional Dynamical Systems in Mechanics and Physics,\", Springer-Verlag, (1988). Google Scholar [24] D. R. Tilley and J. Tilley, \"Superfluidity and Superconductivity,\" Graduate student series in physics 13,, Bristol, (1990). Google Scholar [25] M. Tinkham, \"Introduction to Superconductivity,\", McGraw-Hill, (1975). Google Scholar\n\nshow all references\n\nReferences:\n [1] R. A. Adams, \"Sobolev Spaces,\", Academic Press, (1975). Google Scholar [2] A. V. Babin and M. I. Vishik, \"Attractors of Evolution Equations,\", North-Holland, (1992). Google Scholar [3] V. Berti and S. Gatti, Parabolic-hyperbolic time-dependent Ginzburg-Landau-Maxwell equations,, Quart. Appl. Math., 64 (2006), 617. Google Scholar [4] V. Berti and M. Fabrizio, Existence and uniqueness for a mathematical model in superfluidity,, Math. Meth. Appl. Sci., 31 (2008), 1441. doi:\u00a010.1002\/mma.981. Google Scholar [5] V. Berti, M. Fabrizio and C. Giorgi, Gauge invariance and asymptotic behavior for the Ginzburg-Landau equations of superconductivity,, J. Math. Anal. Appl., 329 (2007), 357. doi:\u00a010.1016\/j.jmaa.2006.06.031. Google Scholar [6] M. Brokate and J. Sprekels, \"Hysteresis and Phase Transitions,\", Springer, (1996). Google Scholar [7] M. Conti and V. Pata, Weakly dissipative semilinear equations of viscoelasticity,, Commun. Pure Appl. Anal., 4 (2005), 705. doi:\u00a010.3934\/cpaa.2005.4.705. Google Scholar [8] Q. Du, Global existence and uniqueness of solutions of the time-dependent Ginzburg-Landau model for superconductivity,, Appl. Anal., 53 (1994), 1. doi:\u00a010.1080\/00036819408840240. Google Scholar [9] A. Eden, C. Foias, B. Nicoalenko and R. Temam, \"Exponential Attractors for Dissipative Evolution Equations,\", John-Wiley, (1994). Google Scholar [10] L. C. Evans, \"Partial Differential Equations,\" Graduate Studies in Mathematics, 19,, American Mathematical Society, (1998). Google Scholar [11] M. Efendiev, A. Miranville and S. Zelik, Exponential attractors for a nonlinear reaction-diffusion system in $\\mathbbR^3$,, C.R. Acad.Sci. Paris Ser. I Math., 330 (2000), 713. doi:\u00a010.1016\/S0764-4442(00)00259-7. Google Scholar [12] M. Fabrizio, Ginzburg-Landau equations and first and second order phase transitions,, Internat. J. Engrg. Sci., 44 (2006), 529. doi:\u00a010.1016\/j.ijengsci.2006.02.006. Google Scholar [13] M. Fabrizio, A Ginzburg-Landau model for the phase transition in Helium II,, Z. Angew. Math. Phys., 61 (2010), 329. doi:\u00a010.1007\/s00033-009-0011-5. Google Scholar [14] P. Fabrie, C. Galusinski, A. Miranville and S. Zelik, Uniform exponential attractors for a singularly perturbed damped wave equation,, Discrete Contin. Dynam. Systems, 10 (2004), 211. doi:\u00a010.3934\/dcds.2004.10.211. Google Scholar [15] J. Fleckinger-Pell\u00e9, H. Kaper and P. Takac, Dynamics of the Ginzburg-Landau equations of superconductivity,, Nonlinear Anal., 32 (1998), 647. doi:\u00a010.1016\/S0362-546X(97)00508-7. Google Scholar [16] S. Gatti, M. Grasselli, A. Miranville and V. Pata, A construction of a robust family of exponential attractors,, Proc. Amer. Math. Soc., 134 (2006), 117. doi:\u00a010.1090\/S0002-9939-05-08340-1. Google Scholar [17] J. K. Hale, \"Asymptotic Behavior of Dissipative Systems,\", Amer. Math. Soc., (1988). Google Scholar [18] H. G. Kaper and P. Takac, An equivalence relation for the Ginzburg-Landau equations of superconductivity,, Z. Angew. Math. Phys., 48 (1997). doi:\u00a010.1007\/s000330050054. Google Scholar [19] K. Mendelssohn, Liquid Helium,, in, XV (1956), 370. Google Scholar [20] R. Nibbi, Some generalized Poincar\u00e9 inequalities and applications to problems arising in electromagnetism,, J. Inequal. Appl., 4 (1999), 283. doi:\u00a010.1155\/S1025583499000405. Google Scholar [21] A. Rodriguez-Bernal, B. Wang and R. Willie, Asymptotic behaviour of the time-dependent Ginzburg-Landau equations of superconductivity,, Math. Meth. Appl. Sci., 22 (1999), 1647. doi:\u00a010.1002\/(SICI)1099-1476(199912)22:18<1647::AID-MMA97>3.0.CO;2-W. Google Scholar [22] Q. Tang Q and S. Wang, Time dependent Ginzburg-Landau superconductivity equations,, Physica D, 88 (1995), 130. Google Scholar [23] R. Temam, \"Infinite-Dimensional Dynamical Systems in Mechanics and Physics,\", Springer-Verlag, (1988). Google Scholar [24] D. R. Tilley and J. Tilley, \"Superfluidity and Superconductivity,\" Graduate student series in physics 13,, Bristol, (1990). Google Scholar [25] M. Tinkham, \"Introduction to Superconductivity,\", McGraw-Hill, (1975). Google Scholar\n [1] Gregory A. Chechkin, Vladimir V. Chepyzhov, Leonid S. Pankratov. Homogenization of trajectory attractors of Ginzburg-Landau equations with randomly oscillating terms. Discrete & Continuous Dynamical Systems - B, 2018, 23 (3) : 1133-1154. doi: 10.3934\/dcdsb.2018145 [2] Hans G. Kaper, Bixiang Wang, Shouhong Wang. Determining nodes for the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 205-224. doi: 10.3934\/dcds.1998.4.205 [3] Dmitry Glotov, P. J. McKenna. Numerical mountain pass solutions of Ginzburg-Landau type equations. Communications on Pure & Applied Analysis, 2008, 7 (6) : 1345-1359. doi: 10.3934\/cpaa.2008.7.1345 [4] Dmitry Turaev, Sergey Zelik. Analytical proof of space-time chaos in Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (4) : 1713-1751. doi: 10.3934\/dcds.2010.28.1713 [5] Noboru Okazawa, Tomomi Yokota. Smoothing effect for generalized complex Ginzburg-Landau equations in unbounded domains. Conference Publications, 2001, 2001 (Special) : 280-288. doi: 10.3934\/proc.2001.2001.280 [6] N. I. Karachalios, H. E. Nistazakis, A. N. Yannacopoulos. Remarks on the asymptotic behavior of solutions of complex discrete Ginzburg-Landau equations. Conference Publications, 2005, 2005 (Special) : 476-486. doi: 10.3934\/proc.2005.2005.476 [7] Yuta Kugo, Motohiro Sobajima, Toshiyuki Suzuki, Tomomi Yokota, Kentarou Yoshii. Solvability of a class of complex Ginzburg-Landau equations in periodic Sobolev spaces. Conference Publications, 2015, 2015 (special) : 754-763. doi: 10.3934\/proc.2015.0754 [8] Bixiang Wang, Shouhong Wang. Gevrey class regularity for the solutions of the Ginzburg-Landau equations of superconductivity. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 507-522. doi: 10.3934\/dcds.1998.4.507 [9] Kolade M. Owolabi, Edson Pindza. Numerical simulation of multidimensional nonlinear fractional Ginzburg-Landau equations. Discrete & Continuous Dynamical Systems - S, 2018, 0 (0) : 835-851. doi: 10.3934\/dcdss.2020048 [10] Yan Zheng, Jianhua Huang. Exponential convergence for the 3D stochastic cubic Ginzburg-Landau equation with degenerate noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (11) : 1-12. doi: 10.3934\/dcdsb.2019075 [11] Bo You, Yanren Hou, Fang Li, Jinping Jiang. Pullback attractors for the non-autonomous quasi-linear complex Ginzburg-Landau equation with $p$-Laplacian. Discrete & Continuous Dynamical Systems - B, 2014, 19 (6) : 1801-1814. doi: 10.3934\/dcdsb.2014.19.1801 [12] Micka\u00ebl Dos Santos, Oleksandr Misiats. Ginzburg-Landau model with small pinning domains. Networks & Heterogeneous Media, 2011, 6 (4) : 715-753. doi: 10.3934\/nhm.2011.6.715 [13] Fanghua Lin, Ping Zhang. On the hydrodynamic limit of Ginzburg-Landau vortices. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 121-142. doi: 10.3934\/dcds.2000.6.121 [14] N. I. Karachalios, Hector E. Nistazakis, Athanasios N. Yannacopoulos. Asymptotic behavior of solutions of complex discrete evolution equations: The discrete Ginzburg-Landau equation. Discrete & Continuous Dynamical Systems - A, 2007, 19 (4) : 711-736. doi: 10.3934\/dcds.2007.19.711 [15] Tianlong Shen, Jianhua Huang. Ergodicity of the stochastic coupled fractional Ginzburg-Landau equations driven by \u03b1-stable noise. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 605-625. doi: 10.3934\/dcdsb.2017029 [16] Iuliana Oprea, Gerhard Dangelmayr. A period doubling route to spatiotemporal chaos in a system of Ginzburg-Landau equations for nematic electroconvection. Discrete & Continuous Dynamical Systems - B, 2019, 24 (1) : 273-296. doi: 10.3934\/dcdsb.2018095 [17] Dingshi Li, Xiaohu Wang. Asymptotic behavior of stochastic complex Ginzburg-Landau equations with deterministic non-autonomous forcing on thin domains. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 449-465. doi: 10.3934\/dcdsb.2018181 [18] Yun Lan, Ji Shu. Dynamics of non-autonomous fractional stochastic Ginzburg-Landau equations with multiplicative noise. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2409-2431. doi: 10.3934\/cpaa.2019109 [19] Dingshi Li, Lin Shi, Xiaohu Wang. Long term behavior of stochastic discrete complex Ginzburg-Landau equations with time delays in weighted spaces. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 5121-5148. doi: 10.3934\/dcdsb.2019046 [20] Leonid Berlyand, Volodymyr Rybalko, Nung Kwan Yip. Renormalized Ginzburg-Landau energy and location of near boundary vortices. 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Local • Parishes
by Moira Cullings
Jackie Schmidt, a parishioner of St. Francis de Sales Parish in Lansing, stands in front of her "Art of Diamonds" collection, which includes an elaborate Epiphany scene on her right. Schmidt has around 63 diamond paintings and a handful of memo books she is selling to raise money for her church. LEAVEN PHOTO BY MOIRA CULLINGS
moira.cullings@theleaven.org
LEAVENWORTH — Walking through Jackie Schmidt's home here is a treat.
Dozens of diamond paintings glisten throughout the house — from the mantle in the living room to the kitchen walls.
Each piece of art was created by Schmidt, and she hopes to sell them all — not for herself, but for St. Francis de Sales in Lansing, her home parish.
"The Lord has blessed me in so many ways," she said, "that I just feel that I owe it to the church.
"I will repay the church for all the good stuff the Lord has done for me."
Since starting her "Art of Diamonds" collection in June 2019, Schmidt has created around 100 print pieces and a handful of memo books.
The diamond paintings portray a variety of subjects — from an eagle flying in front of a mountain range to horses in a pasture to an elaborate Epiphany scene.
The spiritual ones tend to be the most popular, she said.
Schmidt's artistic process begins with ordering a diamond painting art print online — typically from Amazon, Bonanza or Paint with Diamonds.
Each print comes with the necessary supplies, including round or square diamond drills, pens and glue.
The prints are sticky and outlined with marked squares or circles, and the diamonds come in labeled bags that match the spots they're meant for.
Schmidt places each diamond in its correct place using the pen and glue, and when the piece is complete, she takes a rolling pin to it so that it sets in place.
The process can take days, with one of the largest diamond paintings taking her 150 hours to complete.
But the final product is a glimmering, colorful work of art that brightens up the room.
Father Bill McEvoy, pastor of St. Francis de Sales, has seen firsthand the value of Schmidt's handiwork.
Schmidt donated a diamond painting of Mary and the infant Jesus to the parish. It hangs in a religious education room, where pastoral council and finance council meetings take place.
"Jackie, like so many other members of the parish, uses her own unique talents and abilities to further our mission," said Father McEvoy.
"Her art is very uplifting," he added, "and the work is so meticulous, that they are done as acts of love.
"We're blessed for her art and her commitment to the parish, especially during these dark, pandemic days."
Although Schmidt's work is tedious, it's a gratifying way to pass the time, she said.
"I don't realize the hours that I put in," said Schmidt. "My husband [Gene] will be like, 'You've been at this for 16 hours.'
"I cannot wait to get up in the morning and start working."
Schmidt's favorite creation is an image of the Sacred Heart of Jesus.
"As long as I can remember, when I was just a little kid, my mother had that picture in her bedroom," she said. "She's 98. She probably had it for 67 years."
Her mom now lives in a nursing home in Canada, where Schmidt is from, but they haven't been able to see each other in a while because of COVID-19.
Schmidt hopes she'll be able to visit her mom soon, and that she can bring the diamond painting of Jesus with her as a gift.
"If I do sell that one, I'll make her another one," she said. "Or even donate it to the nursing home she's at if something happens to her.
"It would be a memory of my mom."
Father McEvoy is grateful for parishioners like Schmidt who use their gifts in the aid of others, and he hopes her art will be a beacon of light.
"These days, we need reminders of hope," he said. "We need to be lifted up.
"Jackie's bright and colorful art reminds us that we're not alone, and that God is with us, and God will never let us down."
View the 'Art of Diamonds' collection
Jackie Schmidt invites you to view her artwork, two to three people at a time, with exceptions for families. She currently has around 63 pieces and a handful of memo books for sale. If you are interested in seeing the art collection or making a purchase, call or text her at (913) 683-1791. She accepts cash or checks, and all proceeds are currently going toward St. Francis de Sales Parish in Lansing. In the future, she plans on giving all proceeds from the sales to various charitable causes.
Moira Cullings
Moira attended St. Thomas Aquinas High School in Overland Park and Benedictine College in Atchison. She majored in marketing, minored in psychology and played center midfield for the women's soccer team. Moira joined The Leaven staff as a feature writer and social media editor in 2015. After a move to Denver in 2018, Moira resumed her full-time position at The Leaven and continues to write and manage the website and social media channels. Her favorite assignment was traveling to the Holy Land to take photos for a group pilgrimage in 2019.
Moral considerations regarding the new COVID-19 vaccines | {
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{"url":"https:\/\/www.physicsforums.com\/threads\/generalized-velocity-lagrangian.933391\/","text":"# Homework Help: Generalized Velocity: Lagrangian\n\nTags:\n1. Dec 4, 2017\n\n### WWCY\n\n1. The problem statement, all variables and given\/known data\n\nIn this example, I know that I can define the horizontal contribution of kinetic energy to the ball as $\\frac{1}{2}m(\\dot{x} + \\dot{X})^2$.\n\nIn the following example,\n\nMass $M_{x1}$'s horizontal contribution to KE is defined as $\\frac{1}{2}m(\\dot{X} - \\dot{x_1})^2$. Why is this? I have a hunch that it is due to the \"origin\" ($X$ line) $x_1$ and $x_2$ originate from, though I can't exactly put my finger on the exact reason.\n\nAssistance is greatly appreciated!\n\n2. Relevant equations\n\n3. The attempt at a solution\n\nLast edited: Dec 4, 2017\n2. Dec 4, 2017\n\n### kuruman\n\nIt has to do with the way the generalized coordinates are defined. In the top drawing $\\dot{x}$ increases to the right (although not clear from the double arrow) so the velocity relative to the ground is $\\dot{X}+\\dot{x}$. In the second drawing, $x_1$ increases to the left while $X$ increases to the right, so the relative velocity would be $\\dot{X}-\\dot{x_1}$.\n\n3. Dec 4, 2017\n\n### WWCY\n\nThanks for the response,\n\nWould I be right to say that this is also the result of vector addition? Edit: With $x_1$ being defined as positive vector\n\n4. Dec 4, 2017\n\n### kuruman\n\nYou could say that considering that it involves defining cartesian coordinates relative to origin $O_1$ and then adding a cartesian coordinate to define origin $O_1$ relative to new origin $O_2$.","date":"2018-11-18 04:07:37","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 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\": 0.9344028830528259, \"perplexity\": 569.7474641915671}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.3, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2018-47\/segments\/1542039743963.32\/warc\/CC-MAIN-20181118031826-20181118053826-00070.warc.gz\"}"} | null | null |
\section{Introduction}
\vskip0.1cm
In \cite{be1}, Burns and Epstein define a global, biholomorphic, $\mathbf{R}$-valued invariant $\mu$ of a compact, strictly
pseudoconvex $3$-dimensional CR manifold $M$ whose holomorphic tangent bundle is trivial.
As the Burns-Epstein invariant is defined through the transgression form, it depends on the Cartan connection of the CR
structure in a delicate way. Therefore it is not easy to compute the Burns-Epstein invariants for a general CR 3-manifold. In Burns-Epstein's work \cite{be1}, they compute the Burns-Epstein invariant for tangent circle bundles over Riemann surfaces and Reinhardt domain in $\mathbf{C}^2.$ In \cite{be2}, Burns and Epstein raise the following question: ''An interesting question is left open here about the relationship of these invariants to the K$\ddot {\text a}$hler geometry of the interior manifold, and the behavior of developing maps for CR manifolds which are locally CR equivalent to the standard sphere''.
This paper is an attempt to answer the second part of this question.
Namely, we show that for a spherical CR homology sphere, the Burns-Epstein invariant, modulo an integer, is basically a
"topological'' invariant. More precisely, it coincides with minus the Chern-Simons
invariant of the holonomy representation. The main result of our is the development of a cut-and-paste method, inspired from the works of P. Kirk and E. Klassen in gauge theory (\cite{kk1,kk2}), to compute the Burns-Epstein invariant, modulo an integer, of spherical CR homology spheres. We first defined the normal forms of a flat connection near the torus boundary and then prove a result (Theorem 5.1) which expresses the change of the Chern-Simons invariants of a path of normal form flat connections on a manifold with boundary in terms of the boundary holonomy. To compute the Chern-Simons invariant on a closed manifold $M,$ we decompose it as union of manifolds with torus boundary. On each manifold with boundary, we try to connect our original connection to a connection whose Chern-Simons invariant is already known and then use Theorem 5.1 to compute its Chern-Simons invariant.
This method has been applied successfully to compute the Chern-Simons invariants of representations into ${\rm SU}(2), {\rm SL}(2,\C), {\rm SU}(n)$ (\cite{kk1,kk2,nishi}) and the Godbillon-Vey invariants of foliations (\cite{k1}).
The rest of this paper is organized as follows. In the next section, we recall some preliminaries about the universal covering group $G$ of ${\rm U}(2,1)$ and the Burns-Epstein invariant of a CR manifold. In section 3, we show that, up to an integer, the Burns-Epstein invariant of spherical CR homology sphere equals minus the Chern-Simons invariant of its holonomy representation.
Section 4 contains technical results which allow us to define a normal form of a flat $G$ connection on a manifold with boundary. In section 5, We prove Theorem 5.1 which expresses the change of the Chern-Simons invariant of a path of flat connections in terms of the boundary holonomy. This theorem are our tool to compute the Chern-Simons invariant in section 6, where an explicit formula for the Chern-Simons invariant of a Seifert fibered homology sphere is given. As an illustration, we carry out an explicit computation of the Burns-Epstein invariants, modulo an integer, of the homology sphere $\Sigma(2,3,11).$
\noindent
{\bf Acknowledgment.}
The main part of this paper was written during 2005-2006, when the author is supported by a COE Postdoctoral Fellowship from the Graduate School of Mathematical Sciences, University of Tokyo and the JSPS's Kakenhi Grant. The author would like to express his gratitude to the Graduate School of Mathematical Sciences for its support and especially to Professor Takashi Tsuboi his guidance and encouragement during that time.
\section{ Preliminaries}
Recall that the group ${\rm U}(2,1)$ is the matrix group consists of all the $3\times 3$ matrices $A=(a_{ij})_{i,j=1,\dots,3}$ of complex entries such that $JA^\dagger J=A^{-1},$ where $J$ is $diag(1,1,-1)$-the diagonal matrix whose main diagonal are $(1,1,-1)$ and $A^\dagger $ is the complex conjugate of $A.$ Note that ${\rm U}(2,1)$ acts on the open unit ball in $\mathbf{C}^2,$ a model for the complex hyperbolic space $H^2_{\mathbf{C}}$, by :
$$(z,w) \longmapsto (\frac{a_{11}z+a_{12}w+a_{13}}{a_{31}z+a_{32}w+a_{33}},\frac{a_{21}z+a_{22}w+a_{23}}{a_{31}z+a_{32}w+a_{33}} ). $$
This action is transitive and the stabilizer of the origin is isomorphic to ${\rm U}(2) \times {\rm U}(1).$ It follows that the fundamental
group of ${\rm U}(2,1)$ is isomorphic to $\mathbf{Z}\oplus \mathbf{Z}.$
In the following, we will construct the universal covering group
of ${\rm U}(2,1),$ which is denoted by $G$ for short. Let's define
\noindent
$G:=\{(A,\theta_1,\theta_2)\in {\rm U}(2,1)\times \mathbf{R}\times \mathbf{R}\ | \quad \theta_1\equiv \arg(\det(A))\mod 2\pi, \theta_2\equiv \arg(a_{33})\mod 2\pi \}.$
Since $A\in {\rm U}(2,1),$ we find that $|a_{31}|^2+|a_{32}|^2-|a_{33}|^2=-1.$ Therefore $a_{33}\ne 0,$ and the definition makes sense.
The multiplication on $G$ is defined by :
$$(A,\theta_1,\theta_2)(B, \phi_1, \phi_2)= (AB, \theta_1+\phi_1, \theta_2+\phi_2+ \arg(1+\frac{a_{31}b_{13}+a_{32}b_{23}}{a_{33}b_{33}})).$$
Here, $\arg$ is the principal argument which takes value in $(-\frac{\pi}{2},\frac{\pi}{2}).$
Using the same argument as in \cite{hansen}, we can see that the multiplication is well-defined and $G$ is indeed a covering
group of ${\rm U}(2,1).$ To check that $G$ is simply connected, we consider its action on the open unit ball in $\mathbf{C}^2$ through the
action of the first component which lies in ${\rm U}(2,1).$
It is not hard to see that the action is transitive and the stabilizer of the origin is homeomorphic to
${\rm SU}(2)\times\mathbf{R}\times\mathbf{R}$. This implies that homotopically, $G$ is the same as ${\rm SU}(2)$ therefore it is simply connected.
We will also identify the Lie algebra of $G$ with ${\frak {u}(2,1)}-$ the Lie algebra of ${\rm U}(2,1).$
The elements of ${\rm U}(2,1)$ can be divided into 3 types according to their action on the complex hyperbolic space
$H^2_{\mathbf{C}}$(see \cite{chen}). Namely, a matrix is called {\it elliptic} if it has a fixed point in $H^2_{\mathbf{C}}.$ We call it {\it
parabolic} if it has a unique fixed point in $\overline {H^2_{\mathbf{C}}}$ and this lies on $\partial H^2_{\mathbf{C}}.$ And finally, a matrix is
called {\it loxodromic} if it has exactly two fixed points in $\overline {H^2_{\mathbf{C}}}$ which lie on $\partial H^2_{\mathbf{C}}.$ We also classify element of the universal covering group $G$ as elliptic, parabolic or loxodromic if
its image under the projection map $G\rightarrow {\rm U}(2,1)$ is of the corresponding types.
Let $M$ be a smooth, compact, oriented 3-manifold. A \textit{contact structure} on $M$ is a oriented 2-plane field $V=\textrm{ker}\ \alpha,$ where $\alpha$ is an 1-form such that $\alpha\wedge d\alpha$ is nowhere zero.
A \textit{strictly pseudoconvex CR structure} on $M$ is a contact structure $V$ together with a complex structure $J$ on $V.$ Let $V\otimes\mathbf{C}=v\oplus\bar v,$ where $v,\bar v$ are the $i$ and $-i$ eigenspaces of $J$ respectively, then $v$ is called the \textit{holomorphic tangent bundle}.
A CR structure which is locally isomorphic to the standard CR structure on the unit sphere $\mathbf{S}^3\subset \mathbf{C}^2,$ is called a \textit{spherical CR structure.} A spherical CR structure is determined by a pair $(D, \rho),$ where $D: \tilde M\rightarrow \mathbf{S}^3$ is a local isomorphism and $\rho: \pi_1(M) \rightarrow {\rm PU}(2,1) $ is the holonomy representation such that
$ D\circ \gamma =\rho(\gamma) \circ D,\ \text{for all} \ \gamma\in \pi_1(M). $ See \cite{falbel} for more details about spherical CR structures.
We now briefly recalled the definition of the Burns-Epstein invariant. The reader is referred to \cite{be1} for more details.
Let (M,V,J) be a CR 3-manifold with trivial holomorphic tangent bundle and $\pi_Y:Y\rightarrow M$ be its CR structure bundle. Denoted by $\pi$ the Cartan connection form, that is a ${\frak {su}(2,1)}$-valued 1-form on $Y.$ Let $\Pi=d\pi+\pi\wedge\pi$ be its curvature form. Consider a 3-form: $$TC_2(\pi):=\frac{1}{8\pi^2}Tr(\pi\wedge\Pi+\frac{1}{3}\pi\wedge\pi\wedge\pi).$$ The main theorem of Burns-Epstein \cite{be1} says that : there exists a 3-form $\tilde TC_2(\pi)$ on $M,$ which is defined up to an exact form, such that $\pi_Y^*(\tilde TC_2(\pi))= TC_2(\pi).$ Moreover, the integral $\int_M\tilde TC_2(\pi)$ is a biholomorphic invariant of the CR structure on $M.$ For a given CR 3-manifold $(M,V,J),$ this integral is simply denoted by $\mu(M)$ and called the \textit{Burns-Epstein invariant} of $M.$
Since the Burns-Epstein invariant is only defined when the holomorphic tangent bundle is trivial, for simplicity we will restrict ourself to the case of homology spheres so that this condition is automatically fulfilled. With some modifications, the reader may extend our results here to the relative version of the Burns-Epstein on a general $3$-manifold as defined in \cite{cheng}.
\section{Relation to the Chern-Simons invariant}
Next, we will recall the definition of the Chern-Simons invariant of a flat connection associated to a
representation. The reader is referred to \cite{cs} for general facts about the Chern-Simons invariant. For technical reason, we will work with the universal covering group $G.$
Let $\rho:\pi_1(M)\longrightarrow G$ be a representation of the fundamental group of $M.$ Consider the flat $G$ bundle
associated to $\rho:\ E_{\rho}:=\tilde M\times_{\rho}G.$ Let $A$ be the connection form of the flat connection on
$E_{\rho},$ then the Chern-Simons form of $A$ is defined by :
$$CS(A):=\frac{1}{8\pi^2}Tr(A\wedge dA+\frac{2}{3}A\wedge A\wedge A).$$
As we have shown that homotopically $G$ is the same as ${\rm SU}(2),$ standard obstruction theory implies that $E_{\rho}$ is a trivial
bundle. The \textit{Chern-Simons invariant} of $\rho$ is defined by:
$$cs(\rho):=\int_Ms^*(CS(A))\\ \mod \mathbf{Z},$$
where $s$ is a section of $E_{\rho}.$ It is not hard to see that the Chern-Simons invariant is well-defined, since
the difference between two sections is, homologically, some multiples of the fiber and the
Chern-Simons form when restricted to the fiber is the generator of $H^3(G;\mathbf{Z})\cong\mathbf{Z}.$
We can also defined the Chern-Simons invariant for a representation $\rho$ into the group ${\rm U}(2,1), {\rm SU}(2,1)$ or ${\rm PU}(2,1)$ in the same manner as long as the associated bundle $E_{\rho}$ is trivial. However we will get nothing new, since in this case we can lift $\rho$ to a representation $\tilde\rho$
into $G$ and $cs(\rho)\equiv cs(\tilde \rho).$ To show this equality, note that the connection form $A$
is induced from the pullback of the Maure-Cartan form by the projection map $\tilde M\times G\rightarrow G$ and moreover the
Maure-Cartan forms preserve under pullback of the covering maps of Lie groups.
The next observation shows that in the case of a spherical CR structure the Burns-Epstein invariant, modulo an integer, is minus
the Chern-Simons invariant of the holonomy representation. So in this case the Burns-Epstein invariant, modulo an integer, only depends on
the holonomy representation.
\begin{prop}
Let $(M,V,J)$ be a spherical CR homology sphere whose holonomy representation is $\rho:\pi_1(M)\longrightarrow {\rm PU}(2,1)$ then
$$\mu(M)\equiv -cs(\rho)\ \ \mod \mathbf{Z}.$$
\end{prop}
\begin{proof}
Let $D:\tilde M\longrightarrow S^3$ be the developing map for the CR structure on $M.$ Denoted
by $Y_{\tilde M}$ and $Y_{\mathbf{S}^3}$ the CR structure bundles on $\tilde M$ and $\mathbf{S}^3$ respectively.
Consider the commutative diagram below
$$
\begin{diagram}
\node{Y_{\tilde M}}\arrow{e,t}{\mathcal{D}}\arrow{s,l}{\pi_{\tilde M}}
\node{Y_{\mathbf{S}^3}}\arrow{s,r}{\pi_{\mathbf{S}^3}}\\
\node{\tilde M}\arrow{e,t}{D}
\node{\mathbf{S}^3}
\end{diagram}
$$
Let $A$ be the Cartan connection form on $Y_{\mathbf{S}^3},$ then $\mathcal{D}^*(A)$ is the Cartan connection form on
$Y_{\tilde M}.$ As $M$ is a homology sphere, the CR structure bundle of $M$ is trivial, that is, there exist an equivariant
section $s:\tilde M\longrightarrow Y_{\tilde M}.$ Then the equivariant 3-form $s^*(TC_2(\mathcal{D}^*(A)))$ on $\tilde M$ will
induce a 3-form on $M$ which we denote by the same notation. By using the vanishing of the curvature, we see that
$$s^*(TC_2(\mathcal{D}^*(A)))=s^*(-CS(\mathcal{D}^*(A))) =-CS((\mathcal{D}\circ s)^*(A)).$$
On the other hand, the CR structure bundle on the standard sphere can be identified with the Lie group ${\rm SU}(2,1)$ (see \cite{j}) and
the Cartan connection $A$ is the Maure-Cartan form on ${\rm SU}(2,1).$ Under this identification, the equivariant map
$\mathcal{D}\circ s$ can be consider as a section of the bundle $E_{\rho}=\tilde M\times_{\rho}{\rm SU}(2,1).$ It now follows from the
definition of the Burns-Epstein invariant that:
$$\mu(M)\equiv \int_M s^*(TC_2(\mathcal{D}^*(A)))\equiv \int_M -CS((\mathcal{D}\circ s)^*(A)) \equiv -cs(\rho) \mod \mathbf{Z}.$$
\end{proof}
According to \cite{j}, over the standard $3$-sphere the form $\tilde TC_2(A)$ coincides with $-\frac{1}{2\pi^2}dVol,$ where
$dVol$ is the volume form on the unit $3$-sphere. So it
follows from the proof above that:
$$\mu(M)= \int_M (\mathcal{D}\circ s)^*(TC_2(A))=\int_M (\mathcal{D}\circ s)^*(\pi_{\mathbf{S}^3})^*(-\frac{1}{2\pi^2}dVol)$$
$$ =-\frac{1}{2\pi^2}\int_M D^*(dVol).\hskip4.6cm$$
If $\kappa: \tilde M \rightarrow \mathbf{S}^3$ is an equivariant map, considered as a section of the associated bundle $\tilde M\times_\rho \mathbf{S}^3,$ then
$\mu(M)\equiv -\int_M \kappa^*(\frac{1}{2\pi^2}dVol) \ \mod \mathbf{Z}.$ The reason is that two sections $D$ and $\kappa$ are homologically
differed by some multiples of the fiber $\mathbf{S}^3$ and the integral of the form $\frac{1}{2\pi^2}dVol$ over the unit $3$-sphere is $1.$ We get the following corollary:
\begin{corollary}Let $M$ be a spherical CR homology sphere whose holonomy representation
$\rho:\pi_1(M)\longrightarrow {\rm PU}(2,1) $ is reducible then $\mu(M)\equiv 0 \mod \mathbf{Z}.$
\end{corollary}
\begin{proof}As the holonomy representation $\rho$ is reducible, we can find a common fixed point $*\in \mathbf{S}^3$ for all elements in its image. So we can take the constant map as a section of the associated bundle $\tilde M\times_\rho \mathbf{S}^3$ and the corollary follows.
\end{proof}
\section{Normal forms of flat connections near the boundary torus}
On a manifold with boundary, the integral of the Chern-Simons form may depend on the way we choose the section $s.$ In order to define the Chern-Simons invariant on the manifold with boundary we will firstly define explicit normal forms of flat connections near the boundary. We then show that every flat connection can be gauge transformed into a normal form. This will be done by finding explicit form, near the boundary, of the developing map associated to the flat connection.
Let $X$ be a 3-manifold whose boundary $\partial X$ is a torus $T.$ We fixed a pair of meridian and longitude $\mu,\lambda$ on
$T.$ Choose a coordinate $(e^{2\pi ix},e^{2\pi iy})$ on $T$ such that the corresponding map:
\begin{eqnarray*}
\mathbf{R}^2 & \longrightarrow & T \\
(x,y) &\longmapsto &(e^{2\pi ix},e^{2\pi iy})
\end{eqnarray*}
sends the horizontal line to $\mu$ and the vertical line $\lambda.$
Suppose that $T\times [0,1]$ is the
collar neighborhood of $T$ in $X$ and $\{dx,dy,dr\} $ is an oriented basis of 1-forms
on $X$ near $T.$ Here $r$ is the coordinate on $[0,1]$ such that $T\times \{1\}=\partial X$ and we
orient $T$ by the ''outward normal last" convention.
Let $A$ be a flat connection form on a principal $G$-bundle over $X$ with holonomy $\rho.$ As the bundle is trivial, from now
on we will consider $A$ as an ${\frak {u}(2,1)}$-valued $1$-form on $X.$ Recall that the developing map
of $A$ is a map $D_A: \tilde X \longrightarrow G$ such that $D_A(\alpha \circ \tilde x)=\rho(\alpha)\circ D_A(\tilde x)$ for
all $\alpha \in \pi_1(X)$ and $\tilde x \in \tilde X.$
We will follow the scheme in \cite{k1} and define the normal form for a flat $G$ connection on $X$ by dividing into
several cases according to the type of the boundary holonomies.
\noindent (I) Elliptic: suppose that the boundary holonomies $\rho(\mu)$ and $\rho(\lambda)$ are elliptic, then by conjugation we may
assume that:
$\rho(\mu)=(A, 2\pi(\alpha_1+\alpha_2+\alpha_3), 2\pi\alpha_3)$ and $\rho(\lambda)=(B, 2\pi(\beta_1+\beta_2+\beta_3), 2\pi\beta_3).$ Where
$A=\begin{pmatrix}
e^{2\pi i\alpha_1}& 0& 0\\
0& e^{2\pi i\alpha_2}& 0 \\
0& 0 & e^{2\pi i\alpha_3}\end{pmatrix}, \ \ $ $B=\begin{pmatrix}
e^{2\pi i\beta_1}& 0& 0\\
0& e^{2\pi i\beta_2}& 0 \\
0& 0 & e^{2\pi i\beta_3}
\end{pmatrix}$ and
$\alpha_i$ and $\beta_i$ are real numbers.
\noindent
We can see that near the boundary the developing map is given by:
$ D: \mathbf{R}^2\times [0, 1]\longrightarrow G$ such that $D(x,y,r)=(M, 2\pi(\alpha_1x+\alpha_2x+\alpha_3x+\beta_1y+\beta_2y+\beta_3y), 2\pi(\alpha_3x+\beta_3y))$
where $$ M= \begin{pmatrix}
e^{2\pi i(\alpha_1x+\beta_1y)}& 0& 0\\
0& e^{2\pi i(\alpha_2x+\beta_2y)}& 0 \\
0& 0 & e^{2\pi i(\alpha_3x+\beta_3y)}
\end{pmatrix}.$$
It follows that near $\partial X=T,$ we can gauge transform the flat connection $A$ to the form:
$$ D^{-1}dD= \begin{pmatrix}
2\pi i(\alpha_1dx+\beta_1dy)& 0& 0\\
0& 2\pi i(\alpha_2dx+\beta_2dy)& 0 \\
0& 0 & 2\pi i(\alpha_3dx+ \beta_3dy)
\end{pmatrix}.$$
\noindent (II) Loxodromy: It follows from \cite{chen} Lemma 3.2.2 that in this case the boundary holonomy can be conjugated to the form
$\rho(\mu)=(A, 2\pi\theta_1+4\pi\theta_2, 2\pi\theta_2)$ and $\rho(\lambda)=(B, 2\pi\tau_1+4\pi\tau_2, 2\pi\tau_2).$ Where
$$A=\begin{pmatrix}
e^{2\pi i\theta_1}& 0& 0\\
0& e^{2\pi i\theta_2}\cosh u& e^{2\pi i\theta_2}\sinh u \\
0& e^{2\pi i\theta_2}\sinh u & e^{2\pi i\theta_2}\cosh u
\end{pmatrix},$$
$$B=\begin{pmatrix}
e^{2\pi i\tau_1}& 0& 0\\
0& e^{2\pi i\tau_2}\cosh v& e^{2\pi i\tau_2}\sinh v \\
0& e^{2\pi i\tau_2}\sinh v & e^{2\pi i\tau_2}\cosh v
\end{pmatrix}$$
and $\theta_i, \tau_i, u, v$ are real numbers.
The developing map near the boundary is given by $$D(x,y,r)=(M, 2\pi(\theta_1x+\tau_1y) +4\pi(\theta_2x+\tau_2y),2\pi(\theta_2x+\tau_2y))$$ with
$$ M= \begin{pmatrix}
e^{2\pi i(\theta_1x+\tau_1y)}& 0& 0\\
0& e^{2\pi i(\theta_2x+\tau_2y)}\cosh(ux+vy)& e^{2\pi i(\theta_2x+\tau_2y)}\sinh(ux+vy) \\
0& e^{2\pi i(\theta_2x+\tau_2y)}\sinh(ux+vy) & e^{2\pi i(\theta_2x+\tau_2y)}\cosh(ux+vy)
\end{pmatrix}.$$
So near the boundary the connection has the form
$$D^{-1}dD= \begin{pmatrix}
2\pi i(\theta_1 dx+\tau_1 dy) & 0& 0\\
0& 2\pi i(\theta_2 dx+\tau_2 dy)& udx+vdy \\
0& udx+vdy & 2\pi i(\theta_2 dx+\tau_2 dy)
\end{pmatrix} $$
\noindent (III) Parabolic: According to \cite{chen} Theorem 3.4.1
a parabolic element $g \in {\rm U}(2,1) $ has a unique decomposition $g=pe=ep,$ where $p$ is unipotent and $e$ is elliptic.
Furthermore, the classification of parabolic elements of ${\rm U}(2,1)$ can be divided into two cases corresponding to whether the minimal polynomial of the unipotent part $p$ is $(x-1)^3$ or $(x-1)^2.$
If the minimal polynomial of $p$ is $(x-1)^3$ then it can be conjugate to the form:
$ \begin{pmatrix}
1-s& \bar a & s\\
-a & 1& a \\
-s& \bar a & 1+s
\end{pmatrix}e^{2\pi i\alpha} $
, where $\Re(s)= \frac{|a|^2}{2}.$ Note that by further conjugation by an appropriate diagonal matrix, we may assume
that $a$ is real. If $g$ has the unipotent part $p$ with the minimal polynomial $(x-1)^2$
then it can be conjugate to the form:
$$ \begin{pmatrix}
e^{2\pi i\theta_1}(1-ip)& 0& ipe^{2\pi i\theta_1}\\
0& e^{2\pi i\theta_2}& 0 \\
-ipe^{2\pi i\theta_1}& 0 & e^{2\pi i\theta_1}(1+ip)\end{pmatrix}.$$
Therefore, in the following we will divide into two cases.
\noindent Case 1: By conjugation, we may assume that, $$\rho(\mu)=(A, 6\pi\alpha, \arctan(\frac{p}{1+a^2/2})+2\pi\alpha),$$
$$\text{where}\ A=\begin{pmatrix}
1-\frac{a^2}{2}-ip& a & \frac{a^2}{2}+ip\\
-a & 1& a \\
-\frac{a^2}{2}-ip& a & 1+\frac{a^2}{2}+ip
\end{pmatrix}e^{2\pi i\alpha}.$$
As $\rho(\lambda)$ is of the same type and commutes with $\rho(\mu),$ it is not hard to check
that $\rho(\lambda)$ must have the form
$$\rho(\lambda)=(B, 6\pi\beta, \arctan(\frac{q}{1+b^2/2})+2\pi\beta),$$
$$\text{where}\ B=\begin{pmatrix}
1-\frac{b^2}{2}-iq& b & \frac{b^2}{2}+iq\\
-b & 1& b \\
-\frac{b^2}{2}-iq& b & 1+\frac{b^2}{2}+iq
\end{pmatrix}e^{2\pi i\beta}.$$
Note that $a,b,p,q, \alpha, \beta$ are real numbers.
We then can choose the developing map to be
$$D(x,y,r)=(M, 6\pi(\alpha x+\beta y), \arctan(\frac{px+qy}{1+(ax+by)^2/2})+2\pi(\alpha x+\beta y) ),$$
where
$$ M=\begin{pmatrix}
1-\frac{(ax+by)^2}{2}-& (ax+by)& \frac{(ax+by)^2}{2}+\\
i(px+qy)& &i(px+qy)\\
& &\\
-(ax+by)&1 & (ax+by) \\
& &\\
-\frac{(ax+by)^2}{2}&(ax+by)& 1+\frac{(ax+by)^2}{2}+\\
-i(px+qy)&& i(px+qy)
\end{pmatrix}e^{2\pi i(\alpha x+\beta y)}.$$
With straightforward computations, we deduce that the connection form $D^{-1}dD$ near the boundary is
$$ \begin{pmatrix}
-i(pdx+qdy)& (adx+bdy)& i(pdx+qdy)\\
2\pi i(\alpha dx +\beta dy)& &\\
& &\\
-(adx+bdy)& 2\pi i(\alpha dx +\beta dy)& (adx+bdy) \\
& &\\
-i(pdx+qdy)& (adx+bdy) & i(pdx+qdy)+ \\
& &2\pi i(\alpha dx +\beta dy)
\end{pmatrix}.$$
\noindent Case 2: After conjugation, we may assume that $$\rho(\mu)=(A, 4\pi\theta_1+2\pi\theta_2, 2\pi\theta_1+\arctan(p))$$ $$\text{and}\ \rho(\lambda)=(B, 4\pi\tau_1+2\pi\tau_2, 2\pi\tau_1+\arctan(q)).$$
$$ \text{Where}\ A=\begin{pmatrix}
e^{2\pi i\theta_1}(1-ip)& 0& ipe^{2\pi i\theta_1}\\
0& e^{2\pi i\theta_2}& 0 \\
-ipe^{2\pi i\theta_1}& 0 & e^{2\pi i\theta_1}(1+ip)\end{pmatrix}, $$
$$ B=\begin{pmatrix}
e^{2\pi i\tau_1}(1-iq)& 0& iqe^{2\pi i\tau_1}\\
0& e^{2\pi i\tau_2}& 0 \\
-iqe^{2\pi i\tau_1}& 0 & e^{2\pi i\tau_1}(1+iq)\end{pmatrix},$$ and $\theta_i, \tau_i, p, q$ are real numbers. So the developing map is of the form
$$D(x,y,r)=(M, 4\pi(\theta_1 x+\tau_1 y)+ 2\pi(\theta_2 x+\tau_2 y), 2\pi (\theta_1 x+\tau_1 y)+\arctan(px+qy)),$$
where
$$M= \begin{pmatrix}
e^{2\pi i(\theta_1 x+\tau_1 y)}- & 0& i(px+qy)e^{2\pi i(\theta_1 x+\tau_1 y)}\\
i(px+qy)e^{2\pi i(\theta_1 x+\tau_1 y)}&&\\
&&\\
0& e^{2\pi i(\theta_2 x+\tau_2 y)}& 0 \\
&&\\
-i(px+qy)e^{2\pi i(\theta_1 x+\tau_1 y)}& 0 & e^{2\pi i(\theta_1 x+\tau_1 y)}+\\
&& i(px+qy)e^{2\pi i(\theta_1 x+\tau_1 y)} \end{pmatrix}
.$$
After some lengthy computations, we find the connection form $D^{-1}dD$ to be:
$$ \begin{pmatrix}
2\pi i(\theta_1 dx+\tau_1 dy)- & 0& i(pdx+qdy)\\
i(pdx+qdy)&&\\
&&\\
0& 2\pi i(\theta_2 dx+\tau_2 dy)& 0 \\
&&\\
- i(pdx+qdy)& 0 & 2\pi i(\theta_1 dx+\tau_1 dy)+ \\
&& i(pdx+qdy)
\end{pmatrix}.$$
\begin{define} We say that a flat $G$ connection $1$-form on a manifold with torus boundary $X$ is in normal form if, near the boundary, it has one of the forms as in the cases (I), (II) and (III) above.
\end{define}
Note that one connection may have different normal forms as we can add integers to the exponential parameters in the holonomy matrices.
For the computation of the Burns-Epstein invariant, we need to bring connections to the normal forms. The following proposition shows that we can always do so.
\begin{prop} a) Let $A$ be a flat $G$ connection $1$-form on $X.$ Then there exists a gauge transformation which brings $A$ into a normal form.
\label{nf}
\noindent
b) Let $A_t$ be a path of flat $G$ connection $1$-forms on $X.$ Then there exists a path of gauge transformations $g_t$ such that $g_t\cdot A_t$ is in normal form for all $t$.
\end{prop}
\begin{proof}
We have shown above that locally near the torus boundary, we can always gauge transform the flat connection into a normal form. By using standard obstruction theory argument as in \cite{kk2} Proposition 2.3 we can extend the gauge transformation to all of $X$ and prove the proposition.
\end{proof}
The next lemma tells us that the integral of the Chern-Simons form of a normal form flat connection on a manifold with boundary is gauge invariant.
\begin{lemma}
Let $A$ and $B$ be two normal form flat connections on a manifold with torus boundary $X.$ Suppose that:
1. $A$ and $B$ are gauge equivalent.
2. $A$ and $B$ are in normal form and equal near the boundary.
Then $\int_XCS(A)\equiv \int_XCS(B) \mod \mathbf{Z}.$
\end{lemma}
\begin{proof} The proof is similar to the one of \cite{kk2} Theorem 2.4. So we will left it to the reader as an exercise.
\end{proof}
So we may define the \textit{Chern-Simons invariant of a normal form flat connection} $A$ by:
$$cs(A):=\int_X CS(A) \mod \mathbf{Z}.$$
\section{Variations of the Chern-Simons invariants}
In this section, we will prove the main technical tool for computing the Chern-Simons invariant which is the formula for the change of the Chern-Simons invariants of a path of normal form flat connections in terms of the boundary holonomy.
\begin{theorem} Let $A_t$ be a path of normal form flat $G$-connections on a manifold with torus boundary $X.$
Suppose that $\rho_t:\pi_1(X)\longrightarrow G$ is the corresponding path of holonomy representations. Consider the
following cases:
\noindent (I)Elliptic: suppose that $\rho_t(\mu)$ and $\rho_t(\lambda)$ are elliptic elements and the normal form of $A_t$ near the boundary is:
$$\begin{pmatrix}
2\pi i(\alpha_1(t)dx+\beta_1(t)dy)& 0& 0\\
0& 2\pi i(\alpha_2(t)dx+\beta_2(t)dy)& 0 \\
0& 0 & 2\pi i(\alpha_3(t)dx+ \beta_3(t)dy)
\end{pmatrix}$$
then $$cs(A_1)-cs(A_0)\equiv \frac{1}{2}\int_0^1 [(\alpha_1\dot{\beta_1}-\dot{\alpha_1}\beta_1)+ (\alpha_2\dot{\beta_2}-\dot{\alpha_2}\beta_2)
+( \alpha_3\dot{\beta_3}-\dot{\alpha_3}\beta_3)]dt. $$
\noindent (II) Loxodromy: if $\rho_t(\mu)$ and $\rho_t(\lambda)$ are loxodromy elements and the normal form of $A_t$ near the boundary is:
$$\begin{pmatrix}
2\pi i(\theta_1(t) dx+\tau_1(t) dy) & 0& 0\\
0& 2\pi i(\theta_2(t) dx+\tau_2(t) dy)& u(t)dx+v(t)dy \\
0& u(t)dx+v(t)dy & 2\pi i(\theta_2(t) dx+\tau_2(t) dy)
\end{pmatrix} $$
then
$$cs(A_1)-cs(A_0)\equiv \frac{1}{2}\int_0^1 ( \theta_1\dot{\tau_1}-\dot{\theta_1}\tau_1)dt +\int_0^1(\theta_2\dot{\tau_2}-\dot{\theta_2}\tau_2)dt+
\frac{1}{4\pi^2}\int_0^1 (\dot{u}v- u\dot{v})dt.$$
\noindent (III) Parabolic:
\noindent
Case 1: $\rho_t(\mu)$ and $\rho_t(\lambda)$ are parabolic elements whose unipotent parts have minimal polynomial $(x-1)^3.$ Suppose that the normal form of $A_t$ near the boundary is:
$$ \begin{pmatrix}
-i(p(t)dx+q(t)dy)+ & a(t)dx+b(t)dy& i(p(t)dx+q(t)dy)\\
2\pi i(\alpha(t) dx +\beta(t) dy)&\ &\\
&&\\
-(a(t)dx+b(t)dy)& 2\pi i(\alpha(t) dx +\beta(t) dy)& a(t)dx+b(t)dy \\
&&\\
-i(p(t)dx+q(t)dy)& a(t)dx+b(t)dy & i(p(t)dx+q(t)dy)+ \\
\ &\ & 2\pi i(\alpha(t) dx +\beta(t) dy)
\end{pmatrix}$$ then
$$cs(A_1)-cs(A_0)\equiv \frac{3}{2}\int_0^1 ( \alpha\dot{\beta}-\dot{\alpha}\beta)dt.$$
\noindent
Case 2: $\rho_t(\mu)$ and $\rho_t(\lambda)$ are parabolic elements whose unipotent parts have minimal polynomial $(x-1)^2.$ Suppose that the normal form of $A_t$ near the boundary is:
$$ \begin{pmatrix}
2\pi i(\theta_1(t) dx+\tau_1(t) dy)- & 0& i(p(t)dx+q(t)dy)\\
i(p(t)dx+q(t)dy)&\ & \\
&&\\
0& 2\pi i\theta_2(t) dx+& 0 \\
&2\pi i\tau_2(t) dy&\\
&&\\
- i(p(t)dx+q(t)dy)& 0 & 2\pi i(\theta_1(t) dx+\tau_1(t) dy)+ \\
&\ & i(p(t)dx+q(t)dy) \end{pmatrix}$$
then
$$cs(A_1)-cs(A_0)\equiv \int_0^1 (\theta_1\dot{\tau_1}-\dot{\theta_1}\tau_1)dt+ \frac{1}{2}\int_0^1(\theta_2\dot{\tau_2}-\dot{\theta_2}\tau_2)dt.$$
\end{theorem}
\begin{proof}
Consider the path $A_t$ as a connection on $X\times[0, 1],$ then it is a flat connection, that is $F^{A_t}\wedge F^{A_t}\equiv 0.$ by Stokes' Theorem
$$\int_XCS(A_1) - \int_XCS(A_0)-\int_{\partial X\times [0,1]} CS(A_t)=\int_{X\times [0,1]}tr(F^{A_t}\wedge F^{A_t})=0.$$
So $$cs(A_1)-cs(A_0) \equiv \int_{\partial X\times [0,1]} \frac{1}{8\pi^2}tr(A_t\wedge dA_t+\frac{2}{3}A_t\wedge A_t\wedge A_t).$$
We now use this formula and the normal form of the connections to compute the changes of Chern-Simons invariants in each
cases.
\noindent (I) Elliptic: in this case
$tr(A_t\wedge dA_t+\frac{2}{3}A_t\wedge A_t\wedge A_t)=4\pi^2[( \alpha_1\dot{\beta_1}-\dot{\alpha_1}\beta_1) +
(\alpha_2\dot{\beta_2}-\dot{\alpha_2}\beta_2) + (\alpha_3\dot{\beta_3}-\dot{\alpha_3}\beta_3) ]dx\wedge dy\wedge dt.$
So the result follows.
\noindent (II) Loxodromy: after some computation, we get
$tr(A_t\wedge dA_t+\frac{2}{3}A_t\wedge A_t\wedge A_t)=
4\pi^2( \theta_1\dot{\tau_1}-\dot{\theta_1}\tau_1)dx\wedge dy\wedge dt+$
$8\pi^2(\theta_2\dot{\tau_2}-\dot{\theta_2}\tau_2)dx\wedge dy\wedge dt+2(\dot{u}v- u\dot{v})dx\wedge dy\wedge dt.$
We deduce the required formula.
\noindent (III) Parabolic:
\noindent Case 1: straightforward computations show that
$$tr(A_t\wedge dA_t+\frac{2}{3}A_t\wedge A_t\wedge A_t)=12\pi^2(\alpha\dot{\beta}-\dot{\alpha}\beta )dx\wedge dy\wedge dt.$$
So the change in the Chern-Simons invariants is given by the formula.
\noindent Case 2: in this case, we get:
$tr(A_t\wedge dA_t+\frac{2}{3}A_t\wedge A_t\wedge A_t)=8\pi^2(\theta_1\dot{\tau_1}-\dot{\theta_1}\tau_1 )dx\wedge dy\wedge dt +$
$ 4\pi^2(\theta_2\dot{\tau_2}-\dot{\theta_2}\tau_2)dx\wedge dy\wedge dt.$
So the formula follows.
\end{proof}
In the last past of this section, we will study the difference between the Chern-Simons invariants of two different normal form connections in the elliptic case. This result will be used in the computation of the next section. Our result is similar to Theorem 2.5 in \cite{kk2}.
Consider a manifold with toral boundary $X.$ Let A be a normal form flat $G$-connection which has the following form near the boundary $\partial X$:
$$\begin{pmatrix}
2\pi i(\alpha_1dx+\beta_1dy)& 0& 0\\
0& 2\pi i(\alpha_2dx+\beta_2dy)& 0 \\
0& 0 & 2\pi i(\alpha_3dx+ \beta_3dy)
\end{pmatrix}.$$
Let $h:\mathbf{S}^1\rightarrow G $ defined by $h(e^{2\pi i\theta})=(diag(e^{2\pi i\theta},e^{-2\pi i\theta},1),0,0).$ Since $h$ is a nullhomotopic map into $G$ we may find a path $h_t, \ 0\leq t \leq 1 $ such that:
- $h_t$ is constant near $0$ and $1$
- $h_0\equiv 1\in G$ and $h_1=h.$
Next we use the map $h_t$ above to defined two gauge transformations on $X$ as follows. Recall that we denote by $T\times [0,1]$ with the coordinate $(e^{2\pi ix},e^{2\pi iy},r)$ a collar neighbourhood of $\partial X$ in $X$ such that $T\times\{1\}=\partial X.$ We define $g_x:T\times [0,1]\rightarrow G$ and $g_y:T\times [0,1]\rightarrow G$ by $g_x (e^{2\pi ix},e^{2\pi iy},r)=h_r(e^{2\pi ix})$ and
$g_y (e^{2\pi ix},e^{2\pi iy},r)=h_r(e^{2\pi iy}).$
Note that we can extend $g_x$ and $g_y$ to be $1\in G$ outside the collar neighbourhood.
It is not hard to check that $g_x \cdot A$ and $g_y \cdot A$ are also in normal form since near $\partial X$ we have:
$g_x\cdot A=\begin{pmatrix}
2\pi i(\alpha_1+1)dx+& 0& 0\\
2\pi i\beta_1dy&&\\
&&\\
0& 2\pi i(\alpha_2-1)dx+& 0 \\
&2\pi i\beta_2dy&\\
&&\\
0& 0 & 2\pi i(\alpha_3dx+\beta_3dy)
\end{pmatrix}$
$g_y\cdot A=\begin{pmatrix}
2\pi i\alpha_1dx+& 0& 0\\
2\pi i(\beta_1+1)dy&&\\
&&\\
0& 2\pi i\alpha_2dx+& 0 \\
&2\pi i(\beta_2-1)dy& \\
&&\\
0& 0 & 2\pi i(\alpha_3dx+ \beta_3dy)
\end{pmatrix}.$
We now state the following theorem:
\begin{theorem} Suppose that $A$ and $B$ are two gauge equivalent normal form flat $G$ connections on $X$ such that
$A=\begin{pmatrix}
2\pi i(\alpha_1dx+\beta_1dy)& 0& 0\\
0& 2\pi i(\alpha_2dx+\beta_2dy)& 0 \\
0& 0 & 2\pi i(\alpha_3dx+ \beta_3dy)
\end{pmatrix}$ and
$B=\begin{pmatrix}
2\pi i(\alpha_1+m)dx+& 0& 0\\
2\pi i(\beta_1+n)dy&&\\
&&\\
0& 2\pi i(\alpha_2-m)dx+& 0 \\
&2\pi i(\beta_2-n)dy&\\
&&\\
0& 0 & 2\pi i(\alpha_3dx+ \beta_3dy)
\end{pmatrix}$
near the boundary $\partial X,$ for some integers $m$ and $n,$ then:
$$cs(B)-cs(A)\equiv m(\beta_1-\beta_2)/2 - n(\alpha_1-\alpha_2)/2 \mod \mathbf{Z}.$$
\end{theorem}
\begin{proof} By Lemma 4.3, if $g$ is a gauge transformation such that $g\cdot A\equiv B$ near $\partial X$ then $cs(g\cdot A)\equiv cs(B) \mod \mathbf{Z}.$ Therefore, it is enough to prove that
$cs(g_x\cdot A)-cs(A)\equiv (\beta_1-\beta_2)/2\mod \mathbf{Z}$ and $cs(g_y\cdot A)-cs(A)\equiv (\alpha_2-\alpha_1)/2\mod \mathbf{Z}.$
By Proposition 1.27(e) of \cite{freed} the difference between two Chern-Simons forms $CS(g_x\cdot A)-CS(A)$ equals to
$$\frac{1}{8\pi^2}d(Tr(g_x^{-1}Ag_x\wedge g_x^{-1}dg_x)) -\frac{1}{24\pi^2}Tr(g_x^{-1}dg_x\wedge g_x^{-1}dg_x\wedge g_x^{-1}dg_x).$$
It follows from the definition of $g_x$ that $\frac{\partial g_x}{\partial y}=0,$ so the last term in this formula vanishes. By direct computation we deduce that:
$cs(g_x\cdot A)-cs(A)\equiv \frac{1}{8\pi^2}\int_T Tr(g_x^{-1}Ag_x\wedge g_x^{-1}dg_x) = \frac{1}{2}\int_T(\beta_1-\beta_2)dxdy$
$\hskip3cm \equiv (\beta_1-\beta_2)/2.$
By using a similar argument, we can show that the formula for $g_y$ also holds. Thus the Theorem follows.
\end{proof}
\section{Applications}
In this section, we will apply our main theorem to find the Chern-Simons invariants of representations of the Seifert fibered homology sphere. We also give an explicit example, where we use the Chern-Simons invariant to deduce result about the Burns-Epstein invariant.
Let $\Sigma=\Sigma(a_1,\ldots,a_n)$ be a Seifert fibered homology sphere, where $a_i>1$ are pairwise relatively prime integers. We put $a:=a_1\cdots a_n.$ We will denote by $\mathcal{R}^*(\Sigma)$ the space of irreducible representations from $\pi_1(\Sigma)$ to $G.$
The fundamental group of $\Sigma$ is given by :
$$\pi_1(\Sigma) = \{x_1,\ldots, x_n,h\ |h\ \text{is central,}\ \ x_1^{a_1}h^{b_1}= \cdots = x_n^{a_n}h^{b_n}=x_1\ldots x_n=1\}$$
where the $b_i$ are chosen with $\sum_1^n \frac{b_i}{a_i}=\frac{1}{a}.$
Let $\rho$ be an element of $\mathcal{R}^*(\Sigma).$ Since $h$ is in the center of $\pi_1(\Sigma),$ $\rho(h)$ is in the center of $G.$ As $x_i^{a_i}h^{b_i}=1, $ $\rho(x_i)$ is an elliptic element for all $i.$ Suppose that the representation $\rho$ is given by
$$\rho(h)=(diag(e^{2\pi ip_0},e^{2\pi iq_0},e^{2\pi ir_0}),2\pi(p_0+q_0+r_0),2\pi r_0) \ \text{and}$$
$$\rho(x_i) \sim (diag(e^{2\pi ip_i},e^{2\pi iq_i},e^{2\pi ir_i}),2\pi(p_i+q_i+r_i),2\pi r_i).$$ Here we use $\sim$ to denote the conjugacy relation in $G.$
As $\rho(h)$ is central, we have $p_0\equiv q_0\equiv r_0 \mod \mathbf{Z}.$
Since $\rho$ must preserve the relation $x_i^{a_i}h^{b_i}=1, \ i=1\ldots n,$ we deduce that :
(1) \hskip 2cm $a_ip_i+b_ip_0=s_i,$ $(a_iq_i+b_iq_0)=-s_i$ are integers and
(2) \hskip 2cm $a_ir_i+b_ir_0=0.$
\begin{theorem} \label{sf}
The Chern-Simons invariant of $\rho$ is $$cs(\rho)\equiv \frac{1}{2}a((\sum_{i=1}^np_i)^2+(\sum_{i=1}^nq_i)^2+(\sum_{i=1}^nr_i)^2).$$
\end{theorem}
\begin{proof}
We write $\Sigma = X \cup (-S),$ where $X$ is the complement of the $n^{th}$-exceptional fiber and $S$ is the solid torus neighborhood of the $n^{th}$-exceptional fiber. We then find paths of representations on $X$ and $S$ which connect $\rho$ to the trivial representation and apply formula in Theorem 5.1 to compute the Chern-Simons invariants on each $X$ and $S.$
\noindent \textit {Step 1. Computation on $X.$}
Note that $\pi_1(X)$ is obtained from $\pi_1(\Sigma)$ by eliminating the relation $x_n^{a_n}h^{b_n}$ and $X$ has a natural meridian and longitude $\mu=x_n^{a_n}h^{b_n}$ and $\lambda = x_n^{-a_1\ldots a_{n-1}}h^c$ where $c=a_1\ldots a_{n-1}\sum_1^{n-1} \frac{b_i}{a_i}.$
After conjugation, we may assume that
$$\rho(x_n)=(diag(e^{2\pi ip_n}, e^{2\pi iq_n}, e^{2\pi ir_n}),2\pi(p_n+q_n+r_n),2\pi r_n).$$
Since each conjugacy class in $G$ is connected, we can deform $ \rho|_X(x_i)$ within its conjugacy class to the diagonal form for $i=1,\dots,n-1.$
This means that we can find a path of representations
$$\rho_t:\pi_1(X)\longrightarrow G, 0\leq t\leq 1,$$
such that $ \rho_0= \rho|_X,$ $ \rho_t(h)= \rho(h)$ for all $t$ and
$$ \rho_1(x_i)=(diag(e^{2\pi ip_i},e^{2\pi iq_i},e^{2\pi ir_i}),2\pi(p_i+q_i+r_i),2\pi r_i), i=1,\dots,n-1.$$
For this path of representation we have:
$$ \rho_t(x_n)=(diag(e^{2\pi ip(t)}, e^{2\pi iq(t)}, e^{2\pi ir(t)}),2\pi(p(t)+q(t)+r(t)),2\pi r(t)),$$
where $p(0)=p_n, q(0)=q_n, r(0)=r_n$ and
\noindent $p(1)=-\sum_1^{n-1}p_i, q(1)=-\sum_1^{n-1}q_i, r(1)=-\sum_1^{n-1}r_i.$
Therefore, we get:
$$\rho_t(\mu)=(M(t), 2\pi a_n(p(t)+q(t)+r(t))+2\pi b_n(p_0+q_0+r_0),2\pi a_nr(t)+2\pi b_nr_0),$$
$$\text{where}\ M(t)=diag(e^{2\pi i(a_np(t)+b_np_0)}, e^{2\pi i(a_nq(t)+b_nq_0)}, e^{2\pi i(a_nr(t)+b_nr_0)}), \ \text{and}$$
\noindent $\rho_t(\lambda)=(N(t), -2\pi a_1\cdots a_{n-1}(p(t)+q(t)+r(t))+2\pi c(p_0+q_0+r_0),$
$$ -2\pi a_1\cdots a_{n-1}r(t)+2\pi cr_0),\ \text{where}\hskip3.5cm $$
$$N(t)=diag(e^{2\pi i(-a_1\cdots a_{n-1}p(t)+cp_0)}, e^{2\pi i(-a_1\cdots a_{n-1}q(t)+cq_0)}, e^{2\pi i(-a_1\cdots a_{n-1}r(t)+cr_0)}).$$
So by Proposition 4.2 we get a path of normal form ${\frak {u}(2,1)}-$valued flat connections on $X$ which is given near $\partial X$ by :
$$A_t=diag(2\pi i((a_np(t)+b_np_0)dx+(-a_1\cdots a_{n-1}p(t)+cp_0)dy), 2\pi i((a_nq(t)+b_nq_0)dx+$$
$$(-a_1\cdots a_{n-1}q(t)+cq_0)dy), 2\pi i((a_nr(t)+b_nr_0)dx+(-a_1\cdots a_{n-1}r(t)+cr_0)dy)).$$
We now use Theorem 5.1 to compute the difference of Chern-Simons invariants:
$cs(A_1)-cs(A_0)\equiv -\frac{1}{2}\int_0^1(a_nc+a_1\cdots a_{n-1}b_n)(p_0\dot p(t)+q_0\dot q(t))+ r_0\dot r(t))dt$
\hskip2.6cm $=-\frac{1}{2}(a_nc+a_1\cdots a_{n-1}b_n)(p_0p(t)+q_0q(t)+r_0r(t))|_0^1.$
So we arrive at:
$$(*) \qquad cs(A_0)\equiv cs(A_1) -\frac{1}{2}(p_0\sum_{i=1}^np_i+q_0\sum_{i=1}^nq_i+r_0\sum_{i=1}^nr_i).$$
Now, by using relations (1) and (2), we can write
\noindent $A_1=diag(2\pi i(-a_n\sum_{i=1}^{n-1}p_i+b_np_0)dx+2\pi i(a_1\cdots a_{n-1}\sum_{i=1}^{n-1}\frac{s_i}{a_i})dy, 2\pi i(-a_n\sum_{i=1}^{n-1}q_i$
\hskip0.8cm $+b_nq_0)dx- 2\pi i(a_1\cdots a_{n-1}\sum_{i=1}^{n-1}\frac{s_i}{a_i})dy, 2\pi i(a_n\sum_{i=1}^{n-1}r_i+b_nr_0)dx).$
Since $a_1\cdots a_{n-1}\sum_{i=1}^{n-1}\frac{s_i}{a_i}$ is integer, by using Theorem 5.2, we find that:
\noindent $(**) \quad cs(A_1)\equiv cs(A_1')-\frac{1}{2}(a_1\cdots a_{n-1}\sum_{i=1}^{n-1}\frac{s_i}{a_i})(-a_n\sum_{i=1}^{n-1}p_i+b_np_0+$
$\hskip3cm a_n\sum_{i=1}^{n-1}q_i-b_nq_0).$
\noindent Where $A_1'=diag(2\pi i(-a_n\sum_{i=1}^{n-1}p_i+b_np_0)dx, 2\pi i(-a_n\sum_{i=1}^{n-1}q_i+b_nq_0)dx,$
\hskip1.8cm $2\pi i(-a_n\sum_{i=1}^{n-1}r_i+b_nr_0)dx),$ near the boundary $\partial X.$
As the holonomy representation $\rho_1'$ of the $A_1'$ is abelian, it factors through $\pi_1(X)\rightarrow H_1(X)\equiv \mathbf{Z}.$ We find that $\rho_1'(\lambda)=1\in G$ and
\noindent $\rho_1'(\mu)=(diag(e^{2\pi i(-a_n\sum_{i=1}^{n-1}p_i+b_np_0)}, e^{2\pi i(-a_n\sum_{i=1}^{n-1}q_i+b_nq_0)}, e^{2\pi i(-a_n\sum_{i=1}^{n-1}r_i+b_nr_0)}), $
\hskip0.5cm $-2\pi a_n \sum_{i=1}^{n-1} (p_i+q_i+r_i)+2\pi b_n(p_0+q_0+r_0), -2\pi a_n\sum_{i=1}^{n-1}r_i+b_nr_0).$
So we can deform $\rho_1'$ to the trivial representation and get a path of connections $A'_t$ linking $A_1'$ to the trivial connection which obviously has zero Chern-Simons invariant. Applying Theorem 5.1 to the path:
$A'_t =diag(2t\pi i(-a_n\sum_{i=1}^{n-1}p_i+b_np_0)dx, 2t\pi i(-a_n\sum_{i=1}^{n-1}q_i+b_nq_0)dx,$
\hskip1.8cm $2t\pi i(-a_n\sum_{i=1}^{n-1}r_i+b_nr_0)dx),\ 0\leq t\leq 1,$
we conclude that $cs(A'_1)\equiv 0.$ Now combine this with (*) and (**), we get:
$cs(A_0)\equiv -\frac{1}{2}a(\sum_{i=1}^{n-1}\frac{s_i}{a_i})(-\sum_{i=1}^{n-1}p_i+\frac{b_n}{a_n}p_0+\sum_{i=1}^{n-1}q_i-\frac{b_n}{a_n}q_0) -$
$\hskip3.5cm
\frac{1}{2}(p_0\sum_{i=1}^np_i+q_0\sum_{i=1}^nq_i+r_0\sum_{i=1}^nr_i).$
Where $A_0$ is the flat connection form corresponding the representation $\rho|_X.$ Now using (1) we can rewrite this as:
$(***) \qquad cs(A_0)\equiv -\frac{1}{2}a(\sum_{i=1}^{n-1}\frac{s_i}{a_i})(-\sum_{i=1}^{n}p_i+\sum_{i=1}^{n}q_i+2\frac{s_n}{a_n}) -$
$\hskip3.8cm
\frac{1}{2}(p_0\sum_{i=1}^np_i+q_0\sum_{i=1}^nq_i+r_0\sum_{i=1}^nr_i).$
\noindent \textit {Step 2. Computation on the solid torus}
We will denote the connection form corresponding to $\rho|_S$ by $B_0.$ Near the boundary $\partial X$ then $B_0$ coincides with $A_0$ and given by:
\noindent $B_0=diag(2\pi i((a_np_n+b_np_0)dx+(-a_1\cdots a_{n-1}p_n+cp_0)dy)), 2\pi i((a_nq_n+$
\hskip0.3cm $b_nq_0)dx+ (-a_1\cdots a_{n-1}q_n+cq_0)dy)), 2\pi i(-a_1\cdots a_{n-1}r_n+cr_0)dy).$
By (1), the numbers $a_np_n+b_np_0=s_n$ and $a_nq_n+b_nq_0=-s_n$ are integers. So apply Theorem 5.2 we find that
$cs(B_0)\equiv cs(B_1)+\frac{1}{2}s_n( -a_1\cdots a_{n-1}p_n+cp_0+a_1\cdots a_{n-1}q_n-cq_0),$
where
$B_1 = diag(2\pi i(-a_1\cdots a_{n-1}p_n+cp_0)dy, 2\pi i (-a_1\cdots a_{n-1}q_n+$
\hskip2cm $cq_0)dy, 2\pi i(-a_1\cdots a_{n-1}r_n+cr_0)dy)$ near the boundary $\partial X.$
Note that $B_1$ is a connection form corresponding to an abelian representation and it involves only the $dy$ terms. Carrying out a similar computation as we did for the connection $A_1'$ in the previous step, we conclude that $cs(B_1)\equiv 0.$ Therefore we can write
$$cs(B_0)\equiv \frac{1}{2}a\frac{s_n}{a_n}( -p_n+p_0\sum_{i=1}^{n-1}\frac{b_i}{a_i}+q_n-q_0\sum_{i=1}^{n-1}\frac{b_i}{a_i}).$$
Moreover, from (1), we deduce that $$p_0\sum_{i=1}^{n-1}\frac{b_i}{a_i}=\sum_{i=1}^{n-1}\frac{s_i}{a_i}-\sum_{i=1}^{n-1}p_i$$ and that
$$q_0\sum_{i=1}^{n-1}\frac{b_i}{a_i}=-\sum_{i=1}^{n-1}\frac{s_i}{a_i} -\sum_{i=1}^{n-1}q_i.$$ So we arrive at:
$$(****)\qquad cs(B_0)\equiv \frac{1}{2}a\frac{s_n}{a_n}( -\sum_{i=1}^np_i+\sum_{i=1}^nq_i+2\sum_{i=1}^{n-1}\frac{s_i}{a_i}).$$
Now as $\Sigma = X \cup (-S),$ we see that $cs(\rho)=cs(A_0)-cs(B_0).$ Note that the term $a\frac{s_n}{a_n}\sum_{i=1}^{n-1}\frac{s_i}{a_i}$ appearing in (***) and (****) is an integer and therefore can be ignored. What we find is
$cs(\rho)\equiv \frac{1}{2}a(\sum_{i=1}^np_i\sum_{i=1}^n\frac{s_i}{a_i}-\sum_{i=1}^nq_i\sum_{i=1}^n\frac{s_i}{a_i}) $
\hskip3cm $-\frac{1}{2}(p_0\sum_{i=1}^np_i+q_0\sum_{i=1}^nq_i+r_0\sum_{i=1}^nr_i).$
It follows from (1) and (2) that:
$\sum_{i=1}^n\frac{s_i}{a_i}=\sum_{i=1}^np_i+\frac{p_0}{a}=-\sum_{i=1}^nq_i-\frac{q_0}{a}$ and
$r_0=-a\sum_{i=1}^nr_i.$
So, finally, we arrive at the needed formula
$$cs(\rho)\equiv \frac{1}{2}a((\sum_{i=1}^np_i)^2+(\sum_{i=1}^nq_i)^2+(\sum_{i=1}^nr_i)^2).$$
\end{proof}
We can use the above theorem to find \textit {all the possible values} of the Chern-Simons invariants of representations if we know the representation space $\mathcal{R}^*(\Sigma).$ In the general case, this space is still hard to describe in details. Fortunately, for the case of Seifert fibered homology sphere with three singular fibers, the representation space has been studied in \cite{k2} and so we are able to find all the the possible values of the Chern-Simons invariants.
As an illustration, we present an example of the homology sphere $\Sigma(2,3,11).$ Its fundamental group has the following presentation.
{\small
$$\pi_1(\Sigma(2,3,11))= \langle
x_1,x_2,x_3,h| \ h\ \text{central},\ x_1^2h^{-1}=x_2^3h=x_3^{11}h^2=x_1x_2x_3=1
\rangle.$$}
By the computation in \cite{k2}, we know that $\Sigma(2,3,11)$ has five distinct irreducible representations into ${\rm PU}(2,1).$ By homological reason, each ${\rm PU}(2,1)$ representation has a unique lift to a representation into the universal covering group $G.$ By further computations, we obtain the following list of representations into $G.$
\noindent 1) $\rho(x_1) \sim (diag(1,-1,-1), 0,-\pi), \rho(x_2) \sim (diag(1,e^{4\pi i/3},e^{2\pi i/3}),0, \frac{2\pi}{3}),$
$ \rho(x_3) \sim (diag(e^{12\pi i/11},e^{6\pi i/11},e^{4\pi i/11}),0, \frac{4\pi}{11}),\quad \rho(h)\sim (I, 0,-2\pi).$
\noindent 2) $\rho(x_1) \sim (diag(1,-1,-1),0,\pi), \rho(x_2) \sim (diag(e^{2\pi i/3}, 1,e^{4\pi i/3}),0, -\frac{2\pi}{3})),$
$ \rho(x_3) \sim (diag(e^{-12\pi i/11},e^{-6\pi i/11},e^{-4\pi i/11}),0,-\frac{4\pi}{11}),\quad \rho(h)\sim (I, 0,2\pi).$
\noindent 3) $\rho(x_1) \sim (diag(-1,-1, 1), 0, 2\pi), \rho(x_2) \sim (diag(1,e^{4\pi i/3},e^{2\pi i/3}),0,- \frac{4\pi}{3})),$
$ \rho(x_3) \sim (diag(e^{-4\pi i/11},e^{-10\pi i/11},e^{-8\pi i/11}),0,-\frac{8\pi}{11}),\quad \rho(h)\sim (I, 0,4\pi).$
\noindent 4) $\rho(x_1) \sim (diag(-1,-1, 1), 0, -2\pi), \rho(x_2) \sim (diag(e^{2\pi i/3},1,e^{4\pi i/3}),0,\frac{4\pi}{3})),$
$ \rho(x_3) \sim (diag(e^{4\pi i/11},e^{10\pi i/11},e^{8\pi i/11}),0, \frac{8\pi}{11}), \quad \rho(h)\sim (I, 0,-4\pi).$
\noindent 5) $\rho(x_1) \sim (diag(-1,-1, 1), 0, 0), \rho(x_2) \sim (diag(e^{2\pi i/3},e^{4\pi i/3},1),0,0),$
$ \rho(x_3) \sim (diag(e^{-2\pi i/11},e^{2\pi i/11},1),0, 0),\quad \rho(h)\sim (I, 0, 0).$
Using Theorem \ref{sf}, we can find the values of the Chern-Simons invariants of $\Sigma(2,3,11)$ as below.
\begin{center}
\begin{tabular}{cccccc}
Case \qquad &\qquad 1 &\qquad 2 &\qquad 3 &\qquad 4 &\qquad 5 \\
$cs(\rho) \mod \mathbf{Z} $ &\qquad $\frac{13}{66}$ &\qquad $\frac{13}{66} $ &\qquad $ \frac{7}{66}$ &\qquad $\frac{7}{66}$ &\qquad $\frac{25}{66}$
\end{tabular}
\end{center}
Therefore, we can deduce that: \textit{the Burn-Epstein invariant $(\mod \mathbf{Z})$ of any spherical CR structure on $\Sigma(2,3,11)$ with irreducible holonomy representation is one of the values above. }
\begin{remark}
- Little is known about the problem of classification of spherical CR structures on 3-manifolds. On Seifert fibered manifolds, the only work done so far is the classification of the $\mathbf{S}^1$-invariant CR structure by Kamishima and Tsuboi \cite{kt}. We do not know any example of spherical CR structure whose holonomy is the representations given in five cases above.
- Recently, Biquard and Herzlich \cite{bi1} introduce an invariant $\nu$ for strictly pseudoconvex $3$-dimensional CR manifold $M$ and show that their invariant agrees with three times the Burn-Epstein invariant up to a constant. Furthermore, by relating the $\nu$ invariant to a kind of eta invariant, Biquard, Herzlich and Rumin \cite{bi2} are able to give an explicit formula for the $\nu$ invariant of the transverse $\mathbf{S}^1$-invariant CR structure on a Seifert fibered manifold. It would be interesting to work out the relation between the $\nu$ invariant, modulo an integer, and the metric Chern-Simons invariant.
\end{remark}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 286 |
\section{Introduction:}
Three nucleon short-range correlations~(3N-SRCs), in which three nucleons come close together, are unique arrangements in strong interaction
physics. 3N~SRC's have a single nucleon with very large momentum ($\gtrsim 700$~MeV/c) balanced by two nucleons of comparable momenta.
Unlike two-nucleon short-range correlations~(2N-SRCs), 3N-SRCs have never been probed directly through experiment.
As the consequence of the factorization of short-distance effects from low momentum collective phenomena~\cite{FS81,srcrev},
2N- and 3N- SRCs dominate the high momentum component of nuclear wave function which is almost universal
up to a scale factor~(see e.g.\cite{FS81,Atti:2015eda}).
The dynamics of three-nucleon short-range configurations reside at the borderline of our knowledge of nuclear forces making their exploration a testing ground for ``beyond the standard nuclear physics" phenomena such as irreducible three-nucleon forces, inelastic transitions in 3N systems as well as the transition from hadronic to quark degrees of freedom. Their strength is expected to grow faster with the local nuclear density than
the strength of 2N-SRCs~\cite{FS81,srcrev}.
As a result, their contribution will be essential for an understanding of the dynamics of super-dense nuclear matter (see e.g. Ref.~\cite{PanHei:2000}).
Until recently a straightforward experimental probe of 2N- and 3N-SRCs was impossible due to the requirements of high-momentum transfer nuclear reactions being measured in very specific kinematics in which the expected cross sections are very small~(see Ref.\cite{FS81} and references therein). With the advent of the high energy (6~GeV) and high intensity continuous electron accelerator at Jefferson Lab~(JLab) in the late 1990's, an unprecedented exploration of nuclear structure became possible, opening a new window to multi-nucleon SRCs.
\begin{figure}[th]
\vspace{-0.3cm}
\includegraphics[height=2.4cm,width=5.4cm]{figure1.pdf}
\vspace{-0.3cm}
\caption{(a) Geometry of 2N-SRCs, ${\bf p_r}\approx-{\bf p_i}$.
Two configurations of 3N-SRCs: (b) Configuration in which recoil nucleon momenta $\bf {p_{r2}}, \bf {p_{r3}} \sim {\bf - {p_i}/2}$,
(c) configuration in which $p_{r2}\sim p_{r3}\sim p_i$. Here $m_s$ is the invariant mass of the recoiling 2N system.}
\label{src}
\vspace{-0.2cm}
\end{figure}
\section{ Two Nucleon Short Range Correlations~(2N-SRCs)}
The first dedicated study of 2N-SRCs in inclusive, $A(e,e^\prime)X$, high momentum transfer reactions revealed a plateau in the ratios of per nucleon cross sections of heavy nuclei to the deuteron \cite{FSDS} measured at Stanford Linear Accelerator Center~(SLAC) with momentum transfer, $Q^2\gtrsim 2$~GeV$^2$ and Bjorken variable $x> 1.5$. Here $x = {Q^2\over 2m_N q_0}$ with $m_N$ the nucleon mass and $q_0$ the transferred energy to the nucleus, and for a nucleus $A$, $0< x < A$. The observed plateau, largely insensitive to $Q^2$ and $x$, sets the parameter $a_{2}(A,Z)$\cite{FS88} which is the probability of finding 2N-SRCs in the ground state of the nucleus A relative to the deuteron. These plateaus were confirmed in
inclusive cross section ratios of nuclei A to $^3$He\cite{Kim1,Kim2}, at similar kinematics with the magnitude of plateaus taken to be related to the relative probability, ${a_{2}(A,Z)\over a_{2}(^3He)}$. Qualitatively and quantitatively the latter results were in agreement with Ref.\cite{FSDS}. These, together with more recent and dedicated measurements of the nuclear to the deuteron inclusive cross section ratios\cite{Fomin2011}, provided compelling evidence for the sizable ($\sim 20\%$) high momentum component of the ground state nuclear wave function for medium to heavy nuclei originating from 2N-SRCs.
While inclusive processes provided the first evidence for 2N-SRCs and an estimate of their probabilities, $a_2(A,Z)$, the details of correlation dynamics have been obtained
mainly from semi-inclusive experiments in which one or both nucleons from 2N-SRCs were detected.
The first $A(p,ppn)X$ experiments at high momentum transfer were performed at Brookhaven National Laboratory\cite{eip1,eip2}.
The theoretical analysis of these experiments gave the striking result that the probability of finding proton-neutron combinations in 2N-SRCs exceeds by almost a factor of 20 the probabilities for proton-proton and neutron-neutron SRCs\cite{isosrc}.
This result was subsequently confirmed in semi-inclusive electroproduction reactions at JLab\cite{eip3,eip4} and both are understood as arising from the dominance of the tensor component in the NN interaction at distances $|r_1 -r_2| \lesssim 1$~fm \cite{eheppn2,Schiavilla:2006xx}. This reinforced the conclusion that the nucleons have been isolated in SRCs with separations much smaller than average inter-nucleon distances. The dominance of the $pn$ component in 2N-SRCs suggested a new prediction for momentum sharing between high momentum protons and neutrons in asymmetric nuclei\cite{newprops} where the minority component (say protons in neutron rich nuclei) will dominate the high momentum component of the nuclear wave function. This prediction was confirmed indirectly in $A(e,e'p)X$ and $A(e,e'pp)X$ experiments\cite{twofermi} and directly in $A(e,e'p)X$ and $A(e,e'n)X$ processes in which proton and neutron constituents of 2N-SRCs have been probed independently\cite{pndirect,Duer:2018sxh}. The momentum sharing effects also arise from variational Monte-Carlo calculations for light asymmetric nuclei\cite{Wiringa} as well as in model calculations of nuclear wave functions for medium to heavy nuclei\cite{Vanhalst:2014cqa}.
In addition to measuring the isospin content of 2N-SRCs, several experimental analyses\cite{eip2,eip3,Erez18} established a detailed ``geometrical" picture of
2N-SRCs consisting of two overlapping nucleons having relative momentum between $250-650$~MeV/c with back-to-back angular correlations (Fig.{\ref{src}(a))
and with moderate center of mass momentum, $\lesssim 150$~MeV/c, for nuclei ranging from $^4$He to $^{208}$Pb\cite{Erez18}.
Several recent reviews\cite{srcrev,srcprog,Atti:2015eda,arnps,Hen:2016kwk} have documented extensively the recent progress in the investigation of 2N-SRCs in a wide range of atomic nuclei.
\section{Three Nucleon Short Range Correlations~(3N-SRCs)}
Despite an impressive progress achieved in studies of 2N-SRCs the confirmation of 3N-SRCs remains arguable. One signature of 3N-SRCs is the onset of the {\it plateau} in the ratio of inclusive cross sections of nuclei A and $^3$He in the kinematic region of $x>2$ similar to the plateau observed for 2N-SRCs in the region of $1.5 < x < 2$ and discussed above. The first observation of a plateau at $x>2$ was claimed in Ref.\cite{Kim2}. However it was not confirmed by subsequent measurements\cite{Fomin2011,Ye:2017mvo}. The source of this disagreement has been traced to the poor resolution at $x>2$ of the experiment of Ref.\cite{Kim2} which led to bin migration\cite{Higinbotham:2014xna} where events move from smaller to higher $x$.
Additionally, as it will be shown below, the absence of a plateau is related to the the modest invariant momentum transfer, $Q^2$ covered in Ref.~\cite{Kim2}.
\begin{figure}[tb]
\vspace{-0.7cm}
\includegraphics[height=7cm,width=9cm]{figure2.pdf}
\vspace{-1.0cm}
\caption{Kinematics of 3N-SRCs. The surface above the horizontal plane at $\alpha_{3N}=1.6$ defines the kinematics most optimal for
identification of 3N-SRCs in inclusive processes. In this calculation we assumed a minimal mass for $m_S= 2m_N$ which corresponds to
the maximal contribution to the nuclear spectral function with $k_\perp=0$ and $\beta = 1$ (see Eq.(\ref{alpha3N})).}
\vspace{-0.5cm}
\label{3Nkins}
\end{figure}
To quantify the last statement we first need to identify the dominant structure of 3N-SRCs in the nuclear ground state. The problem is that while for 2N-SRCs the correlation geometry is straightforward (two fast nucleons nearly balancing each other, Fig.\ref{src}(a)), in the case of 3N-SRCs the geometry
of balancing three fast nucleons is not unique - ranging from configurations in which two almost parallel spectator nucleons with momenta,
$\sim {\bf - {p_i\over 2}}$ balance the third nucleon with momentum ${\bf p_i}$, Fig.\ref{src}(b)), to the configurations in which all three nucleons have momenta $p_i$ with relative angles $\approx 120^0$
Fig.\ref{src}(c)). The analysis of 3N systems in Ref.\cite{eheppn2} demonstrated that configurations in which two recoil nucleons have the smallest possible mass, $m_S\sim 2m_N$, dominate the 3N-SRC nuclear spectral function at lower excitation energy.
This allows us to conclude \cite{DFSS18} that in inclusive scattering, which integrates over the nuclear excitation energies, the dominant contribution to 3N-SRCs originates from arrangements similar to Fig.\ref{src}(b)) with $m_S\gtrsim 2m_N$.
With the dominant mechanism of 3N-SRCs identified we are able to develop the kinematic requirements to expose 3N correlations in inclusive $eA$ scattering. We use the fact that, due to relativistic nature of SRC configurations, the most natural description is achieved through the light-cone~(LC) nuclear spectral functions\cite{FS88,multisrc} in which the correlated nucleons are described by their nuclear light-cone momentum fractions, $\alpha_i$ and transverse momentum $p_{i,\perp}$. In inclusive scattering one probes the spectral function integrated over the LC momenta of the correlated recoil nucleons, residual nuclear excitation energy and the transverse momentum of the interacting nucleon.
This corresponds to the LC density matrix of the nucleus $\rho_A(\alpha_N)$,
where $\alpha_N$ is the LC momentum fraction of the nucleus carried by the interacting nucleon. It can be shown\cite{FSS15} that $\rho_A(\alpha_N)/\alpha$ is analogous to the partonic distribution function in QCD, $f_i(x)$ where $x$ describes the LC momentum fraction of the nucleon carried by the interacting quark.
To evaluate the LC momentum fraction, $\alpha_{N}$ (henceforth $\alpha_{3N}$) describing the interacting nucleon in the 3N-SRC,
we consider the kinematic condition of quasielastic scattering from a 3N system: $q + 3m_N = p_f + p_{S}$, where $q$, $p_f$ and $p_S$ are the four momenta of
the virtual photon, final struck nucleon and recoil two-nucleon system respectively. One defines the LC momentum fraction of the interacting nucleon, $\alpha_{3N} = 3- \alpha_S$,
where $\alpha_S \equiv 3{E_S - p_S^z\over E_{3N}-p_{3N}^z}$ is the light-cone fraction of the two spectator nucleons in the center of mass of the $\gamma^* (3N)$ system with
$z || q$. Using the boost invariance of
the light-cone momentum fractions one arrives at the following lab-frame expression (see Ref.\cite{DFSS18} for details) :
\begin{eqnarray}
& & \alpha_{3N} = 3 \ - \ {q_- + 3 m_N\over 2 m_N} \left[1 \ \ + \ \ {m_S^2 - m_N^2\over W_{3N}^2} \ \ + \right. \ \ \ \nonumber \\
& & \left.
\sqrt{\left(1 - {(m_S + m_N)^2\over W_{3N}^2}\right)\left(1 - {(m_S - m_N)^2\over W_{3N}^2}\right)}\right],
\label{alpha3N}
\end{eqnarray}
where $W^2_{3N} = (q + 3m_N)^2 = Q^2{3-x\over x} + 9 m_N^2$ and $q_- = q_0 - q$ with $q_0$ and $q$ being energy and momentum
transfer in the lab with $z|| {\bf q}$.
The invariant mass of the spectator 2N system, $m_S^2 = 4{m_N^2 + k_\perp^2\over \beta (2-\beta)}$ where
$\bf {k_\perp}$ is the transverse component of the relative momentum of the 2N system with respect to $\bf {p_S}$
and $\beta$ is the light-front momentum fraction of $p_S$ carried by the spectator nucleon ($0\le \beta\le 2$).
Inclusive reactions integrate over the nuclear spectral function and $k_\perp$ and $m_s$ are not determined experimentally.
The expression for $\alpha_{3N}$, Eq.~(\ref{alpha3N}), makes it possible to identify the kinematical conditions most appropriate for the isolation of 3N-SRCs in
inclusive $A(e,e^\prime)X$ reactions. This is done by identifying the minimal value of $\alpha_{3N}$ above which one expects
the contribution of 3N-SRCs to dominate. First, the threshold can be established from our experience of studying 2N-SRCs.
In this case we know that 2N-SRCs in inclusive processes dominate at $\alpha_N\ge 1.3$ which corresponds to an internal longitudinal momenta of
$\sim 300-350$~MeV/c.
Hence for 3N-SRCs one needs at least $p_{min} \gtrsim 700$~MeV/c, corresponding to $\alpha_{3N} \gtrsim 1.6$,
which will allow two high momentum spectator nucleons to belong to a 3N-SRCs.
This minimal value for $\alpha_{3N}$ is validated by the studies of the fast backward nucleon production in pA scattering within the few-nucleon correlation
model~\cite{FS88} which indicate that the transition from 2N- to 3N- SRCs occurs at $\alpha \sim 1. 6 - 1.7$.
As $\alpha_{3N}$ increases above $1.6$ the contribution of 2N-SRCs is suppressed relative to 3N-SRCs.
This is because as the LC momentum fraction grows, the relative momentum in the 2N system grows much faster than the same quantity in the 3N system.
Thus, in the further discussions we will set
$\alpha_{3N}=1.6$ as the threshold value, above which one expects the 3N-SRCs to dominate in inclusive scattering.
This minimal value for $\alpha_{3N}$ allows us to identify the kinematic parameters most promising for probing 3N-SRCs as illustrated in Fig.~\ref{3Nkins}. The figure shows the relevant kinematics corresponding to the $\alpha_{3N}$ surface being above the $\alpha_{3N}=1.6$ plane. This identifies the $Q^2$ and $x$ domains favorable for probing 3N-SRCs.
In particular, one observes that starting around $Q^2\gtrsim 2.5-3$~GeV$^2$ one gains enough kinematical range in the $x>2$ domain that one
expects to observe 3N-SRCs.
Another advantage of considering 3N-SRCs in terms of $\alpha_{3N}$, is that at sufficiently large $Q^2$ the LC momentum
distribution function $\rho_A(\alpha_{3N})$ is not altered
by final state hadronic interactions~(FSIs). The important feature in the high energy limit is that FSIs redistribute the $p_\perp$ strength in the
nuclear spectral function leaving $\rho_A(\alpha_{3N})$ practically unchanged\cite{ms01,ggraphs,BS15}. In this limit
the distortion of $\alpha_{3N}$ due to FSI can be evaluated as\cite{ms01}:
\begin{equation}
\delta \alpha \approx {x^2\over Q^2}{2m_N E_R\over (1 + {4m_N^2x^2\over Q^2})},
\label{dalpha}
\end{equation}
where $E_R$ is the kinetic energy of the recoil two nucleon system.
The estimates made in Ref.\cite{DFSS18} indicate that for $Q^2 \sim 3$~GeV$^2$ FSI may alter $\alpha_{3N}$ by not more than
$8\%$ which is too small to shift the mean field nucleon, $\alpha_N \approx 1$, to the 3N-SRC domain at $\alpha_{3N}\ge 1.6$.
\section{Signatures of 3N-SRCs}
The cross section in inclusive electron scattering at high $Q^2$ is factorized in the form\cite{FS88}:
\begin{equation}
\sigma_{eA} \approx \sum\limits_N \sigma_{eN}\rho^{N}_A(\alpha_N),
\label{eA}
\end{equation}
where $\sigma_{eN}$ is the elastic electron-bound nucleon scattering cross section and $\rho^N_A(\alpha_N)$ is the
light-front density matrix of the nucleus at a given LC momentum fraction, $\alpha_N$ of the probed nucleon.
This is analogous to the QCD factorization in inclusive deep-inelastic scattering off the nucleon, in which the cross section is a product of
a hard electron-parton scattering cross section and partonic distribution function.
The local property of SRCs suggests that $\rho_A(\alpha_N)$ in the correlation region to be proportional to the light-front density matrix of the two- and three-nucleon systems\cite{FS88,FSDS}. This expectation leads to the prediction of the plateau for the ratios of inclusive cross sections in the SRC region that has been confirmed for 2N-SRCs. Similar to 2N-SRCs for the 3N-SRC one predicts a plateau for the experimental cross section ratios such as:
\begin{equation}
R_{3}(A,Z) = {3\sigma_{A}(x,Q^2)\over A \sigma_{^3He}(x,Q^2)}\left|_{\alpha_{3N}>\alpha^0_{3N}}\right.,
\label{R3}
\end{equation}
where $\alpha^0_{3N}$ is the threshold value for the $\alpha_{3N}$ above which one expects onset of 3N-SRCs (taken as $\sim 1.6$ as described
above).
To quantify the strength of 3N-SRCs we introduce a parameter $a_3(A,Z)$\cite{DFSS18}:
\begin{equation}
a_3(A,Z) = {3\over A} {\sigma_{eA}\over (\sigma_{e^3He} + \sigma_{e^3H})/2},
\label{a3}
\end{equation}
representing an intrinsic nuclear property related to the probability of finding 3N-SRCs in the nuclear ground state.
If a plateau is observed in the 3N-SRC region of $\alpha_{3N}$ then the ratio $R_{3}(A,Z)$ in Eq.(\ref{R3}) can be used to extract
$a_3(A,Z)$ as follows\cite{DFSS18}:
\begin{equation}
a_3(A,Z) = R_3(A,Z) {(2\sigma_{ep} + \sigma_{en})/3\over (\sigma_{ep}+\sigma_{en})/2}.
\label{a3_R3}
\end{equation}
The status of the experimental observation of the scaling in the ratio of Eq.(\ref{R3}) is as follows:
The E02-109 experiment\cite{Arrington:2002ex} provided a high accuracy
ratios, in the 2N-SRC region, at large momentum transfer for a wide range of nuclei\cite{Fomin2011}.
This experiment
covered some part of the 3N-SRC kinematic region with lesser quality of data (see also Refs.\cite{Arrington:2002ex,Fomin:2010ei,FominAIP,FominPHD}),
providing an indication of a plateau in the cross section ratios beginning at $x>2$
once $Q^2$ is sufficiently high.
In Ref.\cite{DFSS18} it was pointed out that the above-mentioned data \cite{Arrington:2002ex,Fomin2011,Fomin:2010ei} suffered from a collapse of
the $^3He$ cross section between $x = 2.68$ and $x = 2.85$ due to difficulties with the subtraction of the Aluminum target walls.
This issue arose from the relatively small diameter of the target cell (4~cm) combined with the fact that $\sigma^{\rm{Al}} \gg \sigma^{^3\rm{He}}$
at large $x$ as $\sigma^{^3\rm{He}}$ must go to 0 at its kinematic limit, $x=3$.
The cross section ratio in Ref.~\cite{Fomin2011} was made possible by the
following: First the inverted ratio $^3$He/$^4$He was formed and then rebinned - combining three bins into one for $x \geq 1.15$.
Subsequently the bins in the inverted ratio that had error bars falling below zero were moved along a truncated gaussian, such that the lower edge of the error bar was at zero.
The ratio was then inverted to give the ratio for $^4$He/$^3$He shown in Figure 3 of Ref.~\cite{Fomin2011} and as the triangles in
Fig.~\ref{ratio4to3} below. The use of a truncated gaussian gave rise to the asymmetric error bars seen
in the ratios.
As an alternative to
the somewhat unconventional procedure above, we have
used the following approach to substitute the $^3$He data of Refs.\cite{Arrington:2002ex,Fomin2011,Fomin:2010ei} in 3N-SRC region:
We have replaced the problematic data between $x = 2.68$ and $x = 2.85$ $(1.6 \leq \alpha_{3N} \leq 1.8)$, point by point,
by employing a y-scaling function $F(y)$\cite{Day:1987az,Sick:1980ey,Day:1990mf} fit to the SLAC data \cite{Day:1979bx,Rock:1981aa}
measured at a comparable $Q^2$.
A simple, two parameter fit $F(y) = a \exp({-bx})$, limited to the range
$1.6 (y =-0.7)\le \alpha_{3N} \le 1.8 (y = -1.1)$ provides a good description of the the SLAC data\cite{DFSS18}.
We preserved the absolute error of the E02019 data set \cite{Arrington:2002ex,Fomin2011,Fomin:2010ei}
rather than the smaller errors from the fit. The fit parameters are $a =0.296$ and $b = 8.241$.
\begin{figure}[th]
\includegraphics[width=0.5\textwidth]{figure3.pdf}
\vspace{-0.5cm}
\caption{The $\alpha_{3N}$ dependence of the inclusive cross section ratios for $^4$He to $^3$He, triangles - JLAB data~\cite{Fomin2011,Fomin:2010ei},
circles - ratios when using a parameterization of SLAC $^3$He cross sections~\cite{Day:1979bx,Rock:1981aa}.
The horizontal line at $1.3\le \alpha_{3N}<1.5$ identifies the magnitude of the 2N-SRC plateau.
The line for $\alpha_{3N}>1.6$ is Eq.(\ref{R3_to_R2sq}) with a $10\%$ error introduced to account for the systematic uncertainty in $a_2(A,Z)$ parameters across
all measurements. The data correspond to $Q^2\approx 2.5$~GeV$^2$ at $x=1, \alpha_{3N} =1$.}
\label{ratio4to3}
\end{figure}
Note that the above approach was first used in Ref.~\cite{FSDS},
which provided the first evidence of 2N-SRCs through cross section ratios in inclusive scattering.
The 2N-SRC results obtained have been confirmed by subsequent precision studies\cite{Kim1,Kim2,Fomin2011}
in which the ratios were measured in single experiment.
It is also worth mentioning that in the case of 2N-SRC the adopted approach was more complicated than the one we employed in the current work.
In Ref.~\cite{FSDS} the data were combined to form the cross section ratios of nuclei ($^3$He, $^4$He, C, Al, Fe and Au) to the deuteron, covering
a range in Q$^2$ from 0.9 to 3.2 (GeV/c)$^2$.
In the current analysis of 3N-SRCs, we worked at a single value of Q$^2 \approx$ 2.7 (GeV/c)$^2$ and, incidentally, the $^3$He data used in 1993 is the same set we employ here. The resulting ratios are displayed as red circles in Fig.~\ref{ratio4to3}.
Fig.~\ref{ratio4to3} presents the results for the cross section ratios obtained within the two above described approaches.
While both give similar results we consider
the replacement of the data points between $x = 2.68$ and $x = 2.85$ $(1.6 \leq \alpha_{3N} \leq 1.8)$ as a best alternative
to the procedure adopted in Ref~\cite{Fomin2011} in part because it allows a consistent treatment of the ratios for all $A $.
In Fig.~\ref{ratio4to3} the plateau
due to 2N-SRCs is clearly visible for $1.3 \le \alpha_{3N} \le 1.5$.
In this region $\alpha_{3N}\approx \alpha_{2N}$\cite{DFSS18}, where $\alpha_{2N}$ is the LC momentum fraction of the nucleon in
the 2N-SRC. Because of this, we refer to the magnitude of this plateau as:
\begin{equation}
R_2(A,Z) = {3\sigma_{A}(x,Q^2)\over A \sigma_{^3He}(x,Q^2)} \left|_{1.3\le \alpha_{3N}\le 1.5}\right. = \frac{a_2(A)}{a_2(^3He)}.\label{eq:R2}
\end{equation}
The horizontal line in the region of $1.3 \le \alpha_{3N} \le 1.5$ is given by the right hand side of Eq.~(\ref{eq:R2}), in which the values of
$a_2(^3He)$ and $a_2(A)$ are taken from the last column of Table II in Ref.~\cite{Arrington:2012ax}, an average of the SLAC, JLAB Hall C and JLAB Hall B results. The magnitude of the horizontal solid line in the region of $1.6 \le \alpha_{3N} \le 1.8$, is the prediction
of $R_{3N}(A,Z)\approx R_{2N}^2(A,Z)$
which will be explained in the next section. We assigned a $10\%$ error to this prediction (dashed lines) related to the uncertainty of
$a_2(A,Z)$ magnitudes across different measurements.
As Fig.\ref{ratio4to3} shows the data at $\alpha_{3N}>1.6$ are consistent with the prediction of the onset of the new plateau in the 3N-SRC region and
that its magnitude} is proportional to $R_{2N}^2$.
With a set of $^3$He data
obtained in the above discussed approach
we are able to estimate the ratios for other nuclei, including, $^9$Be, $^{12}$C, $^{64}$Cu, and $^{197}$Au, albeit with larger uncertainties\cite{DFSS18}.
The large experimental uncertainties in evaluation of the ratios for $^4He$ (Fig.\ref{ratio4to3}) and for heavier nuclei\cite{DFSS18} do not allow us to
claim unambiguously the onset of the plateau at $\alpha_{3N}\ge 1.6$. However one can evaluate the validity of such a plateau by comparing
one- and two- parameter fits to the experimental ratios in the $\alpha_{3N}\ge 1.6$ region.
The one-parameter fit in the 3N-region gives the values ($R_3^{exp}$) of the plateaus as seen in Figure~\ref{ar3ar2}(a) along with our prediction of Eq.~(\ref{R3_to_R2sq}). $R_3^{exp}$ is also listed in Table~\ref{a2R3a3}.
A two-parameter linear fit, returns errors on the parameters nearly as large as the parameters themselves and a correlation matrix
indicating that the second parameter is redundant, providing no additional information.
\section{3N-SRCs and the $\bf pn$ Dominance:}
\label{pndominance}
In Fig.\ref{src}(b) the 3N-SRC is produced in the sequence of two short-range $NN$ interactions in which the nucleon with the
largest momentum interacts with the external probe\cite{multisrc,DFSS18}.
The presence of short-range $NN$ interactions in 3N-SRC configurations tells us that the recently observed $pn$-SRC
dominance\cite{isosrc,eip4,eip3} is critical to our understanding of 3N-SRCs.
For 3N-SRCs one expects that only $pnp$ or $npn$ configurations to contribute, with the $pn$ short-range
interaction playing role of a ``catalyst" in forming a fast interacting nucleon with momentum, $p_i$~(Fig.\ref{src}(b) ).
For example, in the case of $pnp$ configuration, the neutron will play the role of intermediary in furnishing a large momentum transfer
to one of the protons with two successive short range $pn$ interactions.
Quantitatively such a scenario is reflected in the nuclear light-front density matrix in the 3N-SRC domain,
$\rho^N_{A(3N)}(\alpha_N)$, being expressed through the convolution of two $pn$-SRC density matrixes, $\rho^N_{A(pn)}(\alpha,p_\perp)$
as follows:
\vspace{-0.3cm}
\begin{eqnarray}
\rho^N_{A(3N)}(\alpha_N,p_\perp) \approx \sum\limits_{i,j}\int F(\alpha^\prime_i,p_{i\perp},\alpha^\prime_j,p_{j\perp}) \times \ \ \ \ \ \ \ \ \ \ \ &&
\nonumber \\
\rho^N_{A(pn)}\left(\alpha^\prime_i,p^\prime_{i\perp}\right) \rho^N_{A(pn)}\left(\alpha^\prime_j,p^\prime_{j\perp}\right) d\alpha_id^2p_{j\perp}
d\alpha_id^2p_{j\perp}, &&
\label{rho3}
\end{eqnarray}
where $(\alpha^\prime_{i/j}, p^\prime_{i/j\perp})$, are the LC momentum fractions and transverse momenta of spectator nucleons in the center of mass of the $pn$ SRCs.
According to the $pn$ dominance\cite{newprops}:
\begin{equation}
\rho^N_{A(pn)}(\alpha,p_\perp) \approx {a_2(A,Z)\over 2X_N} \rho_d(\alpha,p_\perp),
\label{prop2}
\end{equation}
where $X_N = Z/A$ or $(A-Z)/A$ is the relative fraction of the proton or neutron in the nucleus and $\rho_d(\alpha,p_\perp)$ is the light-front density function of the deuteron at $\alpha\ge 1.3$. The factor $F(\alpha^\prime_i,p_{i\perp},\alpha^\prime_j,p_{j\perp})$ is a smooth function of LC momenta and accounts for the phase factors of nucleons in the intermediate state between the sequential $pn$ interactions with $0< \alpha_{i/j}^\prime<2$.
\begin{figure}[th]
\includegraphics[width=0.45\textwidth]{figure4a.pdf}
\includegraphics[width=0.45\textwidth]{figure4b.pdf}
\caption{(a) The $A$ dependence of the experimental evaluation of $R_3$ compared with the
prediction of Eq.\ref{R3_to_R2sq}. (b) The $A$ dependence of
$a_3(A,Z)$ parameter compared to $a_2(A,Z)$ of Ref.\cite{Fomin2011}. }
\label{ar3ar2}
\vspace{-0.5cm}
\end{figure}
It follows, from Eq.(\ref{rho3}) and the expression of $\rho^N_{A(pn)}(\alpha,p_\perp)$ in Eq.(\ref{prop2}), that the strength of 3N-SRCs is $\propto a_2^2(A,Z)$. This is evident by calculating
$R_3$ in Eq.(\ref{R3}) using the relation~(\ref{eA}) and the conjecture of Eq.(\ref{rho3}), which leads to\cite{DFSS18}:
\begin{equation} R_3(A,Z) = {9\over 8}{(\sigma_{ep}+\sigma_{en})/2\over (2\sigma_{ep} + \sigma_{en})/3}
R^2_2(A,Z) \approx \left( {a_2(A,Z)\over a_2(^3He)}\right)^2,
\label{R3_to_R2sq}
\end{equation}
where $\sigma_{ep} \approx 3\sigma_{en}$ in the considered $Q^2\sim$ 3~GeV$^2$ range. As Fig.\ref{ratio4to3} shows the prediction of $R_3 \approx R_2^2$ is in agreement with the experimental per nucleon cross section ratios of $^4$He to $^3$He targets. There is a similar agreement for other nuclei including $^9$Be, $^{12}$C, $^{64}$Cu and $^{197}$Au\cite{DFSS18}.
\begin{table}[h]
\centering
\caption{Numerical values a$_2$\cite{Arrington:2012ax}, R$_2$ (Eq.~\ref{eq:R2}), R$_3^\textrm{exp}$ (the weighted average in the 3N region) and $a_3$ (Eq.~\ref{a3_R3})).}
\vspace{0.2cm}
\begin{tabular}{|c|c|c|c|c|}\hline
A & a$_2$ & $R_2$ & $R_3^{\textrm{exp}}$ & a$_3$ \\ \hline
3 & 2.13 $\pm 0.04$ & 1 & NA & NA \\ \hline
4 & 3.57 $\pm 0.09$ & $1.68 \pm 0.03$ & 2.74 $\pm 0.24$ & $3.20 \pm 0.28$ \\ \hline
9 & 3.91 $\pm 0.12$ & $1.84 \pm 0.04$ & 3.23 $\pm 0.29$ & $3.77 \pm 0.34$ \\ \hline
12 & 4.65 $\pm 0.14$ & $2.18 \pm 0.04$ & 4.89 $\pm 0.43$ & $5.71 \pm 0,50$ \\ \hline
64 & 5.21 $\pm 0.20$ & $2.45 \pm 0.04$ & 5.94 $\pm 0.52$ & $6.94 \pm 0.77$ \\ \hline
197 & 5.13 $\pm 0.21$ & $2.41 \pm 0.05$ & 6.15 $\pm 0.55$ & $7.18 \pm 0.64$ \\ \hline
\end{tabular}
\vspace{0.2cm}
\label{a2R3a3}
\end{table}
To test the prediction of Eq.(\ref{R3_to_R2sq}) quantitatively
we evaluated the weighted average of $R^{exp}_3(A,Z)$ for $\alpha_{3N}> 1.6$ and compared them with the magnitude of
$({a_2(A,Z)\over a_2({3He})})^2$ in which $a_2(A,Z)$'s are taken from Ref.~\cite{Arrington:2012ax}. The results in which the $^3$He
cross section was taken from the $F(y)$ fit to the SLAC data are presented in Fig.\ref{ar3ar2}(a) and in Table~\ref{a2R3a3}.
They show good agreement with the prediction of Eq.(\ref{R3_to_R2sq}) for the full range of nuclei.
We investigated the sensitivity of the weighted average of $R_3(A,Z)$ on the lower limit of $\alpha_{3N}$ (before rebinning)
and found that the results shown in Fig.~\ref{ar3ar2}(a) remain unchanged within errors which
grow with a larger $\alpha_{3N}>1.6$ cut.
The agreement presented in Fig.\ref{ar3ar2}(a) represents the strongest evidence yet for the presence of 3N-SRCs.
If it is truly due to the onset of 3N-SRCs then one can use the measured $R^{exp}_3$ ratios and Eq.(\ref{a3_R3}) to extract the $a_3(A,Z)$
parameters characterizing the 3N - SRC probabilities in the nuclear ground state. The estimates of $a_3(A,Z)$ and comparisons with
$a_2(A,Z)$ are given in Fig.\ref{ar3ar2}(b) (see also Table~\ref{a2R3a3}).
These comparisons show a faster rise for $a_3(A,Z)$ with $A$, consistent with the expectation
of the increased sensitivity of 3N-SRCs to the local nuclear density\cite{srcrev}. If this result is verified in the future with better quality
data and a wider range of nuclei then the evaluation of the parameter $a_3(A,Z)$ as a function of nuclear density and
proton/neutron asymmetry together with $a_2(A,Z)$ can provide an important theoretical input for the exploration of the dynamics of super dense nuclear matter (see e.g. \cite{Ding:2016oxp}).
\section{Summary} Based on the theoretical analysis of a three-nucleon system we have concluded that the dominating mechanism
of 3N-SRCs in inclusive processes corresponds to the situation in which the recoil mass of the 2N spectator system
is close to a minimum. From that basis we derived a kinematic condition for the onset of 3N-SRCs in inclusive eA scattering
which should result in the observation of a plateau in the ratio of cross sections of heavy to light nuclei, such as, $\frac{3}{A}\frac{\sigma^A}{\sigma^{3He}}$.
The best quality data, available for large enough $Q^2$ (Fig.\ref{ratio4to3}),
indicate a possible onset of such a plateau at $\alpha_{3N}> 1.6$. This first signature of 3N-SRCs is reinforced by the
good agreement with the prediction of the quadratic ($R_3\approx R^2_2$) dependence between the cross section ratios in the 3N-SRCs domain,
$R_3$, and the same ratio measured in the 2N-SRC region, $R_2$. This agreement has allowed us, for the first time,
to extract the parameter $a_3(A,Z)$ characterizing the strength of 3N-SRCs in the ground state wave function of the nucleus.
Further measurements at larger $Q^2$ are necessary to confirm the observation made in this analysis.
Precision data at large $Q^2$ in the 3N-SRC region can be secured in the forthcoming 12~GeV experiment at Jefferson Lab,
E12-06-105\cite{Arrington:2006xx}.
\begin{acknowledgments}
This work is supported by the US Department of Energy grants:
DE-FG02-96ER40950 (DBD), DE-FG02-01ER41172 (MSS) and DE-FG02-93ER40771 (MIS).
\end{acknowledgments}
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 2,291 |
{"url":"https:\/\/support.bioconductor.org\/p\/p132982\/","text":"Beyond dmpFinder for methylation analysis, using limma for covariates\n2\n0\nEntering edit mode\nJeff \u25b4 20\n@acaa643d\nLast seen 4 months ago\nCanada\n\nthe dmpFinder function in minfi is great. But the next question is how do I control for co-variates that clearly effect DNA methylation. For instance, it's well known that smoking effects DNA methylation.\n\nSay I'm determining differences in CpG methylation between schizophrenia patients (who are known to be heavy smokers) and healthy controls. If I collect some arbitrary smoking score from all subjects, how can I incorporate and control for smoking score when I calculate differences in CpG methylation between the two groups?\n\nAttempting to answer my own question, it looks like limma can help me. But I don't see any clear workflows\/pipelines online that explain how to incorporate and control for co-variates in design matricies created for limma.\n\nCan someone point me to a pipeline that can offer me guidance for what I want to do? if not, any general suggestions or tips would be MUCH appreciated.\n\nNOTE: A cross-package Bioconductor workflow for analysing methylation array data by Maksimovic did not really help as she explained paired analysis with methylation data in limma which is not what I want.\n\n**Im a novice to bioconductor btw. I'd say I'm well versed with minfi, but now that I'm trying to figure out how to build models to answer more complex questions I've hit the wall (despite all the time I've spent reading limma documentations and user guides)\n\nminfiDataEPIC arrays limma minfi Epigenetics \u2022 304 views\nADD COMMENT\n0\nEntering edit mode\n@james-w-macdonald-5106\nLast seen 1 day ago\nUnited States\n\nlimma can't really help you (although that's what dmpFinder uses 'under the hood'). You can use limma to analyze each CpG, but then you need to do something to aggregate the signal across genomic regions. Which is what bumphunter does. Both bumphunter and lmFit from limma use a design matrix that you specify with model.matrix.\n\nIf you want bunches of examples of how to specify a design matrix and how to interpret the coefficients, I would look at the limma User's Guide. Or you could look at anything that is intended to show how to use R to do linear modeling. Or just ?model.matrix or ?formula, the latter of which is pretty good.\n\nIf you want more in-depth information, Julian Faraway's book is good as well.\n\nADD COMMENT\n0\nEntering edit mode\n@gordon-smyth\nLast seen 49 minutes ago\nWEHI, Melbourne, Australia\n\nTo analyse covariates in limma, just the the covariate in the design matrices. It works just the same as any multiple regression model. It isn't specifically documented because there's hardly anything to say. It just works as one would expect.\n\nAs James says, limma can't give a complete methylation analysis by itself because it doesn't aggregate results for CpGs into larger regions. However, if you have a methylation workflow for paired data, adapting it to other linear models with covariates should be reasonably straightward.\n\nADD COMMENT\n\nLogin before adding your answer.\n\nTraffic: 202 users visited in the last hour\nHelp About\nFAQ\nAccess RSS\nAPI\nStats\n\nUse of this site constitutes acceptance of our User Agreement and Privacy Policy.\n\nPowered by the version 2.3.6","date":"2021-10-24 04:19:15","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 0, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.19929993152618408, \"perplexity\": 3172.5644506578988}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": false}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-43\/segments\/1634323585837.82\/warc\/CC-MAIN-20211024015104-20211024045104-00358.warc.gz\"}"} | null | null |
# REMNANTS OF ANOTHER AGE
# Remnants of Another Age
_Poems by_
Nikola Madzirov
_Translated from the Macedonian by_
Peggy and Graham W. Reid,
Magdalena Horvat and Adam Reed
_With a Foreword by_
Carolyn Forché
BOA Editions, Ltd. Rochester, NY 2011
Macedonian Text Copyright © 2011 by Nikola Madzirov
English Language Translations Copyright © 2011 by Peggy Reid, Graham W. Reid, Magdalena Horvat, Adam Reed
All rights reserved
Manufactured in the United States of America
First Edition
111213147654321
For information about permission to reuse any material from this book, please contact The Permissions Company at www.permissionscompany.com or e-mail permdude@eclipse.net.
Publications by BOA Editions, Ltd.—a not-for-profit corporation under section 501 (c) (3) of the United States Internal Revenue Code—are made possible with funds from a variety of sources, including public funds from the New York State Council on the Arts, a state agency; the Literature Program of the National Endowment for the Arts; the County of Monroe, NY; the Lannan Foundation for support of the Lannan Translations Selection Series; the Sonia Raiziss Giop Charitable Foundation; the Mary S. Mulligan Charitable Trust; the Rochester Area Community Foundation; the Arts & Cultural Council for Greater Rochester; the Steeple-Jack Fund; the Ames-Amzalak Memorial Trust in memory of Henry Ames, Semon Amzalak and Dan Amzalak; and contributions from many individuals nationwide.
Cover Design: Sandy Knight
Cover Art: "Untitled" by Colleen Buzzard
Interior Design and Composition: Richard Foerster
Manufacturing: McNaughton & Gunn
BOA Logo: Mirko
Library of Congress Cataloging-in-Publication Data
Madzirov, Nikola, 1973-
Remnants of another age / Nikola Madzirov ; translated by Peggy and Graham W.
Reid, Magdalena Horvat and Adam Reed. — 1st ed.
p. cm.
ISBN 978-1-934414-75-0 (alk. paper)
1. Madzirov, Nikola, 1973—Translations into English. 2. Macedonian poetry—Translations into English. I. Reid, Peggy (Margaret) II. Reid,
Graham Wightman. III. Title.
PG1196.23.A27R4613 2011
891.8'191—dc22
2010029674
|
BOA Editions, Ltd.
250 North Goodman Street, Suite 306
Rochester, NY 14607
www.boaeditions.org
A. Poulin, Jr., Founder (1938–1996)
|
---|---|---
# CONTENTS
Foreword
After Us
When Someone Goes Away Everything That's Been Done Comes Back
I Don't Know
Shadows Pass Us By
The Hands of the Clock
Many Things Happened
The Sky Opens
What We Have Said Haunts Us
Flying
When Time Ceases
The Shadow of the World Passes Over My Heart
Things We Want to Touch
Revealing
It Was Spring
Perfection Is Born
We Reveal the Times
Awakening
Fast Is the Century
I Want Only That
I Saw Dreams
Presence
Ruined Homes
Thoughts on the Weather
Everything Is a Caress
What Is to Be Done?
Usual Summer Nightfall
New Lands
Towns That Don't Belong to Us
From Every Scar on My Body
Before We Were Born
Eras of Longing
Two Moons
Light and Dust
Returning
Outside of Time
Days When One Ought to Be Alone
Silence
A Way of Existing
An Involuntary Conquest of Space
The One Who Writes
Home
Separated
Acknowledgments
About the Author
About the Translators
# FOREWORD
It was September 2007, and Nikola Madzirov and I we were walking through the old bazaar on the eastern bank of the Vardar in Skopje, on a cloudless late summer afternoon. This was one of our earliest days together, so he led me to the Isa Bey Mosque and the Church of St. Petka, then through the stalls of the trinket sellers, the old Turkish baths and ancient caravanserais, all the while whispering about poetry. We whispered out of respect for these sacred places, but also because we were walking through one of the world's ruined cities, a city that is again a thriving metropolis, but with a curious silence observed at its center even now, a silence that befalls shoppers and bicyclists, lovers, and students when they reach the epicenter of the earthquake that leveled Skopje in 1963. The clock hands on the ruins of the rail station are still stopped at 5:17, the dawn hour when the station collapsed upon hundreds of school children departing for a holiday.
This hushing of voices interested Nikola Madzirov, as did the legibility of the ruins. He is not from Skopje, but from Strumica in the eastern part of the country across the mountains, near the Bulgarian border—but knowledge of the destruction of Skopje ten years before his birth had been imparted to him in childhood. Strumica is, after all, also built on a river, also on a fault line in a region of seismic shifts, geothermal springs, and the violent ruptures expected at the crossroads of Europe and Asian Minor. He was schooled in this remote region of Macedonia, a country bounded by Albania, Greece, Bulgaria, Serbia, and Kosovo, where mountain roads are still torn up from NATO tanks rolling toward Kosovo during the war.
Madzirov's is a voice aware of provisional existence on the periphery of Europe in a land tiled by ancient mosaics still under excavation, aware that he was born in the former Socialist Republic of Yugoslavia, a nation that no longer exists, at the time still under Marshal Tito's rule. As many young Central Europeans would say, tapping to let fall the long ash of their cigarettes: _and so on, and etcetera._ But Madzirov is not given to cynicism or despair, and much less is he interested in simplistic ideological constructs _that_ offer a future only, without present or past. His poetics spring from a deeper source: if from Strumica, then when the town was called Astraion, the "starry" place, found in the works of Ptolemy and Pliny, and later, the Strumica of Byzantium and the Ottoman Empire, a place of medieval monasteries and frescoed Orthodox churches, ancient necropoli and third-century Roman baths, with their early saunas and fridigaria. In the old city, magara cups have been unearthed, inscribed with motifs from Homer's _Iliad_ , along with plates depicting Artemis and other deities of the hunt. There are also remnants of an early Christian basilica, a feudal tower, and a maiden's well. These are the _Remnants of Another Age,_ in the writings of Strumica's twenty-first century poet, along with all else found is these masterful lyric poems: the dog's attitude toward refugees, the moon's view of executions, the credibility of the dead.
Madzirov calls himself "an involuntary descendent of refugees," referring to his family's flight from the Balkan Wars a century ago: his surname derives from _madzir_ or _majir,_ meaning "people without a home." The idea of shelter and of homelessness, of nomadism, and spiritual transience serves as a palimpsest in these _Remnants,_ at times clearly legible through the surface of other apparent subjects: fast centuries, usual wars, and also bloggers and e-mail and the musical scores of Arvo Pärt, coterminous with walls of fortresses and snow-covered basilicas.
"History is the first border I have to cross," he writes, this poet who "inherited an unnumbered house." His brief against history is clear: "They write about the fall / of empires and epochs but not / of the old man who looks at a toy / dug up by a bulldozer." The world remains, despite its fragility: there are "bumblebees / that carry pollen / between two warring states, and the footprints of our childhood / are covered / by the wheelmarks of a cart." Shelters are distant, as are "moments" and "all the houses I am dreaming of / . . . the voice of my mother" and also "reality," and "we ourselves, we are distant."
So Madzirov gives up rootedness and becomes the poet of chance, of fortuitous encounters, of meetings "like a paper boat and / a watermelon that's been cooling in the river." These are what allow for the unfurling of the human soul in its "secret / migration," calling us to the wilds of fleeting existence, where we wait as if "we're waiting for the wind / like two flags on a border." However, the flags aren't waving, nor are the saints silent, and the borders suggest that we need to be as suspended as snow falling, "snow that doesn't know if it belongs / to the earth or to the air." He writes: "Be a dream, a mezzanine, / sesame seeds at the bottom of the package, /a 'deer' sign by the road, an alphabet / known to only two people—" He advises us to be light, homeless, suspended and awake, warning us that the bright-lit cities of our innocence exist only in atlases. We can move through strange towns, he writes, "unnoticed / . . . like a second earthquake that merely / rearranges what is already ruined." It is good advice for the twenty-first century, where all is not what it seems—where "the rare bird" is not in the branches, but on "the other side of a banknote," and where we might be mistaken, even in grief, "wailing at the wrong graves."
The voice throughout these poems calls ceaselessly to an other, reassuring that we are already "outside of time," that "Nothing exists outside us" and it is this "nothing" that we must comprehend if we are to inhabit the dwelling of impermanence in corporeal and spiritual joy.
—Carolyn Forché
REMNANTS OF ANOTHER AGE
## AFTER US
One day someone will fold our blankets
and send them to the cleaners
to scrub the last grain of salt from them,
will open our letters and sort them out by date
instead of by how often they've been read.
One day someone will rearrange the room's furniture
like chessmen at the start of a new game,
will open the old shoe box
where we hoard pajama-buttons,
not-quite-dead batteries and hunger.
One day the ache will return to our backs
from the weight of hotel room keys
and the receptionist's suspicion
as he hands over the TV remote control.
Others' pity will set out after us
like the moon after some wandering child.
_Ha Mapja H K._
## [WHEN SOMEONE GOES AWAY EVERYTHING
THAT'S BEEN DONE COMES BACK](Contents.html#rch2)
_For Marjan K._
In the embrace on the corner you will recognize
someone's going away somewhere. It's always so.
I live between two truths
like a neon light trembling in
an empty hall. My heart collects
more and more people, since they're not here anymore.
It's always so. One fourth of our waking hours
are spent in blinking. We forget
things even before we lose them—
the calligraphy notebook, for instance.
Nothing's ever new. The bus
seat is always warm.
Last words are carried over
like oblique buckets to an ordinary summer fire.
The same will happen all over again tomorrow—
the face, before it vanishes from the photo,
will lose the wrinkles. When someone goes away
everything that's been done comes back.
## I DON'T KNOW
Distant are all the houses I am dreaming of,
distant is the voice of my mother
calling me for dinner, but I run toward the fields of wheat.
We are distant like a ball that misses the goal
and goes toward the sky, we are alive
like a thermometer that is precise only when
we look at it.
The distant reality every day questions me
like an unknown traveler who wakes me up in the middle of the
journey
saying _Is this the right bus?,_
and I answer _Yes_ , but I mean _I don't know,_
I don't know the cities of your grandparents
who want to leave behind all discovered diseases
and cures made of patience.
I dream of a house on the hill of our longings,
to watch how the waves of the sea draw
the cardiogram of our falls and loves,
how people believe so as not to sink
and step so as not to be forgotten.
Distant are all the huts where we hid from the storm
and from the pain of the does dying in front of the eyes of the
hunters
who were more lonely than hungry.
The distant moment every day asks me
_Is this the window? Is this the life?_ and I say
_Yes,_ but I mean _I don't know,_ I don't know if
birds will begin to speak, without uttering _A sky._
## SHADOWS PASS US BY
We'll meet one day,
like a paper boat and
a watermelon that's been cooling in the river.
The anxiety of the world will
be with us. Our palms
will eclipse the sun and we'll
approach each other holding lanterns.
One day, the wind won't
change direction.
The birch will send away leaves
into our shoes on the doorstep.
The wolves will come after
our innocence.
The butterflies will leave
their dust on our cheeks.
An old woman will tell stories
about us in the waiting room every morning.
Even what I'm saying has
been said already: we're waiting for the wind
like two flags on a border.
One day every shadow
will pass us by.
## THE HANDS OF THE CLOCK
Inherit your childhood
from the photo album.
Transfer the silence
that expands and contracts
like a flock of birds in flight.
Hold in your hands
the irregular snowball
and the drops that run
down the line of life.
Say the prayer
through sealed lips—
the words are seeds falling into a flowerpot.
Silence is learned in the womb.
Try to be born
like the big hand after midnight
and the seconds will overtake you at once.
## MANY THINGS HAPPENED
Many things happened
while the Earth was spinning on
God's finger.
Wires released themselves
from pylons and now
they connect one love to another.
Ocean drops
deposited themselves eagerly
onto cave walls.
Flowers separated
from minerals and set off
following the scent.
From the back pocket pieces of paper
started flying all over our airy room:
irrelevant things which we'd
never do unless
they were written down.
## THE SKY OPENS
I inherited an unnumbered house
with several ruined nests
and cracks in the walls like the veins
of a lover aroused.
It is here the wind sleeps
and the words of condensed
absences. It's summer
and there's a scent of trampled thyme.
The monks finish telling their beads,
the sky opens to create a current of air
in our souls.
The trees are green, we are invisible,
and only thus can they be seen:
our unborn children and the night
which makes the angels
purer still.
## WHAT WE HAVE SAID HAUNTS US
We've given names
to the wild plants
behind unfinished buildings,
given names to all the monuments
of our invaders.
We've christened our children
with affectionate nicknames
taken from letters
read only once.
Afterwards in secret we've interpreted
signatures at the foot of prescriptions
for incurable diseases,
with binoculars we've zoomed in
on hands waving farewell
at windows.
We've left words
under stones with buried shadows,
on the hill that guards the echo
of the ancestors whose names are not
in the family tree.
What we have said without witnesses
will long haunt us.
The winters have piled up in us
without ever being mentioned.
## FLYING
The haze hangs over the city
like the Virgin Mary's bowed head
from a fresco far away.
Satellite dishes talk to
angels
trying to determine tomorrow's weather:
clear, safe, significant
like a calendar with
red dates.
But as soon as the night joins
the shadows to the wall,
you will sneak out towards the branches
like a rare bird
from the other side of a banknote.
## WHEN TIME CEASES
We are the remnants of another age.
That's why I cannot speak
of home, or death
or preordained pains.
Not one illicit digger so far
has found the walls between us,
or the chill in the bones
in the remnants of all the ages.
When time ceases,
then we'll talk about the truth
and the fireflies will form constellations
on our foreheads.
Not one false prophet
foresaw the shattering of a glass
or the touch of two palms—
two great truths, from which
clear water flows.
We are the remnants of another age.
Like wolves in the sights of eternal guilt
we are withdrawing
into the landscapes of tamed solitude.
## THE SHADOW OF THE WORLD PASSES OVER MY HEART
I haven't the courage of a relocated stone.
You'll find me stretched on a damp bench
beyond all army camps and arenas.
I'm empty as a plastic bag
filled with air.
With hands parted and fingers joined
I indicate a roof.
My absence is a consequence
of all recounted histories and deliberate longings.
I have a heart pierced by a rib.
Fragments of glass float through my blood
and clouds hidden behind white cells.
The ring on my hand has no shadow of its own
and is reminiscent of the sun. I haven't the courage
of a relocated star.
Lucian Blaga
## THINGS WE WANT TO TOUCH
Nothing exists outside us:
the reservoirs dry up
just when we thirst
for silence, when nettles
become a healing herb, and the cities
return the dust to the nearest cemetery.
All those black-and-white flowers on the wallpaper
of the homes we've abandoned
blossom among impersonal histories
just when our words
become a nontransferable heritage,
and the things we want to touch
some other person's presence.
We're like a shoe carried off
in a scurry of stray dogs,
we hug each other
like close-twined cables through the hollow bricks
of houses where no one lives.
And it's been like this for a long time now—nothing
exists outside us:
sometimes we call each other
sun, light, angel.
## REVEALING
I haven't belonged to anyone for ages
like a coin fallen from the edge of an old icon.
I am scattered among the strict inheritances and vows
behind the blinds of drawn destinies.
History is the first border I have to cross,
I wait for the voice set apart from the harmony of obedience
that will report how distant I am.
I am like a bronze statue under the city square of stars
above which birds practice their anthems of hope;
I reveal myself like a feather stuck to an eggshell,
which tells of a premature departure and
heralds new life.
Every day my home
secretly changes under the world's tent,
only childhood is like honey
that never lets anything leave a trace in it.
It is usual in the Balkans to put a coin at the edge of an icon as a donation.
## IT WAS SPRING
It was spring when the invader
burned the deeds to the land where we hunted birds,
colorful insects, butterflies
existing only in old biology textbooks.
Many things have changed the world
since then, the world has changed many things in us.
## PERFECTION IS BORN
I want someone to tell me
about the messages in the water in our bodies,
about yesterday's air
in telephone booths,
about flights postponed because of
poor visibility, despite
all the invisible angels on the calendars.
The fan that weeps for tropical winds,
the incense that smells best
as it vanishes—I want someone to tell me about these things.
I believe that when perfection is born
all forms and truths
crack like eggshells.
Only the sigh of gentle partings
can tear a cobweb apart
and the perfection of imagined lands
can postpone the secret
migration of souls.
And what can I do with my imperfect body:
I go and I return, go and return
like a plastic sandal on the waves
by the shore.
## WE REVEAL THE TIMES
We exist when the windows
and the secret documents are open.
We disperse the dust without mentioning
the dead and those they loved undyingly.
We always pack our pajamas
at the bottom of the suitcase
and our shoes are never turned face to face.
We read our letters once
to hide some secret.
With hands stretched out we reveal the times,
stay silent, silent, whisper things
that matter less than the interrupted dream
of a butterfly that lives only for a day.
## AWAKENING
In the temporary embrace
I speak of eternity.
The wind brings us the calls of the church bells
among the feathers where we rest
our sleepy heads.
It's morning. Moist air passes
under the viaducts, clouds part
at a touch, buildings at the swallows' flight,
the farmhands pray for rain that stops,
while the trees give up their leaves
and so the sky grows vaster.
Your hands are soft this morning
and soft is the blossom of the hard almond.
In the nearby church
they have spoken for centuries of a love
that will outlive us.
## FAST IS THE CENTURY
Fast is the century. If I were wind
I would have peeled the bark off the trees
and the facades off the buildings in the outskirts.
If I were gold, I would have been hidden in cellars,
into crumbly earth and among broken toys,
I would have been forgotten by the fathers,
and their sons would remember me forever.
If I were a dog, I wouldn't have been afraid of
refugees, if I were a moon
I wouldn't have been scared of executions.
If I were a wall clock
I would have covered the cracks on the wall.
Fast is the century. We survive the weak earthquakes
watching towards the sky, yet not towards the ground.
We open the windows to let in the air
of the places we have never been.
Wars don't exist,
since someone wounds our heart every day.
Fast is the century.
Faster than the word.
If I were dead, everyone would have believed me
when I kept silent.
## I WANT ONLY THAT
Like a feather my soul
travels between two windows.
God's uncertainty is my path,
your presence is my imperfection.
I bring you petals of cyclamen and peonies
in the apple season,
I squeeze resin from the vulnerable trees
unskillfully into my hands,
I present you with
the fruits of all the branches
we broke as children.
I want you to replace your shield
with a veil of light;
to hide your absence
behind the steam from hot tea
when the whole world cools.
I want your heart to beat
freely behind the bars of your rib cage.
I want only that.
## I SAW DREAMS
I saw dreams that no one remembers
and people wailing at the wrong graves.
I saw embraces in a falling airplane
and streets with open arteries.
I saw volcanoes asleep longer than
the roots of the family tree
and a child who's not afraid of the rain.
Only it was me no one saw,
only it was me no one saw.
## PRESENCE
Put on the space suit of the night
and slice the apple in two
without damaging the seeds.
Stop on the quiet bridge
and let your shadow float away.
Cup your hands above your head
like a crystal wineglass
and wait for the first drops of rain
when the pilotless aircraft depart.
Be a dream, a mezzanine,
sesame seeds at the bottom of the package,
a 'deer' sign by the road, an alphabet
known to only two people—
you and the one who doesn't believe you.
Be alone, but not lonely,
so that the sky can embrace you,
so that you can embrace the lonely earth.
## RUINED HOMES
In the wastebasket, I saw locks of hair
you'd brushed while the birds and the world were waking.
In the mirror, I saw a look
and in that look a lot of homes and skies.
I saw you going towards cities nonexistent in history books
and the bed parting itself into night and day while you were
gone,
the day becoming night, and the night a hiding place.
I don't have a sun
in my eyes, nor plants in my upturned palms.
I will bend the bars that protect gardens
from night travelers. I will cover the day
with the silk scarf from your neck,
with the still flag of the territories that have witnessed our
presence.
Our e-mail letters cannot fade,
our addresses remain the same even when we run away from
here,
from ourselves, from the wideness of our ancient dependence.
I saw someone else writing our names
on walls of fortresses and snow-covered basilicas.
I saw your shadow, too, climbing up my body
as you were climbing down the discovered shelters
after all the official wars.
Since then, every piece of glass blinds me,
every rejected word covers my eyes with silence. I saw.
Our ruined homes were a move of the world,
of the memory, of the memory.
## THOUGHTS ON THE WEATHER
I know that my voice
is influenced by atmospheric conditions,
that my cry depends
on the territory of the great invaders.
I know that in my back pocket
I keep a newspaper clipping—
the weather forecast
in the belief that the rainbow will show again
like a crown of thorns
above the uninhabited hill.
I know that compassion peels away
like the bark of a tree
from which ancient tribes
once built their boats.
Calmness is a belt
that holds history upright.
One should sit down for a moment
and see the sky
in an open tin can
on the shore.
## EVERYTHING IS A CARESS
The snow was folding its wings
over the hills, I was laying my palms
over your body like a tape measure
which unfolds only along the length
of other things.
The universe existed
so that we'd be born in different places,
so that our homeland could be the rainbow
that joins two gardens
which don't know of one another.
And so time went on:
we were raising the fear within us,
while awe was being born in others.
Our shadows were sinking
in poisoned wells,
spoken words were disappearing
and reappearing like shards of glass on a sandy beach—
sharp and shattered.
## WHAT IS TO BE DONE?
To live without reason or necessity,
to embrace the offenders
liberated from love,
to lift the candle from ruined graves
and say a word or two
when there's no wind,
to open the rusty door of the world
and depart with airy footsteps.
To recover from time's dividers
thrust into our own hearts.
1.
2.
## USUAL SUMMER NIGHTFALL
1.
This is what summer nightfall is like:
the adulteress comes onto the balcony
in a silk nightgown that lets through
the trembling of the stars,
a twig drops from the beak of a bird
that falls asleep before it has built its home,
a soldier lowers the flag of the state
with a letter from his mother in his pocket
and atomic tests in the womb of the earth
secretly revive the dead. At that moment someone
quietly interprets Byzantine neumes,
someone else falsifies the exoduses
of the Balkan and the civil wars
in the name of universal truths.
In the factory yards
the statues of participants
in annulled revolutions sleep,
on the symmetrical graves
plastic flowers lose their color
and ordinary ones their shape,
but this peace of the dead
we have parted from
is not ours.
2.
In the village with three lit windows
a fortune-teller foresees only
recoveries, and not illnesses.
The waves throw up bottles enough
to hold the whole sea,
the arrow on the one-way road sign
points to God,
a fisherman rips off a bit of the sky
as he casts his baited line into the river,
some poor child searches for the Little Bear
and the planet he'd like to come from,
in front of the doorstep of the killer with an alibi
a feather attempts to fly.
This is what usual summer nightfall is like.
The town combusts in the redness of the moon
and the fire brigade ladders seem
to lead to heaven, even then when
everyone
is climbing
down
them.
Basic elements of Eastern systems of musical notation prior to the invention of five-line staff notation.
## NEW LANDS
One should scrape the wall
over which dampness has drawn
a map of the new world
and new separations should be applied.
Beneath them, the stones should be
rearranged haphazardly, like
the footprints of a man running from his fears.
One should be
a round mirror in a half-open palm
and reflect others' embraces
as sharp as scissor blades which touch each other
only when there's something to be cut.
New lands should be invented,
so one can walk on water once again.
## TOWNS THAT DON'T BELONG TO US
In strange towns
our thoughts wander calmly
like graves of forgotten circus artists,
dogs bark at dustbins and snowflakes
falling in them.
In strange towns we are unnoticed
like a crystal angel locked in an airless glass case,
like a second earthquake that merely
rearranges what is already ruined.
## FROM EVERY SCAR ON MY BODY
I am a beggar who lacks the courage
to beg charity from himself.
Lines and wounds from all the unfulfilled caresses
intersect on my palms,
from all the unmeasured temperatures on my brow
and the illicit excavations of love.
From every scar on my body
a truth emerges.
I grow and I diminish
together with the day, running fearlessly
towards the depths of origin,
and everything around me is in motion:
the stone becomes a house;
the rock, a grain of sand.
When I stop breathing
my heart beats louder still.
## BEFORE WE WERE BORN
The streets were asphalted
before we were born and all
the constellations were already formed.
The leaves were rotting
on the edge of the pavement,
the silver was tarnishing
on the workers' skin,
someone's bones were growing through
the length of the sleep.
Europe was uniting
before we were born and
a woman's hair was spreading
calmly over the surface
of the sea.
## ERAS OF LONGING
I stand concealed
like a gull waiting for a fish to fly.
Passengers with the same oaths and expectations
come and go on the harbor wall,
the years slide slowly over the sails
like rainwater on a badly leveled path.
The eras of longing end up beyond the horizon,
in the village on the shore where at night
an old woman hides her coins in a kerchief
that once covered her hair.
## TWO MOONS
A woman looked at her reflection
in the town's translucent fences.
Two moons settled in her eyes
while her gaze brought together the ends
of worlds already explored.
Above her the shadows wove moss
on the rooftops,
below her endemic species were dying
of loneliness.
From the hollow
between her hip and her rib cage
light streamed out each night.
## LIGHT AND DUST
In the space between
the four seasons I'll find you,
when children are taken out for a walk,
and souls come back
like dirty dishes in
a workers' canteen.
We are not a religion
and nobody believes in our
holy scriptures.
Our looks hide
in the curtains' folds
which let other people's prayers through
and the falling light.
Will our angels touch
when we hug each other
in the dark, will someone light a candle
to proclaim a kingdom?
We are the light of a burnt match
which turns to dust
when touched.
## RETURNING
I open the door fearfully
to draw a border on the carpet
with the sun rays.
I feel like saying something
but the echo of the unfurnished room
is faster than me.
The sweat on the doorknob is not mine
nor do the lichens on my neck
belong to this world.
I realized myself
in several layers of memories,
my soul is a womb palimpsest
of a distant mother.
Hence the afterthought of returning
and the soft creak of the hinges.
I would expand the space with a step
to multiply the grains of dust
and the hairs that fall down,
always white because of
the light.
## OUTSIDE OF TIME
In the clear expanse of the sky
we wait to see the contours
of our souls' negatives.
We are far away from time.
Look, the buildings are already asleep
on the dried-up seeds
of annual plants,
the kites rest
their tails on the roofs
of our houses, then depart.
We have been living for years
within circled dates, via
agendas of cold joys.
Our ancestors have long been
statues bending their heads
towards the shoulder of each passer-by,
but we are outside of time.
We take eternity and give it back, take it
and give it back . . .
## DAYS WHEN ONE OUGHT TO BE ALONE
It's true the town
sprang up as the consequence of a lie
essential to people,
flowerpots and pets.
(that's how I equip myself
with the necessary justifications)
It's true that all the people
are coming out of all the buildings
(as if there'd been an earthquake)
and with vases in their hands
set out for the meadows.
They return three times sadder
with dust on their hands
and certain sounds
like holes in their memories.
Then
the shared silence again.
## SILENCE
There is no silence in the world.
Monks have created it
to hear the horses every day
and feathers falling from wings.
## A WAY OF EXISTING
Too many rises and falls
are not recorded in the books
that are burned in usual wars.
Has anyone written that crumbs
thrown from a window fall faster
than snowflakes, that waterfalls are merely
victims of their name? They write of the fall
of empires and epochs but not
of the old man who looks at a toy
dug up by a bulldozer.
Traffic-lights cannot stop time
and our uncertainty is just
a way of existence for secrets.
Fear exists in the distance
when soot splits off
from the sparks flying skyward,
but no one so far has written
a tractate on the candles' smoke
that melts into night
or on the drops of wax
that harden on our shoes;
everyone speaks of the flame
that illuminates our faces.
## AN INVOLUNTARY CONQUEST OF SPACE
When we grow apart
the contents of the air will change,
sorrow will descend unheard
on the outer sides of waterspouts
like the shadow of a frightened lizard.
Every waking outside of our bed
will be condemned,
every filling and emptying of the chest
will become an involuntary conquest of space.
Closeness will escape our hands like a drop
from the body of a fish just caught.
When the sun and moon eclipse with a touch,
they are still apart, and everything
becomes night, a false falling asleep of the leaves,
the shadows, the wild animals.
## THE ONE WHO WRITES
You write. About the things that already exist.
And they say you fantasize.
You keep quiet. Like the sunken nets
of poachers. Like an angel
who knows what the night may bring.
And you travel. You forget,
so that you can come back.
You write and you don't want to remember
the stone, the sea, the believers
sleeping with their hands apart.
## HOME
I lived at the edge of the town
like a street lamp whose light bulb
no one ever replaces.
Cobwebs held the walls together,
and sweat our clasped hands.
I hid my teddy bear
in holes in crudely built stone walls
saving him from dreams.
Day and night I made the threshold come alive
returning like a bee that
always returns to the previous flower.
It was a time of peace when I left home:
the bitten apple was not bruised,
on the letter a stamp with an old abandoned house.
From birth I've migrated to quiet places
and voids have clung beneath me
like snow that doesn't know if it belongs
to the earth or to the air.
## SEPARATED
I separated myself from each truth about the beginnings
of rivers, trees, and cities.
I have a name that will be a street of goodbyes
and a heart that appears on X-ray films.
I separated myself even from you, mother of all skies
and carefree houses.
Now my blood is a refugee that belongs
to several souls and open wounds.
My god lives in the phosphorus of a match,
in the ashes holding the shape of the firewood.
I don't need a map of the world when I fall asleep.
Now the shadow of a stalk of wheat covers my hope,
and my word is as valuable
as an old family watch that doesn't keep time.
I separated from myself, to arrive at your skin
smelling of honey and wind, at your name
signifying restlessness that calms me down,
opening the doors to the cities in which I sleep,
but don't live.
I separated myself from the air, the water, the fire.
The earth I was made from
is built into my home.
## ACKNOWLEDGMENTS
Many thanks to the editors of the following publications where these poems first appeared in the United States:
_American Poetry Review, Poetry International, American Literary Review, West Branch, Borderlands, Visions International_ , and _In Our Own Words—A Generation Defining Itself_.
I am deeply thankful to the editors of BOA Editions and to Carolyn Forché, Adam Zagajewski, Tomaz Salamun, Li-Young Lee, Blue Flower Arts, Iowa's International Writing Program (IWP), and Words Without Borders.
## ABOUT THE AUTHOR
Nikola Madzirov (poet, essayist, translator) is one of the most powerful voices of the new European poetry. He was born into a family of Balkan Wars refugees in 1973 in Strumica, R. Macedonia. His poetry has been translated into thirty languages and published in collections and anthologies in the United States, Latin America, Europe, and Asia. For his poetry book _Relocated Stone_ (2007) he received the Hubert Burda Prize for authors born in East Europe (the jury was chaired by Peter Handke and Michael Krüger), and the most prestigious Macedonian poetry prize, the Miladinov Brothers Award at Struga Poetry Evenings. For the book _Locked in the City_ (1999) he was given the Studentski Zbor Award for the best debut, while for the collection of poems _Somewhere Nowhere_ (1999) the Aco Karamanov Prize. Based on his poetry, two short films were shot in Bulgaria and Croatia. The contemporary jazz composer and collaborator of Björk and Lou Reed, Oliver Lake, has composed music based on Madzirov's poems which was performed at the Jazz-Poetry Concert in Pittsburgh in 2008. Nikola Madzirov has participated at many international literary festivals and events in the United States, Latin America, Asia, and Europe and has received several international awards and fellowships: from the International Writing Program (IWP) at the University of Iowa, Literarisches Tandem in Berlin, KulturKontakt in Vienna, the Internationales Haus der Autoren in Graz, the Literatur Haus NÖ in Krems, and Villa Waldberta in Munich. He is one of the coordinators of the world poetry network _lyrikline.org._
## ABOUT THE TRANSLATORS
Peggy Reid, M.A. (Cantab), _Doctor honoris causa_ , Skopje, M.B.E., born Bath, U.K., 1939, taught English at Ss. Cyril and Methodius University, Skopje, Macedonia, for twenty years between 1969 and 2006. Translator/co-translator from Macedonian of novels, poetry, plays and works of nonfiction. Lives in Canterbury, U.K.
Graham W. Reid, M.A., M.B.E. born Edinburgh, 1938. Read English at Trinity College, Cambridge. Taught English for twenty-five years at Ss. Cyril & Methodius University, Skopje, Macedonia. Widely translated both poetry and prose from Macedonian into English. M.A. thesis at Bradford University on _Reflections of Rural-Urban Migration in Contemporary Macedonian Poetry._ Currently lives in Canterbury, Kent, U.K.
Magdalena Horvat (born 1978, Skopje, Macedonia) is the author of two poetry collections: _This is it, your_ (2006) and _Bluish and other poems_ (2010). Among the books she has translated into Macedonian are Sylvia Plath's _The Bell Jar_ and Fiona Sampson's _The Distance Between Us._ She currently lives in Athens, Georgia.
Adam Reed (born 1978, Athens, Georgia) has co-translated/edited several poetry collections, anthologies and works of nonfiction from Macedonian into English. He taught English, Writing and History courses at University American College Skopje, Macedonia, for several years. He currently lives in Athens, Georgia.
The Lannan Translations Selection Series
Ljuba Merlina Bortolani, _The Siege_
Olga Orozco, _Engravings Torn from Insomnia_
GérardMartin, _The Hiddenness of the World_
Fadhil Al-Azzawi, _Miracle Maker_
Sándor Csoóri, _Before and After the Fall: New Poems_
Francisca Aguirre, _Ithaca_
Jean-Michel Maulpoix, _A Matter of Blue_
_Willow, Wine, Mirror, Moon:_
_Women's Poems from Tang China_
Felipe Benítez Reyes, _Probable Lives_
Ko Un, _Flowers of a Moment_
Paulo Henriques Britto, _The Clean Shirt of It_
Moikom Zeqo, _I Don't Believe in Ghosts_
Adonis (Ali Ahmad Sa'id), _Mihyar of Damascus, His Songs_
Maya Bejerano, _The Hymns of Job and Other Poems_
Novica Tadić, _Dark Things_
_Praises & Offenses:_
_Three Women Poets of the Dominican Republic_
Ece Temelkuran, _Book of the Edge_
Aleš Šteger, _The Book of Things_
Nikola Madzirov, _Remnants of Another Age_
For more on the Lannan Translations Selection Series
visit our Web site:
www.boaeditions.org
| {
"redpajama_set_name": "RedPajamaBook"
} | 4,580 |
\section{INTRODUCTION}
Since 1968, Lynden-Bell \cite{LBW68,LBE80,LynB98} has illuminated the concept
of \textit{negative} heat capacity in astrophysical systems. He explained
Antonov's theorem \cite{Ant62} which states (roughly) that a spherical
collection of self-gravitating point masses has no global entropy maximum.
Explanation was needed since the most probable state for such a system is at
the maximum of the Boltzmann entropy. Lynden-Bell and Wood considered a
self-gravitating gas sphere. They calculated the energy $E$ and heat capacity
$C_{V}$ of the isothermal sphere. A graph of the sphere's binding energy vs
density contrast showed a local maximum near an inflection point, and the plot
of specific heat vs density contrast showed a stable branch for
\textit{negative} specific heat.
Negative $C_{V}$ can be understood as follows: the virial theorem for
\textit{inverse square} forces of bounded systems relates the average kinetic
energy and average potential energy as
\[
<K.E.>\ =-\frac{1}{2}<P.E.>.
\]
With total energy $E=\ <K.E.>+<P.E.>$ \ one ha
\[
E=-<K.E.>\ \text{negative,
\]
but for moving particles $K.E.=\frac{3}{2}Nk_{B}T.$ \ It follows that the
specific heat is negativ
\[
C_{V}=\frac{dE}{dT}=-\frac{3}{2}Nk_{B}.
\]
A negative $C_{V}$ system in contact with a large thermal reservoir will have
fluctuations that add energy and make its transient temperature lower, causing
inward heat flow which will drive it to even lower temperatures. Thus negative
$C_{V}$ systems cannot reach thermal equilibrium.
Lynden-Bell coined the name "gravothermal catastrophe" for stellar systems
undergoing such collapse. To describe the collapse, we quote \cite{LynB68}:
"Conductive transfer of heat from the central region will raise the high
central temperature faster than it raises the lower temperature of the outer
parts. No equilibrium is possible; the center continues to contract and get
hotter, sending out heat to the outer parts."
Astrophysical systems where gravothermal catastrophe may occur \cite{BT03} are
older globular clusters with compact cores, and bright elliptical galaxies
with high central density profiles. Lynden-Bell and Eggleton studied the core
collapse of globular clusters. In particular, they calculated self-similar
collapse features of a gravitating gas sphere. One of the things they learned
from their study was, that to model the formation of a central black hole in a
globular cluster, they needed to go beyond self-similarity and study more
dissipative collapse.
In this work, we present an analytic solution \cite{Gla89} with shear and
radial heat flow (SRH) which models many of the features of catastrophic
collapse. The system collapses into a trapped surface with outgoing energy
radiated to a future boundary.
In the following section, the SRH collapse metric with pressure and density is
developed. The results are given in section III and summarized in section IV.
This is followed by two appendices. (A) contains the original SRH solution
with arbitrary functions, and (B) has mass and trapped surface equations.
\section{COLLAPSING SRH FLUID}
The SRH spacetime \cite{Gla89} is divided into an exterior region covered by
the Vaidya metric and a collapsing interior region with spherically symmetric
metri
\begin{equation}
g_{\mu\nu}^{\text{SRH}}dx^{\mu}dx^{\nu}=A^{2}dt^{2}-B^{2}dr^{2}-R^{2
d\Omega^{2} \label{collapse-met
\end{equation}
where $A=A(t,r),$ $B=B(t,r),$ $R=R(t,r),$ and $d\Omega^{2}$ is the metric of
the unit sphere. The stress-energy tensor is given by ($G=c=1$)
\begin{equation}
T^{\mu\nu}=w\hat{u}^{\mu}\hat{u}^{\nu}-p\gamma^{\mu\nu}+q^{\mu}\hat{u}^{\nu
}+\hat{u}^{\mu}q^{\nu}. \label{energy-mom
\end{equation}
$p$ is the isotropic pressure, $w$ is the mass-energy density, $\gamma^{\mu
\nu}=g^{\mu\nu}-\hat{u}^{\mu}\hat{u}^{\nu}$, and $q^{\mu}$ is the radial heat
flow vector orthogonal to $\hat{u}^{\mu}$. Time is comoving with $\hat{u
^{\mu}\partial_{\mu}=A^{-1}\partial_{t}$, $\hat{u}^{\mu}\hat{u}_{\mu}=1$.\ We
use notation of Taub \cite{Tau69} for the mass-energy density $w$. Taub's
purpose was to distinguish mass-energy density from proper mass-density $\rho
$, where $w=\rho(1+\epsilon)$ and $\epsilon$ is specific internal energy.
The original SRH solution, given in Appendix A, provides the following metric
functions and physical scalars (overdots denote $\partial/\partial t$ and
primes denote $\partial/\partial r$).
\begin{subequations}
\begin{align}
R(t,r) & =e^{t/t_{0}}[(\beta_{0}^{2}/2)e^{-4t/t_{0}}+(r+r_{0})^{2
]^{1/2},\label{R-eq}\\
A(t,r) & =\alpha_{0}\dot{R},\\
B(t,r) & =\beta_{0}/R.
\end{align}
Metric functions $A$ and $B$ should be dimensionless and $R$ should have
dimensions of length. We write $A=\alpha_{0}(\dot{R}/c)$ so it is clear that
$\alpha_{0}$ is dimensionless. The other constants have the following
dimensions: $\dim(t_{0})=time,$ \ $\dim(\beta_{0})=length$, \ $\dim
(r_{0})=length$.
The heat flow vector and scalar are given by $4\pi q_{\mu}dx^{\mu}=Qdr$. The
heat flow scalar is
\end{subequations}
\begin{equation}
Q=-\frac{1}{\alpha_{0}}(\frac{r+r_{0}}{R^{3}})e^{2t/t_{0}}. \label{Q-eqn
\end{equation}
The pressure is
\begin{equation}
8\pi p=\frac{(r+r_{0})^{2}e^{4t/t_{0}}}{\beta_{0}^{2}R^{2}}\left[ \frac
{R^{2}+\beta_{0}^{2}e^{-2t/t_{0}}}{R^{2}-\beta_{0}^{2}e^{-2t/t_{0}}}\right]
-\frac{(1+1/\alpha_{0}^{2})}{R^{2}},
\end{equation}
and the mass-energy density is
\begin{equation}
8\pi w=\frac{1}{R^{2}}\left[ 1-\frac{1}{\alpha_{0}^{2}}-\frac{(r+r_{0})^{2
}{\beta_{0}^{2}}e^{4t/t_{0}}\right] -\frac{2}{\beta_{0}^{2}}e^{2t/t_{0}}.
\end{equation}
$w$ is negative whe
\begin{align*}
\left[ \alpha_{0}^{2}-1-\frac{\alpha_{0}^{2}(r+r_{0})^{2}}{\beta_{0}^{2
}e^{4t/t_{0}}\right] -\frac{2\alpha_{0}^{2}R^{2}}{\beta_{0}^{2}}e^{2t/t_{0}}
& <0\\
\alpha_{0}^{2}\beta_{0}^{2}-\beta_{0}^{2}-\alpha_{0}^{2}(r+r_{0
)^{2}e^{4t/t_{0}}-2\alpha_{0}^{2}R^{2}e^{2t/t_{0}} & <0\\
\text{or \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ }\beta_{0}^{2}+3\alpha_{0
^{2}(r+r_{0})^{2}e^{4t/t_{0}} & >0.
\end{align*}
This inequality always holds, and so $w$ is negative.
\section{COLLAPSE RESULTS}
\subsection{End stage of collapse}
For small $\beta_{0}$, $R$ goes a
\begin{equation}
R\simeq e^{t/t_{0}}(r+r_{0}).
\end{equation}
The pressure and mass-energy density, at late times and for small $\beta_{0}
$, go as
\begin{align}
8\pi p & \simeq\frac{e^{2t/t_{0}}}{\beta_{0}^{2}},\label{late-p}\\
8\pi w & \simeq-3\frac{e^{2t/t_{0}}}{\beta_{0}^{2}}. \label{late-w
\end{align}
At late times, the magnitude of the mass-energy density increases
exponentially, and the fluid approaches a photon gas with equation of state
\begin{equation}
p=\frac{1}{3}\mid w\mid.
\end{equation}
Negative mass-energy density $w$ is linked to the \textit{gravothermal
catastrophe}. The collapsing fluid has a negative specific heat which derives
from negative internal energy $\epsilon$. While both $w$ and $\epsilon$ are
negative, the proper mass density, $\rho=w/(1+\epsilon)$, remains positive. To
separately compute $\epsilon$ would require a complete thermodynamic analysis.
One would need a causal relativistic description of thermodynamics such as the
Israel-Stewart `second order' type theory \cite{IS79,Cal98}. This has been
left for future work.
As a model of gravitational dyamics, Chavanis et al \cite{CRS02} studied the
collapse of a gas of self-gravitating Brownian particles in a closed sphere.
For catastrophic collapse in the microcanonical ensemble, they found the mass
density to go a
\begin{equation}
\rho\simeq\rho_{0}(r_{0}/r)^{2.21} \label{brown-part-dens
\end{equation}
and further, in a stability study, they found the perturbation $\delta
\rho/\rho$ has a "core-halo" structure (hinting at central black hole
formation). Without developing the thermodynamics of a collapsing SRH fluid,
the density approximation in Eq.(\ref{brown-part-dens}) can be used to compute
internal energy from $\epsilon=w/\rho-1$. With $w$ at late times given in
Eq.(\ref{late-w}), we fin
\begin{equation}
\epsilon\simeq-const\text{ }\frac{e^{2t/t_{0}}}{\rho_{0}\beta_{0}^{2}
(r/r_{0})^{2.21}-1
\end{equation}
When the density is given a temperature profile (in the canonical ensemble
$\rho\sim T^{-1/2}$), then $\epsilon=\epsilon(T)$ yields negative specific
heat $c_{V}=d\epsilon/dT$.
\subsection{Trapped Surface}
From Eq.(\ref{m-curv}) the Misner-Sharp mass within a $t=const$, $r=const$
2-surface i
\begin{equation}
2m=R\left[ 1+\frac{1}{\alpha_{0}^{2}}-\frac{(r+r_{0})^{2}}{\beta_{0}^{2
}e^{4t/t_{0}}\right] .\label{mass
\end{equation}
Null rays entering and leaving the 2-surface are described by the expansions
of their respective generators. Equations (\ref{div-n}) and (\ref{div-l}) show
that both null generators have non-negative expansions and so a trapped
surface exisits at $R=2m$. At the trapped surface, Eq.(\ref{mass}) sets a
value for constants $\alpha_{0}$ and $\beta_{0}$
\begin{equation}
\frac{\beta_{0}^{2}}{\alpha_{0}^{2}}=(r_{\text{trap}}+r_{0})^{2
e^{4t_{\text{trap}}/t_{0}}.
\end{equation}
The expression for $R$, Eq.(\ref{R-eq}), yield
\begin{equation}
R_{\text{trap}}^{2}=(r_{\text{trap}}+r_{0})\beta_{0}\frac{(2+\alpha_{0}^{2
)}{2\alpha_{0}}.
\end{equation}
We see from Eq.(\ref{Q-eqn}) that, at late times, the heat flow scalar goes to
zer
\begin{equation}
Q\simeq-\frac{1}{\alpha_{0}}\frac{1}{(r+r_{0})^{2}e^{t/t_{0}}}\rightarrow0.
\end{equation}
The heat flow shuts off as the fluid collapses into the trapped surface.
\subsection{Rate of collapse}
The rate of collapse scalar is $\Theta=\nabla_{\mu}\hat{u}^{\mu
=-1/(\alpha_{0}R)$. We again quote Lynden-Bell: "During the gravothermal
catastrophe ... the center continues to constrict and get hotter, giving out
heat to the outer parts, but the temperature difference increases and drives
the collapse onwards still faster." At distances just beyond the trapped
surfac
\begin{equation}
\Theta\simeq-\frac{e^{t/t_{0}}}{\alpha_{0}\beta_{0}}.
\end{equation}
At late times the rate of collapse increases exponentially. This is a
necessary component for a model of catastrophic collapse.
\section{SUMMARY}
An analytic solution of Einstein's equations for dissipative collapse has been
presented. The original SRH solution contains arbitrary functions of time
which have been chosen here to provide an explicit solution. The system
collapses into a trapped surface with outgoing energy radiated to a future
asymptotic boundary. The collapsing fluid has negative mass-energy, which has
been related to negative specific heat. Lynden-Bell has linked collapse with
negative heat capacity to gravothermal catastrophe. The SRH collapse has many
features that model the gravothermal catastrophe.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 4,467 |
Q: Implementing jQuery hoverZoom in AngularJS ng-Repeat I am trying to implement jQuery hoverZoom in my project. It works fine in a normal html jquery app but when i try to run it in angular app using ng-repeat it breaks.
My guess is due to DOM & jQuery. I think I have to create a directive or something.
The working code is:
HTML
<link rel="stylesheet" href="assets/css/zoom_style.css?v=2">
<link rel="stylesheet" href="assets/css/jquery.hoverZoom.css">
<ul class="thumb">
<li><a href="../api/uploads/bjaim/1.jpg"><img alt="Images can have captions" src="../api/uploads/bjaim/1_thumb.jpg" /></a></li>
<li class="noborder"><a href="../api/uploads/bjaim/2.jpg"><img alt="zoomContainer can be styled with CSS" src="../api/uploads/bjaim/2_thumb.jpg" /></a></li>
<li class="redshadow"><a href="../api/uploads/bjaim/3.jpg"><img src="../api/uploads/bjaim/3_thumb.jpg" alt="Images are scaled and centered to the viewport" /></a></li>
<li><a href="../api/uploads/bjaim/bigimage.jpg"><img src="../api/uploads/bjaim/bigimage_thumb.jpg" alt="Images are preloaded and then the animation starts"/></a></li>
<li><a href="../api/uploads/bjaim/5.jpg"><img src="../api/uploads/bjaim/5_thumb.jpg" /></a></li>
<li><a href="../api/uploads/bjaim/6.jpg"><img src="../api/uploads/bjaim/6_thumb.jpg" /></a></li>
<li><a href="../api/uploads/bjaim/7.jpg"><img src="../api/uploads/bjaim/7_thumb.jpg" /></a></li>
</ul>
Javascript
<script src="assets/js/modernizr-1.6.min.js"></script>
<script src="assets/js/jquery.hoverZoom.js"></script>
<script type="text/javascript">
$(document).ready(function(){
$('.thumb img').hoverZoom({speedView:600, speedRemove:400, showCaption:true, speedCaption:600, debug:true, hoverIntent: true, loadingIndicatorPos: 'center'});
});
</script>
But when I chnage the HTML to:
<ul class="thumb">
<li ng-repeat="image in images">
<a ng-href="../api/uploads/{{job}}/{{image}}"><img alt="Images can have captions" ng-src="../api/uploads/{{job}}/{{image}}" /></a>
</li>
</ul>
It displays the images but hover effect is gone.
Please guide.
Not: Tried to replicate the code on jsFiddle
A: Yes you need to create a directive,
.directive('hoverZoom', function() {
return {
link: function(scope, element){
$(element).hoverZoom({speedView:600, speedRemove:400, showCaption:true, speedCaption:600, debug:true, hoverIntent: true, loadingIndicatorPos: 'center'});
}
};
});
And in HTML,
<a hover-zoom ng-href="../api/uploads/{{job}}/{{image}}"><img alt="Images can have captions" ng-src="../api/uploads/{{job}}/{{image}}" /></a>
Depending on your requirement you can try to improve this directive to pass the options for hoverZoom as well. https://docs.angularjs.org/guide/directive
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,482 |
package gov.nih.nci.cabig.caaers.web.ae;
import gov.nih.nci.cabig.caaers.domain.report.Report;
import gov.nih.nci.cabig.caaers.domain.report.ReportVersion;
import gov.nih.nci.cabig.caaers.service.ReportSubmissionService;
import gov.nih.nci.cabig.caaers.web.fields.DefaultInputFieldGroup;
import gov.nih.nci.cabig.caaers.web.fields.InputField;
import gov.nih.nci.cabig.caaers.web.fields.InputFieldAttributes;
import gov.nih.nci.cabig.caaers.web.fields.InputFieldFactory;
import gov.nih.nci.cabig.caaers.web.fields.InputFieldGroup;
import gov.nih.nci.cabig.caaers.web.fields.InputFieldGroupMap;
import gov.nih.nci.cabig.caaers.web.fields.TabWithFields;
import gov.nih.nci.cabig.caaers.web.utils.WebUtils;
import java.util.Map;
import javax.servlet.http.HttpServletRequest;
import org.apache.commons.logging.Log;
import org.apache.commons.logging.LogFactory;
import org.apache.commons.validator.EmailValidator;
import org.springframework.beans.BeanWrapper;
import org.springframework.beans.factory.annotation.Required;
import org.springframework.validation.Errors;
/**
* @author Krikor Krumlian
*/
public class SubmitReportTab extends TabWithFields<SubmitExpeditedAdverseEventCommand> {
protected final Log log = LogFactory.getLog(getClass());
protected ReportSubmissionService reportSubmissionService;
public SubmitReportTab() {
super("Submission", "Recipients", "ae/submitReport");
}
@Override
public InputFieldGroupMap createFieldGroups(SubmitExpeditedAdverseEventCommand command) {
InputFieldGroupMap map = new InputFieldGroupMap();
InputFieldGroup ccReport = new DefaultInputFieldGroup("ccReport");
InputField cc = InputFieldFactory.createTextArea("report.lastVersion.ccEmails", "Cc");
InputFieldAttributes.setColumns(cc, 50);
ccReport.getFields().add(cc);
map.addInputFieldGroup(ccReport);
return map;
}
@Override
public Map<String, Object> referenceData(SubmitExpeditedAdverseEventCommand submitCommand) {
Report report = submitCommand.getReport();
submitCommand.refreshReportDeliveries(report);
return super.referenceData( submitCommand);
}
@Override
protected void validate(SubmitExpeditedAdverseEventCommand command, BeanWrapper commandBean, Map<String, InputFieldGroup> fieldGroups, Errors errors) {
Report report = command.getReport();
ReportVersion lastVersion = report.getLastVersion();
String emailString =lastVersion.getCcEmails();
if (emailString != null) {
String[] emails = emailString.split(",");
for (String email : emails) {
if (!isEmailValid(email)) {
InputField field = fieldGroups.get("ccReport").getFields().get(0);
errors.rejectValue(field.getPropertyName(), "SAE_007", new Object[]{field.getDisplayName()},"Not Valid " + field.getDisplayName());
break;
}
}
}
}
public boolean isEmailValid(String email) {
String trimmedEmail = (email != null) ? email.trim() : email;
return EmailValidator.getInstance().isValid(trimmedEmail);
}
@Override
public void postProcess(HttpServletRequest request, SubmitExpeditedAdverseEventCommand command, Errors errors) {
log.debug("In postProcess");
int targetPage = WebUtils.getTargetPage(request);
if(log.isDebugEnabled()) log.debug("Should process SubmitReportTab:postProcess (back button) ? " + !(targetPage < 2) );
if( targetPage < 2) return; //only process if we are moving forward.
if(log.isDebugEnabled()) log.debug("Should process SubmitReportTab:postProcess (errors) ? " + !errors.hasErrors() );
if(errors.hasErrors()) return;
if(log.isDebugEnabled()) log.debug("Report active ? " + command.getReport().isActive());
if(log.isDebugEnabled()) log.debug("Report submitted already ? " + command.isSubmissionInprogress());
if(command.getReport().isActive() && (!command.isSubmissionInprogress()) ){
reportSubmissionService.submitReport(command.getReport());
command.setSubmissionInprogress(true);
}
}
public ReportSubmissionService getReportSubmissionService() {
return reportSubmissionService;
}
@Required
public void setReportSubmissionService(ReportSubmissionService reportSubmissionService) {
this.reportSubmissionService = reportSubmissionService;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,275 |
Want to Make Indie Films? Meet Cassiah Joski-Jethi!
I've blogged here about my own fitful ventures into the field of indie films. However, today on this blog, I'm featuring an awesome Q&A with talented indie filmmaker Cassiah Joski-Jethi. She's currently crowdfunding a new short film, Kindling. You can check out the campaign here or click on the above photo. Cassiah recently completed another short film, Woman of the Night. And, for what it's worth, I've previously mentioned Cassiah's film, Polly, on yet another blog.
There's a very easy way for anyone to learn to make videos. Just pick up a damn camera and start!
That's essentially what I did when I started making my own original videos for the Vimeo Weekend Challenge. | {
"redpajama_set_name": "RedPajamaC4"
} | 3,288 |
Zakres dynamiki dźwięku – różnica głośności między dźwiękiem najcichszym a najgłośniejszym w sygnale audio. Wartość zakresu jest podawana w decybelach. Zakres dynamiki zależy od rozdzielczości - im większa rozdzielczość, tym większy zakres dynamiki. Jeśli rozdzielczość bitowa jest ograniczona możliwościami przetwornika, to sygnał jest odwzorowywany niedokładnie, a niedokładność ta jest słyszalna przez ludzkie ucho jako szum.
Muzyka zapisana na płycie CD z rozdzielczością 16-bitową ma zakres dynamiki ok. 96 dB. Często jednak w wyniku kompresji dynamiki zakres ten wynosi zaledwie około 12 dB. Kompresja dynamiki to wyrównywanie różnic głośności sygnału dźwiękowego. Polega to na zwiększaniu głośności cichszych fragmentów nagrania oraz pozostawianiu bez zmian, lub wyrównaniu do jednego poziomu, fragmentów głośniejszych. Ułatwia to prawidłowy odbiór muzyki w warunkach, w których cichsze dźwięki mogłyby zostać zagłuszone np. przez niepożądane odgłosy z zewnątrz.
Bibliografia
Elektroakustyka | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 2,291 |
'use strict';
// -----------------------------------------------------------------
// Private members
var _comments = {};
// -----------------------------------------------------------------
// Private methods
function _add_comment(moniker, comment, fn) {
if(!_comments[moniker]) {
_comments[moniker] = [];
}
_comments[moniker].push(comment);
fn();
}
function _get_comments(moniker, fn) {
var results = _comments[moniker] || [];
results.reverse();
fn(null, {
comments_total: results.length,
comments: results
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// Exports
exports = module.exports = {
add_comment: _add_comment,
get_comments: _get_comments
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| {
"redpajama_set_name": "RedPajamaGithub"
} | 1,402 |
Devoke Water is a small lake in the mid-west region of the English Lake District, in the county of Cumbria. It is the largest tarn in the Lake District.
It lies on Birker Fell, 1 km to the west of the road between Ulpha and Eskdale, at an altitude of 770 feet (223 m). It has a depth of .
It can be reached via a bridle track. There is a two-storey stone boathouse-cum-refuge and a ruined stable. Devoke Water has an outlet in the north west, via Linbeck Gill, which joins the River Esk at the hamlet of Linbeck.
The fishing rights to Devoke Water are owned by Millom Anglers and it is stocked with brown trout. It also holds perch.
The Circuit of Devoke Water
One of the chapters of Alfred Wainwright's The Outlying Fells of Lakeland is a circular walk anticlockwise around Devoke Water, starting and finishing on the road to the east. He describes the summits Rough Crag at , Water Crag at , White Pike at , Yoadcastle at , Woodend Height at and Seat How at , and says that "it is predominantly for the mountain prospect that this walk gains a strong recommendation", noting that the view from the summits includes Pillar and nearby fells to the north, the Scafell group to the north north east and the Bowfell group to the north east, as well as the Isle of Man and Sellafield power station. He warns that "Linbeck Gill is uncrossable dryshod after rain".
All six summits are classified as Birketts. Yoadcastle is classed as a Fellranger, being described by Richards in the Coniston volume of his book series. It is among the 21 such summits (originally 18 before the extension of the Lake District) which are not included in Wainwright's main list of 214.
References
Lakes of the Lake District
Fells of the Lake District
Borough of Copeland | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 6,878 |
\section*{}
\vspace{-1cm}
\footnotetext{\textit{$^{a}$~Department of Chemistry, Norwegian University of Science and Technology (NTNU), NO-7491 Trondheim, Norway}}
\footnotetext{\textit{$^{b}$~Scuola Normale Superiore, Piazza dei Cavalieri 7, IT-56126 Pisa, PI Italy; E-mail: tommaso.giovannini@sns.it; henrik.koch@sns.it}}
\footnotetext{\textit{$^{c}$~DTU Chemistry, Technical University of Denmark, DK-2800 Kongens Lyngby, Denmark}}
\footnotetext{\textit{$^{\ddag}$}~Present address: \textit{Scuola Normale Superiore, Piazza dei Cavalieri 7, I-56126 Pisa, Italy}}
\footnotetext{\dag~Electronic Supplementary Information (ESI) available: data related to Fig.~\ref{fig:pna_dip}-\ref{fig:bnz-in-thf_plots}. See DOI: 10.1039/cXCP00000x/}
\section{Introduction}
The response of a molecular system to an external electric field plays a fundamental role in a plethora of technological applications.\cite{helgaker2012recent} In this context, theoretical chemistry can help understand the underlying physics of the different phenomena. Among them, linear response properties are the most basic quantities to be investigated, being the physico-chemical foundation of many different spectroscopic signals.\cite{norman2018principles,amos.1986,jensen2009atomistic,norman1997ab,norman2003polarization} Therefore, the theoretical modeling can have a pivotal role in gaining insight into how molecules and complex systems behave in the presence of electromagnetic radiation.\cite{norman2018principles} Of particular interest are molecular systems embedded in an external environment, being it a solvent or a biological matrix.\cite{cammi2000attempt,pedersen2014damped} In fact, in such cases, the molecular properties of the chromophore, which is usually the target of the study, can be drastically perturbed by the presence of the environment.\cite{egidi2014stereoelectronic}
To face this kind of problems, the most widespread approach is to resort to the so-called focused models.\cite{warshel1976,tomasi2005} In the specific case of solutions, a high level of theory is used for the region of interest---the solute---while the solvent is treated at a less sophisticated level.\cite{mennucci2019multiscale} Thus, within the focused model formalism, the system is usually partitioned into two layers. In particular, the solute is generally described at the quantum mechanical (QM) level, while the treatment of the solvent can range from a continuum,\cite{tomasi2005} to molecular mechanics (MM) approaches,\cite{warshel1976, lin2007qm, senn2009} to a lower-level QM model.\cite{marrazzini2021multilevel,saether2017,svensson1996oniom,govind1999electronic,wesolowski2015,sun2016quantum,olsen2015polarizable} Each of these subcategories encompasses many different methods, where different solute-solvent interactions---e.g. electrostatic effects, polarization, Pauli repulsion, dispersion---are taken into account, and different computational efforts are required. In order to correctly describe strong and specific solute-solvent interactions, the atomistic nature of the environment usually needs to be retained in the modeling.\cite{giovannini2020csr} Among the atomistic models, the most commonly used belong to the family of QM/MM approaches.\cite{senn2009qm} In their basic formulation---the so-called electrostatic embedding---the environment electrostatically perturbs the QM density, but not vice versa.\cite{senn2009qm} To include mutual solute-solvent polarization effects, which might have a huge influence on the QM properties and spectra, polarizable QM/MM approaches are exploited.\cite{curutchet2009electronic,bondanza2020polarizable,olsen2011molecular,olsen2010excited} In this way, both electrostatic and polarization contributions are taken into account. However, non-electrostatic interactions such as Pauli repulsion and dispersion are usually neglected, although they play a crucial role in many complex systems.\cite{giovannini2017disrep,slipchenko2016effective,giovannini2019quantum} To recover a theoretically consistent picture of such interactions, which are intrinsically of quantum nature, quantum embedding approaches can be used.\cite{marrazzini2021multilevel,saether2017,svensson1996oniom,govind1999electronic,wesolowski2015,sun2016quantum,olsen2015polarizable,reinholdt2017polarizable} As mentioned above, these models are based on the description at the QM level---although less sophisticated than the one used for the solute---of at least a part of the environment. This allows for the treatment of solute-solvent Pauli repulsion, and in some cases of dispersion interactions too.\cite{folkestad2019multilevel,myhre2014multi,myhre2016multilevel} Due to the quantum description of a larger part of the system, quantum embedding approaches are generally more computationally demanding than QM/MM methods. This problem can be solved by three-layer approaches, where the largest part of the environment, usually the farthest from the solute, is described by means of classical force fields.\cite{wanko2008effect,olsen2015polarizable, bennie2016projector, nogueira2018effect, macetti2021three,goletto2021combining} In this way, within a small portion of the system most interactions are treated at the QM level, whereas long-range contributions are retained at the classical level only, providing a physically consistent picture.
In this work, we present a computational investigation of linear response properties of two organic systems, namely para-nitroaniline (PNA) and benzonitrile (PhCN), dissolved in dioxane (DIO), acetonitrile (ACN), and tetrahydrofuran (THF). To quantify the solvent effects on such properties, we present a hierarchy of solvation approaches, ranging from common QM/MM methods to three-layer quantum embedding models.
As for QM/MM approaches, we consider both non-polarizable and polarizable frameworks.
The latter is based on the fluctuating charge (FQ) force field,\cite{cappelli2016integrated,giovannini2020csr,giovannini2020pccp} which has been recently parametrized for the selected solvents.\cite{ambrosetti2021quantum}
The three-layer quantum embedding, on the other hand, is based on the multilevel Hartree-Fock (MLHF) method.\cite{saether2017}
Within MLHF, the molecular orbitals (MOs) are partitioned into \textit{active} and \textit{inactive} by means of Cholesky decomposition\cite{beebe1977simplifications, sanchez2010,aquilante2006fast,folkestad2019efficient} coupled with projected atomic orbitals\cite{pulay1983, saebo1993} (PAOs) for the virtual space. The computational advantage of such a method lies in the fact that the active MOs are optimized in the field of the inactive ones, which are kept frozen, but orthogonal to the active space. Therefore, electrostatic and Pauli repulsion (and part of the polarization) active-inactive interactions are automatically taken into account at the HF level. To refine the picture provided by the basic formulation of MLHF, the active and inactive orbitals can be localized in their pre-defined spatial regions by means of an energy-based procedure that we have recently presented.\cite{giovannini2021energy} The model obtained is called MLHF-AB. If such a procedure is applied to an HF optimized wave function, fully accounting for solute-solvent interactions, the resulting MOs are denoted as fragment localized MOs (FLMOs).\cite{giovannini2022fragment} To minimize its computational cost, MLHF(-AB) can be coupled to an external MM layer (MLHF(-AB)/MM).\cite{goletto2021combining}
To calculate the linear response properties, we use the aforementioned two- and three-layer wave functions as the reference for a post-HF description of the solute. In fact, electron correlation has been proven particularly significant for the accurate modeling of both static and dynamic (hyper)polarizabilities.\cite{champagne2005basis, pecul2005density,christiansen1999coupled, kongsted2002qm, kongsted2003linear, hrsak2018polarizable} If the ground state is dominated by a single-determinant wave function, the coupled cluster (CC) hierarchy of methods arguably provides one of the most sophisticated descriptions of electron correlation.\cite{pinkbook} For this reason, coupled cluster is often considered the theoretical golden standard for the prediction of linear response properties, although many other \emph{ab-initio} methods, ranging from density functional theory (DFT) to Møller-Plesset (MP) perturbation theory, have been routinely used for this purpose.\cite{limacher2009accurate, baranowska2013performance, wormer1986analysis, olsen1985linear} Note that, when dealing with excited states and molecular properties, coupled cluster methods typically follow one of two routes: response theory\cite{monkhorst1977calculation, koch1990coupled, pedersen1997coupled} (CCRT) or the equation-of-motion\cite{stanton1993equation, kobayashi1994calculation} (EOM-CC) formalism, which is exploited here. The two frameworks result in identical excitation energies, but differ in the molecular properties, although with a generally small discrepancy.\cite{helgaker2012recent}
The manuscript is organized as follows. In the next section, we detail the theoretical approaches with a focus on the solvation modeling and the calculation of linear response properties at the EOM-CC level of theory. Then, the computational protocol followed in the numerical analysis is presented and applied to the calculation of static and dynamic polarizabilities of PNA and PhCN dissolved in DIO, ACN, and THF. A summary and future perspectives end the manuscript.
\section{Theory}
This section outlines the theoretical basis of the solvation methods employed to compute the polarizabilities at the EOM-CC2 and EOM-CCSD levels, which are also briefly described. In particular, we briefly recall the theory of non-polarizable QM/MM and polarizable QM/FQ, together with that of the three-layer MLHF-AB/MM method.
\subsection{Non-polarizable QM/MM and polarizable QM/FQ}
As stated above, QM/MM methods rely on the partitioning of the total energy of the system into a QM ($E_{\text{QM}}$) and an MM ($E_{\text{MM}}$) contribution\cite{senn2009qm}
\begin{equation}
E = E_{\text{QM}} + E_{\text{MM}} + E^{\text{int}}_{\text{QM/MM}}\ ,
\end{equation}
where $E^{\text{int}}_{\text{QM/MM}}$ is the QM/MM interaction energy. In electrostatic embedding, $E^{\text{int}}_{\text{QM/MM}}$ is limited to the purely electrostatic interaction, whereas in polarizable embedding, a polarization contribution is also included. In the former, each MM atom is endowed with a fixed charge. In the latter, if the FQ force field is exploited, the charge assigned to each atom can vary as a response to the QM potential. Since both the non-polarizable QM/MM and polarizable QM/FQ depend only on charges, the QM/MM interaction energy reads\cite{giovannini2020csr}
\begin{equation}
\label{eq:qmtip3p}
E^{\text{int}}_{\text{QM/MM}} = \sum_i q_i V_i(\mathbf{D})\ ,
\end{equation}
where $V_i(\mathbf{D})$ is the QM potential due to the QM part acting on the i$th$ MM charge ($q_i$). While in electrostatic embedding such charges are fixed, in QM/FQ their values are obtained by solving the following set of linear equations\cite{giovannini2020csr,giovannini2020pccp}
\begin{equation}
\label{eq:linearqmfq}
\mathbf{M}\mathbf{q}_{\lambda} = -\mathbf{C}_Q - \mathbf{V}(\mathbf{D}),
\end{equation}
where $\mathbf{q}_\lambda$ collects the FQ charges $q$ and suitable Lagrangian multipliers that ensure charge conservation and $\mathbf{M}$ is a matrix containing charge-charge interactions and Lagrangian blocks. The right hand side consists instead of $\mathbf{C}_Q$, which takes into account atomic electronegativities and the total charge constraints, and $\mathbf{V}(\mathbf{D})$---the QM potential. %
In both QM/MM approaches, the QM Fock matrix $F_{\mu\nu}$ (in the atomic orbital basis $\{\chi_\mu\}$) is modified by the inclusion of the QM/MM interaction\cite{giovannini2020csr}
\begin{equation}
F_{\mu\nu} = h_{\mu\nu} + G_{\mu\nu}(\mathbf{D}) + \sum_i q_i V_{i,\mu\nu},
\label{eq:fock-qmmm}
\end{equation}
where $h_{\mu\nu}$ and $G_{\mu\nu}$ are the one- and two-electron matrix elements. This additional term is fixed in non-polarizable QM/MM, whereas it varies at each self consistent field (SCF) step in QM/FQ, because the charges $q$ depend on the QM density.
\subsection{Multilevel Hartree-Fock}
In the MLHF model,\cite{saether2017} the total density matrix ($\mathbf{D}$) of the system is decomposed into an active ($\mathbf{D}^A$) and an inactive ($\mathbf{D}^B$) component. Under this assumption, the total energy of a system described at the HF level can be written as
\begin{equation}
\begin{aligned}
E^{\mathrm{TOT}} = &\mathrm{Tr}\mathbf{h}\mathbf{D}^A + \frac{1}{2}\mathrm{Tr}\mathbf{D}^A\mathbf{G}(\mathbf{D}^A) + \mathrm{Tr}\mathbf{D}^A\mathbf{G}(\mathbf{D}^B) \\
+ &\mathrm{Tr}\mathbf{h}\mathbf{D}^B + \frac{1}{2}\mathrm{Tr} \mathbf{D}^B\mathbf{G}(\mathbf{D}^B) + h_{\text{nuc}},
\label{eq:mlhf_en}
\end{aligned}
\end{equation}
where $\mathbf{h}$ and $\mathbf{G}$ are the one- and two-electron matrices, and $h_{\text{nuc}}$ is the nuclear repulsion energy. In MLHF, only $\mathbf{D}^A$ is iteratively optimized, whereas $\mathbf{D}^B$ is kept fixed during the SCF optimization. Thus, the last three terms in Eq. \ref{eq:mlhf_en}, i.e. the inactive energy and the nuclear repulsion, are constant throughout the procedure. Additionally, the minimization is performed in the MO space of the active part only, thus intrinsically reducing the computational cost of a full HF description. As a consequence, the MLHF Fock matrix elements take the following form
\begin{equation}
F_{\mu\nu} = h_{\mu\nu} + G_{\mu\nu}(\mathbf{D}^A) + G_{\mu\nu}(\mathbf{D}^B),
\label{eq:mlhf_f}
\end{equation}
where $G_{\mu\nu}(\mathbf{D}^B)$ describes the interaction between the active and inactive parts, and is indeed a one-electron term in the Fock matrix, because $\mathbf{D}^B$ is fixed. Within the MLHF framework the electrostatic, Pauli repulsion, and part of the polarization contributions between the active and inactive parts are described at the HF level.
To further reduce the computational cost associated with an MLHF description, MLHF can be coupled with an additional MM layer, yielding the MLHF/MM method introduced in Ref. \citenum{goletto2021combining}. Between the MLHF and MM parts, the interaction is described at the purely electrostatic level, as in Eq. \ref{eq:qmtip3p}. In this case, $\mathbf{D}$ is obtained as the sum of the active and inactive density matrices. The MM layer can be equivalently treated at the non-polarizable or polarizable FQ level. In the latter case, in Eq. \ref{eq:linearqmfq} $\mathbf{D}$ refers to the total density matrix ($\mathbf{D}^A + \mathbf{D}^B$). Finally, the MLHF Fock matrix is modified by the coupling with the external MM layer as
\begin{equation}
F_{\mu\nu} = h_{\mu\nu} + G_{\mu\nu}(\mathbf{D}^A) + G_{\mu\nu}(\mathbf{D}^B) + \sum_i q_i V_{i,\mu\nu}.
\label{eq:mlhf_mm_f}
\end{equation}
After converging the MLHF(/MM) wave function, the active and inactive MOs can be localized in their specific spatial regions by using an energy-based localization of the MOs. In the resulting MLHF-AB approach,\cite{giovannini2021energy} the $\mathbf{h}$ contribution is separated in terms of the kinetic operator and the $A$, $B$, and interaction ($AB$) electron-nuclei potentials. Hence, the total energy can be rewritten as
\begin{equation}
\begin{aligned}
E^{\mathrm{TOT}} & = \underbrace{\mathrm{Tr} \mathbf{h}^A\mathbf{D}^A + \frac{1}{2}\mathrm{Tr} \mathbf{D}^A\mathbf{G}(\mathbf{D}^A) + h^{A}_{nuc}}_{E_A} \\
& +\underbrace{\mathrm{Tr}\mathbf{h}^B\mathbf{D}^B + \frac{1}{2}\mathrm{Tr} \mathbf{D}^B\mathbf{G}(\mathbf{D}^B) + h^{B}_{nuc}}_{E_B} \\
& + \underbrace{\mathrm{Tr} \mathbf{V}^B\mathbf{D}^A + \mathrm{Tr}\mathbf{V}^A\mathbf{D}^B + \mathrm{Tr}\mathbf{D}^A \mathbf{G}(\mathbf{D}^B) + h^{AB}_{nuc}}_{E_{AB}}.
\end{aligned}
\label{eq:mlhf-ab_en}
\end{equation}
The MOs of the A and B fragments are localized by means of a minimization of the $E_A + E_B$ energy in the space spanned by the occupied MOs of the two fragments. This means that the total density is not changed and that such a minimization is equivalent to a maximization of the repulsion energy $E_{AB}$. In this way, the occupied MOs of both fragments are localized in their specific spatial regions.
If a full HF optimization is performed before the localization procedure, FLMOs are obtained,\cite{giovannini2022fragment} yielding the HF\textsubscript{FLMOs} approach. Note that, if an additional MM layer is included in the modeling, it does not affect the minimization procedure: the total density matrix remains fixed, and so does the MLHF-MM interaction.
The localization procedure outlined here makes MLHF-AB a promising tool for the calculation of local properties, such as dipole moments\cite{giovannini2021energy} and polarizabilities.
\subsection{EOM-CC2 and -CCSD linear polarizabilities}
After the SCF convergence is reached with any of the aforementioned approaches (QM/MM, MLHF-AB/MM, and HF$_{\text{FLMOs}}$/MM), the polarizabilities are computed at the EOM-CC2 and -CCSD level of theory by restraining the coupled cluster treatment to the QM (in QM/MM) or to the active part (in MLHF-AB/MM and HF$_{\text{FLMOs}}$/MM) only.
The coupled cluster wave function is expressed as the exponential parametrization\cite{pinkbook}
\begin{equation}
\ket{\mathrm{CC}} = e^{T}\ket{\mathrm{HF}},
\end{equation}
where $\ket{\mathrm{HF}}$ is the reference Hartree-Fock wave function and $T$ is the cluster operator---i.e., the sum of all the excitation operators $\tau_{\nu}$ weighted by their amplitudes $t_{\nu}$
\begin{equation}
\begin{aligned}
T =& T_1 + T_2 + ...\\
=& \sum_{\nu_1} t_{\nu_1} \tau_{\nu_1} + \sum_{\nu_2} t_{\nu_2} \tau_{\nu_2} + ...
\label{eq:t}
\end{aligned}
\end{equation}
Here, $\nu_n$ refers to the $n$-th electronic excitation. In the CC2\cite{christiansen1995second} and CCSD\cite{purvis1982full} models, the cluster operator $T$ is truncated after double excitations. The difference between the two models lies in the fact that in CC2 the double excitation component, $T_2$, is treated perturbatively. In CCSD, the doubles amplitudes are considered to infinite order, while the CC2 doubles are considered through first order only. The doubles amplitude equations of the two models (for a $T_1$-transformed Hamiltonian, $\bar{H}=e^{-T_1}He^{T_1}$) take the form\cite{christiansen1995second}
\begin{align}
&\mathrm{CCSD:} \quad &\bra{\nu_2}\bar{H}+[\bar{H},T_2]+\frac{1}{2}[[\bar{H},T_2],T_2]\ket{\mathrm{HF}} = 0\\
&\mathrm{CC2:} \quad &\bra{\nu_2}\bar{H}+[F,T_2]\ket{\mathrm{HF}} = 0
\end{align}
while the singles equations remain the same for CC2 as in CCSD:
\begin{equation}
\bra{\nu_1}\bar{H}+[\bar{H},T_2]\ket{\mathrm{HF}} = 0
\end{equation}
where
\begin{equation}
\ket{\nu} = {\tau}_{\nu}\ket{\mathrm{HF}}, \hspace{1em}
\bra{\nu} = \bra{\mathrm{HF}}{\tau}^{\dagger}_{\nu}.
\end{equation}
As a result, CC2 scales as $N^5$ compared to CCSD which is a $N^6$ model.
The CC2 model has a structure that is compatible with the exact linear response functions, and the CC2 response properties of a molecule can thus be computed within the same framework and formalism as CCSD.\cite{christiansen1995second}
Molecular response properties arise from the perturbation of an unperturbed system by an external time-periodic field, and can be expressed in terms of response functions. The second-order (linear) response to an external electric field gives rise to the frequency-dependent electronic polarizability.
In the EOM-CC formalism,\cite{stanton1993coupled, stanton1993equation} the ground and excited states are explicitly parametrized as
\begin{equation}
\ket{k} = e^{{T}} \sum_{\nu\ge0} R^k_{\nu} \ket{\nu}, \hspace{1em}
\bra{\bar{k}} = \sum_{\nu\ge0}\bra{\nu}L^k_{\nu} e^{-{T}}.
\label{eq:eom-es}
\end{equation}
Inserting the EOM states of eq.\ref{eq:eom-es} in the exact-state linear response function,\cite{linderberg2004propagators}
and considering the response of the components of the dipole moment operator $\mu$, the EOM-CC linear electronic polarizability, $\alpha$, reads\cite{kobayashi1994calculation, pedersen1997coupled}
\begin{equation}
\alpha_{XY}(\omega) = \frac{1}{2}P^{XY}\left(\bm{\eta}^{X}\bm{t}^{Y}(\omega) + \bm{\eta}^{X}\bm{t}^{Y}(-\omega)\right),
\end{equation}
where the permutation operator $P^{XY}$ performs an interchange of the operators $X$ and $Y$, $\omega$ is the frequency of the external field, $X$ and $Y$ are cartesian components of $\mu$ and, for a generic operator $\cal{O}$,\cite{pawlowski2015response,faber2019rixseom}
\begin{align}
\eta^{\cal{O}}_\nu &=
\left(\bra{\mathrm{HF}}+\bra{\bar{t}}\right) \bar{\cal{O}}\tau_\nu\ket{\mathrm{HF}} - \bar{t}_\nu\mel{\mathrm{HF}}{{\bar{\cal{O}}}}{\mathrm{HF}}\\
\label{eq:eom-etax}
&=
\left(\bra{\mathrm{HF}}+\bra{\bar{t}}\right) [\bar{\cal{O}},\tau_\nu]\ket{\mathrm{HF}}
+ \sum_{\mu>\nu}\bar{t}_\mu\mel{\mu}{\tau_\nu\bar{\cal{O}}}{\mathrm{HF}} - \bar{t}_\nu(\bar{t}\,\xi^{\cal{O}}),
\end{align}
where we have introduced the similarity transformed operator $\bar{\cal{O}} = e^{-T}{\cal{O}} e^{T}$, and $\bra{\bar{t}} = \sum_\mu\bar{t}\bra{\mu}$ with $\bm{\bar{t}}$ being the ground state Lagrangian multipliers.
The amplitude response vectors $\bm{t}^{\cal{O}}(\omega)$ are obtained by solving the linear equations
\begin{equation}
(\mathbf{A} - \omega\mathbf{I})\bm{t}^{\cal{O}}(\omega) = - \bm{\xi}^{\cal{O}},
\end{equation}
where
\begin{equation}
\xi^{\cal{O}}_\nu = \bra{\nu} \bar{\cal{O}} \ket{\mathrm{HF}},
\end{equation}
and $\mathbf{A}$ is the \textit{coupled cluster Jacobian matrix} with elements
\begin{equation}
A_{\nu\mu} = \bra{\nu} [\bar{H}, \hat{\tau}_{\mu}] \ket{\mathrm{HF}}.
\end{equation}
\section{Computational details}
\begin{figure}
\centering
\includegraphics[width=0.7\linewidth]{2_solutes-solvents.png}
\caption{Molecular structures of the solutes (para-nitroaniline, PNA, and benzonitrile, PhCN) and solvents (1,4-dioxane, DIO, acetonitrile, ACN, and tetrahydrofuran, THF).}
\label{fig:solutes-solvents}
\end{figure}
In this work, we select different organic molecules dissolved in non-aqueous environments, namely para-nitroaniline dissolved in 1,4-dioxane (PNA-in-DIO) and benzonitrile dissolved in both acetonitrile (PhCN-in-ACN) and tetrahydrofuran (PhCN-in-THF)---see Fig. \ref{fig:solutes-solvents}. Such systems have been selected because their measured linear polarizabilities have been previously reported in the literature.\cite{wortmann1993deviations, alvarado2003solvent}
In order to correctly sample the solute-solvent phase-space, classical molecular dynamics (MD) simulations are performed for both PNA and PhCN dissolved in the different environments. In the case of PNA-in-DIO, the MD simulation has been performed by following the procedure recently proposed in Ref. \citenum{ambrosetti2021quantum}. Similarly, for both PhCN-in-ACN and PhCN-in-THF, the General Amber Force Field (GAFF)\cite{wang2004development} is used to describe the solute and solvents, for which charges and parameters are obtained by using the RESP charge-fitting method\cite{bayly1993well} and the Antechamber package,\cite{wang2006automatic} respectively. Optimized CAM-B3LYP/aug-cc-pVDZ geometries are used to generate the force field parameters with the initial solvent effects incorporated by means of the polarizable continuum model.\cite{tomasi2005} PhCN is kept frozen during the MD runs, similarly to PNA (see Ref. \citenum{ambrosetti2021quantum}). This choice is justified by their planar and rigid structure, and also avoids any potentially poor description of dihedral distributions by the classical force field.\cite{kjellgren2018importance,giovannini2018effective}
All simulations are performed using the GROMACS package.\cite{gromacs} Following a similar methodology as in Ref. \citenum{ambrosetti2021quantum}, our systems consist of a single molecule of PhCN surrounded by thousands of solvent molecules and enclosed in a simulation box of \SI{7}{\nano\meter} size. They are minimized for 500 steps, prior to a \SI{2}{\nano\second} equilibration in the isothermal-isobaric ensemble, keeping the temperature (\SI{300}{\kelvin}) and the pressure (\SI{1}{atm}) constant by means of a velocity-rescaling method,\cite{bussi2007} with a coupling constant of \SI{0.1}{\pico\second}, and the Berendsen barostat,\cite{berendsen1986practical} with a coupling constant of \SI{2.0}{\pico\second}, respectively. Values of \SI{9.7e-5}{} and \SI{9.6e-5}{\per\bar} are used for the isothermal compressibilities of THF and ACN, respectively. Afterward, an NVT production stage of \SI{10}{\nano\second} is performed in order to have a well-equilibrated system before extracting representative configurations. The LINCS algorithm\cite{hess1997lincs} is used to constrain all bonds of the solute molecule. The particle-mesh Ewald (PME) algorithm\cite{darden1993pme} is employed to handle long-range electrostatic interactions. Van der Waals and short-range electrostatic interactions are truncated with a smoothed \SI{1.4}{\nano\meter} spherical cutoff. The equations of motion are integrated with a \SI{2}{\femto\second} time step.
A set of 20 snapshots is selected from the production stage of each MD simulation. The time separation between them (\SI{100}{\pico\second} for PNA-in-DIO and \SI{500}{\pico\second} for PhCN-in-ACN and PhCN-in-THF) is large enough to ensure that they are uncorrelated.\cite{reinholdt2018modeling, harczuk2015frequency, skoko2020simulating, puglisi2019interplay}
A droplet with a spherical shape of radius \SI{20}{\angstrom} centered on the solute is cut.
Note that the number of selected frames is enough to guarantee the convergence of the results (see Fig. S1-S3 and S14-S15 in the Electronic Supplementary Information -- ESI\dag).
The geometries of all the frames studied in this work can be found in Ref.\citenum{geometries}.
For each extracted snapshot, the linear polarizability is then calculated by describing the whole system at different levels of theory, defined within a hierarchical ladder: (i) the solute is described at the QM level, whereas the environment is described by means of electrostatic (QM/EE) or polarizable embedding (by exploiting the FQ force field -- QM/FQ); (ii) The solute and the closest solvent molecules are included in the MLHF-AB region, while the remaining solvent molecules are described at the FQ level. The solvent molecules within a range of \SI{2.5}{\angstrom} (PNA-in-DIO), \SI{2.75}{\angstrom} (PhCN-in-THF) and \SI{3.5}{\angstrom} (PhCN-in-ACN) from any atom of the solute are included in the inactive MLHF-AB calculations, whereas the solute molecule represents the active part. Such an approach is called MLHF-AB/FQ in what follows. The same solvent molecules and the solute represent the two regions described at the HF$_{\text{FLMOs}}$ level in the HF$_{\text{FLMOs}}$/FQ approach. The partitioning of the spherical snapshots at the different levels of theory is graphically depicted in Fig. \ref{fig:pna-in-dio_levels}, by taking PNA-in-DIO as a representative example.
For MLHF-AB/FQ calculations, the protocol outlined in Ref. \citenum{goletto2021combining} is followed. A superposition of molecular densities\cite{neugebauer2005merits} is used as a starting guess. While the active MO virtual space is constructed at the beginning of the calculation by means of orthonormalized\cite{lowdin1970} PAOs,\cite{pulay1983, saebo1993} the active occupied space is firstly determined by a limited Cholesky decompositon algorithm,\cite{beebe1977simplifications, sanchez2010, aquilante2006fast, folkestad2019efficient} and then iteratively adjusted by maximizing the interaction energy $E_{AB}$ in Eq. \ref{eq:mlhf-ab_en}. The number of active MOs is selected to be equal to the correct number of occupied MOs of the active region.
In order to calculate the linear polarizability of each snapshot, the solute is described at the EOM-CC2 or EOM-CCSD level with the aug-cc-pVDZ basis set, by using the HF/EE, HF/FQ, MLHF-AB/FQ, and HF$_{\text{FLMOs}}$/FQ reference wave functions. The basis set is selected by following Refs. \citenum{egidi2014benchmark, cuesta2004polarizabilities, alparone2011theoretical, alparone2013linear}. For the MLHF-AB/FQ and HF$_{\text{FLMOs}}$/FQ reference wave functions, the solvent molecules are described with the cc-pVDZ basis set. In CC/EE, GAFF atomic charges are used for the EE region.\cite{wang2004development, wang2006automatic} For the FQ layer in CC/FQ, CC-in-MLHF-AB/FQ and CC-in-HF$_{\text{FLMOs}}$/FQ, the atomic electronegativity and chemical hardness parameters have been taken from Ref. \citenum{ambrosetti2021quantum}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{pna-in-dio_levels.png}
\caption{Graphical depiction of a snapshot of PNA-in-DIO as partitioned in the CC/FQ and CC/EE (top) and CC-in-MLHF-AB/FQ and CC-in-HF\textsubscript{FLMOs}/FQ (bottom) approaches.
The atoms in blue are treated at the CC level, by using HF as reference wavefunction;
the atoms in orange are treated either at the HF level---in HF\textsubscript{FLMOs}---or as the inactive MLHF part---in MLHF-AB; the atoms in grey are treated at the MM level.}
\label{fig:pna-in-dio_levels}
\end{figure}
For each snapshot, we calculate the static and dynamic isotropic electronic part of the polarizabilities, which is given by
\begin{equation}
\alpha^{\text{iso}} = \frac{1}{3} (\alpha_{xx} + \alpha_{yy} + \alpha_{zz})
\end{equation}
In the static case, a reorientation term $\alpha_\mu$ is added to the purely electronic term to yield the total static polarizability, $\alpha_{0}^{\mathrm{tot}}$:
\begin{align}
\alpha_{0}^{\mathrm{tot}} & = \alpha^\text{iso}_0 + \alpha_\mu.\label{eq:alpha_tot_static} \\
\alpha_{\mu} & = \frac{|\mu|^2}{3k_B T},\label{eq:alphamu}
\end{align}
where $\mu$ is the molecular dipole moment, $k_B$ is the Boltzmann constant and $T$ is the temperature.\cite{liptay1982determination, cammi2000attempt}
The final isotropic polarizability is obtained by averaging the results computed for each snapshot. It is worth noting that local field effects induced on the active part by the polarization of solvent under external radiation are not considered in this work, although they might affect computed linear response properties.\cite{list2016local,egidi2014benchmark,tomasi2002molecular} All the calculations are performed with a locally modified version of the electronic structure program $e^T$.\cite{eT_jcp} The $e^T$ default thresholds are used for the optimization of the reference and coupled cluster ground state wave functions, as well as for the dipole moments and EOM-CC polarizabilities. The threshold for the Cholesky decomposition of the two-electron repulsion integrals is set to $10^{-4}$.
\section{Numerical Results}
All the aforementioned methods are used to calculate the linear polarizabilities of PNA-in-DIO, PhCN-in-ACN and PhCN-in-THF. In this section, the computed results are analyzed in terms of the different physico-chemical solute-solvent interactions introduced by the different methods. The accuracy and robustness of the approaches are then tested by comparing the computed data with the available experimental results.\cite{wortmann1993deviations, alvarado2003solvent}
\subsection{PNA-in-DIO}
Let us discuss the case of PNA-in-DIO. PNA has been the focus of a large variety of theoretical\cite{daniel1990nonlinear, karna1991nonlinear, champagne1996vibrational,egidi2014benchmark} and experimental\cite{millefiori1977electronic, carsey1979systematics, woodford1997solvent} investigations. It is characterized by a push-pull electronic structure, presenting an electron-acceptor and an electron-donor functional groups on the opposite sides of a $\pi$-conjugated aromatic structure (see also Fig. \ref{fig:solutes-solvents}). Such a feature implies that its optical properties are strongly influenced by solvent effects,\cite{cammi2000attempt,cammi1998solvent, painelli2001linear,cammi2003multiconfigurational,giovannini2019fqfmulinear,kosenkov2010solvent} making PNA a perfect candidate for studying the performances of the different theoretical approaches.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{pna-in-dio_dip.png}
\caption{PNA-in-DIO snapshot-to-snapshot differences between CC-in-HF\textsubscript{FLMOs}/FQ, CC-in-MLHF-AB/FQ, CC/FQ and CC/EE ground state dipole moments results.}
\label{fig:pna_dip}
\end{figure}
In order to highlight the different solute-solvent physico-chemical interactions taken into account by the different investigated approaches, we analyze the results computed for each snapshot (see Fig. \ref{fig:pna_dip}, \ref{fig:pna_stat_pol} and \ref{fig:pna_dyn_pol}).
In Fig. \ref{fig:pna_dip}a, the differences between the dipole moments computed at the CC/FQ and CC/EE levels are graphically depicted. It can be noticed that CC/FQ dipole moments are larger than CC/EE ones, independently of the solute-solvent configurations, i.e. the snapshots. This is due to the inclusion of polarization effects, described by means of the FQ force field, which increase the magnitude of solute-solvent interactions, and consequently the computed dipole moments. Remarkably, such an increase highly varies as a function of the snapshot, ranging from 1 to about 2.2 Debye, and yields an increase of about $17$-$18\%$ on average.
Moving to the three layers approaches, Fig. \ref{fig:pna_dip}b depicts the CC-in-HF$_{\text{FLMOs}}$/FQ and CC-in-MLHF-AB/FQ $\Delta\mu$ results as a function of the snapshot.
This case provides a different picture.
In fact, a variability both in magnitude and in sign is reported between the two approaches. However, for most snapshots, the CC-in-HF$_{\text{FLMOs}}$/FQ approach predicts larger dipole moments as compared to CC-in-MLHF-AB/FQ. Such a finding can be explained by considering that within the HF$_{\text{FLMOs}}$/FQ reference, all the solute-solvent interactions are fully accounted for at the HF level. At the MLHF-AB/FQ level, on the contrary, only part of the solute-solvent polarization effects are taken into account, and the inactive MOs---those belonging to the solvent---are not fully optimized. As a consequence, for most snapshots, the full account of polarization of CC-in-HF$_{\text{FLMOs}}$/FQ yields large dipole moments.
The negative deviations can be instead related to the optimization of the MOs of the inactive part, which enhances Pauli repulsion effects.
Finally, we compare the results obtained by exploiting CC-in-HF$_{\text{FLMOs}}$/FQ and CC-in-MLHF-AB/FQ to the CC/FQ values, as reported in Fig. \ref{fig:pna_dip}c and d, respectively.
While CC-in-MLHF-AB/FQ generally yields a decrease of the dipole moment, a variability in intensity and sign is reported for CC-in-HF$_{\text{FLMOs}}$/FQ with respect to CC/FQ.
The numerical results can again be discussed in light of the physico-chemical interactions included in the different approaches. In the CC/FQ approach, the solute-solvent interactions are limited to electrostatics and polarization, whereas in both the three-layer methods Pauli repulsion effects are also taken into consideration. On the one hand, this explains the average decrease of the dipole moment reported for the two quantum embedding methods. On the other hand, the variability depicted in Fig. \ref{fig:pna_dip}d can suggest that for some snapshots the CC/FQ approach is not able to fully account for the solute-solvent polarization effects.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{pna-in-dio_stat_pol.png}
\caption{PNA-in-DIO snapshot-to-snapshot differences between CC-in-HF\textsubscript{FLMOs}/FQ, CC-in-MLHF-AB/FQ, CC/FQ and CC/EE results for the electronic component of the static polarizability.}
\label{fig:pna_stat_pol}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{pna-in-dio_dyn_pol.png}
\caption{PNA-in-DIO snapshot-to-snapshot differences between CC-in-HF\textsubscript{FLMOs}/FQ, CC-in-MLHF-AB/FQ, CC/FQ and CC/EE electronic dynamic polarizability results (\SI{589}{nm}).}
\label{fig:pna_dyn_pol}
\end{figure}
We now move to the differences in the static and dynamic polarizabilities, which are graphically depicted as a function of the snapshot in Fig. \ref{fig:pna_stat_pol} and \ref{fig:pna_dyn_pol}, respectively.
The above discussion for the dipole moments is generally valid also for these linear-response properties, but with some noticeable exceptions.
In fact, the differences between CC/FQ--CC/EE, and CC-in-HF$_{\text{FLMOs}}$/FQ--CC-in-MLHF-AB/FQ polarizabilities follow the same trends reported for $\mu$.
In particular, the inclusion of polarization in CC/FQ yields an increase of both static and dynamic polarizabilities, with a larger effect on the dynamic one.
The full account of polarization effects in HF$_{\text{FLMOs}}$ provides an overall increase of the computed properties, albeit with some negative values.
However, the effect of using HF$_{\text{FLMOs}}$ in place of MLHF-AB is much smaller than the effect of using FQ in place of EE (on average a $0.2$-$0.4\%$ increase vs. a $4$-$7\%$ increase).
The differences between the quantum embedding models and QM/FQ (see Fig. \ref{fig:pna_stat_pol}c-d and \ref{fig:pna_dyn_pol}c-d) are negative for all the snapshots, showing that the inclusion of Pauli repulsion effects provides a general confinement of the active density.
As a consequence, both the static and the dynamic polarizability values decrease by $2$-$3\%$, on average.
\begin{table*}
\centering
\begin{tabular}{l c c c c c}
\hline
Method & $\mu[\mathrm{D}]$ & $\alpha_{0}^{\mathrm{iso}}\big[\frac{\mathrm{cm}^3}{\mathrm{mol}}\big]$ & $\alpha^{\mu}\big[\frac{\mathrm{cm}^3}{\mathrm{mol}}\big]$ & $\alpha_{0}^{\mathrm{tot}}\big[\frac{\mathrm{cm}^3}{\mathrm{mol}}\big]$ & $\alpha_{0}^{\mathrm{exp}}\big[\frac{\mathrm{cm}^3}{\mathrm{mol}}\big]$ \\
\hline
CC2 in vacuo & $ 6.7$ & $10.2$ & $221.7$ & $231.9$ & \\
CCSD in vacuo & $ 6.9$ & $ 9.6$ & $230.2$ & $239.9$ &\\
\hline
CC2/EE & $ 9.1$ & $10.7$ & $403.8$ & $414.5$ & \multirow{8}{*}{$404\pm 6$}\\
CC2/FQ & $10.8$ & $11.1$ & $565.5$ & $576.6$ & \\
CC2-in-MLHF-AB/FQ & $10.5$ & $10.8$ & $536.4$ & $547.2$ & \\
CC2-in-HF\textsubscript{FLMOs}/FQ & $10.8$ & $10.8$ & $573.9$ & $584.8$ & \\
CCSD/EE & $ 9.1$ & $10.0$ & $406.1$ & $416.1$ &\\
CCSD/FQ & $10.7$ & $10.3$ & $558.6$ & $568.9$ &\\
CCSD-in-MLHF-AB/FQ & $10.4$ & $10.1$ & $532.4$ & $542.5$ &\\
CCSD-in-HF\textsubscript{FLMOs}/FQ & $10.8$ & $10.1$ & $569.4$ & $579.5$ & \\
\hline
\end{tabular}
\caption{Calculated PNA-in-DIO dipole moments ($\mu$), isotropic electronic static polarizabilities ($\alpha_{0}^{\text{iso}}$), reorientation terms ($\alpha^\mu$), and total static polarizabilities ($\alpha_{0}^\text{tot}$). The experimental reference\cite{wortmann1993deviations} is also provided, along with CC2 and CCSD \textit{in vacuo} values.}
\label{tab:pna_static}
\end{table*}
\begin{table}
\centering
\begin{tabular}{l c c}
\hline
Method & $\alpha_{\omega}^{\mathrm{iso}}\big[\frac{\mathrm{cm}^3}{\mathrm{mol}}\big]$ & $\alpha_{\omega}^{\mathrm{exp}}\big[\frac{\mathrm{cm}^3}{\mathrm{mol}}\big]$ \\
\hline
CC2 in vacuo & $11.0$ & \\
CCSD in vacuo & $10.3$ & \\
\hline
CC2/EE & $11.9 $ & \multirow{8}{*}{$14.1 \pm 0.4$
} \\
CC2/FQ & $12.6 $ & \\
CC2-in-MLHF-AB/FQ & $12.2 $ & \\
CC2-in-HF\textsubscript{FLMOs}/FQ & $12.3 $ &\\
CCSD/EE & $10.9 $ & \\
CCSD/FQ & $11.5 $ & \\
CCSD-in-MLHF-AB/FQ & $11.2 $ & \\
CCSD-in-HF\textsubscript{FLMOs}/FQ & $11.2 $ &\\
\hline
\end{tabular}
\caption{Calculated PNA-in-DIO dynamic polarizabilities (\SI{589}{nm}). The experimental reference\cite{wortmann1993deviations} is also provided, along with CC2 and CCSD \textit{in vacuo} values.}
\label{tab:pna_dyn}
\end{table}
Finally, let us move to comment on the averaged results, which can be compared to the available experimental data. In Table \ref{tab:pna_static} the averaged isotropic values of both the electronic static polarizability ($\alpha_0^{\text{iso}}$) and the dipole moment obtained with the different theoretical methods are reported, together with the experimental value from Ref. \citenum{wortmann1993deviations}. To better quantify the solvent effects on the computed properties, Table \ref{tab:pna_static} also lists the \textit{vacuo} values calculated at the CC2 and CCSD levels. The total static polarizability, $\alpha_{0}^{\text{tot}}$, is computed using Eq.~\ref{eq:alpha_tot_static}, so it consists of an orientationally averaged polarizability ($\alpha_\mu$) and of the purely electronic contribution $\alpha^{\text{iso}}_0$.
The CC2 and CCSD results mainly differ in two respects: while CC2 $\alpha^{\text{iso}}_{0}$ is larger than the CCSD counterpart, the opposite holds for the dipole moments in the gas phase. Solvent effects can be appreciated by comparing the gas-phase results with those obtained by including a description of the embedding. In particular, an overall increase of both the static polarizability ($4$-$8\%$) and the dipole moments ($33$-$61\%$) is highlighted. Such a trend is compatible with what has been reported in the literature for similar systems.\cite{mikkelsen1994solvent, cammi2003multiconfigurational, targema2013molecular} While the CC/EE approaches provide the smallest values of $\mu$ ($\sim$ \SI{1.5}{D} away from all other methods) and $\alpha^{\text{iso}}_0$, QM/FQ reports the largest $\alpha^{\text{iso}}_0$, and CC-in-HF$_{\text{FLMOs}}$/FQ the largest $\mu$. All the trends between the values computed with different solvation approaches follow what has already been pointed out for Fig. \ref{fig:pna_dip}--\ref{fig:pna_dyn_pol}. Indeed, CC/FQ $\mu$ results are larger than CC/EE $\mu$, and a similar trend is reported for CC-in-HF\textsubscript{FLMOs}/FQ as compared to CC-in-MLHF-AB/FQ. Such a result is primarily due to the inclusion (in CC/FQ) and full accounting (in CC-in-HF\textsubscript{FLMOs}/FQ) of polarization.
When considering $\alpha^{\text{iso}}_{0}$, the aforementioned trends remain valid.
In addition, we note that the use of the multilevel wave function as a reference decreases $\alpha^{\text{iso}}_{0}$ with respect to the CC/MM data.
This is in agreement with Fig. \ref{fig:pna_stat_pol} and \ref{fig:pna_dyn_pol}, and is due to the quantum confinement of the solute density as a result of the solute-solvent Pauli repulsion contributions introduced by the multilevel modeling. The differences between CC2 and CCSD dipole moment results are small, regardless of the solvation model employed.
When comparing to the experimental reference in Ref.~\citenum{wortmann1993deviations} (see Table \ref{tab:pna_static}), it is worth remarking that the largest contribution to $\alpha^{tot}_0$ is given by $\alpha^{\mu} \propto \mu^2$.
Hence, the computed values of the total static polarizability strongly depend on the numerical values of $\mu$. Indeed, CC/EE reports the smallest $\alpha^{tot}_0$, whereas the largest $\alpha^{tot}_0$ values are given by CC-in-HF\textsubscript{FLMOs}/FQ, which has a similar performance as CC-in-MLHB-AB/FQ, and CC/FQ. The differences between CC/EE and the other solvation methods range between $34$ and $45\%$. This is primarily due to the differences in the computed dipole moments, as the $\alpha^{\text{iso}}_0$ only slightly affects the final computed property.
Since CC/EE $\alpha^{tot}_0$ is closest to the experimental value, the most important error source for the other methods lies in the overestimation of $\mu$, which is enhanced when the polarization is included in the modeling, and lowered by the Pauli repulsion effects introduced in multilevel methods.
A small difference in $\mu$ is reflected in large differences in $\alpha^\mu$, and consequently in $\alpha_{0}^\text{tot}$. In fact, as commented above, $\alpha_{0}^\text{tot}$ is calculated as the sum of $\alpha_{0}^\text{iso}$ and $\alpha^\mu$, and our results show that $\alpha^\mu$ is approximately 50 times larger than $\alpha_0^\text{iso}$.
To remove the dependency of the results on the computed dipole moments, we move to dynamic polarizabilities, for which $\alpha_\mu = 0$. Therefore, the computed $\alpha^{\text{iso}}_\omega$ can be directly compared to the experimental values, which are given in Table \ref{tab:pna_dyn}. Similar trends as reported for the static electronic polarizability (see Table \ref{tab:pna_static}) can be observed also for the frequency-dependent case. The inclusion of solvent effects in the modeling increases the computed values independently of the exploited method, ranging between a $6\%$ (CCSD/EE) and a $15\%$ (CC2/FQ) shift. The gap between CCSD and CC2 results is larger as compared to the static polarizability, with the CCSD results being lower than CC2 by up to $9\%$ (CC/FQ). The trends between the different approaches directly follow those discussed in the static case: CC/FQ $\alpha^{\text{iso}}_{\omega}$ are larger than the corresponding CC/EE values, and the same is generally valid between CC-in-HF$_{\text{FLMOs}}$/FQ and CC-in-MLHF-AB/FQ, due to the inclusion and the full accounting of polarization effects, respectively. Similarly to the static case, CC-in-HF$_{\text{FLMOs}}$/FQ and CC-in-MLHF-AB provide very similar results, indicating that the MLHF-AB/FQ method is able to account for most of the polarization effects. On the contrary, by moving from CC/FQ to the multilevel methods the value of the computed property decreases ($\sim 2$-$3\%$), because the Pauli repulsion yields a confinement effect in the reference wave function.
Comparing the results in Table \ref{tab:pna_dyn} with the experimental counterpart, we first note that, differently from the static case, all the computed values are smaller. The CC/EE results present the largest deviations, while the CC/FQ ones are closest to experiment. As commented above, CC2 polarizabilities are generally larger than CCSD ones, thus resulting in a better agreement with the experimental reference. However, it is worth noting that CC2 might reportedly overestimate linear polarizabilities,\cite{christiansen1999frequency, salek2005comparison} probably due to an overestimation of the dispersion interaction.\cite{rocha2009linear, schmies2011structures, kolaski2013aromatic} Considering the high level of theory employed in this work, the systematic underestimation of all methods could be explained by the fact that our model discards the zero-point correction and the pure vibrational contribution to the linear response properties. While the latter plays a negligible role in determining dynamic polarizabilities (due to the unfavourable dependence on the external frequency), studies at the DFT level have shown that the former can increase the purely electronic contribution by up to $6\%$.\cite{egidi2014stereoelectronic} Considering that the discrepancies with the experimental counterpart range between $1.5$ ($\sim 10\%$, with CC2/FQ) and $3.2$ ($\sim 22\%$, with CCSD/EE) cm$^3$/mol, an overall agreement with the experiment can be reported for almost all methods. Finally, it is worth noting that our modeling neglects explicit terms arising from polarization contributions in response equations, which may enhance the computed linear polarizabilities, as reported in similar contexts.\cite{cammi2012coupled, caricato2018coupled, caricato2019coupled, caricato2020coupled}
\subsection{PhCN-in-ACN and PhCN-in-THF}
We now move to the case of PhCN-in-ACN and PhCN-in-THF polarizabilities, which have been studied from the experimental point of view in Ref. \citenum{alvarado2003solvent} as a function of the external frequency (ranging from \SI{1.553}{} to \SI{2.295}{\per\micro\meter}). We again model solvent effects by means of CC/EE, CC/FQ and CC-in-MLHF-AB/FQ; the CC-in-HF\textsubscript{FLMOs}/FQ methods have not been included in the comparison, as in the previous section the differences with CC-in-MLHF-AB/FQ in the polarizabilities have been found to be negligible, but the method is associated with a higher computational cost.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{bnz-in-acn_bars_pol2.png}
\caption{PhCN-in-ACN snapshot-to-snapshot differences between CC-in-MLHF-AB/FQ, CC/FQ and CC/EE electronic dynamic polarizability results (\SI{1.696}{\per\micro\meter}).}
\label{fig:bnz-in-acn_bars}
\end{figure}
The snapshot-to-snapshot differences CC/FQ--CC/EE and CC-in-MLHF-AB/FQ--CC/FQ are depicted in Fig. \ref{fig:bnz-in-acn_bars} and \ref{fig:bnz-in-thf_bars} for the PhCN-in-ACN and PhCN-in-THF systems, respectively. The differences are presented for a specific frequency equal to \SI{1.696}{\per\micro\m}, which corresponds to the experimental sodium D line. The plots for all the other frequencies considered in this work are reported in Fig. S4-S13 of the ESI\dag, and show similar trends.
For PhCN-in-ACN (see Fig. \ref{fig:bnz-in-acn_bars}), the differences between CC/FQ and CC/EE are negligible, ranging between $\pm$ 0.02 cm$^3$/mol. Additionally, the differences strongly depend on the specific solute-solvent configurations and display a sign alternation, thus averaging out in the final property. Note that, in contrast to what has been discussed in the previous section (see Fig. \ref{fig:pna_dyn_pol}), the sign alternation indicates that the parametrization of the EE modeling overestimates electrostatics, similarly to other non-polarizable force fields.\cite{mobley2007comparison} The mutual polarization between PhCN and ACN appears to play a minor role in the solute-solvent interaction, as compared to PNA-in-DIO system. On the other hand, the introduction of the intermediate MLHF-AB layer between the coupled cluster and FQ regions lowers the polarizabilities results in all the snapshots, with differences ranging from 0.2 to 0.4 cm$^3$/mol. This indicates once again the confinement effects provided by the accounting for solute-solvent Pauli repulsion.
\begin{figure}
\centering
\includegraphics[width=\linewidth]{bnz-in-thf_bars_pol2.png}
\caption{PhCN-in-THF snapshot-to-snapshot differences between CC-in-MLHF-AB/FQ, CC/FQ and CC/EE electronic dynamic polarizability results (\SI{1.696}{\per\micro\meter}).}
\label{fig:bnz-in-thf_bars}
\end{figure}
For PhCN-in-THF, the CC/FQ values are smaller than the CC/EE values in all snapshots. Such a finding is opposite to the PNA-in-DIO case, and shows that the parametrization exploited in the EE force field is including (overestimating) electrostatic effects. However, the CC/FQ--CC/EE differences only reach the $1\%$ of the total value of the polarizability. Therefore the THF polarization, albeit numerically more significant than that of ACN (see Fig. \ref{fig:bnz-in-acn_bars}), does not play a significant role for such systems. On the contrary, the Pauli repulsion interactions introduced by the MLHF-AB layer have again a larger influence on the polarizabilities, lowering the results by approximately $3\%$ (and numerically by 0.1--0.25 cm$^3$/mol).
\begin{figure}
\centering
\includegraphics[width=\linewidth]{bnz-in-acn_plot_sigma}
\caption{Calculated PhCN-in-ACN polarizabilities as a function of the external frequency. Standard error bars at the 68\% confidence interval are plotted. PhCN \textit{in vacuo} polarizabilities are also given, along with the experimental benchmark from Ref. \citenum{alvarado2003solvent}. $\text{Exp}^a$ employs the Lorentz local field correction, while $\text{Exp}^b$ employs the Onsager local field correction.}
\label{fig:bnz-in-acn_plots}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{bnz-in-thf_plot_sigma}
\caption{Calculated PhCN-in-THF polarizabilities as a function of the external frequency. Standard error bars at the 68\% confidence interval are plotted. PhCN \textit{in vacuo} polarizabilities are also given, along with the experimental benchmark from Ref. \citenum{alvarado2003solvent}. $\text{Exp}^a$ employs the Lorentz local field correction, while $\text{Exp}^b$ employs the Onsager local field correction.}
\label{fig:bnz-in-thf_plots}
\end{figure}
The computed averaged values of the PhCN-in-ACN static/dynamic polarizabilities are plotted as a function of the external frequency in Fig. \ref{fig:bnz-in-acn_plots} (see Table S1 in the ESI$\dag$ for the numerical data), together with the \textit{in vacuo} data and the experimental results reproduced from Ref. \citenum{alvarado2003solvent}. In particular, we report two different experimental references, which are obtained by applying the Lorentz ($\text{Exp}^a$) and Onsager ($\text{Exp}^b$) local field corrections to the measured refractive indexes. Indeed, it is worth remarking that the reported data are not the measured quantities---that is, the refractive indexes---, but the electronic part of the polarizability extracted from them. Therefore, the reference data are associated with an intrinsic systematic error related to the approach exploited to extrapolate a microscopic quantity (the polarizability) from a macroscopic one (the refractive index).
In Fig. \ref{fig:bnz-in-acn_plots}, the averaged polarizabilities computed by all the different methods are shown to follow the same trend with respect to the external frequency. In particular, CC/EE and CC/FQ results are almost identical, whereas the inclusion of the MLHF-AB layer lowers the polarizabilities by approximately $3\%$. Indeed, with respect to the \textit{in vacuo} results, CC/EE and CC/FQ yield an increase in the polarizabilities, albeit with a negligible deviation ($< 1\%$). On the other hand, the values obtained at the CC-in-MLHF-AB/FQ level are $2$ to $3\%$ lower than the corresponding CC results \textit{in vacuo}. Thus, depending on how the environment is treated, solvent effects shift the polarizabilities to opposite directions. As the main difference between the approaches lies in the Pauli repulsion between PhCN and ACN being mostly taken into account in CC-in-MLHF-AB/FQ, this contribution appears to have a significant influence on the results. CCSD polarizabilities are approximately $4\%$ smaller than the corresponding CC2 values with all the multiscale methods, following the same trend observed \textit{in vacuo}.
We now move to the comparison with the experimental reference.\cite{alvarado2003solvent} Regardless of the choice of local field correction, being it Lorentz or Onsager, the experimental results are lower than the computed values. In particular, $\text{Exp}^b$ and $\text{Exp}^a$ polarizabilities differ of about $1\%$, the former being the lowest. The best agreement with the experiment is obtained by using CCSD-in-MLHF-AB/FQ, which presents $1$-$3\%$ and $2$-$4\%$ deviations from $\text{Exp}^a$ and $\text{Exp}^b$, respectively. In particular, CCSD-in-MLHF-AB/FQ reports its largest discrepancy with the experiment for the static polarizability. Here, the values in Ref. \citenum{alvarado2003solvent} are not recovered from the experimental permittivity, but obtained by extrapolation with a Cauchy-type dispersion curve fit, which might introduce further inaccuracy. The worst agreement with the experiment is given by CC2/EE and CC2/FQ, which deviate from the experiment by $9\%$ and $12\%$, respectively. Remarkably, among the considered methods, only the three-layer CC-in-MLHF-AB/FQ approaches shift the computed polarizabilities from the corresponding \textit{in vacuo} results towards the experiment.
Fig. \ref{fig:bnz-in-thf_plots} depicts the averaged polarizability values (see Table S2 in the ESI$\dag$ for the numerical data), the \textit{in vacuo} reference, and experimental benchmark\cite{alvarado2003solvent} for the PhCN-in-THF system, plotted again as a function of the external frequencies. Similarly to the previous case, CC/EE and CC/FQ results are very similar, with CC/FQ providing computed polarizabilities less than $1\%$ smaller than CC/EE. This is in agreement with the results reported in Fig. \ref{fig:bnz-in-thf_bars}. An additional decrease is given by CC-in-MLHF-AB/FQ, with values $1$-$2\%$ smaller than CC/FQ. The CC2 and CCSD values show the same trend with respect to a different treatment of the environment, with CCSD providing polarizabilities $4$-$5\%$ smaller than CC2. The same behaviour is observed for the \textit{in vacuo} results. When comparing to the \textit{in vacuo} reference, the CC/FQ values are almost identical, with deviations $<0.3\%$. While being very similar, the CC/EE polarizabilities are slightly larger, even if the differences fall below $1\%$. On the other hand, the CC-in-MLHF-AB/FQ methods lower the polarizabilities by $2\%$ with respect to the \textit{in vacuo} results.
Additionally, Fig. \ref{fig:bnz-in-thf_plots} shows that the experimental references are once again smaller than the computed values. The comparison between the methods follows the same trends as in PhCN-in-ACN, with CCSD-in-MLHF-AB/FQ being the closest (with a $2$-$4\%$ deviation) and CC2/EE being the furthest (with a $9$-$11\%$ deviation) from the experiment. While starting from the \textit{in vacuo} reference CC/EE goes in the wrong direction with respect to the experiment, both CC/FQ and CC-in-MLHF-AB/FQ lower the polarizabilities towards the experiment.
\section{Summary and conclusions}
We have presented a computational investigation of linear polarizabilities of organic moieties embedded in non-aqueous solvents, employing different strategies to model solvent effects. We have considered a hierarchy of solvation approaches, ranging from common QM/MM methods to three-layer approaches based on a multilevel partitioning of the reference wave function. In particular, we have considered both non-polarizable and polarizable QM/MM approaches, the latter based on the FQ force field, suitably parameterized for the selected solvents. The three-layer approaches are instead based on a partitioning of the system into three portions: an active region (the solute), an inactive region (the solvent molecules closest to the solute), and an MM region (the rest of the solvent), treated by means of the FQ force field. In this way, the external MM layer accounts for long range interactions with an electrostatic and polarization nature. The MLHF-AB reference wave function introduces the electron repulsion effects between solute and solvent, as well as an approximated HF treatment of the polarization.
The approaches have been applied to the calculation of static and dynamic linear polarizabilities (at the CC2 and CCSD levels) of the PNA-in-DIO, PhCN-in-ACN, and PhCN-in-THF systems. To sample the solute-solvent phase-space, the calculations have been run on various snapshots extracted from classical MD simulations. Overall, this protocol gives a consistent physical description of the properties and interactions at work in a complex environment. The computed results have been rationalized in terms of the different solute-solvent physico-chemical interactions modeled by each solvation approach and compared with the available experimental data. In all cases, we have obtained an overall good agreement with the reference measurements, in particular when Pauli repulsion effects, which are introduced in the three-layer approaches, are taken into account.
To further increase the agreement with experimental results, the three-layer model could be further improved to include dispersion effects between the solute and the solvent. Also, the quality of long-range electrostatics and polarization effects can be increased by including an additional source of polarization in the FQ force field in terms of fluctuating dipoles (FQF$\mu$) to account for anisotropic interactions.\cite{giovannini2019fqfmu,giovannini2019fqfmuir,marrazzini2020calculation} Finally, the protocol can be extended to the treatment of higher-order properties, such as first-hyperpolarizabilities ($\beta$).
\section*{Conflicts of interest}
There are no conflicts to declare.
\section*{Acknowledgements}
We thank Eirik F. Kjønstad, Sarai D. Folkestad and Alexander C. Paul for their contributions to the $e^T$ code. J.H.A. acknowledges Sonia Coriani (DTU) for discussions. We acknowledge funding from the Marie Sklodowska-Curie Interational Training Network "COSINE - COmputational Spectroscopy In Natural sciences and Engineering", Grant Agreement No. 765739, and from the Research
Council of Norway through the FRINATEK project 275506, TheoLight. We acknowledge computing resources through UNINETT Sigma2---the National Infrastructure for High Performance Computing and Data Storage in Norway, through project number NN2962k.
We also acknowledge Chiara Cappelli (SNS) for computing resources, and the Center for High Performance Computing (CHPC) at SNS for providing the computational infrastructure.
\balance
| {
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} | 2,818 |
{"url":"https:\/\/news.ycombinator.com\/item?id=19335936","text":"Maybe some people actually like a graphical UI that does not require learning arcane syntax or keyboard shortcuts, provides a passing simile to actual paper and allows them to be productive (if not proficient)?As for \"anything significant\", even OP shows one file per chapter. That's very likely what most users of word processors do, I'd imagine, even though Word (for example) had master\/child documents support in the late nineties.\n\n No. I get that people think that, but it's a mistaken belief. Word processors are at least as arcane, but they disguise and hide their arcane bits. That is, word processors give an illusion of being easier, but really they end up much more complicated.Graphical UI? There are dozens of text editors that work with TeX and most of them provide graphical UIs, with menus and buttons similar to a word processor. In fact, as soon you want to do anything evenly mildly 'advanced', word processors end up being visibly more complicated.In a word processor, I can click the 'bold' button, or press Ctrl-b to switch into 'bold mode', or highlight some text, and use the button\/keyboard shortcut to bold that text.In a TeX editor, I can also click the 'bold' button or press a shortcut to auto-create a LaTeX bold environment \\textbf{} with my cursor placed in-between the {}s. Alternatively, I can highlight text, e.g 'my text', and click the bold button or press the shortcut and the editor will wrap `\\textbf{}' arount the text, producing '\\textbf{my text]'.Up to this point the two approaches are equivalent. But now say that I want to make all instances of 'important phrase' bold. With the TeX\/editor approach, it's just like any other search and replace, I tell the editor to replace all instances of 'important phrase' with '\\textbf{important phrase}'. In the word processor, I have to figure out how to click into an advanced search-and-replace and choose something about replace\/add styles etc.In LaTeX, for something complicated, I can figure it out and write my function(s) for it, which are easily re-usable. In a word processor, what one 'knows' in the case of doing something complicated is a series of mouse clicks through menus - which is not only more arcane than an explicit function, but is likely to be disrupted by version changes.\n The vast majority of users will forever be beginners - they'll learn a few techniques and wont bother with the rest - simply because that's not the only thing they do. For such users, a canned, ready-to-understand UI that looks like paper is pragmatically better than providing the capability to do whatever they want at the cost of becoming a power user in word processing.People use spaces for tabs and newlines instead of page breaks despite there being solutions for them even on mechanical typewriters. As long as it works for them, that's ok.\n Making a full fledged latex document is as easy as making a plain text file with the following content:\\documentclass{article} \\begin{document}Your whole life story goes here.\\end{document}That's fscking difficult.\n\nSearch:","date":"2019-05-21 22:17:17","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\": 0.6741837859153748, \"perplexity\": 2579.442346294203}, \"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-2019-22\/segments\/1558232256571.66\/warc\/CC-MAIN-20190521202736-20190521224736-00007.warc.gz\"}"} | null | null |
package beta
import (
"context"
"crypto/sha256"
"encoding/json"
"fmt"
"time"
"google.golang.org/api/googleapi"
"github.com/GoogleCloudPlatform/declarative-resource-client-library/dcl"
)
type WorkflowTemplate struct {
Name *string `json:"name"`
Version *int64 `json:"version"`
CreateTime *string `json:"createTime"`
UpdateTime *string `json:"updateTime"`
Labels map[string]string `json:"labels"`
Placement *WorkflowTemplatePlacement `json:"placement"`
Jobs []WorkflowTemplateJobs `json:"jobs"`
Parameters []WorkflowTemplateParameters `json:"parameters"`
DagTimeout *string `json:"dagTimeout"`
Project *string `json:"project"`
Location *string `json:"location"`
}
func (r *WorkflowTemplate) String() string {
return dcl.SprintResource(r)
}
// The enum WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigPrivateIPv6GoogleAccessEnum.
type WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigPrivateIPv6GoogleAccessEnum string
// WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigPrivateIPv6GoogleAccessEnumRef returns a *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigPrivateIPv6GoogleAccessEnum with the value of string s
// If the empty string is provided, nil is returned.
func WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigPrivateIPv6GoogleAccessEnumRef(s string) *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigPrivateIPv6GoogleAccessEnum {
v := WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigPrivateIPv6GoogleAccessEnum(s)
return &v
}
func (v WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigPrivateIPv6GoogleAccessEnum) Validate() error {
if string(v) == "" {
// Empty enum is okay.
return nil
}
for _, s := range []string{"PRIVATE_IPV6_GOOGLE_ACCESS_UNSPECIFIED", "INHERIT_FROM_SUBNETWORK", "OUTBOUND", "BIDIRECTIONAL"} {
if string(v) == s {
return nil
}
}
return &dcl.EnumInvalidError{
Enum: "WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigPrivateIPv6GoogleAccessEnum",
Value: string(v),
Valid: []string{},
}
}
// The enum WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinityConsumeReservationTypeEnum.
type WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinityConsumeReservationTypeEnum string
// WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinityConsumeReservationTypeEnumRef returns a *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinityConsumeReservationTypeEnum with the value of string s
// If the empty string is provided, nil is returned.
func WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinityConsumeReservationTypeEnumRef(s string) *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinityConsumeReservationTypeEnum {
v := WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinityConsumeReservationTypeEnum(s)
return &v
}
func (v WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinityConsumeReservationTypeEnum) Validate() error {
if string(v) == "" {
// Empty enum is okay.
return nil
}
for _, s := range []string{"TYPE_UNSPECIFIED", "NO_RESERVATION", "ANY_RESERVATION", "SPECIFIC_RESERVATION"} {
if string(v) == s {
return nil
}
}
return &dcl.EnumInvalidError{
Enum: "WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinityConsumeReservationTypeEnum",
Value: string(v),
Valid: []string{},
}
}
// The enum WorkflowTemplatePlacementManagedClusterConfigMasterConfigPreemptibilityEnum.
type WorkflowTemplatePlacementManagedClusterConfigMasterConfigPreemptibilityEnum string
// WorkflowTemplatePlacementManagedClusterConfigMasterConfigPreemptibilityEnumRef returns a *WorkflowTemplatePlacementManagedClusterConfigMasterConfigPreemptibilityEnum with the value of string s
// If the empty string is provided, nil is returned.
func WorkflowTemplatePlacementManagedClusterConfigMasterConfigPreemptibilityEnumRef(s string) *WorkflowTemplatePlacementManagedClusterConfigMasterConfigPreemptibilityEnum {
v := WorkflowTemplatePlacementManagedClusterConfigMasterConfigPreemptibilityEnum(s)
return &v
}
func (v WorkflowTemplatePlacementManagedClusterConfigMasterConfigPreemptibilityEnum) Validate() error {
if string(v) == "" {
// Empty enum is okay.
return nil
}
for _, s := range []string{"PREEMPTIBILITY_UNSPECIFIED", "NON_PREEMPTIBLE", "PREEMPTIBLE"} {
if string(v) == s {
return nil
}
}
return &dcl.EnumInvalidError{
Enum: "WorkflowTemplatePlacementManagedClusterConfigMasterConfigPreemptibilityEnum",
Value: string(v),
Valid: []string{},
}
}
// The enum WorkflowTemplatePlacementManagedClusterConfigWorkerConfigPreemptibilityEnum.
type WorkflowTemplatePlacementManagedClusterConfigWorkerConfigPreemptibilityEnum string
// WorkflowTemplatePlacementManagedClusterConfigWorkerConfigPreemptibilityEnumRef returns a *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigPreemptibilityEnum with the value of string s
// If the empty string is provided, nil is returned.
func WorkflowTemplatePlacementManagedClusterConfigWorkerConfigPreemptibilityEnumRef(s string) *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigPreemptibilityEnum {
v := WorkflowTemplatePlacementManagedClusterConfigWorkerConfigPreemptibilityEnum(s)
return &v
}
func (v WorkflowTemplatePlacementManagedClusterConfigWorkerConfigPreemptibilityEnum) Validate() error {
if string(v) == "" {
// Empty enum is okay.
return nil
}
for _, s := range []string{"PREEMPTIBILITY_UNSPECIFIED", "NON_PREEMPTIBLE", "PREEMPTIBLE"} {
if string(v) == s {
return nil
}
}
return &dcl.EnumInvalidError{
Enum: "WorkflowTemplatePlacementManagedClusterConfigWorkerConfigPreemptibilityEnum",
Value: string(v),
Valid: []string{},
}
}
// The enum WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigPreemptibilityEnum.
type WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigPreemptibilityEnum string
// WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigPreemptibilityEnumRef returns a *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigPreemptibilityEnum with the value of string s
// If the empty string is provided, nil is returned.
func WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigPreemptibilityEnumRef(s string) *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigPreemptibilityEnum {
v := WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigPreemptibilityEnum(s)
return &v
}
func (v WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigPreemptibilityEnum) Validate() error {
if string(v) == "" {
// Empty enum is okay.
return nil
}
for _, s := range []string{"PREEMPTIBILITY_UNSPECIFIED", "NON_PREEMPTIBLE", "PREEMPTIBLE"} {
if string(v) == s {
return nil
}
}
return &dcl.EnumInvalidError{
Enum: "WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigPreemptibilityEnum",
Value: string(v),
Valid: []string{},
}
}
// The enum WorkflowTemplatePlacementManagedClusterConfigSoftwareConfigOptionalComponentsEnum.
type WorkflowTemplatePlacementManagedClusterConfigSoftwareConfigOptionalComponentsEnum string
// WorkflowTemplatePlacementManagedClusterConfigSoftwareConfigOptionalComponentsEnumRef returns a *WorkflowTemplatePlacementManagedClusterConfigSoftwareConfigOptionalComponentsEnum with the value of string s
// If the empty string is provided, nil is returned.
func WorkflowTemplatePlacementManagedClusterConfigSoftwareConfigOptionalComponentsEnumRef(s string) *WorkflowTemplatePlacementManagedClusterConfigSoftwareConfigOptionalComponentsEnum {
v := WorkflowTemplatePlacementManagedClusterConfigSoftwareConfigOptionalComponentsEnum(s)
return &v
}
func (v WorkflowTemplatePlacementManagedClusterConfigSoftwareConfigOptionalComponentsEnum) Validate() error {
if string(v) == "" {
// Empty enum is okay.
return nil
}
for _, s := range []string{"COMPONENT_UNSPECIFIED", "ANACONDA", "DOCKER", "DRUID", "FLINK", "HBASE", "HIVE_WEBHCAT", "JUPYTER", "KERBEROS", "PRESTO", "RANGER", "SOLR", "ZEPPELIN", "ZOOKEEPER"} {
if string(v) == s {
return nil
}
}
return &dcl.EnumInvalidError{
Enum: "WorkflowTemplatePlacementManagedClusterConfigSoftwareConfigOptionalComponentsEnum",
Value: string(v),
Valid: []string{},
}
}
type WorkflowTemplatePlacement struct {
empty bool `json:"-"`
ManagedCluster *WorkflowTemplatePlacementManagedCluster `json:"managedCluster"`
ClusterSelector *WorkflowTemplatePlacementClusterSelector `json:"clusterSelector"`
}
type jsonWorkflowTemplatePlacement WorkflowTemplatePlacement
func (r *WorkflowTemplatePlacement) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacement
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacement
} else {
r.ManagedCluster = res.ManagedCluster
r.ClusterSelector = res.ClusterSelector
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacement is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacement *WorkflowTemplatePlacement = &WorkflowTemplatePlacement{empty: true}
func (r *WorkflowTemplatePlacement) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacement) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacement) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedCluster struct {
empty bool `json:"-"`
ClusterName *string `json:"clusterName"`
Config *WorkflowTemplatePlacementManagedClusterConfig `json:"config"`
Labels map[string]string `json:"labels"`
}
type jsonWorkflowTemplatePlacementManagedCluster WorkflowTemplatePlacementManagedCluster
func (r *WorkflowTemplatePlacementManagedCluster) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedCluster
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedCluster
} else {
r.ClusterName = res.ClusterName
r.Config = res.Config
r.Labels = res.Labels
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedCluster is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedCluster *WorkflowTemplatePlacementManagedCluster = &WorkflowTemplatePlacementManagedCluster{empty: true}
func (r *WorkflowTemplatePlacementManagedCluster) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedCluster) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedCluster) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfig struct {
empty bool `json:"-"`
StagingBucket *string `json:"stagingBucket"`
TempBucket *string `json:"tempBucket"`
GceClusterConfig *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfig `json:"gceClusterConfig"`
MasterConfig *WorkflowTemplatePlacementManagedClusterConfigMasterConfig `json:"masterConfig"`
WorkerConfig *WorkflowTemplatePlacementManagedClusterConfigWorkerConfig `json:"workerConfig"`
SecondaryWorkerConfig *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfig `json:"secondaryWorkerConfig"`
SoftwareConfig *WorkflowTemplatePlacementManagedClusterConfigSoftwareConfig `json:"softwareConfig"`
InitializationActions []WorkflowTemplatePlacementManagedClusterConfigInitializationActions `json:"initializationActions"`
EncryptionConfig *WorkflowTemplatePlacementManagedClusterConfigEncryptionConfig `json:"encryptionConfig"`
AutoscalingConfig *WorkflowTemplatePlacementManagedClusterConfigAutoscalingConfig `json:"autoscalingConfig"`
SecurityConfig *WorkflowTemplatePlacementManagedClusterConfigSecurityConfig `json:"securityConfig"`
LifecycleConfig *WorkflowTemplatePlacementManagedClusterConfigLifecycleConfig `json:"lifecycleConfig"`
EndpointConfig *WorkflowTemplatePlacementManagedClusterConfigEndpointConfig `json:"endpointConfig"`
GkeClusterConfig *WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfig `json:"gkeClusterConfig"`
MetastoreConfig *WorkflowTemplatePlacementManagedClusterConfigMetastoreConfig `json:"metastoreConfig"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfig WorkflowTemplatePlacementManagedClusterConfig
func (r *WorkflowTemplatePlacementManagedClusterConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfig
} else {
r.StagingBucket = res.StagingBucket
r.TempBucket = res.TempBucket
r.GceClusterConfig = res.GceClusterConfig
r.MasterConfig = res.MasterConfig
r.WorkerConfig = res.WorkerConfig
r.SecondaryWorkerConfig = res.SecondaryWorkerConfig
r.SoftwareConfig = res.SoftwareConfig
r.InitializationActions = res.InitializationActions
r.EncryptionConfig = res.EncryptionConfig
r.AutoscalingConfig = res.AutoscalingConfig
r.SecurityConfig = res.SecurityConfig
r.LifecycleConfig = res.LifecycleConfig
r.EndpointConfig = res.EndpointConfig
r.GkeClusterConfig = res.GkeClusterConfig
r.MetastoreConfig = res.MetastoreConfig
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfig *WorkflowTemplatePlacementManagedClusterConfig = &WorkflowTemplatePlacementManagedClusterConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigGceClusterConfig struct {
empty bool `json:"-"`
Zone *string `json:"zone"`
Network *string `json:"network"`
Subnetwork *string `json:"subnetwork"`
InternalIPOnly *bool `json:"internalIPOnly"`
PrivateIPv6GoogleAccess *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigPrivateIPv6GoogleAccessEnum `json:"privateIPv6GoogleAccess"`
ServiceAccount *string `json:"serviceAccount"`
ServiceAccountScopes []string `json:"serviceAccountScopes"`
Tags []string `json:"tags"`
Metadata map[string]string `json:"metadata"`
ReservationAffinity *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinity `json:"reservationAffinity"`
NodeGroupAffinity *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigNodeGroupAffinity `json:"nodeGroupAffinity"`
ShieldedInstanceConfig *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigShieldedInstanceConfig `json:"shieldedInstanceConfig"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigGceClusterConfig WorkflowTemplatePlacementManagedClusterConfigGceClusterConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigGceClusterConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigGceClusterConfig
} else {
r.Zone = res.Zone
r.Network = res.Network
r.Subnetwork = res.Subnetwork
r.InternalIPOnly = res.InternalIPOnly
r.PrivateIPv6GoogleAccess = res.PrivateIPv6GoogleAccess
r.ServiceAccount = res.ServiceAccount
r.ServiceAccountScopes = res.ServiceAccountScopes
r.Tags = res.Tags
r.Metadata = res.Metadata
r.ReservationAffinity = res.ReservationAffinity
r.NodeGroupAffinity = res.NodeGroupAffinity
r.ShieldedInstanceConfig = res.ShieldedInstanceConfig
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigGceClusterConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigGceClusterConfig *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfig = &WorkflowTemplatePlacementManagedClusterConfigGceClusterConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinity struct {
empty bool `json:"-"`
ConsumeReservationType *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinityConsumeReservationTypeEnum `json:"consumeReservationType"`
Key *string `json:"key"`
Values []string `json:"values"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinity WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinity
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinity) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinity
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinity
} else {
r.ConsumeReservationType = res.ConsumeReservationType
r.Key = res.Key
r.Values = res.Values
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinity is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinity *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinity = &WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinity{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinity) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinity) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigReservationAffinity) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigNodeGroupAffinity struct {
empty bool `json:"-"`
NodeGroup *string `json:"nodeGroup"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigGceClusterConfigNodeGroupAffinity WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigNodeGroupAffinity
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigNodeGroupAffinity) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigGceClusterConfigNodeGroupAffinity
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigGceClusterConfigNodeGroupAffinity
} else {
r.NodeGroup = res.NodeGroup
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigNodeGroupAffinity is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigGceClusterConfigNodeGroupAffinity *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigNodeGroupAffinity = &WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigNodeGroupAffinity{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigNodeGroupAffinity) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigNodeGroupAffinity) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigNodeGroupAffinity) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigShieldedInstanceConfig struct {
empty bool `json:"-"`
EnableSecureBoot *bool `json:"enableSecureBoot"`
EnableVtpm *bool `json:"enableVtpm"`
EnableIntegrityMonitoring *bool `json:"enableIntegrityMonitoring"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigGceClusterConfigShieldedInstanceConfig WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigShieldedInstanceConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigShieldedInstanceConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigGceClusterConfigShieldedInstanceConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigGceClusterConfigShieldedInstanceConfig
} else {
r.EnableSecureBoot = res.EnableSecureBoot
r.EnableVtpm = res.EnableVtpm
r.EnableIntegrityMonitoring = res.EnableIntegrityMonitoring
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigShieldedInstanceConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigGceClusterConfigShieldedInstanceConfig *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigShieldedInstanceConfig = &WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigShieldedInstanceConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigShieldedInstanceConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigShieldedInstanceConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigGceClusterConfigShieldedInstanceConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigMasterConfig struct {
empty bool `json:"-"`
NumInstances *int64 `json:"numInstances"`
InstanceNames []string `json:"instanceNames"`
Image *string `json:"image"`
MachineType *string `json:"machineType"`
DiskConfig *WorkflowTemplatePlacementManagedClusterConfigMasterConfigDiskConfig `json:"diskConfig"`
IsPreemptible *bool `json:"isPreemptible"`
Preemptibility *WorkflowTemplatePlacementManagedClusterConfigMasterConfigPreemptibilityEnum `json:"preemptibility"`
ManagedGroupConfig *WorkflowTemplatePlacementManagedClusterConfigMasterConfigManagedGroupConfig `json:"managedGroupConfig"`
Accelerators []WorkflowTemplatePlacementManagedClusterConfigMasterConfigAccelerators `json:"accelerators"`
MinCpuPlatform *string `json:"minCpuPlatform"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigMasterConfig WorkflowTemplatePlacementManagedClusterConfigMasterConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigMasterConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigMasterConfig
} else {
r.NumInstances = res.NumInstances
r.InstanceNames = res.InstanceNames
r.Image = res.Image
r.MachineType = res.MachineType
r.DiskConfig = res.DiskConfig
r.IsPreemptible = res.IsPreemptible
r.Preemptibility = res.Preemptibility
r.ManagedGroupConfig = res.ManagedGroupConfig
r.Accelerators = res.Accelerators
r.MinCpuPlatform = res.MinCpuPlatform
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigMasterConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigMasterConfig *WorkflowTemplatePlacementManagedClusterConfigMasterConfig = &WorkflowTemplatePlacementManagedClusterConfigMasterConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigMasterConfigDiskConfig struct {
empty bool `json:"-"`
BootDiskType *string `json:"bootDiskType"`
BootDiskSizeGb *int64 `json:"bootDiskSizeGb"`
NumLocalSsds *int64 `json:"numLocalSsds"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigMasterConfigDiskConfig WorkflowTemplatePlacementManagedClusterConfigMasterConfigDiskConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfigDiskConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigMasterConfigDiskConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigMasterConfigDiskConfig
} else {
r.BootDiskType = res.BootDiskType
r.BootDiskSizeGb = res.BootDiskSizeGb
r.NumLocalSsds = res.NumLocalSsds
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigMasterConfigDiskConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigMasterConfigDiskConfig *WorkflowTemplatePlacementManagedClusterConfigMasterConfigDiskConfig = &WorkflowTemplatePlacementManagedClusterConfigMasterConfigDiskConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfigDiskConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfigDiskConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfigDiskConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigMasterConfigManagedGroupConfig struct {
empty bool `json:"-"`
InstanceTemplateName *string `json:"instanceTemplateName"`
InstanceGroupManagerName *string `json:"instanceGroupManagerName"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigMasterConfigManagedGroupConfig WorkflowTemplatePlacementManagedClusterConfigMasterConfigManagedGroupConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfigManagedGroupConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigMasterConfigManagedGroupConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigMasterConfigManagedGroupConfig
} else {
r.InstanceTemplateName = res.InstanceTemplateName
r.InstanceGroupManagerName = res.InstanceGroupManagerName
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigMasterConfigManagedGroupConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigMasterConfigManagedGroupConfig *WorkflowTemplatePlacementManagedClusterConfigMasterConfigManagedGroupConfig = &WorkflowTemplatePlacementManagedClusterConfigMasterConfigManagedGroupConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfigManagedGroupConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfigManagedGroupConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfigManagedGroupConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigMasterConfigAccelerators struct {
empty bool `json:"-"`
AcceleratorType *string `json:"acceleratorType"`
AcceleratorCount *int64 `json:"acceleratorCount"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigMasterConfigAccelerators WorkflowTemplatePlacementManagedClusterConfigMasterConfigAccelerators
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfigAccelerators) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigMasterConfigAccelerators
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigMasterConfigAccelerators
} else {
r.AcceleratorType = res.AcceleratorType
r.AcceleratorCount = res.AcceleratorCount
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigMasterConfigAccelerators is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigMasterConfigAccelerators *WorkflowTemplatePlacementManagedClusterConfigMasterConfigAccelerators = &WorkflowTemplatePlacementManagedClusterConfigMasterConfigAccelerators{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfigAccelerators) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfigAccelerators) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigMasterConfigAccelerators) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigWorkerConfig struct {
empty bool `json:"-"`
NumInstances *int64 `json:"numInstances"`
InstanceNames []string `json:"instanceNames"`
Image *string `json:"image"`
MachineType *string `json:"machineType"`
DiskConfig *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigDiskConfig `json:"diskConfig"`
IsPreemptible *bool `json:"isPreemptible"`
Preemptibility *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigPreemptibilityEnum `json:"preemptibility"`
ManagedGroupConfig *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigManagedGroupConfig `json:"managedGroupConfig"`
Accelerators []WorkflowTemplatePlacementManagedClusterConfigWorkerConfigAccelerators `json:"accelerators"`
MinCpuPlatform *string `json:"minCpuPlatform"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigWorkerConfig WorkflowTemplatePlacementManagedClusterConfigWorkerConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigWorkerConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigWorkerConfig
} else {
r.NumInstances = res.NumInstances
r.InstanceNames = res.InstanceNames
r.Image = res.Image
r.MachineType = res.MachineType
r.DiskConfig = res.DiskConfig
r.IsPreemptible = res.IsPreemptible
r.Preemptibility = res.Preemptibility
r.ManagedGroupConfig = res.ManagedGroupConfig
r.Accelerators = res.Accelerators
r.MinCpuPlatform = res.MinCpuPlatform
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigWorkerConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigWorkerConfig *WorkflowTemplatePlacementManagedClusterConfigWorkerConfig = &WorkflowTemplatePlacementManagedClusterConfigWorkerConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigWorkerConfigDiskConfig struct {
empty bool `json:"-"`
BootDiskType *string `json:"bootDiskType"`
BootDiskSizeGb *int64 `json:"bootDiskSizeGb"`
NumLocalSsds *int64 `json:"numLocalSsds"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigWorkerConfigDiskConfig WorkflowTemplatePlacementManagedClusterConfigWorkerConfigDiskConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigDiskConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigWorkerConfigDiskConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigWorkerConfigDiskConfig
} else {
r.BootDiskType = res.BootDiskType
r.BootDiskSizeGb = res.BootDiskSizeGb
r.NumLocalSsds = res.NumLocalSsds
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigWorkerConfigDiskConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigWorkerConfigDiskConfig *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigDiskConfig = &WorkflowTemplatePlacementManagedClusterConfigWorkerConfigDiskConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigDiskConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigDiskConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigDiskConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigWorkerConfigManagedGroupConfig struct {
empty bool `json:"-"`
InstanceTemplateName *string `json:"instanceTemplateName"`
InstanceGroupManagerName *string `json:"instanceGroupManagerName"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigWorkerConfigManagedGroupConfig WorkflowTemplatePlacementManagedClusterConfigWorkerConfigManagedGroupConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigManagedGroupConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigWorkerConfigManagedGroupConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigWorkerConfigManagedGroupConfig
} else {
r.InstanceTemplateName = res.InstanceTemplateName
r.InstanceGroupManagerName = res.InstanceGroupManagerName
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigWorkerConfigManagedGroupConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigWorkerConfigManagedGroupConfig *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigManagedGroupConfig = &WorkflowTemplatePlacementManagedClusterConfigWorkerConfigManagedGroupConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigManagedGroupConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigManagedGroupConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigManagedGroupConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigWorkerConfigAccelerators struct {
empty bool `json:"-"`
AcceleratorType *string `json:"acceleratorType"`
AcceleratorCount *int64 `json:"acceleratorCount"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigWorkerConfigAccelerators WorkflowTemplatePlacementManagedClusterConfigWorkerConfigAccelerators
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigAccelerators) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigWorkerConfigAccelerators
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigWorkerConfigAccelerators
} else {
r.AcceleratorType = res.AcceleratorType
r.AcceleratorCount = res.AcceleratorCount
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigWorkerConfigAccelerators is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigWorkerConfigAccelerators *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigAccelerators = &WorkflowTemplatePlacementManagedClusterConfigWorkerConfigAccelerators{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigAccelerators) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigAccelerators) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigWorkerConfigAccelerators) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfig struct {
empty bool `json:"-"`
NumInstances *int64 `json:"numInstances"`
InstanceNames []string `json:"instanceNames"`
Image *string `json:"image"`
MachineType *string `json:"machineType"`
DiskConfig *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigDiskConfig `json:"diskConfig"`
IsPreemptible *bool `json:"isPreemptible"`
Preemptibility *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigPreemptibilityEnum `json:"preemptibility"`
ManagedGroupConfig *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigManagedGroupConfig `json:"managedGroupConfig"`
Accelerators []WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigAccelerators `json:"accelerators"`
MinCpuPlatform *string `json:"minCpuPlatform"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfig WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfig
} else {
r.NumInstances = res.NumInstances
r.InstanceNames = res.InstanceNames
r.Image = res.Image
r.MachineType = res.MachineType
r.DiskConfig = res.DiskConfig
r.IsPreemptible = res.IsPreemptible
r.Preemptibility = res.Preemptibility
r.ManagedGroupConfig = res.ManagedGroupConfig
r.Accelerators = res.Accelerators
r.MinCpuPlatform = res.MinCpuPlatform
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfig *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfig = &WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigDiskConfig struct {
empty bool `json:"-"`
BootDiskType *string `json:"bootDiskType"`
BootDiskSizeGb *int64 `json:"bootDiskSizeGb"`
NumLocalSsds *int64 `json:"numLocalSsds"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigDiskConfig WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigDiskConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigDiskConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigDiskConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigDiskConfig
} else {
r.BootDiskType = res.BootDiskType
r.BootDiskSizeGb = res.BootDiskSizeGb
r.NumLocalSsds = res.NumLocalSsds
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigDiskConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigDiskConfig *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigDiskConfig = &WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigDiskConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigDiskConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigDiskConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigDiskConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigManagedGroupConfig struct {
empty bool `json:"-"`
InstanceTemplateName *string `json:"instanceTemplateName"`
InstanceGroupManagerName *string `json:"instanceGroupManagerName"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigManagedGroupConfig WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigManagedGroupConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigManagedGroupConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigManagedGroupConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigManagedGroupConfig
} else {
r.InstanceTemplateName = res.InstanceTemplateName
r.InstanceGroupManagerName = res.InstanceGroupManagerName
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigManagedGroupConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigManagedGroupConfig *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigManagedGroupConfig = &WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigManagedGroupConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigManagedGroupConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigManagedGroupConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigManagedGroupConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigAccelerators struct {
empty bool `json:"-"`
AcceleratorType *string `json:"acceleratorType"`
AcceleratorCount *int64 `json:"acceleratorCount"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigAccelerators WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigAccelerators
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigAccelerators) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigAccelerators
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigAccelerators
} else {
r.AcceleratorType = res.AcceleratorType
r.AcceleratorCount = res.AcceleratorCount
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigAccelerators is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigAccelerators *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigAccelerators = &WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigAccelerators{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigAccelerators) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigAccelerators) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecondaryWorkerConfigAccelerators) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigSoftwareConfig struct {
empty bool `json:"-"`
ImageVersion *string `json:"imageVersion"`
Properties map[string]string `json:"properties"`
OptionalComponents []WorkflowTemplatePlacementManagedClusterConfigSoftwareConfigOptionalComponentsEnum `json:"optionalComponents"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigSoftwareConfig WorkflowTemplatePlacementManagedClusterConfigSoftwareConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigSoftwareConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigSoftwareConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigSoftwareConfig
} else {
r.ImageVersion = res.ImageVersion
r.Properties = res.Properties
r.OptionalComponents = res.OptionalComponents
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigSoftwareConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigSoftwareConfig *WorkflowTemplatePlacementManagedClusterConfigSoftwareConfig = &WorkflowTemplatePlacementManagedClusterConfigSoftwareConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigSoftwareConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigSoftwareConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigSoftwareConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigInitializationActions struct {
empty bool `json:"-"`
ExecutableFile *string `json:"executableFile"`
ExecutionTimeout *string `json:"executionTimeout"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigInitializationActions WorkflowTemplatePlacementManagedClusterConfigInitializationActions
func (r *WorkflowTemplatePlacementManagedClusterConfigInitializationActions) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigInitializationActions
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigInitializationActions
} else {
r.ExecutableFile = res.ExecutableFile
r.ExecutionTimeout = res.ExecutionTimeout
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigInitializationActions is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigInitializationActions *WorkflowTemplatePlacementManagedClusterConfigInitializationActions = &WorkflowTemplatePlacementManagedClusterConfigInitializationActions{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigInitializationActions) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigInitializationActions) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigInitializationActions) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigEncryptionConfig struct {
empty bool `json:"-"`
GcePdKmsKeyName *string `json:"gcePdKmsKeyName"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigEncryptionConfig WorkflowTemplatePlacementManagedClusterConfigEncryptionConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigEncryptionConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigEncryptionConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigEncryptionConfig
} else {
r.GcePdKmsKeyName = res.GcePdKmsKeyName
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigEncryptionConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigEncryptionConfig *WorkflowTemplatePlacementManagedClusterConfigEncryptionConfig = &WorkflowTemplatePlacementManagedClusterConfigEncryptionConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigEncryptionConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigEncryptionConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigEncryptionConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigAutoscalingConfig struct {
empty bool `json:"-"`
Policy *string `json:"policy"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigAutoscalingConfig WorkflowTemplatePlacementManagedClusterConfigAutoscalingConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigAutoscalingConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigAutoscalingConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigAutoscalingConfig
} else {
r.Policy = res.Policy
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigAutoscalingConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigAutoscalingConfig *WorkflowTemplatePlacementManagedClusterConfigAutoscalingConfig = &WorkflowTemplatePlacementManagedClusterConfigAutoscalingConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigAutoscalingConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigAutoscalingConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigAutoscalingConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigSecurityConfig struct {
empty bool `json:"-"`
KerberosConfig *WorkflowTemplatePlacementManagedClusterConfigSecurityConfigKerberosConfig `json:"kerberosConfig"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigSecurityConfig WorkflowTemplatePlacementManagedClusterConfigSecurityConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigSecurityConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigSecurityConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigSecurityConfig
} else {
r.KerberosConfig = res.KerberosConfig
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigSecurityConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigSecurityConfig *WorkflowTemplatePlacementManagedClusterConfigSecurityConfig = &WorkflowTemplatePlacementManagedClusterConfigSecurityConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecurityConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecurityConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecurityConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigSecurityConfigKerberosConfig struct {
empty bool `json:"-"`
EnableKerberos *bool `json:"enableKerberos"`
RootPrincipalPassword *string `json:"rootPrincipalPassword"`
KmsKey *string `json:"kmsKey"`
Keystore *string `json:"keystore"`
Truststore *string `json:"truststore"`
KeystorePassword *string `json:"keystorePassword"`
KeyPassword *string `json:"keyPassword"`
TruststorePassword *string `json:"truststorePassword"`
CrossRealmTrustRealm *string `json:"crossRealmTrustRealm"`
CrossRealmTrustKdc *string `json:"crossRealmTrustKdc"`
CrossRealmTrustAdminServer *string `json:"crossRealmTrustAdminServer"`
CrossRealmTrustSharedPassword *string `json:"crossRealmTrustSharedPassword"`
KdcDbKey *string `json:"kdcDbKey"`
TgtLifetimeHours *int64 `json:"tgtLifetimeHours"`
Realm *string `json:"realm"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigSecurityConfigKerberosConfig WorkflowTemplatePlacementManagedClusterConfigSecurityConfigKerberosConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigSecurityConfigKerberosConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigSecurityConfigKerberosConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigSecurityConfigKerberosConfig
} else {
r.EnableKerberos = res.EnableKerberos
r.RootPrincipalPassword = res.RootPrincipalPassword
r.KmsKey = res.KmsKey
r.Keystore = res.Keystore
r.Truststore = res.Truststore
r.KeystorePassword = res.KeystorePassword
r.KeyPassword = res.KeyPassword
r.TruststorePassword = res.TruststorePassword
r.CrossRealmTrustRealm = res.CrossRealmTrustRealm
r.CrossRealmTrustKdc = res.CrossRealmTrustKdc
r.CrossRealmTrustAdminServer = res.CrossRealmTrustAdminServer
r.CrossRealmTrustSharedPassword = res.CrossRealmTrustSharedPassword
r.KdcDbKey = res.KdcDbKey
r.TgtLifetimeHours = res.TgtLifetimeHours
r.Realm = res.Realm
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigSecurityConfigKerberosConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigSecurityConfigKerberosConfig *WorkflowTemplatePlacementManagedClusterConfigSecurityConfigKerberosConfig = &WorkflowTemplatePlacementManagedClusterConfigSecurityConfigKerberosConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecurityConfigKerberosConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecurityConfigKerberosConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigSecurityConfigKerberosConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigLifecycleConfig struct {
empty bool `json:"-"`
IdleDeleteTtl *string `json:"idleDeleteTtl"`
AutoDeleteTime *string `json:"autoDeleteTime"`
AutoDeleteTtl *string `json:"autoDeleteTtl"`
IdleStartTime *string `json:"idleStartTime"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigLifecycleConfig WorkflowTemplatePlacementManagedClusterConfigLifecycleConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigLifecycleConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigLifecycleConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigLifecycleConfig
} else {
r.IdleDeleteTtl = res.IdleDeleteTtl
r.AutoDeleteTime = res.AutoDeleteTime
r.AutoDeleteTtl = res.AutoDeleteTtl
r.IdleStartTime = res.IdleStartTime
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigLifecycleConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigLifecycleConfig *WorkflowTemplatePlacementManagedClusterConfigLifecycleConfig = &WorkflowTemplatePlacementManagedClusterConfigLifecycleConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigLifecycleConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigLifecycleConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigLifecycleConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigEndpointConfig struct {
empty bool `json:"-"`
HttpPorts map[string]string `json:"httpPorts"`
EnableHttpPortAccess *bool `json:"enableHttpPortAccess"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigEndpointConfig WorkflowTemplatePlacementManagedClusterConfigEndpointConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigEndpointConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigEndpointConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigEndpointConfig
} else {
r.HttpPorts = res.HttpPorts
r.EnableHttpPortAccess = res.EnableHttpPortAccess
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigEndpointConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigEndpointConfig *WorkflowTemplatePlacementManagedClusterConfigEndpointConfig = &WorkflowTemplatePlacementManagedClusterConfigEndpointConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigEndpointConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigEndpointConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigEndpointConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfig struct {
empty bool `json:"-"`
NamespacedGkeDeploymentTarget *WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfigNamespacedGkeDeploymentTarget `json:"namespacedGkeDeploymentTarget"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigGkeClusterConfig WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigGkeClusterConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigGkeClusterConfig
} else {
r.NamespacedGkeDeploymentTarget = res.NamespacedGkeDeploymentTarget
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigGkeClusterConfig *WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfig = &WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfigNamespacedGkeDeploymentTarget struct {
empty bool `json:"-"`
TargetGkeCluster *string `json:"targetGkeCluster"`
ClusterNamespace *string `json:"clusterNamespace"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigGkeClusterConfigNamespacedGkeDeploymentTarget WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfigNamespacedGkeDeploymentTarget
func (r *WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfigNamespacedGkeDeploymentTarget) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigGkeClusterConfigNamespacedGkeDeploymentTarget
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigGkeClusterConfigNamespacedGkeDeploymentTarget
} else {
r.TargetGkeCluster = res.TargetGkeCluster
r.ClusterNamespace = res.ClusterNamespace
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfigNamespacedGkeDeploymentTarget is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigGkeClusterConfigNamespacedGkeDeploymentTarget *WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfigNamespacedGkeDeploymentTarget = &WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfigNamespacedGkeDeploymentTarget{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfigNamespacedGkeDeploymentTarget) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfigNamespacedGkeDeploymentTarget) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigGkeClusterConfigNamespacedGkeDeploymentTarget) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementManagedClusterConfigMetastoreConfig struct {
empty bool `json:"-"`
DataprocMetastoreService *string `json:"dataprocMetastoreService"`
}
type jsonWorkflowTemplatePlacementManagedClusterConfigMetastoreConfig WorkflowTemplatePlacementManagedClusterConfigMetastoreConfig
func (r *WorkflowTemplatePlacementManagedClusterConfigMetastoreConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementManagedClusterConfigMetastoreConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementManagedClusterConfigMetastoreConfig
} else {
r.DataprocMetastoreService = res.DataprocMetastoreService
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementManagedClusterConfigMetastoreConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementManagedClusterConfigMetastoreConfig *WorkflowTemplatePlacementManagedClusterConfigMetastoreConfig = &WorkflowTemplatePlacementManagedClusterConfigMetastoreConfig{empty: true}
func (r *WorkflowTemplatePlacementManagedClusterConfigMetastoreConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementManagedClusterConfigMetastoreConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementManagedClusterConfigMetastoreConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplatePlacementClusterSelector struct {
empty bool `json:"-"`
Zone *string `json:"zone"`
ClusterLabels map[string]string `json:"clusterLabels"`
}
type jsonWorkflowTemplatePlacementClusterSelector WorkflowTemplatePlacementClusterSelector
func (r *WorkflowTemplatePlacementClusterSelector) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplatePlacementClusterSelector
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplatePlacementClusterSelector
} else {
r.Zone = res.Zone
r.ClusterLabels = res.ClusterLabels
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplatePlacementClusterSelector is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplatePlacementClusterSelector *WorkflowTemplatePlacementClusterSelector = &WorkflowTemplatePlacementClusterSelector{empty: true}
func (r *WorkflowTemplatePlacementClusterSelector) Empty() bool {
return r.empty
}
func (r *WorkflowTemplatePlacementClusterSelector) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplatePlacementClusterSelector) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobs struct {
empty bool `json:"-"`
StepId *string `json:"stepId"`
HadoopJob *WorkflowTemplateJobsHadoopJob `json:"hadoopJob"`
SparkJob *WorkflowTemplateJobsSparkJob `json:"sparkJob"`
PysparkJob *WorkflowTemplateJobsPysparkJob `json:"pysparkJob"`
HiveJob *WorkflowTemplateJobsHiveJob `json:"hiveJob"`
PigJob *WorkflowTemplateJobsPigJob `json:"pigJob"`
SparkRJob *WorkflowTemplateJobsSparkRJob `json:"sparkRJob"`
SparkSqlJob *WorkflowTemplateJobsSparkSqlJob `json:"sparkSqlJob"`
PrestoJob *WorkflowTemplateJobsPrestoJob `json:"prestoJob"`
Labels map[string]string `json:"labels"`
Scheduling *WorkflowTemplateJobsScheduling `json:"scheduling"`
PrerequisiteStepIds []string `json:"prerequisiteStepIds"`
}
type jsonWorkflowTemplateJobs WorkflowTemplateJobs
func (r *WorkflowTemplateJobs) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobs
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobs
} else {
r.StepId = res.StepId
r.HadoopJob = res.HadoopJob
r.SparkJob = res.SparkJob
r.PysparkJob = res.PysparkJob
r.HiveJob = res.HiveJob
r.PigJob = res.PigJob
r.SparkRJob = res.SparkRJob
r.SparkSqlJob = res.SparkSqlJob
r.PrestoJob = res.PrestoJob
r.Labels = res.Labels
r.Scheduling = res.Scheduling
r.PrerequisiteStepIds = res.PrerequisiteStepIds
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobs is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobs *WorkflowTemplateJobs = &WorkflowTemplateJobs{empty: true}
func (r *WorkflowTemplateJobs) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobs) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobs) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsHadoopJob struct {
empty bool `json:"-"`
MainJarFileUri *string `json:"mainJarFileUri"`
MainClass *string `json:"mainClass"`
Args []string `json:"args"`
JarFileUris []string `json:"jarFileUris"`
FileUris []string `json:"fileUris"`
ArchiveUris []string `json:"archiveUris"`
Properties map[string]string `json:"properties"`
LoggingConfig *WorkflowTemplateJobsHadoopJobLoggingConfig `json:"loggingConfig"`
}
type jsonWorkflowTemplateJobsHadoopJob WorkflowTemplateJobsHadoopJob
func (r *WorkflowTemplateJobsHadoopJob) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsHadoopJob
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsHadoopJob
} else {
r.MainJarFileUri = res.MainJarFileUri
r.MainClass = res.MainClass
r.Args = res.Args
r.JarFileUris = res.JarFileUris
r.FileUris = res.FileUris
r.ArchiveUris = res.ArchiveUris
r.Properties = res.Properties
r.LoggingConfig = res.LoggingConfig
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsHadoopJob is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsHadoopJob *WorkflowTemplateJobsHadoopJob = &WorkflowTemplateJobsHadoopJob{empty: true}
func (r *WorkflowTemplateJobsHadoopJob) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsHadoopJob) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsHadoopJob) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsHadoopJobLoggingConfig struct {
empty bool `json:"-"`
DriverLogLevels map[string]string `json:"driverLogLevels"`
}
type jsonWorkflowTemplateJobsHadoopJobLoggingConfig WorkflowTemplateJobsHadoopJobLoggingConfig
func (r *WorkflowTemplateJobsHadoopJobLoggingConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsHadoopJobLoggingConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsHadoopJobLoggingConfig
} else {
r.DriverLogLevels = res.DriverLogLevels
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsHadoopJobLoggingConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsHadoopJobLoggingConfig *WorkflowTemplateJobsHadoopJobLoggingConfig = &WorkflowTemplateJobsHadoopJobLoggingConfig{empty: true}
func (r *WorkflowTemplateJobsHadoopJobLoggingConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsHadoopJobLoggingConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsHadoopJobLoggingConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsSparkJob struct {
empty bool `json:"-"`
MainJarFileUri *string `json:"mainJarFileUri"`
MainClass *string `json:"mainClass"`
Args []string `json:"args"`
JarFileUris []string `json:"jarFileUris"`
FileUris []string `json:"fileUris"`
ArchiveUris []string `json:"archiveUris"`
Properties map[string]string `json:"properties"`
LoggingConfig *WorkflowTemplateJobsSparkJobLoggingConfig `json:"loggingConfig"`
}
type jsonWorkflowTemplateJobsSparkJob WorkflowTemplateJobsSparkJob
func (r *WorkflowTemplateJobsSparkJob) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsSparkJob
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsSparkJob
} else {
r.MainJarFileUri = res.MainJarFileUri
r.MainClass = res.MainClass
r.Args = res.Args
r.JarFileUris = res.JarFileUris
r.FileUris = res.FileUris
r.ArchiveUris = res.ArchiveUris
r.Properties = res.Properties
r.LoggingConfig = res.LoggingConfig
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsSparkJob is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsSparkJob *WorkflowTemplateJobsSparkJob = &WorkflowTemplateJobsSparkJob{empty: true}
func (r *WorkflowTemplateJobsSparkJob) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsSparkJob) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsSparkJob) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsSparkJobLoggingConfig struct {
empty bool `json:"-"`
DriverLogLevels map[string]string `json:"driverLogLevels"`
}
type jsonWorkflowTemplateJobsSparkJobLoggingConfig WorkflowTemplateJobsSparkJobLoggingConfig
func (r *WorkflowTemplateJobsSparkJobLoggingConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsSparkJobLoggingConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsSparkJobLoggingConfig
} else {
r.DriverLogLevels = res.DriverLogLevels
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsSparkJobLoggingConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsSparkJobLoggingConfig *WorkflowTemplateJobsSparkJobLoggingConfig = &WorkflowTemplateJobsSparkJobLoggingConfig{empty: true}
func (r *WorkflowTemplateJobsSparkJobLoggingConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsSparkJobLoggingConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsSparkJobLoggingConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsPysparkJob struct {
empty bool `json:"-"`
MainPythonFileUri *string `json:"mainPythonFileUri"`
Args []string `json:"args"`
PythonFileUris []string `json:"pythonFileUris"`
JarFileUris []string `json:"jarFileUris"`
FileUris []string `json:"fileUris"`
ArchiveUris []string `json:"archiveUris"`
Properties map[string]string `json:"properties"`
LoggingConfig *WorkflowTemplateJobsPysparkJobLoggingConfig `json:"loggingConfig"`
}
type jsonWorkflowTemplateJobsPysparkJob WorkflowTemplateJobsPysparkJob
func (r *WorkflowTemplateJobsPysparkJob) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsPysparkJob
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsPysparkJob
} else {
r.MainPythonFileUri = res.MainPythonFileUri
r.Args = res.Args
r.PythonFileUris = res.PythonFileUris
r.JarFileUris = res.JarFileUris
r.FileUris = res.FileUris
r.ArchiveUris = res.ArchiveUris
r.Properties = res.Properties
r.LoggingConfig = res.LoggingConfig
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsPysparkJob is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsPysparkJob *WorkflowTemplateJobsPysparkJob = &WorkflowTemplateJobsPysparkJob{empty: true}
func (r *WorkflowTemplateJobsPysparkJob) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsPysparkJob) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsPysparkJob) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsPysparkJobLoggingConfig struct {
empty bool `json:"-"`
DriverLogLevels map[string]string `json:"driverLogLevels"`
}
type jsonWorkflowTemplateJobsPysparkJobLoggingConfig WorkflowTemplateJobsPysparkJobLoggingConfig
func (r *WorkflowTemplateJobsPysparkJobLoggingConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsPysparkJobLoggingConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsPysparkJobLoggingConfig
} else {
r.DriverLogLevels = res.DriverLogLevels
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsPysparkJobLoggingConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsPysparkJobLoggingConfig *WorkflowTemplateJobsPysparkJobLoggingConfig = &WorkflowTemplateJobsPysparkJobLoggingConfig{empty: true}
func (r *WorkflowTemplateJobsPysparkJobLoggingConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsPysparkJobLoggingConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsPysparkJobLoggingConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsHiveJob struct {
empty bool `json:"-"`
QueryFileUri *string `json:"queryFileUri"`
QueryList *WorkflowTemplateJobsHiveJobQueryList `json:"queryList"`
ContinueOnFailure *bool `json:"continueOnFailure"`
ScriptVariables map[string]string `json:"scriptVariables"`
Properties map[string]string `json:"properties"`
JarFileUris []string `json:"jarFileUris"`
}
type jsonWorkflowTemplateJobsHiveJob WorkflowTemplateJobsHiveJob
func (r *WorkflowTemplateJobsHiveJob) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsHiveJob
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsHiveJob
} else {
r.QueryFileUri = res.QueryFileUri
r.QueryList = res.QueryList
r.ContinueOnFailure = res.ContinueOnFailure
r.ScriptVariables = res.ScriptVariables
r.Properties = res.Properties
r.JarFileUris = res.JarFileUris
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsHiveJob is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsHiveJob *WorkflowTemplateJobsHiveJob = &WorkflowTemplateJobsHiveJob{empty: true}
func (r *WorkflowTemplateJobsHiveJob) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsHiveJob) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsHiveJob) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsHiveJobQueryList struct {
empty bool `json:"-"`
Queries []string `json:"queries"`
}
type jsonWorkflowTemplateJobsHiveJobQueryList WorkflowTemplateJobsHiveJobQueryList
func (r *WorkflowTemplateJobsHiveJobQueryList) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsHiveJobQueryList
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsHiveJobQueryList
} else {
r.Queries = res.Queries
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsHiveJobQueryList is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsHiveJobQueryList *WorkflowTemplateJobsHiveJobQueryList = &WorkflowTemplateJobsHiveJobQueryList{empty: true}
func (r *WorkflowTemplateJobsHiveJobQueryList) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsHiveJobQueryList) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsHiveJobQueryList) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsPigJob struct {
empty bool `json:"-"`
QueryFileUri *string `json:"queryFileUri"`
QueryList *WorkflowTemplateJobsPigJobQueryList `json:"queryList"`
ContinueOnFailure *bool `json:"continueOnFailure"`
ScriptVariables map[string]string `json:"scriptVariables"`
Properties map[string]string `json:"properties"`
JarFileUris []string `json:"jarFileUris"`
LoggingConfig *WorkflowTemplateJobsPigJobLoggingConfig `json:"loggingConfig"`
}
type jsonWorkflowTemplateJobsPigJob WorkflowTemplateJobsPigJob
func (r *WorkflowTemplateJobsPigJob) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsPigJob
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsPigJob
} else {
r.QueryFileUri = res.QueryFileUri
r.QueryList = res.QueryList
r.ContinueOnFailure = res.ContinueOnFailure
r.ScriptVariables = res.ScriptVariables
r.Properties = res.Properties
r.JarFileUris = res.JarFileUris
r.LoggingConfig = res.LoggingConfig
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsPigJob is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsPigJob *WorkflowTemplateJobsPigJob = &WorkflowTemplateJobsPigJob{empty: true}
func (r *WorkflowTemplateJobsPigJob) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsPigJob) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsPigJob) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsPigJobQueryList struct {
empty bool `json:"-"`
Queries []string `json:"queries"`
}
type jsonWorkflowTemplateJobsPigJobQueryList WorkflowTemplateJobsPigJobQueryList
func (r *WorkflowTemplateJobsPigJobQueryList) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsPigJobQueryList
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsPigJobQueryList
} else {
r.Queries = res.Queries
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsPigJobQueryList is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsPigJobQueryList *WorkflowTemplateJobsPigJobQueryList = &WorkflowTemplateJobsPigJobQueryList{empty: true}
func (r *WorkflowTemplateJobsPigJobQueryList) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsPigJobQueryList) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsPigJobQueryList) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsPigJobLoggingConfig struct {
empty bool `json:"-"`
DriverLogLevels map[string]string `json:"driverLogLevels"`
}
type jsonWorkflowTemplateJobsPigJobLoggingConfig WorkflowTemplateJobsPigJobLoggingConfig
func (r *WorkflowTemplateJobsPigJobLoggingConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsPigJobLoggingConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsPigJobLoggingConfig
} else {
r.DriverLogLevels = res.DriverLogLevels
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsPigJobLoggingConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsPigJobLoggingConfig *WorkflowTemplateJobsPigJobLoggingConfig = &WorkflowTemplateJobsPigJobLoggingConfig{empty: true}
func (r *WorkflowTemplateJobsPigJobLoggingConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsPigJobLoggingConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsPigJobLoggingConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsSparkRJob struct {
empty bool `json:"-"`
MainRFileUri *string `json:"mainRFileUri"`
Args []string `json:"args"`
FileUris []string `json:"fileUris"`
ArchiveUris []string `json:"archiveUris"`
Properties map[string]string `json:"properties"`
LoggingConfig *WorkflowTemplateJobsSparkRJobLoggingConfig `json:"loggingConfig"`
}
type jsonWorkflowTemplateJobsSparkRJob WorkflowTemplateJobsSparkRJob
func (r *WorkflowTemplateJobsSparkRJob) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsSparkRJob
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsSparkRJob
} else {
r.MainRFileUri = res.MainRFileUri
r.Args = res.Args
r.FileUris = res.FileUris
r.ArchiveUris = res.ArchiveUris
r.Properties = res.Properties
r.LoggingConfig = res.LoggingConfig
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsSparkRJob is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsSparkRJob *WorkflowTemplateJobsSparkRJob = &WorkflowTemplateJobsSparkRJob{empty: true}
func (r *WorkflowTemplateJobsSparkRJob) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsSparkRJob) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsSparkRJob) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsSparkRJobLoggingConfig struct {
empty bool `json:"-"`
DriverLogLevels map[string]string `json:"driverLogLevels"`
}
type jsonWorkflowTemplateJobsSparkRJobLoggingConfig WorkflowTemplateJobsSparkRJobLoggingConfig
func (r *WorkflowTemplateJobsSparkRJobLoggingConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsSparkRJobLoggingConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsSparkRJobLoggingConfig
} else {
r.DriverLogLevels = res.DriverLogLevels
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsSparkRJobLoggingConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsSparkRJobLoggingConfig *WorkflowTemplateJobsSparkRJobLoggingConfig = &WorkflowTemplateJobsSparkRJobLoggingConfig{empty: true}
func (r *WorkflowTemplateJobsSparkRJobLoggingConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsSparkRJobLoggingConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsSparkRJobLoggingConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsSparkSqlJob struct {
empty bool `json:"-"`
QueryFileUri *string `json:"queryFileUri"`
QueryList *WorkflowTemplateJobsSparkSqlJobQueryList `json:"queryList"`
ScriptVariables map[string]string `json:"scriptVariables"`
Properties map[string]string `json:"properties"`
JarFileUris []string `json:"jarFileUris"`
LoggingConfig *WorkflowTemplateJobsSparkSqlJobLoggingConfig `json:"loggingConfig"`
}
type jsonWorkflowTemplateJobsSparkSqlJob WorkflowTemplateJobsSparkSqlJob
func (r *WorkflowTemplateJobsSparkSqlJob) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsSparkSqlJob
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsSparkSqlJob
} else {
r.QueryFileUri = res.QueryFileUri
r.QueryList = res.QueryList
r.ScriptVariables = res.ScriptVariables
r.Properties = res.Properties
r.JarFileUris = res.JarFileUris
r.LoggingConfig = res.LoggingConfig
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsSparkSqlJob is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsSparkSqlJob *WorkflowTemplateJobsSparkSqlJob = &WorkflowTemplateJobsSparkSqlJob{empty: true}
func (r *WorkflowTemplateJobsSparkSqlJob) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsSparkSqlJob) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsSparkSqlJob) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsSparkSqlJobQueryList struct {
empty bool `json:"-"`
Queries []string `json:"queries"`
}
type jsonWorkflowTemplateJobsSparkSqlJobQueryList WorkflowTemplateJobsSparkSqlJobQueryList
func (r *WorkflowTemplateJobsSparkSqlJobQueryList) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsSparkSqlJobQueryList
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsSparkSqlJobQueryList
} else {
r.Queries = res.Queries
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsSparkSqlJobQueryList is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsSparkSqlJobQueryList *WorkflowTemplateJobsSparkSqlJobQueryList = &WorkflowTemplateJobsSparkSqlJobQueryList{empty: true}
func (r *WorkflowTemplateJobsSparkSqlJobQueryList) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsSparkSqlJobQueryList) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsSparkSqlJobQueryList) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsSparkSqlJobLoggingConfig struct {
empty bool `json:"-"`
DriverLogLevels map[string]string `json:"driverLogLevels"`
}
type jsonWorkflowTemplateJobsSparkSqlJobLoggingConfig WorkflowTemplateJobsSparkSqlJobLoggingConfig
func (r *WorkflowTemplateJobsSparkSqlJobLoggingConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsSparkSqlJobLoggingConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsSparkSqlJobLoggingConfig
} else {
r.DriverLogLevels = res.DriverLogLevels
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsSparkSqlJobLoggingConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsSparkSqlJobLoggingConfig *WorkflowTemplateJobsSparkSqlJobLoggingConfig = &WorkflowTemplateJobsSparkSqlJobLoggingConfig{empty: true}
func (r *WorkflowTemplateJobsSparkSqlJobLoggingConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsSparkSqlJobLoggingConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsSparkSqlJobLoggingConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsPrestoJob struct {
empty bool `json:"-"`
QueryFileUri *string `json:"queryFileUri"`
QueryList *WorkflowTemplateJobsPrestoJobQueryList `json:"queryList"`
ContinueOnFailure *bool `json:"continueOnFailure"`
OutputFormat *string `json:"outputFormat"`
ClientTags []string `json:"clientTags"`
Properties map[string]string `json:"properties"`
LoggingConfig *WorkflowTemplateJobsPrestoJobLoggingConfig `json:"loggingConfig"`
}
type jsonWorkflowTemplateJobsPrestoJob WorkflowTemplateJobsPrestoJob
func (r *WorkflowTemplateJobsPrestoJob) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsPrestoJob
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsPrestoJob
} else {
r.QueryFileUri = res.QueryFileUri
r.QueryList = res.QueryList
r.ContinueOnFailure = res.ContinueOnFailure
r.OutputFormat = res.OutputFormat
r.ClientTags = res.ClientTags
r.Properties = res.Properties
r.LoggingConfig = res.LoggingConfig
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsPrestoJob is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsPrestoJob *WorkflowTemplateJobsPrestoJob = &WorkflowTemplateJobsPrestoJob{empty: true}
func (r *WorkflowTemplateJobsPrestoJob) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsPrestoJob) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsPrestoJob) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsPrestoJobQueryList struct {
empty bool `json:"-"`
Queries []string `json:"queries"`
}
type jsonWorkflowTemplateJobsPrestoJobQueryList WorkflowTemplateJobsPrestoJobQueryList
func (r *WorkflowTemplateJobsPrestoJobQueryList) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsPrestoJobQueryList
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsPrestoJobQueryList
} else {
r.Queries = res.Queries
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsPrestoJobQueryList is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsPrestoJobQueryList *WorkflowTemplateJobsPrestoJobQueryList = &WorkflowTemplateJobsPrestoJobQueryList{empty: true}
func (r *WorkflowTemplateJobsPrestoJobQueryList) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsPrestoJobQueryList) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsPrestoJobQueryList) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsPrestoJobLoggingConfig struct {
empty bool `json:"-"`
DriverLogLevels map[string]string `json:"driverLogLevels"`
}
type jsonWorkflowTemplateJobsPrestoJobLoggingConfig WorkflowTemplateJobsPrestoJobLoggingConfig
func (r *WorkflowTemplateJobsPrestoJobLoggingConfig) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsPrestoJobLoggingConfig
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsPrestoJobLoggingConfig
} else {
r.DriverLogLevels = res.DriverLogLevels
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsPrestoJobLoggingConfig is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsPrestoJobLoggingConfig *WorkflowTemplateJobsPrestoJobLoggingConfig = &WorkflowTemplateJobsPrestoJobLoggingConfig{empty: true}
func (r *WorkflowTemplateJobsPrestoJobLoggingConfig) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsPrestoJobLoggingConfig) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsPrestoJobLoggingConfig) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateJobsScheduling struct {
empty bool `json:"-"`
MaxFailuresPerHour *int64 `json:"maxFailuresPerHour"`
MaxFailuresTotal *int64 `json:"maxFailuresTotal"`
}
type jsonWorkflowTemplateJobsScheduling WorkflowTemplateJobsScheduling
func (r *WorkflowTemplateJobsScheduling) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateJobsScheduling
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateJobsScheduling
} else {
r.MaxFailuresPerHour = res.MaxFailuresPerHour
r.MaxFailuresTotal = res.MaxFailuresTotal
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateJobsScheduling is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateJobsScheduling *WorkflowTemplateJobsScheduling = &WorkflowTemplateJobsScheduling{empty: true}
func (r *WorkflowTemplateJobsScheduling) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateJobsScheduling) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateJobsScheduling) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateParameters struct {
empty bool `json:"-"`
Name *string `json:"name"`
Fields []string `json:"fields"`
Description *string `json:"description"`
Validation *WorkflowTemplateParametersValidation `json:"validation"`
}
type jsonWorkflowTemplateParameters WorkflowTemplateParameters
func (r *WorkflowTemplateParameters) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateParameters
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateParameters
} else {
r.Name = res.Name
r.Fields = res.Fields
r.Description = res.Description
r.Validation = res.Validation
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateParameters is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateParameters *WorkflowTemplateParameters = &WorkflowTemplateParameters{empty: true}
func (r *WorkflowTemplateParameters) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateParameters) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateParameters) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateParametersValidation struct {
empty bool `json:"-"`
Regex *WorkflowTemplateParametersValidationRegex `json:"regex"`
Values *WorkflowTemplateParametersValidationValues `json:"values"`
}
type jsonWorkflowTemplateParametersValidation WorkflowTemplateParametersValidation
func (r *WorkflowTemplateParametersValidation) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateParametersValidation
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateParametersValidation
} else {
r.Regex = res.Regex
r.Values = res.Values
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateParametersValidation is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateParametersValidation *WorkflowTemplateParametersValidation = &WorkflowTemplateParametersValidation{empty: true}
func (r *WorkflowTemplateParametersValidation) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateParametersValidation) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateParametersValidation) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateParametersValidationRegex struct {
empty bool `json:"-"`
Regexes []string `json:"regexes"`
}
type jsonWorkflowTemplateParametersValidationRegex WorkflowTemplateParametersValidationRegex
func (r *WorkflowTemplateParametersValidationRegex) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateParametersValidationRegex
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateParametersValidationRegex
} else {
r.Regexes = res.Regexes
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateParametersValidationRegex is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateParametersValidationRegex *WorkflowTemplateParametersValidationRegex = &WorkflowTemplateParametersValidationRegex{empty: true}
func (r *WorkflowTemplateParametersValidationRegex) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateParametersValidationRegex) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateParametersValidationRegex) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
type WorkflowTemplateParametersValidationValues struct {
empty bool `json:"-"`
Values []string `json:"values"`
}
type jsonWorkflowTemplateParametersValidationValues WorkflowTemplateParametersValidationValues
func (r *WorkflowTemplateParametersValidationValues) UnmarshalJSON(data []byte) error {
var res jsonWorkflowTemplateParametersValidationValues
if err := json.Unmarshal(data, &res); err != nil {
return err
}
var m map[string]interface{}
json.Unmarshal(data, &m)
if len(m) == 0 {
*r = *EmptyWorkflowTemplateParametersValidationValues
} else {
r.Values = res.Values
}
return nil
}
// This object is used to assert a desired state where this WorkflowTemplateParametersValidationValues is
// empty. Go lacks global const objects, but this object should be treated
// as one. Modifying this object will have undesirable results.
var EmptyWorkflowTemplateParametersValidationValues *WorkflowTemplateParametersValidationValues = &WorkflowTemplateParametersValidationValues{empty: true}
func (r *WorkflowTemplateParametersValidationValues) Empty() bool {
return r.empty
}
func (r *WorkflowTemplateParametersValidationValues) String() string {
return dcl.SprintResource(r)
}
func (r *WorkflowTemplateParametersValidationValues) HashCode() string {
// Placeholder for a more complex hash method that handles ordering, etc
// Hash resource body for easy comparison later
hash := sha256.New().Sum([]byte(r.String()))
return fmt.Sprintf("%x", hash)
}
// Describe returns a simple description of this resource to ensure that automated tools
// can identify it.
func (r *WorkflowTemplate) Describe() dcl.ServiceTypeVersion {
return dcl.ServiceTypeVersion{
Service: "dataproc",
Type: "WorkflowTemplate",
Version: "beta",
}
}
func (r *WorkflowTemplate) ID() (string, error) {
if err := extractWorkflowTemplateFields(r); err != nil {
return "", err
}
nr := r.urlNormalized()
params := map[string]interface{}{
"name": dcl.ValueOrEmptyString(nr.Name),
"version": dcl.ValueOrEmptyString(nr.Version),
"create_time": dcl.ValueOrEmptyString(nr.CreateTime),
"update_time": dcl.ValueOrEmptyString(nr.UpdateTime),
"labels": dcl.ValueOrEmptyString(nr.Labels),
"placement": dcl.ValueOrEmptyString(nr.Placement),
"jobs": dcl.ValueOrEmptyString(nr.Jobs),
"parameters": dcl.ValueOrEmptyString(nr.Parameters),
"dag_timeout": dcl.ValueOrEmptyString(nr.DagTimeout),
"project": dcl.ValueOrEmptyString(nr.Project),
"location": dcl.ValueOrEmptyString(nr.Location),
}
return dcl.Nprintf("projects/{{project}}/locations/{{location}}/workflowTemplates/{{name}}", params), nil
}
const WorkflowTemplateMaxPage = -1
type WorkflowTemplateList struct {
Items []*WorkflowTemplate
nextToken string
pageSize int32
resource *WorkflowTemplate
}
func (l *WorkflowTemplateList) HasNext() bool {
return l.nextToken != ""
}
func (l *WorkflowTemplateList) Next(ctx context.Context, c *Client) error {
ctx, cancel := context.WithTimeout(ctx, c.Config.TimeoutOr(0*time.Second))
defer cancel()
if !l.HasNext() {
return fmt.Errorf("no next page")
}
items, token, err := c.listWorkflowTemplate(ctx, l.resource, l.nextToken, l.pageSize)
if err != nil {
return err
}
l.Items = items
l.nextToken = token
return err
}
func (c *Client) ListWorkflowTemplate(ctx context.Context, project, location string) (*WorkflowTemplateList, error) {
ctx = dcl.ContextWithRequestID(ctx)
ctx, cancel := context.WithTimeout(ctx, c.Config.TimeoutOr(0*time.Second))
defer cancel()
return c.ListWorkflowTemplateWithMaxResults(ctx, project, location, WorkflowTemplateMaxPage)
}
func (c *Client) ListWorkflowTemplateWithMaxResults(ctx context.Context, project, location string, pageSize int32) (*WorkflowTemplateList, error) {
ctx, cancel := context.WithTimeout(ctx, c.Config.TimeoutOr(0*time.Second))
defer cancel()
// Create a resource object so that we can use proper url normalization methods.
r := &WorkflowTemplate{
Project: &project,
Location: &location,
}
items, token, err := c.listWorkflowTemplate(ctx, r, "", pageSize)
if err != nil {
return nil, err
}
return &WorkflowTemplateList{
Items: items,
nextToken: token,
pageSize: pageSize,
resource: r,
}, nil
}
func (c *Client) GetWorkflowTemplate(ctx context.Context, r *WorkflowTemplate) (*WorkflowTemplate, error) {
ctx = dcl.ContextWithRequestID(ctx)
ctx, cancel := context.WithTimeout(ctx, c.Config.TimeoutOr(0*time.Second))
defer cancel()
// This is *purposefully* supressing errors.
// This function is used with url-normalized values + not URL normalized values.
// URL Normalized values will throw unintentional errors, since those values are not of the proper parent form.
extractWorkflowTemplateFields(r)
b, err := c.getWorkflowTemplateRaw(ctx, r)
if err != nil {
if dcl.IsNotFound(err) {
return nil, &googleapi.Error{
Code: 404,
Message: err.Error(),
}
}
return nil, err
}
result, err := unmarshalWorkflowTemplate(b, c, r)
if err != nil {
return nil, err
}
result.Project = r.Project
result.Location = r.Location
result.Name = r.Name
c.Config.Logger.InfoWithContextf(ctx, "Retrieved raw result state: %v", result)
c.Config.Logger.InfoWithContextf(ctx, "Canonicalizing with specified state: %v", r)
result, err = canonicalizeWorkflowTemplateNewState(c, result, r)
if err != nil {
return nil, err
}
if err := postReadExtractWorkflowTemplateFields(result); err != nil {
return result, err
}
c.Config.Logger.InfoWithContextf(ctx, "Created result state: %v", result)
return result, nil
}
func (c *Client) DeleteWorkflowTemplate(ctx context.Context, r *WorkflowTemplate) error {
ctx = dcl.ContextWithRequestID(ctx)
ctx, cancel := context.WithTimeout(ctx, c.Config.TimeoutOr(0*time.Second))
defer cancel()
if r == nil {
return fmt.Errorf("WorkflowTemplate resource is nil")
}
c.Config.Logger.InfoWithContext(ctx, "Deleting WorkflowTemplate...")
deleteOp := deleteWorkflowTemplateOperation{}
return deleteOp.do(ctx, r, c)
}
// DeleteAllWorkflowTemplate deletes all resources that the filter functions returns true on.
func (c *Client) DeleteAllWorkflowTemplate(ctx context.Context, project, location string, filter func(*WorkflowTemplate) bool) error {
listObj, err := c.ListWorkflowTemplate(ctx, project, location)
if err != nil {
return err
}
err = c.deleteAllWorkflowTemplate(ctx, filter, listObj.Items)
if err != nil {
return err
}
for listObj.HasNext() {
err = listObj.Next(ctx, c)
if err != nil {
return nil
}
err = c.deleteAllWorkflowTemplate(ctx, filter, listObj.Items)
if err != nil {
return err
}
}
return nil
}
func (c *Client) ApplyWorkflowTemplate(ctx context.Context, rawDesired *WorkflowTemplate, opts ...dcl.ApplyOption) (*WorkflowTemplate, error) {
ctx, cancel := context.WithTimeout(ctx, c.Config.TimeoutOr(0*time.Second))
defer cancel()
ctx = dcl.ContextWithRequestID(ctx)
var resultNewState *WorkflowTemplate
err := dcl.Do(ctx, func(ctx context.Context) (*dcl.RetryDetails, error) {
newState, err := applyWorkflowTemplateHelper(c, ctx, rawDesired, opts...)
resultNewState = newState
if err != nil {
// If the error is 409, there is conflict in resource update.
// Here we want to apply changes based on latest state.
if dcl.IsConflictError(err) {
return &dcl.RetryDetails{}, dcl.OperationNotDone{Err: err}
}
return nil, err
}
return nil, nil
}, c.Config.RetryProvider)
return resultNewState, err
}
func applyWorkflowTemplateHelper(c *Client, ctx context.Context, rawDesired *WorkflowTemplate, opts ...dcl.ApplyOption) (*WorkflowTemplate, error) {
c.Config.Logger.InfoWithContext(ctx, "Beginning ApplyWorkflowTemplate...")
c.Config.Logger.InfoWithContextf(ctx, "User specified desired state: %v", rawDesired)
// 1.1: Validation of user-specified fields in desired state.
if err := rawDesired.validate(); err != nil {
return nil, err
}
if err := extractWorkflowTemplateFields(rawDesired); err != nil {
return nil, err
}
initial, desired, fieldDiffs, err := c.workflowTemplateDiffsForRawDesired(ctx, rawDesired, opts...)
if err != nil {
return nil, fmt.Errorf("failed to create a diff: %w", err)
}
diffs, err := convertFieldDiffsToWorkflowTemplateDiffs(c.Config, fieldDiffs, opts)
if err != nil {
return nil, err
}
// TODO(magic-modules-eng): 2.2 Feasibility check (all updates are feasible so far).
// 2.3: Lifecycle Directive Check
var create bool
lp := dcl.FetchLifecycleParams(opts)
if initial == nil {
if dcl.HasLifecycleParam(lp, dcl.BlockCreation) {
return nil, dcl.ApplyInfeasibleError{Message: fmt.Sprintf("Creation blocked by lifecycle params: %#v.", desired)}
}
create = true
} else if dcl.HasLifecycleParam(lp, dcl.BlockAcquire) {
return nil, dcl.ApplyInfeasibleError{
Message: fmt.Sprintf("Resource already exists - apply blocked by lifecycle params: %#v.", initial),
}
} else {
for _, d := range diffs {
if d.RequiresRecreate {
return nil, dcl.ApplyInfeasibleError{
Message: fmt.Sprintf("infeasible update: (%v) would require recreation", d),
}
}
if dcl.HasLifecycleParam(lp, dcl.BlockModification) {
return nil, dcl.ApplyInfeasibleError{Message: fmt.Sprintf("Modification blocked, diff (%v) unresolvable.", d)}
}
}
}
// 2.4 Imperative Request Planning
var ops []workflowTemplateApiOperation
if create {
ops = append(ops, &createWorkflowTemplateOperation{})
} else {
for _, d := range diffs {
ops = append(ops, d.UpdateOp)
}
}
c.Config.Logger.InfoWithContextf(ctx, "Created plan: %#v", ops)
// 2.5 Request Actuation
for _, op := range ops {
c.Config.Logger.InfoWithContextf(ctx, "Performing operation %T %+v", op, op)
if err := op.do(ctx, desired, c); err != nil {
c.Config.Logger.InfoWithContextf(ctx, "Failed operation %T %+v: %v", op, op, err)
return nil, err
}
c.Config.Logger.InfoWithContextf(ctx, "Finished operation %T %+v", op, op)
}
return applyWorkflowTemplateDiff(c, ctx, desired, rawDesired, ops, opts...)
}
func applyWorkflowTemplateDiff(c *Client, ctx context.Context, desired *WorkflowTemplate, rawDesired *WorkflowTemplate, ops []workflowTemplateApiOperation, opts ...dcl.ApplyOption) (*WorkflowTemplate, error) {
// 3.1, 3.2a Retrieval of raw new state & canonicalization with desired state
c.Config.Logger.InfoWithContext(ctx, "Retrieving raw new state...")
rawNew, err := c.GetWorkflowTemplate(ctx, desired)
if err != nil {
return nil, err
}
// Get additional values from the first response.
// These values should be merged into the newState above.
if len(ops) > 0 {
lastOp := ops[len(ops)-1]
if o, ok := lastOp.(*createWorkflowTemplateOperation); ok {
if r, hasR := o.FirstResponse(); hasR {
c.Config.Logger.InfoWithContext(ctx, "Retrieving raw new state from operation...")
fullResp, err := unmarshalMapWorkflowTemplate(r, c, rawDesired)
if err != nil {
return nil, err
}
rawNew, err = canonicalizeWorkflowTemplateNewState(c, rawNew, fullResp)
if err != nil {
return nil, err
}
}
}
}
c.Config.Logger.InfoWithContextf(ctx, "Canonicalizing with raw desired state: %v", rawDesired)
// 3.2b Canonicalization of raw new state using raw desired state
newState, err := canonicalizeWorkflowTemplateNewState(c, rawNew, rawDesired)
if err != nil {
return rawNew, err
}
c.Config.Logger.InfoWithContextf(ctx, "Created canonical new state: %v", newState)
// 3.3 Comparison of the new state and raw desired state.
// TODO(magic-modules-eng): EVENTUALLY_CONSISTENT_UPDATE
newDesired, err := canonicalizeWorkflowTemplateDesiredState(rawDesired, newState)
if err != nil {
return newState, err
}
if err := postReadExtractWorkflowTemplateFields(newState); err != nil {
return newState, err
}
// Need to ensure any transformations made here match acceptably in differ.
if err := postReadExtractWorkflowTemplateFields(newDesired); err != nil {
return newState, err
}
c.Config.Logger.InfoWithContextf(ctx, "Diffing using canonicalized desired state: %v", newDesired)
newDiffs, err := diffWorkflowTemplate(c, newDesired, newState)
if err != nil {
return newState, err
}
if len(newDiffs) == 0 {
c.Config.Logger.InfoWithContext(ctx, "No diffs found. Apply was successful.")
} else {
c.Config.Logger.InfoWithContextf(ctx, "Found diffs: %v", newDiffs)
diffMessages := make([]string, len(newDiffs))
for i, d := range newDiffs {
diffMessages[i] = fmt.Sprintf("%v", d)
}
return newState, dcl.DiffAfterApplyError{Diffs: diffMessages}
}
c.Config.Logger.InfoWithContext(ctx, "Done Apply.")
return newState, nil
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 333 |
\section{Introduction}
The bacterial chromosome is highly structured over a wide range of length-scales~\cite{Wang2014, Gruber2014a, Badrinarayanan2015, Kleckner2014}.
Indeed, microscopy experiments have revealed a remarkable degree of sub-cellular organization of the chromosome across many species, including \textit{Bacillus subtilis}, \textit{Caulobacter crescentus} and \textit{Escherichia coli}~\cite{Viollier2004, Marbouty2015, Wiggins2010, Wu2018,Sullivan}.
The structural organization of chromosomes was further exposed by recent Hi-C experiments measuring the contact probability between pairs of loci on the chromosome~\cite{Le2013,Wang2015a, Wang2017a, Marbouty2015, Tran2017}.
On small length-scales, the chromosome appears to be organized in so-called Chromosome Interaction Domains: genomic domains with above-average contact probabilities between pairs of loci. On large length-scales, the most prominent feature is the emergence of a cross-diagonal in Hi-C maps spanning the whole length of the genome, indicating anomalously high contact probabilities between opposing pairs of DNA-loci positioned on the left and right chromosomal arms. Such a cross-diagonal is observed in \textit{B.~subtilis} and \textit{C.~crescentus}~\cite{Wang2015a,Wang2017a,Le2013, Umbarger2011}, but not in \textit{E.~coli}~\cite{Lioy2018}.
The measured cross-diagonal indicates a robust juxtaposition of the left and right arms of the chromosome~\cite{Le2013,Wang2017a,Wang2015a, Umbarger2011}, an organizational feature that is important for faithful chromosome segregation~\cite{Gruber2009, Marbouty2015, Gruber2014, Sullivan, Wang2014b}. This juxtaposed organization is largely controlled by the highly conserved ATPase SMC condensin~\cite{Wang2015a,Wang2017a, Marbouty2015}. While much is known about condensin at the molecular level~\cite{Soh2015,Burmann2013,Hirano2006a, Diebold-Durand2017, Eeftens2016, Kamada2017}, it is unclear how small numbers of condensins (3--30 per chromosome~\cite{Wilhelm2015})
are capable of collectively organizing the chromosome over such a range of length-scales. Thus, the physical principles underlying the juxtaposed organization of the chromosome remain elusive.
The functional capability of SMC condensins to organize the chromosome into a juxtaposed state crucially depends on two factors:\,(i)\,the presence of a specific loading site on the chromosome~\cite{Wang2015a, Minnen2016}, and (ii)\,the ability of condensin to bind and hydrolyze ATP~\cite{Minnen2016, Wilhelm2015}.
In \textit{B. subtilis}, the loading site is established by a large nucleoprotein complex composed of ParB proteins bound around \textit{parS} close to the origin of replication (\textit{ori})~\cite{Broedersz2014, Mohl1997, surtees2003plasmid, Livny2007}.
Condensins are recruited to this $\mathrm{ParB}S$\, region, and from there propagate deep into the bulk of the DNA polymer~\cite{Minnen2016, Wilhelm2015}. Removing the loading site results in a loss of the cross-diagonal in Hi-C maps, and adding additional loading sites disrupts the cross-diagonal~\cite{Wang2015a, Wang2017a}.
Moreover, the translocation of condensins away from the loading site depends on their ATPase activity; condensin mutants that cannot bind ATP only weakly associate with DNA, and mutants that do not hydrolyze ATP do not efficiently propagate away from the $\mathrm{ParB}S$\, loading site (SI~\ref{sec:empirical-metrics} and~\cite{Minnen2016, Wilhelm2015}). With a ring-like topology of $25-50\unit{nm}$ in diameter, the condensin ring is large enough to trap a DNA loop by threading a DNA duplex through it~\cite{Eeftens2016, Burmann2017, Soh2015,Wilhelm2015,Hirano2006a,Diebold-Durand2017, Burmann2017, Burmann2013}. It has been proposed that the possibility of condensins to trap DNA-loops would enable them to align the chromosomal arms by progressively extruding DNA-loops from the origin to the terminus region~\cite{Wang2015a, Wang2017a, Marbouty2015}.
Important clues on the role of ATPase activity in SMC condensin come from \textit{in vitro} single-molecule experiments. These experiments revealed that \textit{Saccharomyces cerevisiae} condensin is a molecular motor, and performs active translocation over DNA duplexes~\cite{Terekawa2017}. Kymographs showed that the direction of movement of yeast condensin is \textit{a priori} random, and switches direction after a typical time-scale; in other words, yeast condensin performs persistent random motion over DNA. In fact, recent single-molecule experiments have revealed that such yeast condensin performs active loop extrusion~\cite{Ganji2018}.
Similar experiments have not yet observed such motor activity for bacterial condensin~\cite{Kim2016}. Nevertheless, it has been widely speculated that the ATPase activity of bacterial condensin is also directed towards motor activity~\cite{Eeftens2017, Burmann2017, Wilhelm2015, Diebold-Durand2017}.
In this picture, condensin would actively extrude DNA-loops, possibly by feeding DNA duplexes through its ring-like structure~\cite{Sanborn, Fudenberg2016, Brackley2016, Alipour2012}. In contrast, other models have been proposed in which the ATPase activity of condensin is directed towards regulating its association with DNA~\cite{Wilhelm2015, Minnen2016, Brackley2016, Brackley2018}. Thus, it remains an open question whether bacterial condensin also acts as a loop extruding enzyme, or whether the ATPase activity primarily regulates the DNA-recruitment of condensins.
To elucidate the role of condensin ATPase activity on bacterial chromosome organization, we develop two minimal models for condensin--DNA interactions, where we describe SMC condensin as a slip-link that non-topologically traps a DNA-loop (Fig.~\ref{fig:Microscopic-reactions}). In these models, we analyze the complex interplay between the molecular-scale dynamics of SMC condensins and large-scale chromosome organization. In the most basic model, condensin activity is assumed to be directed towards regulating its DNA-recruitment, while the dynamics of condensin slip-links on the DNA is diffusive.
Although the motion of individual slip-links on the DNA is purely diffusive, their active recruitment to DNA results in non-equilibrium collective motion of slip-links. Interestingly, we find that such \textit{diffusive slip-links} can organize the chromosome into a juxtaposed state, but not within physiological constraints. Next, we expand the model to include \textit{motor slip-links} that perform persistent random motion on DNA, as observed for yeast condensin\textit{in vitro}~\cite{Terekawa2017}. We find that these motor slip-links are much more effective in organizing the entire chromosome. In particular, our motor slip-link model requires at least 2--3 orders of magnitude fewer condensins to organize the chromosome than in the diffusive slip-link model.
In addition, the development of the juxtaposed state exhibits sub-diffusive dynamics in the diffusive slip-link model, in contradiction with the rapid re-organization observed \textit{in vivo}~\cite{Wang2017a}. We show that such a fast re-organization of the chromosome can be achieved by motor activity in the form of active loop extrusion. More generally, we provide a quantitative model to address key questions in bacterial chromosome organization, such as the role of an exclusive loading site, cell confinement, motor activity and interaction of SMC condensin with other DNA-bound factors.
\begin{figure}[t!]
\includegraphics[width=\linewidth]{figures/1.model_outline/microscopic_reactions_small.pdf}
\caption{ {\small Schematic of slip-link model for SMC condensin--DNA interactions. \textbf{Left}: The bacterial chromosome has a circular topology. SMC condensins are loaded onto the DNA by the ParB\textit{S} complex located near the origin of replication (\textit{ori}, green). \textbf{Right}: Microscopic reactions in our model with associated rate constants. Condensins are represented as slip-links, which either perform diffusive or more persistent, motorized movement (inset). Our computational method simulates the stochastic interplay between DNA dynamics and slip-link positioning.}
\label{fig:Microscopic-reactions}
}
\end{figure}
\section{Model}
Our model of condensin--DNA interactions (Fig.~\ref{fig:Microscopic-reactions}) contains two ingredients: (i)\,a circular DNA polymer, and (ii)\,multiple SMC condensins that interact with the DNA. We employ a lattice polymer with a lattice constant set by the persistence length of DNA (SI~\ref{sec:Explanation-of-KMC}). Condensins are modeled as slip-links: elastic rings that trap a DNA-loop by encircling two DNA duplexes. Importantly, we consider both diffusive and motor slip-links. Diffusive slip-links move randomly (stepping rate $k_0$) over the DNA (Fig.~\ref{fig:Microscopic-reactions}, inset); motor slip-links perform persistent random motion (translocation rate $k_\mathrm{motor}$) with a persistence time-scale $\tau_\mathrm{switch} = k_\mathrm{switch}^{-1}$. These persistent dynamics enable motor slip-links to actively extrude DNA loops.
To simulate the dynamics of both the DNA polymer and the slip-links, we developed a Kinetic Monte-Carlo (KMC) algorithm (SI~\ref{sec:Explanation-of-KMC}). Our KMC algorithm simulates the Rouse dynamics of DNA~\cite{Weber2010, Doi1986}, the associated stochastic motion of slip-links on the DNA, as well as the microscopic reactions in which slip-links bind to (rate $k_+$) or unbind from (rate $k_-$) the DNA. For simplicity, we assume instantaneous slip-link binding $k_+ \rightarrow \infty$, justified by the relatively fast cytosolic diffusion of condensin~\cite{Stigler2016a}.
Additionally, we fix a maximum number of slip-links $N_p$ that can bind to the DNA.
\section*{Results}
\subsection{Diffusive slip-links with a specific loading site can organize the chromosome}
In the simplest implementation of our model, there is no specific loading site: diffusive slip-links can bind and unbind anywhere on the DNA (Fig.~\ref{fig:overview-stationary-results}a, $\Delta\mathrm{ParB}/S$\,). This assumption results in a homogeneous binding probability $p_p(i)$ (Fig.~\ref{fig:overview-stationary-results}a, bottom). Importantly, in the $\Delta\mathrm{ParB}/S$\, scenario, all microscopic reactions are fully reversible, implying that the system relaxes into thermodynamic equilibrium. In equilibrium, small loops are strongly favored, owing to the increasing entropic cost of loop formation with larger loop size. This tendency to form small loops is reflected in the loop diagrams (Fig.~\ref{fig:overview-stationary-results}a, top). Indeed, the loop sizes trapped by the slip-links are consistent with the equilibrium loop-size distribution (SI~\ref{sec:diff-slip-links-contact-distribution}). Furthermore, in this scenario the contact maps are structureless and only exhibit a single main diagonal (Fig.~\ref{fig:overview-stationary-results}a, middle), as for a random polymer. These results demonstrate that our KMC model with reversible microscopic reactions evolves towards thermodynamic equilibrium. Thus, although the slip-links can easily bind over the full extent of the polymer, they do not organize the chromosome.
\begin{figure}[t!]
\includegraphics[width=0.9\linewidth]{figures/2.results_overview/results_overview_w_outlined_embedded_figs_w_slashes_small.pdf}
\caption{
{\small Steady-state organization of a circular chromosome by diffusive and motor slip-links. For all sub-figures (a)--(h): top panels show a loop diagram that illustrates the loop structure induced by slip-links, middle panels the contact map $p_c(i,j)$, and bottom panels the binding profile $p_p(i)$ of slips-links along the DNA. \textbf{Diffusive slip-links, (a)--(d)}: (a) lacks a specific loading site ($\Delta\mathrm{ParB}/S$\,), (b) $k_- = 10^{-6}k_0$, (c)--(d) $k_- = k_0$ with $N_p=5$ in (c) and $N_p=25$ in (d). \textbf{Motor slip-links, (e)--(g)} (e) lacks a specific loading site ($\Delta\mathrm{ParB}/S$\,), (f) $k_{\rm switch}=10^{-6}k_\mathrm{motor}$, (g) $k_{\rm switch}=10^{-1} k_\mathrm{motor}$. All simulations were performed with a circular polymer polymer of length $N_m=80$. Maximum number of polymer-bound slip-links from (a)--(g) are respectively $N_p = 16,20,5,25,2,2,2$. \textbf{Legend, (h)}: We use the fraction of nested loops ($\theta$, see SI~\ref{sec:measurement-of-metrics-from-simulation-data}), cross-diagonal length ($\hat X_c$, see SI~\ref{sec:measurement-of-metrics-from-simulation-data}) and the slip-link propagation distance ($\hat X_p=\sqrt {\mathrm{var} \ p_p(i) }$) to quantify polymer organization. The slip-link loading site (when present) is positioned in the center of the contact map, so that the axes run from
$-\tfrac{1}{2}N_m\ldots+\tfrac{1}{2}N_m$. }
\label{fig:overview-stationary-results}
}
\end{figure}
We next investigate how the presence of a slip-link loading site~\cite{Minnen2016, Wilhelm2015, Wang2015a, Wang2017a, Gruber2009} impacts steady-state chromosome organization. Note, while recruitment of slip-links in our model is exclusive to \textit{ori}, unbinding can occur anywhere on the DNA. This implies that the reactions involving slip-link binding/unbinding are partially irreversible. Hence, detailed balance is broken~\cite{Gnesotto2018}, and the system may no longer evolve towards thermodynamic equilibrium. Nevertheless, for slow dissociation kinetics (small $k_-$), we observe an unstructured contact map (Fig.~\ref{fig:overview-stationary-results}b; SI~\ref{sec:diff-slip-links-contact-distribution}), similar to the equilibrium system lacking a specific loading site (Fig.~\ref{fig:overview-stationary-results}a; SI~\ref{sec:diff-slip-links-contact-distribution}). Moreover, the loop diagrams again show that the slip-links mostly encircle small loops (Fig.~\ref{fig:overview-stationary-results}b, top). In sum, we see that, although detailed balance is broken on the level of slip-link binding/unbinding, the diffusive slip-links with slow dissociation kinetics do not appear to organize the DNA polymer.
Interestingly, increasing the dissociation kinetics of slip-links results in dramatically different contact map. We observe the emergence of a prominent cross-diagonal, which either disperses away from the loading site (Fig.~\ref{fig:overview-stationary-results}c) or remains clearly resolved over the whole polymer (Fig.~\ref{fig:overview-stationary-results}d), depending on the density of slip-links. Interestingly, the loop diagrams for these systems exhibit a topology distinct from the equilibrium configuration; slip-links trap DNA-loops in a cooperative, nested manner. Movies of the loop diagrams and contact maps clearly demonstrate that these nested loops propagate away from the loading site, dynamically driving a juxtaposition of the two polymer arms (SI movies\,3a--b; SI~\ref{section:relaxatin-juxtaposed-organization}). This dynamical arm--arm alignment is a distinct out-of-equilibrium phenomenon, and thus requires the exclusive binding of slip-links to the loading site. Thus, our observations show that diffusive slip-links with fast dissociation kinetics in conjunction with a loading site can, in principle, generate a non-equilibrium polymer organization similar to that found in living cells~\cite{Wang2015a,Wang2017a, Marbouty2015, Le2013}.
To investigate the role of slip-link kinetics on loop topology more quantitatively, we employ a metric that captures essential topological differences between the loop network of random and juxtaposed polymers (compare Fig.~\ref{fig:overview-stationary-results}b to Figs.~\ref{fig:overview-stationary-results}c--d). To this end, we define the order parameter $\theta \in [0,1] $ as the fraction of nested loops (Fig.~\ref{fig:overview-stationary-results}h and SI~\ref{sec:measurement-of-metrics-from-simulation-data}). We observe a sigmoidal relation between $\theta$ and the dissociation rate $k_-$ for diffusive slip-links, with $\theta$ transitioning from low to high values with increasing $k_-$ (Fig.~\ref{fig:stationary-phenomology}, blue).
This surprising link between $k_-$ and $\theta$ can ultimately be traced to collective interactions between slip-links: steric hindrance between slip-links drives loops away from the loading site, resulting in ballistic collective motion of slip-links (SI Movie 3b). The lifetime $\tau_{\mathrm{NS}}$ of nested loops depends on both the velocity of the ballistic movement and on the polymer length $N_m$. We argue below that the increase in $\tau_{\mathrm{NS}}$ due to these two factors quantitatively accounts for the increase of $\theta$ with $k_-$.
We find that the characteristic dissociation rate at the inflection point of $\theta$ coincides with the transition from a random polymer to a juxtaposed organization with a cross-diagonal in contact maps ($d_1$ vs. $d_2$ in Fig.~\ref{fig:stationary-phenomology}). In addition, we observe that the equilibrium implementation of diffusive slip-links that bind non-specifically to the DNA ($\Delta\mathrm{ParB}/S$\,) yields $\theta \approx 0$ ($d_3$ in Fig.~\ref{fig:stationary-phenomology}), confirming our previous observation that the loading site is necessary for creating large, nested loops (SI~\ref{sec:diff-slip-links-contact-distribution}). Thus, in the presence of a loading site, diffusive slip-links appear to drive a dynamical transition between phases of weak and strong nesting of DNA-loops, and this transition is crucial to establish the juxtaposed state of the chromosome.
Importantly, in our model with diffusive slip-links, we find that having many nested loops is a necessary, but not sufficient condition to organize the polymer into a juxtaposed state (SI~\ref{sec:Relationship-between-Xc-Xp-and-theta}). These loops also need to propagate into the bulk of the polymer. Indeed, we find that the propagation of diffusive slip-links is a density-driven process: the nested loops only propagate over the full length of the polymer for very high slip-link densities (SI~\ref{sec:universal-scaling-propagation-length}). This can be clearly seen in the binding profiles $p_p(i)$; at low slip-link densities, the binding profile is sharply peaked around the loading site (Fig.~\ref{fig:overview-stationary-results}c, bottom), whereas this peak broadens as we increase the slip-link density (Fig.~\ref{fig:overview-stationary-results}d, bottom).
\begin{figure}
\includegraphics[width=\linewidth]{figures/3.stationary_phenomology/3_order_parameter_and_Xp_small.pdf}
\caption{{\small Loop network topology controls polymer organization. The fraction of nested loops $\theta$ of diffusive slip-links (blue) is shown as a function of the dissociation rate $k_-$ (in units of the slip-link diffusion rate $k_0$), and for motor slip-links (red) as a function of the switching rate $k_\mathrm{switch}$ (in units of the slip-link translocation rate $k_\mathrm{motor}$). Error bars represent the standard deviation $\sqrt{\mathrm{var}{\theta}}$. A mean-field model yields an estimate for $\theta$ of diffusive slip-links (blue curve). Representative loop diagrams and contact maps are indicated for different parameter choices, labeled $d_1$--$d_3$ for diffusive links and $m_1$--$m_3$ for motor slip-links. Also shown are $\theta$ for diffusive ($d_3$) and motor ($m_3$) slip-links with non-specific slip-link binding ($\Delta\mathrm{ParB}/S$\,). Polymer lengths $N_m$ and slip-link numbers $N_p$ are $N_m=40, N_p=10$ (diffusive slip-links) and $N_m=80,N_p=2$ (motor slip-links).}
\label{fig:stationary-phenomology}
}
\end{figure}
\subsubsection{Irreversible slip-link binding acts as a kinetic filter for nested loops}\label{subsubsec:mean-field-model}
To understand the impact of loop lifetime on $\theta$ more quantitatively, we model the loop topology of the slip-links as $N_p$ independent two-state systems: (i) self-loop (S) with a lifetime $\tau_\mathrm{S}$; (ii) nested loop (NS) with a lifetime $\tau_{\mathrm{NS}}$. From this mean-field perspective, the fraction of nested loops $\langle \theta \rangle$ is simply the weighted lifetime of a nested loop:
\[
\left\langle \theta\right\rangle \approx \dfrac{p_\mathrm{NS}\tau_{\mathrm{NS}}}{p_\mathrm{NS}\tau_{\mathrm{NS}}+p_\mathrm{S}\tau_{\mathrm{S}}},
\]
where $p_\mathrm{S}$ and $p_\mathrm{NS}$ are the probabilities for a diffusive slip-link to enclose a self and nested loop respectively after loading to \textit{ori}. The lifetime of a self-loop is $\tau_{\mathrm{S}}\approx k_{-}^{-1}$. We estimate the lifetime of a nested loop by $ \tau_{\mathrm{NS}}\approx \tfrac{1}{2}N_{m} / \left\langle v\right\rangle + k_-^{-1},$ where $\left\langle v\right\rangle $ is the mean velocity of a slip-link moving through the bulk of the polymer. Since the nested loops propagate ballistically in the high density phase (Fig. \ref{fig:velocity-of-tracer-slip-links}; SI movie 3b), there exists a well-defined velocity $\left\langle v\right\rangle $ that depends on the system size. In particular, we empirically find for $\phi_p = 0.4$ that $\langle v \rangle \approx c k_0 /N_m^2$ where $k_0$ is the slip-link movement attempt rate and $c\approx 9$ (Fig.~\ref{fig:velocity-of-tracer-slip-links}). This $1/N_m^2$ scaling is distinct from the $1/N_m$ scaling observed in the Simple Symmetric Exclusion Process \cite{Derrida2007}, likely due to polymer loop entropy that impedes the movement of slip-links away from the loading site.
\begin{figure}
\includegraphics[width=8cm]{figures/SI.maximum_theta_val_scaling_w_system_size/ea120_velocity_scaling_w_annotations_small.pdf}
\caption{{\small Diffusive slip-links propagate ballistically over a polymer in the juxtaposed state with a velocity $\langle v \rangle \sim 1/N_m^2.$ \textbf{Main panel}: Averaged velocities of tracer slip-links $\langle v \rangle$ (markers) for different system sizes $N_m$ (error bars: twice the standard error in the mean). The dashed line $\langle v \rangle \approx 9k_0/ N_m^2$ is shown together with data computed in the high-density and fast-dissociation regime ($\phi_{p}=0.4, k_- = k_0$).
\textbf{Inset}: Trajectories of diffusive slip-links $l(t)$ (thin, gray) with ensemble average $\langle l(t) \rangle$ (thick, blue; shaded region is the standard deviation), where $l(t)$ is the distance traversed by a tracer slip-link at time $t$ after loading onto the polymer.}
\label{fig:velocity-of-tracer-slip-links} }
\end{figure}
In sum, our estimate for $\langle\theta\rangle$ is
\begin{equation}
\left\langle \theta\right\rangle \approx \dfrac{\tfrac{1}{3}C \left( \tfrac{1}{2}cN_{m}^{3}k_{0}^{-1} + k_-^{-1} \right)}{\tfrac{1}{3}C \left( \tfrac{1}{2}cN_{m}^{3}k_{0}^{-1} + k_-^{-1} \right)+(1- \tfrac 1 3 C)k_{-}^{-1}}. \label{eq:mft-theta}
\end{equation}
From this, we determine that nested loops start to dominate the loop topology from a characteristic dissociation rate $k_-^\star \sim N_m^{-3}$.
The dependency of $\theta$ on $k_-$ reveals that the irreversible loading mechanism functions as a \emph{kinetic filter}: The fast dissociation kinetics filters out self-loops, only allowing nested loops to propagate through the system.
Since $c$ in Eq.~\eqref{eq:mft-theta} can be determined from a measurement of $\langle v \rangle$, this form for $\langle \theta \rangle$ does not contain any free fit-parameters, and collapses data for various $N_{m}, k_-$ onto a single master curve (Fig.\,\ref{fig:theta-vs-kmin-data-collapse}).
\begin{figure}
\includegraphics[width=8cm]{figures/SI.maximum_theta_val_scaling_w_system_size/ea119_theta_vs_kMin_data_collapse_small.pdf}
\caption{{\small A single master curve approximately describes the fraction of nested loops. \textbf{Main panel}: Loop parameter $\langle \theta \rangle$ (markers) for different system sizes $N_m$ as indicated (error bars: $\sqrt{ \mathrm{var} \theta}$) together with mean-field (Eq.~\eqref{eq:mft-theta}) estimate (solid curves). All data were measured in the high-density regime ($\phi_{p}=0.4$ for $N_m=20,40,200$ and $\phi_{p}=0.38$ for $N_m=100$).
\textbf{Inset}: The mean-field estimate for $\theta$ (Eq.~\eqref{eq:mft-theta}) collapses the data onto a single master-curve $x/(1+x)$. The reduced parameters are: $\tilde \theta = (\theta - \theta_\mathrm{min})/(\theta_\mathrm{max}-\theta_\mathrm{min}),\theta_\mathrm{max}=1, \theta_\mathrm{min} = \tfrac 1 3 C$ and $x = \tfrac{C}{3-C} ( \tfrac{1}{2}cN_{m}^{3}k_-/k_{0} + 1 )$ with $x \sim N_m^3 k_-$ for $N_m\gg 1$. \label{fig:theta-vs-kmin-data-collapse}} }
\end{figure}
\subsection{Motor slip-links are highly effective in organizing the chromosome, even at low densities}
Motivated by recent observations of motor activity of yeast condensin in single-molecule experiments~\cite{Terekawa2017, Ganji2018}, we next explore how such activity impacts the ability of slip-links to organize the chromosome. In our model, motor slip-links are assumed to perform persistent random motion (Fig.~\ref{fig:Microscopic-reactions}, inset). Such persistent slip-links actively extrude loops~\cite{Ganji2018,Sanborn, Fudenberg2016, Brackley2016, Alipour2012}. The active dynamics of motor slip-links is characterized by the \textit{switching rate} $k_\mathrm{switch}$. For $k_\mathrm{switch} \rightarrow 0$, the motor slip-links never reverse direction, whereas for $k_\mathrm{switch} \gg k_\mathrm{motor}$ they behave as diffusive slip-links. We find that motor slip-links with small $k_\mathrm{switch}$ organize a system-spanning cross-diagonal (Fig.~\ref{fig:overview-stationary-results}f), whereas the cross-diagonal retracts for increasing $k_\mathrm{switch}$ (Fig.~\ref{fig:overview-stationary-results}g).
Overall, we observe that the persistence of such motor slip-links renders them much more effective at producing the cross-diagonal (Figs.~\ref{fig:overview-stationary-results}f--g). Even for low slip-link densities, a system-spanning cross-diagonal is formed together with an extended binding profile (Fig.~\ref{fig:overview-stationary-results}f). Indeed, motor slip-links can readily drive the chromosome into a state with a high degree of loop nesting. Even in the absence of a loading site ($m_3$ in Fig.~\ref{fig:stationary-phenomology}), motor slip-links efficiently create nested loops. However, the degree of loop nesting $\theta$ is sensitive to $k_\mathrm{switch}$ (Fig.~\ref{fig:stationary-phenomology}, red). We observe a decline of $\theta$ with increasing $k_\mathrm{switch}$, although $\theta$ remains above 50\% even if the motor slip-link on average switches direction with each step ($k_\mathrm{switch}=k_\mathrm{motor}$). Since a high degree of loop nesting is necessary for establishing the juxtaposed state (SI~\ref{sec:Relationship-between-Xc-Xp-and-theta}), the rate of directional switching must remain sufficiently small for the motor slip-links to organize the chromosome.
Importantly, in the absence of a specific loading site, motor slip-links still extrude large loops and efficiently propagate along the polymer, as reflected in the higher contact probability away from the main diagonal (Fig.~\ref{fig:overview-stationary-results}e). However, there is no breaking of translational symmetry by a loading site, resulting in a leveled time-averaged contact map.
Interestingly, we observe that systems without a loading site can organize \textit{transiently} into a juxtaposed state (SI~\ref{sec:motors-without-ori} and SI movies\,7a--b), but the location of the fold diffuses randomly over the polymer.
\begin{figure}[h!]
\includegraphics[width=0.73\linewidth]{figures/3.stationary_phenomology/4_phase_diagrams_small.pdf}
\caption{
{\small Propagation length of slip-links and associated arm--arm juxtaposition \textbf{(a)} Scaled propagation length $\hat x_p \equiv \hat X_p/\tfrac{1}{2}N_m$ as a function of slip-link density $\phi_p$ for diffusive (blue) and motor (red) slip-links.
Indicated are also experimental measurements from~\cite{Minnen2016} for wild-type cells (``WT'', red star) and mutants whose SMC condensins have suppressed ATPase activity (``ATP-mutant'', blue star) (SI~\ref{sec:empirical-metrics}).
\textbf{(b)}: State diagram of scaled cross-diagonal length $\hat x_c = \hat X_c/\tfrac{1}{2}N_m$ for diffusive slip-links as a function of dissociation rate $k_-$ (in units of $k_0$) and $\phi_p$. \textbf{(c)}:\,State diagram of scaled cross-diagonal length $\hat x_c$ for motor slip-links as a function of the switching rate $k_{\rm switch}$ (in units of $k_\mathrm{motor}$) and $\phi_p$. Both (b--c) have $N_m=100$. Marker ``exp.'': is an estimate of WT behavior (Table \ref{TABLE}). Markers $m_1$ (SI movie\,10a) and $m_2$ (SI movie\,10b) are respectively at low and high $k_\mathrm{switch}$. Hatched area indicates values that we did not reach computationally.
We define $\hat x _c$ as the 75$^\mathrm{th}$ percentile of the cross-diagonal contacts (SI~\ref{sec:measurement-of-metrics-from-simulation-data}).}
\label{fig:stationary-phenomology-2}
}
\end{figure}
\subsection{Condensins need motorized movement to efficiently propagate into the bulk of the chromosome}
\textit{In vivo} experiments have demonstrated that condensins propagate far from their loading site~\cite{Minnen2016, Wilhelm2015}. To quantify the distribution of condensins on the chromosome in our model, we measure the extent $\hat X_p$ of slip-link propagation as the standard deviation of the binding profile $p_p(i)$ (Fig.~\ref{fig:overview-stationary-results}h). We eliminate the system-size dependence by considering the scaled propagation length $ \hat x _p = \hat X_p / \tfrac 1 2 N_m$ (SI~\ref{sec:universal-scaling-propagation-length}) as a function of slip-link density $\phi_p = 2 N_p / N_m$.
The scaled propagation length of diffusive slip-links only approaches the \textit{in vivo} value, $\hat x_p \approx 0.5$ (\cite{Minnen2016, Wilhelm2015} and SI~\ref{sec:empirical-metrics}), when we use slip-link densities~$\gtrsim\!\! 10\%$ in our simulations (Fig.~\ref{fig:stationary-phenomology-2}a, blue). Importantly, this slip-link density would correspond to thousands of condensins on the chromosome, 2--3 orders of magnitude more than reported \textit{in vivo}~\cite{Wilhelm2015}. This further illustrates that diffusive slip-links are not efficient at forcing the nested loops into the bulk of the polymer at low densities. In contrast, motor slip-links propagate over the full length of the polymer at all slip-link densities we considered (red data in Fig.~\ref{fig:stationary-phenomology-2}), as observed experimentally in cells (\cite{Minnen2016, Wilhelm2015} and SI~\ref{sec:empirical-metrics}).
Our results are summarized in a ``state diagram'' (Figs.~\ref{fig:stationary-phenomology-2}b--c), indicating the scaled extent $\hat x_c = \hat X_c/\tfrac 1 2 N_m$ of the cross-diagonal in contact maps.
\textit{Diffusive} slip-links require both fast dissociation kinetics as well as a high slip-link density to bring the polymer into the juxtaposed state (Fig.~\ref{fig:stationary-phenomology-2}b). For high densities of \textit{motor} slip-links $\phi_p \gtrsim 10\%$ and low $k_\mathrm{switch}$, the slip-links readily juxtapose the DNA polymer (e.g. $m_1$ in Fig.~\ref{fig:stationary-phenomology-2}c). Contrarily, for increasing $k_\mathrm{switch}$, motor slip-links antagonize each other's translocation (e.g. $m_2$ in Fig.~\ref{fig:stationary-phenomology-2}c), impeding the collective propagation of slip-links away from the loading site, thereby resulting in a reduced $\hat x_c$ (compare $m_1$, $m_2$ in Fig.~\ref{fig:stationary-phenomology-2}c; SI movies\,10a--b). In the limit $k_\mathrm{switch} \gg k_\mathrm{motor}$, motor slip-links effectively behave as diffusive slip-links with enhanced unbinding kinetics, placing them in the fast dissociation regime (data for $k_\mathrm{switch} \gtrsim 10 k_\mathrm{motor}$ in Fig.~\ref{fig:stationary-phenomology-2}c).
In contrast, for $\phi_p \lesssim 10\%$, motor slip-links only require $k_\mathrm{switch}$ to be sufficiently low. We estimate that wild-type condensin is indeed in this slow switching regime (``exp.'' in Fig.~\ref{fig:stationary-phenomology-2}c).
In sum, our simulations indicate that condensins at physiological densities can drive nested loops into the bulk of the polymer, crucial for establishing arm--arm alignment, only if they perform motorized, persistent motion.
\subsection{SMC condensin requires motor activity to rapidly re-organize the chromosome}
SMC induction experiments revealed that condensin can propagate from the loading site into the bulk of the DNA, thereby organizing an entire bacterial chromosome in a timespan of only $ T_\mathrm{WT} \approx 24 \unit{min}$ (SI~\ref{sec:empirical-metrics} and~\cite{Wang2017a}). Based on these experiments, we estimate that condensins translocate away from their loading site with a velocity of $\approx 300 \unit{nm/s}$. To understand the origin of these remarkably fast dynamics, we compute time-traces $X_p (t)$ of the width of the slip-link binding profile for our minimal models. From these traces, we extract a typical propagation time $T$ for slip-links to establish a steady-state binding profile on a DNA polymer of physical length $L = a N_m$ (Fig.~\ref{fig:dynamic-phenomenology}, inset), where $a\approx 50 \unit{nm}$ is the size of one monomer (Table~\ref{TABLE}).
The propagation time of diffusive slip-links scales strongly with DNA length: $T \sim L^x$, with $x\approx 2.5$ (Fig.~\ref{fig:dynamic-phenomenology}, blue). This scaling differs from simple diffusive motion ($x=2$, black in Fig.~\ref{fig:dynamic-phenomenology}), which we attribute to the loop-entropic forces that impede slip-link movement away from their loading site. Based on this observed scaling of $T$, we estimate that diffusive slip-links propagate several orders of magnitude slower over the DNA than observed in live cells (Fig.~\ref{fig:dynamic-phenomenology}, ``WT''). In contrast, the propagation time of motor slip-links exhibits ballistic scaling, $T = L/v$ (Fig.~\ref{fig:dynamic-phenomenology}, red), where $v$ is the effective translocation velocity.
Interestingly, our model prediction of motor slip-links, $T = v L $, is remarkably close to the observed propagation time \textit{in vivo} (Fig.~\ref{fig:dynamic-phenomenology}, ``WT''). Indeed, recent single-molecule experiments have revealed that yeast condensin can extrude DNA loops with a velocity of up to $425 \unit{nm/s}$~\cite{Ganji2018}. These data combined with our simulations, strongly indicate that rapid re-organization of the chromosome by SMC condensin requires fast and active loop extrusion.
\begin{figure}
\includegraphics[width=\linewidth]{figures/4.dynamic_phenomenology/ea086_time_scale_versus_system_size_real_units_using_velocity_square_small.pdf}
\caption{{\small Propagation dynamics of slip-links and comparison with wild-type cells. For diffusive slip-links (blue), we quantify the propagation time-scale $T$ by an exponential fit to the propagation length $X_p (t)$ (legend, top). For motor slip-links (red), we computed the average translocation velocity $v$ and define $T = L/v$ (legend, bottom). %
Diffusive slip-links exhibit sub-diffusive scaling $T \sim L^{2.5}$ (blue, dashed); diffusive scaling $T\sim L^2$ is shown for comparison (black, dashed). The extrapolation of the simulated data for motor slip-links (red) is in accord with \textit{in vivo} data (``WT'', see SI~\ref{sec:empirical-metrics}). Simulation units were converted to real units as described in SI~\ref{sec:empirical-metrics}.}
\label{fig:dynamic-phenomenology}
}
\end{figure}
\subsubsection{DNA relaxation dynamics can limit slip-link velocity}
What is the relationship between the slip-link translocation attempt rate $k_\mathrm{motor}$, the monomer diffusion attempt rate $k_0$, the polymer size $N_m$, and the effective motorized slip-link translocation velocity $v$? To answer this question, we distinguish two regimes: a \textit{fast} relaxation regime $k_\mathrm{motor} \ll k_0$ and a \textit{slow} relaxation regime $k_\mathrm{motor} \gg k_0$. In both regimes, for a stiff slip-link to make a step, the two polymer bonds in the direction of movement need to be parallel (Fig. S\ref{fig:illustration-counting-argument}a). If we denote the prior probability of observing these two polymer bonds to be parallel by $C \approx 3/16$ (Fig. S\ref{fig:illustration-counting-argument}b), then the effective rate of bond--bond alignment is $k_0^\mathrm{eff} \approx C k_0$. In our estimate for the rate $k_0^\mathrm{eff}$ we neglect the force that motor slip-links might exert on these bonds, because \emph{in vitro} experiments indicate that the stalling force of yeast condensin is very small \cite{Ganji2018}. Hence, the characteristic rate $k_\mathrm{motor}^\star$ that sets the transition from the fast relaxation to the slow relaxation regime occurs at $k_\mathrm{motor}^\star = k_0^\mathrm{eff}$. Consistent with the prediction that $k_\mathrm{motor}$ only depends on \textit{local} kinetics, our data shows that $k_\mathrm{motor}$ is independent of the system size (Fig. \ref{fig:motor-dynamics-combined}a).
In the fast relaxation regime, the rate-limiting factor is $k_\mathrm{motor}$. In this case, the velocity $v$ of slip-links is approximated by $v_\mathrm{fast}(k_\mathrm{motor}) \approx C \ell k_\mathrm{motor}$. In the slow relaxation regime, $k_\mathrm{motor} > k_\mathrm{motor}^\star$, polymer relaxation becomes the rate-limiting factor, so that $v_\mathrm{max} \approx v_\mathrm{fast}(k_\mathrm{motor}^\star) \approx C^2 \ell k_0 $.
This argument suggests that the scaling form of dimensionless variables $\tilde v \equiv v / (\ell k_0)$, $\tilde{k} \equiv k_\mathrm{motor} / k_0 $ will collapse the data of $k_\mathrm{motor}, v, k_0$ onto a universal curve $\tilde v = C \tilde{k}$ for $\tilde{k} < C$ and $\tilde v = C^2$ for $ \tilde{k} \geq C$. Indeed, our numerical data is well-described by this scaling form (Fig.\,S\ref{fig:motor-dynamics-combined}b). By combining \textit{in vitro} with \textit{in vivo} empirical data, we estimate that SMC condensin in \textit{B. subtilis} is well within the fast relaxation regime $k_\mathrm{motor} < 10^{-5} k_0 \ll k_0 $.
\begin{figure}
\centering
\includegraphics[width=8cm]{figures/4.dynamic_phenomenology/ea115_only_two_panels.pdf}
\caption{{\small DNA relaxation dynamics limits the maximum velocity of motor slip-links. \textbf{(a):} The average motor velocity (error bars: standard deviation) for various translocation attempt rates (turquoise to dark blue, $k_\mathrm{motor} = 100\ldots 0.01$) and various system sizes $N_m$. For clarity, we shifted the datapoints for decreasing $k_\mathrm{motor}$; the value of $N_m$ used is always the datapoint corresponding to $k_\mathrm{motor}=100$ (turquoise). \textbf{(b):} Rescaled motor velocity as a function of relative motor rate for the data shown in panel a). Propagation of wild-type condensins is within the ``fast relaxation'' regime ($\tilde{k}=6.8\times 10^{-6}$) using data from Table \ref{TABLE}.}}
\label{fig:motor-dynamics-combined}
\end{figure}
\section{Discussion}
Our computational framework reveals the basic physical requirements for condensins to collectively organize the bacterial chromosome as observed in live cells~\cite{Wang2015a, Wang2017a, Marbouty2015, Le2013}. In the presence of a specific loading site and with physiologically relevant numbers of condensins, we find that motor activity is required to robustly and rapidly generate a system-size spanning juxtaposition of the chromosomal arms. In contrast, purely diffusive condensins would require more kinetic fine-tuning and unphysiologically high copy numbers to organize the chromosome.
Our minimal model for the action of SMC condensin as a motor slip-link accounts for several key observations, including the rapid development of the juxtaposed state~\cite{Wang2017a} and the crucial role of the $\mathrm{ParB}S$\, nucleoprotein complex as a specific loading site~\cite{Wang2015a, Wang2017a, Gruber2009, Marbouty2015}. Without a well-defined loading site ($\Delta\mathrm{ParB}/S$\,), motor slip-links still transiently organize the polymer into a juxtaposed state. However, the chromosomal fold diffuses along the chromosome, resulting in a structureless time-averaged contact map with enhanced long-range contacts (Fig.~\ref{fig:overview-stationary-results}e and SI~\ref{sec:motors-without-ori}). These theoretically predicted behaviors may account for observations in \textit{E. coli}, where the action of the SMC complex MukBEF does not appear to involve an exclusive loading site~\cite{Lioy2018}. Hi-C maps of \textit{E. coli} do not display a cross-diagonal, but rather an elevated contact probability at large length-scales~\cite{Lioy2018}, in line with our simulations (Fig.~\ref{fig:overview-stationary-results}e and SI~\ref{sec:motors-without-ori}). Indeed, our model predicts that motor slips-links efficiently create nested loops ($m_3$ in Fig.~\ref{fig:stationary-phenomology}) even in the absence of a loading site, resulting in enhanced long-range contacts (Fig.~\ref{fig:overview-stationary-results}e and Fig.~S\ref{fig:contact-distribution-motor-wo-loading-site}). Although the function of such a change in polymer organization is unclear, local chromosome organization by SMC proteins has been linked to transcription regulation in various bacteria such as \textit{E. coli} and \textit{C. crescentus}~\cite{Lioy2018, Le2013}.
Experiments of SMC condensin propagation in \textit{B. subtilis} suggest that two condensin complexes might link together in a hand-cuff topology, with each of the two condensins in the dimer actively extruding a separate DNA duplex~\cite{Wang2017a, Tran2017}. In our model, motor slip-links extrude DNA in a symmetrical fashion, as expected for condensins in a hand-cuff configuration: both sides of a slip-link move over a separate DNA duplex with the same translocation rate. However, in recent \textit{in vitro} assays, single yeast condensin complexes actively extrude DNA loops asymmetrically: one end of the complex appears anchored at a DNA locus, while the opposite end actively translocates over DNA~\cite{Ganji2018}. Interestingly, contact maps of such asymmetric motor slip-links contain a star-shaped pattern around the loading site (SI~\ref{sec:asymmetric-loop-extrusion}), a feature that is also visible in Hi-C maps of \textit{B. subtilis}~\cite{Wang2015a, Marbouty2015}. This suggests that there is at least some fraction of condensins performing asymmetric translocation. An equally tantalizing explanation is that the movement of one of the condensins in a dimer is impeded by other DNA-bound factors, thereby forcing a condensin-dimer to propagate asymmetrically. Indeed, there is growing evidence that the movement of SMC complexes can be antagonized by oncoming transcription factors~\cite{Wang2015a, Tran2017, Wang2017a}.
It has also been suggested that the cylindrical geometry of many bacteria, combined with DNA--cell-pole tethering, facilitates chromosome organization~\cite{Buenemann2010, MathiasBuenemann2011}. However, our simulations indicate that the effect of confinement alone on chromosome organization is weak, and hence cannot be solely responsible for the juxtaposed organization observed in live cells (SI~\ref{section:cross-diagonal-from-cell-asymmetry}). This is in line with the observation that mutants lacking SMC condensin, but with the \textit{ori}-proximal loci still tethered to a cell-pole, also lack the cross-diagonal~\cite{Wang2017a}.
Our simulations further indicate that even purely diffusive slip-links with fast dissociation kinetics can induce arm-arm alignment, but at low slip-link densities this organized state remains localized near the loading site (SI~\ref{section:relaxatin-juxtaposed-organization}).
For a given number of condensins (3--30 per chromosome \cite{Wilhelm2015}), however, the slip-link density depends on DNA length. Therefore, we expect that a physiological number of non-motorized condensins can organize a mini-chromosome or plasmid of tens of microns in length. Indeed, plasmids are known to bind SMCs~\cite{Kumar2006} and contain $\mathrm{ParB}S$\, nucleoprotein complexes that could act as a condensin loading site~\cite{Funnell2016}.
Finally, our computational model can be used to unravel the function of juxtaposed organization in faithful chromosome segregation~\cite{Wang2013a,Wang2014, Hirano2016, Gruber2009, Gruber2014, Marbouty2015, Nolivos2013a}. More broadly, we provide a framework to elucidate the role of loop-extruding enzymes and ATPases~\cite{Brackley2018, Brackley2016, Hirano2016, Gruber2014a, Lioy2018} on chromosome organization.
\section*{Author Contributions}
C.A.M. and C.P.B. designed research; C.A.M. performed research; C.A.M. and C.P.B. analyzed data; C.A.M. and C.P.B. wrote the paper.
\section*{Data Accessibility}
Original data of contact maps, polymer configurations and slip-link dynamics displayed in this paper are available upon request in the form of HDF5 data files.
\section*{Funding Statement}
This work was supported by the German Excellence Initiative via the program ``NanoSystems Initiative Munich'' (NIM) and the Deutsche Forschungsgemeinschaft (DFG) Grants TRR174 and GRK2062/1.
\section*{Acknowledgments}
We kindly thank S. Gruber, D. Rudner, E. Frey, J. Messelink, T. Krueger, F. Mura, F. Gnesotto, L. van Buren, G. Dadushnavili, and A. Le Gall for their insights on this work and stimulating discussions.
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Apple Pursued BMW i3 For Its Own Car (Rumor)
July 24, 2015 Zachary Shahan 0 Comments Apple, Apple electric car, Apple self driving car, BMW, BMW i3
If one Apple electric car story this week wasn't enough for you, here's another. Rumor has it Apple wanted to use the BMW i3 (or at least part of it) as the base for its own electric car. Apple CEO Tim Cook (and other high-rollers at Apple) even visited BMW's Leipzig (Germany) factory where the BMW i3 is assembled.
Photo by Zachary Shahan | CleanTechnica | EV Obsession (CC BY-SA 4.0)
German magazine Manager (h/t Green Car Reports) says that talks have broken off regarding this possibility, but it's not clear why. Plus, let's be honest, this is all just a rumor for now. Though, it seems that Manager has a pretty good reputation — it ain't no Gawker or Telegraph.
Naturally, the BMW i3 has a few things going for it that Apple could have found appealing. It's the most efficient car on the US market. It utilizes lightweight, cutting-edge carbon fiber. It is super green, making use of recycled plastic and bamboo, among other things. And it has gotten fairly good reviews (full disclosure: that's my giddy review — click it!).
Furthermore, if it were used in taxi/chauffeur services, the BMW i3's suicide doors would be a handy way to ensure people don't jump into the backseat or jump out of it when not desired — come on, that is a legitimate benefit!
Since we're off in no-man's land with regards to speculation now, perhaps I'll wrap up this story.
I think the evidence is getting pretty darn clear that Apple has been working on an electric car of some sort. Whether it still is (I would think so, given this recent hire), and the various details of what would be so special about the car (self-driving? integrated with everything Apple? flying? for the 0.01%?), are still up in the air. Hopefully we'll have more information soon. And heck, maybe Apple could just tell us what's going on?!
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is the director of CleanTechnica, the most popular cleantech-focused website in the world, and Planetsave, a world-leading green and science news site. He has been covering green news of various sorts since 2008, and he has been especially focused on solar energy, electric vehicles, and wind energy since 2009. Aside from his work on CleanTechnica and Planetsave, he's the founder and director of Solar Love, EV Obsession, and Bikocity. To connect with Zach on some of your favorite social networks, go to ZacharyShahan.com and click on the relevant buttons.
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Q: Queries to Create a Basic Threshold Analysis Report in MS Access In my attempt to create a MS Access report for a basic threshold analysis I have created 3 queries that are sequential. The performance is awful given how inefficient they are and I am also not getting the intended results from the final query (explained below). Any suggestions to clean this up would be greatly appreciated.
The first query (Day_Period_Bins_qry1) runs as attended and sums my data into various day period 'bins', which are constructed working backwards from the date entered in a form (Main_frm!Date_prompt_txt). The query (below) is extremely obtuse, but it is what I got figured out
'Category' as N_group,
Data_tbl.Item_Category as Item,
Sum(IIf([Data_tbl].[occ_date]<=[Forms]![Main_frm]![Date_prompt_txt] And [Data_tbl].[occ_date]>=DateAdd("d",-6,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 7Day_Curnt,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-7,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-13,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 7Day_1,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-14,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-20,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 7Day_2,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-21,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-27,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 7Day_3,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-28,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-34,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 7Day_4,
Sum(IIf([Data_tbl].[occ_date]<=[Forms]![Main_frm]![Date_prompt_txt] And [Data_tbl].[occ_date]>=DateAdd("d",-27,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 28Day_Curnt,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-28,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-55,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 28Day_1,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-56,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-83,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 28Day_2,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-84,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-111,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 28Day_3,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-112,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-139,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 28Day_4,
Sum(IIf([Data_tbl].[occ_date]<=[Forms]![Main_frm]![Date_prompt_txt] And [Data_tbl].[occ_date]>=DateAdd("d",-83,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 84Day_Curnt,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-84,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-167,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 84Day_1,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-168,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-251,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 84Day_2,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-252,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-335,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 84Day_3,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-336,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-419,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 84Day_4,
Sum(IIf([Data_tbl].[occ_date]<=[Forms]![Main_frm]![Date_prompt_txt] And [Data_tbl].[occ_date]>=DateAdd("d",-363,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 364Day_Curnt,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-364,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-727,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 364Day_1,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-728,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-1091,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 364Day_2,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-1092,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-1455,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 364Day_3,
Sum(IIf([Data_tbl].[occ_date]<=DateAdd("d",-1456,[Forms]![Main_frm]![Date_prompt_txt]) And [Data_tbl].[occ_date]>=DateAdd("d",-1819,[Forms]![Main_frm]![Date_prompt_txt]),1,Null)) AS 364Day_4
FROM Data_tbl
GROUP BY Data_tbl.Item_Category
There are two other queries union-ed to the above query that query item_type and item_category.
The second query (Columns_qry2) puts the data from the above query into columns in order for the third column to perform some calculations
Select
N_group,
Item,
nz([7Day_Curnt],0) as nCount,
'7Day' as Day_Period,
'Current' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
Select
N_group,
Item,
Nz( [7Day_1],0) as nCount,
'7Day' as Day_Period,
'1' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
SELECT N_group,
Item,
Nz( [7Day_2],0) as nCount,
'7Day' as Day_Period,
'2' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
SELECT
N_group,
Item,
Nz( [7Day_3],0) as nCount,
'7Day' as Day_Period,
'3' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
SELECT N_group,
Item,
Nz( [7Day_4],0) as nCount,
'7Day' as Day_Period,
'4' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
Select
N_group,
Item,
nz([28Day_Curnt],0) as nCount,
'28Day' as Day_Period,
'Current' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
SELECT
N_group,
Item,
Nz( [28Day_1],0) as nCount,
'28Day' as Day_Period,
'1' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
SELECT
N_group, Item,
Nz( [28Day_2],0) as nCount,
'28Day' as Day_Period,
'2' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
SELECT
N_group, Item,
Nz( [28Day_3],0) as nCount,
'28Day' as Day_Period,
'3' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
SELECT
N_group, Item,
Nz( [28Day_4],0) as nCount,
'28Day' as Day_Period,
'4' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
Select
N_group,
Item,
nz([84Day_Curnt],0) as nCount,
'84Day' as Day_Period,
'Current' as Day_bin
FROM Day_Period_Bins_qry1
Union All
SELECT
N_group,
Item,
Nz( [84Day_1],0) as nCount,
'84Day' as Day_Period,
'1' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
SELECT N_group,
Item,
Nz( [84Day_2],0) as nCount,
'84Day' as Day_Period,
'2' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
SELECT
N_group, Item,
Nz( [84Day_3],0) as nCount,
'84Day' as Day_Period,
'3' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
SELECT N_group,
Item, Nz( [84Day_4],0) as nCount,
'84Day' as Day_Period,
'4' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
Select
N_group,
Item,
nz([364Day_Curnt],0) as nCount,
'364Day' as Day_Period,
'Current' as Day_bin
FROM Day_Period_Bins_qry1
Union All
SELECT
N_group,
Item,
Nz( [364Day_1],0) as nCount,
'364Day' as Day_Period,
'1' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
SELECT N_group, Item,
Nz( [364Day_2],0) as Count,
'364Day' as Day_Period,
'2' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL
SELECT
N_group, Item,
Nz( [364Day_3],0) as nCount,
'364Day' as Day_Period,
'3' as Day_bin
FROM Day_Period_Bins_qry1
UNION ALL SELECT N_group, Item,
Nz( [364Day_4],0) as nCount,
'364Day' as Day_Period,
'4' as Day_bin
FROM Day_Period_Bins_qry1;
The last query is unfinished and in its third or fourth iteration. Currently, I am having problems figuring out how to get the columns to only perform calculations within the same day_period (i.e. calculate only the average for [7day] bins to compare against the current 7day count. Previously, a simple where clause was added (where day_bin = '7day) and repeated in union queries for the other day periods, which would have given me the results I want, but adding one more union query was the log that broke the camels back, causing a crash.
SELECT
Columns_qry2.N_group AS N_group,
Columns_qry2.item AS Item,
Columns_qry2.Day_period,
Columns_qry2.nCount AS Current_Ct,
Round(Avg(Columns_qry2.[nCount]),2) AS Avg_Prev4_Bins,
Round((Day_Period_bins_qry1.[7Day_Curnt]-[Avg_Prev4_Bins])/StDevP([nCount]),2) AS Z_score,
Round((Day_Period_bins_qry1.[7Day_Curnt]-Avg([nCount]))/Avg([nCount]),2) AS PercentChg_From_BinAvg_Prev4Bins,
Round(Day_Period_Bins_qry1.[7Day_Curnt]/(Select SUM(Day_Period_Bins_qry1.[7Day_Curnt]) FROM Day_Period_Bins_qry1),2) AS Percent_of_Total_Current,
Round(Avg(Day_Period_bins_qry1.[7Day_1]/(Select SUM(Day_Period_bins_qry1.[7Day_1]) FROM Day_Period_bins_qry1)+Day_Period_bins_qry1.[7Day_2]/(Select SUM(Day_Period_bins_qry1.[7Day_2]) FROM Day_Period_bins_qry1)+Day_Period_bins_qry1.[7Day_3]/(Select SUM(Day_Period_bins_qry1.[7Day_3]) FROM Day_Period_bins_qry1)+Day_Period_bins_qry1.[7Day_4]/(Select SUM(Day_Period_bins_qry1.[7Day_4]) FROM Day_Period_bins_qry1))/4,2) AS Avg_Percent_of_Tot_prev4_Bins,
Round(([Percent_of_Total_Current]-Avg_Percent_of_Tot_prev4_Bins)/Avg_Percent_of_Tot_prev4_Bins,4) AS PercentChg_Percent_of_Total
FROM Columns_qry2
INNER JOIN Day_Period_bins_qry1
ON Day_Period_bins_qry1.item = Columns_qry2.Item
GROUP BY Columns_qry2.N_group, Columns_qry2.item, Columns_qry2.Day_period, Columns_qry2.nCount, Day_Period_bins_qry1.[7Day_Curnt];
A: While not a complete answer but one that needs testing and adjustment on your end, consider below re-formulation of your process. Specifically, consider keeping your data long by joining to a static lookup table of your needed DateAdd values, then join with date calculation, and aggregate with Day_bin added to grouping.
Table (myDateRangeTable)
Start_Num End_Num Day_Period Day_Bin
0 -6 '7Day' 'Curnt'
-7 -13 '7Day' '1'
-14 -20 '7Day' '2'
-21 -27 '7Day' '3'
-28 -34 '7Day' '4'
0 -27 '28Day' 'Curnt'
-28 -55 '28Day' '1'
-56 -83 '28Day' '2'
-84 -111 '28Day' '3'
-112 -139 '28Day' '4'
0 -83 '84Day' 'Curnt'
-84 -167 '84Day' '1'
-168 -251 '84Day' '2'
-252 -335 '84Day' '3'
-336 -419 '84Day' '4'
0 -363 '364Day' 'Curnt'
-364 -727 '364Day' '1'
-728 -1091 '364Day' '2'
-1092 -1455 '364Day' '3'
-1456 -1819 '364Day' '4'
Base Query 1
Below cross joins above with original data table to pair match every combination of data table row with above selections. Then, counts any matches of date range. If original table is very large below may take some time to process. Consider making it a temp table if that is the case by adding an INTO [myTempTable] just before FROM. Below should replicate your union query.
SELECT 'Category' as N_group,
d.Item_Category as Item,
m.Day_Period,
m.Day_Bin,
SUM(d.[occ_date] BETWEEN DateAdd("d", m.Start_Num, [Forms]![Main_frm]![Date_prompt_txt])
AND DateAdd("d", m.End_Num, [Forms]![Main_frm]![Date_prompt_txt])
) AS Date_Count
FROM Data_tbl d,
myDateRangeTable m
GROUP BY d.Item_Category,
m.Day_Period,
m.Day_Bin;
Final Query (myFinalQuery)
Below is an incomplete query, combining unit level and aggregate levels by Day_Period. Consider adjusting for needed formulas. The remaining challenges are to calculate fields across rows for different Day_Bin counts. This can require a nested IIF expression or join to a crosstab. Note parentheses are required with multiple joins in MS Access' SQL dialect.
SELECT
b.N_Group,
b.Item,
b.Day_Period,
b.Day_Bin,
Round((b.[Day_Count] - agg1.[Avg_Prev4_Bins]) / agg1.Std_Prev4_Bins, 2)
AS Z_score,
Round((b.[Day_Count] - agg1.[Avg_Prev4_Bins])) / b.[Day_Count], 2)
AS PercentChg_From_BinAvg_Prev4Bins,
Round(IIF(b.Day_Bin = 'Current', b.[Day_Count] / agg2.curr_SumDayCount,
IIF(b.Day_Bin = '1', b.[Day_Count] / agg2.d1_SumDayCount,
IIF(b.Day_Bin = '2', b.[Day_Count] / agg2.d2_SumDayCount,
IIF(b.Day_Bin = '3', b.[Day_Count] / agg2.d3_SumDayCount,
IIF(b.Day_Bin = '4', b.[Day_Count] / agg2.d4_SumDayCount, NULL)
)
)
)
),
2) AS Percent_of_Total,
...
FROM
((myBaseQuery AS b
INNER JOIN
(SELECT Day_Period,
AVG(Day_Count) AS Avg_Prev4_Bins,
StDevP([Day_Count]) AS Std_Prev4_Bins
FROM myBaseQuery
GROUP BY Day_Period) AS agg1
ON agg1.Day_Period = b.Day_Period)
INNER JOIN
(SELECT Day_Period,
SUM(IIF(Day_Bin = 'Current', Day_Count, NULL)) AS curr_SumDayCount,
SUM(IIF(Day_Bin = '1', Day_Count, NULL)) AS d1_SumDayCount,
SUM(IIF(Day_Bin = '2', Day_Count, NULL)) AS d2_SumDayCount,
SUM(IIF(Day_Bin = '3', Day_Count, NULL)) AS d3_SumDayCount,
SUM(IIF(Day_Bin = '4', Day_Count, NULL)) AS d4_SumDayCount
FROM myBaseQuery
GROUP BY Day_Period) AS agg2
ON agg2.Day_Period = b.Day_Period)
Crosstab Query
Below is example of crosstab to be used as needed for unit level counts on same row to be joined to above and then have pivoted columns included in formulas with aggregate counterparts.
TRANSFORM SUM(Day_Count) AS SumDayCount
SELECT Day_Period
FROM myBaseQuery
GROUP BY Day_Period
PIVOT Day_Bin IN ([Curnt], [1], [2], [3], [4])
Hopefully some part of above is helpful!
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 8,773 |
\section{Introduction and the proof}
In recent years, there is a growing interest in applying information theoretic arguments to combinatorics and theoretical computer science. For example, Fox's new proof \cite{fox2011new} of the graph removal lemma, Tao's solution \cite{tao2016erdHos,tao2016logarithmically} of the Erd{\"o}s discrepancy problem, and the application of information theory to communication complexity by Braverman et al. \cite{braverman2013information}. For more discussion, see the surveys \cite{entropyCount1,entropyCount2,wolf2017some,braverman2014interactive}. The purpose of this note is to give a short information theoretic proof of Chang's lemma, an important result and tool in both additive combinatorics and theoretical computer science.
For every function $f: \{-1, 1\}^n \to \mathbb{R}$, and every $S \subseteq [n]$, define $\widehat{f}(S) = \Ex_{x\sim\{-1,1\}^n} f(x) \chi_S(x)$, where $\chi_S(x) = \prod_{i \in S} x_i$. The numbers $\widehat{f}(S)$ are called \textit{Fourier coefficients} of $f$, and $f(x) = \sum_{S \subseteq [n]} \widehat{f}(S) \chi_S(x)$ is called the \textit{Fourier expansion} of $f$.
Given these definitions, Chang's lemma states the following.
\begin{theorem}[Chang's lemma, \cite{chang2002polynomial}] \label{lem:chang}
Let $A \subseteq \{-1, 1\}^n$ have density $\alpha = \frac{|A|}{2^n}$. Let $f =1_A$ denote the characteristic function of $A$, that is $f(x) = 1$ if $x \in A$, and $f(x) = 0$ if $x \not\in A$. Then,
\[
\sum_{i=1}^n \widehat{f}(\{i\})^2 \le 2 \alpha^2 \ln \frac{1}{\alpha}.
\]
\end{theorem}
The original Chang's lemma \cite{chang2002polynomial} is stated for more general groups than $\{-1, 1\}^n$, and its proof relies on Rudin's inequality. Our proof is inspired by \cite{impagliazzo2014entropic}, where a proof for $\{-1, 1\}^n$ is given using entropy and Taylor expansion. However, our proof is shorter and more direct by replacing the Taylor expansion with Pinsker's inequality.
Let $p$ and $q$ be two probability distributions on a finite space $\Omega$. The Shannon entropy of $p$ is defined as $\ent(p) = - \sum_{x \in \Omega} p(x) \ln p(x)$.
The Kullback-Leibler divergence from $q$ to $p$ is defined as
$D(p||q) = \sum_{x \in \Omega} p(x) \ln \frac{p(x)}{q(x)}$, assuming $p(x) = 0$ whenever $q(x) = 0$. Observe that $D(p||q) = \ent(q) - \ent(p)$ if $q$ is the uniform distribution. Let $\norm{\cdot}_1$ denote the $L_1$ norm: $\norm{g}_1 = \sum_{x} |g(x)|$. Pinsker's inequality states $D(p||q) \ge \frac{1}{2} \norm{p-q}_1^2$.
\begin{proof}[Proof of Theorem \ref{lem:chang}]
Let $p$ be the uniform distribution on the set $A$, and $q$ be the uniform distribution on $\{-1, 1\}^n$. For every $i \in [n]$, denote the corresponding marginal distribution $p_i$ of $p$ as the pair $p_i =(\alpha_i, 1-\alpha_i)$ where $\alpha_i = \Pr[x_i=1 | x \in A]$. The marginal distributions of $q$ are $q_i = (1/2, 1/2)$, i.e., they are uniform distributions on $\{-1,1\}$. As the marginals of $q$ are independent, we have $\ent(q) = \sum_{i=1}^n \ent(q_i)$. Observe
\begin{align} \label{eq:eq1}
\begin{split}
\widehat{f}(\{i\})^2
&=\left(\Ex_{x} f(x) x_i \right)^2 = \alpha^2 (\alpha_i -(1-\alpha_i))^2 \\
&= \alpha^2 \left(\left|\alpha_i -\frac{1}{2}\right|+ \left|1-\alpha_i -\frac{1}{2}\right|\right)^2=\alpha^2 \norm{p_i - q_i}_1^2.
\end{split}
\end{align}
By the subadditivity of Shannon entropy and Pinsker's inequality,
\begin{align} \label{eq:eq2}
\begin{split}
\ln \frac{1}{\alpha}
&= D(p||q) = \ent(q) - \ent(p) \\
&\ge \sum_{i=1}^n \Big(\ent(q_i) - \ent(p_i) \Big)
= \sum_{i=1}^n D(p_i || q_i) \ge \frac{1}{2}\sum_{i=1}^n \norm{p_i - q_i}_1^2.
\end{split}
\end{align}
Combining \eqref{eq:eq1} and \eqref{eq:eq2} gives the desired bound.
\end{proof}
\section{Concluding remarks}
Firstly, let $W^k = \sum_{|S| = k} \widehat{f}(S)^2$. In the analysis of boolean functions, Chang's Lemma is also called as the \emph{level-$1$ inequality} (see \cite{o2014analysis}), since it gives an upper bound for $W^1$. There is a generalization of Chang's lemma that states $\sum_{|S| \le k} \widehat{f}(S)^2 \le (\frac{2e}{k} \ln (1/\alpha))^k \alpha^2$ whenever $k \le 2 \ln(1/\alpha)$. This is called the \emph{level-$k$ inequality} in \cite{o2014analysis}. \emph{Can our argument be generalized to give a simple proof of the level-$k$ inequality?} On the one hand, the level-$k$ inequality \cite{o2014analysis} can be derived from hypercontractivity which adopts some entropic proofs (see \cite{hyp1,hyp2,hyp3}). This indicates some hope. On the other hand, the level-$k$ inequality only holds for sets $A$ with small density depending on $k$ for every $k \ge 2$. However, it is unclear how this constraint on the density can appear in an informational argument. For example, Pinsker's inequality does not have any constraint.
Secondly, we show that the inequality $D(p||q) \ge \sum_{i=1}^n D(p_i||q_i)$ that appears in \eqref{eq:eq2} can be generalized to the case whenever $q$ is a product distribution. Let $\Omega = \Omega_1 \times \Omega_2 \times \cdots \times \Omega_n$ be a finite product space. Let $p$ and $q$ be two probability distributions on $\Omega$, and let $p_i$ and $q_i$ denote the marginal distribution of $p$ and $q$ on $\Omega_i$, respectively.
\begin{lemma}[supadditivity] \label{lem:divergence}
If $q$ is a product distribution, then
\[
D(p || q) \ge \sum_{i=1}^n D(p_i || q_i).
\]
\end{lemma}
\begin{proof}
By induction, it suffices to prove it for $n=2$. Now suppose $n=2$. For notational clarity, denote $p$ by $p(X,Y)$ where $(X,Y) \in \Omega_1 \times \Omega_2$, and similarly for $q$. By the chain rule of divergence $D(p(X,Y)||q(X,Y)) = D(p(X)||q(X)) + D(p(Y|X) || q(Y|X))$. Hence, it suffices to show that $D(p(Y|X) || q(Y|X)) \ge D(p(Y)||q(Y))$. By the definition of divergence, one has
\begin{align*}
& D(p(Y|X) || q(Y|X)) - D(p(Y)||q(Y)) \\
&= \sum_{(x,y) \in \Omega_1 \times \Omega_2} p(x,y) \ln \frac{p(y|x)}{q(y|x)}
- \sum_{y \in \Omega_2} p(y)\ln\frac{p(y)}{q(y)} \\
&= \sum_{(x,y) \in \Omega_1 \times \Omega_2} p(x,y) \ln \frac{p(y|x) q(y)}{q(y|x) p(y)} \\
&= \sum_{(x,y) \in \Omega_1 \times \Omega_2} p(x,y) \ln \frac{p(y|x)}{p(y)}
= \sum_{(x,y) \in \Omega_1 \times \Omega_2} p(x,y) \ln \frac{p(x,y)}{p(x)p(y)} \ge 0,
\end{align*}
where the last step follows from the log sum inequality.
\end{proof}
One can apply Lemma \ref{lem:divergence} directly in \eqref{eq:eq2} without using Shannon entropy. We point out that the supadditivity of the Kullback-Leibler divergence in Lemma \ref{lem:divergence} is not necessarily true if $q$ is not a product distribution. Let $p(X,Y), q(X,Y)$ be two distributions given by
\[
p =
\begin{pmatrix}
1/4 & 1/4 \\
1/4 & 1/4
\end{pmatrix},
\quad
q =
\begin{pmatrix}
1/4-3\epsilon & 1/4+\epsilon \\
1/4+\epsilon & 1/4+\epsilon
\end{pmatrix},
\]
where $-1/4 < \epsilon < 1/12$. In particular, $p$ is a product distribution. We will choose $\epsilon$ such that $q$ is not a product distribution. The marginal distributions are: $p(X) =(1/2, 1/2)$, $p(Y) = (1/2, 1/2)$, and $q(X) =(1/2-2\epsilon, 1/2+2\epsilon)$, $q(Y) = (1/2-2\epsilon, 1/2+2\epsilon)$. Let $\epsilon = 0.01$, using Wolfram Mathematica,
\[
D(p(X,Y) || q(X,Y)) \approx 0.0025 > D(p(X) || q(X)) + D(p(Y) || q(Y)) \approx 0.0016.
\]
Let $\epsilon = -0.2$, using Wolfram Mathematica,
\[
D(p(X,Y) || q(X,Y)) \approx 0.90 < D(p(X) || q(X)) + D(p(Y) || q(Y)) \approx 1.02.
\]
\section*{Acknowledgements}
Both authors wish to thank Hamed Hatami for stimulating and helpful discussions. We thank the anonymous referees for their comments which improved the presentation. The authors are supported by an NSERC funding.
| {
"redpajama_set_name": "RedPajamaArXiv"
} | 7,336 |
Woll ist der Familienname folgender Personen:
Alexander Woll (* 1963), deutscher Sportwissenschaftler und Hochschullehrer
Artur Woll (1923–2020), deutscher Wirtschaftswissenschaftler
Balthasar Woll (1922–1996), deutsches Mitglied der Waffen-SS
Bencie Woll (* 1950), britisch-US-amerikanische Soziolinguistin
Deborah Ann Woll (* 1985), US-amerikanische Schauspielerin
Dieter Woll (1933–2012), deutscher Romanist
Erna Woll (1917–2005), deutsche Komponistin und Kirchenmusikerin
Felicitas Woll (* 1980), deutsche Schauspielerin
Helmut Woll (* 1950), deutscher Wirtschaftspädagoge
Irene Woll-Schumacher (* 1943), deutsche Soziologin und Hochschullehrerin
Katharina Woll (* 1984), deutsche Filmregisseurin und Drehbuchautorin
Karl August Woll (1834–1893), deutscher Lyriker
Matthew Woll (1880–1956), US-amerikanischer Gewerkschafter
Woll ist der deutsche Name von:
La Bresse, Gemeinde im Département Vosges, Frankreich
Siehe auch:
Wol
Wöll | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,161 |
Q: Fix SSH (client and server) on Mac OS X Mountain Lion I have upgraded my Mac to Mountain Lion. Now that its complete I cannot SSH into any server for more than a few seconds, and I cannot receive any SSH connections.
I can only assume something from the upgrade has caused the issue. What can I do to diagnose the error and fix it? I would like to connect to other systems using the user name jjasonclark and I would like others to connect to my system with user name remotepair.
I don't have any logs about ssh in /var/log/system.log. I don't have a file called /var/log/secure.log. And greping through syslog -d /var/log/asl for ssh shows only those times I edited the /etc/sshd_config file.
When others try to connect to me (with the -vvv option) their client will hang on this message.
debug1: expecting SSH2_MSG_KEX_DH_GEX_GROUP
Example connection from my machine to others ssh -vvv jjasonclark.analoganalytics.com
OpenSSH_5.9p1, OpenSSL 0.9.8r 8 Feb 2011
debug1: Reading configuration data /usr/local/Cellar/openssh/5.9p1/etc/ssh_config
debug2: ssh_connect: needpriv 0
debug1: Connecting to jjasonclark.analoganalytics.com [184.106.115.234] port 22.
debug1: Connection established.
debug3: Incorrect RSA1 identifier
debug3: Could not load "/Users/jjasonclark/.ssh/id_rsa" as a RSA1 public key
debug1: identity file /Users/jjasonclark/.ssh/id_rsa type 1
debug1: identity file /Users/jjasonclark/.ssh/id_rsa-cert type -1
debug1: identity file /Users/jjasonclark/.ssh/id_dsa type -1
debug1: identity file /Users/jjasonclark/.ssh/id_dsa-cert type -1
debug1: identity file /Users/jjasonclark/.ssh/id_ecdsa type -1
debug1: identity file /Users/jjasonclark/.ssh/id_ecdsa-cert type -1
debug1: Remote protocol version 2.0, remote software version OpenSSH_4.3
debug1: match: OpenSSH_4.3 pat OpenSSH_4*
debug1: Enabling compatibility mode for protocol 2.0
debug1: Local version string SSH-2.0-OpenSSH_5.9
debug2: fd 3 setting O_NONBLOCK
debug3: load_hostkeys: loading entries for host "jjasonclark.analoganalytics.com" from file "/Users/jjasonclark/.ssh/known_hosts"
debug3: load_hostkeys: found key type RSA in file /Users/jjasonclark/.ssh/known_hosts:37
debug3: load_hostkeys: found key type RSA in file /Users/jjasonclark/.ssh/known_hosts:38
debug3: load_hostkeys: loaded 2 keys
debug3: order_hostkeyalgs: prefer hostkeyalgs: ssh-rsa-cert-v01@openssh.com,ssh-rsa-cert-v00@openssh.com,ssh-rsa
debug1: SSH2_MSG_KEXINIT sent
debug1: SSH2_MSG_KEXINIT received
debug2: kex_parse_kexinit: ecdh-sha2-nistp256,ecdh-sha2-nistp384,ecdh-sha2-nistp521,diffie-hellman-group-exchange-sha256,diffie-hellman-group-exchange-sha1,diffie-hellman-group14-sha1,diffie-hellman-group1-sha1
debug2: kex_parse_kexinit: ssh-rsa-cert-v01@openssh.com,ssh-rsa-cert-v00@openssh.com,ssh-rsa,ecdsa-sha2-nistp256-cert-v01@openssh.com,ecdsa-sha2-nistp384-cert-v01@openssh.com,ecdsa-sha2-nistp521-cert-v01@openssh.com,ssh-dss-cert-v01@openssh.com,ssh-dss-cert-v00@openssh.com,ecdsa-sha2-nistp256,ecdsa-sha2-nistp384,ecdsa-sha2-nistp521,ssh-dss
debug2: kex_parse_kexinit: aes128-ctr,aes192-ctr,aes256-ctr,arcfour256,arcfour128,aes128-cbc,3des-cbc,blowfish-cbc,cast128-cbc,aes192-cbc,aes256-cbc,arcfour,rijndael-cbc@lysator.liu.se
debug2: kex_parse_kexinit: aes128-ctr,aes192-ctr,aes256-ctr,arcfour256,arcfour128,aes128-cbc,3des-cbc,blowfish-cbc,cast128-cbc,aes192-cbc,aes256-cbc,arcfour,rijndael-cbc@lysator.liu.se
debug2: kex_parse_kexinit: hmac-md5,hmac-sha1,umac-64@openssh.com,hmac-sha2-256,hmac-sha2-256-96,hmac-sha2-512,hmac-sha2-512-96,hmac-ripemd160,hmac-ripemd160@openssh.com,hmac-sha1-96,hmac-md5-96
debug2: kex_parse_kexinit: hmac-md5,hmac-sha1,umac-64@openssh.com,hmac-sha2-256,hmac-sha2-256-96,hmac-sha2-512,hmac-sha2-512-96,hmac-ripemd160,hmac-ripemd160@openssh.com,hmac-sha1-96,hmac-md5-96
debug2: kex_parse_kexinit: none,zlib@openssh.com,zlib
debug2: kex_parse_kexinit: none,zlib@openssh.com,zlib
debug2: kex_parse_kexinit:
debug2: kex_parse_kexinit:
debug2: kex_parse_kexinit: first_kex_follows 0
debug2: kex_parse_kexinit: reserved 0
debug2: kex_parse_kexinit: diffie-hellman-group-exchange-sha1,diffie-hellman-group14-sha1,diffie-hellman-group1-sha1
debug2: kex_parse_kexinit: ssh-rsa,ssh-dss
debug2: kex_parse_kexinit: aes128-ctr,aes192-ctr,aes256-ctr,arcfour256,arcfour128,aes128-cbc,3des-cbc,blowfish-cbc,cast128-cbc,aes192-cbc,aes256-cbc,arcfour,rijndael-cbc@lysator.liu.se
debug2: kex_parse_kexinit: aes128-ctr,aes192-ctr,aes256-ctr,arcfour256,arcfour128,aes128-cbc,3des-cbc,blowfish-cbc,cast128-cbc,aes192-cbc,aes256-cbc,arcfour,rijndael-cbc@lysator.liu.se
debug2: kex_parse_kexinit: hmac-md5,hmac-sha1,hmac-ripemd160,hmac-ripemd160@openssh.com,hmac-sha1-96,hmac-md5-96
debug2: kex_parse_kexinit: hmac-md5,hmac-sha1,hmac-ripemd160,hmac-ripemd160@openssh.com,hmac-sha1-96,hmac-md5-96
debug2: kex_parse_kexinit: none,zlib@openssh.com
debug2: kex_parse_kexinit: none,zlib@openssh.com
debug2: kex_parse_kexinit:
debug2: kex_parse_kexinit:
debug2: kex_parse_kexinit: first_kex_follows 0
debug2: kex_parse_kexinit: reserved 0
debug2: mac_setup: found hmac-md5
debug1: kex: server->client aes128-ctr hmac-md5 none
debug2: mac_setup: found hmac-md5
debug1: kex: client->server aes128-ctr hmac-md5 none
debug1: SSH2_MSG_KEX_DH_GEX_REQUEST(1024<1024<8192) sent
debug1: expecting SSH2_MSG_KEX_DH_GEX_GROUP
My /etc/syslog.conf
# Note that flat file logs are now configured in /etc/asl.conf
install.* @127.0.0.1:32376
My /etc/sshd_config
Protocol 2
SyslogFacility AUTHPRIV
LogLevel INFO
PermitRootLogin no
PubkeyAuthentication yes
PasswordAuthentication no
PermitEmptyPasswords no
ChallengeResponseAuthentication no
ClientAliveInterval 120
ClientAliveCountMax 3
AcceptEnv LANG LC_*
AllowUsers jjasonclark remotepair
My /etc/ssh_config
Host *
SendEnv LANG LC_*
ServerAliveInterval 60
A: Solved!
It was a weird one to solve this issue. It turned out that I needed to reset my PRAM. It must have stored some details that needed to change with the upgrade.
Apple has a article on how to reset the PRAM. You reboot and hold cmd+opt+p+r until you hear the a 2nd time.
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 5,180 |
World United States Pennsylvania Unionville
Read and rate Travel Journal Entries for Unionville, Pennsylvania, United States
Sep 2, 2009 - Valley Forge/Hopewell Furnace
Sept. 2--We drove the Honda about 40 miles to Valley Forge National Historical Park. This was the winter encampment for the Continental Army from December 1777 until May 1778. Upon arrival, the soldiers had to build their own huts. There was plenty of hardwood around, and they were given specific measurements and drawings. About 2000 huts were built, laid out in parallel lines along military avenues. The troops also constructed miles of trenches, five earthen forts called readouts, and a bridge over the Schuylkill River. Disease was the...
Where is the Bus? 2009
Sep 1, 2009 - Philadelphia, day 3
Sept. 1--We spent another full day in Philadelphia checking out the Old City. We took a Duck tour. These ducks were made for the military in WWII as a landing craft. The tour lasted about 70 minutes and drove us around the old city, and then into the Delaware River. We ate lunch at the City Tavern. The original building was torn down some time ago, and a new one was built in the 1980's I think. Drawings were used to make it as authentic as possible, and it is on the original location. The meals are what you would find in a tavern of that...
Aug 31, 2009 - Philadelphia, day 2
Aug. 31--We walked around what is known as Independence National Historic Park. This is the birthplace of our nation. We saw the Liberty Bell, Independence Hall (which became the name after the First Continental Congress met), Ben Franklin's home, Congress Hall, Old City Hall, etc. England ruled over the colonies, and it taxed everything. The colonists were over here working hard, clearing land, farming, etc, and had no say in what was happening to them. Taxation without representation. That was the focus for independence. Now don't think...
The bus is in Unionville, PA parked in the KOA ========================================================================= Drivers Log: 138 miles this leg, 7592 miles total. How we got here -- See Google Map "Drive to Unionville, PA" A to B 138 miles Stamps in New York 10, 45 stamps total ========================================================================= Saturday night there was a fireworks display on the Hudson River, just in front of the RV Park. It was nice of them to give us a sendoff. The drive to Unionville was easy. Two tolls:... | {
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Q: create a file with 2 columns and each from various file how can i get the following output using perl or awk or is it possible in linux command?
file1:
1
2
2
4
file2:
5
6
7
8
Desired output:
1 5
2 6
2 7
4 8
A: with command paste
paste file1 file2
A: A more flexible solution for when the lines in the file aren't fixed width would be with pr:
$ pr -mtw 10 file1 file2
1 5
2 6
3 7
4 8
Changing file1 to containing variable width lines:
$ cat file1
The number 1
two
3
The last number is the number four
# With pr two columns are output
$ pr -mt file1 file2
The number 1 5
two 6
3 7
The last number is the number four 8
# Paste simply inserts a tab which doesn't format the output in two columns
$ paste file1 file2
The number 1 5
two 6
3 7
The last number is the number four 8
A: To be more precise:
paste command just prints it out. To save it to a new file the command should be:
paste file1 file2 > outputFile
outputFile now contains both columns.
A: You can also do this with GNU parallel:
parallel --xapply echo '{1}' '{2}' :::: file1 :::: file2
| {
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Grand Piano est un thriller espagnol d'Eugenio Mira sorti en 2013.
Synopsis
, à Chicago. Tom Selznick, un pianiste de renom qui s'était retiré de la scène depuis 5 ans en raison de son trac paralysant, revient pour un concert unique à l'occasion du premier anniversaire de la mort de son mentor Patrick Godureaux. Lorsqu'il s'apprête à jouer sur le piano de son mentor (un Bösendorfer Imperial), il voit une menace de mort écrite sur sa partition. Des instructions l'ordonnent de récupérer une oreillette cachée dans sa loge durant une pause. Pouvant communiquer discrètement de vive voix, Selznick entend un psychopathe lui expliquer que, au cas où il ne donnerait pas la meilleure prestation de sa vie, sa femme Emma, une célèbre actrice présente dans le public, serait assassinée.
Par la suite, Selznick apprend que cet homme, un serrurier, cherche en fait à récupérer une clef, cachée dans le piano, qui permet d'ouvrir un coffre contenant la fortune de Patrick Godureaux. Pour ce faire, Selznick doit réussir à jouer parfaitement un morceau d'une grande difficulté, composé par Godureaux : La Cinquette…
Fiche technique
Titre original : Grand Piano
Réalisation : Eugenio Mira
Scénario : Damien Chazelle
Direction artistique : Javier Alvariño
Décors : Jaime Anduiza
Costumes : Patricia Monné
Montage : José Luís Romeu
Musique : Víctor Reyes
Photographie : Unax Mendía
Son : Albert Manera
Production : Rodrigo Cortés et Adrián Guerra
Sociétés de production : Antena 3, Nostromo Pictures et Telefonica Producciones
Pays d'origine :
Langue : Anglais
Durée : 90 minutes
Format : Couleurs - 35 mm - 1.85:1 - Son Dolby numérique
Genre : Thriller
Dates de sortie :
:
: (directement en vidéo)
Distribution
Elijah Wood (V. F. : Alexandre Gillet) : Tom Selznick
John Cusack (V. F. : Bernard Gabay) : Clem
Kerry Bishé (V. F. : Élisabeth Ventura) : Emma Selznick
Tamsin Egerton : Ashley
Allen Leech : Wayne
Don McManus : Reisinger
Dee Wallace : Marjorie Green
Alex Winter : l'assistant
Distinctions
Récompense
International Film Music Critics Association Awards 2013 : Meilleure musique d'un film d'aventure/thriller/action
Nominations
Box-office
Réception critique
Voir aussi
Articles connexes
Liens externes
Notes et références
Film espagnol sorti en 2013
Thriller (film) espagnol
Film tourné à Chicago
Film se déroulant à Chicago
Film tourné dans les îles Canaries | {
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Longines continues to ride on its expertise in the world of sports with new iterations in several key collections including the HydroConquest, Record and Conquest V.H.P. Chronograph. These models combine both technical excellence and traditional elegance that are hallmarks of the winged hourglass brand – featuring an assertive rendition to each of its iconic pieces while remaining true to the elegance of Longines watches.
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HydroConquest is a diving watch that doubles as a dress watch, perfect for work and play.
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For the ladies, the Record collection carries an exceptional aura of elegance. It is the bridge that celebrates the timelessness and innovative philosophy of the brand. Its new variations in rose gold and in stainless steel/rose gold bring together a unique femininity to the brand. The use of rose gold elevates the resolutely classic collection, making them appeal to men and women with discerning taste in watches.
In this modern age where people are constantly on the go, a timekeeper that offers elegance with a sporty spirit makes perfect sense – and here's where Longines comes into play with timepieces that do just that. | {
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National Boss's Day is around the corner! Wish your boss a happy Boss's Day with a festive bouquet of flowers instead of just a Boss's Day card! Pike Creek Flower & Gift offers same day and next day flower delivery to Wilmington, DE and nationwide! | {
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{"url":"https:\/\/mathematica.stackexchange.com\/questions\/16375\/how-to-differentiate-formally\/16390","text":"# How to differentiate formally?\n\nI have been wrapping my head around this for a while now and I have not found a solution so far. I want to work with an arbitrary number of variables in Mathematica and use some built in functions. To make things more specific for starters I want to do the following. Define a sum with $n$ summands each containing a new variable x[i] (in the $i$-th summand):\n\nsum[n_] = Sum[i x[i], {i, 1, n}]\n\n\nThen I want to differentiate the expression with respect to some x[i] like:\n\nD[sum[n], x[2]]\n\n\nMathematica returns $0$ instead of $2$. However if I supply a specific $n$ like:\n\nD[sum[2], x[2]]\n\n\neverything works fine. I thought about using Assumption for $n$, but with no success so far. How can I do that right?\n\n\u2022 Evaluating D[sum[n], x[2]] is essentially the same as evaluating D[Sum[n, y], x[2]], where y is value-free, which is nothing D recognizes as depending on x[2]. \u2013\u00a0m_goldberg Dec 15 '12 at 16:10\n\u2022 This question is closely related to and might even be considered a duplicate of this one \u2013\u00a0m_goldberg Dec 15 '12 at 16:25\n\u2022 Extending Mathematica to allow sums of symbolic length really would require extended it to allow lists of symbolic length; only then could one properly worry about differentiating such objects. But how to make a robust, general design for such an extension is hardly obvious and raises all sorts of issues, e.g.: Should Infinity to an acceptable value for n? What would the effect be upon processing speed for specific-length objects? \u2013\u00a0murray Dec 15 '12 at 17:41\n\u2022 I've changed your title to something more specific. please change it to something else if you think it's better \u2013\u00a0acl Dec 15 '12 at 19:39\n\u2022 For formal differentiation, what I've noticed is if it's in a tuxedo it's probably a guy, and if it's in a gown it's usually a gal. But this is only a rough guideline, and anyway times have changed. \u2013\u00a0Daniel Lichtblau Dec 15 '12 at 22:15\n\nsum[n_] := Sum[i x[i], {i, 1, n}]\n\nx \/: D[x[i_], x[j_], NonConstants -> {x}] :=\nKroneckerDelta[i, j]\n\nD[sum[n], x[2], NonConstants -> x]\n\n\n$\\begin{cases} 2 & n>1 \\\\ 1-n & \\text{True} \\end{cases}$\n\nThe trick here is the use of the NonConstants option of the derivative operator. This then has to be combined with a definition stating that the x[i] are independent variables for the purposes of this differentiation (hence the KroneckerDelta on the second line).\n\nEdit: more discussion\n\nAnd here is another cool result, completely symbolic:\n\nAssuming[m \u2208 Integers, D[sum[n], x[m], NonConstants -> x]]\n\n\n$\\left( \\begin{array}{cc} \\{ & \\begin{array}{cc} m & m\\geq 1 \\\\ 0 & \\text{True} \\\\ \\end{array} \\\\ \\end{array} \\right)-\\left( \\begin{array}{cc} \\{ & \\begin{array}{cc} m & m>2\\land m\\geq n+1 \\\\ n+1 & m=2\\land n=1 \\\\ \\end{array} \\\\ \\end{array} \\right)$\n\nThis isn't easy to absorb, but it works if you check it with specific examples by doing\n\ncondition = %;\n\nSimplify[condition \/. m -> 10]\n\n\n$\\begin{cases} 10 & n>9 \\\\ 0 & \\text{True} \\end{cases}$\n\nIn summary, it's worth pointing out that a lot of symbolic differentiation tasks can be achieved by using either NonConstants specifications in D or conversely using Constants specifications in Dt.\n\n\u2022 This is really helpful. Thank you very much. \u2013\u00a0Wizard Dec 22 '12 at 17:03\n\u2022 It might be worth mentioning that when doing this with more than one variable (say $x$ and $y$), it is necessary to have them Sorted in the NonConstants rule for the UpValue of the variables, otherwise it won't find a match when trying to evaluate D (and leave it unevaluated). \u2013\u00a0Joel Bosveld Jun 17 '15 at 7:44\n\u2022 sorry if this a super newbie question, but why does it say $1-n$ for...true? :\/ What does that condition even mean? \u2013\u00a0Pinocchio Aug 7 '15 at 19:58\n\u2022 @Pinocchio True is the entry that humans would call \"otherwise\" - it's the default alternative if none of the above holds. \u2013\u00a0Jens Aug 7 '15 at 20:05\n\u2022 @Jens I see what true means now (weird, the programmer could have just said otherwise in a print statement...). But, how is the derivative wrt x[2] be 1 - n for n < 1? for n<1 the sum doesn't even exist...so wouldn't technically it be the derivative of a non-existent sum (so zero), so the derivative of zero is zero? Or am I completely off? \u2013\u00a0Pinocchio Aug 7 '15 at 20:10\n\nI did some computation of formal derivatives a while back which might be of interest in this context (though keep in mind that this is anything but bullet proof! it does work for the cases I bothered to check though).\n\nClear[a]; Format[a[k_]] = Subscript[a, k]\n\n\nLet us say we have an objective function which is formally a function of the vector a[i]\n\nQ = Sum[Log[Sum[a[r] Subscript[B, r][Subscript[x, i]], {r, 1, p}]\/\nSum[a[r] , {r, 1, p}]], {i, 1, n}]\n\n\nLet us define a couple of rules for formal differentiation as follows\n\nClear[d];\nd[Log[x_], a[k_]] := 1\/x d[x, a[k]]\nd[Sum[x_, y__], a[k_]] := Sum[d[x, a[k]], y]\nd[ a[k_] b_., a[k_]] := b \/; FreeQ[b, a]\nd[ a[q_] b_., a[k_]] := b Subscript[\u03b4, k, q] \/; FreeQ[b, a]\nd[ c_ b_, a[k_]] := d[c, a[k]] b + d[b, a[k]] c\nd[ b_ + c_, a[k_]] := d[c, a[k]] + d[b, a[k]]\nd[Subscript[\u03b4, r_, q_], a[k_]] := 0\nd[x_, a[k_]] := 0 \/; FreeQ[x, a]\nd[G_^n_, a[k_]] := n G^(n - 1) d[G , a[k]] \/; ! FreeQ[G, a]\nd[Exp[G_], a[q_]] := Exp[G] d[G , a[q]] \/; ! FreeQ[G, a]\n\n\n\nAnd a rule to deal with Kroneckers\n\nds = {Sum[a_ + b_, {s_, 1, p_}] :> Sum[a, {s, 1, p}] + Sum[b, {s, 1, p}],\nSum[ y_ Subscript[\u03b4, r_, s_], {s_, 1, p_}] :> (y \/. s -> r),\nSum[ y_ Subscript[\u03b4, s_, r_], {s_, 1, p_}] :> (y \/. s -> r),\nSum[ Subscript[\u03b4, s_, r_], {r_, 1, p_}] :> 1,\nSum[\u03b4[i_, k_] \u03b4[j_, k_] y_. , {k_, n_}] -> \u03b4[i, j] (y \/. k -> i),\nSum[a_ b_, {r_, 1, p_}] :> a Sum[b, {r, 1, p}] \/; NumberQ[a],\nSum[a__, {r_, 1, p_}] :> Sum[Simplify[a], {r, 1, p}] }\n\n\nThen, for instance, the gradient of Q with respect to one of the a[k] reads\n\ngrad = d[Q, a[k]] \/. ds \/\/ Simplify;\n\n\nSimilarly the tensor of second derivatives w.r.t. a[k] and a[s] is given by\n\nhess = d[d[Q, a[k]], a[s]] \/. ds \/\/ Simplify\n\n\nAs a less trivial example let us consider the 4th order derivatives of Q\n\n d[d[d[d[Q, a[k]], a[s]], a[m]], a[t]]; \/. ds \/\/ Simplify\n\n\nFor the problem at hand we check easily that\n\n Q = Sum[r a[r] , {r, 1, p}];\n\ngrad = d[Q, a[k]] \/\/ Simplify;\n\n\nreturns k as it should\n\nEDIT\n\nThis process can be made a bit more general, say, on this Objective function\n\n Q = 1\/2 Sum[(Sum[a[r] Subscript[B, r, i][a[q]], {r, 1, p}] -\nSubscript[y, i])^2, {i, 1, n}]\n\n\nwhich depends non linearly on a[k] via B.\n\nAll we need is to add a new rule for d\n\nd[H_[a[q_]],\na[k_]] := (D[H[x] , x] \/. x -> a[k] ) Subscript[\u03b4, k, q]\n\n\ngrad = d[Q, a[k]] \/\/ Simplify;\nhess = d[d[Q, a[k]], a[s]];\n\n\n\nhess \/. ds \/\/ Simplify\n\n\nAs a other example, let us look at a parametrized entropy distance,\n\n Q = -Sum[(Sum[a[r] Subscript[B, r, i], {r, 1, p}]\/\nSubscript[y, i]) Log[(Sum[a[r] Subscript[B, r, i], {r, 1, p}]\/\nSubscript[y, i])], {i, 1, n}]\n\n\nwe can compute its Hessian while mapping twice the sum rule\n\n Map[# \/. ds &, d[d[Q, a[k]], a[s]] \/. ds]\n\n\nAs a final example, consider a Poisson likelihood\n\n Q = Sum[Log[Exp[-a[k]] a[k]^Subscript[y, k]\/Subscript[y, k]!], {k, 1, n}]\n\n\nso that\n\n grad = d[Q, a[k]] \/\/ Simplify\n\n\nand\n\n hess =d[d[Q, a[k]], a[s]] \/. ds \/\/ Simplify\n\n\nOf course these algebraic rules are not bullet proof but illustrate nicely the way mathematica handles new grammar.\n\n\u2022 @Nasser have you seen this? \u2013\u00a0Mr.Wizard Dec 15 '12 at 19:51\n\u2022 unfortunately, i can not replicate these results in Mathematica 10.2. I would be appreciative if someone could try to reproduce this and\/or suggest edits. \u2013\u00a0Eric Brown Sep 11 '15 at 1:53\n\u2022 I also cannot replicate these results using Mathematica 11 \u2013\u00a0MaPo Jan 26 '17 at 15:01\n\u2022 I have added one rule to solve the pb on the last two example. It now works for Mathematica 10.3. \u2013\u00a0chris Jan 26 '17 at 19:18\n\nStarting in M11.1, this works:\n\nsum[n_] = Sum[i x[i],{i,1,n}];\n\nD[sum[n],x[2]] \/\/InputForm\n\n\nPiecewise[{{2, n >= 2}}, 0]\n\n\u2022 I've got M11, and I still get a result of 0. \u2013\u00a0clr66 Jul 23 '17 at 1:27\n\u2022 Works for me in Mathematica 11.1.1 on MacOS Sierra. \u2013\u00a0Wizard Jul 24 '17 at 15:13\n\u2022 @clr66 I should have said starting in M11.1. \u2013\u00a0Carl Woll Jul 24 '17 at 15:44\n\u2022 Finally managed to upgrade to M11.1.1 and attempted the following: Assuming[{1<=j<=n,Element[j,Integers]},D[Sum[i f[i],{i,1,n}],f[j]]] (* j *) So far, so good. However, I also get the following: Assuming[{1<=j<=n,Element[j,Integers]},D[Sum[i g[f[i]],{i,1,n}],f[j]]] (* Sum[i*KroneckerDelta[i, j]*Derivative[1][g][f[i]], {i, 1, n}] *) This answer is technically correct, although it would be nice to get a clean result of j g'[f[j]] Is there a way to get the KroneckerDelta to return this result? \u2013\u00a0clr66 Aug 2 '17 at 19:34\n\nOne solution might be ...\n\nMethod 1.\n\nDefine some variables:\n\nx = Table[Unique[], {5}];\n\n\nForm the inner product and differentiate:\n\nD[Inner[Times, x, Range@Length@x], x[[2]]]\n\n\n2\n\nOr if you prefer it in a functional form:\n\nsum[n_] := Inner[Times, x[[1 ;; n]], Range@n] \/; ( Length@x >= n)\n\nD[sum[4], x[[3]]]\n\n\n3\n\nMethod 2.\n\nYou could take a different approach, but you need to be aware that it leaks variables of the form {x1,x2,x3,...} out into the general context:\n\nsum[n_] := Inner[Times, Symbol[\"x\" <> ToString@#] & \/@ Range@n, Range@n]\n\nsum[5]\n\n\nx1 + 2 x2 + 3 x3 + 4 x4 + 5 x5\n\nD[sum3[4], x3]\n\n\n3\n\n\u2022 This is exactly what I was trying to avoid. The exact amount of variables should not have to be specified. \u2013\u00a0Wizard Dec 22 '12 at 17:04\n\nI like image_doctor's solution better, but how about using Array and looking for the position using that index each time? Like this:\n\nxx = Array[x, 10, 1];\nsum[n_] := Times[List @@ xx , Range[10]]\n\nsum[n]\n(* {x[1], 2 x[2], 3 x[3], 4 x[4], 5 x[5], 6 x[6], 7 x[7], 8 x[8], 9 x[9], 10 x[10]} *)\n\n\nNow\n\nsum[n_] := Times[List @@ xx^4, Range[10]]\nD[sum[n], xx[[5]]][[5]]\n(* 20 x[5]^3 *)\n\n\nand\n\nsum[n_] := Times[List @@ xx, Range[10]]\nD[sum[n], xx[[2]]][[2]]\n\n(* 2 *)\n\n\nIt is kinda clumsy though. The problem, of course, is that one can't treat\n\nas a single variable as we do on paper. P.S., I tried to see if one can do this in Maple and could not do it directly as you wanted. Had to do a hack as above. (But I do not know Maple much, though.)\n\n\u2022 Nasser, if it matters to you, your \"gravatar\" is changing along with your I.P. address. You can make it a fixed one by entering an email address in your profile. This email is only visible to moderators and furthermore it doesn't need to be a real one anyway. Also see: mathematica.stackexchange.com\/q\/13525\/121 \u2013\u00a0Mr.Wizard Dec 15 '12 at 19:42\n\nA possible solution is to use the fact that Mathematica can easily take the derivative of an actual sum, but has problems with the symbolic one. In order to take the derivative on j-th term of a sum, we extract from the sum few terms centered around j. The number of extracted terms is equal to 2*dj+1. There is one drawback of this solution that you can always differentiate on j-th term. This method works with subscripted variables as well.\n\ns1 = Sum[p[k]*a[k + 1] + p[k + 1]*a[k], {k, 1, nn}];\nderivSum[s0_, xj_, dj_] := Block[{k, j}, D[Sum[s0[[1]], {k, j - dj, j + dj}], xj]];\nderivSum[s1, p[j], 4]\n(* a[-1+j]+a[1+j] *)\n\n\\$Version\n\n(* \"11.1.1 for Mac OS X x86 (64-bit) (April 18, 2017)\" *)\n\nsum[n_] = Sum[i x[i], {i, 1, n}];\n\nAssuming[Element[{n, m}, Integers] && n >= m >= 1, D[sum[n], x[m]]]\n\n(* m *)\n\nAssuming[Element[{n, m}, Integers] && n >= 1 && m >= 1,\nD[sum[n], x[m]] \/\/ Simplify]","date":"2020-10-27 02:51:34","metadata":"{\"extraction_info\": {\"found_math\": true, \"script_math_tex\": 0, \"script_math_asciimath\": 0, \"math_annotations\": 0, \"math_alttext\": 0, \"mathml\": 0, \"mathjax_tag\": 0, \"mathjax_inline_tex\": 1, \"mathjax_display_tex\": 0, \"mathjax_asciimath\": 1, \"img_math\": 0, \"codecogs_latex\": 0, \"wp_latex\": 0, \"mimetex.cgi\": 0, \"\/images\/math\/codecogs\": 0, \"mathtex.cgi\": 0, \"katex\": 0, \"math-container\": 0, \"wp-katex-eq\": 0, \"align\": 0, \"equation\": 0, \"x-ck12\": 0, \"texerror\": 0, \"math_score\": 0.3270876109600067, \"perplexity\": 4137.122913530389}, \"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-2020-45\/segments\/1603107893011.54\/warc\/CC-MAIN-20201027023251-20201027053251-00594.warc.gz\"}"} | null | null |
<?php
# Generated by the protocol buffer compiler. DO NOT EDIT!
# source: google/cloud/shell/v1/cloudshell.proto
namespace Google\Cloud\Shell\V1;
use Google\Protobuf\Internal\GPBType;
use Google\Protobuf\Internal\RepeatedField;
use Google\Protobuf\Internal\GPBUtil;
/**
* Request message for
* [StartEnvironment][google.cloud.shell.v1.CloudShellService.StartEnvironment].
*
* Generated from protobuf message <code>google.cloud.shell.v1.StartEnvironmentRequest</code>
*/
class StartEnvironmentRequest extends \Google\Protobuf\Internal\Message
{
/**
* Name of the resource that should be started, for example
* `users/me/environments/default` or
* `users/someone@example.com/environments/default`.
*
* Generated from protobuf field <code>string name = 1;</code>
*/
private $name = '';
/**
* The initial access token passed to the environment. If this is present and
* valid, the environment will be pre-authenticated with gcloud so that the
* user can run gcloud commands in Cloud Shell without having to log in. This
* code can be updated later by calling AuthorizeEnvironment.
*
* Generated from protobuf field <code>string access_token = 2;</code>
*/
private $access_token = '';
/**
* Public keys that should be added to the environment before it is started.
*
* Generated from protobuf field <code>repeated string public_keys = 3;</code>
*/
private $public_keys;
/**
* Constructor.
*
* @param array $data {
* Optional. Data for populating the Message object.
*
* @type string $name
* Name of the resource that should be started, for example
* `users/me/environments/default` or
* `users/someone@example.com/environments/default`.
* @type string $access_token
* The initial access token passed to the environment. If this is present and
* valid, the environment will be pre-authenticated with gcloud so that the
* user can run gcloud commands in Cloud Shell without having to log in. This
* code can be updated later by calling AuthorizeEnvironment.
* @type array<string>|\Google\Protobuf\Internal\RepeatedField $public_keys
* Public keys that should be added to the environment before it is started.
* }
*/
public function __construct($data = NULL) {
\GPBMetadata\Google\Cloud\Shell\V1\Cloudshell::initOnce();
parent::__construct($data);
}
/**
* Name of the resource that should be started, for example
* `users/me/environments/default` or
* `users/someone@example.com/environments/default`.
*
* Generated from protobuf field <code>string name = 1;</code>
* @return string
*/
public function getName()
{
return $this->name;
}
/**
* Name of the resource that should be started, for example
* `users/me/environments/default` or
* `users/someone@example.com/environments/default`.
*
* Generated from protobuf field <code>string name = 1;</code>
* @param string $var
* @return $this
*/
public function setName($var)
{
GPBUtil::checkString($var, True);
$this->name = $var;
return $this;
}
/**
* The initial access token passed to the environment. If this is present and
* valid, the environment will be pre-authenticated with gcloud so that the
* user can run gcloud commands in Cloud Shell without having to log in. This
* code can be updated later by calling AuthorizeEnvironment.
*
* Generated from protobuf field <code>string access_token = 2;</code>
* @return string
*/
public function getAccessToken()
{
return $this->access_token;
}
/**
* The initial access token passed to the environment. If this is present and
* valid, the environment will be pre-authenticated with gcloud so that the
* user can run gcloud commands in Cloud Shell without having to log in. This
* code can be updated later by calling AuthorizeEnvironment.
*
* Generated from protobuf field <code>string access_token = 2;</code>
* @param string $var
* @return $this
*/
public function setAccessToken($var)
{
GPBUtil::checkString($var, True);
$this->access_token = $var;
return $this;
}
/**
* Public keys that should be added to the environment before it is started.
*
* Generated from protobuf field <code>repeated string public_keys = 3;</code>
* @return \Google\Protobuf\Internal\RepeatedField
*/
public function getPublicKeys()
{
return $this->public_keys;
}
/**
* Public keys that should be added to the environment before it is started.
*
* Generated from protobuf field <code>repeated string public_keys = 3;</code>
* @param array<string>|\Google\Protobuf\Internal\RepeatedField $var
* @return $this
*/
public function setPublicKeys($var)
{
$arr = GPBUtil::checkRepeatedField($var, \Google\Protobuf\Internal\GPBType::STRING);
$this->public_keys = $arr;
return $this;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 5,466 |
var gulp = require('gulp'),
concat = require('gulp-concat'),
aliasify = require('aliasify'),
browserify = require('browserify'),
source = require('vinyl-source-stream'),
buffer = require('vinyl-buffer'),
uglify = require('gulp-uglify'),
watch = require('gulp-watch'),
batch = require('gulp-batch'),
sass = require('gulp-sass'),
glob = require('glob');
var PATH = {
input : './src/js/**/*.js',
outputPath : './static/js/',
outputFile : 'creports.js',
lib : './node_modules/',
watchJS : ['src/js/**', 'node_modules/**/*.js'],
watchSCSS : ['node_modules/**/*.{sass,css}', 'src/scss/**/*.scss'],
}
var libs = [
PATH.lib + 'd3',
PATH.lib + 'topojson',
PATH.lib + 'datamaps/dist',
PATH.lib + 'jquery/dist',
PATH.lib + 'lodash',
PATH.lib + 'normalize.css',
PATH.lib + 'backbone',
PATH.lib + 'backbone.marionette/lib'
];
gulp.task('watch', function () {
watch(PATH.watchJS, function () {
gulp.start('browserify');
});
watch(PATH.watchSCSS, function () {
gulp.start('sass');
});
});
gulp.task('sass', function () {
return gulp.src('src/scss/main.scss')
.pipe(sass())
.pipe(concat('main.css'))
.pipe(gulp.dest('./static/css'));
});
gulp.task('fonts', function () {
return gulp.src('src/fonts/**/*').pipe(gulp.dest('./static/fonts'));
});
gulp.task('default', ['fonts', 'sass', 'browserify', 'watch']);
gulp.task('browserify', function() {
return browserify(glob.sync(PATH.input), {
paths: libs})
.transform(aliasify, {global: true})
.bundle()
.pipe(source(PATH.outputFile))
.pipe(buffer())
// .pipe(uglify())
.pipe(gulp.dest(PATH.outputPath))
});
| {
"redpajama_set_name": "RedPajamaGithub"
} | 6,094 |
The history of Arsenal Football Club from 1966 to the present day covers the third, fourth, and fifth periods of success in Arsenal's history, including three Doubles, a Cup Double, and success in European football, and an unbeaten league season.
Following Bertie Mee's appointment in 1966, Arsenal won the Inter-Cities Fairs Cup, their first European trophy, in 1969–70, and their first League and FA Cup double in 1970–71. The Double-winning side, however, was soon broken up and the following decade was characterised by a series of near misses: Arsenal lost three FA Cup finals (1971–72, 1977–78, and 1979–80) and the 1979–80 Cup Winners' Cup final on penalties. The club's only success during this time was an FA Cup win in 1978–79 against Manchester United.
After stagnation in the 1980s, the return of former player George Graham as manager in 1986 brought a third period of glory. Arsenal won the League Cup in 1986–87, the Football League Centenary Trophy in 1988, two League title wins in 1988–89 and 1990–91, the FA Cup and League Cup double in 1992–93 and a second European trophy, the Cup Winners' Cup, in 1993–94. However, Graham's reputation was tarnished when it was revealed that he had taken kickbacks for signing certain players and he was sacked in 1995.
Arsenal's fifth period of success came with the appointment of Arsène Wenger in 1996. Under him, Arsenal won a second league and cup double in 1997–98 and then a third in 2001–02. In addition, the club were victorious in the 2002–03 and 2004–05 FA Cups, and won the Premier League in 2003–04 without losing a single match. In 2005–06 became the first London club to reach the UEFA Champions League Final, but lost 2–1 against FC Barcelona. During the following close season, they left their longstanding home of Highbury to the new Emirates Stadium nearby. They started in their new home with a seven year trophy drought, followed by winning 3 FA Cups in the next four seasons.
The first Double (1966–76)
Following the dismissal of Billy Wright in the summer of 1966, Arsenal appointed physiotherapist Bertie Mee as his successor. The move that brought surprise to some, not least Mee himself, who requested that he be able to return to his old role as physio if being manager had not worked out after 12 months. With assistant Dave Sexton, Mee brought a more professional approach to the club and promoted talent from within; Arsenal's youth team had won the FA Youth Cup in 1966, and talented attacking players such as Charlie George, John Radford, Peter Simpson and Ray Kennedy graduated to the first team.
Mee complemented this attacking ability with some more experienced heads; captain Frank McLintock at centre half marshalled a strong defence, while the hard-tackling Peter Storey filled the defensive midfield position. The team showed early signs of promise, reaching two successive League Cup finals, in 1968 and 1969. Both times the Gunners went home empty-handed. The first time Arsenal lost to Don Revie's Leeds United 1–0 in a dour match of few chances, Terry Cooper grabbing the only goal.
The second League Cup loss was an infamous upset – Arsenal lost 3–1 to Third Division side Swindon Town. Eight of the team had been struck by flu that had led to the postponement of Arsenal's previous League fixture, and Arsenal had only reached extra time thanks to a late goalkeeping error that had allowed Bobby Gould to score. In extra time, Don Rogers scored twice as Arsenal searched for a winner. However, that season was not a total disaster for Arsenal; they had also finished fourth, which won them a place in Europe for the 1969–70 season.
In turn, this led to the club collecting their first silverware in seventeen years and also their first European trophy, the 1969–70 Inter-Cities Fairs Cup. Arsenal beat Ajax 3–1 on aggregate in the semi-finals, and then staged a famous comeback against Anderlecht in the final. Arsenal were 3-0 down after 74 minutes of the first leg at Stade Émile Versé, but Ray Kennedy got a late away goal to give the Gunners a glimmer of hope; in the second leg in front of a packed Highbury, inspired by captain Frank McLintock, Arsenal won 3–0 with goals from John Radford, Eddie Kelly and Jon Sammels, to win the tie 4–3 on aggregate.
The same season, Arsenal had only finished 12th in the league, perhaps distracted by their European campaign, and did not look like league contenders. Yet the following season, 1970–71, Arsenal went on to become only the second club of the 20th century to win the FA Cup and League Double, the club's first. After a bright start Arsenal looked to be out of the title chase with a 5–0 loss to Stoke City in September. However, Arsenal recovered and put in a strong run (they did not lose again in the league until January), and as the season closed they became involved in a tight race with Leeds United.
Arsenal were pushed all the way – after being defeated 1–0 by Leeds in April, they needed to beat or draw 0–0 with North London rivals Tottenham Hotspur at White Hart Lane on the last day of the season to take the title on goal average. An 87th-minute goal by Ray Kennedy gave Arsenal a 1–0 lead and despite Spurs' desperate attempts for an equaliser Arsenal hung on to win and take the title. In the meantime, Arsenal had also reached the FA Cup Final, following a titanic semi-final battle with Stoke which saw them come from 2–0 down to force a replay and eventual victory. In the Final, five days after the win at Tottenham, Arsenal beat Liverpool 2–1 at Wembley; Arsenal went 1–0 down early in extra time, before Eddie Kelly's 101st-minute equaliser from close range. Ten minutes later, Charlie George scored the winner from the edge of the penalty area to win the game, and the Double, for Arsenal.
The Double proved to be a premature high point of a decade characterised by a string of near-misses. Despite signing World Cup winner Alan Ball for a club record £220,000 in the close season, Arsenal began 1971–72 badly, losing three matches in August, and were forced to play catch-up for the rest of the season, ultimately finishing fifth. Their debut in the European Cup started encouragingly, but they were knocked out in the quarter-finals by a Johann Cruyff-inspired Ajax, who went on to win the trophy as part of a hat-trick of European titles. Arsenal also reached the FA Cup Final for the second year in a row; in a repeat of the 1968 League Cup Final they lost 1–0 to Leeds United, in an ugly match of few real chances.
Arsenal finished as First Division runners-up in 1972–73, but within a year the Double-winning side had been broken up, and Mee was unable to build a new team in its place. The club's form declined sharply, finishing 16th in 1974–75 and 17th in 1975–76, their lowest in more than forty years, which prompted Mee's resignation. Tottenham manager Terry Neill, a former Arsenal player, was appointed in his place, even though he had never got Spurs anywhere beyond mid-table, to become Arsenal's youngest-ever manager.
Four cup finals under Neill (1976–80)
Arsenal moved back into the top half of the table, inspired in part by the emergence of Irish superstar Liam Brady. Brady formed part of a large Irish contingent at Highbury, which included Pat Rice, Frank Stapleton, Pat Jennings Sammy Nelson, John Devine and the young David O'Leary. Further to this were experienced signings such as Malcolm Macdonald and Alan Hudson, as well as the return of Don Howe, who had been part of the backroom staff when the Double was won, to the Arsenal coaching setup.
Although they could not challenge the League dominance of Liverpool at the time, towards the end of the decade they proved their mettle in the FA Cup. Arsenal reached three finals in a row (1978, 1979, and 1980), but won only one, the 1979 final against Manchester United. Largely inspired by Brady, Arsenal went 2–0 up through Brian Talbot and Frank Stapleton and looked to be coasting to victory; with five minutes to go, United scored twice in quick succession to level the match. Extra time loomed, but Alan Sunderland converted Graham Rix's cross in injury time to secure a famous 3–2 win.
The next season, 1979–80, proved to be cruel as Arsenal played a record-breaking 70 matches and reached two cup finals, only to end the season empty-handed. Arsenal were favourites to beat Second Division West Ham United in the FA Cup final, but lost 1–0 to a Trevor Brooking header. Meanwhile, they had also reached the Cup Winners' Cup final against Valencia, after Paul Vaessen's goal had given them a famous victory over Juventus in the semi-finals; the final finished goalless and Arsenal lost on penalties, with Brady and Rix having their efforts saved.
Slight decline (1980–86)
Liam Brady left Arsenal for Juventus in the summer of 1980, and the team entered another barren spell. They continued to finish in the top four at the start of the 1980s, but never really looked like winning the title, and they could not rediscover their FA Cup form either – aside from 1982–83 when Arsenal reached both cup semi-finals only to be knocked out in both by Manchester United. Neill struggled to control his team at times; throughout his tenure, he had fallings-out with many of his players (including Hudson and Macdonald) and he was unable to contain the drinking culture within the squad. His signings to replace the departed Brady and Stapleton failed to make the same impact, and towards the end of Neill's reign the club suffered several embarrassing cup defeats; this included losing to part-timers K.F.C. Winterslag in the 1981–82 UEFA Cup and Third Division Walsall in the 1983–84 League Cup. Neill was sacked in December 1983, soon after the latter result.
Don Howe, Neill's assistant, succeeded him but he could not get the side anywhere near a trophy either. Although Arsenal managed to finish sixth and seventh under him, they never seriously challenged for the title (although they did briefly top the league in October 1984) and were dumped out of the 1984–85 FA Cup by Third Division York City. The fans were getting increasingly disillusioned with the club's muddling performances and attendances started to dip beneath 20,000. In March 1986, after hearing the board had approached FC Barcelona coach Terry Venables as his replacement, Howe resigned. Steve Burtenshaw was briefly caretaker manager but the club decided to look to outside for Howe's long-term successor.
The Arsenal board of directors did contact Scottish club Aberdeen with a view to offering the job to their manager Alex Ferguson (while also drawing up an offer to Millwall manager George Graham to become assistant manager of Arsenal), but Ferguson rejected the offer. However, Ferguson did cross the border six months later to succeed Ron Atkinson at Manchester United.
The George Graham years (1986–95)
In May 1986, Millwall manager George Graham, a former Arsenal player, was appointed as Howe's long-term replacement, and it was the beginning of a new era of success at Highbury. Graham gradually sold off most of the older players and replaced them with new signings and players promoted from the youth team, while imposing much stricter discipline than his predecessors, both in the dressing room and on the pitch. Arsenal's form immediately improved, so much so that the club were top of the League at Christmas 1986.
Players like Kenny Sansom, Steve Williams, Tommy Caton, Charlie Nicholas and Gus Caesar were gradually discarded and a new-look Arsenal side featured players including Lee Dixon, Nigel Winterburn, Steve Bould, David Rocastle, Alan Smith, Tony Adams and Paul Merson.
Though Arsenal finished fourth in Graham's first season in charge (having led the First Division for most of the winter), Arsenal did win the League Cup, in a campaign marked by comebacks. Arsenal faced Tottenham Hotspur in the semi-finals; after losing 1–0 at Highbury in the first leg and conceding a second goal in the first half of the second leg at White Hart Lane, Arsenal scored twice through Viv Anderson and Niall Quinn to draw 2–2 on aggregate and force a replay; in the replay Spurs went 1–0 up, only for Arsenal to come back again with late goals from Ian Allinson and David Rocastle to win. The final against Liverpool was a repeat performance; after Arsenal had gone 1–0 down to an Ian Rush goal, two Charlie Nicholas goals brought Arsenal their first League Cup triumph and their first major trophy for eight years. However, UEFA voted to continue the ban on English clubs in European competitions that was imposed in the wake of the Heysel disaster in 1985 for a third season, and this meant that Arsenal were unable to compete in the 1987–88 UEFA Cup.
While Arsenal lost the League Cup final the following year in a shock 3–2 defeat to Luton Town and dipped to sixth place in the league, their League form steadily improved afterwards, thanks largely to a revamped defence which consisted of Lee Dixon, Nigel Winterburn, Steve Bould and Tony Adams, which formed the basis of Arsenal's successes for a decade or more. Until he left in 1993, long-serving defender David O'Leary remained an important member of the squad who frequently appeared as a substitute and filled in whenever the younger members of the back four were unavailable. However, during this time Graham's Arsenal were not a purely defensive side; Graham also employed capable midfielders such as David Rocastle, Michael Thomas and Paul Merson, and striker Alan Smith, whose prolific goalscoring regularly brought him more than 20 goals in most of the eight seasons he spent at the club.
In Graham's third season (1988–89), the club won the Football League Centenary Trophy before winning their first League title since 1971, in highly dramatic fashion. Having led the League since Christmas, Arsenal were overtaken by Liverpool after losing to Derby County and drawing at home to Wimbledon in May. Arsenal had seemingly thrown away the title, but the final game of the season, on 26 May, was against Liverpool at Anfield; Arsenal needed to win by two goals to take the title; Liverpool had already won the FA Cup and were favourites to complete the Double. Alan Smith scored for Arsenal early in the second half to make it 1–0, but as time ticked by Arsenal struggled to get a second, and with 90 minutes gone on the clock, Arsenal still needed another goal and it looked as though the league title would be staying at Anfield. But, with only seconds to go, a Smith flick-on found Michael Thomas surging through the Liverpool defence; the young midfielder lifted the ball over Bruce Grobbelaar and into the net, giving Arsenal the title.
Arsenal did not retain the title the following season; they finished fourth in 1989–90 and fell behind champions Liverpool, runners-up Aston Villa and third-placed Tottenham Hotspur in the title challenge. They also failed to make their mark in the cups, and the post-Heysel ban on English clubs in European competition was still in force at that time, so Arsenal were unable to represent England in the European Cup. The ban was lifted at the end of the season, though Liverpool (the team present at the Heysel disaster) had to serve an extra year.
Graham prepared for another title challenge by signing goalkeeper David Seaman and Swedish winger Anders Limpar in the summer of 1990; both players proved vital as Arsenal retook the title in 1990–91, despite two major setbacks. Arsenal had two points deducted in November 1990 after ten of their players were involved in a brawl with Manchester United players in a match at Old Trafford, and captain Tony Adams was sentenced to four months' imprisonment for drink driving just before Christmas. Despite these setbacks, Arsenal lost only one league match all season and finished seven points clear of Liverpool at the end of what had for most of the season been a two-horse race for the title. They also reached the FA Cup semi-finals, where they faced Tottenham Hotspur; Paul Gascoigne scored with a free kick from 30 yards after just five minutes and Tottenham ran home 3–1 winners, dashing hopes of a unique second Double.
In September 1991, Arsenal paid a club record of £2.5million for Crystal Palace striker Ian Wright, who would go on to spend seven years at the club and become their all-time leading goalscorer in the process. The 1991–92 season saw the club's first entry in the European Cup for 20 years. The European venture went badly; Arsenal were knocked out by Benfica in the second round and failed to make the lucrative group stage. The season went from bad to worse when the Gunners were knocked out of the FA Cup by lowly Wrexham, though Arsenal recovered to finish fourth in the League. The ban on English clubs in European competitions had been lifted two years earlier, but Arsenal missed out on a UEFA Cup place as English clubs were gradually being phased back into European competitions and at this stage only the second and third placed teams were qualifying for the UEFA Cup.
During the 1992 close season the club acquired Danish midfielder John Jensen, who had just won Euro 92 with Denmark, scoring a goal in their victory over Germany in the final. Jensen's arrival coincided with the departure of fellow midfielder David Rocastle to Leeds United, the defending league champions. Around this point, Graham altered his tactics; he became more defensive and turned out far less attack-minded sides, which depended mainly on goals from Wright rather than the whole team. Between 1986–87 and 1991–92 Arsenal averaged 66 League goals a season (scoring 81 in 1991–92), but between 1992–93 and 1994–95 they only averaged 48; this included just 40 in 1992–93, when the club finished 10th in the inaugural season of the FA Premier League, scoring fewer than any other team in the division, though they had briefly topped the table in November.
They were founder members of the FA Premier League on its launch for the 1992–93 season. They lost their first ever Premiership game 4-2 after taking a 2–0 lead over underdogs Norwich City at Highbury; Norwich, among the pre-season relegation favourites, were actively involved in the title race and finished third, whereas Arsenal (among the pre-season title favourites) finished 10th.
Arsenal's form in the cups was much better than in the league, and in 1992–93 they became the first side to win the FA Cup and League Cup double. In the League Cup final, Arsenal faced Sheffield Wednesday; a Merson-inspired Arsenal side came from 1–0 down to win 2–1 thanks to a Steve Morrow goal. In the FA Cup, Arsenal beat Spurs 1–0 in the semi-finals (avenging their defeat of 1991), and played Sheffield Wednesday in the final, just as they had done in the League Cup final a few weeks earlier. The game ended 1–1 and went to a replay; Wright opened the scoring for Arsenal but Chris Waddle equalised. Extra time came, and still no goal broke the deadlock until the 120th minute, when Andy Linighan powered home a header from a corner to win the match and the cup double for Arsenal.
In 1993–94, Arsenal won their second European trophy; a side missing key players (John Jensen and Martin Keown were injured, while Ian Wright was suspended), beat favourites and holders Parma 1–0 in the Cup Winners' Cup final in Copenhagen, with a tight defensive performance and Alan Smith's 21st-minute goal from a left foot volley. The 1994 Cup Winners' Cup proved to be George Graham's last trophy at the club; the following February the Scot was sacked after nearly nine years in charge, after it was discovered he had accepted an illegal £425,000 payment from Norwegian agent Rune Hauge following Arsenal's 1992 acquisition of John Jensen, one of Hauge's clients.
In the weeks before Graham was sacked, he made three major signings for Arsenal. Glenn Helder, a Dutch winger signed from Vitesse, was a regular first-team player for more than a year after joining the club but was then loaned to Benfica before permanently departing in October 1997 to join NAC Breda back in the Netherlands. Chris Kiwomya, an attacking midfielder signed from Ipswich Town, scored 3 goals in 17 matches before the end of the season but never played for the club again, finally departing in 1998 to sign for Queens Park Rangers. Nineteen-year-old Welsh striker John Hartson was signed from Luton Town and occupied the first-team place vacated by the injury-hit Alan Smith, who retired from playing months later. However, Hartson was not a regular player the following season and was sold to West Ham United in 1997.
George Graham's final season at Arsenal was also the final season at the club for several of the club's key players. Alan Smith, one of his first signings, was forced into retirement by injury several months later. Paul Davis, the club's longest-serving player, was given a free transfer at the season's end, having found his first team opportunities increasingly limited towards the end of his time at Arsenal. Swedish midfielder Stefan Schwarz was sold to Fiorentina that summer after just one season at Highbury. Striker Kevin Campbell, who had struggled to establish himself as a regular player in spite of some impressive performances over five seasons, was sold to Nottingham Forest. Winger Jimmy Carter, who had failed to establish himself as a regular player in four seasons at Arsenal, was sold to Portsmouth.
Bruce Rioch: The interregnum (1995–96)
Assistant manager Stewart Houston took charge until the end of the 1994–95 season. Arsenal finished 12th in the Premier League. However, they did reach the Cup Winners' Cup final again, after a titanic semi-final against Sampdoria, which they won on penalties after drawing 5–5 on aggregate. Arsenal faced Real Zaragoza in the final; Esnáider scored for the Spaniards and John Hartson equalised for Arsenal. The game was heading to a 1–1 draw and penalties, before midfielder Nayim struck from 40 yards in the 120th minute, in virtually the last kick of the game. David Seaman, who had been Arsenal's hero in the semi-final shootout, could not backpedal fast enough and only got a hand to the ball as it went in.
In June 1995, Arsenal appointed Bruce Rioch, who had just guided Bolton Wanderers to the League Cup final and promotion to the top flight, as manager. He (briefly) broke the English transfer record by paying Internazionale £7.5 million for Dutch striker Dennis Bergkamp, and the new signing formed an impressive partnership with Ian Wright. Arsenal reached the League Cup semi-finals and finished fifth in the Premiership at the end of 1995–96, securing a place in the following season's UEFA Cup and giving hope for an eventual title challenge. However, the Rioch era ended abruptly: in August 1996, just before the start of the new season, Rioch was sacked after a dispute over transfer funds with the board of directors, triggering a couple of months' turmoil at the club. Stewart Houston was once again put in temporary charge; he remained at the helm for a month, before resigning to take over at QPR. Youth team coach Pat Rice held the fort for several games, before making way for the Frenchman Arsène Wenger at the end of September.
Wenger's arrival and two Doubles (1996–2003)
The team immediately improved under Wenger's management, coming third and winning a UEFA Cup place in 1996–97, missing out on second (and a Champions League spot) on goal difference. Wenger rebuilt the Arsenal squad with a crop of French players who were largely unknown in the UK. Patrick Vieira had been signed on Wenger's recommendation before he had officially taken up the reins, and Wenger added Nicolas Anelka and Emmanuel Petit, as well as Dutch winger Marc Overmars in the summer of 1997. Wenger melded the new arrivals with some of the "old guard", retaining Adams, Dixon, Winterburn, Keown and Bould, and he kept Pat Rice on as assistant manager.
Wenger got his first silverware, and became the first foreign manager to win the English league, the following season, when he steered the side to their second double. It had looked like Arsenal were out of the title race by December after losing 3–1 at home to Blackburn, but they overcame a twelve-point deficit to overtake Manchester United; a 4–0 home win over Everton on 3 May won the title with two matches to spare. On 16 May, Arsenal beat Newcastle United 2–0 in the FA Cup final to complete the double. To top it off, the same season Ian Wright broke Cliff Bastin's goalscoring record, bringing his tally to 185 goals before leaving the club in the summer of 1998.
Despite the signing of Freddie Ljungberg in 1998 and Thierry Henry a year later, a more barren period followed for Arsenal over the next few years, though they came close several times. Arsenal led the League for much of 1998–99, until a 1–0 loss to Leeds United allowed Manchester United to overtake them; Arsenal beat Aston Villa on the last day of the season but United's victory over Spurs meant they took the title. To rub it in further, Arsenal also lost the last ever FA Cup semi-final replay to Manchester United; Dennis Bergkamp had missed a penalty in normal time, and Ryan Giggs scored the winner in extra time after a mazy solo run through the Arsenal defence. Arsenal's return to the Champions League for the first time in seven years was also unsuccessful, as they failed to get past the group stage.
Arsenal came second again in 1999–2000; this time, there was never any real title race and Arsenal finished the season 18 points behind winners Manchester United. Arsenal had another poor season in the Champions League, finishing third in their group; this won them a consolation place in the UEFA Cup, and Arsenal got all the way to the final, where they faced Galatasaray in Copenhagen, the scene of their 1994 Cup Winners' Cup triumph. The match was a tepid affair, a 0–0 draw with few chances; it went to penalties and Arsenal lost after Davor Šuker and Patrick Vieira missed their spot-kicks.
Arsenal again finished second in 2000–01, this time ten points behind Manchester United; the title race had been as good as over since February, when Arsenal lost 6–1 at Old Trafford. Arsenal's season gave priority to the Cups and Europe. They beat Spurs in the semi-finals and met Liverpool in the final at the Millennium Stadium, Cardiff; Arsenal dominated most of the match, and were denied a goal by the arm of defender Stéphane Henchoz, which went unpunished. Arsenal finally did go 1–0 up through Ljungberg but succumbed to two late Michael Owen goals and lost 2–1. In Europe, Arsenal made it to the Champions League quarter-finals for the first time since 1972, only to be eliminated on the away goals rule by eventual finalists Valencia.
By now, Wenger had been forced to rebuild much of the Double-winning side of 1998; Anelka, Overmars and Petit had all left for Spanish clubs in return for hefty fees, while age was finally catching up with the famous back line; Bould and Winterburn had already left, and Adams and Dixon would only last another season before retiring. In their place, Wenger signed the likes of Sol Campbell and Lauren in defence, as well as promoting Ashley Cole from the youth ranks. In midfield, Wenger added the talismanic Robert Pires and signed his compatriot Sylvain Wiltord in attack, while in the meantime Thierry Henry had adapted to the English game to become one of the Premiership's best strikers.
Attack was definitely Arsenal's forté as they won a record-equalling third Double in 2001–02 season; the Gunners were the only team to score in every game of the Premiership season, and went unbeaten in domestic away games. After an initially tight title race (just three points separated the top four in February), Arsenal pulled away from the pack with a 13-game winning streak, finishing seven points ahead of runners-up Liverpool. Arsenal secured the title in the penultimate match of the season with a 1–0 win over Manchester United at Old Trafford, the goal coming from Wiltord. The previous weekend, Arsenal had wrapped up their eighth FA Cup, beating Chelsea 2–0 with goals from Ray Parlour and Freddie Ljungberg.
In 2002–03, Arsenal became the first club in more than 20 years to retain the FA Cup, with a 1–0 victory against Southampton thanks to a Pires goal. Their joy was soured by the fact that they narrowly missed out on retaining the Premiership title. Arsenal had led eventual winners Manchester United by eight points at one stage, but their form collapsed late on in the season; they drew 2–2 away to Bolton Wanderers after leading 2–0, and then lost 3–2 at home to Leeds United a week later, which gave United the title.
The "Invincibles" and a Champions League Final (2003–06)
Little did they know it at the time, but the defeat to Leeds would be Arsenal's last in the League for over a year. 2003–04 was a record-breaking season for Arsenal, as they won the Premiership unbeaten (26 wins, 12 draws, 0 defeats), finishing a clear 11 points ahead of second-place Chelsea. They became only the second team to do so, the first having been Preston North End in 1888–89. Their rivals for the title gained revenge in other competitions – Arsenal were defeated in the Champions League quarter-finals and FA Cup semi-finals by Chelsea and Manchester United, respectively, in successive matches. Faced with the potential collapse of their season, Arsenal recovered from being 1–0 and 2–1 behind to Liverpool in their next league match to win 4–2, thanks to a Thierry Henry hat-trick, and went on to win the league with a 2–2 draw away to Tottenham Hotspur, mimicking their success in 1971.
Arsenal were unable to retain the title in 2004–05, finishing second, 12 points behind a record-breaking Chelsea side. However, the Gunners did stretch their unbeaten run to 49 consecutive matches, an English league football record; the record was equalled with a dramatic 5–3 win over Middlesbrough (Arsenal having trailed 3–1 shortly after half-time) and then surpassed with a 3–0 win over Blackburn Rovers in August 2004, before their unbeaten season was ended with a 2–0 away defeat by Manchester United. This defeat arguably upset the team's form and they fell away from title contention before recovering with a late flourish to finish second, sealed with a 7–0 drubbing of Everton. Champions League glory eluded them again, with the club getting knocked out 3–2 on aggregate by Bayern Munich in the second round. Arsenal did not end the season empty-handed; they came away with their third FA Cup in four years, winning 5–4 on penalties after a 0–0 draw where they played ten versus eleven against Manchester United.
Weakened by the sale of captain Patrick Vieira to Juventus in the summer of 2005, Arsenal's 2005–06 season was comparatively disappointing domestically and the club failed to challenge for any trophies at home. In the league, their poor away form dogged them and despite recording some impressive wins at home (5–0 over Aston Villa, and 7–0 over Middlesbrough), Arsenal spent much of the latter stages of the season in fifth place or lower, and looked set to miss out on the Champions League for the first time since 1997. However, they won their last three matches of the season, culminating in a 4–2 victory over Wigan Athletic in the last ever match at Highbury; coupled with Tottenham Hotspur's loss at West Ham the same day, this meant Arsenal pipped Spurs to fourth place and a Champions League spot.
In contrast to their domestic form, Arsenal's form in Europe in 2005–06 was much stronger; they reached the UEFA Champions League final for the first time in their history, becoming the first London club ever to do so. Arsenal finished top of their group unbeaten, above Ajax, Thun and Sparta Prague against whom Thierry Henry scored two goals on away to become the all-time record goalscorer for Arsenal; in the knockout stages they beat Real Madrid (becoming the first British team to beat Madrid at the Santiago Bernabéu Stadium), Juventus and then Villarreal to reach the final, setting a competition record of ten matches without conceding a goal in the process. In the final, against Barcelona, Arsenal were reduced to ten men early on when goalkeeper Jens Lehmann was sent off for a professional foul; nevertheless they were the ones who scored first, Sol Campbell scoring with a header from a free kick in the 37th minute. Arsenal desperately defended their lead, but two late goals from Samuel Eto'o and Juliano Belletti meant Barcelona ran out 2–1 winners.
Move to the Emirates and Trophy Drought (2006-13)
Arsenal had been highly successful in the 1990s and 2000s, but Highbury's capacity was limited to only 38,500 in the post-Taylor report era; virtually every match was sold out and the club were unable to maximise matchday revenue. With expansion of Highbury ruled impossible, in 1999, Arsenal announced plans to move to nearby Ashburton Grove; construction started in December 2002 with the demolition of buildings on the site, and in July 2006, the new Emirates Stadium opened, ready for the start of the 2006–07 season.
Arsenal took a little time to get used to their new surroundings and as early as November, manager Arsène Wenger conceded that his side was unlikely to make a serious challenge for the title. Dogged by poor away form throughout the season, Arsenal eventually finished fourth, level on points with third-placed Liverpool. With a team largely filled with reserve and younger players, they reached the League Cup Final, which they lost 2–1 to Chelsea. They were less successful in other competitions, however, being knocked out early on in both the Champions League and FA Cup.
The move to the Emirates had transitional effects on the pitch as well as off of it, as Arsenal lost a number of the Invincibles side, including Robert Pires, Dennis Bergkamp, Lauren, Ashley Cole and Sol Campbell in 2006, and a year later club captain and all-time record scorer Thierry Henry departed for Barcelona. Arsene Wenger was building a new side, with the likes of Cesc Fabregas, Emmanuel Adebayor and Theo Walcott being drafted into the side. The young team made a strong bid for the title in 2007/08, completing a club-record unbeaten run of 28 games and leading the league until February, when an injury to striker Eduardo proved a turning point, as the Gunners finished third. The side made two more bids for the title, in 2010 and 2011, but both campaigns followed similar patterns; Arsenal would be involved in the race deep into the season before suffering an untimely loss of form. The club also began making stronger Champions League challenges, reaching the quarter-finals in 2008 and 2010 and the semi-finals in 2009.
However, the young side ultimately failed to win a trophy; the closest the Gunners came was a League Cup final defeat to Birmingham City in 2011. As a result, the side began to break up-Arsenal lost Fabregas and Samir Nasri in 2011 and captain Robin van Persie in 2012, and failed to really replace them, and in 2011-12 and 2012-13 spent the seasons fighting for the top four instead of the title, as they struggled to compete with the financial advantages held by Manchester United, Chelsea and Manchester City. However, the side still had not dropped below fourth in the final table since 1996, and were capable still of excellent performances. However, in Europe, their fortunes were waning-Arsenal were eliminated in the last-16 stage of the Champions League for seven years running between 2011 and 2017, though in 2011 did record a famous first-leg win over Barcelona.
End of the Drought and Wenger's departure (2013-18)
Arsenal's nine-year trophy drought came to an end in 2014, as they won the FA Cup for the fifth time under Wenger. A new era had begun with the signing of German superstar Mesüt Ozil from Real Madrid in summer 2013 for 40 million to break the club's transfer record. The Gunners had a fine start to the league season, leading the table until February before once again they lost their form in the key months to finish fourth; however in the FA Cup Arsenal reached the final without travelling outside London, and, despite going 2-0 down inside eight minutes to Hull City, turned it around to win 3-2 and end their trophy drought. The Gunners retained the cup with a 4-0 thumping of Aston Villa in 2015, having signed Chilean striker Alexis Sanchez from Barcelona in the summer, similarly to Ozil's signing. In 2016 Arsenal record their highest league finish at the Emirates, finishing second behind 5000-1 winners Leicester City, having once again led the league in February.
In 2017 they won the FA Cup for a record thirteenth time-and a record seventh under Wenger, who became the most successful manager in the competitions history, but fell out of the top four for the first time since 1996, finishing fifth to end their 19-year run in the UEFA Champions League. In 2017-18, the Gunners were knocked out of the UEFA Europa League at the semi-final stage, and reached the League Cup final for the third time under Wenger, losing 3-0 to Manchester City, but endured an eventful off-pitch season; they lost Theo Walcott, Olivier Giroud and Alexis Sanchez in January and twice broke the clubs transfer record, for Alexandre Lacazette in the summer and Gabonese Pièrre-Emerick Aubameyang in January. Most significantly though, Arsène Wenger announced his resignation after 22 years in charge, leaving as the clubs longest-serving and most successful manager. A month afterwards, the club announced that former Paris-Saint Germain manager Unai Emery would be taking over in a new role of "head coach", instead of manager.
Post-Wenger Years: 2018-present
Despite early promise, Unai Emery's Arsenal reign was ultimately short-lived, as he was sacked just 18 months into the job. A 22-game unbeaten run at the start of the season had fans optimistic about the clubs future under the Spaniard, but they failed to return to the Champions League, finishing fifth in the league. In the Europa League, Arsenal reached the final, their first in Europe in 13 years, but lost 4-1 to Chelsea in Baku. A nine-game winless run in October–November of the next season resulted in Emery being dismissed from his position as head coach of the first team by the Arsenal board.
After a three-week caretaker stint from assistant manager and former Invincible Freddie Ljungberg, Arsenal hired former club captain Mikel Arteta as head coach, with Arteta having been previously working as assistant with Pep Guardiola at Manchester City. However, less than three months into Arteta's reign, the season was suspended due to the COVID-19 pandemic. After the season's resumption 100 days later, with all matches played behind closed doors, Arsenal secured a disappointing 8th in the league standings, but made up for it with a record-extending 14th FA Cup win, which came just 28 games into Arteta's reign; Pierre-Emerick Aubameyang, installed as captain in November, scored twice in a 2-1 win over Chelsea in the final, although no fans were in attendance.
Despite displaying some early-season promise, Arsenal once again finished 8th in the league in 2020-21, and were eliminated in the semi-finals of the Europa League, meaning they would compete the 2021-22 season not in any European competition for the first time since 1995-96. In April 2021 the club was one of the twelve founding members of the breakaway European Super League, but was one of the first to withdraw after heavy public backlash. In the 2021-22 season, Arsenal found themselves bottom of the league at the end of August, but recovered to challenge for the top four spots. In May, they held pole position over local rivals Tottenham but eventually finished fifth, and exited both cup competitions in January, to Nottingham Forest in the FA Cup third round and Liverpool in the EFL Cup semi-finals.
Footnotes
References
Further reading
Arsenal
History 1966-Present
Arsenal | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 7,422 |
import json
import logging
import os
import time
import unittest2 as unittest
from mock import Mock, patch, call
import trends.db as db
import trends.tweets as tweets
import trends.exceptions as exceptions
logging.basicConfig(filename='log_test_trends.txt',level=logging.DEBUG)
class Test(unittest.TestCase):
def setUp(self):
self.tweets = tweets.Tweets('tweets.pid')
@patch.object(time, 'time')
@patch.object(logging, 'exception')
def test_stream_filter(self, time_mock, logging_mock):
time_mock.side_effect = Exception()
logging_mock.return_value = None
self.tweets.persons = [{'name': 'test_namé'}]
self.tweets.stream = Mock()
self.tweets.stream.filter.side_effect = Exception()
self.assertRaises(Exception,
self.tweets.stream_filter)
self.tweets.stream.filter.assert_called_once_with(track=['test_name'])
if __name__ == '__main__':
unittest.main()
| {
"redpajama_set_name": "RedPajamaGithub"
} | 8,619 |
Q: JComponent - set transparent background I have on a panel a JTable and a JLabel. I set it a background image but I can't get rid of that grey background from the table and label.
How can I do it?
A: Try doing .setOpaque(false); on your JTable and JLabel
| {
"redpajama_set_name": "RedPajamaStackExchange"
} | 733 |
support@writingservicesonline.com
Roper Technologies will receive front-line education from Thome Bravo
Thoma Bravo, a leading technology-focused private equity firm, announced that it has entered into a definitive agreement to sell Frontline Education, a leading provider of educational software, to Roper Technologies in an all-cash transaction valued at approximately $3.725 billion.
The transaction is expected to close in the fourth quarter, subject to regulatory approvals and customary closing conditions, according to a press release issued by the company.
Frontline Education is a leading provider of school administration software, offering solutions for human capital management, student and special programs, and business operations with powerful analytics to increase productivity and improve overall performance for K-12 administrators. Frontline partners with school systems to deliver tools, data and insights that support greater efficiency and productivity, enabling school leaders to spend more time and resources implementing strategies that drive teacher effectiveness, student success and district excellence.
Frontline's broad portfolio includes solutions for proactive recruiting and hiring, absence and time management, professional growth, student information systems, special education, special programs, Medicaid reimbursement, school health management, inventory control and asset management, payroll, benefits and financial management , and analytics solutions that help district leaders access their data to make more informed decisions for the benefit of their students and communities.
Commenting on the development, Holden Spaht, CEO of Thoma Bravo, said in a statement,
"There is a great sense of pride and accomplishment as we reflect on our partnership with Frontline. Over the past five years, we have worked closely with the management team to significantly grow the company's ARR and build a market-leading integrated software and analytics platform developed specifically for the unique challenges they face K-12 administrators and educators. Not only are we very proud of Frontline's financial success, but also its positive impact on the K-12 education system, and we look forward to watching the company continue to grow and advance its important mission under Roper's ownership."
Thoma Bravo acquired Frontline in 2017, and during the five-year partnership, the company leveraged its specialized operating model and deep sector expertise to enable Frontline to drive profitable growth and extend its market leadership in the K-12 education sector.
Mark Gruzin, CEO of Frontline Education, said,
"We deeply appreciate the Thome Bravo partnership over the past five years and the countless contributions of our dedicated team members, who have enabled us to deliver an expanded portfolio of mission-critical solutions for our clients. Roper's acquisition of Frontline Education represents the next phase of our journey, as we remain true to our culture and mission of partnering with K-12 schools in their pursuit of excellence."
Welcoming Frontline to the Roper family, Neil Hunn, President and CEO of Roper Technologies, said,
"Frontline is a great business with clear niche leadership, a proven track record of strong organic and inorganic growth, excellent cash conversion and an outstanding management team that will thrive as part of Roper. The acquisition of Frontline demonstrates Roper's disciplined capital deployment strategy that focuses on identifying high-quality technology companies which are market leaders that will improve Roper's cash flow. We are excited to welcome Frontline to the Roper family."
Roper Technologies is a component of the S&P 500 and Fortune 500. The company operates market-leading companies that design and develop vertical software and technology products for a variety of defensible market niches. It uses a disciplined, analytical and process-driven approach to redeploy its excess free cash flow towards high-quality acquisitions.
JP Morgan Securities LLC, Jefferies LLC and Macquarie Capital served as financial advisors and Kirkland & Ellis LLP as legal advisor to Thoma Bravo and Frontline Education.
Author: Stjepan Soulunii
I'm not a student anymore, but I like to learn. He is not a teacher, but he cares about how students are taught. I'm not an educator, but I want everyone to be educated. I'm not a social worker, but I want to see change. I am not a reformer, but I always want to see a better world. The author believes that only healthy education can bring a better future, a better world, and technology can help achieve a lot in this field.
The latest EdTech news in your inbox
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{if(f.fbq)return;n=f.fbq=function(){n.callMethod?
n.callMethod.apply(n,arguments):n.queue.push(arguments)};
if(!f._fbq)f._fbq=n;n.push=n;n.loaded=!0;n.version='2.0';
n.queue=[];t=b.createElement(e);t.async=!0;
t.src=v;s=b.getElementsByTagName(e)[0];
s.parentNode.insertBefore(t,s)}(window, document,'script',
'https://connect.facebook.net/en_US/fbevents.js');
Categories Education Post navigation
Toronto-based LumiQ is raising C$5 million to expand its CPA training podcast platform in the US
Renaissance Acquires Whole Child MTSS Illuminate Education Management Platform
© 2023 Writing services • Built with GeneratePress | {
"redpajama_set_name": "RedPajamaCommonCrawl"
} | 9,747 |
IC 2868 — галактика типу * (зірка) у сузір'ї Лев.
Цей об'єкт міститься в оригінальній редакції індексного каталогу.
Посилання
IC 2868 в оригінальному новому загальному каталозі
IC 2868 в оригінальному новому загальному каталозі
http://www.seds.org/~spider/ngc/revngcic.cgi?IC+2868
IC 2868 в базі SIMBAD
IC 2868 в базі Vizier
IC 2868 в базі NASA Extragalactic Database
Бази даних про об'єкти NGC/IC
IC 2868
IC 2868
IC 2868 | {
"redpajama_set_name": "RedPajamaWikipedia"
} | 3,777 |
<?php
namespace Gojira\Api\Endpoint;
use Gojira\Api\Client\ClientInterface;
/**
* Abstract base endpoint class
*
* @package Gojira\Api\Endpoint
* @author Toan Nguyen <me@nntoan.com>
*/
abstract class BaseEndpoint implements EndpointInterface
{
/**
* @var ClientInterface
*/
protected $apiClient;
/**
* BaseEndpoint constructor.
*
* @param ClientInterface $apiClient
*/
public function __construct(ClientInterface $apiClient)
{
$this->apiClient = $apiClient;
}
}
| {
"redpajama_set_name": "RedPajamaGithub"
} | 513 |
{"url":"https:\/\/matplotlib.org\/3.3.1\/gallery\/subplots_axes_and_figures\/broken_axis.html","text":"# Broken Axis\u00b6\n\nBroken axis example, where the y-axis will have a portion cut out.\n\nimport numpy as np\nimport matplotlib.pyplot as plt\n\nnp.random.seed(19680801)\n\npts = np.random.rand(30)*.2\n# Now let's make two outlier points which are far away from everything.\npts[[3, 14]] += .8\n\n# If we were to simply plot pts, we'd lose most of the interesting\n# details due to the outliers. So let's 'break' or 'cut-out' the y-axis\n# into two portions - use the top (ax1) for the outliers, and the bottom\n# (ax2) for the details of the majority of our data\nfig, (ax1, ax2) = plt.subplots(2, 1, sharex=True)\n\n# plot the same data on both axes\nax1.plot(pts)\nax2.plot(pts)\n\n# zoom-in \/ limit the view to different portions of the data\nax1.set_ylim(.78, 1.) # outliers only\nax2.set_ylim(0, .22) # most of the data\n\n# hide the spines between ax and ax2\nax1.spines['bottom'].set_visible(False)\nax2.spines['top'].set_visible(False)\nax1.xaxis.tick_top()\nax1.tick_params(labeltop=False) # don't put tick labels at the top\nax2.xaxis.tick_bottom()\n\n# Now, let's turn towards the cut-out slanted lines.\n# We create line objects in axes coordinates, in which (0,0), (0,1),\n# (1,0), and (1,1) are the four corners of the axes.\n# The slanted lines themselves are markers at those locations, such that the\n# lines keep their angle and position, independent of the axes size or scale\n# Finally, we need to disable clipping.\n\nd = .5 # proportion of vertical to horizontal extent of the slanted line\nkwargs = dict(marker=[(-1, -d), (1, d)], markersize=12,\nlinestyle=\"none\", color='k', mec='k', mew=1, clip_on=False)\nax1.plot([0, 1], [0, 0], transform=ax1.transAxes, **kwargs)\nax2.plot([0, 1], [1, 1], transform=ax2.transAxes, **kwargs)\n\nplt.show()\n\n\nKeywords: matplotlib code example, codex, python plot, pyplot Gallery generated by Sphinx-Gallery","date":"2021-08-03 01:02:00","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\": 0.5505722761154175, \"perplexity\": 6959.643791641996}, \"config\": {\"markdown_headings\": true, \"markdown_code\": true, \"boilerplate_config\": {\"ratio_threshold\": 0.18, \"absolute_threshold\": 10, \"end_threshold\": 15, \"enable\": true}, \"remove_buttons\": true, \"remove_image_figures\": true, \"remove_link_clusters\": true, \"table_config\": {\"min_rows\": 2, \"min_cols\": 3, \"format\": \"plain\"}, \"remove_chinese\": true, \"remove_edit_buttons\": true, \"extract_latex\": true}, \"warc_path\": \"s3:\/\/commoncrawl\/crawl-data\/CC-MAIN-2021-31\/segments\/1627046154408.7\/warc\/CC-MAIN-20210802234539-20210803024539-00073.warc.gz\"}"} | null | null |
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